toward massive detection of planets around m …
TRANSCRIPT
TOWARD MASSIVE DETECTION OF PLANETS AROUND M DWARFS USING THERADIAL VELOCITY TECHNIQUE
By
JI WANG
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2012
c⃝ 2012 Ji Wang
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I dedicate it to my parents and my wife
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ACKNOWLEDGMENTS
I would like to thank my parents, who have given all they have to raise and educate me.
They taught me how to become a good man, a man with skills, faith and integrity. I want to
thank my advisor, Dr. Jian Ge, for his valuable advices and continuous support throughout
my graduate study at the University of Florida. I learned from him the 3-R standard, i.e.,
responsive, reliable and responsible. The 3-R standard has become my guidance and will
keep benefiting me in my future career. He taught me how to become a leader in the field
of Astronomy with vision and determination. I want to thank Dr. Eric Ford for his advices on
academic life. During our collaboration on the eccentricity paper, I have learned the entire
process of publishing a paper. It is his patience and meticulousness that set me an example
of how to become a responsible advisor and an exceptional researcher. I want to thank Dr.
Xiaoke Wan for his help in my laboratory work. He is a precise and careful experimentalist.
He taught me basics on optics and hands-on experience in the lab. I also thank him for his
advices on how to survive as a Chinese in the US.
I thank my wife, He Huang, for her unconditional support for me. If I have to give a rea-
son for her support, that is love, the most amazing thing in the world. It is her encouragement
that motivates me to strive for more and makes me to believe that I can do better. It is her
sacrificing support at home that lessens my burden as a family member and frees up more
of my time that is devoted to research. She is the person that made me start to realize the
power of love and the faith that together we can overcome every adversity and achieve one
goal and then another in life.
There are numerous teachers that I want to express my gratitudes to. Mr. Zhenlong Ou,
the math teacher in my junior school, he taught me to be more aggressive and hardworking
when I naively thought that I can success just with my smartness. May he rest in peace! Mrs.
Xiaomei Tang is my Chinese teacher in junior school. I want to thank her for her voluntary
tutoring and free dinners when I was preparing the entrance exam for high school. She is
more like a mother to me than a teacher who is willing to take care of everything for me. Mr.
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Zhifang Li is my high school math teacher. I want to thank her for motivating me to be the
best student by letting me know that I am not her favorite student. She also taught me that
study should not be everything for a person, there are other things that make my life colorful
and diversified such as music, sports and so on. It became more valuable an advice after I
was admitted in the University of Science and Technology of China (USTC), where I found
myself living a happier life than most of my classmates. I would like to thank my advisors at
USTC, Dr. Tinggui Wang, Dr. Fuzhen Chen and Dr. Xu Kong. They are excellent professors
in astronomy research and would be good examples for my upcoming path in astronomy.
The entire ET group led by Dr. Jian Ge has helped me a lot in finishing my dissertation.
I would like to thank former member Dr. Suvrath Mahadevan, who is now an assistant
professor at PennState University. He is always there whenever I need help on M dwarf
planet science no matter how busy he is. I thank Dr. Jullian van Eyken for his valuable
discussion on DFDI theory. I want to specially thank Dr. Justin Crepp, who is going to start
a new chapter of his life at the University of Notre Dame as an assistant professor. I thank
him for the valuable experience I gained from our laboratory work for high-contrast imaging
of exoplanet. I thank him for the good time we had at UF, Pasadena and Yellowstone.
I also want to thank him for his generous offer when I was looking for a job. I thank Dr.
Scott Fleming, Dr. Nathan De Lee and Dr. Brian Lee for his advices on my projects and
observation proposals. I wish all the group members the best in their future researches and
lives.
During my six years’ study at the University of Florida, I am so thankful to have many
friends accompanied. I thank Pengcheng Guo, aka PC, for all his help when I first came and
settled down here and the good time playing pool, badminton and basket ball and swimming.
He taught me to be a gentleman and to be less emotional no matter what happens. Dr. Bo
Zhao came here two years ahead of me and treated me like his little brother. I thank Dan Li
for being the dedicated fishing partner. Dr. Peng Jiang is an elder alumni from USTC who
keeps me motivated and I am always encouraged by him to think deeper into a problem. I
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thank Liang Chang, aka, Liang ge, for all the time we spent together only trying to kill time.
I thank Dr. Jiwei Xie and his wife for their funny stories. I thank Bo Ma for always being the
guy who is made fun of. We would be less happier without your existence. PC, me, Peng,
Jiwei and Bo made the old Astronomy Five, the basketball team that defeated other teams
”effortlessly”. With part of the old team leaving and new blood coming in, the new Astronomy
Five is formed with members being PC, me, Bo, Rui and Shuo. The new Astronomy Five
is younger and more sophisticated in game execution and it is on its way to be the ”best”
basketball team in the Chinese community within Gainesville area.
Besides my Chinese friends, I would like to thank my international friends (US friends
included). I thank Soungchul and Mark for all the lunches we had together. We had such a
good time in discussing recent projects, keeping each other motivated and exchanging ideas
on topics that interested all of us. I thank Craig for keeping me fit by pushing me to finish
one more uphill stairs and then another in stadium running. I thank Catherine for helping me
out of trouble in registration and departmental affairs. I thank Dr. Cullen Blake for valuable
discussions and reference letters. I thank Dr. Robert Wittenmyer for his mental support from
Australia, the other end of the Earth. I thank Hali Jakeman for proofreading all my papers and
for her help during Stephanie’s qualifying exam.
I would like thank my friends growing up with, Zhi Zeng, Li Zhou and Liqin Huang, for
their generosity in treating meals every time I go back to China. I cannot thank enough
because I am such a blessed person.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
CHAPTER
1 BACKGROUND AND MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1 The Chronicle of Exoplanets Search . . . . . . . . . . . . . . . . . . . . . . . 171.2 Detection Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.1 The Radial Velocity Technique . . . . . . . . . . . . . . . . . . . . . . 171.2.2 The Transit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.3 Direct Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.4 The Timing Technique . . . . . . . . . . . . . . . . . . . . . . . . . . 181.2.5 Microlensing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.6 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Conventional Spectrograph and the Dispersed Fixed Delay Interferometer . . 191.4 Major Questions To Be Answered . . . . . . . . . . . . . . . . . . . . . . . . 201.5 Planets Around M Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.5.1 Essential Facts About M-dwarf . . . . . . . . . . . . . . . . . . . . . . 241.5.2 M-dwarf Planets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.5.3 Surveys and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 FUNDAMENTAL PERFORMANCE OF THE DFDI METHOD . . . . . . . . . . . . 28
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Methodology of Calculating Photon-limited RV Measurement Uncertainty . . . 33
2.2.1 Photon-limited RV Uncertainty of DE . . . . . . . . . . . . . . . . . . 342.2.2 Photon-limited RV Uncertainty of DFDI . . . . . . . . . . . . . . . . . 35
2.3 Comparison between DE and Optimized DFDI . . . . . . . . . . . . . . . . . 362.3.1 Optimized DFDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3.2 Influence of Spectral Resolution R . . . . . . . . . . . . . . . . . . . . 392.3.3 Influence of Detector Pixel Numbers . . . . . . . . . . . . . . . . . . . 442.3.4 Influence of Multi-Object Observations . . . . . . . . . . . . . . . . . 472.3.5 Influence of Projected Rotational Velocity V sin i . . . . . . . . . . . . 50
2.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.4.1 Q Factors for DFDI, DE and FTS . . . . . . . . . . . . . . . . . . . . 502.4.2 Application of DFDI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7
3 COMPREHENSIVE SIMULATIONS FOR HABITABLE PLANET SEARCH IN THENIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Simulation Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 High Resolution Synthetic Spectra . . . . . . . . . . . . . . . . . . . . 583.2.2 RV Calibration Sources . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.3 Stellar Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.4 Telluric Lines Contamination . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.3.1 RV Calibration Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . 663.3.2 Optimal Spectral Band For RV Measurements . . . . . . . . . . . . . 67
3.3.2.1 Stellar Spectral Quality . . . . . . . . . . . . . . . . . . . . . 683.3.2.2 Stellar Spectral Quality + Stellar Rotation . . . . . . . . . . . 693.3.2.3 Stellar Spectral Quality + RV Calibration Source . . . . . . . . 723.3.2.4 Stellar Spectral Quality + RV Calibration Source + Atmosphere 743.3.2.5 Comparisons to Previous Work . . . . . . . . . . . . . . . . . 76
3.3.3 Current Precision vs. Signal of an Earth-like Planet in Habitable Zone 793.3.3.1 Stellar Spectral Quality . . . . . . . . . . . . . . . . . . . . . 803.3.3.2 Stellar Spectral Quality + RV Calibration Source + Atmosphere 823.3.3.3 Stellar Spectral Quality + RV Calibration Source + Stellar
Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4 PLANET SEARCH AROUND M DWARFS . . . . . . . . . . . . . . . . . . . . . . 92
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1.1 Current Status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1.2 Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.2.1 Atmophsere . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1.2.2 Wavelength Calibration Sources . . . . . . . . . . . . . . . . 93
4.2 Tackling Adversities in NIR RV Measurement . . . . . . . . . . . . . . . . . . 934.2.1 Software Advancement . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2.1.1 Precise Telluric Lines Removal . . . . . . . . . . . . . . . . . 934.2.1.2 Binary Mask Cross Correlation . . . . . . . . . . . . . . . . . 95
4.2.2 Hardware Advancement . . . . . . . . . . . . . . . . . . . . . . . . . 964.3 M-dwarf Planet Search and Characterization-Results . . . . . . . . . . . . . 99
4.3.1 Telluric Line RV Stability . . . . . . . . . . . . . . . . . . . . . . . . . 994.3.2 RV Measurements of a Reference Star-GJ 411 . . . . . . . . . . . . . 100
4.4 M-dwarf Planet Search and Characterization-Future Works . . . . . . . . . . 1014.4.1 Searching For Planets Around M Dwarfs with EXPERT . . . . . . . . . 1014.4.2 Multi-Band Study of Radial Velocity Induced by Stellar Activity with
EXPERT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.4.3 Mid-Late Type M Dwarf Planet Survey Using FIRST . . . . . . . . . . 107
4.4.3.1 Science Justification . . . . . . . . . . . . . . . . . . . . . . . 1084.4.3.2 Target Selection . . . . . . . . . . . . . . . . . . . . . . . . . 109
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4.4.3.3 Planet Yield Prediction . . . . . . . . . . . . . . . . . . . . . 110
5 ACCURATE GROUP DELAY MEASUREMENT FOR RV INSTRUMENTS USINGTHE DFDI METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 GD Measurement Using White Light Combs . . . . . . . . . . . . . . . . . . 119
5.2.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.2.2 Data Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2.3 GD Measurement Results . . . . . . . . . . . . . . . . . . . . . . . . 1215.2.4 GD Measurement Error Analysis . . . . . . . . . . . . . . . . . . . . . 124
5.3 GD Calibration: Observing an RV Reference Star . . . . . . . . . . . . . . . 1275.3.1 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.3.2 GD Calibration Precision . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.4 Implementation of Measured GD in Astronomical Observations . . . . . . . . 1285.5 Summaries and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5.1 Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.5.2 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.5.2.1 White Light Comb (WLC) Method . . . . . . . . . . . . . . . . 1305.5.2.2 Reference Star (RS) Method . . . . . . . . . . . . . . . . . . 1325.5.2.3 A Future M-Dwarf Survey With the DFDI Method . . . . . . . 132
6 ECCENTRICITY DISTRIBUTION FOR SHORT-PERIOD EXOPLANETS . . . . . . 134
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.2.1 Bayesian Orbital Analysis of Individual Planet . . . . . . . . . . . . . . 1356.2.2 Γ Analysis of Individual Systems . . . . . . . . . . . . . . . . . . . . . 137
6.3 Results for Individual Planets . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.3.1 Comparison: Standard MCMC and References . . . . . . . . . . . . . 1416.3.2 Comparison: Standard MCMC and Γ Analysis . . . . . . . . . . . . . 1436.3.3 Discussion of Γ Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 144
6.4 Tidal Interaction Between Star and Planet . . . . . . . . . . . . . . . . . . . . 1476.5 Eccentricity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1626.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
7 SUMMARY, CONCLUSION AND CONTRIBUTION . . . . . . . . . . . . . . . . . . 169
7.1 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697.2 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.3 Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1707.4 Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.5 Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
9
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10
LIST OF TABLES
Table page
2-1 OPD choice as a function of R and V sin i at different Teff . . . . . . . . . . . . . . 38
2-2 Power Law Index χ as a function of Spectral Resolution R (Teff = 2400K ) . . . . . 41
2-3 Spectral Resolution and wavelength coverage on a given detector . . . . . . . . . 46
2-4 Q ′′ comparison of DFDI and DE as a function of V sin i . . . . . . . . . . . . . . . 50
3-1 Definition of observational bandpasses . . . . . . . . . . . . . . . . . . . . . . . . 61
3-2 RV uncertainties caused by calibration sources at different spectral resolutions . . 68
3-3 Photon-limited RV uncertainties based on stellar spectral quality at different spec-tral resolutions for different spectral types . . . . . . . . . . . . . . . . . . . . . . . 70
3-4 Spectral Type, corresponding Teff, and typical stellar rotation V sin i . . . . . . . . . 72
3-5 Comparison of Q factors from our results to Bouchy et al. (2001) . . . . . . . . . . 77
3-6 Comparison of predicted RV precision between our results to Reiners et al. (2010) 78
3-7 Comparison of predicted RV precision between our results to Rodler et al. (2011) . 79
3-8 Required S/N for detection of an Earth-like Planet in the HZ as a function of spec-tral type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3-9 Two examples of telluric contamination . . . . . . . . . . . . . . . . . . . . . . . . 83
3-10 Prediction vs. HARPS observation . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5-1 GD measurement results as a function of spectrum number (GD(#) = C0 +C1 · #+C2 · #2) and standard deviation (δGD) at different frequencies (ν) . . . . . . . . . . 127
5-2 MARVELS predicted RV uncertainty (at an average S/N of 100) vs. Teff . . . . . . 128
5-3 Comparison between two methods of GD measurement and calibration . . . . . . 130
6-1 Comparison of Eccentricities Calculated From Different Methods . . . . . . . . . . 148
6-2 Two-sample K-S test result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6-3 Bayesian analysis results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6-4 Catalog of Short-Period Single-Planet Systems . . . . . . . . . . . . . . . . . . . . 165
11
LIST OF FIGURES
Figure page
1-1 Illustration of conventional spectrograph and the DFDI method . . . . . . . . . . . 21
1-2 Illustration of Doppler sensitivity for a conventional spectrograph . . . . . . . . . . 22
1-3 Illustration of Doppler sensitivity for the DFDI method . . . . . . . . . . . . . . . . 23
2-1 DFDI layout diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2-2 DFDI illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2-3 Examples of M-dwarf spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2-4 Power spectrum of a M-dwarf spectrum . . . . . . . . . . . . . . . . . . . . . . . . 32
2-5 Optimal OPD vs. V sin i and R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2-6 Q factor gain vs. spectral resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2-7 Q factor vs. spectral resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2-8 Improvement of QDFDI over QDE vs. spectral resolution . . . . . . . . . . . . . . . . 43
2-9 Comparison of Q ′IRET and Q ′DE at different spectral resolutions . . . . . . . . . . . . 46
2-10 Comparison of Q ′′DFDI and Q ′′DFDI,R=100,000 at different R . . . . . . . . . . . . . . . . 48
2-11 Comparison of Q ′′DE and Q ′′DE,R=100,000 at different spectral resolutions . . . . . . . . 49
2-12 QDFDI vs. V sin i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3-1 Comparisons between synthetic and observed M-dwarf spectra . . . . . . . . . . . 59
3-2 RV calibration uncertainties vs. spectral resolutions . . . . . . . . . . . . . . . . . 67
3-3 RV precision based on spectral quality factor . . . . . . . . . . . . . . . . . . . . . 69
3-4 RV precision based on spectral quality factor and typical stellar rotation . . . . . . . 71
3-5 RV precision based on spectral quality factor and RV calibration uncertainties . . . 73
3-6 RV precision considering spectral quality factor, RV calibration uncertainties andtelluric contamination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3-7 The percentage contribution of RV uncertainty induced by telluric contamination . . 77
3-8 RV precisions considering spectral quality factor at a S/N of 425 . . . . . . . . . . 81
3-9 RV precision (R=120,000) considering spectral quality factor (S/N=425), RV cali-bration uncertainties and telluric contamination . . . . . . . . . . . . . . . . . . . . 83
12
3-10 RV precision considering spectral quality factor (S/N=425), RV calibration uncer-tainties and stellar noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3-11 RMS error of Keplerian orbit fitting for planets detected by HARPS since 2004 . . . 87
4-1 Comparison between two spectra before and after removing telluric lines . . . . . . 94
4-2 Comparison between an observed stellar spectra (GJ 411, telluric lines removed)and a synthetic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4-3 Telluric line removal residual is 2.7% . . . . . . . . . . . . . . . . . . . . . . . . . 96
4-4 An example of binary mask template . . . . . . . . . . . . . . . . . . . . . . . . . 97
4-5 Application of the sine source as an absorption cell . . . . . . . . . . . . . . . . . 99
4-6 Application of the sine source as an emission lamp for simultaneous wavelengthcalibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
4-7 Sin source demonstration experiment . . . . . . . . . . . . . . . . . . . . . . . . . 101
4-8 Telluric lines RV stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4-9 RV measurements for GJ 411 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4-10 RV measured in I band using DEM of EXPERT for KEP 11859158 . . . . . . . . . 106
4-11 RV measured in V band using DEM of EXPERT for KEP 11859158 . . . . . . . . . 107
4-12 V and J band distribution for FIRST survey targets . . . . . . . . . . . . . . . . . . 110
4-13 Teff distribution for FIRST survey targets . . . . . . . . . . . . . . . . . . . . . . . 111
4-14 Predicted RV measurement precision for the FIRST survey . . . . . . . . . . . . . 112
4-15 The predicted survey completeness contours based on observation strategy andRV precision for the pessimistic case . . . . . . . . . . . . . . . . . . . . . . . . . 113
4-16 The predicted survey completeness contours based on observation strategy andRV precision for the baseline case . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5-1 Illustration of the DFDI method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5-2 Simulated WLCs of an interferometer . . . . . . . . . . . . . . . . . . . . . . . . . 120
5-3 Phase of simulated WLCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5-4 The normalized flux and visibility (γ) as a function of frequency . . . . . . . . . . . 122
5-5 Top and side view of an individual fiber beam feeding of the MARVELS interfer-ometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
13
5-6 White light combs phase as a function of frequency . . . . . . . . . . . . . . . . . 124
5-7 Measured group delay as a function fiber number . . . . . . . . . . . . . . . . . . 125
5-8 GD as a function of frequency at different fiber numbers . . . . . . . . . . . . . . . 126
5-9 RVs of HIP 14810 (barycentric velocity not corrected) over a period of 70 days . . . 129
6-1 Examples of how credible intervals of standard MCMC analysis are calculated us-ing posterior distribution of e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6-2 Contours of posterior distribution in h and k space for HD 68988 . . . . . . . . . . 139
6-3 Γ as a function of eccentricity e for HD 68988 . . . . . . . . . . . . . . . . . . . . . 141
6-4 Comparison among standard MCMC analysis, Γ andlysis and previous references . 142
6-5 Cumulative distributions functions (CDFs) of eccentricities from different methods . 146
6-6 Distribution of short-period single-planet systems in (e,τage/τcirc ) space . . . . . . . 150
6-7 Cumulative distribution function of eccentricity . . . . . . . . . . . . . . . . . . . . 154
6-8 Marginalized probability density functions of parameters for analytical eccentricitydistribution with a mixture of exponential and Rayleigh pdfs . . . . . . . . . . . . . 157
6-9 Marginalized probability density functions of parameters for analytical eccentricitydistribution with a mixture of exponential and uniform pdf . . . . . . . . . . . . . . 158
6-10 Cumulative distributions functions (cdf) of eccentricities from different methods . . . 159
6-11 Distribution of short-period single-planet systems in period-eccentricity space . . . 162
14
Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
TOWARD MASSIVE DETECTION OF PLANETS AROUND M DWARFS USING THERADIAL VELOCITY TECHNIQUE
By
Ji Wang
August 2012
Chair: Jian GeMajor: Astronomy
M dwarfs are the least massive but the most common type of stars in the solar neigh-
borhood. Discoveries of M dwarf planets would lead to a complete understanding of planet
formation and evolution around stars of different types. Radial velocity (RV) technique is one
of the leading technologies that detect exoplanets and the RV technique favors detection of
planet in the habitable zone of an M dwarf. The dispersed fixed delay interferometer (DFDI)
method is one branch of the RV technique that uses an interferometer to boost Doppler
sensitivity of a spectrograph with a given resolution. I systematically studied the comparison
between the DFDI method and the traditional high-resolution spectrograph method (the DE
method). My work provides a guidance for future exoplanet survey: 1, a survey of a large
sample of stars should adopt the DFDI method, which enables both adequate RV precision
and high survey efficiency; 2, high precision low-mass exoplanet search should adopt the
DE method with a high resolution spectrograph. I concluded that NIR observation of mid-late
type M dwarfs is the most realistic and likely approach in the search for a habitable Earth-like
planet. Current M dwarf planet survey in the NIR faces two severe challenges, telluric con-
tamination and lack of a precise wavelength calibration source. I developed the binary mask
cross correlation technique and the telluric standard star method in the NIR to eliminate
telluric contamination. I also developed an precise and stable wavelength calibration source,
i.e., the Sine source, which provides a promising candidate for NIR wavelength calibrator. All
these software and hardware developments will pave the way for the next wave of massive
15
detection of M dwarf planets using instruments such as the FIRST. The MARVELS project
is multi-object planet survey with the DFDI method. Owing to my work in MARVELS inter-
ferometer group delay calibration, over 250 binary and a dozen of brown dwarfs have been
discovered and they provide a valuable insight of formation and evolution of low-mass stellar
companion and brown dwarf. I have envisioned an M-dwarf planet survey and provided a
concept study of such survey.
16
CHAPTER 1BACKGROUND AND MOTIVATION
1.1 The Chronicle of Exoplanets Search
The search for planets around other stars (i.e., exoplanets) was first proposed by Struve
(1952). van de Kamp (1963) claimed a detection an astrometric wobble of Barnard’s star
which is later disputed by Gatewood & Eichhorn (1973). Griffin (1973) demonstrated
∼30 m · s−1 stellar radial velocity measurement precision using telluric lines for wavelength
reference. It was not until 1992 that the first exoplanet around a pulsar was detected by
using the pulsar timing technique (Wolszczan & Frail, 1992). The first exoplanet around a
main sequence star was 51 Pegasi discovered by Mayor & Queloz (1995). More discoveries
of exoplanets quickly followed (Marcy & Butler, 1996), and the field of exoplanet started to
gain tremendous attention since then. As of Feb 2012, there are more than 700 exoplanets
detected by a variety of methods1 .
1.2 Detection Methods
1.2.1 The Radial Velocity Technique
In a star-planet system, a star and a planet orbit around their common center of mass.
The wobble of the star can be detected by measuring the stellar velocity along line of sight,
i.e., the radial velocity (RV). RV is usually measured by monitoring tiny shifts of stellar
absorption lines with a high-resolution spectrograph. Most (more than 70%) of currently
known exoplanets are discovered by the RV technique2 . RV measurements precision
of ∼1 m · s−1 has been routinely obtained (Bouchy et al., 2009; Howard et al., 2010b)
with instruments such as HARPS (Mayor et al., 2003) and HIRES (Vogt et al., 1994). In
comparison, Jupiter causes 12.5 m · s−1 RV wobble while the Earth induces 10 cm · s−1
annual RV variation in our solar system. With a increasing time baseline, exoplanets with
1 http://exoplanet.eu/
2 http://exoplanets.org/
17
semi-major axis of ∼6 AU (orbital period∼5000 day) have been detected (Fischer et al., 2008;
Jones et al., 2010).
1.2.2 The Transit Method
Star brightness drops when an object blocks a portion of star light during a transit
event. A periodical star dimming event may be indicative of an orbiting exoplanet. The transit
method is powerful tool in studying properties of an exoplanet such as radius, absorption
and transmission spectrum of atmosphere and so on. Ground-based transiting observation
has yielded many discoveries including a super-Earth around a low mass star (Charbonneau
et al., 2009). A super-Earth refers to a planet with mass roughly 2-10 times of the Earth.
Space-based transiting planets survey such as CoRot (Baglin, 2003) and Kepler (Borucki
et al., 2011a,c) overcome the atmospheric turbulence-induced noise in precision photometric
measurements. Many interesting exoplanet systems are discovered including super-
Earths (Batalha et al., 2011; Leger et al., 2009), habitable planet (Borucki et al., 2012) and
Earth-like planets in a multiple system (Gautier et al., 2011).
1.2.3 Direct Imaging
Exoplanets can be directly imaged by both ground-based and space-based telescopes.
Because of the high brightness contrast between a star and an exoplanet, it is extremely
difficult to directly image an close-in exoplanet. Therefore, most of currently known directly-
imaged exoplanets are more than 10 AU away host stars. The directly-imaged exoplanets
complement exoplanets discovered by the RV technique and with the transit method in the
discovery parameter space.
1.2.4 The Timing Technique
Some astronomical phenomena exhibit stringent repeatability in time domain. The
existence of an exoplanet may be betrayed by a small but detectable perturbation of the
repeatability. For example, Wolszczan & Frail (1992) discovered 3 planets around a pulsar
PSR-1257-12 by measuring the timing variation of its rotation period, Qian et al. (2012)
made a tentative discovery of a Jupiter-like planet around a binary star with the eclipsing
18
binary timing technique. Similar idea was proposed in the measurement of transit timing
variation (Ford & Holman, 2007; Holman & Murray, 2005), and this technique has also
resulted in several discoveies, e.g., Ballard et al. (2011).
1.2.5 Microlensing
Microlensing is predicted by Einstein with his general theory of relativity. When a field
star is aligned with the line of sight between the Earth and a distant star, the brightness of
the distant star experiences a sudden boost because of the field star (lensing star). More
interestingly, if the lensing star has an orbiting planet, a finer spike due to the planet will
be observed together with major spike due to the lensing star in a continuous photometry
measurement. The microlensing method is capable of detecting an exoplanet at a great
distance that is out of reach for other detection methods, which are typically limited within
several hundred parsec away from the Earth. One latest detection by this method is reported
by Batista et al. (2011)
1.2.6 Astrometry
Star position is periodically perturbed by a surrounding planet. A precise astrometry
measurement of a nearby star would be a good candidate method for detecting an exoplanet.
Like the direct imaging method it complements the RV technique and the transit method in
the discovery parameter space for its sensitivity to planets far away from host stars. Unlike
the RV and the transit methods, astrometry can independently measure planet mass without
the aid from other methods. However, no exoplanet has been detected by this method yet.
1.3 Conventional Spectrograph and the Dispersed Fixed Delay Interferometer
Two major methods exist for the RV technique for exoplanet detection, conventional
spectrograph Marcy & Butler (1996); Mayor & Queloz (1995) and the Dispersed Fixed Delay
Interferometer (DFDI) method (Erskine, 2003; Erskine & Ge, 2000; Ge, 2002; Ge et al.,
2002). The RV is obtained by directly measuring the movement of stellar spectral lines
along the dispersion direction for a conventional spectrograph (see bottom of Fig. 1-1).
In comparison, the DFDI measures the movement of vertical flux distribution in order to
19
measure RV (see top right of Fig. 1-1). The vertical flux distribution is created by stellar
absorption lines and white light combs generated by a fixed delay interferometer.
Stellar absorption lines are usually very narrow (with a line width of ∼0.1 Å ), a high
resolution spectrograph with R ≥ 50, 000 is usually required to resolve those lines in order
to precisely determined the line movement. Such a high resolution spectrograph requires a
long light path and therefore large and expensive. It is difficult to realize such design under
tight financial budget and space constraint for a new equipment. In contrast, the fixed delay
interferometer in a DFDI instrument provides an extra spectral resolving power and enables
an DFDI instrument with low spectral resolution to have a equivalent Doppler sensitivity
with a conventional spectrograph at a high resolution. Therefore, a DFDI instrument can be
made at a lower cost and more compact. What’s more, the multi-object capability of a DFDI
instrument makes it attractive for future large area Doppler planet surveys (Ge, 2002; Wang
et al., 2011).
Erskine (2003) used a simple example to illustrate the advantage of the DFDI method
over a conventional spectrograph. I will briefly introduce the illustration and its main con-
clusion. Reader can refer to his original paper for more details. For a conventional spectro-
graph, as illustrated in Fig. 1-2, an intrinsic absorption line is blurred by the limited spectral
resolution. The reaction function to a small frequency shift due to Doppler shift is the slope
of the blurred profile. According to Erskine (2003), the S/N for a fixed Doppler shift of ∆νD is√
2n∆νDHi/√Ai (Ao/Ai )3/2. In contrast, for the DFDI method with a fixed delay interferometer
added into the optical path prior to the dispersing element (as illustrated in Fig. 1-3), the
dominant measurable becomes line depth change instead of line centroid movement. The
S/N for a fixed Doppler shift for the DFDI method is, according to Equation 10 in Erskine
(2003),√n∆νDHi/2
√Ai (Ao/Ai )1/2. Therefore, the ratio of S/N induced by a fixed Doppler
shift, (S/N)DE/(S/N)DFDI is 2√
2/(Ao/Ai ). For example, Ao is ∼1 Å at 5000 Å at R=5,000,
(S/N)DFDI is a factor of 3.5 higher than (S/N)DE for a fixed Doppler shift.
20
Figure 1-1. Top left: a sample spectrum of the DFDI method. The spectrum is composed of astellar absorption line and white light combs generated by a fixed delayinterferometer. Top right: Flux distribution of vertical direction. Bottom: fluxdistribution of horizontal direction, i.e., dispersion direction.
1.4 Major Questions To Be Answered
Facing the number and the diversity of current known exoplanets, we cannot help
wondering many questions. Among them, two sequential questions may be the most
frequent. How common are exoplanets and how common is life on other exoplanets? With
continuously advancing technologies, we begin to have a reasonable handling of the first
question. For the second one, we may not be able to answer until a population of habitable
exoplanets has been discovered. However, we start seeing the tip of iceberg after a couple of
potential habitable worlds are detected (Borucki et al., 2012; Charbonneau et al., 2009).
21
Figure 1-2. Illustration of Doppler sensitivity for a conventional spectrograph (Fig. 6 inErskine (2003)). a) Intrinsic absorption line. b) An absorption line after blurringdue to limited spectral resolution. c) Reaction function to a small frequency shiftdue to Doppler effect, i.e., the slope of line profile in b).
Planet occurrence rate is a complicated issue involving many dependences such as
stellar metallicity, planet and stellar mass. There is a well-established planet-metallicity corre-
lation indicating that occurrence rate rises from 3% for [Fe/H]≤0 to 25% for [Fe/H]≥+0.4 (Fis-
cher & Valenti, 2005). As measurement precision keeps increasing, a population of exoplan-
ets, such as super-Earths and sub-Neptunes (∼ 2 ≤ M ≤∼ 20M⊕), starts to be probed.
This population together with other planets with lower mass are what is called low-mass ex-
oplanets. The study of planet occurrence has been focused on low-mass exoplanets around
solar-type stars. Scientists using HARPS (Mayor et al., 2003) estimated a 30-50% planet
occurrence for super-Earths (Lovis et al., 2009; Mayor et al., 2009a; Udry, 2010). Another
22
Figure 1-3. Illustration of Doppler sensitivity for the DFDI method (Fig. 6 in Erskine (2003)).a) Intrinsic absorption line. b) Interferometer combs. c) Product of intrinsicabsorption line and interferometer combs. d) After blurring, combs are smoothedand become half continuum, ”Bite area” becomes a flux dip. e) Reaction functionto a small frequency shift due Doppler effect. Unlike a conventional spectrograph,flux dip as shown in d) is a more dominant measurable than flux change alongdispersion direction.
23
RV survey yielded a slightly lower occurrence rate ∼20%, which is reported by Howard
et al. (2010b). The occurrence rate is even lower, i.e., 13±0.8% (Howard et al., 2011) or
19% (Youdin, 2011), according to recently-released Kepler data (Borucki et al., 2011c).
Discrepancies among different surveys spurs speculations and efforts to explain (Wolfgang
& Laughlin, 2011), however, a complete understanding would not be obtained until we fully
understand the bias and completeness of each survey and strive for better measurement
precision in the future.
Johnson et al. (2010a) studied the correlation between planet occurrence and stellar
mass and found a positive correlation characterized as a rise from 3% at 0.5M⊙ to 14% at
2.0M⊙. Occurrence rate of low-mass exoplanets around stars other than solar-type stars
is rarely mentioned until very recently. Bonfils et al. (2011b) found that super-Earths are
abundant around M dwarfs (∼35%) and the occurrence rate for habitable planets is 41+54−13%
for a M-dwarf sample in their survey. In comparison, Howard et al. (2010b) found this
number to be 23+16−10% for solar-type stars. The large error bars from their reports indicate the
amount of effort required for constraining the low-mass planet occurrence rate as a function
of stellar mass.
Other questions are as interesting as, if not more interesting than, those mentioned
above. For example, what is the mass distribution of exoplanet? This question helps us
to distinguish between planets and other objects such as brown dwarfs and stars and to
explain the observed ”brown dwarf desert” (Marcy et al., 2005). What is the eccentricity
distribution of exoplanets and what can be inferred from it? What is the statistics of multiple
planetary systems? All these statistical informations help to constrain and refine theoretical
models (Ida & Lin, 2005; Mordasini et al., 2009) that eventually provide complete and
accurate picture of planet formation and evolution.
1.5 Planets Around M Dwarfs
With hundreds of exoplanets discovered, searching for low-mass exoplanets has been
gaining increasing attention. A planet of given mass and semi-major axis would produce
24
larger RV signal around a M dwarf than around a solar-type star. The fact makes M dwarfs
interesting targets in RV surveys for low-mass exoplanets.
1.5.1 Essential Facts About M-dwarf
In astronomy, most stars are classified as O, B, A, F, G, K and M according to their
spectral features such as spectral energy distribution and spectral line characteristics. F, G
and K stars are usually referred to as solar-type stars. M stars have the lowest stellar mass
(M≤∼0.45 M⊙) and the lowest effective temperature (Teff ≤∼3700 K) among all classified
stars. Because of low Teff and thus low luminosity (L≤∼0.08 L⊙), M stars are usually called
M dwarfs because of their positions on a H-R diagram (lower stellar mass end). M dwarfs
spend a very long time on the main sequence, which is even longer compared to the age
of the universe. Therefore, M stars on the main sequence have not evolved to an advanced
evolutionary stages.
