research on u cephei

RESEARCH ON U CEPHEI: AN INTERACTING BINARY STAR 1

Research on U Cephei: An Interacting Binary Star

Kristen N. Lauria

Advised by Dr. Phillip Reed

Department of Physical Sciences

Kutztown University

2011


Acknowledgements

I would like to thank the many people who helped me achieve my goal of writing this

paper. First and foremost, I would like to thank my husband, Jim Lauria, for his unwavering

support and encouragement during the entire process of my academic career, including

researching and writing this paper. I would also like to thank my children, Charlie and Cate,

and my parents, Susan and Fred Roehs, who have given so much of their time and their time

with me to help me pursue my dreams.

I would also like to thank the many who gave their professional help in this process

including my capstone advisor, Dr. Phillip Reed, without whom I would not have a project.

He taught me all the knowledge needed to learn this daunting field of binary stars and put up

with my many delays and difficulties. Dr. Reed was also kind enough to allow me access to

equipment and software needed to perform the research and countless hours of checking my

research and making sure I was coming to the correct conclusions.

The Tzec Maun Foundation, which provided me with free access to professional

grade teslescopes also deserves my gratitude. Without this access, I doubt this project would

have gotten off the ground. I also would like to thank Dr. David Cohen and Dr. Eric Jensen

at Swarthmore University who provided me with data that was invaluable to this project.

Lastly, I would like to thank my advisors, professors, fellow students, and friends

who taught me much about physics and life. I would also like to give a special thank you to

Erin Lease for teaching me what dedication and determination really are and to Rachel

Massie for putting up with my ever-changing life. It takes a village to raise a child and a

university to write a capstone project. Thank you to all who helped. You are very much

appreciated.


Table of Contents Acknowledgements..............................................................................................................2 Table of Contents ................................................................................................................3 Introduction ........................................................................................................................4 Purpose of the Study .............................................................................................................5 Statement of the Problem ......................................................................................................6 Summary ..............................................................................................................................6 Technical Review ................................................................................................................7 Introduction ..........................................................................................................................7 Stellar Evolution ...................................................................................................................7 Binary Stars ..........................................................................................................................9 Tools and Equations............................................................................................................10 Ephemeris ...........................................................................................................................11 Light Curve .........................................................................................................................14 Summary ............................................................................................................................14 Methodology......................................................................................................................14 Introduction ........................................................................................................................14 Data Collection ...................................................................................................................15 Tzec Maun Foundation .......................................................................................................16 Swarthmore College............................................................................................................17 Kutztown University ............................................................................................................18 Processing Data...................................................................................................................18 Summary ............................................................................................................................22 U Cephei ............................................................................................................................23 Introduction ........................................................................................................................23 Charts and Figures ..............................................................................................................24 Calculations ........................................................................................................................27 Summary ............................................................................................................................28 New Model of U Cephei ....................................................................................................28 Introduction ........................................................................................................................28 New Results ........................................................................................................................29 Comparison and Error Analysis...........................................................................................30 Summary ............................................................................................................................31 Conclusion .........................................................................................................................31 Introduction ........................................................................................................................31 Directions for Future Research ............................................................................................32 Summary ............................................................................................................................33 Reference List....................................................................................................................34 Glossary of Astronomical Terms......................................................................................36


Research on U Cephei: An Interacting Binary Star

"It is not so very important for a person to learn facts. For that he does not really need a college. He can learn them from books. The value of an education in a liberal

arts college is not the learning of many facts but the training of the mind to think something that cannot be learned from textbooks."

- Albert Einstein

Introduction

For my capstone project at Kutztown University I was given a rather broad range of

subjects on which to produce a project that would capture the essence of the thorough

education I received. When given this project requirement as a transfer student with only two

years of study ahead of me, I found myself quite at a loss for ideas. Luckily, however, the

staff of the Physics department was able to quickly point me in the direction of Dr. Phillip

Reed who was conducting research on interacting binary stars, a continuation of this doctoral

thesis research on R Arae.

Dr. Reed and I began our research for this paper in the fall of 2009 by going over the

basics of binary stars. My own experience in this field was limited and therefore Dr. Reed

focused his attentions on teaching me the basics of the field of astronomy. Since my research

would involve interacting binary stars specifically, we then focused on the mechanics of

interacting binary stars and how they are studied in a technical field. In addition to this, the

topics of telescopes, software, and data interpretation also needed to be discussed in order to

be able to procure, process, and interpret the data accurately.