According to the law of black body radiation, M dwarfs are generally faint and emit the
bulk of energy in the near infrared (NIR). There are few M dwarfs that can be seen by naked
eyes despite the fact that more than 70% solar neighborhood stars are M dwarfs (Henry,
1998).
1.5.2 M-dwarf Planets
RV signal increases with planet mass and inversely with square root of stellar mass and
semi-major axis. The equation is given by Zechmeister et al. (2009):
K =
√G
1 − e2
m sin i√
(M +m)a= 28.4 km · s−1 · m sin i
MJup
(M⊙M
AUa
)1/2
, (1–1)
where K is RV amplitude, G is the universal gravitational constant, e is eccentricity, m is
planet mass, i is orbital inclination with i=90 when seen edge-on,M is stellar mass and a is
semi-major axis.
As considerable interest has been focused on searching for planets in the habitable
zone (HZ), M dwarfs become promising targets for several reasons:
25
• According to Equation 1–1, for give m and a, K is larger because of low stellar mass ofa M dwarf
• Because of low luminosity of M dwarfs, HZ, defined as a region around a star whereliquid water may exist, is closer in compared to solar-type stars whose stronger radi-ation pushes HZ further away. For example, HZ around the Sun is ∼1 AU, where theEarth orbit is. In comparison, HZ is typically 0.03-0.4 AU for a M dwarfs (Zechmeisteret al., 2009)
• Theoretical work has shown that low-mass planets should be common around Mdwarfs (Ida & Lin, 2005)
1.5.3 Surveys and Results
There are several RV surveys targeting M-dwarf planets (Bean et al., 2010; Blake
et al., 2010; Bonfils et al., 2011b; Clubb et al., 2009; Endl et al., 2006; Zechmeister et al.,
2009). The results of these surveys indicate that, while gas giants are rare (Endl et al., 2006;
Zechmeister et al., 2009), low-mass planets may be abundant around M dwarfs (Bonfils
et al., 2011b). 21 planets around 15 M dwarfs have been detected by the RV technique.
The first exoplanet around a M dwarf was GJ 876 b discovered in 1998 by Delfosse
et al. (1998); Marcy et al. (1998). The second planet GJ 876 c around the star was an-
nounced by Marcy et al. (2001), which is another gas giant in resonance with the one
previously discovered. This type of resonance, commonly found for small objects such as
moons and astroids, is known for gas giants for the first time. In 2004, another gas giant
HD 41004B b was discovered by (Zucker et al., 2004). The HD 41004 binary star system is
unique because it has a brown dwarf orbiting the faint companion and a planet orbiting the
bright companion. The discoveries of gas giants around low-mass M dwarfs pose challenges
to planet formation and evolution theory and motivate theorists to upgrade and refine models.
The first Neptune-size exoplanet was found by Butler et al. (2004) with a minimum mass
m sin i of 21 M⊕. The third planet around GJ 876 d, which is a ∼7.5 M⊕ super-Earth, was
discovered by Rivera et al. (2005) in 2005. With the discovery of the fourth planet, GJ 876
e (Rivera et al., 2010), which is Neptune-sized, GJ 876 system becomes the first known
Sun-analog with two outside gas giants shepherding an inside low-mass planet and the
26
outermost one to be a Neptune-size planet. GJ 581 b was found by Bonfils et al. (2005) in
the same year, which is the prelude for a series of subsequent discoveries of other mem-
bers c,d (Udry et al., 2007) and e (Mayor et al., 2009b) in the system. Bonfils et al. (2007)
detected a planet with a minimum mass of 11 M⊕, GJ 674 b. According to their analysis, they
found evidence of the existence of planet-metallicity correlation for M dwarfs. In the same
year, Johnson et al. (2007a) found the third M-dwarf giant planet with a long period (P=692.9
d). They found giant planet occurrence rate is correlated with host star mass even after cor-
recting for metallicity. Forveille et al. (2009) added one super-Earth in the discovery list after
correcting the claim made by Endl et al. (2008). The planet found by Bailey et al. (2009),
GJ 832 b, hold the record for the longest orbital period and the lowest host star metallicity
among current census of M-dwarf planets. GJ 1214 b is the first planet around a M dwarf
discovered with the transit method and later confirmed with the RV technique (Charbonneau
et al., 2009). Haghighipour et al. (2010) found a saturn-mass planet, HIP 57050 b. In 2010,
three more giant planet discoveries, HIP 79431 b (Apps et al., 2010), GJ 649 b (Johnson
et al., 2010b) and GJ 179 b (Howard et al., 2010a), were announced. Bonfils et al. (2011a)
detected another short-period super-Earth GJ 3634 b.
The discovery list will keep being replenished as exoplanet scientists continue to push-
ing the detection limit for surveys using a variety of methods. For example, 1235 planetary
candidates were announced after Kepler releases its first 4 months of data (Borucki et al.,
2011c). There are M-dwarf planet candidates waiting to be confirmed by other methods such
as the RV technique.
27
CHAPTER 2FUNDAMENTAL PERFORMANCE OF THE DFDI METHOD
2.1 Introduction
The popular Doppler instruments are based on the cross-dispersed echelle spectrogaph
design, which we called the direct echelle (DE) method. In this method, the RV signals are
extracted by directly measuring the centroid shift of stellar absorption lines. The fundamental
photon-limited RV uncertainty using the DE method has been studied and reported by
several research groups (e.g., Bouchy et al. (2001); Butler et al. (1996)). While DE is the
most widely adopted method in precision RV measurements, a totally different RV method
using a dispersed fixed delay interferometer (DFDI) has also demonstrated its capability
in discovering exoplanets (Fleming et al., 2010; Ge et al., 2006b; Lee et al., 2011). In this
method, the RV signals are derived from phase shift of the interference fringes created by
passing stellar absorption spectra through a Michelson type interferometer with fixed optical
path difference (OPD) between the two interferometer arms (Erskine, 2003; Erskine & Ge,
2000; Ge, 2002; Ge et al., 2002). The stellar fringes are separated by a post-disperser, which
is typically a medium-resolution spectrograph. Doppler sensitivity of DFDI can be optimized
by carefully choosing the optical path difference of the interferometer. The DFDI method is
promising for its low cost, compact size and potential for multi-object capability (Ge, 2002).
van Eyken et al. (2010) discussed the theory and application of DFDI in details. However,
its fundamental limit for Doppler measurements has not been well studied before. In this
chapter, I will introduce a method to calculate photon-noise limited Doppler measurement
uncertainty in the near infrared (NIR) wavelength region, where we plan to apply the DFDI
method for launching a Doppler planet survey around M dwarfs.
NIR Doppler planet surveys are very important to address planet characteristics around
low mass stars, especially M dwarfs. M dwarfs emit most of their photons in the NIR region.
Due to the lack of NIR Doppler techniques, only a few hundreds bright M dwarfs have been
searched for exoplanets using optical DE instruments (Blake et al., 2010; Clubb et al., 2009;
28
Endl et al., 2006; Zechmeister et al., 2009). To date, only about 20 exoplanets around M
dwarfs have been discovered compared to more than 700 exoplanets discovered around
solar type stars (i.e., FGK stars) despite of the fact that M dwarfs account for 70% stars in
local universe. Nonetheless, searching for planets around M dwarfs is essential to answer
questions such as the dependence of planetary properties on the spectral type of host
stars. In addition, the smaller stellar mass of M dwarfs favors detection of rocky planets in
habitable zone (HZ) using the RV technique. However, the stellar absorption lines in NIR
are not as sharp as those in the visible band. Recent study by Reiners et al. (2010) shows
that precision RV measurements can only reach better Doppler precision in the NIR than in
visible wavelength for M dwarfs with stellar types later than M4. In this chapter, I will report
results from our study on fundamental limits with the NIR Doppler technique using the DFDI
method.
The theory of DFDI has been discussed by several papers (Erskine, 2003; Ge, 2002;
van Eyken et al., 2010; Wang et al., 2011). Readers may refer to previous references for
more detailed discussion. DFDI is realized by coupling a fixed delay interferometer with a
post-disperser (Fig. 5-1). The resulting fringing spectrum is recorded on a 2-D detector. The
formation of the final fringing spectrum is illustrated in Fig. 2-2. B(ν, y ) is a mathematical
representation of the final image formed at the 2-D detector and it is described by the
following equation:
B(ν, y ) =[S0(ν)hν× IT (ν, y )
]⊗ LSF (ν,R), (2–1)
where S0(ν) is the intrinsic stellar spectrum and ν is optical frequency. S0 is divided by hν
to convert energy flux into photon flux. IT is the intensity transmission function (Equation
2–2), y is the coordinate along the slit direction which is transverse to dispersion direction, ⊗
represents convolution and LSF is the line spread function of the post-disperser which is a
function of ν and spectral resolution R . In Equation 2–2: γ is visibility for a given frequency
channel, the ratio of half of the peak-valley amplitude and the DC offset, which is determined
by stellar flux S0(ν); c is the speed of light; and τ is the optical path difference (OPD) of the
29
!"#$%
&'(%)'*$+,-"(
.'($%/$%01$($%
203(+4"35$%3$%
667
,5$*(%08%)59
Figure 2-1. A schematic layout of an RV instrument using the DFDI method.
! " #! Q
" $ #"# $Q
" $ #% $Q
Q
$
Figure 2-2. DFDI Illustration. S0(ν) is a stellar spectrum; IT (ν, y ) is interferometertransmission; B(ν, y ) is the image taken at a 2-D detector. ν is optical frequencyand y is coordinate of slit direction. DFDI measures RV by monitoring phase shiftof stellar absorption line fringes in the y direction (the slit direction).
30
interferometer which is designed to be tilted along the slit direction such that several fringes
are formed along each ν channel (Middle, Fig. 2-2). We assume the LSF is a gaussian
function (Equation 2–3), ∆ν = ν/R/2.35 because we assume that one resolution element is
equal to the FWHM of a spectral line.
IT = 1 + γ(ν) · cos[2πντ(ν, y )c
], (2–2)
LSF (ν0, ∆ν) =1
2πν2 exp[− (ν − ν0)2
2∆ν2
]. (2–3)
Figure 2-3. Examples of synthetic M dwarf stellar spectra (V sin i=0 km · s−1), which aregenerated by PHOENIX (Allard et al., 2001; Hauschildt et al., 1999). The toppanel shows the spectra between 0.8-1.35 µm, the bottom panel shows anenlarged spectral region around 1177 nm showing stellar line profiles.
Figure 2-3 shows high-resolution (0.005 Å spacing) synthetic spectra of M dwarfs with
solar metallicity (Allard et al., 2001; Hauschildt et al., 1999). Teff ranges from 2400K to
31
Figure 2-4. Top: power spectrum of derivatives of stellar spectrum, F [dS0/dν], withSRF (ρ) = F [LSF (ν,R)] for different R overplotted, where LSF is line spreadfunction; Bottom: F [dS0/dν] shifted by ∆ρ = 20mm using a fixed-delayMichelson interferometer. The SRFs for different R are overplotted.
3100K, and log g is 4.5. No rotational broadening is added in the spectrum. Most absorption
lines are shallow with FWHMs of several tenths of an Å. Since RV information is embedded
in the slope of an absorption line, sharp and deep lines contain more RV information than
broad and shallow lines. Mathematically, the slope is the derivative of flux as a function
of optical frequency, i.e., dS0/dν. The power spectrum of dS0/dν is obtained by Fourier
transform. According to properties of Fourier transform, F [dS0/dν] = (iρ) · F [S0], where
F manifests Fourier transform, i is the unit of imaginary number and ρ is the representation
of ν/c in Fourier space. We plot F ([dS0/dν] in Fig. 2-4 as well as the spectral response
function (SRF), which is F [LSF ]. SRF at R = 5, 000 drops drastically toward high spatial
32
frequency (high ρ value) such that it misses most of the RV information contained in stellar
spectrum. As R increases, SRF gradually increases toward high ρ where the bulk of RV
information is stored. A spectrograph with R of ∼100,000 is capable of nearly completely
extracting RV information. Unlike DE, DFDI can shift F [dS0/dν] by an amount determined by
the OPD of the interferometer (Erskine, 2003). For example, Fig. 2-4 also shows the power
spectrum of F ([dS0/dν] of a fringing spectrum obtained with a DFDI instrument with a 20
mm optical delay, which shifts F [dS0/ν] by 20 mm. In this case, RV information has been
shifted from the original high spatial frequencies to low spatial frequencies which can be
resolved by a spectrograph with a low or medium R in DFDI.
2.2 Methodology of Calculating Photon-limited RV Measurement Uncertainty
In the DE method, an efficient way based on a spectral quality factor (Q) was introduced
by Bouchy et al. (2001) to calculate the fundamental uncertainty in the Doppler measure-
ments. The Q factor is a measure of spectral profile information within a given wavelength
region considered for Doppler measurements. Here we develop a similar method to calculate
Q values for the DFDI method. Instead of representing the spetral line profile information in
the DE method, the Q factor in our DFDI method represents stellar fringe profile information.
We use high resolution (0.005 Å spacing) synthetic stellar spectra generated by PHOENIX
code(Allard et al., 2001; Hauschildt et al., 1999) because observed spectra of low mass
stars do not have high enough resolution and broad effective temperature coverage. Reiners
et al. (2010) have conducted several comparisons between synthetic spectra generated by
PHOENIX and the observed spectra. They concluded that the synthetic spectra are accurate
enough for RV measurement uncertainty calculation. We used synthetic stellar spectra of
solar abundance with Teff ranging from 2400K to 3100K (corresponding spectral type from
M9V to M4V) and a surface gravity log g of 4.5. The Q factor is calculated for a series of
10 nm spectral slices from 800 nm to 1350 nm. We artificially broaden spectra with V sin i
from 0 km · s−1 to 10 km · s−1 assuming a limb darkening index of 0.6, which is a typical value
for an M dwarf. We convolve the rotational broadening profile with each spectral slice of
33
10 nm to obtain a rotationally-broadened spectrum. We assume a Gaussian LSF which is
determined by spectral resolution R (Equation 2–3). After artificial rotational broadening and
LSF convolution, we rebin each spectral slice according to 4.2 pixels per resolution element
(according to the optical design of IRET by Zhao et al. (2010)) to generate the final 2D image
on a detector based on which we compute the Q factor.
2.2.1 Photon-limited RV Uncertainty of DE
Bouchy et al. (2001) described a method of calculating the Q factor for the DE method.
We briefly introduce the method here and the reader can refer to Bouchy et al. (2001) for
more details. Let S0(ν) designate an intrinsic stellar spectrum. A0, a digitalized and calibrated
spectrum, is considered as a noise-free template spectrum for differential RV measurement,
which is related to S0(ν) via the following equation:
A0(i ) =S0(ν)hν⊗ LSF (ν), (2–4)
where i is pixel number and S0 is divided by hν to convert energy flux into photon flux.
Another spectrum A is taken at a different time with a tiny Doppler shift, which is small
relative to the typical line width of an intrinsic stellar absorption. Assuming that the two
spectra have the same continuum level, Doppler shift is given by:
δv
c=δν
ν, (2–5)
where c is speed of light and ν is optical frequency. The overall RV uncertainty for the entire
spectral range is given by (Bouchy et al., 2001):
δvrmsc
= Q−1 ·∑i
A0(i )
−1/2
=1
Q√Ne−, (2–6)
where Q is defined as:
Q ≡
∑i
W (i )∑i
A0(i )
1/2
, (2–7)
34
andW (i ) is expressed as:
W (i ) =
[∂A0(i)∂ν(i)
]2ν(i )2
A(i ). (2–8)
The Q factor is independent of photon flux and represents extractable Doppler information
given an intrinsic stellar spectrum and instrument spectral resolution R . According to
Equation 2–6, we can calculate photon-limited RV uncertainty given the Q factor and photon
flux Ne− =∑i
A0(i ) within the spectral range.
2.2.2 Photon-limited RV Uncertainty of DFDI
A new method of calculating the Q factor for DFDI is developed and discussed here.
After a digitalization process, a 2-D flux distribution expressed by Equation 2–1 is recorded
on a 2-D detector in DFDI. The digitalization process involves distributing photon flux into
each pixel according to: 1) pixels per resolution element (RE); 2) spectral resolution; 3)
number of fringes along slit. B0(i , j), which is a noise-free template, is then calculated.
B(i , j) is a frame taken at a different time with a tiny Doppler shift. i is the pixel number along
the dispersion direction, and j is the pixel number along the slit direction. The observable
intensity change at a given pixel (i , j) in DFDI is expressed by:
B(i , j) − B0(i , j) =∂B0(i , j)∂ν(i )
δν(i )
=∂B0(i , j)∂ν(i )
· δvc· ν(i ). (2–9)
The Doppler shift is measured by monitoring the intensity change at a given pixel in the
equation:δv
c=B(i , j) − B0(i , j)∂B0(i ,j)∂ν(i) · ν(i )
. (2–10)
Frame B0 is assumed to be a noise-free template and the noise of frame B is the quadratic
sum of the photon noise and the detector noise σD :
Brms(i , j) =√B(i , j) + σ2
D. (2–11)
35
Equation 2–11 is approximated under photon-limited conditions as Brms(i , j) =√B(i , j).
Therefore, the RV uncertainty at pixel (i , j) is given by:
δvrms(i , j)c
=
√B(i , j)
∂B0(i ,j)∂ν(i) · ν(i )
. (2–12)
The overall RV uncertainty for the entire spectral range is given by:
δvrmsc
=
∑i ,j
[δvrms(i , j)c
]−2−1/2
≡∑i ,j
W (i , j)
−1/2
≡ Q−1 ·∑i ,j
B0(i , j)
−1/2
=1
Q√Ne−, (2–13)
where
W (i , j) ≡
(∂B0(i ,j)∂ν(i)
)2
ν(i )2
B(i , j), (2–14)
and
Q ≡[∑i ,j
W (i , j)∑i ,j
B0(i , j)
]1/2
. (2–15)
Equation 2–15 calculates the Q factor for the DFDI method, which is also independent of
flux and represents the Doppler information that can be extracted with the DFDI method.
According to Equation 2–13, we can calculate photon-limited RV uncertainty given the Q
factor and photon flux Ne− =∑i ,j
B0(i , j) within the spectral range.
2.3 Comparison between DE and Optimized DFDI
2.3.1 Optimized DFDI
Optical Path Difference (OPD) of a fixed delay interferometer is a crucial parameter that
affects the Doppler sensitivity of a DFDI instruments (Ge, 2002). An optimized OPD can
36
help increase the instrument Doppler sensitivity. We calculate the optimal OPD for spectra of
various Teff and V sin i at different spectral resolutions (Table 2-1). We assume a wavelength
range from 800 nm to 1350 nm and an OPD range from 10mm to 41mm with a step size of
1mm in the calculation as described in §2.2.2. Optimal OPD is selected as the one which
results in the highest Q factor value. Increasing V sin i or decreasing R naturally broadens
absorption lines, decreasing the coherence length of each stellar absorption line (Ge,
2002). Consequently, our simulations show in general that the optimal OPD decreases with
increasing V sin i or decreasing R values (Fig. 2-5). We also note that Teff influence on
optimal OPD is not significant.
Figure 2-5. Optimal OPD correlation with V sin i (left) and spectral resolution R (right).Optimal OPD for R=20,000 (solid), 50,000 (dotted), 80,000 (dashed) are used onthe left panel. V sin i=2 (solid), 5 (dotted), 10 (dashed) km · s−1 are assumed onthe right panel. Complete results of optimal OPD can be found in Table 2-1. Teff
influence on optimal OPD is not significant, Teff=2800 K is adopted in the plot.
37
Tabl
e2-
1.O
PD
choi
ceas
afu
nctio
nofR
andV
sini
atdi
ffere
ntT
eff
RV
sini
[km·s−1
]0
12
34
56
78
910
5,00
019
,27,
2919
,19,
2819
,19,
1915
,15,
1715
,15,
1513
,13,
1311
,12,
1210
,11,
1110
,10,
1010
,10,
1010
,10,
1010
,000
21,2
3,27
19,2
1,26
19,1
9,20
17,1
7,17
15,1
5,15
13,1
3,13
11,1
2,12
11,1
1,11
10,1
0,10
10,1
0,10
10,1
0,10
15,0
0021
,23,
2621
,22,
2319
,20,
2017
,17,
1715
,15,
1514
,14,
1413
,13,
1312
,11,
1111
,11,
1111
,10,
1010
,10,
1020
,000
22,2
3,26
21,2
3,24
20,2
1,21
19,1
9,19
17,1
7,17
15,1
5,15
14,1
4,14
13,1
3,13
12,1
2,12
11,1
1,11
11,1
1,11
25,0
0023
,24,
2623
,24,
2521
,22,
2219
,20,
2019
,18,
1817
,17,
1616
,15,
1514
,14,
1414
,13,
1314
,13,
1212
,12,
1230
,000
24,2
6,26
24,2
5,26
23,2
3,24
21,2
1,21
19,1
9,19
18,1
7,17
17,1
6,16
16,1
5,15
14,1
4,14
14,1
4,14
14,1
4,14
35,0
0026
,26,
2826
,26,
2624
,24,
2522
,22,
2321
,21,
2019
,19,
1919
,18,
1717
,17,
1716
,16,
1616
,16,
1616
,16,
1540
,000
27,2
8,28
27,2
7,28
26,2
6,26
24,2
4,24
22,2
2,22
21,2
0,20
19,1
9,19
19,1
8,18
19,1
8,18
19,1
8,18
19,1
8,16
45,0
0029
,29,
3028
,28,
2927
,27,
2825
,25,
2624
,23,
2322
,22,
2122
,20,
2022
,20,
2022
,20,
1822
,18,
1822
,18,
1850
,000
30,3
0,31
29,3
0,30
28,2
8,28
27,2
6,26
24,2
4,24
24,2
2,23
22,2
2,22
22,2
2,22
22,2
2,22
22,2
2,22
24,2
2,22
55,0
0032
,32,
3231
,31,
3230
,30,
3027
,28,
2827
,26,
2624
,24,
2424
,24,
2424
,24,
2224
,24,
2224
,24,
2224
,24,
2460
,000
32,3
2,34
32,3
2,33
32,3
1,32
29,2
8,28
27,2
7,26
27,2
6,26
24,2
4,24
24,2
4,24
24,2
4,24
24,2
4,24
24,2
4,24
65,0
0034
,34,
3534
,34,
3432
,32,
3232
,30,
3029
,28,
2827
,27,
2627
,26,
2627
,24,
2624
,24,
2624
,24,
2424
,24,
2470
,000
35,3
5,36
35,3
5,36
34,3
2,34
32,3
2,32
32,3
0,30
32,2
8,28
27,2
8,28
27,2
8,26
27,2
4,26
24,2
4,26
24,2
4,26
75,0
0037
,36,
3737
,36,
3735
,35,
3532
,32,
3232
,32,
3232
,32,
3232
,32,
2832
,32,
2832
,32,
3232
,32,
3232
,24,
3280
,000
37,3
7,39
37,3
7,37
37,3
6,36
35,3
4,35
35,3
2,32
32,3
2,32
32,3
2,32
32,3
2,32
35,3
2,32
35,3
2,32
35,3
2,32
85,0
0040
,39,
4140
,39,
3937
,37,
3737
,36,
3635
,32,
3235
,32,
3235
,32,
3235
,32,
3235
,32,
3235
,32,
3235
,32,
3290
,000
40,4
1,41
40,4
1,41
40,3
7,39
37,3
6,36
37,3
6,36
37,3
6,36
35,3
2,36
35,3
2,36
37,3
7,36
37,3
7,36
37,3
7,36
95,0
0041
,41,
4141
,41,
4140
,41,
4140
,37,
3737
,37,
3637
,37,
3637
,37,
3637
,37,
3637
,37,
3737
,37,
3737
,37,
3710
0,00
041
,41,
4141
,41,
4140
,41,
4140
,40,
4137
,37,
3737
,37,
3637
,37,
3637
,37,
3737
,37,
3737
,37,
3737
,37,
3710
5,00
041
,41,
4141
,41,
4141
,41,
4140
,41,
4140
,40,
4137
,37,
3737
,37,
3737
,37,
3737
,37,
3737
,37,
3737
,37,
3711
0,00
041
,41,
4141
,41,
4140
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,37,
4137
,37,
3737
,37,
3737
,37,
3737
,37,
3711
5,00
041
,41,
4141
,41,
4141
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,37,
4137
,37,
3737
,37,
3737
,37,
3712
0,00
041
,41,
4141
,41,
4141
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,37,
4137
,37,
3737
,37,
3737
,37,
3712
5,00
041
,41,
4141
,41,
4141
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,37,
4137
,37,
3737
,37,
3737
,37,
3713
0,00
041
,41,
4141
,41,
4140
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,37,
4137
,37,
3737
,37,
3737
,37,
3713
5,00
041
,41,
4141
,41,
4140
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,37,
4137
,37,
3737
,37,
3737
,37,
3714
0,00
041
,41,
4141
,41,
4140
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,41,
4137
,37,
4137
,37,
3737
,37,
3714
5,00
041
,41,
4141
,41,
4140
,41,
4140
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,41,
4137
,41,
4137
,41,
4115
0,00
041
,41,
4141
,41,
4140
,41,
4140
,41,
4140
,41,
4140
,41,
4137
,41,
4137
,41,
4137
,41,
4137
,41,
4137
,41,
41
38
We investigate how the Q factor is affected if OPD is deviated from the optimal value.
We calculate the Q factor when the actual OPD is deviated from the optimal OPD by 5mm.
We choose the lower value of the two Qs from the 5 mm deviation from the optimal delay
(both positive and negative sides) as Qdeviated. We plot the ratio of Qoptimal and Qdeviated as
a function of spectral resolution R in Fig. 2-6. We found that deviating OPD by 5mm does
not result in severe degradation of the Q factor. The maximum degradation is 1.115 and
occurs at R of 5,000 and V sin i of 5 km · s−1 for a star with Teff of 2800 K. The degradation
can be compensated by increasing the integration time by 24% (1.1152) to reach the same
photon-limited Doppler precision according to Equation 2–13. The degradation becomes
smaller as R increases. As shown in Fig. 2-4, DFDI shifts the power spectrum of dS0/dν
by an amount determined by the interferometer OPD so that the SRF has a reasonable
response at a region where most of the RV information is contained. The SRF broadens as
R increases. Therefore, it can still recover most of the RV information in a stellar spectrum
even if OPD is deviated from the optimal value. At low and medium resolutions (5,000
to 20,000), Q factor degradation becomes larger as V sin i increases. This is because
rotational broadening removes the high frequency signal from a stellar spectrum, which
makes the region containing most of the RV information more sensitive to the choice of OPD
as the power spectrum distribution becomes narrower due loss of high ρ component.
2.3.2 Influence of Spectral Resolution R
In theory, a spectrograph with an infinitely high resolution would be able to extract all
the RV information contained in a stellar spectrum. However, in practice, it is impossible to
completely recover the RV information with a spectrograph with a finite spectral resolution
whose spectral response function drops at the high spatial frequency end. Although the
power spectrum of the derivative of the stellar spectrum is shifted to the low frequency
region where most of the RV information is carried, the power spectrum is still broad in the
spatial frequency (ρ) domain (see Fig. 2-4). Therefore, high R can help to extract more RV
information. In a wavelength coverage from 800 nm to 1350 nm, we calculate Q values for
39
Figure 2-6. Ratio of Qoptimal and Qdeviated as a function of spectral resolution, where Qoptimal isQ factor at optimal OPD and Qdeviated is Q factor when OPD is deviated fromQoptimal by 5 mm. Different line styles represent different V sin i while colorsindicate different Teff.
stellar spectra with V sin i of 0 ,2 ,5 and 10 km · s−1 at different R (5,000 to 150,000 with
a step of 5,000) in order to investigate the dependence of Q on R (Fig. 2-7). We find that
more RV information (higher Q factor) can be extracted as R increases. Q factors for DFDI
and DE converge at high R because the spectral response function is wide enough in the ρ
domain to cover the region rich in RV information, not affected by the power spectrum shifting
involved in DFDI. In addition, the Q factor at a given R increases as Teff drops from 3100K to
2400K, which is largely due to stronger molecular absorption features in the I, Y and J bands
(see Fig. 2-3).
40
Table 2-2. Power Law Index χ as a function of Spectral Resolution R (Teff = 2400K )DFDI DE
V sin iR
5,000 20,000 50,000 5,000 20,000 50,000[km · s−1] -20,000 -50,000 -150,000 -20,000 -50,000 -150,000
0
χ
0.62 0.59 0.31 1.08 0.93 0.442 0.63 0.56 0.27 1.07 0.89 0.385 0.62 0.45 0.16 1.01 0.69 0.21
10 0.58 0.28 0.09 0.87 0.39 0.10
We divide R into three regions, low resolution (5,000 to 20,000), medium resolution
(20,000 to 50,000) and high resolution (50,000 to 150,000). We use a power law to fit Q for
both DFDI and DE as a function of R . The power indices χ of three regions for Teff = 2400K
are presented in Table 2-2. At low R region, χ remains roughly a constant for 0 km · s−1
≤ V sin i ≤ 5 km · s−1, but it drops for stars with V sin i of 10 km · s−1 indicating stellar
absorption lines begin to be resolved even at low R . At higher R regions, χ decreases as
V sin i increases, a reduced value of χ implies diminishing benefit brought by increasing R .
Stellar absorption lines are broadened by stellar rotation, and they are resolved at a certain
R beyond which increasing R does not significantly gain Doppler sensitivity. Overall, χ for
DE is larger than that of DFDI, especially for low and medium R . In other words, QDFDI is
less sensitive to a change of R , and the DFDI instrument can extract relatively more Doppler
information at low or medium spectral resolution than the DE method. For example, for
slow rotators (V sin i=2 km · s−1) at the low R region (R=5,000-20,000), QDFDI ∝ R0.63.
Doppler sensitivity δvrms is inversely proportional to two factors: Q and√Ne− according to
Equation 2–6 and 2–13, where Ne− is the total photon count collected by the CCD detector.
Ne− ∝ (S/N)2 · Npixel, where S/N is the average signal to noise ratio per pixel, and Npixel is
total number of pixels. Note that Ne− ∝ R if the wavelength coverage, S/N per pixel and the
resolution sampling are fixed. Therefore, δvrms ∝ R−0.63−0.5 = R−1.13 for DFDI. In comparison,
δvrms ∝ R−1.57 for DE given the same wavelength coverage and S/N per pixel. The power law
is consistent with the previous theoretical work by Ge (2002) and Erskine (2003).
41
Figure 2-7. Q factor as a function of spectral resolution. (left: Teff = 2400K ; right:Teff = 3100K . Open circles represent QDFDI; crosses represent QDE; solid linesare best power-law fits for QDFDI; dashed lines are best power-law fits for QDE)
We compare Q factors for both DFDI and DE at given R values, and the results are
shown in Fig. 2-8. For very slow rotators (0 km · s−1 ≤ V sin i ≤ 2 km · s−1), the advantage
of DFDI over DE is obvious at low and medium R (5,000 to 20,000) because the center of
the power spectrum of the derivative of the stellar spectrum is at a high frequency domain
which cannot be covered in DE due to the limited frequency response range of its SRF at low
and medium R . The improvement of DFDI is ∼3.1 times (R=5,000), ∼2.4 times (R=10,000)
and ∼1.7 times (R=20,000) respectively. In other words, optimized DFDI with R of 5,000,
10,000 and 20,000 is equivalent to DE with R of 16,000, 24,000 and 34,000 respectively in
terms of Doppler sensitivity for the same wavelength coverage, S/N per pixel and spectral
sampling (otherwise, see more discussions in §2.3.3, the gain with the DFDI would be more
42
significant for a fixed detector size and exposure time). Overall, DE with the same spectral
resolution as DFDI at R=5,000-20,000 requires ∼3-9 times longer exposure time to reach the
same Doppler sensitivity as DFDI if both instruments have the same wavelength coverage
and same detection efficiency (i.e., Ne− is the same). The improvement of DFDI at R=20,000-
50,000 is not as noticeable as at the low R range. The difference between DFDI and DE
becomes negligible when R is over 100,000. In other words, the advantage of the DFDI over
DE gradually disappears as R reaches high resolution domain (R > 50, 000). In addition, the
improvement for relatively faster rotators (5 km · s−1 ≤ V sin i ≤ 10 km · s−1) with DFDI is less
significant than it is for slow rotators.
Figure 2-8. Improvement of QDFDI over QDE as a function of spectral resolution.
43
2.3.3 Influence of Detector Pixel Numbers
In the NIR, detector pixel number is typically smaller than the optical detector. Further-
more, the total cost for an NIR array is much higher than an optical detector with the same
pixel number. In the foreseeable future, detector size may be one of the major limitations
for Doppler sensitivity improvement. We study the impact of the limited detector resource
on the Doppler measurement sensitivity. Using the same detector resource, we find that it
is fair to compare their Doppler performance for the same target with the same exposure
to understand strength and weakness for each method although DFDI and DE are totally
different Doppler techniques. According to Equation 2–6 and 2–13, we define a new merit
function,
Q ′ = Q ·√Ne−, (2–16)
to study photon-limited Doppler performance for both methods with the same detector size.
Note that the newly defined merit function is directly related to photon-limited RV uncertainty,
i.e., inverse proportionality. Ne− is calculated by Equation 2–17:
Ne− =F∗ · η · Stel · texp
2.512mJ, (2–17)
in which F∗ is the photon flux in the wavelength coverage region ∆λ of an mJ = 0 star with
the unit of photons · s−1 · cm−2; η is instrument total throughput; Stel is the effective surface area
of the telescope; texp is the time of exposure; and mJ is the J band magnitude.
Here we use IRET (Zhao et al., 2010) as an example to illustrate strengths of the DFDI
method for Doppler measurements. IRET adopts the DFDI method and has a wavelength
coverage from 800 nm to 1350 nm and a spectral resolution of 22,000. For a fixed detec-
tor size (i.e., total number of detector pixels) and fixed number of pixels to sample each
resolution element, the total wavelength coverage of a Doppler instrument, ∆λ, is
∆λ =Npix
Porder· λcR · NS
, (2–18)
44
where Npix is total number of pixels available on a CCD detector and NS is the number of
pixels per resolution element, λc is the central wavelength and Porder is the number of pixels
sampling each pixel channel between spectral orders1 . Equation 2–18 shows that ∆λ is
inversely proportional to R . Table 2-3 gives the relation of R and ∆λ assuming Npixel, Porder
and NS as constants. λc is set to be 1000 nm because it is approximately the center of the
Y band. We calculated the ratio of Q ′DFDI and Q ′DE in which we use the photon flux of a star
with a Teff of 2400K (Fig. 2-9). Q ′IRET is consistently higher than Q ′DE regardless of R of the
DE instrument. In other words, IRET is able to achieve lower photon-limited RV uncertainty
compared to a DE instrument with the same detector. The result seems to be different from
the conclusion we drew in §2.3.2, in which we compare Q factors of the same R and ∆λ and
reached a conclusion that DFDI with an R of 22,000 is equivalent to DE with an R of 35,000
(a factor of 1.6 gain) for the same wavelength coverage and the same detection efficiency
(Fig. 2-7). The key difference between this case and the earlier case is the fixed detector
resource instead of fixed total collected photon numbers. Since lower spectral resolution
allows to cover more wavelengths, more photons will be collected for the same instrument
detection efficiency for both DFDI and DEM. Note that Q ′ consists of two components, Q
and Ne− . For a given number of pixels on the detector, Ne−,DFDI is higher than Ne−,DE due
to the larger wavelength coverage. In addition, QDFDI(∆λDFDI) is more than QDE(∆λDE).
Consequently, we see in Fig. 2-9 that Q ′IRET is higher than Q ′DE at all R of a DE instrument.
Figure 2-9 also shows that the minimum of Q ′DFDI/Q′DE is dependent of V sin i . The ratio of
Q ′ reaches a minimum (Q ′DE reaches a maximum) around an R of 50,000 for slow rotators
(V sin i ≤ 5 km · s−1). It increases at the low R end because the spectrograph has not yet
1 In principle, each frequency channel for both DFDI and DE instruments can be designedidentical. In practice, the DFDI instrument tends to use ∼ 20 pixels to sample fringes in theslit direction, which is only for measurement convenience, not a requirement. In fact, Muir-head et al. (2011) has demonstrated a phase-stepping method which does not require sam-ple fringes in the slit direction.)
45
resolved stellar absorption lines. On the other hand, the ratio increases at the high R end
because of fewer photons (see Table 2-3). For fast rotators (V sin i=10 km · s−1), the ratio
reaches a minimum around R of 30,000.
Table 2-3. Spectral Resolution and wavelength coverage on a given detectorR ∆λ λmin − λmax
(nm) (nm)25,000 480 800−128030,000 400 800−120040,000 300 850−115050,000 240 880−112060,000 200 900−111070,000 170 910−108080,000 150 920−1070
Figure 2-9. Comparison of Q ′IRET and Q ′DE at different R . Note that Q ′ = Q ·√Ne− . Different
color represents different rotational velocity.
46
For a slow rotator (V sin i=2 km · s−1) at low R region (R=5,000-20,000), QDFDI ∝ R0.63.
Since Doppler sensitivity δvrms is inversely proportional to two factors: Q and√Ne− according
to Equation 2–6 and 2–13, the Doppler sensitivity becomes nearly independent of spectral
resolution for the DFDI method (∝ R−0.13) if the detection size (or total number of pixels) is
fixed. This indicates that we can use quite moderate resolution spectrograph to disperse the
stellar fringes produced by the interferometer in a DFDI instrument while maintaining high
Doppler sensitivity. This opens a major door for multi-object Doppler measurements using
the DFDI method as proposed by Ge (2002). In comparison, the Doppler sensitivity for the
DE method still strongly depends on spectral resolution for a fixed number of detector pixels
(∝ R−0.57), indicating that higher spectral resolution will offer better Doppler sensitivity.
2.3.4 Influence of Multi-Object Observations
As discussed in §2.3.2 and 2.3.3, the DFDI instrument can be designed to have a
moderate resolution spectrograph coupled with a Michelson type interferometer. Moderate
spectral resolution allows a single order spectrum or a few order spectra to cover a broad
wavelength region in the NIR region while keeping the Doppler sensivitiy similar to a high
resolution DE design which requires a large detector array to cover spectra from a single
target. This indicates that the DFDI method has much greater potential for accommodating
multiple targets on the same detector as proposed by Ge (2002) than a DE instrument. In
order to evaluate the potential impact of multi-object DFDI instruments, we redefine the merit
function Q ′′ as:
Q ′′ = Q ·√Ne− · Nαobj , (2–19)
where Nobj is the number of objects that can be monitored simultaneously, and α is the index
of importance for multi-object observations. From the perspective of photon count and S/N,
multi-object observations are equivalent to an increase of Ne− , and thus α is 0.5. However,
from an observational efficiency point of view, Q ′′ should be proportional to Nobj because the
more objects are observed simultaneously, the quicker the survey is accomplished, and α is
therefore equal to 1.
47
Figure 2-10. Comparison of Q ′′DFDI and Q ′′DFDI,R=100,000 at different R . Note thatQ ′′ = Q ·
√Ne− · Nαobj . The maximum of each curve is indicated by filled circle.
Different color represents different rotational velocity, the same as Fig. 2-9.
We assume a detector that covers from 800 nm to 1350 nm at R=100,000 so that we
can use the Q factors obtained in §2.3.2. Ne− is a constant since we assume identical ∆λ.
Nobj is inversely proportional to the number of pixels per object which is proportional to
spectral resolution R (Equation 2–18). Note that we do not require Nobj to be an integer
because we can, in principle, fit a fraction of spectrum on a detector to make full use of the
detector. Q ′′s for both DFDI and DE are calculated. Figure 2-10 shows the ratio of Q ′′ and
Q ′′R=100,000 for DFDI under two different assumptions of α. For α=0.5, i.e., increase of Nobj is
equivalent to photon gain, only a slight improvement is achieved if the detector is used for
multi-object observations at lower resolution than 100,000. In comparison, from a survey
efficiency point of view (i.e., α=1), we see a factor of ∼4-6 times boost of Q ′′ in multi-object
48
Figure 2-11. Comparison of Q ′′DE and Q ′′DE,R=100,000 at different R . Note thatQ ′′ = Q ·
√Ne− · Nαobj . The maximum of each curve is indicated by filled circle.
Different color represents different rotational velocity, the same as Fig. 2-9.
observations. The truncation at R=5,000 is due to a practical reason that a lower resolution
than 5,000 is rarely used in planet survey using RV techniques. On the other hand, similar
calculation is also conducted for the DE method (Fig. 2-11), in which we find that high
resolution single object observation is an optimal operation mode for DE from a perspective
of photon gain (α=0.5). At α=1, the increase of Q ′′ is a factor of ∼3 at the most.
We compare the maximum of Q ′′ for both DFDI and DE at different V sin i in Table 2-4.
At α=0.5, the advantage of DFDI over DE is ∼1.1 for a wide range of V sin i . In other words,
from the photon gain point of view, there is no significant difference between DFDI and DE
in multi-object RV instruments. However, from the survey efficiency point of view (α=1), we
see a factor of 3 boost of Q ′′ in DFDI for slow rotators (V sin i ≤2km · s−1), suggesting 9
49
Table 2-4. Q ′′ comparison of DFDI and DE as a function of V sin iDFDI DE
V sin i [km · s−1] Roptimal Q ′′DFDI Roptimal Q ′′DE Q ′′DFDI/Q′′DE
α=0.5
0 50,000 7502 110,000 6623 1.0652 50,000 6806 75,000 6001 1.1345 25,000 4979 50,000 4424 1.125
10 15,000 3394 25,000 2996 1.133
α=1.0
0 5,000 26384 30,000 8705 3.0312 5,000 24297 25,000 8490 2.8625 5,000 18929 15,000 7884 2.401
10 5,000 12877 10,000 7239 1.779
times faster in terms of survey speed. For fast rotators (i.e., V sin i=10km · s−1), the boost
drops to 1.78. Our study confirms that the DFDI method has an advantage for multi-object
RV measurements over the DE method as suggested by Ge (2002).
2.3.5 Influence of Projected Rotational Velocity V sin i
Projected rotational velocity V sin i broadens stellar absorption lines and thus reduces
the Q factor. We carry out simulations calculating Q factors of different V sin i (0 km · s−1 ≤
V sin i ≤10 km · s−1) at R=150,000 and various Teff. We assume a wavelength range from
800 nm to 1350 nm. The results are shown in Fig. 2-12. The Q factor decreases as V sin i
increases. It is clear that slow rotators would be better targets to reach higher photon-limited
RV precision because the spectrum of a slow rotator contains more Doppler information.
2.4 Summary and Discussion
2.4.1 Q Factors for DFDI, DE and FTS
We develop a new method of calculating photon-limited Doppler sensitivity of an
instrument adopting the DFDI method. We conduct a series of simulations based on
high resolution synthetic stellar spectra generated by PHOENIX code(Allard et al., 2001;
Hauschildt et al., 1999). In simulations, we investigate the correlations of Q and other
parameters such as OPD of the interferometer, spectral resolution R and stellar projected
rotational velocity V sin i . We find that optimal OPD increases with increasing R and
decreasing V sin i . Empirically, the optimal OPD is chosen such that the density of the
50
Figure 2-12. QDFDI as a function of V sin i .
interference combs matches with the line density of the stellar spectrum. Based on the
simulation results, the optimal OPD is determined as the one that maximizes the Q factor. In
fact, optimal OPDs found from empirical way and from numerical simulation are consistent
with each other. For example, for V sin i=0 km·s−1 and R=50,000, simulation gives an optimal
OPD of 30 mm. The interference comb density of an interferometer with OPD of 30 mm is
∼0.3 Å at 1000 nm, which indeed matches the width of a typical absorption line after spectral
blurring with R of 50,000. An independent method to calculate photon-noise limited Doppler
measurement uncertainty in the optical is being developed, and the results will be reported
in a separate paper (Jiang et al., 2011). We have compared results from both methods and
confirmed that both independent methods produce essentially the same results for both
optical and NIR Doppler measurements.
51
We investigate how the Q factor is affected if OPD is deviated from the optimal value
and find that a deviated OPD (5mm) does not result in a significant Q factor degradation,
which is mitigated as R increases. We find that the Q factor increases with R for both DFDI
and DE, and eventually converge at very high R (R ≥100,000). The convergence of DFDI
and DE methods is a natural consequence because the measurement method does not
make a difference after the spectral resolution becomes extremely high. In addition, Q
factors at a given R increase as Teff drops from 3100K to 2400K, which is due to stronger
molecular absorption features in NIR (see Fig. 2-3). The Q factor decreases as V sin i
increases because stellar rotation broadens the absorption lines, leading to less sensitive
measurement.
We compare Q factors for both DFDI and DE at a given R . For slow rotators (0 km · s−1 ≤
V sin i ≤ 2 km · s−1), DFDI is more advantageous over DE at low and medium R (5,000 to
20,000) for the same wavelength coverage ∆λ. The improvement of DFDI compared to DE
is ∼3.1 (R=5,000), ∼2.4 (R=10,000) and ∼1.7 (R=20,000), respectively. In other words,
optimized DFDI with R of 5,000, 10,000 and 20,000 are equivalent in Doppler sensitiv-
ity to DE with R of 16,000, 24,000 and 34,000, respectively. The improvement of DFDI
at R 20,000 to 50,000 is not as noticeable as at low R range. The difference between
DFDI and DE becomes negligible when R is over 100,000. For relatively faster rotators
(5 km · s−1 ≤ V sin i ≤ 10 km · s−1), the improvement with DFDI is less obvious than it is for
very slow rotators. DFDI has strength when the spectral lines in a stellar spectrum are not
resolved by a spectrograph, which is the case for low and medium resolution spectrograph.
Under such conditions, the fixed delay interferometer provides additional resolving powers for
the system. After the lines are fully resolved by the spectrograph itself, the interferometer in
the system becomes dispensable, which is the reason why we see the convergence of DFDI
and DE at very high spectral resolution.
Fundamental performance of a Fourier-transform spectrometer (FTS) in the application
of Doppler measurements has been discussed by Maillard (1996). There are similarities
52
between the FTS and the DFDI method, for example: 1, both methods use the interferom-
eter as a fine spectral resolving element; 2, RV is measured by monitoring the temporal
phase change at a fixed OPD of the interferometer. In DFDI method, OPD is scanned
in each frequency channel because of two relatively tilted mirrors, and the resolution of
the post-disperser in DFDI is chosen to ensure a reasonable fringe visibility. Therefore,
the DFDI method is a extended version of the FTS method with a low-medium resolution
post-disperser. However, one major difference between these two methods is that the in-
terferometer itself is used as a spectrometer by OPD scanning in the FTS method while an
additional spectrograph is employed in the DFDI method. The advantage of introducing an
additional spectrograph into the system is that the visibility (or fringe contrast) is no longer
limited by the bandpass as in the FTS case, which is the reason that the DFDI method
can be applied in broad-band Doppler measurements. Mosser et al. (2003) discussed the
possibility of an FTS working in broad band by introducing a low resolution post-disperser
and concluded that the FTS method is inferior (by a factor between 1 and 2) to DE method
even after employing an post-disperser. This conclusion should be accepted with cautions
because they compared an FTS with a post-disperser (R=1200) with a DE instrument with a
much higher spectral resolution (R=84,000), which is not necessarily a fair comparison.
We define new merit functions (Equations 2–16 and 2–19) to objectively evaluate
Doppler performance for both DFDI and DE methods. For Q ′, the merit function for single
object observation, we find that Q ′DFDI is consistently higher than Q ′DE regardless of the R of
the DE instrument under the constraint of total number of pixels, i.e., both the DFDI and DE
instrument adopt the same NIR detector. The DE instrument requires using a larger detector
in order to reach the same wavelength coverage as the DFDI instrument. Note that the above
conclusion is based on the assumption that the number of pixels per spectral order are the
same for DFDI and DE. In practice, a DFDI instrument uses ∼20 pixels to sample spatial
direction, i.e., the direction transverse to dispersion direction, while ∼5 pixels are usually
used to sample spatial direction in a DE instrument. However, the ∼20 pixels sampling is
53
not a requirement for DFDI but rather for the convenience of data reduction. Normally, ∼7
pixels sample one spatial period of a stellar fringe, which in principle are adequate based on
a phase-stepping algorithm provided by Erskine (2003).
If the same detector is used, the spare part of the detector in DFDI can be used
for multi-object observations. Consequently, in addition to single-object instrument, we
also investigate Q ′′, a merit function for multi-object RV measurement for both DFDI and
DE. Different conclusions are reached depending on different value of α, an index of the
importance of multi-object observation. From a pure photon gain point of view, DFDI and
DE instruments have similar Q ′′ values with Q ′′DFDI slightly better than Q ′′DE (a factor of ∼1.1).
From a survey efficiency point of view, a DFDI multi-object instrument is 9 times faster than
its counterpart using DE for slow rotating stars (0 km · s−1 ≤ V sin i ≤ 2 km · s−1) and ∼4 times
faster for fast rotators (V sin i ≥ 10 km · s−1).
2.4.2 Application of DFDI
There may be other practical concerns about the instrument using the DFDI method,
most of them are due to the relative low spectral resolution compared to current DE instru-
ments. First of all, an absolute wavelength calibration for a DFDI instrument is not as precise
as a DE instrument with a higher spectral resolution. For example, at a spectral resolution
of 22,000 for IRET, a line profile with a FWHM of ∼0.45 Åin Y band is expected. Following
the method described in Butler et al. (1996), it corresponds to 136.4 m · s−1 RV uncertainty
at a S/N of 100 if only one spectral line is used. Ramsey et al. (2010) proposed to use a
U-Ne emission lamp as a wavelength calibration source and it has approximately ∼500
lines in Y band according to their measurement. Therefore, after all the lines in Y band are
considered, ∼6 m · s−1 RV uncertainty is introduced in the process of absolute wavelength
calibration. In comparison, a DE instrument at R of 110,000 causes ∼1.2 m · s−1 RV uncer-
tainty in an absolute wavelength calibration. However, an absolute wavelength solution is
only required for the DE method in order to measure RV drift due to instrument instability,
which is measured in a different method in a DFDI instrument. It is similar to a stellar RV
54
measurement, the difference is that the object is switched from a star to an wavelength cali-
bration source. Vertical fringe movement of absorption or emission lines of an RV calibration
source is measured instead of centroid movement measurement in a DE instrument. In this
case, a DFDI instrument (e.g., IRET, R=22,000) is equivalent to a DE instrument with R of
37,000 in terms of Doppler measurement precision (see §2.3.2). Therefore, instrument RV
drift calibration process introduces an RV uncertainty of ∼3.5 m · s−1 for IRET in the example
of a U-Ne lamp calibration source. In addition, the RV uncertainty can be further reduced
by increasing S/N and number of measurement. Secondly, at a low spectral resolution, it is
challenging to perform spectral line profile analysis and thus it requires high-resolution follow-
up in order to confirm or exclude a possible detection. Last but not least, in a binary case
in which the observed spectrum is blended, two approaches can be used for identification:
1, from measured RV, if the flux ratio is small, similar to the planet companion case, then
the lower mass companion can be identified in the measured RV curve even though small
flux contamination exists; if the flux ratio is about unity, indicating strong flux contamination,
a large RV scattering is expected because this case is not considered and modeled in the
current data reduction pipeline; 2, from measured spectrum, even the observed spectrum
is a 2-D fringing spectrum in DFDI, we can still de-fringe the spectrum into a 1-D traditional
spectrum, on which special treatment can be performed to quantify the blending such as
TODCOR (Zucker et al., 2003).
55
CHAPTER 3COMPREHENSIVE SIMULATIONS FOR HABITABLE PLANET SEARCH IN THE NIR
Discovering an Earth-like exoplanet in habitable zone is an important milestone for
astronomers in search of extra-terrestrial life. While the radial velocity (RV) technique re-
mains one the most powerful tools in detecting and characterizing exo-planetary systems,
we calculate the uncertainties in precision RV measurements considering stellar spectral
quality factors, RV calibration sources, stellar noise and telluric contamination in different ob-
servational bandpasses and for different spectral types. We predict the optimal observational
bandpass for different spectral types using the RV technique under a variety of conditions.
We compare the RV signal of an Earth-like planet in the habitable zone (HZ) to the near
future state of the art RV precision and attempt to answer the question: How close are we to
detecting Earth-like planet in the HZ using the RV technique?
3.1 Introduction
The fundamental photon-noise RV uncertainties have been discussed in several pre-
vious papers (Bouchy et al., 2001; Butler et al., 1996). However, only intrinsic properties of
stellar spectra are discussed in their works while no detailed calculation of RV uncertainties
introduced by the calibration sources. In a recent paper, Reiners et al. (2010) considered
the uncertainties in the NIR caused by RV calibration sources, i.e., a Th-Ar lamp and an
Ammonia gas absorption cell. However, these two RV calibration sources can not be com-
pletely representative of the calibration sources used and proposed in current and planned
Doppler planet survey in the NIR. For example, there are other emission lamps available in
the NIR for RV calibration such as a U-Ne lamp as proposed by Mahadevan et al. (2010).
In addition, other gas absorption cells besides the Ammonia cell have been proposed in
the NIR (Mahadevan & Ge, 2009; Valdivielso et al., 2010). Futhermore, in Reiners et al.
(2010), the calculation of RV calibration uncertainty of a gas absorption cell assumes a 50
nm band width in K band, and then the uncertainty was applied to other NIR bands, which is
purely hypothetical. Therefore, a more comprehensive and detailed study of RV calibration
56
uncertainties is necessary at different observational bandpasses in the NIR in the search
of planets around cool stars. On the other hand, in the visible, even though the current RV
precision is not limited by the RV calibration source such as a Th-Ar lamp or an Iodine ab-
sorption cell, a better understanding of their performances under the photon-limited condition
helps us discern a stage in which the RV calibration source becomes the bottle neck as RV
precision keeps improving. Rodler et al. (2011) recently investigated RV precision achiev-
able for M and L dwarfs, but did not quantitatively discussed the influence of RV calibration
sources and stellar noise on precision Doppler measurement.
Stellar noise is a significant contributor to RV uncertainty budget, which falls into three
categories: p-mode oscillation, spots and plagues, and granulations. P-mode oscillation
usually produces an RV signature with a period of several minutes. The oscillation mode
has been relatively well studied by previous work (e.g., Carrier & Bourban (2003); Kjeldsen
et al. (2005)). Exposure time of 10-15 min is proposed in order to smooth the RV signature
induced by p-mode oscillation (Dumusque et al., 2011). Spots and plagues induced RV
signal has been discussed by several papers (e.g., Desort et al. (2007); Lagrange et al.
(2010); Meunier et al. (2010); Reiners et al. (2010)). Meunier et al. (2010) concluded that
the photometric contribution of plages and spots should not prevent detection of Earth-mass
planets in the HZ given a very good temporal sampling and signal-to-noise ratio. Granulation
is considered to be the major obstacle in detection of Earth planets in the HZ because
it produces an RV signal with an amplitude of 8∼10 m · s−1 based on observation on the
Sun (Meunier et al., 2010). In addition, there is by far no good method of removing the RV
noise from this phenomenon. Dumusque et al. (2011) provided a model of noise contribution
in RV measurements based on precision RV observation on stars of different spectral type
and at different evolution stages.
The telluric lines from the Earth’s atmosphere are usually masked out in calculations
of the photon-limited RV uncertainties in NIR (Reiners et al., 2010; Rodler et al., 2011).
Although Wang et al. (2011) proposed a method to quantitatively estimate the influence of
57
atmosphere removal residual on precision Doppler measurement using the Dispersed Fixed
Delay Interferometer (DFDI) method (Erskine, 2003; Ge, 2002; van Eyken et al., 2010), no
attempt has ever been made for precision Doppler measurements using a high-resolution
Echelle spectrograph. In practice, the telluric lines are not masked out, but instead modeled
and removed. Therefore, a quantitative way of estimating the RV uncertainties produced by
the residual of telluric line removal is necessary before we fully understand the performance
of an RV instrument. In the visible band, the estimation of telluric line contamination is
equally important as higher RV precision is required in the search of lower-mass planets
around solar type stars.
We address two basic questions in this chaper after considering a variety of factors
including stellar spectrum quality, RV calibration precision, stellar noise and atmosphere
contamination: 1, which observational bandpass is optimal to conduct precision Doppler
measurements for stars of different spectral types; 2, is current RV precision adequate for
detecting Earth-like planets in the HZ in the most optimistic scenario, in which the star is the
least active and telluric lines are perfectly modeled and removed. The methods and findings
of this study will provide insights to the design and optimization of a planned or ongoing
precision Doppler planet survey. In addition, it also helps us to access at what stage we are
in the search of Earth-like planets in the HZ.
3.2 Simulation Methodology
3.2.1 High Resolution Synthetic Spectra
Because observed stellar spectra do not have high enough spectral resolution and
broad effective temperature coverage, we decide to use high resolution synthetic stellar
spectra in the calculation of photon-limited RV uncertainty. For solar type stars, i.e., FGK type
stars (3750 K≤ Teff ≤7000 K), we adopt the spectra with a 0.02 Å sampling from Coelho
et al. (2005). For M dwarfs, (2400 K≤ Teff ≤3500 K), we use high-resolution (0.005 Å sam-
pling) synthetic stellar spectra generated by PHOENIX code(Allard et al., 2001; Hauschildt
58
et al., 1999). Reiners et al. (2010) conducted several comparisons between synthetic spec-
tra generated by PHOENIX and observed spectra in NIR. They concluded that the synthetic
spectra are accurate enough for the purpose of simulations. For more massive stars (7000
K< Teff ≤9600 K), we also use the synthetic spectra with 0.005 Å sampling generated
by PHOENIX. Throughout the chapter, we assume a metallicity of solar abundance and a
surface gravity log g of 4.5 for main sequence stars. We assume a Gaussian line spreading
function (LSF) which is determined by spectral resolution(R). After an artificial rotational
line broadening using a kernel provided by Gray (1992) and an LSF convolution, we rebin
each spectral slice according to 4.0 pixels per resolution element (RE) to generate the a one-
dimensional spectrum. In comparison, the sampling rate is 3.2 pixel/RE for HARPS (Mayor
et al., 2003) and 3.5 pixel/RE for HIRES (Vogt et al., 1994).
We compare the synthetic spectra to the observed ones in the visible to ensure that
the synthetic spectra are good approximations of observed stellar spectra (Bagnulo et al.,
2003). Comparison in NIR requires carefully removing telluric lines from the observed
stellar spectra, which is beyond the scope of my work. Figure 3-1 shows comparisons of
synthetic spectra and the observed high resolution (R=80,000) stellar spectra from Bagnulo
et al. (2003). The comparison spans a wide range of spectral types from M6V to A5V.
The synthetic spectrum of an A5V star matches well the an observed one with an RMS of
0.02. As the features in a stellar spectrum increases due to cooler Teff and slower stellar
rotation, the RMS increases due to an increasing complexity of comparison and imprecise
spectral line modeling. The RMS of difference is 0.05 and 0.04 for an F8V and a G2V star.
It get worse in the comparison for a K5V star, in which the RMS is 0.10. And the RMS of
difference is 0.05 for an M6V star. The results from the comparisons between synthetic
and observed spectra indicate the difficulty in modeling the spectra of cool stellar objects.
Although not perfect, the synthetic spectra are able to reproduce majority of the features in
the observed spectra. Therefore, we decide to use the synthetic spectra in our calculation
59
Figure 3-1. Comparisons between synthetic and observed spectra. Black lines representobserved spectra and red lines are synthetic spectra after rotational linebroadening and LSF convolution at R=80,000. The Teff and V sin i are chosenaccording to the spectral type and line width empirically, they are not necessarilythe best-fit parameters for the observed spectra. The chosen Teff and V sin i are,from top to bottom, 9000 K and 80.0 km · s−1 for HD 39060 (A5V), 6250 K and 4.5km · s−1 for HD 30562 (F8V), 5750 K and 6.0 km · s−1 for HD 14802 (G2V), 4750 Kand 4.0 km · s−1 for HD 10361 (K5V), 2900 K and 10.0 km · s−1 for HD 34055(M6V). The difference between observed and synthetic spectrum is also plottedat the bottom of each panel with RMS of difference.
60
of RV uncertainty. Photon-limited RV measurement uncertainty is calculated based on the
method discussed in §2.2.1.
3.2.2 RV Calibration Sources
RV calibration sources are important in precision Doppler measurements because
they not only provide wavelength solutions but also help track drift due to instrument in-
stabilities. The RV uncertainties due to calibrations must be considered if we want to fully
understand the performance of an RV instrument. We consider the photon-limited uncer-
tainties introduced by RV calibration sources based on their spectral quality factors. Two
types of calibration sources have been successfully applied in RV measurements in the
visible bands: 1), a Th-Ar emission lamp (Lovis & Pepe, 2007); 2), an Iodine gas absorption
cell (Butler et al., 1996). Searching for planets in NIR using the RV technique has already
been conducted by several groups (Bean et al., 2010; Blake et al., 2010; Figueira et al.,
2010b; Mahadevan et al., 2010; Muirhead et al., 2011) and several high resolution NIR spec-
trographs will be put into use in the foreseeable future (Ge et al., 2006a; Quirrenbach et al.,
2010). We limit the discussions in the emission lamps and gas absorption cells although
there are other candidates for RV calibration sources, for examples, laser combs (Li et al.,
2008; Steinmetz et al., 2008), which are unfortunately very expensive and not yet readily
available, and interferometer calibration sources as proposed by Wildi et al. (2010) and Wan
& Ge (2010)
In the following discussions, the RV calibration sources are categorized by the obser-
vational bandpass in which they are applied. The corresponding wavelength range for each
observational bandpass is given in Table 3-1.
In B band, a Th-Ar lamp is a suitable calibration source. The lines list of a Th-Ar
lamp from Lovis & Pepe (2007) is adopted in my work. Only Thorium lines are used in
the calculation because the instability of Argon lines is at the order of ∼10 m · s−1, which
is not stable enough for high precision Doppler measurements. A Iodine absorption cell
is assumed in V band for RV calibration, a Th-Ar lamp is also considered in this band
61
Table 3-1. Definition of observational bandpasses used in this study: center wavelength andwavelength range
Band λ0 λmin − λmax(nm) (nm)
B 450 400−500V 545 500−590R 660 590−730Y 1020 960−1080J 1220 1110−1330H 1580 1480−1680K 2275 2170−2380
for comparison. We obtained a high resolution spectrum (R ≥200,000) using the Coude
Spectrograph at Kitt peak for a Iodine cell with a 6-inch light path at 60 C. Note that an
iodine cell spectrum is superimposed on a stellar spectrum (Butler et al., 1996), the S/N of
RV calibration is thus determined by the S/N of the continuum of a stellar spectrum. This
case is called Superimposing in this chapter. Howerver, for very stable instruments, there
are other ways of calibrating the non-stellar drift including spatial (Mayor et al., 2003) and
temporal approaches (Lee et al., 2011). In a spatial approach, the light from a star and a
Th-Ar lamp is fed onto nearby but different parts of CCD by two separate fibers (We call
this case Non-Common Path in the chapter). In Bracketing method, on the other hand, RV
calibrations are conducted right before and after a stellar exposure in a temporal approach.
In both cases, the S/N of RV calibration is not dependent on stellar flux. The disadvantage is,
however, the light from a calibration source does not pass through the instrument in exactly
the same path or at the same time as the light from a star. We choose a Th-Ar lamp as the
RV calibration source in R band, where strong Argon lines exist that saturate the CCD. Since
we exclude Argon lines in RV calibration uncertainty calculation, a more practical result when
Argon lines are considered is expected to be worse unless a CCD with higher dynamic range
is used.
In Y and J band, a U-Ne emission lamp is proposed by Mahadevan et al. (2010), we
use a lines list of Uranium provided by Stephen Redman (Redman et al., 2011). In H band,
62
a series of absorption cells is proposed by Mahadevan & Ge (2009), in which a mixture of
gas cells including H13C 14N, 12C2H2, 12CO, and 13CO creates a series of absorption lines
that spans over 120 nm of the H band. Bean et al. (2010) demonstrated that an Ammonia
absorption cell is a good candidate for calibration source in K band. Therefore, we assume
an Ammonia cell in the calculation of RV calibration uncertainty in the K band. Valdivielso
et al. (2010) proposed a gas absorption cell with the mixture of acetylene, nitrous oxide,
ammonia, chloromethanes, and hydrocarbons covering most of the H and K bands. We do
not consider this cell in this study since a detailed lines list of the cell is not available.
3.2.3 Stellar Noise
Stellar noise is a significant contributor to RV uncertainty budget, therefore we devote
the following part to discuss a method of quantifying its influence on precision Doppler
measurement. Granulation is considered to be the major obstacle in detection of Earth
planets in the HZ because it produces an RV signal with an amplitude of 8∼10 m · s−1 based
on observation on the Sun (Meunier et al., 2010). In addition, there is by far no good method
of removing the RV noise from this phenomenon. Dumusque et al. (2011) provided a model
of noise contribution in RV measurements based on precision RV observation on stars of
different spectral type and at different evolution stages. We adopt this model and quantify
the RV uncertainty contribution of granulation based their measurement of three stars, i.e.,
α Cen A (G2V), τ Ceti (G8V), and α Cen B (K1V). The sum of three exponentially decaying
functions represents a power spectrum density function with contributions from granulation,
meso-granulation and super-granulation, using the values given in Table 2 from Dumusque
et al. (2011). An RV RMS error due to granulation is then calculation based on Equation
(6) in their paper assuming a 100-day (300-day) consecutive observation for K (G) type
star with an optimal strategy found in the paper, i.e., three measurements per night of 10
min exposure each, 2 h apart. The total length of consecutive observation is roughly in
accordance with the orbital period of a planet in the HZ. We find that the RV RMS error due
63
to granulation is 0.55, 1.05 and 1.05 m · s−1 for a K1V, G8V and G2V star respectively. These
number are going to be used later in this study to estimate a total RV uncertainty.
Detailed study of RV uncertainty induced by stellar noise has so far been limited in K
and G type stars due to practical concerns such as stellar photon flux and stellar activity.
Despite their intrinsic faintness and relative higher level of stellar activity due to fast rotation
and deep convection zone, M dwarfs are among primary targets in search of planets in the
HZ. The RMS fitting error of orbit of GJ 674 b (Bonfils et al., 2007) is 0.82 m · s−1 after RV
noise due to a stellar spot is modeled and removed, providing an good target for Earth-like
planet search with an upper limit of other stellar noise contribution of 0.82 m · s−1, if we
interpret the RMS fitting error is due to an combination of instrument instability, photon-
noise and other source of stellar noise. We adopt the model proposed by Dumusque et al.
(2011) to estimate the RV RMS error due granulation phenomenon for an M dwarf using
the parameters for a K or G star. Aware of the caveat of different stellar type, we find the
RMS error is 0.52, 1.07 and 1.04 m · s−1 using the parameters for a K1V, G8V or G2V star.
50-day consecutive observation is assumed with an optimal strategy described in Dumusque
et al. (2011). The theoretical calculation of granulation-induced RV RMS error is worse than
observation of GJ 674 b using parameters for G stars, suggesting the parameters for G
stars are not representative of optimistic scenario in observation of M dwarfs. Therefore, we
use 0.52 m · s−1, which is a result of using the parameters for K stars, as an estimation of
granulation-induced RV RMS error for an M dwarf in an optimistic case.
3.2.4 Telluric Lines Contamination
Ground-based observations are prone to contamination by telluric lines. Precision
Doppler measurements in the NIR requires a significant level of disentanglement of stellar
absorption lines and telluric lines. As RV precision keeps improving, Doppler measurements
in the visible band such as B , V and R band should also consider telluric lines, because
the contamination of them will no longer be negligible. The quantification of telluric line
contamination has been discussed by Wang et al. (2011) in the context of Dispersed Fixed
64
Delay Interferometer method (Erskine, 2003; Ge, 2002; van Eyken et al., 2010), here we
present a generalization of the method for the case of Echelle spectrograph, which is a more
conventional application. Atmospheric transmission (AT) is calculated by a service provided
by spectralcalc .com based on a method described in Gordley et al. (1994).