Over the following year, specifically from late fall of 2009 to the winter of 2010, I

worked to increase my knowledge in the area of interacting binary stars. I also got hands on


training in operating a telescope by taking data using a remote telescope. Then I learned how

to process and interpret the data accurately utilizing software programs. Taking the data was

a unique experience, knowing that I was controlling a telescope as big as my house on the

other side of the country to look at objects light-years away from Earth that would have been

nearly impossible to study in any detail only a century ago. These data sets, each having up

to several hundred images, needed to be processed one image at a time, making me appreciate

the advent of computers.

By the spring of 2011 I had compiled enough data to interpret and produce usable

findings. The results came in the form of a new time of primary minimum, new ephemeris,

and new parameters for each star such as temperature and radius. Others, who wish to

continue researching this star or others like it, can utilize these conclusions, helping

researchers better understand the life cycle of interacting binary stars. It is my hope that this

paper will lead to a long career in astrophysics, a field I had not really considered prior to the

start of this project.

Purpose of the Study

Many papers on U Cep have been published in the last century, mainly in the early

part of the twentieth century. This binary star has received minimal attention when in

contrast to the time shortly after it was discovered in 1880. The last published data that I was

able to locate was collected in the early 1990s and thus this star was a prime candidate for my

research project having been so overlooked in recent years. When I first began the research

I even hoped to publish a paper in a scientific journal with the information that was gathered.


Although I have not yet accomplished this goal, I hope to have an undergraduate paper on

this topic published in the next year or two.

Scientifically, the purpose of this study has been to accumulate data that will allow

me to determine a new ephemeris and new parameters for the stars that make up U Cep. I

hoped that since U Cep was known to be an interacting binary star, the lapse in time since it

had last been studied was significant enough for a real change to have occurred in the system.

This change would verify that U Cep was, in fact, an interacting binary star and by

comparing the old data to the information I had collected, I could mathematically determine

the amount of mass transfer taking place.

Statement of the Problem

In order to better understand the principles of interacting binary stars, many different

stars must be thoroughly studied over a long period of time. One star that has been studied

previously is U Cep, which has a long record of data that can be used in comparison studies.

Since it has been over 15 years since the last major study of U Cep was performed, the

results of this study can be used to determine changes that have occurred within the U Cep

system.

Summary

Although the scope of the original project assignment was broad, I decided to focus

on performing research in the field of astronomy. Narrowing the options further, I resolved

to study one star in depth enough to really produce usable data that might be used by other

scientists in the future, furthering our knowledge of stellar evolution. This decision rested on

the idea that learning to do scientific research would not only help my current studies by


acquiring additional knowledge in the field of physics, but also let me see what a future in

graduate school and the field of astronomy as a career might be like.

Technical Review

Introduction

When I began this project I did not understand much more than the basics about stars,

binary stars, or celestial objects in general. In order to accurately perform the observations,

process the data, and draw conclusions, it was necessary to acquire some foundational

knowledge. I gleaned this information from many sources including textbooks, papers

published in scientific journals, and from Dr. Reed. So that the reader may understand what I

found in my studies, I will first lay out some of the basics that a person would need in order

to comprehend what I found and why I drew certain conclusions. The list of references also

provides several great resources if further study of interacting binaries is of interest.

Stellar Evolution

The life of a star actually begins, like everything else, before it is actually a star. The

correct conditions must be present in the universe for a protostar to form. A protostar is “the

formation of pre-nuclear-burning objects…from interstellar molecular clouds.” (Carroll &

Ostlie, 2007, p. 412) This is how the material for a star is gathered, but before nuclear fusion

begins taking place. It is also this fusion that creates the pressure that overcomes the

gravitational pull and keeps the star from imploding in on itself.

The initial mass of the star is arguably the most important factor that sets the stage for

the lifecycle the star will follow. This helps researchers determine whether the star will

become a white dwarf, neutron star, or black hole. ("Chandra :: Educational Materials ::


Stellar Evolution," 2008) A smaller star may not have enough mass to maintain fusion of

anything more than hydrogen, leading to a brown dwarf. If the mass is too high, the

temperature can rise so high that it disintegrates through a process called pair-instability

supernova.