The following equation describes the flux distribution on a CCD detector if telluric
absorption lines are considered:
F (ν) =[S0(ν)hν× AT (ν)
]⊗ LSF
=
[S0(ν)hν× (1 − α × AA(ν))
]⊗ LSF
=
[S0(ν)hν
]⊗ LSF +
[− S0(ν)hν× α × AA(ν)
]⊗ LSF
= FS(ν) + FN(ν), (3–1)
where S0(ν) is stellar energy flux, which is converted into photon flux by being divided by hν,
AT is the atmospheric transmission function, AA is the atmospheric absorption function, and
α is a parameter describing the level of telluric line removal as a first-order estimation.
In Equation 3–1, photon flux distribution on the detector, F , is comprised of a signal
component FS and a noise component FN . Ideally, we require that the detector flux change,
δF , is entirely due to the stellar RV change δvS . However, δF is also partly induced by telluric
line shift δvN resulting from random atmospheric motions. Therefore, both δvS and δvN
contribute to δF . We have two sets of RV measurements, δvS + σ(0, δvrms,S) for stellar RV
and δvN + σ(0, δvrms,N) for RV induced by the Earth’s atmosphere, where σ(0, δ) represents
random numbers following a gaussian distribution with a mean of 0 and a standard deviation
of δ. δvrms,S is the photon-limited measurement error for component FS and δvrms,N is the
photon-limited measurement error for component FN . We weigh the final RV measurement
with the inverse square of photon-limited RV uncertainties of these two components, which is
expressed by the following equation:
δv =(δvS + σ(0, δvrms,S)) · δv−2
rms,S + (δvN + σ(0, δvrms,N)) · δv−2rms,N
δv−2rms,S
+ δv−2rms,N
, (3–2)
65
In practical Doppler measurements, δvS consists two components, stellar RV and Earth’s
barycentric RV. Depending on the position of the Earth in its orbit, there is an offset between
δvS and δvN , which is the Earth’s barycentric velocity. The Earth’s barycentric motion has a
semi-amplitude of 30 km · s−1. Statistically, observed star has an annually-varying RV with
a semi-amplitude of on-average 21.21 km · s−1. We artificially shift a stellar spectrum by
an amount less than 21.21 km · s−1 in order to generate a offset between stellar spectrum
and AA spectrum. δvrms,S and δvrms,N are then calculated for FS and FN . We choose the
median of δvrms,N to represent a typical δvrms,N value from calculations based on different
input barycentric velocities. We further assume that observed star has a constant RV (i.e., no
differential RV), and FN has an RV fluctuation with an RMS of δvN,ATM because of the Earth’s
turbulent atmosphere. The measured RV uncertainty δv is equal to:
δvrms =(δvrms,S) · δv−2
rms,S + (δv 2N,ATM + δv
2rms,N)1/2 · δv−2
rms,N
δv−2rms,S
+ δv−2rms,N
, (3–3)
In reality, RV uncertainty of FN is not dominated by photon-noise, instead, it is dominated by
atmospheric behaviors such as wind, molecular column density change, etc. Figueira et al.
(2010a) used HARPS archive data and found that O2 lines are stable to a 10 m · s−1 level
over 6 years. However, long term stability of telluric lines (over years) becomes worse if we
take into consideration other gas molecules such as H2O and CO2. The uncertainty induced
by atmospheric telluric lines is transferred to δvrms via Equation 3–3. In order to calculate the
final RV uncertainty, δvrms , we need to calculate photon-limited RV uncertainty δvS,rms and
δvN,rms according to Equation 2–6, in which two terms need to be calculated: Q and Ne− . The
spectral quality factors (QS and QN) for the two components (FS and FN) from Equation 3–1
are calculated based on Equation 2–7. Ne−,S and Ne−,N , the photon flux of FS and FN are
calculated based on stellar type, magnitude, exposure time, instrument specifications and
telluric absorption properties. Note the ratio of Ne−,S and Ne−,N remains constant as long as
atmospheric absorption stays unchanged because telluric line absorption is imprinted on the
stellar spectrum.
66
The method described above provides a quantitative way of answering the questions
such as: 1), how the RV uncertainty is correlated with different levels of residual of telluric
line removal; 2), what the contribution of RV uncertainty due to telluric contamination is in the
final RV error budget in each different observational bandpass.
3.3 Results
3.3.1 RV Calibration Uncertainty
RV calibration sources are used to track the drift that is not caused by the stellar reflex
motion due to an unseen companion. A emission lamp or a gas absorption cell is usually
used for such purpose. We calculate the RV uncertainties brought by the calibration sources
themselves based on their spectral properties. The RV calibration sources in different
observational bandpasses are discussed in §3.2.2. For gas absorption cells, we assume
a continuum level of 30,000 ADU (within the typical linear range) on a CCD with a 16-bit
dynamic range, which corresponds to a S/N of 425 if the gain is at 6 electron/ADU. For a
emission lamp, we assume that the strongest line in the spectral region has a peak flux of
30,000 ADU. Note that 4 pixels per resolution element is assumed throughout the paper.
Figure 3-2 shows the RV calibration uncertainties as a function of observational bandpass
at different spectral resolutions. Note that there are currently two successful calibration
sources in V band, i.e., a Th-Ar lamp (asterisk) and an Iodine cell (square). Therefore, both
are considered and plotted for V band in Fig. 3-2. The results shown in Fig. 3-2 are also
summarized in Table 3-2. Gas absorption cells usually offer higher calibration precision than
emission lamps because of denser lines distribution and on-average higher S/N. However,
this conclusion depends on the calibration methods, it is usually the case for Non-Common
Path and Bracketing method while it is not always true in Superimposing scheme, which will
be discussed later in this section.
3.3.2 Optimal Spectral Band For RV Measurements
The optimal observational bandpass for precision RV measurements depends on the
quality of a stellar spectrum (Q factor), photon flux (S/N), RV calibration uncertainty, the
67
Figure 3-2. RV calibration uncertainties as a function of observational bandpass at differentspectral resolutions (color-coded). Squares in V band represent Iodine cellmethod and asterisks in V band represent Th-Ar lamp method. Refer to Table3-2 for RV calibration sources in different observational bandpasses.
Table 3-2. RV uncertainties caused by calibration sources at different spectral resolutions.R B V R Y J H K
Th-Ara Th-Ar, Iodineb Th-Ar U-Nec U-Ne Mixed celld Ammoniae
(m · s−1) (m · s−1) (m · s−1) (m · s−1) (m · s−1) (m · s−1) (m · s−1)20,000 0.78 0.90, 0.33 1.0 3.0 2.4 1.4 0.5740,000 0.65 0.74, 0.21 0.81 2.0 1.6 0.75 0.3060,000 0.54 0.60, 0.15 0.63 1.4 1.1 0.50 0.2180,000 0.46 0.50, 0.12 0.51 1.1 0.87 0.38 0.17100,000 0.38 0.41, 0.10 0.43 0.90 0.70 0.31 0.15120,000 0.33 0.36, 0.08 0.36 0.76 0.59 0.27 0.14
Note. — a: Lovis & Pepe (2007); b: Butler et al. (1996); c: Redman et al. (2011); d: Mahadevan & Ge (2009);e: Bean et al. (2010).
68
severity of telluric line contamination and other factors. We will consider different situations
in the following discussion. We assume a S/N (per pixel) of 100 at the center of Y band
(i.e., λ=1020 nm) at R=60,000, the S/N in other observational bandpass varies with stellar
spectral energy distribution (SED) and spectral resolution accordingly. The S/N reported in
this paper is at the center of each observational bandpass (see Table 3-1) unless otherwise
specified. We will investigate the optimal observational bandpass for precision Doppler
measurements given the same exposure time, the same telescope aperture and the same
instrument throughput (independent of wavelength).
3.3.2.1 Stellar Spectral Quality
We start with the simplest case in which the RV uncertainty is only determined by
the stellar spectral quality factor and the SED of a star. In other words, the RV calibration
source is perfect and no uncertainty is introduced when calibrating out the non-stellar drift.
In addition, telluric lines are perfectly removed from the observed stellar spectrum. Table 3-3
summarizes the obtainable RV precisions and the S/Ns at three different spectral resolutions,
i.e., 20,000, 60,000 and 120,000. An example of R = 120, 000 is plotted in Fig. 3-3. We
find the optimal observational bandpass is B band for a wide range of spectral types from K
to A. The optimal observational bandpass for an M dwarf is either in R band or in K band.
More specifically, R band is optimal for an early-to-mid-type M dwarf while K band for an
late-type M dwarf. The finding remains the same for a wide range of spectral resolutions from
20,000 to 120,000. The RV uncertainty for another spectral type or at a different S/N can be
obtained by either interpolating or scaling based on the results in Table 3-3.
3.3.2.2 Stellar Spectral Quality + Stellar Rotation
Stellar rotation broadens the absorption lines in a stellar spectrum, resulting in less
Doppler sensitivity. It is therefore necessary to consider the stellar rotation in the discussion
of photon-limited RV uncertainty. Typical values of rotation velocities of different spectral
types are obtained based on the measurement results from Jenkins et al. (2009) for M
dwarfs and Valenti & Fischer (2005) for FGK stars. In addition, typical rotational velocities
69
Tabl
e3-
3.P
hoto
n-lim
ited
RV
unce
rtai
ntie
sba
sed
onst
ella
rspe
ctra
lqua
lity
atdi
ffere
ntsp
ectra
lres
olut
ions
ford
iffer
ents
pect
ral
type
s,av
erag
eS
/Npe
rpix
elis
repo
rted
inpe
rent
hese
sS
pec.
Type
BV
RY
JH
K(m·s−1
)(m·s−1
)(m·s−1
)(m·s−1
)(m·s−1
)(m·s−1
)(m·s−1
)R
=20,
000
A5V
4.7(
211.
7)8.
3(20
9.0)
21.5
(198
.3)
23.9
(173
.2)
22.6
(155
.1)
62.9
(126
.8)
...(..
.)F5
V3.
8(16
6.9)
6.3(
177.
1)12
.8(1
82.1
)23
.3(1
73.2
)20
.9(1
64.2
)27
.2(1
48.9
)...
(...)
G5V
3.3(
126.
4)5.
1(14
5.6)
8.6(
163.
3)20
.8(1
73.2
)15
.9(1
71.9
)14
.9(1
68.0
)...
(...)
K5V
3.7(
88.8
)4.
8(11
0.9)
7.5(
141.
0)17
.3(1
73.2
)13
.5(1
81.8
)11
.2(1
99.8
)...
(...)
M5V
14.7
(26.
8)11
.7(4
4.0)
9.0(
66.2
)12
.8(1
73.2
)12
.0(2
03.0
)9.
5(20
5.6)
6.8(
191.
6)M
9V28
.2(9
.3)
18.4
(17.
9)13
.8(2
6.6)
8.2(
173.
2)4.
6(25
0.5)
5.4(
246.
3)2.
7(25
0.9)
R=6
0,00
0A
5V1.
5(12
2.2)
2.7(
120.
7)6.
9(11
4.5)
10.0
(100
.0)
9.6(
89.5
)24
.6(7
3.2)
...(..
.)F5
V1.
1(96
.4)
1.8(
102.
3)3.
6(10
5.1)
8.8(
100.
0)7.
9(94
.8)
9.4(
86.0
)...
(...)
G5V
1.0(
73.0
)1.
6(84
.0)
2.5(
94.3
)6.
9(10
0.0)
5.7(
99.3
)5.
6(97
.0)
...(..
.)K
5V1.
2(51
.3)
1.5(
64.0
)2.
3(81
.4)
5.9(
100.
0)5.
1(10
5.0)
4.6(
115.
3)...
(...)
M5V
4.6(
15.5
)3.
5(25
.4)
2.5(
38.2
)4.
8(10
0.0)
4.4(
117.
2)3.
8(11
8.7)
2.4(
110.
6)M
9V9.
3(5.
4)5.
9(10
.4)
4.2(
15.3
)3.
3(10
0.0)
1.8(
144.
6)2.
0(14
2.2)
1.0(
144.
9)
R=1
20,0
00A
5V1.
1(86
.4)
2.0(
85.3
)5.
1(81
.0)
8.0(
70.7
)7.
9(63
.3)
18.9
(51.
8)...
(...)
F5V
0.7(
68.2
)1.
2(72
.3)
2.4(
74.3
)6.
4(70
.7)
5.9(
67.0
)6.
5(60
.8)
...(..
.)G
5V0.
6(51
.6)
1.1(
59.4
)1.
6(66
.7)
4.6(
70.7
)3.
9(70
.2)
4.0(
68.6
)...
(...)
K5V
0.8(
36.3
)1.
0(45
.3)
1.5(
57.6
)4.
0(70
.7)
3.5(
74.2
)3.
0(81
.6)
...(..
.)M
5V3.
1(11
.0)
2.3(
18.0
)1.
5(27
.0)
3.3(
70.7
)3.
1(82
.9)
2.8(
83.9
)1.
8(78
.2)
M9V
6.4(
3.8)
4.1(
7.3)
2.8(
10.8
)2.
3(70
.7)
1.3(
102.
3)1.
5(10
0.6)
0.8(
102.
4)
70
Figure 3-3. RV precision (R=120,000) based on spectral quality factor as a function ofobservational bandpass for different spectral types from M9V to A5V(color-coded). Average S/N per pixel is also shown in the plot, see Table 3-3 forresults at other spectral resolutions. K band RV uncertainties are not calculatedfor stars with Teff higher than 3500 K because they are usually observed in thevisible band at current stage.
for early type stars such as A stars are extrapolated from values of solar type stars. Table
3-4 summarizes the spectral types and the corresponding Teff and Vsin i used in the paper.
A trend of increasing Vsin i is seen as spectral type moves either to early type end (F
and A) or late type end (M). After considering typical stellar rotation velocities for different
spectral types (as shown in Fig. 3-4, R=120,000), F and A stars are not suitable targets for
precision Doppler measurements because of their intrinsic high stellar rotation. M dwarfs
RV uncertainties are getting worse than non-rotating case, but 2−5 m · s−1 RV precision are
expected in optimal cases, i.e., R band for M5V and K band for M9V. For K and G stars, sub
71
m · s−1 precision is reached under photon-limited condition even after considering typical
stellar rotation.
Figure 3-4. RV precision (R=120,000) based on spectral quality factor and typical stellarrotation as a function of observational bandpass for different spectral types fromM9V to A5V (color-coded). Average S/N per pixel is the same as shown in Fig.3-3.
3.3.2.3 Stellar Spectral Quality + RV Calibration Source
The above RV precisions considering typical stellar rotation broadening are good
indicators for RV planet surveys in which a population of stars is observed with a distribution
of stellar rotations. However, in the search for an Earth-like planet, a different approach
is taken in which stars with favorable properties for Doppler measurements are assigned
higher priority in observation. The properties usually include slow stellar rotation and low
stellar activity. Therefore, we will reduce stellar rotation in the following discussion since we
72
Table 3-4. Spectral Type, corresponding Teff, and typical stellar rotation V sin iSpectral Type Teff Vsin i
(K) (km · s−1)A0V 9600 131.0A2V 9000 108.0A5V 8400 85.5A8V 7800 62.5F0V 7400 47.5F2V 7000 32.0F5V 6750 23.0F8V 6250 6.5G0V 6000 3.5G2V 5750 2.2G5V 5500 1.7G8V 5250 1.8K0V 5000 1.9K2V 4750 1.8K5V 4500 2.0K8V 4000 2.5M0V 3750 2.8M2V 3500 2.8M5V 3100 3.9M8V 2600 6.8M9V 2400 8.0
emphasize discovery of an Earth-like planet. After considering the uncertainties brought
by an RV calibration source, RV uncertainties in Fig. 3-3 degrade to those in Fig. 3-5
(R = 120, 000). Two scenarios of calibration are considered: Superimposing (Dotted), and
Non-Common Path and Bracketing (Solid). The difference between these two is whether the
S/N depends on the stellar flux. In the Superimposing case, because the absorption cell is in
the light path of the stellar flux, the continuum of the resulting Iodine absorption spectrum is
determined by the the continuum flux of a star. Consequently, the RV calibration uncertainty
is strongly dependent on the incoming stellar flux. In the comparison of the two cases in
Fig. 3-5, we see the Non-Common Path and the Bracketing methods always introduce less
uncertainty in RV calibration than the Superimposing method. The major reason for that is
the S/N in the former case may be optimized by adjusting the source intensity (Non-Common
73
Path) or the exposure time (Bracketing). The main conclusion about optimal observational
bandpass from §3.3.2.1 remains unchanged.
Figure 3-5. RV precision (R=120,000) based on spectral quality factor and RV calibrationuncertainties as a function of observational bandpass for different spectral typesfrom M9V to A5V (color-coded). Average S/N per pixel is the same as shown inFig. 3-3. Squares in V band represent Iodine cell method and asterisks in Vband represent Th-Ar lamp method. Dotted lines show result from Superimposingcases and solid lines for Non-Common Path and Bracketing cases. Refer to Table3-2 for RV calibration sources in different observational bandpasses.
3.3.2.4 Stellar Spectral Quality + RV Calibration Source + Atmosphere
The optimal band for Doppler measurements is in the NIR (K band in particular) for
stars with spectral types later than M5 from previous discussions in this paper. However, one
important element is missing in the discussion, which is the contamination from the telluric
lines in the Earth’s atmosphere, which is a severe problem in the NIR observation. The
74
quantitative analysis of telluric line contamination is introduced in §3.2.4 and we apply that
method in estimating the RV uncertainty brought by the telluric contamination. We confine
our discussions for late-type M dwarfs since NIR observation does not gain advantage for
other spectral types earlier than M5V.
Figure 3-6 shows an example of how RV uncertainty for an M9V star changes with
observational bandpass under different values of α (i.e., level of telluric line removal, see
Equation 3–1). 1 indicates no telluric line removal and 0 indicates complete removal of
telluric lines (see Equation 3–1). RV fluctuation of 10 m · s−1 due to random atmospheric
movement is assumed in the calculation. Bracketing RV calibration is assumed in the
calculation. There are several points worth noting in this plot: 1), different observational
bandpasses are affected differently by telluric lines, the significance of telluric line contam-
ination is indicated by the span of RV uncertainties at different α values. For example, B
band is the least affected by telluric lines because the RV uncertainties in B band at different
levels of telluric line removal remain roughly the same, while J, H and K bands suffer severe
telluric line contamination because any small change of α results in significant change of RV
uncertainty. 2), If there is no attempt of removing telluric lines from observed stellar spectrum
(purple in Fig. 3-6), there is no advantage in observing late-type M dwarfs in NIR, RV uncer-
tainty is dominated by Earth’s atmosphere behavior in the NIR. In this case, the optimal band
is V and R band. Only when α ≤0.01, i.e., more than 99% telluric line strength is removed,
the advantage of observing late-type M dwarfs in the NIR becomes obvious, at a factor of ∼3
improvement.
In practice, there have been several examples in which telluric line modeling and
removal is demonstrated to be successful. Vacca et al. (2003) achieved maximum deviations
of less than 1.5% and RMS deviations of less than 0.75% with R=2000 and S/N≥100 using
a telluric standard star nearby the science target star. Bean et al. (2010) has shown that
the RMS deviation is as low as 0.7% after using a 3-component model (Stellar spectrum,
telluric absorption and Ammonia absorption) to fit an observed NIR spectrum. In both cases,
75
Figure 3-6. RV precision (R=120,000) considering spectral quality factor, RV calibrationuncertainties and telluric contamination for an M9V star as a function of α, i.e.,telluric line removal level (color-coded). 1 indicates no telluric line removal and 0indicates complete removal of telluric lines. Bracketing RV calibration is assumedfor the results shown in the plot. Squares in V band represent Iodine cell methodand asterisks in V band represent Th-Ar lamp method. Refer to Table 3-2 for RVcalibration sources in different observational bandpasses.
an α value of better than 0.01 has been demonstrated showing great potential of precision
Doppler measurement in the NIR band.
Figure 3-7 shows the percentage contribution of RV uncertainty introduced by telluric
contamination at different α values. If no telluric line removal is performed, the RV uncer-
tainty in the NIR is dominated by those caused by telluric contamination, i.e., the percentage
contributions are more than 87.9% in Y , J, H and K band. In comparison, the percentage
contribution of telluric contamination induced RV uncertainty is 2.9%, 3.1% and 53.0% in B ,
76
Y and R band respectively. As α decreases, i.e., more strength of telluric lines is removed,
less RV uncertainty is contributed to the final RV uncertainty budget. However, there is still a
significant fraction (more than 70%) of RV uncertainty contributed by telluric contamination
in J, H and K band even after 90% of telluric line strength is removed. The percentage
contribution drops below 10% throughout considered observational bandpasses when more
than 99.9% strength is removed. To sum up the discussion, RV uncertainty is dominated
by telluric contamination in the NIR band. Therefore, telluric line removal in the NIR is a
necessary step to reduce the telluric contamination and extract more of Doppler information
intrinsically carried by a stellar spectrum.
Figure 3-7. The percentage contribution of RV uncertainty induced by telluric contaminationas a function of observational bandpass. Different α values are indicated bycolors. 1 indicates no telluric line removal and 0.001 indicates 99.9% effectiveremoval of telluric lines.
77
3.3.2.5 Comparisons to Previous Work
There are works that have been previously done in attempts to understand the funda-
mental photon-limited RV uncertainties based on high resolution synthetic stellar spectra.
Bouchy et al. (2001) calculated Q factors for a set of synthetic stellar spectra for solar type
dwarf stars. We restrict the comparison to spectra with the same turbulence velocity (Vt).
Since the spectra for solar type stars in our study have a Vt of 1.0 km · s−1, we only compare
the results from spectra with Vt of 1.0 km · s−1 in Bouchy et al. (2001). Table 3-5 summarizes
a comparison of our results to those from Bouchy et al. (2001). The Q factors from our study
are generally 10−15% lower if no stellar rotation is considered, i.e., V sin i=0 km · s−1. It may
be due a different sampling rate in the synthetic spectra, 0.005 Å in Bouchy et al. (2001)
and 0.02 Å in our paper. More fine features are seen in a spectrum with higher sampling
rate and thus more Doppler information is contained. At low stellar rotation rate (V sin i=4
and 8 km · s−1), our results agree with theirs within 6%, which is improved compared to
non-rotating case because the fine features are smoothed out by stellar rotation. For fast ro-
tators, i.e., V sin i=12 km · s−1, 10% difference is seen in the worst case, for which a different
limb-darkening value might be responsible.
Table 3-5. Comparison of Q factors from our results to Bouchy et al. (2001) (numbers inparenthathes).
Teff logg Vt Vsin i(K) (cm · s−1) (km · s−1) 0 km · s−1 4 km · s−1 8 km · s−1 12 km · s−1
4500 4.5 1.0 30238(34940) 17235(17080) 8700(8440) 5793(5380)5000 4.5 1.0 30001(33405) 16607(16140) 8305(7815) 5432(4930)5500 4.5 1.0 26892(30375) 14858(14700) 7397(7020) 4700(4385)
Reiners et al. (2010) investigated the precision that can be reached in RV measure-
ments for stellar objects cooler than solar type stars in the NIR. The treatment of telluric lines
in their calculations was to block the regions where the telluric absorption is over 2% and
30 km · s−1 in the vicinity. Following the method described in their paper, we calculated the
fraction of the wavelength range affected by telluric contamination in V , Y , J and H band,
the results are 2.4%, 22.7%, 60.0% and 50.6%. In comparison, the results are 2%, 19%,
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55% and 46% for V , Y , J and H band in their paper. The difference may be caused by a
different atmospheric absorption used in calculation.
Both Reiners et al. (2010) and we reach the same conclusion that NIR RV measure-
ments start to gain advantage over visible bands for mid-to-late-type M dwarfs. However, we
predict that Y and H band are similar in terms of giving the highest RV precision among V ,
Y , J and H bands, while it is found in their paper that Y band is the optimal band consider-
ing stellar spectrum quality and telluric line masking. Note that we adopt the definition of V ,
Y , J and H bands according to Reiners et al. (2010) in comparisons. Table 3-6 summarizes
our calculations of RV precision can be reached for an M9V star (Teff = 2400K ) in compari-
son to the results in their paper as well as the S/N obtained in each observational bandpass.
In further examination, we compare our results in J and H band and find that RV precisions
in H band are in general better than those in J band. It is explained by the wider wavelength
coverage and richer absorption features in H band for an M9V star. On the contrast, the
improvement of RV precision in H band is not seen in the comparison of J and H band in the
results from Reiners et al. (2010). In addition, we report better RV precisions in V band by a
factor of 1.5.
Table 3-6. Comparison of predicted RV precision (in the unit of m · s−1) between our results toReiners et al. (2010) for an M9 dwarf (Teff=2400 K, V sin i=0 km · s−1). TheFollowing results are calculated based on a telluric masking treatment in whichtelluric lines with more than 2% absorption depth and 30 km · s−1 within its vicinityare masked out.
R S/N This study Reiners et al. (2010)V Y J H V Y J H V Y J H
60000 12 100 134 128 5.1 3.0 4.9 2.7 8.0 2.2 4.6 4.080000 10 86 116 111 4.2 2.5 4.1 2.2 6.2 1.7 3.5 3.5100000 9 77 104 99 3.9 2.4 3.8 2.0 5.3 1.5 2.9 3.3
We have also conducted similar calculation to Rodler et al. (2011) for an M9.5 dwarf
(Teff=2200 K, V sin i=5 km · s−1) and the comparison is presented in Table 3-7. Instead of
finding Y band, we find K band gives the highest RV precision in telluric-contamination-free
case, while H band gives the highest RV precision in the case where telluric lines with an
79
absorption depth more than 3% are masked out in RV calculation. We also notice that our
predicted RV precisions are less sensitive to spectral resolution. Note that stellar absorption
lines typically become resolved by spectrograph after spectral resolution goes over 50,000,
we do not expect RV precision increases steeply as spectral resolution goes well beyond
50,000.
Table 3-7. Comparison of predicted RV precision (in the unit of m · s−1) between our results toRodler et al. (2011) for an M9.5 dwarf (Teff=2200 K, V sin i=5 km · s−1). Case A isfor complete and perfect removal of telluric contamination; Case B is for the casein which telluric lines with absorption depth of ≥3% were masked out.
R S/N This study Rodler et al. (2011)Y J H K Y J H K Y J H K
Case A
20000 139 180 171 152 16.4 12.2 8.7 5.1 22.2 25.5 22.8 27.940000 98 127 121 108 9.4 6.9 5.4 3.2 6.9 8.7 7.8 10.860000 80 104 99 88 7.7 5.5 4.6 2.8 4.2 5.7 5.1 3.880000 70 90 85 76 6.8 5.1 4.4 2.6 3.3 4.0 3.8 5.1
Case B20000 139 180 171 152 18.8 19.8 13.9 16.4 24.2 29.7 29.1 39.340000 98 127 121 108 10.5 10.9 8.5 9.5 8.7 12.2 11.9 17.360000 80 104 99 88 8.6 8.7 7.0 7.8 5.4 7.1 6.8 10.780000 70 90 85 76 7.6 8.0 6.7 7.2 3.8 5.2 5.1 7.9
3.3.3 Current Precision vs. Signal of an Earth-like Planet in Habitable Zone
One of the most intriguing tasks in exoplanet science is to search and characterize
Earth-like planets in the HZ. Over the past two decades, great advances have been seen
but we have not yet discovered another Earth. We are trying to answer several questions in
the following discussion: 1), is it possible to detect an Earth-like planet in the HZ using the
RV technique? 2), If so, at what S/N in which observational bandpass and for which spectral
type? 3), Based on the current available RV calibration sources and knowledge of stellar
noise, is it practical to detect an Earth-like planet in the HZ in the most optimistic case?
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3.3.3.1 Stellar Spectral Quality
We first consider an ideal situation in which the RV precision is only determined by the
Q factor of a stellar spectrum. The highest S/N per pixel obtainable for a single exposure
is 425 (assuming 30,000 ADU and a gain of 6 electron/ADU). Figure 3-8 shows the RV
precisions obtainable at spectral resolution of 120,000 for different spectral types in different
observational bandpasses. Overplotted are RV signals of an Earth-like planet in the inner
(dashed) and outer edge (dash-dotted) of the HZ of a star with a certain spectral type (color
coded). The position of the HZ is calculated based on Kasting et al. (1993). The position
of the HZ gets closer to the host star as stellar temperature and luminosity drops. The RV
signal is enhanced by both the decreasing distance to the star and the decreasing stellar
mass. Earth-like planet is detectable in every observational bandpass at a S/N as high as
425 for M dwarfs. B and V band bear the highest probability for K stars and B band is the
sweet sopt for G stars. Predicted RV precisions are not adequate to detect the signal of an
Earth-like planet in the HZ around F and A stars with single exposure on a current typical
CCD with 16-bit dynamic range. Table 3-8 summarizes the S/N required for detection of an
Earth-like planet in the HZ as a function of spectral type, in which we assume that a detection
is possible when the RV precision is equal to the signal. Even though it only require a S/N
of 17 for an M9V star in B band to detect an habitable Earth-like planet, the exposure time
could be as long as 1 hour even at the Keck telescope for a J=6 M9V star and there is no
such bright late M type star in the sky. In addition, only 25 M stars are available with J band
magnitude less than 6 (Lepine & Shara, 2005). In comparison, ∼1 min exposure time at Keck
will obtain a S/N of 175 for a B=8 star, which is adequate for detecting habitable Earth-like
planet around a K5V star. 10% instrument throughput is assumed in the above calculations.
3.3.3.2 Stellar Spectral Quality + RV Calibration Source + Atmosphere
Current RV precision is not only restricted by the photon-limited uncertainty determined
by a stellar spectrum, but also by the uncertainties brought by an RV calibration source and
the telluric contamination from the Earth’s atmosphere. Figure 3-9 shows the RV precisions
81
Figure 3-8. RV precisions (R=120,000) considering spectral quality factor at a S/N of 425 asa function of observational bandpass for different spectral types from M9V to A5V(color-coded). Overplotted are RV signals of an Earth-like planet in the inner(dashed) and outer edge (dash-dotted) of the HZ.
Table 3-8. Required S/N for detection of an Earth-like Planet in the HZ as a function ofspectral type
Spectral Band HZ propertiesB V R Y J H K m ain aout vin vout
(M⊙) (AU) (AU) (m · s−1) (m · s−1)A5V 3483 6087 14724 20389 17827 35053 ... 1.82 2.893 4.167 0.03 0.03F5V 837 1547 3088 8042 7003 6976 ... 1.30 1.614 2.325 0.06 0.05G5V 318 625 1053 3149 2660 2648 ... 0.87 0.720 1.037 0.11 0.09K5V 175 280 535 1710 1570 1496 ... 0.64 0.387 0.558 0.18 0.15M5V 47 58 58 328 359 330 193 0.19 0.068 0.098 0.78 0.65M9V 17 22 22 118 99 108 58 0.10 0.034 0.049 1.51 1.26
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taking into consideration of Q factors, RV calibration uncertainties and telluric contamination.
Bracketing calibration (at a S/N of 425) is considered in the calculations in which the S/N of
calibration is not determined by the continuum of the observed star. Two cases are discussed
for telluric contamination, in one case no telluric removal is attempted (solid) while in the
other case 99.9% of telluric line strength is removed (dotted). For the visible bands (i.e., B , V
and R band), RV precision in B band is barely affected by telluric contamination (10 m · s−1
random RV of telluric lines is assumed in the calculations) but limited by the RV calibration
uncertainty due to a Th-Ar lamp. There are two RV calibration sources considered in the V
band, a Th-Ar lamp and a Iodine absorption cell, the latter one provides higher calibration
precision in the Bracketing case.
Even though only 2.4% of the wavelength range is affected by telluric line contamination
(§3.3.2.5), handling telluric lines is still very important. The RV uncertainty budget of a K5V
star in Table 3-9 shows an example in which the RV precision is worse than detection limit if
no telluric line removal is involved while it is below the detection limit in the 99.9% removal
case (α=0.001). This example address the importance of telluric line removal in the search of
an Earth-like planet even in the visible band where telluric contamination is less severe than
the NIR band. After 99.9% of telluric line strength is removed, the RV uncertainty of a K5V
star is dominated by the spectral quality of a K5V star and the RV calibration source.
Table 3-9. Two examples of telluric contaminationSpectral Type Bandpass α δvS,rms δvATM,rms δvcal δvrms δvHZ
(m · s−1) (m · s−1) (m · s−1) (m · s−1) (m · s−1)
K5V V1.0
0.110.19
0.080.24
0.180.001 0.01 0.14
M5V K1.0
0.337.97
0.147.98
0.780.001 0.17 0.39
In comparison, in the NIR (Y , J, H and K band), the RV uncertainties are dominated
by telluric contamination, resulting RV precisions at 5−10 m · s−1 that are not adequate
in habitable Earth-like planet detections. A similar example of how telluric contamination
83
Figure 3-9. RV precision (R=120,000) considering spectral quality factor (S/N=425), RVcalibration uncertainties and telluric contamination as a function of observationalbandpass for different spectral types from M9V to G5V (color-coded). Overplottedare RV signals of an Earth-like planet in the inner (dashed) and outer edge(dash-dotted) of the HZ. Solid lines represent non-telluric-removal cases whiledotted lines represent cases in which 99.9% of the strength of telluric lines isremoved. Squares in V band represent Iodine cell method and asterisks in Vband represent Th-Ar lamp method. Refer to Table 3-2 for RV calibration sourcesin different observational bandpasses.
raises the floor of RV uncertainty is also given for an M5V star in K band in Table 3-9. In the
99.9% removal case, RV uncertainties in the NIR are no longer mainly dominated by telluric
contamination, but by spectral quality factor. To sum up, telluric line removal is an important
and indispensable step toward the discovery of an Earth-like planet even in the visible band.
After telluric lines are successfully removed from observed stellar spectrum, the RV precision
is limited by the uncertainty caused by stellar spectral quality and RV calibration sources.
84
After completely removing the telluric contamination, we compare our prediction of
RV uncertainties and what is reported from HARPS instrument (Mayor et al., 2003). An
example of HD 47186 (Bouchy et al., 2009) is given in Table 3-10. HD 47186 is a G5V star
with a V sin i of 2.2 km · s−1, the best achievable RV precision for this star is 0.3 m · s−1 at a
S/N of 250 according to Bouchy et al. (2009). Our prediction indicates that an RV precision
of 0.24 m · s−1 is possible to achieve at the same S/N for the same wavelength coverage.
The difference may come from those uncounted factors in our calculation, for example,
stellar noise. However, our prediction of RV precision is within 20% to precision from real
observation.
Table 3-10. Prediction vs. HARPS observation. a: HARPS observation of HD 47186from Bouchy et al. (2009), the best achievable RV precision is at a S/N of 250 forthis G5V star with V sin i of 2.2 km · s−1; b: our RV uncertainties prediction for thisstar assuming the same S/N, spectral type, observation bandpasses and stellarrotation.
Bandpass δvS,rms δvcal δvrms(m · s−1) (m · s−1) (m · s−1)
HD47186a B+V+R ... ... 0.30
G5Vb
B 0.14 0.33 0.36V 0.29 0.36 0.46R 0.48 0.36 0.60
B+V+R 0.12 0.20 0.24
3.3.3.3 Stellar Spectral Quality + RV Calibration Source + Stellar Noise
Assuming the telluric lines are perfectly measured and removed, we consider the
obtainable RV precision based on stellar spectral quality, RV calibration precision and stellar
noise. Stellar noise of different spectral type is estimated in §3.2.3 based on Dumusque
et al. (2011). Figure 3-10 shows predicted RV precision, it is clear that the RV precision for
G and K type stars is not adequate for detecting Earth-like planet in the HZ after stellar noise
is taken into consideration. However, it is still possible to detect Earth-like planets around
M dwarfs because of relatively larger RV signal. For an M5V star, visible band and K band
85
provide adequate precision for an Earth-like planet detection while all bandpasses allow an
Earth-like planet detection for an M9V star.
Figure 3-10. RV precision (R=120,000) considering spectral quality factor (S/N=425), RVcalibration uncertainties and stellar noise as a function of observationalbandpass for different spectral types from M9V to G5V (color-coded).Overplotted are RV signals of an Earth-like planet in the inner (dashed) andouter edge (dash-dotted) of the HZ. HZs of G and K type stars are not plottedbecause they are out of reach based on the predicted RV precision. Squares inV band represent Iodine cell method and asterisks in V band represent Th-Arlamp method. Refer to Table 3-2 for RV calibration sources in differentobservational bandpasses.
In order to compare our predicted RV precision to observations, we choose 69 planets
detected by HARPS since 2004 after an instrument upgrade (Mayor et al., 2003) and plot
the RMS errors of Keplerian orbit fitting as a function Teff (Fig. 3-11). The minima of three
subsets (corresponding to G, K and M stype stars) are found based on Teff. In comparison,
86
the predicted RV predictions (after combining results from B , V and R bands) for a G5V,
K5V and M5V star are plotted as open diamonds. Since the predicted RV precision is based
on an optimistic case, we compare the predictions with the RMS minima we find in the
observation. For M dwarfs, the minimum of RMS errors is found at 0.8 m · s−1 with GJ 674
b (Bonfils et al., 2007). In comparison, our prediction is 0.62 m · s−1 considering stellar
spectral quality factor, RV calibration error and stellar noise. If ∼0.5 m · s−1 instrumental
uncertainty as mentioned in Bonfils et al. (2007) is added in quadrature, our prediction is
well matched with HARPS M dwarfs observation in the best case scenario. Lovis et al. (2006)
reported 0.64 m · s−1 RMS errors for a planet system of a K0V star (i.e., HD 69830), which
is consistent with our prediction of 0.65 m · s−1. It is plotted in the bin with Teff between
5000 K and 6000 K because reported Teff of 5385 K. For G type stars, Bouchy et al. (2009)
reported 0.91 m · s−1 RMS error for HD 47186 b and c, a planetary system around a G5V
star. In comparison, we predict a total RV uncertainty of 1.1 m · s−1. The overestimation
of RV measurement uncertainty is possibly due to an overestimation of stellar noise or an
increasing S/N because of multiple measurements in real observation.
We predict a total RV measurement uncertainty of 0.62, 0.65 and 1.1 m · s−1 for spectral
type M5V, K5V and G5V considering stellar spectral quality, RV calibration and stellar noise.
According to the calculations in §3.2.3, RV uncertainty due to stellar noise is 0.52, 0.55 and
1.05 m ·s−1 for the above three types of stars, accounting for 70.3%, 71.6% and 91.1% of total
RV measurement uncertainty. Based on comparisons of our predictions and observation,
we therefore conclude that stellar noise is one major contributor in error budget of precision
Doppler measurement. M dwarfs should be the primary targets in search of Earth-like
planets in the HZ. Unlike G and K stars, the RV signal of Earth-like planets in the HZ is not
overwhelmed by stellar noise for M dwarfs in the most optimistic case.
3.4 Summary and Discussion
We provide a method of practically estimating the photon-limited RV precision based
on the spectral quality factor, stellar rotation, RV calibration uncertainty, stellar noise and
87
Figure 3-11. RMS error of Keplerian orbit fitting for planets detected by HARPS since 2004.Predicted RV precisions considering stellar noise for different stellar types areoverplotted as open diamonds.
telluric line contamination. The methodology described and the results presented in this
paper can be used for design and optimization of planned and ongoing precision Doppler
planet surveys. For pure consideration of stellar spectral quality without artificial rotationally
broadening the absorption line profile, the optimal band for RV planet search is B band for a
wide range of spectral types from K to A, while it is R or K band for mid-late type M dwarfs.
Nevertheless, the above conclusion remains unchanged after considering typical stellar
rotation of each spectral type. However, F and A stars become unsuitable for precision RV
measurements because of typically fast stellar rotation. We confirm the finding in Reiners
et al. (2010) that the NIR Doppler measurements gain advantage for mid-late M dwarfs.
However, instead of finding Y band as the optimal band considering stellar spectrum quality
88
and telluric masking, we find that both Y and H bands give the highest RV precision among
V , Y , J and H bands. In a comparison to Rodler et al. (2011), we find K band is the optimal
band for precision Doppler measurement in a telluric-free case and H band is optimal in a
telluric-masking case, while they found Y band gives the highest RV precision in both cases.
Fundamental photon-limited RV precision for evolved stars has been discussed by Jiang
et al. (2011), which is valuable for ongoing RV planet search around retired stars discussed
in Johnson et al. (2007b).
We also consider the uncertainties brought by current available RV calibration sources
at different spectral resolutions (Fig. 3-2). Sub m · s−1 calibration precision can be reached for
each observational bandpass. Note that the Q factors may change as gas pressure, length of
light path and temperature changes. The precision also depends on the methods used in the
RV calibration. We categorized the calibration methods into several cases: Superimposing, in
which the calibration spectrum is imprinted onto a stellar spectrum; Non-Common Path and
Bracketing, in which the calibration is conducted either spatially or temporally. The former
method depends on stellar flux while the latter one can only be applicable for very stable
instruments. There are other calibration sources we have not included into the discussions in
this study, for example, laser combs (Li et al., 2008; Steinmetz et al., 2008), the Fabry-Perot
calibrator (Wildi et al., 2010) and the Monolithic Michelson Interferometer (Wan & Ge, 2010).
Once they become more economically affordable or more technically ready, the RV precision
will be greatly improved in the future.
For the first time we have quantitatively estimated the uncertainty caused by the residual
of telluric contamination removal for high resolution echelle spectroscopy method. Depending
on the telluric absorption, different observational bandpasses are affected differently. B
band is the least sensitive to telluric contamination because there are barely any telluric
absorption features in B band. However, the NIR bands suffer the most in precision RV
measurements because the stellar absorption lines and telluric lines are mixed together
severely in this spectral region. Only when α ≤0.01, i.e., more than 99% of strength of telluric
89
lines is removed, the advantage of NIR observation of mid-late type M dwarfs begins to show,
which is a factor of 3 improvement. This quantitative method in estimating the RV uncertainty
induced by telluric contamination can be easily adapted to other problems, for example,
estimating the moon light contamination.
Besides telluric line removal, telluric line masking has also been discussed in several
of previous studies (Reiners et al., 2010; Rodler et al., 2011; Wang et al., 2011). In Reiners
et al. (2010), telluric absorption with depth more than 2% and 30 km · s−1 in the vicinity is
blocked out when measuring RV. Based on this blocking criterium, the photon-limited RV
uncertainty, δvrms,S (refer to Equation 3–3) , for an M9V star at R=100,000 is 3.9, 2.2, 3.9, 2.2
m · s−1 in V , Y , J and H band respectively (see Table 3-6). In comprison, δvrms,N is 71.3,
5.8, 6.5, 3.7 m · s−1 in V , Y , J and H band respectively. Except for V band, δvrms,S and
δvrms,N are at the same order of magnitude, and the uncertainty caused by telluric absorption
cannot be neglected even though that the spectral region with any telluric absorption of
more than 2% is blocked. If more strict criterium of telluric line masking is applied, fewer
photons are considered in measuring the RV, which effectively increases the photon-limited
RV uncertainty. In order to reach photon-limited RV precision predicted by pure consideration
of spectral Q factor, telluric removal should be applied in which telluric contamination is
measured or modeled and then removed from measured stellar spectrum.
RV uncertainty due to stellar granulation is taken into consideration in this paper. High
frequency (∼min) stellar noise such as p-mode oscillations usually have a RV amplitude of
0.1 to 4.0 m · s−1 (Schrijver & Zwaan, 2000) and they can be averaged out within typical
10−15 exposure time. RV uncertainties due to low frequency (10−100 day) stellar noise such
as stellar spots have been discussed in recent papers, for example, Desort et al. (2007)
and Reiners et al. (2010). The amplitudes of spot-induced RV range from one to several
hundred m · s−1. Since stellar spot induced RV uncertainties are periodic and therefore can
be modeled and removed, however, the amplitude of residual is unknown at this stage.
90
We compare the RV precision based on stellar spectral quality and the signal of an
Earth-like planet in the HZ of a star with a certain spectral type. We find that it is likely to
detect a habitable Earth-like planet around G,K and M stars while it is too demanding to
detect one around F and A stars. B band is the optimal band for G and K stars and K band
for M dwarfs. After considering practical issues such as telluric contamination, we find that,
except for B band, every observational bandpass is affected by telluric contamination to
some extent. The major RV measurement uncertainty comes from telluric contamination,
which overwhelms the RV signal of an habitable Earth-like planet around G and K stars.
Surprisingly, telluric contamination becomes an issue in V band even there is only 2.4%
of spectral region affected by telluric lines. After telluric lines are removed at a very high
level, i.e., α ≤0.001, the error from RV calibration becomes the major contributor of Doppler
measurement uncertainty. After stellar noise (granulation only) is taken into consideration,
which is dominant contributor to RV uncertainty, M dwarfs become the only type of star that is
suitable for the search for Earth-like planets in the HZ.
The RV precision in the discussion of habitable Earth detectability considers four factors:
stellar spectral quality, RV calibration uncertainty, stellar noise and telluric contamination.
However, the discussion of stellar noise should be treated with great caution for several
reasons: 1), stellar noise is not very well understood and characterized at this stage; 2), it
is different from case to case and therefore it is difficult to draw a general conclusion; 3), a
habitable planet search is different from a planet survey, the targets are chosen in favor of
detection at the best case scenario, for example, high stellar flux, slow stellar rotation, low
stellar activity and low stellar noise and so on. Therefore, the stellar noise assumed in this
study is in the best case scenario according to current theory and observation. In addition,
we assume the highest signal within linear range (30,000 ADU) for current typical CCD
(16-bit dynamic range) in single exposure in the discussion, note that the S/N can also be
improved by multiple independent measurements. The S/Ns required for Earth-like planet
detections are provided in Table 3-8 based on stellar spectral quality. Please note that the HZ
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changes over time as the luminosity of the host star changes. It also depends on properties
of a planet such as atmosphere composition, albedo and orbit. The purpose of discussion
in this paper is to provide a basic idea of the comparison of current best obtainable RV
precision to a typical RV signal of an habitable Earth-like planet.
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CHAPTER 4PLANET SEARCH AROUND M DWARFS
4.1 Introduction
4.1.1 Current Status
As of May 2011, there are 35 planets in 28 planetary systems of M dwarfs. Radial
velocity technique is the most productive method in M-dwarf planet search with discoveries
of 21 planets in 15 systems. Microlensing ranks the second with 12 planet detections in 11
systems. Transiting method has by far detected 2 planets around M dwarfs.
Giant planet occurrence rate for M dwarfs is generally thought to be lower than that for
solar-type stars. Bonfils et al. (2011b) found a low frequency (f ) of giant planet around M
dwarfs, f ≤ 1% for P=1-10 day and f = 2+3−1% for P=10-100 day, P denotes orbit period. In
comparison, Cumming et al. (2008) found that the frequency is ∼10% for solar-type stars.
On the other hand, low-mass planets are frequently detected despite of an adverse detection
bias. Bonfils et al. (2011b) found that super-Earths (m sin i = 1 − 10M⊕) are abundant around
M dwarfs with a frequency of ∼35%. Given the frequency of super-Earth around M dwarfs,
there are many planets awaiting for discoveries.
4.1.2 Challenges
M dwarfs emit the bulk their energy in the near infrared (NIR), they are thus much more
brighter in the NIR than in the optical wavelengths. NIR observation provides a promising
way of detecting planets around M dwarfs. However, there are several obstacles that prevent
us from making discoveries.
4.1.2.1 Atmophsere
Ground-based NIR RV measurement is severely affected by the Earth’s atmosphere,
which consists absorption lines of many species including H2O, O2, CO, CO2 and so on
(telluric lines). The RV of these species is correlated with the motion of the atmosphere such
as wind and turbulence. Measured RV of an M dwarf is therefore affected if telluric lines are
not properly removed from observed spectrum.
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4.1.2.2 Wavelength Calibration Sources
Another fact that has limited the improvement of RV precision in the NIR is the lack of
a stable and precise wavelength calibration source. A wavelength calibration source provide
a caliber (absolute wavelength calibration) for those obtained stellar spectra and enables
exclusion of instrument instability and measurement of stellar RV. Unlike those matured
wavelength calibration sources in the visible bands such as an Iodine cell (Butler et al.,
1996) and a Th-Ar emission lamp (Lovis & Pepe, 2007), the quest for a suitable wavelength
calibration source in the NIR remains a wide open question. Please refer to §3.2.2 for a more
detailed and complete review of the field of wavelength calibration sources.
4.2 Tackling Adversities in NIR RV Measurement
4.2.1 Software Advancement
4.2.1.1 Precise Telluric Lines Removal
Telluric lines exit and some times populate in the NIR part of an observed stellar
spectrum. Stellar RV will not be precisely measured unless telluric lines are carefully
removed from an observed stellar spectrum. Several ways of telluric removing scheme have
been proposed and practiced including telluric line forward modeling (Bean et al., 2010)
and observing a telluric standard star (Vacca et al., 2003). The former method relies on a
synthetic telluric absorption spectrum to forward model an observed spectrum together with
spectra of an stellar template and an absorption cell that provides an absolute wavelength
solution. It is computationally intense and the cross talk between different components
in the model is difficult to fully understand. The latter method relies on an observation of
a telluric standard star, usually a fast-rotating early type star whose intrinsic spectrum is
almost featureless. Telluric line spectrum can be obtained based on the observed spectrum
of an telluric standard star. This method requires more observational time but reduces the
complexity in data reduction process. I adopted the telluric standard star method to remove
telluric lines from an observed spectrum.
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Figure 4-1 show an example of an observed stellar spectrum (GJ 411) contaminated
with telluric lines (mostly water lines) in the NIR region between 8130 Å and 8270 Å . Most
of the observed lines are not associated with the star but formed by the Earth’s atmosphere.
The comparison between a telluric-line-removed stellar spectrum and a synthetic stellar
spectrum can be seen in Fig. 4-2. At this stage, a precise measurement of stellar RV
becomes possible. To further investigate the telluric removal level, I divide these two spectra
and remove outliers due to mismatches between an observed spectrum and a synthetic
spectrum. I found the residual after telluric line removal has an rms of 0.027 indicating ∼97%
of telluric line strength has been successfully removed.
Figure 4-1. Comparison between two spectra before (black) and after (red) removing telluriclines.
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Figure 4-2. Comparison between an observed stellar spectra (black GJ 411, telluric linesremoved) and a synthetic spectrum (red).
4.2.1.2 Binary Mask Cross Correlation
Stellar RV can be measured with the so-called cross correlation function (CCF) method
in which a fully wavelength-calibrated stellar spectrum is multiplied with a series of Doppler-
shifted template spectra. The maximum of the CCF corresponds the most likely stellar RV.
The template spectra can be obtained either from observation or from theoretical high-
resolution synthetic spectra. I adopted the binary mask cross correlation technique (Baranne
et al., 1996; Pepe et al., 2002; Queloz, 1995). The template spectra are generated from
synthetic stellar spectra and each absorption line has a boxed shape whose width and depth
are determined by the actual line width and line depth. The advantage of the binary mask
cross correlation technique is that it makes it possible to select specific lines of interests
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Figure 4-3. Telluric line removal residual is 2.7% after ejecting points that are caused bymismatches between an observed spectrum and a synthetic spectrum (markedby red asteriks).
while discarding potential contaminating spectral region. It is particularly useful when dealing
with spectra taken in the NIR which is severely contaminated by the telluric lines. This
technique is also useful to eliminate noise source contributed by weak absorption lines in the
low S/N case. Figure 4-4 shows an example of a wavelength-calibrated stellar spectrum in
the NIR (black) and a binary mask template (red) used in the cross correlation process.
4.2.2 Hardware Advancement
There is a lack of precise and stable wavelength calibration source in the NIR although
many candidates have been proposed (Bean et al., 2010; Li et al., 2008; Mahadevan & Ge,
2009; Redman et al., 2011; Steinmetz et al., 2008; Wildi et al., 2010). Suitable gas cells such
as an Iodine cell in the optical wavelengths are hard to find in the NIR and the precision is
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Figure 4-4. Red: an example of binary mask template. Black: wavelength-calibrated stellarspectrum.
limited by temperature and species contamination. Sources such as emission lamps are
facing aging problem. What’s more, their spectra are irregular and non-uniform, which poses
challenges in data reduction process.
Etalon can be used for precise wavelength calibration (Wildi et al., 2010). However, the
high finesse mirror limits its operating band to be relatively narrow. Laser frequency combs
technique (Li et al., 2008; Steinmetz et al., 2008) is widely believed to be the next generation
wavelength calibration source in the NIR, but it is still too expensive and inmatured to be
incorporated with an astronomical Doppler instrument.
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We are developing a Michelson interferometer (namely the sine source) that can be
used as a precise and stable wavelength calibration source. When compared to previously-
mentioned candidate sources, it has many advantages:
1. It is precise enough to provide a calibration precision of better than 10 cm · s−1whichwould enable detection of an Earth-like planet in the habitable zone of a solar-type star.
2. Its spectral stability is solely dependent upon its thermal stability. This feature reducesthe complexity when designing the interferometer and its enclosure.
3. It has a very broad wavelength coverage from optical wavelengths to the NIR with aoperation bandwidth of more than 800 nm. The wide wavelength coverage allows us tosimultaneously include more spectral region to increase S/N.
4. The spectral feature is uniformed and periodical which enables fast and precise dataprocessing.
5. It is compact with a dimension of 2 × 2 × 2 inch for its optics components. Whenequipped with a thermal enclosure, its dimension is 4 × 4 × 4 inch.
The applications of the sine source is flexible. It can be used either as an absorption
cell (Fig. ??) or as an emission lamp source (Fig. 4-6). In the former case the star light goes
through the sine source before recorded by a CCD. This application is similar to the Iodine
absorption cell method currently being used at the HIRES for the Keck telescope, but the
wavelength region with calibration is dramatically increased because of the broad wavelength
coverage of the sine source. However, the Earths telluric lines have to be considered in the
NIR. In the latter case where the sine source is used as an emission lamp source, star light
and the emission spectrum of the sine source are directed to the detector by two nearby but
separated fibers. This case is similar to the calibration scheme adopted by HARPS using a
Th-Ar emission lamp, but data reduction is expected to be simplified because of the simple
spectrum output of the sine source.
We have conducted demonstration experiments both in the lab and at the observatory
using the EXPERT instrument at the KPNO 2.1m telescope. Figure 4-7 shows the compar-
ison between two difference wavelength calibration sources. One is the sine source, the
other one is a Th-Ar lamp. In a 2-day experiment, we have demonstrated that the measured
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Figure 4-5. Application of the sine source as an absorption cell. Obtained spectrum is amultiplication of stellar spectrum, atmosphere absorption and sine sourcespectrum.
instrument drift (in m·s−1) from the two sources over time is consistent with each other at 10.7
m · s−1 level which is at the level of predicted photon-noise limited measurement error. The
sine source provides an overall more precise measurement and thus a better wavelength
calibrator.
4.3 M-dwarf Planet Search and Characterization-Results
4.3.1 Telluric Line RV Stability
It has long been wondered how stable the Earth’s telluric lines are and different RV
stabilities of different species. Figueira et al. (2010a) has studied O2 lines RV stability using
HARPS data and concluded that O2 lines are as stable as 10 m · s−1 over 6 years and the
intrinsic stability of O2 lines is even higher (2-3 m · s−1) when a simple physical atmosphere
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Figure 4-6. Application of the sine source as an emission lamp for simultaneous wavelengthcalibration. Stellar spectrum and sine source spectrum are arranged next to eachother.
model is considered. Bean et al. (2010) mentioned that H2O lines is as stable as 20
m · s−1 over half year. The small wavelength coverage (one spectral order, 8130-8270Å for
H2O and 6870-6935Å for O2) and unknown systematic error prevent us from studying the
absolute RV stability of these two species. However, the relative stability can be studied by
subtracting one RV measurement result from the other. Figure 4-8 shows the relative RV
stability between H2O and O2 lines and the RV scatter rms is 18.3 m · s−1 over 10 days. The
result is consistent with previous studies by Figueira et al. (2010a) and Bean et al. (2010).
The implication is that telluric lines can be used as a wavelength calibration source at ∼10-20
m · s−1 precision level.
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Figure 4-7. Top: Monitored RV drift over 2-day period. Results from Sin Source are plotted inblack with error bars, and the results from Th-Ar lamp are in red. The median ofmeasurement error is 3.8 m · s−1 for Sin Source and 18.4 m · s−1 for Th-Ar lamp.Bottom: The difference between results from Sin Source and Th-Ar lamp. Thetwo methods track each other with an RV RMS of 10.7 m · s−1.
4.3.2 RV Measurements of a Reference Star-GJ 411
GJ 411 is a known RV stable star with an RV scatter rms less than 7 m · s−1 according
to Endl et al. (2006). We have shown in Fig. 4-9 that the RV rms is 24.7 m · s−1 with a Th-Ar
lamp wavelength calibration and 40.6 m · s−1 with telluric lines as a wavelength calibration
source. Both results are 1.5 times worse than photon-noise limited prediction. The results
are expected to be improved significantly with larger wavelength coverage.
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Figure 4-8. Telluric water lines RV stability compared to O2 lines RV stability.
4.4 M-dwarf Planet Search and Characterization-Future Works
4.4.1 Searching For Planets Around M Dwarfs with EXPERT
The endeavor of searching for exoplanets has led to discoveries of over 400 planets
around stars of spectral type A-M1 . Only 16 (4.0%) planets has been found around M type
stars despite of the fact that they make up more than 70% of the galaxy including solar
neighbors. As of today, planets search programs around M type stars have resulted in
relative low detection rate compared to solar type stars (Cumming et al., 2008; Endl et al.,
2006; Zechmeister et al., 2009). Part of the reason is that RV measurement precision is
1 http://exoplanet.eu/
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Figure 4-9. RV measurements for GJ 411. Black: results using a Th-Ar lamp as a wavelengthcalibration source. Red: results using telluric lines as a wavelength calibrationsource.
relatively worse for M type stars, making it difficult to detect Neptune-like or lower mass
planets. On the other hand, the RV measurement is precise enough to detect close-in
Jupiter-like planets. The relative low detection rate indicates the low frequency of gas giant
around M type stars compared to solar type stars. Endl et al. (2006) gives 1σ upper limit
on frequency of gas giant around M type stars of < 1.27%. Butler et al. (2006) estimates
planets fraction of 1.8% ± 1.0% for planets masses over 0.4 MJ . Johnson et al. (2007a) finds
this fraction 1.8% for stellar mass range from 0.1-0.7 M⊙, and Cumming et al. (2008) found
that M dwarfs are 10 times unlikely to harbor a gas giant within a 2000-day orbit compared
to solar type stars. In contrast, Bonfils et al. (2007)found that planets less massive than ∼25
M⊕ are significantly more frequent around M dwarfs which supports the prediction that the
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frequency of Neptune-like planets are higher around M type stars than G type stars (Ida &
Lin, 2005; Kennedy & Kenyon, 2008; Kennedy et al., 2006). Among 16 planets discovered
around M type stars, 3 (18.8%) has been found more massive than 1 MJ and another 3
(18.8%) planets between 0.6 MJ and 1MJ . These Jupiter-like exoplanets pose challenges
to traditional core-accretion model in the way that classical core accretion model has severe
problem with forming gas giant planets due to less massive protoplanetary disk (Laughlin
et al., 2004), while competing gravitational instability model can effectively form Jupiter-like
planets around M type stars (Boss, 2006). More observations are needed to constrain
planetary formation theory and discoveries of planet around M type stars will help us address
the question from statistical perspective.
We are proposing to use EXPERT(EXtremely high Precision ExtrasolaR planet Tracker)
DEM(Direct Echelle Mode) at KPNO 2.1m telescope to conduct precise RV(radial Velocity)
measurements of 41 mid-late M type stars of spectral range between M3.5 and M6. We have
increased the mid-late M type stars sample by a factor of 1.5 compared to similar RV surveys
combined in the past (Cumming et al., 2008; Endl et al., 2006; Zechmeister et al., 2009). It
will provide robust statistical constraints on the frequency of close-in Jupiter-like planets and
Neptune-like planets around mid-late M type stars. Simulations show that we will be able to
obtain better than 3 m/s RV precision in photon-noise limit case in 10 min exposure for J=7
target. In comparison, a 20 M⊕ planet in a edge-on orbit with a of 0.1 AU (p∼20 day) around
a star of 0.3 M⊙ produces RV signal with semi-amplitude of 10 m/s.
M type stars emit the bulk of energy around 1 µm, so it may be advantageous observing
them in NIR(near infrared). Practically, we are inevitably facing two major obstacles going
into NIR: (1) telluric lines contamination; (2) lack of source for absolute wavelength cali-
bration. We will observe bright fast rotating early type stars(mostly A type) as telluric lines
standard star and try to remove the contamination of telluric lines from M type star spectrum.
Telluric lines contaminating portion in spectrum will be masked during data reduction if no
telluric standard star is available.
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4.4.2 Multi-Band Study of Radial Velocity Induced by Stellar Activity with EXPERT
Searching for earth-like habitable exoplanet has long been pursued by planet hunters.
However, it is extremely difficult to achieve due to the low RV signal (0.1 m · s−1) for solar type
star in contrast with current RV precision in visible band (∼ 1 m · s−1). On the other hand,
planets around late-type stars (i.e. M dwarfs) induce relatively larger stellar reflex motion
due to lower stellar mass, thus higher RV signal. Therefore, M dwarfs become favorable
targets in the search of habitable planets. However, visible band observation of M dwarfs
is difficult due to the intrinsic faintness of the objects. Therefore, RV measurements in near
infrared (NIR) is becoming an increasingly interesting field. Meanwhile, it is claimed that RV
jitters due to stellar activities are reduced in NIR, which becomes anther advantage of NIR
RV measurements. The argument appears to be observationally confirmed in a number
of cases (Bean et al., 2010; Huelamo et al., 2008; Martın et al., 2006; Prato et al., 2008).
However, in above observations, the measurements in visible band and in NIR were not
conducted simultaneously, in which we cannot rule out the possibility that the stars observed
were experiencing a less active period during the NIR observation. Therefore, simultaneous
measurements are required to confirm the trend of decreasing RV jitter toward longer
wavelength, i.e., NIR.
Desort et al. (2007) studied the RV induced by a star spot and gave an empirical
correlation between RV amplitude in visible and other parameters such as spectral type, spot
size and V sin i . Reiners et al. (2010) carried it further to compare the RV amplitude induced
by a star spot in the visible and NIR band. According to the simulation, they found that the
RV amplitude in Y band is at least twice smaller than that in V band for hot star spot. It is
an theoretical support for the trend of decreasing RV in NIR which needs confirmation from
multi-band simultaneous observation.
We have demonstrated the short-term RV precision in I band (λ0=806 nm, ∆λ=149 nm)
by observing a photometric stable star, KEP 11859158. The RMS is 57.6 m · s−1 for 7 days’
continuous observation (Fig. 4-10). We used Telluric lines as wavelength and RV calibration
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reference since the lines of Th-Ar lamp are very sparse in the same wavelength region.
We also removed the telluric lines from stellar spectra using a nearby telluric line standard
star which is observed at almost the same time as the observation of KEP 11859158. We
suspect that the scattering of the RV measurements is due to the intrinsic instability of telluric
lines (i.e., wind, water vapor density, etc.). We have developed a new RV calibration source,
i.e., an RV-calibration interferometer which produces dense lines over large wavelength cov-
erage including B, V, R, I and Y band. In-lab demonstration has shown that RV calibrations in
V band using Th-Ar lamp and with the RV-calibration interferometer track each other and the
RV calibration with the interferometer shows much smaller RMS scattering than Th-Ar lamp
(3.8 m · s−1 VS. 18.4 m · s−1, Fig. 4-7). After the recent installation of the RV-calibration inter-
ferometer in Feb 2011, the RV precision beyond 0.7 µm is expected to be greatly improved.
21.9 m · s−1 RV RMS scatter with the same star has been reached in the V band using DEM
of EXPERT (Fig. 4-11).
Most objects with RV measured both in V and NIR band show RV jitter or amplitude
more than 300 m · s−1 in V band. Plus, spectral types of the objects span from brown
dwarfs (Martın et al., 2006) to active young stars (Prato et al., 2008). If the RV jitter is indeed
smaller in NIR than in V band by a factor of at leasst two as predicted by Reiners et al.
(2010), then it is observable using EXPERT. Furthermore, the question of multi-band RV
jitter difference dependence on spectral type will for the first time be answered if the sample
of targets is carefully chosen. The finding of the proposal will provide insights for future
habitable earth-like exoplanet searching mission in NIR, helping understand the RV jitter of
stars of different spectral types.
4.4.3 Mid-Late Type M Dwarf Planet Survey Using FIRST
We proposes to conduct a pilot survey of 50 J≤8 nearby M dwarfs for exoplanets with
the 2-m Automatic Spectroscopic Telescope (AST) in 2013-2015 with a new generation
cryogenic high-resolution (R=60,000, 0.8-1.35 micron) near IR (NIR) cross-dispersed echelle
spectrograph. This instrument, called FIRST, is scheduled to see first light at the AST 2m
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Figure 4-10. RV measured in I band using DEM of EXPERT for KEP 11859158, aphotometric stable star. The RV RMS is 57.6 m · s−1.
telescope in June 2013. This pilot survey is designed to tune the new instrument and the
data pipeline to get ready for launching a NIR high precision Doppler exoplanet survey of 215
M dwarfs.
4.4.3.1 Science Justification
Historically, exoplanet searches have focused on stars with masses similar to that
of our Sun. Lower mass stars have received less attention in large part because they
are intrinsically faint and cool, emitting most of their light at NIR wavelengths where our
observational techniques are less well developed. Radial velocity searches have probed
the early M dwarfs (mass≥0.4 Solar mass) and have clearly indicated that short-period
giant planet companions to early M dwarfs are rare (Endl et al., 2006; Johnson et al.,
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Figure 4-11. RV measured in V band using DEM of EXPERT for KEP 11859158, aphotometric stable star. The RV RMS is 21.9 m · s−1.
2007a). However, planetary companions to stars at the peak of the stellar mass function and
below (mass≤0.4 Solar mass) remain essentially unexplored. These small stars represent
perhaps our best opportunity to detect Earth-mass planets, including those orbiting in the
Habitable Zone (HZ), given current levels of RV precision. At the same time, these planets
will necessarily be in the solar neighborhood, making them amongst the most important
targets for future space-based efforts to directly image Earth-like planets and to study their
atmospheres.
There is mounting evidence that sub-Neptune mass planets, including the Super-Earths,
may be very common in orbit around low-mass stars. While Kepler observes only a small
number of low-mass stars, it provides evidence that the planet mass function increases
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toward smaller planetary and stellar mass. An analysis by Howard et al. (2011) indicates that
up to 30% of early M dwarfs have Super- Earth sized planets with orbital periods less than
50 days. Analysis of the HARPS RV data by Bonfils et al. (2011b) finds similarly high values
for the rate of occurrence of super-Earths orbiting early M dwarfs with periods between 10
and 100 days (35%) as well evidence for a significant population with orbital periods less
than 10 days (36%). The detection of a system of Mars-size companions orbiting a late M
dwarf in Kepler data by Muirhead et al. (2012), one of only a handful of late-M dwarfs in the
Kepler field, provides further circumstantial evidence that these types of companions are very
common.
M dwarfs later than M4 are of great scientific interest. For these stars, the mass, size,
and temperature of the stars begin to rapidly decrease. To date, most exoplanet searches
targeting M dwarfs have been conducted at visible wavelengths (Bonfils et al., 2011b; Endl
et al., 2006; Johnson et al., 2007a) for stars with spectral type earlier than M4 because of
intrinsic faintness of mid-late type M dwarfs. There are only 12 M4 or later type stars with
V≤12 north of -30 degrees (Reid & Gizis, 1997). For comparison, there are about 300 nearby
stars M4 or later with J≤9 (Lepine & Shara, 2005). Therefore, for the latest types of stars, an
observing program must operate in the NIR.
The current state-of-the-art for NIR RV detection of planets around late M dwarfs
has been demonstrated with the VLTs CRIRES with moderate simultaneous wavelength
coverage (364 ) using an ammonia gas cell for calibration (Bean et al., 2010). Long-term
( 6 months) RV precisions of 5 m/s have been demonstrated with this system. With this
precision, and the observing time available at this facility, searches for giant planets orbiting
late-M dwarfs can be carried out. The precision that will be delivered by FIRST (better than 4
m/s) and the cadence enabled by TSUs AST make this system a logical next step for survey
of low-mass planets around mid-late type M dwarfs.