Some events can change the path a star will take during its lifetime. One of these

events is the transfer of mass from one star to another in a binary system. The change in

mass can alter the structure of either of the stars causing them to become stars they might not

have become with their initial mass. Some mass from binary stars is also lost to the

interstellar medium, which can lead to the formation of a new star altogether, beginning the

stellar lifecycle anew.


Figure 1: The Hertzprung-Russell Diagram (Courtesy Pearson Education, Addison Wesley)

As you can see on the H-R diagram above, stars are divided into spectral classes

labeled O through M. Our sun is a G class star, which is determined by it’s chemical make-

up and, therefore, surface temperature. It also falls into the main-sequence section of the

diagram, as most observed stars do during some point in their life cycle. The mass of the sun

is the baseline measurement from which all stars mass are measured, being equal to one solar

mass ( ) This spectral classification is related to the method of spectroscopy; which is to

study a star using the star’s spectrum, determining atoms and ions are present in a star. In

contrast, photometry was used as the method of data collection in this project; which is to

study a star using its visible light by recording its apparent magnitude.

Binary Star

Binary stars are a different kind of animal altogether from single stars. Although the

stars themselves are exactly the same as their single star counterparts, the life that binary

stars lead can be quite different. For example, although most stars eventually grow large

enough to lose mass to the surrounding environment as discussed previously. In the case of a

binary this mass is not lost entirely to the interstellar medium, but rather it can be captured by

the gravitational pull of the nearby sister star. This mass then changes how the two stars

interact, changing the rotational element, temperature, class of star, and many other

parameters.

The specific type of binary star involved in this study of U Cep can be classified as

an eclipsing binary system, which means that the orbital plane lies along the observer’s line

of sight and will, therefore, periodically cause one star to be obscured by the other. The

inclination in this case is actually lying at 86.3°, which makes the orbital plane lie close to


the observer’s line of sight. Since U Cep belongs to the eclipsing binary class, this allows for

data to be taken using visual techniques rather than the use of spectroscopic or other, more

complicated methods.

Two further classification for U Cep is that of interacting and Algol. The

categorization of interacting is due to the fact that, as the name implies, mass is being

transferred between the two stars of the system. There are binary stars that do not interact

and, as such, they follow different lives than do interacting binaries. Algol is a descriptor

meaning that the two stars are normal, or main-sequence, and are in a semi-detached

configuration. According to Carroll and Ostlie, active Algols “provide laboratories for

studying rapid (short-lived) stages of stellar and binary star evolution. These systems are

important for studying accretion processes and accretion disks.” (2007, p. 672)

One important point to note is that in a binary system there are, obviously, two stars

that must be referred to without confusion. For this purpose, the small, hotter star is called

the primary, or 1°, and the larger, cooler star is referred to as the secondary, or 2°. Burnett

and Etzel previously classified the stars of U Cep as follows: the primary was a spectral type

of B8 V; the secondary was a G8 III. (1993)

Tools and Equations

The following are several tools that are necessary to study an interacting, eclipsing

binary star such as U Cep. The ephemeris is an equation that allows researchers to calculate

when certain periods of the phase occur both in the past and in the future. This is useful

when planning dates to perform observations so that the researcher is able to take a wide

sample of data and also make sure to get key points during the cycle such as the primary and

secondary eclipses. The light curve is a graphical representation of the period of the binary


star. This light curve lets researchers see how the primary and secondary eclipses compare

on a curve normalized to one.

Ephemeris

An important part of the calculations used throughout this project was how to

determine and use the ephemeris of U Cep. The ephemeris is a tool used to determine when

any point in the phase of the star will occur with precision. In an interacting binary system

like U Cep, where the masses of the stars change over time, the ephemeris also changes.

These changes can be seen in Figure 2, which models what the possible evolutionary paths

that a low-mass binary star, defined as the combined mass is between one solar mass and 11

solar masses, such as U Cep might take. As can be seen in the figure, the effect that the mass

has on the period is quite dramatic. A change in mass in either star or both can make the

period grow shorter or longer and also, in the case of eclipsing binaries, can change whether

the primary is completely obscured during the primary eclipse.