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4.4.3.2 Target Selection
Our targets were selected from the following catalogs: Gliese Catalog of Nearby
Stars; Gliese Catalog of Nearby Stars cross identified with 2MASS (Stauffer et al., 2010);
ROSAT All-Sky Survey: Nearby Stars (Hunsch et al., 1999). The selection was based on the
following criteria:
• J≤10 and dec≥-20
• MV ≤8.7 and V-K≥3.5
• Ratio between X ray luminosity and bolometric luminosity, RX ≤-3.0
215 M dwarfs were selected with the above criteria (Note: before we launch the survey,
we will use the 2MASS catalog to reject additional M dwarfs with a J≤14 stellar companion
within 5 arcsec and replace them with slightly fainter M dwarfs. During the survey, we
will reject spectroscopic binaries after 3 RV measurements from our targets and replace
them with new survey targets). Based on the empirical equation of rotation velocity vs. RX
in Kiraga & Stepien (2007), we expect 87% of our M dwarfs with rotational velocity less than
5 km/s. Therefore, most of them are inactive stars, which can help to minimize the RV jitters
caused by stellar activities although the jitter level is significantly reduced in NIR (Ma & Ge,
2012; Reiners et al., 2010). Figure 4-12 and Figure 4-13 show the number distribution in the
J and V bands of the FIRST M dwarf survey targets and the effective temperature distribution
of the survey targets.
4.4.3.3 Planet Yield Prediction
A Monte-Carlo simulation is used to estimate planet yield of the survey with FIRST.
For each selected star, an RV measurement precision is calculated based on its apparent
magnitude and spectral type. Figure 4-14 shows RV measurement precision in two cases.
In the baseline case, we assume total measurement error consists of 1.5 times photon-
noise measurement uncertainty, 0.5 m/s wavelength calibration error, 98% telluric line
masking error. The baseline case represents a reasonable scenario for the FIRST survey.
In the pessimistic case, we assume total measurement error consists of a 3 m/s unknown
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Figure 4-12. V and J band distribution for FIRST survey targets.
systematic error in addition to error sources assumed in the baseline case. The pessimistic
case represents a worst case scenario for the FIRST survey. Stellar mass is estimated from
its absolute magnitude. Once RV precision and stellar mass is known, we can generate
a detectability plot on mass-period space. More specifically, for a given planet mass and
orbital period, we generate a RV curve of 100 days from which 24 RV points are randomly
drawn to form a RV data set, eccentricity distribution follows that from (Wang & Ford, 2011).
The data set is then analyzed by a detection code based on periodogram, if the peak of the
periodogram agrees with the input period and the false alarm probability (FAP) is less than
1/1000, then we mark it as a detection. This test is repeated 100 times for each given planet
mass and period. Therefore, there is a detectability/completeness plot for each star. The
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Figure 4-13. Teff distribution for FIRST survey targets. 3180 K is used to divide early andmid-late type M dwarfs in the sample.
survey completeness plot (Fig. 4-15 and Fig. 4-16) is the average of completeness plots of
all selected stars.
To estimate planet yield, for example, estimated number of detected super Earths (1-10
M⊕), the planet yield is calculated by x · η · N, where x is the median of completeness in
the region with 1≤Msini≤10M⊕ and 1≤Period≤100 day, η=0.35 (Bonfils et al., 2011b) is the
frequency of Super-Earth around M dwarfs with 1-100 day period, and N is the sample size.
In total, 23 planets are expected to be detected for the pessimistic case including 5 super-
Earths, 2 giant planet (m sin i ≥ 100M⊕) and 16 intermediate-mass planets (10-100 M⊕). For
the baseline case, 30 planets are expected to be detected including 10 super-Earths, 2 giant
planets and 18 intermediate-mass planets.
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Figure 4-14. Predicted RV measurement precision for the FIRST survey. Black dotsrepresent baseline case (1.5 times photon-noise+calibration error+98% telluricmasking error) and red dots represent pessimistic case (1.5 timesphoton-noise+calibration error+98% telluric masking error+3.0 m/s unknownsystematic error). Dashed line represents the best precision achieved byHARPS M dwarf planet survey.
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Figure 4-15. The predicted survey completeness contours based on observation strategyand RV precision for the pessimistic case.
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Figure 4-16. The predicted survey completeness contours based on observation strategyand RV precision for the baseline case.
116
CHAPTER 5ACCURATE GROUP DELAY MEASUREMENT FOR RV INSTRUMENTS USING THE DFDI
METHOD
5.1 Introduction
As of Apr 2012, there are over 700 discovered exoplanets, and most of them are
detected by the radial velocity (RV) technique1 . RV precision of 1 m · s−1 has been rou-
tinely achieved (Bouchy et al., 2009; Howard et al., 2010b) with instruments such as
HARPS (Mayor et al., 2003) and HIRES (Vogt et al., 1994), which are cross-dispersed
echelle spectrographs. While cross-dispersed echelle spectrographs are commonly used in
instruments for precision RV measurements, a method using a dispersed fixed delay inter-
ferometer (DFDI) has offered an alternative method (Fleming et al., 2010; Ge et al., 2006b;
Lee et al., 2011). In this method, a Michelson-type interferometer is used in combination with
a moderate resolution spectrograph, RV signals are then extracted from phase shift of inter-
ference fringes of stellar absorption lines (Erskine, 2003; Erskine & Ge, 2000; Ge, 2002; Ge
et al., 2002). The details about the DFDI theory and applications are discussed in van Eyken
et al. (2010) and Wang et al. (2011). Instrument adopting the DFDI method has demon-
strated advantages such as low cost, compact size and multi-object capability (Fleming et al.,
2010; Ge, 2002; Ge et al., 2006b; Lee et al., 2011; Wisniewski et al., 2012).
The MARVELS (Multi-object APO Radial Velocity Exoplanet Large-area Survey) (Ge
et al., 2009) is a ground-based Doppler survey with the main goal of obtaining a large-scale,
statistically well-defined sample of giant planets. It has operated since 2008 until 2014. It
will search for gaseous planets around 11,000 stars that have orbital periods ranging from
hours to 2 years, and are between 0.5 and 10 Jupiter masses. It has completed observation
of 3,300 stars with over 94,000 RV data points, i.e., on average 28 data points per star. Over
250 binaries and a dozen of brown dwarfs have been detected from the survey.
1 http://exoplanet.eu/; http://exoplanets.org/
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In the DFDI method, a fixed delay interferometer (Wan et al., 2011, 2009) plays a
crucial role in creating stellar spectral fringes for high precision RV measurements (Erskine,
2003; Ge, 2002). The Doppler sensitivity can be optimized by carefully choosing the group
delay (GD) of the interferometer (Wang et al., 2011). More specifically, GD of an interferom-
eter should be chosen such that the spatial frequency of white light combs (WLCs) matches
with that of a stellar spectrum after rotational broadening. GD is defined by the following
equation:
GD(ν) = − 12π· dϕ
dν, (5–1)
where ϕ is phase shift and ν is optical frequency. The interferometer in a DFDI instrument is
usually designed to be field-compensated to minimize the influence of input beam instabil-
ity (Wan et al., 2009; Wang et al., 2010). It is realized by carefully selecting glass materials
and thicknesses of two second surface mirrors such that their virtual images are overlapped.
Because glasses are used in the optical paths, ϕ does no longer linearly change with fre-
quency, therefore GD is dependent on optical frequency. An inaccurate GD measurement
may significantly limit the RV measurement accuracy (Barker & Schuler, 1974; van Eyken
et al., 2010).
In practice, there may be several methods of measuring GD:
1. Calculate GD based on glass refractive indices using Sellmeier equation and thick-nesses from manufacturer specification.
2. Forward model the spectrum of a known spectral source, such as an Iodine cell or aTh-Ar lamp.
3. Measure phase and frequency using a while light source, such as a tungsten lamp.
4. Calibrate GD using a source with known velocity.
Method 1 is straightforward but may lack of adequate precision because of uncertainty
in parameters in Sellmeier equation and manufacturer tolerance for glass thickness. Method
2 holds great promise for accurately determining GD but there some current practical issues
118
preventing us from adopting this method (see more detailed discussion in §5.5.2.1). We will
use Method 3 and 4 to measure GD of an interferometer in this paper.
!"#
!"$
!%!
&%
&
Figure 5-1. Illustration of the DFDI method. Tilted lines represent interference combsgenerated by an interferometer. Vertical line represents an stellar absorption line(solid: original position with a frequency of ν0; dashed: shifted position with afrequency of ν).
In the DFDI method, GD determines the phase-to-velocity (PV) scale, the proportion-
ality between the measured phase shift and the velocity shift. Since the DFDI method is
realized by coupling a fixed delay interferometer with a post-disperser, the resulting fring-
ing spectrum−stellar absorption lines superimposing on the WLCs−is recorded on a CCD
detector (illustrated in Fig. 5-1). The fringe phase is expressed by the following equation:
ϕ(ν, y ) =2π · τ(ν, y ) · ν
c, (5–2)
119
where y is the coordinate along slit direction, which is transverse to dispersion direction,
τ is the optical path difference (OPD) of an interferometer and c is the speed of light. Two
mirrors (arms) of the interferometer are designed to be tilted towards each other along the
slit direction such that several fringes are formed along each ν channel. The intersection
of a stellar absorption line and a WLC moves (from Po to P in Fig. 5-1) if there is a shift of
an absorption line due to a change of stellar RV. Consequently, a small change of ϕ in the
dispersion direction, ∆ϕx , is induced:
∆ϕx =dϕ
dv· ∆v = dϕ
dν· dνdv· ∆v
=dϕ
dν· νc· ∆v = Γ · ∆v , (5–3)
where Γ is defined as phase-to-velocity scale (PV scale). It is determined by the GD of an
interferometer, which becomes explicit if Equation 5–1 and 5–3 are combined:
Γ = −2π · GD · νc. (5–4)
At resolutions typically adopted by the DFDI method (5, 000 ≤ R ≤ 20, 000), stellar lines
(line width∼0.1) are not resolved and a measurement of ∆ϕx is extremely difficult. Instead,
∆ϕy , phase shift along y direction can be measured, which is equal to ∆ϕx if the combs
generated by an interferometer are parallel to each other. This is a good approximation at
very high orders of interference. The advantage of measuring ∆ϕy instead of ∆ϕx is seen
from Fig. 5-1, in which the physical shift in the ν direction is amplified in y direction, the
amplification rate is determined by the relative angle between the interferometer combs and
a stellar absorption line. Therefore, ∆ϕy is relatively easier to measure compared to ϕx and it
is measured by fitting a well-sampled periodical flux signal along the y direction in the DFDI
method. Compared to conventional high-resolution Echelle method, the number of freedom
for the DFDI method in the fitting process is much less and small Doppler phase shift can be
relatively easier detected with a simple functional form, i.e., a sinusoidal function. However,
we want to point out that while the DFDI method provides a boost in instrument Doppler
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sensitivity, the Doppler sensitivity is not strongly dependent on the amplification rate because
flux slope decreases as amplification rate increases, which negates the gain of phase slope.
5.2 GD Measurement Using White Light Combs
5.2.1 Method
MARVELS (Multi-object Apache Point Observatory Radial Velocity Exoplanet Large-
area Survey) instrument covers a wavelength range from 500 nm to 570 nm and uses
a post-dispersive grating with a spectral resolution of 11,000 after a fixed delay interfer-
ometer (Ge et al., 2009). A Th-Ar emission lamp and an iodine absorption cell serve as
wavelength calibration sources. The instrument setup of MARVELS (Ge et al., 2009; Wan
et al., 2009) is similar to the equipments that measure GD as described in Kovacs et al.
(1995) and Amotchkina et al. (2009), in which a white light interferometer (WLI) is combined
with a post-disperser. However, the OPD is scanned by a moving picomotor in Amotchk-
ina et al. (2009) while it is realized by two relatively tilted arms in the WLC method using
MARVELS instrument. WLCs are generated by the interferometer when fed by a white light
source (e.g., a tungsten lamp). ϕ(ν), the phase of each frequency channel ν, is measured
and then unwrapped to remove ambiguity of 2π. GD is then derived by taking the derivative
of ϕ(ν) according to Equation 5–1.
Fig. 5-2 shows an example WLCs created by an interferometer with a fixed delay of
4 mm. The combs is created by an input continuum modulated with frequency due to con-
structive and destructive interference. The phase is determined by Equation 5–2. The phase
can be measured with a Fourier-transform-based algorithm described by (Rochford & Dyer,
1999): the signal H(ν) is obtained by firstly removing the negative Fourier components of
F (ν)−the flux distribution with frequency−and then conducting an inverse Fourier transform.
The phases ϕ(ν) are obtained by calculating and unwrapping the arguments of H(ν). Fig. 5-3
shows the unwrapped phase measured from flux distribution in Fig. 5-2. GD can be deter-
mined by measuring the derivative of unwrapped phase with respect to frequency according
to GD definition (Equation 5–1).
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Figure 5-2. Simulated WLCs of an interferometer with a fixed delay of 4mm. Flux ismodulated with frequency due to constructive and destructive interference.
5.2.2 Data Reduction
Standard spectroscopy reduction procedures are performed with an IDL data reduction
pipeline dedicated to MARVELS. Figure 5-4 shows an example of normalized flux as a
function of frequency for a processed spectrum. A zoom-in sub-plot shows the WLCs
produced by frequency modulation of the interferometer. Visibility, defined as the ratio of
half of peak-valley value to the DC offset, increases with frequency in the red part of the
spectrum. The increasing visibility in the blue end of the spectrum is not physical but caused
by an increasing photon noise and our algorithm of visibility calculation.
The fringe phases as a function of ν are calculated by the Hilbert transform tech-
nique (Rochford & Dyer, 1999) described in §5.2.1. We find that the phase change between
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Figure 5-3. Phase of simulated WLCs. Phase can be calculated by Fourier-transform-basedalgorithm described in §5.2.2.
pixels exceeds π in the blue part and therefore the phase unwrap cannot be successfully
applied, so we decide to use only part of the spectrum with a pixel range from 1800 to 3800
for phase unwrapping. A third-order polynomial is used to fit ϕ as a function ν. GD is then
calculated according to its definition (Equation 5–1).
5.2.3 GD Measurement Results
The top view and side view of the MARVELS interferometer are shown in Fig. 5-5. 60
fibers are mounted and each creates two spectra, one is picked from the forwarding beam
and the other one is from the returning beam (see Fig. 5-5). In total, 120 spectra are formed,
allowing us to measure GD at 60 positions on the interferometer along vertical (slit) direction.
Each position corresponds to a fiber number. There are 24 pixels along the slit direction
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Figure 5-4. The normalized flux and visibility (γ) as a function of frequency of a tungstenspectrum taken with MARVELS. The solid line is the normalized flux and filledcircles represent visibilities in different frequency channels.
for each spectrum. We chose 15 rows in the middle to measure GD because of relatively
higher photon flux, and thus smaller photon noise in the middle region of the spectrum.
The top panel of Fig. 5-6 shows phase measurement results for center row as a function
of frequency at different fiber numbers. Phase fitting residual (shown on the bottom panel
of Fig. 5-6, RMS=0.9 rad) is consistent with photon-noise limited measurement error (see
§5.2.4 for details). GD for a particular fiber number is obtained by averaging the results of
GD measurements for those rows associated with the fiber. Figure 5-7 shows the results
at ν=550 THz as a function of fiber number. Note that the two arms of the interferometer
are intentionally tilted to each other and the 60 fibers are evenly mounted along the slit
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direction. The measured GDs should gradually vary with fiber number. We use a second-
order polynomial to fit the GD variation with the fiber number. The fitting residual has an
RMS of 0.0046 ps. Figure 5-8 shows fitted GD as a function of frequency for different fibers.
GD varies 0.15 ps (0.6%) across measurement range from 540 to 565 THz. Ignoring GD
dependence of frequency would result in 180 m · s−1 measurement offset between two
ends of measurement range (assuming a true RV of 30,000 m · s−1, which is a typical stellar
RV value due to the Earth’s barycentric motion). Table 5-1 provides the polynomial fitting
coefficients of GD vs. fiber number at different frequencies within measurement range.
Figure 5-5. Top: top view of an individual fiber beam feeding of the MARVELS interferometer.Two spectra are formed by one fiber. One (Slit A) is from the returning beam armwhile the other one (Slit B) is from the forwarding beam arm. Bottom: side view ofthe fiber array beam feeding of the MARVELS interferometer. There are 60 fibersyielding 120 spectra. Note the exaggerated wedge angle of the shown secondsurface mirror, GD gradually changes along the vertical direction.
125
Figure 5-6. Top: white light combs phase as a function of frequency at different fiberlocations. Bottom: phase residual after third-order polynomial fitting.
5.2.4 GD Measurement Error Analysis
Two physical parameters, ϕ and ν, are measured in the experiment. The uncertainty
of the ϕ measurement is ∼0.8 rad under photon-noise limited condition assuming a S/N of
120 and a typical fringe visibility of 1.5%. The uncertainty due to the wavelength calibration
is ∼0.002 THz (0.02Å). We conduct a bootstrapping process to investigate the uncertainty
of GD caused by the measurement uncertainties of ϕ and ν. We add gaussian noises with
standard deviation of measurement errors to both ϕ and ν and calculate the group delay.
We run 1000 iterations for bootstrapping in order to estimate the uncertainty of GD. The
median of the relative error of GD measurements, δGD/GD, is 4.4 × 10−5. In comparison,
the median of the relative GD measurement error is ∼ 1.8 × 10−4 after smoothing by fitting
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Figure 5-7. Measured group delay as a function fiber number. Filled circles are measuredresults, solid line represents the best second-order polynomial fitting with anRMS fitting error of 0.0046 ps. Measurement results can be found in Table 5-1 atother frequencies.
a polynomial to GD variation with the fiber number. This number does not agree with the
relative GD error predicted by the bootstrapping experiment. We suspect that the uncounted
error in the bootstrapping simulation comes from image distortion due to optics which the
data pipeline has not fully corrected for, e.g., spectrum curvature, spectral line slant, etc.
In a 2-D spectrum as illustrated in Fig. 5-1, the phase shift between adjacent pixels along
slit direction is ∼0.6 rad, and the phase shift between each wavelength chanel is ∼2.5 rad.
An imperfect spectrum curvature tracing tends to shift pixel in the slit direction while an
imperfect slant correction can affect pixel shifting in both slit and dispersion directions. The
range of unwrapped phase is ∼4000 rad. For one fiber, if a gradually-changing phase error
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Figure 5-8. GD as a function of frequency at different fiber numbers. GD measurementresults vs. frequency and fiber numbers can be found in Table 5-1.
is introduced by the data pipeline when correcting for the optical distortion, for example, 0.4
rad deviation from true value at one end while no deviation at the other end, a relative error
of GD would be caused with an estimation of 0.4/4000 = 1 × 10−4. If different fibers are
treated independently, which is the case for the MARVELS data reduction pipeline, then this
gradually-changing phase error, introduced by imperfect optical distortion correction, may
explain the standard deviation error we see after the polynomial fitting for GDs as a function
of fiber number.
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Table 5-1. GD measurement results as a function of spectrum number(GD(#) = C0 + C1 · # + C2 · #2) and standard deviation (δGD) at differentfrequencies (ν)
ν[THz] C0 C1 C2 δGD[ps]540.0000 -2.5195975233e+01 4.1224184409e-03 -1.3537701123e-05 0.0066542.0000 -2.5216854974e+01 4.8064043246e-03 -2.3438315106e-05 0.0070544.0000 -2.5219400268e+01 3.8524896642e-03 -8.1355543995e-06 0.0057546.0000 -2.5232735338e+01 3.5346857088e-03 -2.6247780042e-06 0.0055548.0000 -2.5246276599e+01 3.5171101494e-03 -3.3394045215e-06 0.0047550.0000 -2.5259435353e+01 3.3877963986e-03 -1.7477210448e-06 0.0046552.0000 -2.5270775315e+01 2.8029235840e-03 9.6338400722e-06 0.0066554.0000 -2.5286851225e+01 3.5335302125e-03 -4.0412480867e-06 0.0093556.0000 -2.5295872014e+01 3.4000221425e-03 -2.5813725398e-07 0.0067558.0000 -2.5303654859e+01 2.3432563708e-03 1.8389109388e-05 0.0087560.0000 -2.5319347252e+01 2.6464319719e-03 1.2493738694e-05 0.0075
5.3 GD Calibration: Observing an RV Reference Star
5.3.1 Method
A deviated PV scale would result in an inaccurate velocity measurement given the same
amount of fringe phase shift:
∆ϕ = ∆v ′ · Γ′ = ∆v · Γ, (5–5)
where ∆v ′ represents a measured velocity shift while ∆v represents a true velocity shift.
Combining Equation 5–4 and 5–5, we obtain the following equation from which GD can be
calculated by using the measured velocity shift of an object with a known velocity.
GD =GD′ · ∆v′
∆v. (5–6)
This approach is similar to that of Barker & Schuler (1974), but the difference is that the latter
applied correction for discrete laser frequencies while we seek corrections for a continuous
frequency distribution.
In order to realize the method of GD calibration using an RV reference star, we need to:
1), assume a GD′ that is close to the true value of GD; 2), measure velocity shift ∆v ′ based
on an assumed GD′; 3), know the true value of the velocity shift of an RV reference star.
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Table 5-2. MARVELS predicted RV uncertainty (at an average S/N of 100) vs. Teff
Teff[K ] 4500 5000 5500 6000 6500 7000δv ′[m · s−1] 1.9 2.3 2.7 3.1 3.5 4.0
5.3.2 GD Calibration Precision
The calibration precision using an RV reference star is determined by measurement
error of ∆v ′:
δGD =GD′ · δv′∆v
, (5–7)
where δv ′ is RV measurement uncertainty. Wang et al. (2011) provided a method of calculat-
ing photon-limited RV uncertainty for the DFDI method. Under photon-noise limited condition,
we expect the GD calibration error to be determined by the photon-limited RV uncertainty
within the instrument band width, which is provided in Table 5-2. In Equation 5–7, GD′ is usu-
ally estimated to be within a few percent of true GD, ∆v is statistically ∼10,000 m · s−1 given
a uniform reference star distribution and a quarter year observational availability. Therefore,
relative error of GD measurement is ∼ 2 × 10−4 for an RV reference star with a Teff of 4500 K.
5.4 Implementation of Measured GD in Astronomical Observations
We use an RV reference star, HIP 14810 (V=8.5), as an example to show the RV
measurement results after implementation of the newly measured GD using the WLC
method. HIP 14810 is a star known to harbor 3 planets and its RV jitter is estimated to be 2
m · s−1 (Wright et al., 2009). After RV changes due to instrument drift, the Earth’s barycentric
motion and orbiting planets are removed, RV RMS error is 17.13 m · s−1 but has not reached
the predicted photon-limited RV uncertainty (4.8 m · s−1, S/N=80 with a half wavelength
coverage from 535 to 565 nm). RV RMS error is expected to be further reduced after the
data pipeline is improved in the future.
We also examine the reference star GD calibration method. We use one spectral
block within measurement range centering at 550 THz (540-560 THz) and set GD′ to be
an arbitrary value of -23.873 ps. The measured RVs (barycentric velocity not corrected)
are shown in Fig. 5-9. After applying correction according to Equation 5–6, we find that the
130
GD is -25.107±0.027 ps. In comparison, GD measurement result of fiber number 51 (the
fiber for HIP 14810) using the WLC method gives -25.091±0.005 ps (refer to Table 5-1). We
confirm that the GDs measured by these two methods are consistent with each other at 68%
significance level.
Figure 5-9. Top: measured (∆v ′) and true (∆v ) RVs of HIP 14810 (barycentric velocity notcorrected) over a period of 70 days. Bottom: the ratio of ∆v ′ and ∆v as a functionof time.
Many binary and brown dwarf discoveries are made owing to the well calibrated
interferometer GD. As of June 2012, more than 250 binaries have been discovered and a
dozen of brown dwarfs have been discovered by MARVELS and later confirmed by follow-
up observations conducted at other observatories (Fleming et al., 2010; Lee et al., 2011;
Wisniewski et al., 2012).
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Table 5-3. Comparison between two methods of GD measurement and calibrationWLC RS
Spectrum Coverage Half FullS/N ∼15 100 for V∼8a
Current precision 4.6 × 10−3 ∼ 9.3 × 10−3 ps 0.027 psCurrent RV errorb ∼ 2 m · s−1 10.8 m · s−1
Potential precision ∼ 3.1 × 10−4 ps ∼ 5 × 10−3 psPotential RV errorb ∼ 0.1 m · s−1 ∼ 2 m · s−1
Dependence on observation × XDependence on pipeline X X
Note. — a: assuming MARVELS throughput; b: RV error is calculated assuming a true velocity shift (∆v ) of10,000 m · s−1 according to Equation 5–6.
5.5 Summaries and Discussions
5.5.1 Summaries
The PV scale is an important parameter in the DFDI method that translates a measured
phase shift to an RV shift, and is determined by the group delay (GD) of an interferometer.
We have provided and discussed two methods of GD measurement and calibration: 1),
GD measurement using white light combs (WLCs) generated by the interferometer in a
DFDI Doppler instrument; 2), GD calibration using an RV reference star (RS). Table 5-3
summarizes the main results and the comparison between these two methods. The accuracy
of GD measurement is sufficient for current RV precision achieved with instruments using the
DFDI method (Fleming et al., 2010; Lee et al., 2011; Muirhead et al., 2011). However, higher
measurement and calibration precision is required in the near future as higher RV precision
is achieved by DFDI instruments in search for exoplanets. RS and WLC methods can serve
as complementary methods of GD measurement and calibration for DFDI instruments.
5.5.2 Discussions
5.5.2.1 White Light Comb (WLC) Method
The GD measurement using WLCs created by the interferometer provides a direct way
of calibrating the PV scale. In the region where combs are visible, effective S/N is relatively
132
low (∼15) because of low comb visibility (1.5%). In addition, GD cannot be measured in the
region where combs are not visible, which limits the application of this method. We are able
to measure GD in a region that accounts for half of the spectrum coverage. Extrapolation
beyond the measurement range may result in large uncertainties. The major issue facing
the method is that the data reduction pipeline may have introduced unknown errors while
correcting optical distortions such as spectrum curvature and slant.
In principle, we can use a tungsten lamp with an iodine cell or a Th-Ar lamp instead of
a tungsten lamp in order to increase the fringe visibility. However, there are some practical
concerns that hinder us from applying the above solutions: 1) line blending, because of low
spectral resolution, many spectral lines can not be resolved and it is not certain at this stage
how line blending affects phase measurement; 2) illumination correction, which is required
to correct for illumination profile in slit direction in order to properly measure the phase. We
adopt a self-illumination correction procedure in the pipeline which requires a certain contin-
uum level to be successfully achieved. Th-Ar is less affected by line blending if a careful line
selection process is involved, but it does not have enough continuum level for self-illumination
correction. An experiment is being conducted in which a second interferometer is used to
improve the visibility of WLCs so that GD is more precisely measured at a higher effective
S/N for a wider frequency coverage.
When compared to previous work in the field of GD measurement, Amotchkina et al.
(2009) achieved a measurement precision of 1 × 10−4 ps. Our measurement of GD has a
typical accuracy of ∼ 6 × 10−3 ps (limited by effective S/N and systematic errors), which
is more than an order of magnitude lower. However, it is shown in §5.2.4 that substantial
improvement would be able to be achieved once the data reduction pipeline has a better
handle of optical distortion. Wan et al. (2010) measured GD for the MARVELS interferometer
using a scanning WLI method and achieved a precision of 0.6 × 10−5 ∼ 2.4 × 10−5 ps,
which is more than two order of magnitude better than the results in this paper. However,
there are practical concerns using GD measurement results from the scanning WLI method
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because they are not measured in situ, therefore it is not easy to associate a position in a
WLI measurement to a fiber position.
5.5.2.2 Reference Star (RS) Method
The GD calibration using an RV reference star (RS) is a self-calibrating process and has
the potential of achieving a high calibration precision if the following requirements are met:
1), the RV reference star has a large velocity shift during a observation window; 2), the RV
reference star is bright; 3), the data reduction pipeline is able to produce the photon-limited
RV precision. In addition, GD is practically measured within a certain band width:
GD(ν) =
∫∆ν
GD(ν)ω(ν)dν∫∆νω(ν)dν
, (5–8)
where ω(ν) is weight function. The band width, ∆ν, should be small such that the dispersion
effect is negligible. The limitations stated above prevent us from precisely determining the PV
scale at the position of each fiber because not every fiber has a continuous observation on
a bright known RV reference star. For MARVELS, the brightest RV reference star available
has V mag of 8 and the resulting S/N is ∼100 per pixel. However, the RS method is a very
promising approach for a single-object DFDI instrument because only one bright reference
star is required in the field. We are planning to apply this method in calibrating GD of another
DFDI instrument (EXPERT) at KPNO 2.1m telescope (Ge et al., 2010). Note that the S/N
can be further increased by conducting multiple independent measurements and increasing
instrument throughput.
5.5.2.3 A Future M-Dwarf Survey With the DFDI Method
The MARVELS survey concept can be adopted by a future M-dwarf planet survey using
the DFDI method. As the planets around M dwarfs are gaining increasingly more attention, a
survey of nearby M dwarfs is important in order to completely understand planet occurrence
rate for M dwarfs and its dependency on stellar metallicity, stellar mass and activity and so
on. This type survey is ideal for large area spectroscopic telescopes such the SDSS 2.5m
telescope and the LAMOST telescope. There are about 15,000 mid-late type M dwarfs
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with J magnitude less than 12 in the northern sky (Wang et al., 2011). Considering about 6
square degree field of view for these telescopes, there are 5 such stars in the field of view.
According to previous study on the multi-object case, a resolution of 5,000 would be optimal
for a multi-object DFDI survey (see §2.3.4). At such resolution, we expect 125 m/s RV
precision for a J=12 star and 8 m/s for a J=6 star. With this design concept and instrument
specification, a 1 K by 1 K near infrared detector if sufficient for 800 to 1350 nm wavelength
coverage.
135
CHAPTER 6ECCENTRICITY DISTRIBUTION FOR SHORT-PERIOD EXOPLANETS
6.1 Introduction
The discovery of exoplanets has significantly advanced our understanding of formation
and evolution of planetary system (Marcy & Butler, 1996; Mayor & Queloz, 1995; Wolszczan,
1994). As of February 2011, over 500 exoplanets have been discovered including 410 sys-
tems detected by radial velocity (RV) technique1 . The eccentricity distribution of exoplanets
is very different from that of solar system. For sufficiently short-period planets, it is expected
that tidal circularization would lead to nearly circular orbits. Yet, several short-period planets
appear to have eccentric orbits. Several mechanisms (e.g. planet scattering, Kozai effect)
have been proposed to explain the observed eccentricity distribution (Ford & Rasio, 2008;
Juric & Tremaine, 2008; Takeda & Rasio, 2005; Zhou & Lin, 2007). This chapter aims to
improve our understanding of the true eccentricity distribution and its implications for orbital
evolution.
The Bayesian approach offers a rigorous basis for determining the posterior eccentricity
distribution for individual system. The Bayesian method is particularly advantageous relative
to traditional bootstrap method when the orbital eccentricity is poorly constrained by RV
data (Ford, 2006). Ford (2006) discussed eccentricity estimation using Markov Chain
Monte Carlo (MCMC) simulation in the framework of Bayesian inference theory and found
a parameter set that accelerates convergence of MCMC for low eccentricity orbit. For a
population of planets on nearly circular orbits, eccentricity estimates for planets on circular
orbit are biased resulting in overestimation of orbital eccentricities (Zakamska et al., 2011).
Further complicating matters, the population of known exoplanets is not homogeneous, and
the observed eccentricity distribution is affected by the discovery method, selection effects
and data analysis technique.
1 http://exoplanet.eu/; http://exoplanets.org/
136
6.2 Method
We select all the systems with: 1) a single known planet discovered with the radial
velocity technique as of April 2010; 2) an orbital period of less than 50 days; and 3) a publicly
available radial velocity data set. We exclude planets discovered by the transit technique
in order to avoid complications due to selection effects (Gaudi et al., 2005). We perform an
orbital analysis on each system in our sample using: 1) a standard MCMC analysis (§2.1)
and 2) a new method, Γ analysis (described in §2.2). We focus on the eccentricity estimation
for each planet since the eccentricity is an important indication of orbital evolution and tidal
interaction.
6.2.1 Bayesian Orbital Analysis of Individual Planet
We performed a Bayesian analysis of the published radial velocity observation using a
model consisting of one low-mass companion following a Keplerian orbit. If a long-term RV
trend is included in the original paper reporting the RV data or if a linear trend of more than 1
m · s−1 · yr−1 is apparent, then we add to the model a constant long-term acceleration due to
distant planetary or stellar companion.
We calculate a posterior sample using the Markov Chain Monte Carlo (MCMC) tech-
nique as described in Ford (2006). Each state in the Markov chain is described by the
parameter set θ = P,K , e,ω,M0,Ci ,D,σj , where P is orbital period, K is the velocity
semi-amplitude, e is the orbital eccentricity, ω is the argument of periastron, M0 is the mean
anomaly at chosen epoch τ, Ci is constant velocity offset (subscript i indicates constant for
different observatory), D is the slope of a long-term linear velocity trend, and σj is the ”jitter”
parameter. The jitter parameter describes any additional noise including both astrophysical
noises, e.g., stellar oscillation, stellar spots (Wright et al., 2005) and any instrument noise
not accounted for in the reported measurement uncertainties. The RV perturbation of a host
star at time tk due to a planet on Keplerian orbit and possible perturbation is given by
vk,θ = K · [cos(ω + T ) + e · cos(ω)] +D · (tk − τ), (6–1)
137
where T is the true anomaly which is related to eccentric anomaly E via the relation,
tan(T
2
)=
√1 + e1 − e tan
(E
2
). (6–2)
The eccentric anomaly is related to the mean anomaly M via Kepler’s equation,
E (t) − e · sin[E (t)] = M(t) −M0 =2πP
(t − τ). (6–3)
We choose priors of each parameter as described in Ford & Holman (2007). The prior
is uniform in logarithm of orbital period. For K and σj we use a modified Jefferys prior in the
form of p(x) ∝ (x + xo)−1 (Gregory, 2005) with Kmin = σj ,min = 0.1 m · s−1. Priors are uniform
for: e (0 ≤ e ≤ 1), ω andM (0 ≤ ω,M < 2π), Ci and D. We verified that the parameters in θ
did not approach the limiting values. We assume each observation results in a measurement
drawn from normal distribution centered at the true velocity, resulting in a likelihood (i.e.,
conditional probability of making the specified measurements given a particular set of model
parameters) of
p(v |θ,M) ∝∏k
exp[−(vk,θ − vk)2/2σ2k ]
σk, (6–4)
where vk is radial velocity at time tk , and vk,θ is the model velocity at time tk given the model
parameters θ. Noise σk consists of two parts. One component is from the observation
uncertainty σk,obs reported in the radial velocity data, and the other is the jitter, σj , which
accounts for any unforseen additional noise including instrument instability and stellar jitter.
The two parts are added in quadrature in order to generate σk . We calculate the Gelman-
Rubin statistic, R , to test for nonconvergence of Markov chains.
We perform a MCMC analysis for RV data set of each system in the sample and obtain
posterior samples of h and k , where h = e cosω and k = e sinω. This parameterization
has been shown to be more effective in description of the eccentricity distribution for low
eccentric orbits (Ford, 2006). We take steps in h and k and adjust the acceptance rate
138
according to the Jacobian of the coordiante transformation, so as to maintain a prior that is
uniform in e and ω. Mean values, h and k , from posterior samples of h and k are adopted
to calculate eMCMC using the equation eMCMC =√h2 + k2. The posterior distribution of e
is not always Gaussian distribution especially near e ∼0. Therefore, it is not appropriate
to calculate the uncertainty of e using the equation of error propagation in which gaussian
noise is assumed. We use posterior distribution of e to infer the credible interval of e. The
boundaries of the region where 68% posterior samples populate are adopted as elower
and eupper (Table 6-1). Figure 6-1 illustrates two examples of how the credible intervals are
inferred for HD 68988 (eccentric orbit) and HD 330075 (circular orbit).
Figure 6-1. Examples of how credible intervals of standard MCMC analysis are calculatedusing posterior distribution of e. Grey region contains 68% of total number ofposterior samples of e.
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6.2.2 Γ Analysis of Individual Systems
A fully Bayesian analysis of the population of exoplanet eccentricities would be com-
putationally prohibitive due to the large number of dimensions. Therefore, we develop a
hybrid Bayesian-frequentist method to evaluate the significance of a non-zero eccentricity
measurement. We combine a bootstrap style approach of generating and analyzing synthetic
data sets with MCMC analysis of each synthetic data set to obtain a frequentist confidence
level for each eccentricity that accounts for biases. First, we perform the standard MCMC
analysis described in §2.1 on the real RV data set and adopt the mean value of each orbital
parameter in θ except e. We generate a series of simulated radial velocity data sets at dif-
ferent values of e. The adopted K is scaled accordingly to K ∝ (1 − e2)−0.5. The simulated
radial velocity data has the same number of observations, and each simulated observation
takes place at exactly the same time and the same mean anomaly as the real observation.
Gaussian noise with standard deviation of σk (§2.1) are added to simulated radial velocity
data sets at different eccentricities. Each simulated RV data set has the same reported RV
measurement uncertainties as the real RV observations. Standard MCMC analysis is then
performed on each of the simulated RV data sets.
For both real and simulated data sets, we construct a two-dimensional histogram
using the posterior samples in (h, k) space to approximate a two-dimensional posterior
distribution for h and k , dr (hi , kj ) and ds(hi , kj ), where i and j denote bin indices, and the
subscripts r and s denote the real and simulated data set. We compare the distribution for
each simulated data set ds(hi , kj ) to the distribution for real radial velocity data set dr (hi , kj ).
To quantify the similarity between ds(hi , kj ) and dr (hi , kj ), we calculate the statistic defined
as Γ = [N∑i=1
N∑j=1
(ds(hi , kj ) − µs) · (dr (hi , kj ) − µr )]/[σsσr (N2 − 1)], where N is the number
of bins in h or k dimension, µ and σ represent mean and standard deviation. In other
words, the Γ statistic is obtained by cross-correlating two posterior distributions in h and k
space. Figure 6-2 illustrates the process by which we obtain Γ for the case of HD 68988.
If ds(hi , kj ) matches dr (hi , kj ), we expect to obtain a Γ value that approaches unity (blue
140
and red contours). If the samples differ significantly then Γ decreases towards zero (red
and green contours). For each eccentricity, we simulated 21 radial velocity data sets and
compare ds(hi , kj ) with dr (hi , kj ) to obtain 21 Γ statistics between simulated and real RV data.
We choose the median value Γ as an indicator of overall similarity at given eccentricity.
Figure 6-2. Contours of posterior distribution in h and k space for HD 68988 (Solid-68% ofsample points included; dashed-95% of sample points included; dotted-99% ofsample points included). Red contours are posterior distribution for realobservation, green contours are for simulated RV data set with e=0.00, and bluecontours are for simulated RV data set with e=0.13. Eccentricity of HD 68988 is0.1250±0.0087 according to Butler et al. (2006).
Based on above analysis of simulated radial velocity data with different input eccentric-
ities e, we obtain a relationship between Γ and e, i.e. Γ(e). We use a high-order polynomial
to interpolate for Γ(e). We define e to be the eccentricity at which Γ(e) reaches maximum
and we interpret e as an estimator of eccentricity. For HD 68988 (Fig. 6-3), input eccentricity
141
ranges from 0.00 to 0.29 with step size of 0.01. Γ(e) reaches maximum at e = 0.134. We
estimate statistical confidence level of e using every pair of posterior eccentricity samples
calculated from the data sets that are generated assuming the same eccentricity. Consider
the example of HD 68988 again: 21 sets of posterior distribution in h and k space, ds(hi , kj ),
are obtained. Comparison between each pair gives a Γ statistic between simulated RV data.
Σ20i=1 = 210 Γ statistics in total for simulated RV data sets are calculated at eccentricity of
0.13 and 68.1% of pairs (143 out of 210) have a Γ statistic greater than 0.3714. We define
this value, Γc,0.68, as the critical Γ value for HD 68988 at 68% confidence level for e = 0.13.
Therefore, if Γ statistic obtained in comparison between ds(hi , kj ) and dr (hi , kj ) is less than
Γc,0.68, we argue that the eccentricity inferred from simulated RV data set is not consistent
with the observed eccentricity of the system at 68% confidence level. In the case where e is
located between grids of simulated e values, we calculate Γc,0.68 at e using interpolation of
Γc,0.68 at nearby e values. We use a high-order polynomial to approximate the discrete data
Γ(e). The polynomial is later used to infer e, lower and upper limit of eccentricity. For HD
68988, Γc,0.68 is 0.3663 at e = 0.134 after interpolation. Using the relationship between Γ and
e (Fig. 6-3), we look for the e values corresponding to Γc,0.68 as estimators of the lower and
upper limit for eccentricity of the planet system at a 68% confidence level. In HD 68988, we
obtained e = 0.134+0.040−0.040 using Γ analysis. In comparison, we have obtained e = 0.119+0.025
−0.022
using a standard MCMC analysis and Butler et al. (2006) reported e = 0.125 ± 0.009.
6.3 Results for Individual Planets
In our sample of 50 short-period single-planet systems, we successfully analyzed 42
systems using Γ analysis, and 46 systems using standard MCMC analysis (Table 6-1).
All the error are based on a 68% confidence level (Γ analysis) or a 68% credible interval
(MCMC). The unsuccessful cases in standard MCMC analysis are HD 189733, HD 219828,
HD 102195 and GJ 176. In HD 189733, most of the RV data points were taken during
observation of Rossiter-McLaughlin effect, which is not modeled here. MCMC analysis fails
for HD 219828 and GJ 176 because of low signal to noise ratio (K=7 m · s−1 and nobs=20
142
Figure 6-3. Γ as a function of eccentricity e for HD 68988. Open circles are results fromsimulations, solid line is the result of polynomial fitting. The long dashed line isthe critical threshold, Γc,0.68 at the 68% confidence level.
for HD 219828 and K=4.1 m · s−1 and nobs=57 for GJ 176). RV data points of HD 102195
were taken at 3 observatories and MCMC analysis is complicated using different observatory
offsets. In addition to the above, Γ analysis was unsuccessful for HD 162020, GJ 86, HD
17156 and HD 6434. Since Γ analysis involves generating simulated RV data, the limited
number of observations and partial phase coverage for these systems can cause poor
convergence for some simulated data sets. These limitations become more severe for
systems with high eccentricity (e.g. HD 17156, e = 0.684) since phase coverage is more
important for eccentric orbits.
143
Figure 6-4. Top: comparison between standard MCMC analysis (blue) and previousreferences (red); Bottom: comparison between standard MCMC analysis (blue)and Γ analysis (green). The systems where disagreements take place aremarked with a number: 1, HD 149026; 2, τ Boo; 3, HD 195019.
6.3.1 Comparison: Standard MCMC and References
In Fig. 6-4 (top), we compare results from two sources, standard MCMC analysis and
previous references. In most cases the two methods provide similar results. We found there
are 3 systems for which the eccentricity estimates are not consistent, i.e., the published
eccentricity error bar does not overlap the 68% credible interval from our MCMC analysis.
They are HD 149026, τ Boo, HD 195019. For HD 149026, Sato et al. (2005) set the
eccentricity to be zero when fitting the orbit. In contrast, we treat eccentricity as a variable
and the standard MCMC method found that the 68% credible interval for the systems
mentioned above does not include zero. In addition, HD 149026 b is a known transiting
144
planet and transit photometry provides additional constraints on eccentricity which we
have not included (Charbonneau, 2003). Knutson et al. (2009) measured ∆tΠ = 20.9+7.2−6.2
minute(2.9σ) for HD 149026 which is inconsistent with zero eccentricity, because e ≥
e cosω ≃ π2P∆tΠ, where P is period and ∆tΠ is the deviation of secondary eclipse from
midpoint of primary transits. Standard MCMC results for other two systems (τ Boo and HD
195019) are not consistent with those from previous references even though eccentricity was
treated as variable in previous references. Butler et al. (2006) report e = 0.023 ± 0.015 for τ
Boo and e = 0.014 ± 0.004 for HD 195019. On the contrary, standard MCMC analysis gives
e=0.0787+0.0382−0.0246 for τ Boo and e=0.0017+0.0049
−0.0017 for HD 195019 (See Table 6-1).
6.3.2 Comparison: Standard MCMC and Γ Analysis
Figure 6-4 (bottom) compares the results from standard MCMC analysis and Γ analysis.
We find that the 68% credible/confidence intervals for the two methods overlap in the cases
where there are discrepancies between standard MCMC analysis and previous references
(see § 6.3.1). The confidence interval from the Γ analysis is generally larger than the credible
interval from a standard MCMC analysis. The larger uncertainty from Γ analysis is likely due
to the analysis accounting for the uncertainty in each velocity observation twice, first when
generating synthetic data sets and a second time when analyzing the simulated data. Thus,
the Γ analysis results in slightly larger uncertainty in eccentricity estimation.
In order to understand the behavior of standard MCMC analysis and Γ analysis for
planets on nearly circular orbits, we conduct an additional experiment generating many
synthetic data sets where each system has a single planet on a circular orbit. We assume
that they are observed at the same times and with the same RV measurement precisions
as actual RV data sets. In order to understand the bias of each method for nearly circular
systems, we compare the output eccentricities and their uncertainties. Using standard
MCMC analysis, we find that 76.4 ± 2.9% of the simulated data sets are consistent with zero
using a 68% credible interval, and 23.6 ± 1.6% of the simulated data sets have 68% credible
intervals that do not include zero. In contrast, for 16.8 ± 1.4% of the simulated data sets, the
145
Γ analysis does not result in a 68% confidence interval that includes zero. In both cases,
more than 68% of data sets are consistent with a circular orbit at a 68% level using either
method. Using the Γ analysis, 6.8 ± 3.0% more simulated data sets are consistent with a
circular orbit than based on the standard MCMC analysis. This confirms our intuition that the
Γ analysis is a less biased method for analyzing systems at very small eccentricity. Thus, the
Γ analysis may be a useful tool in assessing the significance of a measurement of a small
non-zero eccentricity. In particular, we find 5 cases (11.4%) in which elower=0 for Γ analysis
even though elower for MCMC is greater than zero (e.g., HD 46375, HD 76700, HD 7924, HD
168746, HD 102117).
A similar experiment is conducted except that an eccentricity of 0.2 is assigned to each
system instead of zero eccentricity. Again, we assess the accuracy of the two methods
by comparing the input and output eccentricities. When using the MCMC method, we find
that the 68% credible interval for the eccentricity does not include the input eccentricity
for 26.0 ± 2.1% of simulated data sets. When using the Γ method, we find that the 68%
confidence interval for the eccentricity does not include the input eccentricity for 18.9 ± 1.8%
of simulated data sets. Again, there is a larger fraction of results from the MCMC method
that are not consistent with the input at a sizable eccentricity, indicating Γ analysis is less
likely to reject the correct eccentricity. We also investigate the bias of the two methods at a
significant eccentricity (i.e. e=0.2). In the cases where the output eccentricities are consistent
with the input, we find that 47.9 ± 3.3% of the output eccentricities are below 0.2 while
52.1 ± 3.5% of outputs are over 0.2 for MCMC method indicating the MCMC method is not
biased at a sizable eccentricity, which agrees with the finding from Zakamska et al. (2011).
In comparison, Γ analysis is also an unbiased analysis with 49.9 ± 3.3% below input and
50.1 ± 3.3% exceeding input. Therefore, we find no evidence for significant bias of either
method for data sets with a significant eccentricity.
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6.3.3 Discussion of Γ Analysis
The different methods for analyzing Doppler observations are complimentary. Bayesian
methods and MCMC in particular are routinely used to sample from the posterior distribution
for the Keplerian orbital parameters for a given system. However, the analysis of an exo-
planet population is more complicated than simply performing a Bayesian analysis of each
system. To illustrate this point, consider a population of planets that all have exactly circular
orbits. Due to measurement uncertainties and finite sampling, the ”best-fit” eccentricity for
each system will be non-zero. Similarly, since eccentricity is a positive-definite quantity, the
analysis of each system will result in a posterior distribution that has significant support for
e > 0. This property remains even if one combines many point estimates (e.g., ”best-fit”),
frequentist confidence intervals or Bayesian posterior distributions. While the posterior
distribution for the orbital parameters represents the best possible analysis of an individual
system, the inevitable bias for nearly circular orbits is a potential concern for population
analyses. Therefore, it is important to apply different methods for population analyses (e.g.,
Hogg et al. (2010); Zakamska et al. (2011)).
Since we intend to investigate the potential role of tidal effects on the eccentricity
distribution of short-period planets, we developed a hybrid technique to assess the sensitivity
of our results to bias in the posterior distribution for planets with nearly circular orbits. This
hybrid technique (Γ analysis) involves performing Bayesian analyses of each individual
planetary system along with several simulated data sets, each of varying eccentricity.
The MCMC analysis of each data set allows us to account for the varying precision of
eccentricity measurements depending on the velocity amplitude, measurement precision,
number of observations and phase coverage. We assess the extent of the eccentricity
biases by performing the same analysis on simulated data sets with known eccentricity.
We compare the posterior densities for the actual data set to the posterior density for each
of the simulated data sets to determine which input model parameters are consistent with
the observations. We can construct frequentist confidence intervals based on Monte Carlo
147
simulations (i.e., comparing the posterior distributions for the synthetic data sets to each
other).
The basic approach of the Γ analysis is similar to likelihood-free methods more com-
monly used in approximate Bayesian computation. In this case, we do have a likelihood
which allows us to sample from the posterior probability distribution using standard MCMC.
We compare the posterior densities calculated for several simulated data sets to the posterior
density of the actual observations, so as to assess the accuracy and bias of the standard
MCMC analysis.
Figure 6-5. Cumulative distributions functions (CDFs) of eccentricities from differentmethods. The solid red line is for eMCMC , adopted from MCMC method (Table6-1). The dotted red line is similar to MCMC, but any eccentricity with elower of 0for a 95% credible interval is assigned to 0. The blue lines are for Γ analysis,where the solid line is for eΓ from the Γ analysis (Table 6-1) and the dotted line issimilar, but any eccentricity with elower of 0 for a 95% confidence interval isassigned to 0.
148
The problem of biased eccentricity estimators for nearly circular orbits is familiar from
previous studies of binary stars. In particular, Lucy & Sweeney (1971) investigated the
possibility of mistakenly assigning an eccentric orbit to a circular spectroscopic binary due
to inevitable measurement errors. As many spectroscopic binaries may have been affected
by tidal circularization, they suggested assigning a circular orbit to any system for which the
eccentricity credible interval contained 0. When studying a population of systems for which
circular orbits are common, this approach significantly reduces the chance of erroneously
concluding the system has a non-zero eccentricity. One obvious disadvantage of this
approach is that it would lead to a negative bias for systems where the eccentricity is of order
σ/K , where σ is the typical measurement precision and K is the velocity amplitude. For
binary stars, σ/K may be small enough that this is not a significant concern. For exoplanets,
where σ/K may be as small as ∼ 2 − 3, such a procedure would result in a significant
negative bias for many systems. The Γ analysis offers an alternative approach, which may
be particularly useful when analyzing the eccentricity distribution of a population of planetary
systems.
For the sake of comparison, we consider a modernized version of the Lucy & Sweeney
(1971) approach which is based on the posterior distribution from a standard MCMC anal-
ysis or the confidence interval from our Γ analysis. We construct a histogram or cumulative
distribution of the eccentricities for a population of systems, using a single summary statistic
for each system: the posterior mean for the standard MCMC analysis or the eccentricity
that maximizes the Γ statistic. Following Lucy & Sweeney (1971), we adopt an eccentricity
of zero for any system for which the 95% significance level (Γ method) or the 95% credible
interval (MCMC) includes e = 0. The cumulative distribution functions of the eccentricities
using different methods are plotted in Fig. 6-5. Based on the generalized Lucy & Sweeney
(1971) approach, ∼81% (70%) of the short-period planet systems in our sample are consis-
tent with circular orbits using the Γ analysis (standard MCMC analysis). Clearly, the Lucy &
Sweeney (1971) approach results in a large fraction being assigned a circular orbit, largely
149
due to the choice of a 95% threshold. The fraction assigned a circular orbit is sensitive to
the size of the credible interval used when deciding whether to set each eccentricity to zero.
There is no strong justification for the choice of the 95% threshold (as opposed to 68% or
99.9% threshold) and tuning the threshold to agree with other methods negates the primary
advantage of the Lucy & Sweeney (1971) method, that it requires no additional computa-
tions. Therefore, we do not recommend using the Lucy & Sweeney (1971) approach to learn
about the eccentricity distribution for a population when σ/K is not large.
6.4 Tidal Interaction Between Star and Planet
Several factors affect the eccentricity distribution of short-period planets including
tidal interaction between host star and planet and possible perturbation of an undetected
companion. We will discuss how these factors affect the eccentricity distribution and whether
the effect is observable.
In order to understand the influence of tidal interaction on eccentricity distribution, we
first divide our sample into two subsets, one subset contains systems with a long-term RV
trend while the other subset contains systems that do not show a long-term RV trend (Fig.
6-6). The systems that are noted with a long-term velocity trend include 51 PEG, BD -10
3166, GJ 436, GJ 86, HD 107148, HD 118203, HD 149143, HD 68988, HD 7924, HD 99492
and τ Boo. We further divide the no-trend subset into two groups, one group is distinguished
by τage/τcirc ≥ 1, and the other group is distinguished by τage/τcirc < 1, where τcirc is
tidal circularization time scale and τage is the age of the host star. We investigate whether
there is a significant difference in the eccentricity distribution between these two groups
as expected if tidal interaction is an important factor in shaping eccentricity distribution.
Following Matsumura et al. (2008), we estimate τcirc using:
τcirc =2
81Q ′p
n
Mp
M∗
(a
Rp
)5[Q ′pQ ′∗
(MpM∗
)2(R∗Rp
)5
F∗ + Fp
]−1
, (6–5)
where the subscripts p and ∗ denote planet and star,M is mass, R is radius, a is semi-major
axis, Q ′ is modified tidal quality factor and n = [G (M∗ + Mp)/a3]1/2 is the mean motion.
150
Table 6-1. Comparison of Eccentricities Calculated From Different MethodsName Ref. MCMC Γ
eref δeref eMCMC elower eupper eΓ elower eupperHD 41004 B 0.081 0.012 0.058 0.000 0.109 0.000 0.000 0.085HD 86081 0.008 0.004 0.058 0.008 0.166 0.013 0.001 0.098
HD 189733 0.000 0.000 ... ... ... ... ... ...HD 212301 0.000 ... 0.015 0.000 0.051 0.034 0.000 0.091
GJ 436 0.159 0.052 0.191 0.146 0.248 0.223 0.128 0.295HD 63454 0.000 ... 0.018 0.000 0.038 0.003 0.000 0.017
HD 149026 0.000 ... 0.192 0.118 0.270 0.182 0.050 0.344HD 83443 0.012 0.023 0.007 0.000 0.020 0.006 0.000 0.037HD 46375 0.063 0.026 0.052 0.030 0.084 0.065 0.000 0.121
HD 179949 0.022 0.015 0.014 0.000 0.024 0.013 0.000 0.042τ Boo 0.023 0.015 0.079 0.054 0.117 0.086 0.034 0.139
HD 330075 0.000 ... 0.019 0.000 0.087 0.008 0.000 0.095HD 88133 0.133 0.072 0.076 0.000 0.127 0.086 0.000 0.175HD 2638 0.000 ... 0.041 0.000 0.076 0.005 0.000 0.042
BD -10 3166 0.019 0.023 0.010 0.000 0.030 0.019 0.000 0.064HD 75289 0.034 0.029 0.021 0.000 0.043 0.000 0.000 0.063
HD 209458 0.000 ... 0.008 0.000 0.016 0.005 0.000 0.018HD 2198281 0.000 ... ... ... ... ... ... ...
HD 76700 0.095 0.075 0.062 0.003 0.104 0.045 0.000 0.104HD 149143 0.000 ... 0.012 0.000 0.022 0.009 0.000 0.014HD 102195 0.000 ... ... ... ... ... ... ...
51 Peg 0.013 0.012 0.007 0.000 0.014 0.006 0.000 0.020GJ 6742 0.100 0.020 0.070 0.000 0.147 0.047 0.000 0.131
HD 49674 0.087 0.095 0.050 0.000 0.119 0.000 0.000 0.073HD 109749 0.000 ... 0.045 0.000 0.070 0.042 0.000 0.143HD 7924 0.170 0.160 0.119 0.022 0.224 0.054 0.000 0.297
HD 118203 0.309 0.014 0.293 0.264 0.328 0.297 0.218 0.367HD 68988 0.125 0.009 0.118 0.096 0.143 0.134 0.094 0.174
HD 168746 0.107 0.080 0.079 0.025 0.139 0.086 0.000 0.155HD 1852693 0.276 0.037 0.276 0.242 0.314 0.279 0.175 0.389HD 162020 0.277 0.002 0.277 0.274 0.279 ... ... ...
GJ 1764 0.000 ... ... ... ... ... ... ...HD 130322 0.011 0.020 0.007 0.000 0.052 0.031 0.000 0.086HD 108147 0.530 0.120 0.526 0.429 0.624 0.556 0.302 0.698HD 4308 0.000 0.010 0.068 0.000 0.123 0.060 0.000 0.111
GJ 86 0.042 0.007 0.042 0.034 0.051 ... ... ...HD 99492 0.050 0.120 0.036 0.000 0.125 0.056 0.000 0.115HD 27894 0.049 0.008 0.024 0.000 0.045 0.029 0.000 0.093HD 33283 0.480 0.050 0.458 0.401 0.523 0.483 0.386 0.544
HD 195019 0.014 0.004 0.002 0.000 0.007 0.006 0.000 0.014HD 102117 0.121 0.082 0.068 0.038 0.121 0.052 0.000 0.119
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Table 6-1. ContinuedName Ref. MCMC Γ
eref δeref eMCMC elower eupper eΓ elower eupperHD 171565 0.684 0.013 0.683 0.672 0.691 ... ... ...
HD 6434 0.170 0.030 0.159 0.124 0.202 ... ... ...HD 192263 0.055 0.039 0.026 0.000 0.055 0.015 0.000 0.118HD 117618 0.420 0.170 0.352 0.233 0.511 0.381 0.212 0.538HD 224693 0.050 0.030 0.031 0.000 0.055 0.019 0.000 0.089HD 436916 0.140 0.020 0.090 0.055 0.133 0.104 0.021 0.182ρ Crb 0.057 0.028 0.048 0.000 0.069 0.055 0.000 0.123
HD 456527 0.380 0.060 0.434 0.371 0.496 0.443 0.299 0.588HD 107148 0.050 0.170 0.028 0.000 0.141 0.073 0.000 0.221References:1 Melo et al. (2007);2 Bonfils et al. (2007);3 Johnson et al. (2006);4 Forveille et al. (2009);5 Barbieri et al. (2009);6 da Silva et al. (2007);7 Santos et al. (2008);eref and δeref are from Butler et al. (2006) if otherwise noted.
152
Figure 6-6. Distribution of short-period single-planet systems in (e,τage/τcirc ) space. Filledcircles are systems showing no linear RV trend and open circles are systemsshowing long-term linear RV trends. Different colors indicate differentcombinations of Q ′∗ and Q ′p
Matsumura et al. (2008) adopted 106 as a typical value for Q ′∗ for short period planetary
system host stars and considered Q ′p ranging from 105 to 109. We use Q ′∗ = [106, 107] and
Q ′p = [105, 107, 109] in our analysis. The factors F∗ and Fp are defined in the following two
equations:
F∗ = f1(e2) − 1118f2(e2)
Ω∗,rotn, (6–6)
Fp = f1(e2) − 1118f2(e2)
Ωp,rot
n, (6–7)
153
where Ω is rotational frequency. For short-period planets one could set Ωp,rot/n = 1
based on the assumption that all the planets in our sample have been synchronized since
τsynch ∼ 10−3τcirc (Rasio et al., 1996). In order to check whether our conclusion is sensitive to
the choice of Ω∗,rot/n, we conduct calculations with other Ω∗,rot/n values in which we choose
stellar rotation period to be 3, 30, and 70 days for all the stars. We find that this range for
Ω∗,rot/n does not change the conclusions in the chapter. Therefore, for future discussion, we
adopt Ω∗,rot/n = 0.67, which results in stellar rotation periods consistent with typical values
from 3 to 70 days (Matsumura et al., 2008). The uncertainties in Ω∗,rot/n are accounted for
by our subsequent data analysis (Equation. 6–10). And f1 and f2 are approximated by the
equations:
f1(e2) = (1 +154e2 +
158e4 +
564e6)/(1 − e2)13/2, (6–8)
f2(e2) = (1 +32e2 +
18e4)/(1 − e2)5. (6–9)
Planet radius Rp is estimated based on Fortney et al. (2007). We assume that the
planet and host star are formed at the same epoch. We assume that planet structure is
similar to Jupiter with a core mass fraction of 25M⊕/MJ = 7.86%. Radii of GJ 436 b and
HD 149026 b are adopted from reference papers because there is a factor of ≥2 difference
between observed values (Torres et al., 2008) and theoretically calculated values. Stellar
radius and age estimations are obtained from the following sources with descending priority:
1, Takeda et al. (2007); 2, nsted .ipac .caltech.edu; 3, exoplanet.eu. The calculated τcirc
values are presented in Table 6-4 in addition to the results of MCMC analysis of individual
system and other stellar and planetary properties.
We use the eccentricity posterior samples for each system for which the standard
MCMC analysis was successful (i.e., results of §2.1) to construct the eccentricity samples
of two groups separated by τage/τcirc . We note that there are considerable uncertainties in
the estimation of τage and τcirc , so τage/τcirc > 1 does not necessarily mean that the actual
154
system age is larger than the actual circularization time. We consider the sensitivity of our
results to these uncertainties by adopting a probability function:
ρ(τage
τcirc) =
1 − 0.5 exp
[− η · ( τage
τcirc− 1)
]if τageτcirc≥ 1
0.5 exp[− η · ( τcirc
τage− 1)
]if τageτcirc< 1
(6–10)
where η is a parameter tuning the confidence of τage and τcirc estimation. For example, if
τage/τcirc = 2 and η = 1, then ρ(τage/τcirc ) = 0.816, meaning there is 81.6% chance that
the system is from the group of τage/τcirc ≥ 1 because of the uncertainties in τage and τcirc
estimation. Therefore, we take 81.6% of the eccentricity posterior samples of the system
to construct eccentricity sample for the group of τage/τcirc ≥ 1 and the remaining 18.4%
eccentricity posterior samples to construct eccentricity sample for group of τage/τcirc < 1. The
η parameter reflects our confidence in τage and τcirc estimation. If we are not very confident in
the estimation of τage and τcirc , then we set η to a small value approaching zero, so half of the
eccentricity posterior samples for each system are assigned to the group with τage/τcirc ≥ 1
and the the other half are assigned to the group with τage/τcirc < 1. After constructing the
eccentricity sample for the two groups, we use two-sample K-S test to test the null hypothesis
that these two samples from two groups were drawn from the same parent distribution.
The results (Table 6-2) show that we are unable to exclude the null hypnosis at a low p
value (statistic of two-sample K-S test) because of the small effective sample size (N ′∼8).
If ∆max = 0.2, where ∆max is the maximum difference between cumulative distribution
functions of two groups, we can exclude the null hypnosis at p = 0.05 only if N ′ is more than
44. In comparison, our current sample size is inadequate to draw a statistically significant
conclusion on whether or not the the groups are from the same parent distribution. However,
we do see a hint of a difference between cumulative distribution functions of two groups (Fig.
6-7 left), there are more systems with low-eccentricity for the group with τage/τcirc < 1, which
is a consequence of tidal circularization. We also find that the conclusion is unchanged for a
wide range of η, Q ′∗, and Q ′p values.
155
Figure 6-7. The left panel compares the cumulative distribution function of eccentricity for twogroups of planets: 1) τage/τcirc ≥ 1 (solid line) and 2) τage/τcirc < 1 (dotted line).The right panel compares cumulative distribution functions of eccentricity for twosubsets of planets: planets without a long-term RV slope (dashed line) andplanets with a long-term RV slope (dash-dotted line).
We conduct similar test for two subsets distinguished by whether or not a long-term
velocity trend is recognized and find the similar result that our current sample size is inade-
quate to draw a statistically significant conclusion on whether or not the the groups are from
the same parent distribution. Again, we see a hint of a difference between cumulative distri-
bution functions of two subsets (Fig. 6-7 right) although it is not statistically significant. There
are more systems with low-eccentricity for subsets showing no sign of external perturbation.
The maximum difference between the cumulative distribution functions ∆max is 0.123, N ′ is
8.37 and K-S statistic is 0.999. In that case, we need an effective sample size of 119 in order
156
Table 6-2. Two-sample K-S test result. ∆max is the maximum difference between cumulativedistribution functions of eccentricity for two groups separated by τage/τcirc = 1. N ′
is the effective sample size, calculated by (N1 · N2)/(N1 + N2), p is the significancelevel at which two-sample K-S test rejects the null hypothesis that the twoeccentricity samples are from the same parent distribution; η is a parameter tuningthe confidence of τage and τcirc estimation.Q ′p 105 107 109
Q ′∗ = 106
η=1000∆max 0.088 0.175 0.086N ′ 7.57 8.76 7.57p 1.00 0.93 1.00
η=1∆max 0.103 0.160 0.093N ′ 7.72 8.72 7.02p 1.00 0.97 1.00
η=0.001∆max 0.043 0.027 0.061N ′ 8.59 8.50 8.08p 1.00 1.00 1.00
Q ′∗ = 107
η=1000∆max 0.088 0.286 0.190N ′ 7.57 8.42 2.80p 1.00 0.43 1.00
η=1∆max 0.103 0.259 0.168N ′ 7.72 8.36 2.81p 1.00 0.56 1.00
η=0.001∆max 0.040 0.027 0.065N ′ 8.61 8.39 7.67p 1.00 1.00 1.00
to make a statistically significant conclusion (p=0.05). In other cases, larger sample size is
required since ∆max is less.
We have shown that any difference in eccentricity distribution depending on expected
time scale for tidal circularization or the presence of additional bodies capable of exciting
inner planet’s eccentricity is not statistically significant, although this may be a consequence
of small effective sample size. The data are also consistent with the argument that both
factors play roles in affecting the eccentricity distribution.
157
6.5 Eccentricity Distribution
We seek an analytical function that is able to approximate the observed eccentricity
distribution for short period single planetary systems in the framework of Bayesian inference.
For this purpose, we first exclude systems showing long-term RV trends to reduce the
effect of perturbation on the estimated eccentricity distribution. We also assume that the
distribution of τage/τcirc in our sample is representative of short-period single-planet systems.
Using the posterior samples of eccentricity from standard MCMC analysis, we obtain
an observed eccentricity probability density function (pdf) f (e) by summing the posterior
distributions together. While not statistically rigorous, this provides a simple summary of
our results. Logarithmic binning is adopted because the shape of f (e) at low eccentricity
is of particular interest. The uncertainty σ(e) for each bin is set by assuming a Poisson
distribution. Then, we use a brute-force Bayesian analysis to find the most probable values
of parameters for the candidate eccentricity pdf f ′(e) that approximates the observed
eccentricity pdf. In the observed eccentricity pdf, there is a pile-up in small eccentricities near
zero and a scatter of nonzero eccentricity less than 0.8. Therefore, we use a mixture of two
distributions: an exponential pdf fexpo(e, λ) = (1/λ) · exp(−e/λ) to represent the pile-up of
small eccentricity near zero and either a uniform distribution or a Rayleigh distribution (Juric
& Tremaine, 2008) to represent the population with sizable eccentricities. We assume a
uniform distribution for parameters in prior Σ(θ), where θ is vector containing the parameters
for f ′(e). Our results are not sensitive to the choice of priors. We adopt Poisson likelihood
for each bin in the form of fPoisson(n; ν) = (νn · exp(−ν))/n!, i.e., L(ei |θ) = fPoisson(f ′(ei |θ) ·
N; f (ei ) · N), where N is total number of posterior samples. The value of f ′(ei |θ) · N is
rounded if it is not an integer. The posterior distribution of θ is calculated as p(θ|e) =[ M∏i
Σ(θ)L(ei |θ)] 1M/ ∫ [ M∏
i
Σ(θ)L(ei |θ)] 1M d θ , whereM is the number of bins. We have
explored a range of bin size from 10 to 40 bins in logarithmal space and conclude that
the results of Bayesian analysis do not change significantly with choice of bin size. Since
158
the plausible values of parameters for f ′(e) are limited, we do not have to explore a large
parameter space. Therefore, a brute-force Bayesian analysis is practical.
We apply Bayesian analysis to three different planet populations for different η values
(η=0.001,1,1000) assuming Q ′∗ = 107 and Q ′P = 107: 1) systems without a long-term RV
slope and τage/τcirc ≥ 1; 2) systems without long-term RV slope and τage/τcirc < 1; 3) the
union of 1) and 2). The results of Bayesian analysis are presented in Table 6-3.
Figure 6-8. Marginalized probability density functions of parameters for analytical eccentricitydistribution with a mixture of exponential and Rayleigh pdfs. Uncircularizedsystems (group 2) are represented by colored solid lines while circularizedsystems (group 1) are represented by colored dashed lines. Different colorsindicate different η values, red: η = 0.001; green: η = 1; blue: η = 1000. Solidblack lines are for union of group 1 and 2.
Figure 6-8 shows marginalized probability density of different parameters for analytical
eccentricity distribution with a mixture of exponential and Rayleigh distributions (f ′(e) =
159
α · fexpo(e, λ) + (1 − α) · frayl (e,σ)). α is the fraction of exponential distribution component and
frayl (e,σ) represents Rayleigh distribution with the form of frayl (e,σ) = (e/σ2) · exp (−e2/2σ2).
Group 1 (potentially circularized system) is represented by dashed lines of different colors
indicating different values of η while group 2 (systems unlikely to have been circularized) is
represented by colored solid lines. Group 1 and group 2 are well mixed if η=0.001, i.e., a
loose constraint is applied on the boundary of τage/τcirc = 1. When adopting η=0.001, the
marginalized posterior pdfs for both groups 1 and 2 approach the marginalized pdf for group
3 (solid black lines; the union of group 1 and 2, including all planets without a velocity slope).
When we adopt larger η values (η=1 or 1000), α, the fraction of the exponential component
of the pdf is greater for group 1 (Fig. 6-8 top: blue and green dashed lines) than group 2 (Fig
6-8 top: blue and green solid lines). This is consistent with the hypothesis that significant tidal
circularization affected group 1. For group 1, the fraction of the exponential component of the
pdf is consistent with unity, implying that the eccentricity distribution for planets from group
1 can be described by an exponential pdf. In comparison, there is a substantial fraction of
Rayleigh component (40%) for the analytical function describing eccentricity distribution for
uncircularized systems (i.e., group 2). Similar conclusions are also drawn for the analytical
eccentricity distribution with a mixture of exponential and uniform distribution (Fig. 6-9) in the
form of f ′(e) = α · fexpo(e, λ) + (1 − α) · funif (e, β), where funif (e, β) is uniform distribution with
lower boundary of 0 and upper boundary of β.
Figure 6-10 shows the cumulative distribution of the eccentricities sample based on
summing the posterior eccentricity samples of each system (green solid line) and the CDFs
of the two analytic functions (blue dotted and dashed) with the parameters that maximize the
posterior probability (see Table 6-3, subset 3). For comparison, the cumulative distribution of
eMCMC from standard MCMC analysis (Table 6-1) is shown in red. The difference between
the red and the green line cannot be distinguished at 0.05 significant level in a K-S test
(N ′=46).
160
Figure 6-9. Marginalized probability density functions of parameters for analytical eccentricitydistribution with a mixture of exponential and uniform pdf. Uncircularized systems(group 2) are represented by colored solid lines while circularized systems (group1) are represented by colored dashed lines, red: η = 0.001; green: η = 1; blue:η = 1000. Different colors indicate different η values. Solid black lines are forunion of group 1 and 2.
161
Figure 6-10. Cumulative distributions functions (cdf) of eccentricities from different methods.MCMC: most probable eccentricities, eMCMC , adopted from MCMC method(Table 6-1); MCMC, sum: eccentricities by summing up posterior distributionsamples of each system; AF1: cdf of the analytical function with the mostprobable parameters, f ′(e) = α · fexpo(e, λ) + (1 − α) · frayl (e,σ); AF2: cdf of theanalytical function with the most probable parameters,f ′(e) = α · fexpo(e, λ) + (1 − α) · funif (e, β).
162
Tabl
e6-
3.B
ayes
ian
anal
ysis
resu
lts.
Gro
up1:
syst
ems
with
outl
ong-
term
RV
slop
ean
dτ age/τcirc≥
1;gr
oup
2:sy
stem
sw
ithou
tlo
ng-te
rmR
Vsl
ope
andτ age/τcirc<
1;gr
oup
3:un
ion
of1
and
2.N
umbe
rsin
brac
keta
reun
cert
aint
ies
ofth
ela
sttw
odi
gits
.Th
efraction
colu
mn
repo
rts
the
perc
enta
geof
tria
lsin
whi
chec
cent
ricity
test
ing
sam
ples
gene
rate
dby
anal
ytic
alfu
nctio
nca
nno
tbe
diffe
rent
iate
dfro
mth
eob
serv
edec
cent
ricity
sam
ple
at0.
05co
nfide
nce
leve
l.
ηsu
bset
f′ (e
)=α·fexpo(e,λ
)+(1−α
)·f rayl
(e,σ
)f′ (e
)=α·fexpo(e,λ
)+(1−α
)·f unif
(e,β
)λ
[×10−2
]α
[×10−1
]σ
[×10−1
]fra
ctio
nλ
[×10−2
]α
[×10−1
]β
[×10−1
]fra
ctio
n
0.00
11
6.83
(84)
7.61
(40)
3.38
(36)
100.
0%7.
08(8
2)7.
35(3
8)7.
56(4
1)10
0.0%
26.
67(5
9)7.
56(3
2)3.
15(2
2)10
0.0%
6.86
(59)
7.32
(33)
7.06
(23)
100.
0%
1.00
01
7.71
(58)
9.78
(20)
0.83
(54)
99.6
%7.
96(5
9)10
.0(1
2)1.
8(11
)99
.9%
25.
55(5
3)6.
10(3
1)3.
19(1
6)10
0.0%
5.55
(57)
5.66
(34)
7.12
(18)
100.
0%
1000
.1
7.41
(72)
8.9(
12)
0.95
(24)
99.9
%7.
62(5
6)9.
2(11
)2.
10(5
7)10
0.0%
25.
24(4
9)5.
95(3
0)3.
27(1
6)10
0.0%
5.11
(53)
5.44
(33)
7.20
(19)
100.
0%
36.
67(4
7)7.
54(2
5)3.
22(1
9)10
0.0%
6.89
(47)
7.33
(26)
7.22
(20)
100.
0%
163
In order to check whether the analytical function with the most probable parameters is
an acceptable approximation to the eccentricity distribution for short-period single-planet
systems, we generate test samples from the analytical functions and compare the resulting
eccentricity samples to observations. For each test sample there are N ′ eccentricities
following the distribution of the analytical function f ′(e |θ), where N ′ is the effective sample
size. Each eccentricity is perturbed by an simulated measurement error that follows the
distribution of posterior samples for our analysis of the actual observations. The test sample
is then compared to the observed eccentricity samples obtained by standard MCMC
analysis using two-sample K-S test. We report in Table 6-3 the percentage of trials in which
eccentricity samples generated by our analytical function can not be differentiated from the
observed eccentricity sample at 0.05 confidence level. All the candidate analytical functions
we have tested are able to reproduce the observed eccentricity distribution in more than
99.6% of the trials at 0.05 confidence level. We conclude that the best-fit analytical function
is an adequate approximation to observed eccentricity distribution. We also compare to
analytical eccentricity distribution used in Shen & Turner (2008) f ′(e) ∼ [1/(1 + e)a − e/2a]
in which a=4, although it is not specifically for short period single planetary systems. Similar
to what we did in previous test, we found that in 39.0% of the tests, the eccentricity samples
generated by analytical functions can not be differentiated from the observed eccentricity
sample at 0.05 confidence level.
From the results of Bayesian analysis, there is a clear difference in α, the fraction
of exponential distribution, between group 1 and 2, suggesting the role played by tidal
circularization. Group 1 with τage/τcirc ≥ 1 shows more planets with near-zero eccentricities
(α ∼ 75%) as compared to group 2 (α ∼ 55%) with τage/τcirc < 1 (Table 6-3). Since the
eccentricity samples tested were perturbed by measurement errors, the analytical function
we found can be interpreted as an approximation of the underlying eccentricity distribution of
short-period single-planet systems. However, the actual parameter values and uncertainties
in the analytical function are dependent upon the quality of observation and the number of
164
systems in the sample of short-period single planets. As the measurement precision and
the sample size improve, we will be able to better constrain the values of parameters in the
analytical function which approximates the underlying eccentricity distribution.
6.6 Discussion
Figure 6-11. Distribution of short-period single-planet systems in our sample inperiod-eccentricity space (top: p ≤ 10 day; bottom: p > 10 day). Systems thatare not consistent with zero eccentricity according to Γ analysis (filled circles) orMCMC analysis (open circles) are marked with corresponding names. We alsoinclude transiting planets (marked as cross) for comparison. Transiting systemsnot consistent with zero eccentricity are marked with numbers: 1, WASP-18; 2,WASP-12; 3, WASP-14; 4, HAT-P-13; 5, WASP-10; 6, XO-3; 7, WASP-6;8,WASP-17; 9, CoRoT-5; 10, HAT-P-11; 11, HAT-P-2.
The median eccentricity of short-period single-planet systems in our sample is 0.088.
When compared to median eccentricity for all the detected exoplanets 0.15, it suggests that
the population may be affected by tidal circularization. We use eccentricity estimated by
165
Γ analysis in the following discussion since it is a less biased method for accessing small
eccentricity. Figure 6-11 shows the period-eccentricity correlation. We would expect planets
with sufficient short period to be tidally circularized. While this is generally true, there are 3
(17.6%) planets with P < 4d and non-circular orbits: GJ 436, HD 149026 and τ Boo. For
GJ 436, it is suspected that an outer companion may be pumping the eccentricity (Maness
et al., 2007; Ribas et al., 2008). Similarly, observations of τ Boo b are inconsistent with
circular orbit, but might be explained by the perturbation of an unseen companion indicated
by a long-term RV linear trend (Butler et al., 2006). Secondary eclipse timing indicates that
the eccentricity of HD 149026 is quite small but inconsistent with zero (Knutson et al., 2009).
Considering planets with orbital periods up to 10 days, there are 4 additional systems that
are not circularized. HD 118203, HD 68988 and HD 185269 might be due to perturbations
by additional bodies in the system (Butler et al., 2006) while the non-zero eccentricity of
HD 162020 b may be attributed to a different formation mechanism (Udry et al., 2002).
With period longer than 10 days, there are 8 (44.4%) planets with τage/τcirc > 1 which
have non-zero eccentricity and no detected long-term linear trend. In comparison, there
are 8 (28.6%) eccentric planets with orbital periods less than 10 days. The increasing
fraction of recognizably eccentric orbits as period increases is suggestive of decreasing
tidal circularization effect, but large sample of planets is required to draw firm conclusion.
The discovery of over 705 planets candidates by the Kepler mission presents an excellent
opportunity to analyze the eccentricity distribution of short-period planets (Borucki et al.,
2011b; Ford et al., 2008). We briefly consider transiting planets. We note that 11 of 58
(19.0%) transiting systems as of June 2010 are not consistent with zero eccentricity, and
the orbital periods for all transiting planets but 3 (i.e. CoRoT-9b, HAT-P-13c HD 80606b)
are less than 10 days. We infer that the tidal circularization process might be effective for
isolated planets with orbital period of less than 10 days. Alternatively, the planet formation
and migration processes for short-period giant planet may naturally lead to a significant
fraction of nearly circular orbits, even before tidal effect takes place.
166
It is worth noting that HD 17156 b (Fischer et al., 2007) has one of the most eccentric
orbits among short-period planet systems in spite the fact that τage/τcirc could be well over 10
(Fig. 6-6, red). However, Barbieri et al. (2009) found no indications of additional companions
based on observations of directing imaging, RV and astrometry measurement. Anglada-
Escude et al. (2010) investigated the possibility that a 2:1 resonant orbit can be hidden by an
eccentric orbital solution. It is interesting to explore such possibility on this particular system
to solve the discrepancy of high eccentricity and τage to τcirc ratio. Another possibility is that
the system is in the process of circularization that began well after the star and planet formed
(e.g., due to planets scattering). However, we are cautious in drawing conclusions since
τage/τcirc could be less than unity (Fig. 6-6, green and blue).
6.7 Conclusion
We apply standard MCMC analysis for 50 short-period single-planet systems and
construct a catalog of orbital solutions for these planetary systems (Table 6-4). We find
general agreement between MCMC analysis and previous references with the primary
exception being cases where eccentricity was held fixed in previous analysis. We develop a
new method to test the significance of non-circular orbits (Γ analysis), which is better suited
to performing population analysis. We find the eccentricity estimations from Γ analysis are
consistent with results from both standard MCMC analysis and previous references.
Our results suggest that both tidal interactions and external perturbations may play
roles in shaping the eccentricity distribution of short-period single-planet systems but large
sample sizes are needed to provide sufficient sensitivity to make these trends statistically
significant. We identify two analytical functions that approximate the underlying eccentricity
distribution: 1) mixture of an exponential distribution and a uniform distribution and 2) a
mixture of an exponential distribution and a Rayleigh distribution. We use Bayesian analysis
to find the most probable values of parameters for the analytical functions given the observed
eccentricities (Table 6-3). The analytical functions can be interpreted as the underlying
167
distribution of eccentricities for short-period single-planet systems. Thus, the analytical
functions can be used in the future theoretical works or as priors for eccentricity distribution.
168
Table 6-4. Catalog of Short-Period Single-Planet SystemsName P K e ω M0
(day) (m · s−1) (deg) (deg)
HD 41004B 1.32363±8.91594e-05 4599.21±337.57 0.0580+0.0511−0.0580 149.0±72.2 119.5±72.1
HD 86081 1.99809±0.00677822 189.65±12.09 0.0575+0.1080−0.0501 -27.4±71.5 64.8±74.5
HD 1897331 2.2186±0.0005 205±6 0±0.0002 90 270HD 212301 2.24571±0.000147699 57.26±3.01 0.0147+0.0365
−0.0147 172.8±85.6 56.0±85.6GJ 436 2.64394±9.85041e-05 18.07±1.03 0.1912+0.0571
−0.0449 -5.6±15.3 -66.6±14.5HD 63454 2.81747±0.000382247 63.19±1.82 0.0177+0.0203
−0.0177 -122.9±76.2 25.7±76.3HD 149026 2.87807±0.00146571 54.63±11.90 0.1918+0.0777
−0.0743 114.2±25.9 10.9±22.5HD 83443 2.98572±5.30373e-05 56.00±1.05 0.0070+0.0135
−0.0070 117.3±82.2 134.0±82.0HD 46375 3.02358±6.44902e-05 33.67±0.81 0.0524+0.0320
−0.0229 113.7±34.3 -53.2±34.3HD 179949 3.0925±3.30046e-05 112.62±1.77 0.0104+0.0099
−0.0104 -170.7±63.4 42.4±63.3τ Boo 3.31249±3.12595e-05 469.59±14.86 0.0787+0.0382
−0.0246 -141.6±25.0 24.4±24.9HD 88133 3.41566±0.000841134 34.13±3.57 0.0761+0.0508
−0.0761 -2.8±60.4 -39.5±60.1HD 2638 3.43752±0.00823876 66.26±2.83 0.0407+0.0351
−0.0407 126.9±78.6 -123.4±77.7BD -10 3166 3.4878±0.000104858 60.53±1.44 0.0104+0.0192
−0.0104 -14.6±83.9 36.6±83.7HD 75289 3.50928±7.2946e-05 54.84±1.87 0.0211+0.0217
−0.0211 136.3±73.9 -162.2±74.1HD 209458 3.52472±2.81699e-05 84.30±0.88 0.0082+0.0078
−0.0082 43.8±68.4 92.5±68.5HD 330075 3.6413±0.00187111 97.84±8.79 0.0187+0.0684
−0.0187 38.2±91.6 35.2±91.7HD 2198282 3.833±0.0013 7±0.5 0 0 0
HD 76700 3.97101±0.000203194 27.24±1.31 0.0616+0.0426−0.0587 12.3±54.3 45.6±53.8
HD 149143 4.07206±0.000320041 149.28±1.65 0.0123+0.0093−0.0115 -155.2±55.9 -150.9±55.9
HD 1021953 4.1138±0.000557 63±2 0 0 051 PEG 4.2308±3.72905e-05 55.65±0.53 0.0069+0.0066
−0.0069 54.1±72.3 85.1±72.3GJ 674 4.6944±0.00182591 9.46±1.09 0.0700+0.0766
−0.0700 0.5±71.2 77.6±71.1HD 49674 4.94739±0.000974925 11.78±1.18 0.0495+0.0691
−0.0495 -96.1±79.3 82.4±79.3HD 109749 5.23921±0.000935533 28.49±1.12 0.0451+0.0250
−0.0451 72.3±55.8 -155.1±55.8HD 7924 5.39785±0.00096697 3.74±0.44 0.1186+0.1050
−0.0970 25.0±55.6 62.1±54.8HD 118203 6.13322±0.00129898 213.75±6.67 0.2943+0.0342
−0.0298 -27.3±5.5 -2.9±4.8HD 68988 6.27699±0.0002195 184.63±4.69 0.1187+0.0246
−0.0216 32.4±11.2 61.3±10.8HD 168746 6.40398±0.000979461 28.41±1.38 0.0791+0.0595
−0.0541 14.8±47.2 -147.2±47.3HD 185269 6.83796±0.00119146 89.39±4.20 0.2758+0.0386
−0.0334 173.3±6.4 -99.4±5.9HD 162020 8.42826±7.76104e-05 1808.84±5.26 0.2765+0.0023
−0.0025 -151.2±0.3 68.3±1.0GJ 1764 8.783±0.0054 4.1±0.52 0 0 0
HD 130322 10.7086±0.00184045 108.10±7.31 0.0068+0.0455−0.0068 128.3±93.9 -10.4±93.8
HD 108147 10.8984±0.00316767 24.60±3.62 0.5161+0.0979−0.0966 -54.4±14.0 -68.0±9.9
HD 4308 15.5646±0.0213556 4.20±0.34 0.0682+0.0547−0.0682 -166.9±60.6 -13.0±59.5
GJ 86 15.765±0.000382114 376.64±2.79 0.0416+0.0092−0.0073 -93.7±12.4 -76.8±12.0
HD 99492 17.0495±0.00525091 8.39±0.98 0.0364+0.0891−0.0364 -138.6±93.6 -137.6±93.5
HD 27894 18.0059±0.0163248 56.88±1.75 0.0240+0.0211−0.0240 131.3±66.5 -60.0288±66.5
HD 33283 18.1801±0.00830584 24.48±2.41 0.4576+0.0656−0.0569 156.0±9.6 -60.0±7.4
HD 195019 18.2018±0.000595221 269.70±1.60 0.0017+0.0049−0.0017 -127.4±47.4 -175.5±47.3
HD 102117 20.8210±0.01006465 10.20±0.91 0.0685+0.0529−0.0647 137.6±72.0 -9.5±72.5
HD 17156 21.2178±0.00371004 279.88±8.43 0.6829+0.0080−0.0106 121.3±1.1 -156.7±1.2
HD 6434 21.9975±0.0127604 34.30±1.52 0.1589+0.0434−0.0345 155.4±15.8 -14.2±15.5
HD 192263 24.3546±0.00507675 51.13±2.62 0.0256+0.0297−0.0256 -147.0±78.9 -56.7±78.9
HD 117618 25.8221±0.0155045 12.25±1.70 0.3524+0.1583−0.1192 -105.6±22.9 -123.3±21.1
HD 224693 26.732±0.0245934 39.73±1.54 0.0313+0.0236−0.0313 11.6±68.6 162.5±68.9
HD 43691 36.9916±0.0350715 125.98±4.06 0.0897+0.0432−0.0346 91.3±26.2 16.7±24.9
ρ Crb 39.8459±0.00917865 65.25±2.20 0.0476+0.0218−0.0476 -62.1±42.2 -172.6±42.5
HD 45652 43.6896±0.105825 35.76±2.84 0.4339+0.0625−0.0632 83.2±12.8 77.7±10.2
HD 107148 48.6168±3.59204 10.01±4.15 0.0279+0.1135−0.0279 126.5±90.8 -50.1±88.8
169
Table 6-4. ContinuedName τ trend Jitter Nobs M∗ R∗
(day) (m · s−1 · d−1) (m · s−1) (MS ) (RS )HD 41004B 2452532.699 ...±... 2761.45±168.90 149 0.40 0.40HD 86081 2453753.2 ...±... 32.13±5.74 26 1.21 1.22
HD 1897331 2454037.612 ...±... 15 86 ... ...HD 212301 2453388.9 ...±... 8.53±1.78 23 1.05 1.19
GJ 436 2452992.1 0.0037±0.0015 0.41±1.46 55 0.41 0.46HD 63454 2453238.057 ...±... 5.70±1.22 26 0.80 0.78
HD 149026 2453545.35 ...±... 2.41±3.26 17 1.30 1.50HD 83443 2452248.9 ...±... 3.12±1.67 51 1.00 1.02HD 46375 2451920.7 ...±... 3.28±0.60 50 0.92 0.94
HD 179949 2452419.1 ...±... 9.44±1.06 88 1.21 1.22τ Boo 2450529.2 -0.051±0.0036 94.30±8.13 98 1.35 1.33
HD 88133 2453180.0 ...±... 5.67±1.61 21 1.20 1.93HD 2638 2453323.282 ...±... 5.48±4.87 28 0.93 1.01
BD -10 3166 2451844.7 0.005±0.0026 4.00±1.74 31 1.01 0.84HD 75289 2452593.9 ...±... 4.73±1.73 30 1.21 1.28
HD 209458 2452499.3 ...±... 3.27±0.86 64 1.14 1.14HD 330075 2452968.399 ...±... 24.59±5.00 21 0.70 0.90
HD 2198282 2453898.63 ...±... 1.7 27 ... ...HD 76700 2452655.1 ...±... 1.35±4.21 35 1.13 1.34
HD 149143 2453413.1 0.027±0.0056 0.48±1.96 17 1.20 1.61HD 1021953 2453895.96 ...±... 6.1 59 ... ...
51 PEG 2450404.4 -0.0045±0.00046 0.27±0.91 256 1.09 1.18GJ 674 2453823.784 ...±... 3.55±0.55 32 0.35 0.46
HD 49674 2452308.9 ...±... 3.56±0.82 39 1.06 0.95HD 109749 2453426.3 ...±... 0.31±1.10 20 1.21 1.28
HD 7924 2454096.65 0.35±0.07 2.59±0.22 93 0.83 0.78HD 118203 2453351.2 0.14±0.0166 23.08±3.96 43 1.23 2.15HD 68988 2452441.3 -0.065±0.0056 13.30±2.34 28 1.18 1.14
HD 168746 2452510.8 ...±... 0.41±1.62 16 0.93 1.04HD 185269 2453795.0 ...±... 7.70±2.10 30 1.28 1.88HD 162020 2451672.02 ...±... 11.02±3.43 46 0.78 0.74
GJ 1764 2454399.8 ...±... 2.5 57 ... ...HD 130322 2452430.2 ...±... 10.25±3.79 12 0.88 0.85HD 108147 2452407.0 ...±... 8.65±1.49 54 1.19 1.25
HD 4308 2453338.121 ...±... 0.58±0.78 41 0.90 0.92GJ 86 2452199.4 -0.260±0.0029 10.67±1.62 42 0.77 0.80
HD 99492 2452523.85 0.0035±0.00066 3.96±0.51 86 0.86 0.76HD 27894 2453344.278 ...±... 4.48±1.07 20 0.75 0.90HD 33283 2453560.0 ...±... 0.27±0.94 24 1.24 1.20
HD 195019 2451844.0 ...±... 10.50±1.22 154 1.07 1.38HD 102117 2452931.7 ...±... 0.57±1.73 44 1.11 1.26HD 17156 2454111.21 ...±... 3.68±0.64 34 1.24 1.63HD 6434 2451753.933 ...±... 7.39±1.15 130 0.79 0.57
HD 192263 2451867.6 ...±... 6.99±1.31 31 0.81 0.77HD 117618 2452838.0 ...±... 3.30±1.20 57 1.09 1.20HD 224693 2453607.2 ...±... 1.00±2.78 23 1.33 1.70HD 43691 2454046.7 ...±... 10.35±2.58 36 1.38 1.92ρ Crb 2451181.1 ...±... 0.65±2.79 26 1.00 1.28
HD 45652 2453692.66 ...±... 8.38±2.20 45 0.83 1.04HD 107148 2452799.9 0.003±0.0012 3.53±1.04 35 1.14 1.12
170
Table 6-4. ContinuedName MP RP τcirc τage RV
(MJ ) (RJ ) (Gyr) (Gyr) ref.HD 41004B 18.40 1.06 0.26 6.32 7HD 86081 1.50 1.08 0.56 6.21 8
HD 1897331 ... ... ... ... 9HD 212301 0.40 1.07 0.28 5.90 11
GJ 436 0.07 0.38 5.57 6.00 6HD 63454 0.39 1.06 0.67 1.00 12
HD 149026 0.36 0.65 3.46 2.00 6HD 83443 0.40 1.04 1.10 11.68 6HD 46375 0.23 1.02 0.67 11.88 6
HD 179949 0.92 1.05 2.90 2.56 6τ Boo 4.13 1.06 4.12 1.64 6
HD 88133 0.30 1.00 1.49 9.56 6HD 2638 0.48 1.04 2.38 3.00 12
BD -10 3166 0.46 1.03 2.70 1.84 6HD 75289 0.47 1.03 3.03 3.28 6
HD 209458 0.69 1.05 4.07 2.44 6HD 330075 0.62 1.06 1.97 6.21 13
HD 2198282 ... ... ... ... 2HD 76700 0.23 0.99 2.80 9.84 6
HD 149143 1.33 1.05 7.01 7.60 14HD 1021953 ... ... ... ... 3
51 PEG 0.47 1.03 6.61 6.76 6GJ 674 0.04 1.13 0.21 0.55 15
HD 49674 0.10 0.98 3.25 3.56 6HD 109749 0.28 0.99 12.04 10.30 14
HD 7924 0.03 1.05 0.61 0.88 16HD 118203 2.14 1.05 3.68 4.60 17HD 68988 1.86 1.05 80.15 3.40 6
HD 168746 0.25 0.99 20.31 12.40 6HD 185269 0.94 1.04 19.34 4.20 8HD 162020 15.00 0.98 148.00 0.76 19
GJ 1764 ... ... ... ... 4HD 130322 1.09 1.04 670.08 10.80 6HD 108147 0.26 0.98 8.43 3.20 6
HD 4308 0.05 1.00 169.24 8.68 20GJ 86 3.91 1.05 6701.62 8.48 6
HD 99492 0.11 0.96 692.89 1.80 21HD 27894 0.62 1.02 3620.33 3.90 12HD 33283 0.33 0.99 203.12 3.20 8
HD 195019 3.69 1.05 4051.07 9.32 6HD 102117 0.17 0.95 3205.49 9.40 6HD 17156 3.20 1.04 19.10 8.00 10HD 6434 0.40 1.00 4393.14 6.85 22
HD 192263 0.64 1.02 14630.88 2.56 6HD 117618 0.18 0.95 1435.34 5.68 6HD 224693 0.71 1.03 23506.41 2.00 8HD 43691 2.49 1.04 42743.17 2.80 18ρ Crb 1.09 1.03 166705.71 7.64 6
HD 45652 0.47 1.01 12199.36 5.00 23HD 107148 0.21 0.96 122900.60 5.60 6
171
Table 6-4. ContinuedReferences: 1 Bouchy et al. (2005); 2 Melo et al. (2007); 3 Ge et al. (2006b); 4 Forveille et al. (2009);5 Maness et al. (2007); 6 Butler et al. (2006); 7 Zucker et al. (2004); 8 Johnson et al. (2006);9 Winn et al. (2006); 10 Winn et al. (2009); 11 Lo Curto et al. (2006);12 Moutou et al. (2005);13 Pepe et al. (2004); 14 Fischer et al. (2006); 15 Ge et al. (2006b); 16 Bonfils et al. (2007);17 Howard et al. (2009); 18 da Silva et al. (2006); 19 da Silva et al. (2007); 20 Udry et al. (2002);21 Udry et al. (2006); 22 Marcy et al. (2005);23 Mayor et al. (2004); 24 Santos et al. (2008)Stellar radius and age estimations are obtained from the following sources with descending priority:1, Takeda et al. (2007);2, nsted.ipac.caltech.edu;3, exoplanet.eu.τcirc is calculated assuming Q ′∗ = 107 and Q ′p = 107
172
CHAPTER 7SUMMARY, CONCLUSION AND CONTRIBUTION
My dissertation works pave the way for massive detections of M-dwarf planets. More
specifically, I compare two existing RV techniques, i.e., the DE method and the DFDI method,
and find their own strength in exoplanet search. I answer the question whether and how to
detect a habitable Earth-like planet. I take effort to overcome challenges in NIR RV precision
measurement. I help design and understand the scientific deliverable of a M-dwarf planet
survey. I take part into a multi-object planet survey project using the DFDI method and
help solve instrumentation problem which leads to fruitful discoveries. I envision a multi-
object M-dwarf planet survey to complete our understand of planet formation around stars
with a broad mass spectrum. In addition to my main work on M dwarf planet, I present my
collaboration work with Dr. Eric Ford on the eccentricity distribution of short-period planets.
7.1 Chapter 2
I conducted comprehensive comparison between the DE and the DFDI method. In order
to ensure a fair and quantitative comparison, I invented a method of calculating the Q factor
for the DFDI method. This method is a natural extension of the work of Bouchy et al. (2001)
in which they developed a method of calculating the Q factor for the DE method. Comparison
between the DE method and the DFDI method has been done by several previous work
such as Erskine (2003); Erskine & Ge (2000); Ge (2002); Ge et al. (2002); van Eyken et al.
(2010), but it has never been done as systematic and complete as my work described in this
dissertation. I have considered many cases such as the same wavelength coverage case,
the same detector case and the multi-object case. My work provides a guidance for future
exoplanet survey: 1, a survey of a large sample of stars should adopt the DFDI method,
which enables both adequate RV precision and high survey efficiency; 2, high precision
low-mass exoplanet search should adopt the DE method with a high resolution spectrograph.
173
7.2 Chapter 3
Following the conclusion of Chapter 2, I investigated into the high precision case for
which the DE method with a high resolution spectrograph is suitable. I was trying to answer
the question if we are able to detect a habitable Earth-like planet given spectral properties
and state-of-the-art RV technique. The theoretical framework is given by Bouchy et al.
(2001) that provides the method to estimate RV precision for a given spectrum at a given
S/N. There are several previous papers on this topic such as Reiners et al. (2010); Rodler
et al. (2011), but my work is the first of its kind in which I have considered all the dominant
factors that contribute to RV precision errors. They include stellar properties such as spectral
SED, line properties and rotational velocity, wavelength calibration error, stellar noise and
telluric contamination. I concluded that NIR observation of mid-late type M dwarfs is the most
realistic and likely approach in the search for a habitable Earth-like planet.
7.3 Chapter 4
There are several challenges in NIR precision RV measurement. I have conducted
pioneering work using the EXPERT spectrograph (R=27,000) at the KNNO 2.1m telescope.
My work includes observation, data analysis and hardware development. For observation, I
has worked with other members in the ET group for proposal writing and nightly observation.
I have two proposals as PI accepted for which a total of 14 nights were awarded. For data
analysis, I have used a telluric standard star method described by Vacca et al. (2003) to
remove telluric contamination from observed stellar spectrum and demonstrated 2.7%
removal residual. I have for the first time introduced the binary mask cross correlation
technique into the NIR RV measurement data analysis. This technique has only been used
for the optical wavelengths before by Baranne et al. (1996). For hardware development, I
was mainly working with Dr. Xiaoke Wan on the development of the sine source. We have
together conducted feasibility demonstration in lab and at the KPNO 2.1m telescope, I was
partially involved in experiment and data acquisition and mainly in charge of data analysis.
The sine source has been demonstrated a stable and precise wavelength calibration source
174
and further demonstration experiment is required. I was heavily involved in the FIRST
survey proposal. My contribution is mainly on target selection and planet yield simulation. I
predicted that 30 planets will be detected including 10 super-Earths, 2 giant planets and 18
intermediate-mass planets.
7.4 Chapter 5
The other major conclusion from Chapter 2 is that a survey of large sample stars should
adopt the DFDI method at a low-medium spectral resolution. The MARVELS project is planet
survey that follows this conclusion. This project takes enormous effort in design, instrument
building, commissioning, operating and data analysis. My major contribution to this project is
instrumentation and interferometer calibration. The RV error due to interferometer is limited
within 2 m · s−1. Owing to my work in MARVELS interferometer group delay calibration, over
250 binary and a dozen of brown dwarfs have been discovered and they provide a valuable
insight of formation and evolution of low-mass stellar companion and brown dwarf. I was
awarded the Architect status for the MARVELS project in 2012 for my contribution to this
project and will be guaranteed data access. I have envisioned an M-dwarf planet survey and
provided a concept study of such survey.
7.5 Chapter 6
This chapter reports my collaboration work with Dr. Eric Ford on the eccentricity dis-
tribution of short-period planet. We apply standard MCMC analysis for 50 short-period
single-planet systems and construct a catalog of orbital solutions for these planetary sys-
tems. We develop a new method to test the significance of non-circular orbits (Γ analysis),
which is better suited to performing population analysis. Our results suggest that both tidal
interactions and external perturbations may play roles in shaping the eccentricity distribution
of short-period single-planet systems but large sample sizes are needed to provide sufficient
sensitivity to make these trends statistically significant.
175
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BIOGRAPHICAL SKETCH
Ji Wang was born in Guilin, a beautiful city located in the mountainous area in Southern
China. 18 years after happily living in Guilin, he moved to the University of Science and
Technology of China (USTC) in Hefei to pursue his dream of being a scientist and studying
abroad. At USTC, the best university in China, he started to learn and to appreciate the
concept of hard work and smart thinking. He built a solid foundation for study and research
at USTC for the coming years before being admitted into the Astronomy program at the
University of Florida. He spent 6 good years in Florida, where he studied hard but exercised
harder. Thanks to all the sport facilities on campus, he became an all-around athletic who
is good at basket ball, tennis, golf, swimming, long distance running. Now he is ready for his
next stop, Yale University where he will work as a post-doc in the Department of Astronomy.
Hopefully in a few years he will find a tenure track position somewhere either in the US or
back in China.
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