Before actually calculating an ephemeris, one other piece of information must be

understood. The calendars in use by society has changed many times over the last several

thousand years, but for experiments to all use the same time-scale, a universal time must be

utilized. This universal time is known as a Julian Date (JD). For this experiment I used

Heliocentric Julian Dates (HJD) for our calculations, although Geocentric Julian Dates (GJD)

can also be used in scientific research. The main difference between the two is that the HJD

uses the sun, or the


Figure 2: Possible evolutions of a low mass binary like U Cep. (Courtesy Icko Iben Jr, Alexander Tutukov and Don Dixon)


center of gravity of the solar system, as the frame of reference, whereas the GJD uses the

center of the earth as the frame of reference. This difference can cause a change of as much

as 16 minutes to the date, skewing calculations, which is why I chose to use HJD for this

project.

The Julian calendar progresses in a linear fashion with strictly numbers, similar to the

Star Date used in Star Trek. The calendar begins on January 1, 4713 BCE at noon universal

time. An example calculation for the date of graduation this year, which is May 7, 2011 at

2pm local time (7pm Universal Time), would be HJD 2455689.291667. ("Julian Date

Converter," n.d.)

The ephemeris utilizes this time notation and shows researchers the cyclic pattern of

an eclipsing binary star. Below is the equation that shows how an ephemeris is written. The

letter E represents the epoch, and for the primary eclipse this number would be an integer

number. By performing the calculation one is able to determine the times of primary

minimum indefinitely. It can also be used to calculate the time of any other part of the period

by setting the epoch equal to a number between 0 and 1 which represents that portion of the

period, such as .5 for the secondary eclipse in a circular (or nearly circular) orbit. After the

first calculation to determine the time of a specific phase, the period should be multiplied by

integers to determine subsequent times of that phase. HJD0 is the time of primary eclipse and

P represents the period of the binary system.


Light Curve

The light curve is an important part of the study of an interacting, eclipsing binary

star. Using the light curve, a sample of which can be seen later in the U Cephei section, the

researcher can utilize software that will let him calculate many parameters for the star such as

radii, temperature, mass, etc. This light curve is built upon the initial data and shows the

apparent magnitude of the star versus the phase, with the primary minimum centered at 0

phase and the secondary minimum centered at .5 phase in the case of U Cep. For the light

curve produced in this experiment, the Wilson-Devinney code was used.

Summary

In summary, it can be said that there is so much to know about visual, eclipsing,

interacting Algol-type binaries like U Cep that the best method for learning how they operate

and change is to study them. This project has been an enormous learning experience for me,

both from books and from literally looking at the data and trying to determine just what it all

means. There is still so much unknown in astrophysics that books can only take you so far,

after that a person just has to use a strong base of physical comprehension, find a

knowledgeable mentor, and keep at it.

Methodology

Introduction

During the process of conducting this research it was very important that, as in most

experiments, the data be taken accurately and methodically. If the data is not taken and

processed accurately, it will skew the results. If you contaminate the data upstream, the

conclusions downstream will be even more erroneous. The data must be taken methodically


so that all the data is taken in the same manner and to foster reproducibility. Without this,

the experiment cannot be considered scientifically valid.

Data Collection

To achieve these ends, the process of data collection was carefully controlled. The

data taken by either Dr. Reed or myself was taken with precision using software that

controlled the telescope and took images at specific intervals. All of the information

regarding the time, location, and other pertinent details was captured by the computer and

linked to the image taken. Anything that was missing or entered incorrectly could be

manually fixed during the data processing phase.

The method described above lent a certain amount of reproducibility to the project.

Dr. Reed and I developed a written procedure that provided the parameters needed to

properly program the computer controlling the telescope. The base procedure is relatively

standard in the field of photometric analysis. The details of filter type, length of exposure,

and times of observation were determined to best achieve our goals.

Several facilities were enlisted to take data, another method that can help account for

error. By dividing the time between several telescopes, problems such as slight calibration

errors, atmospheric conditions, and user error can be greatly reduced. To this end, I utilized

equipment at three different locations, two of which were located 2000 miles apart. Dr.

Reed, myself, and Dr. Cohen and Dr. Jensen from Swarthmore College in Swarthmore, PA

took the data over a period of about a year.


Tzec Maun Foundation

For the data that I took, I utilized free access to a telescope through the Tzec Maun

Foundation. Dr. Reed had previously acquired an account with the foundation that would

allow him access to several professional grade telescopes with charge-coupled device (CCD)

cameras. Using a RC Optical Systems 16" Ritchey-Chretien equipped with a CCD camera,

Dr. Reed took data of a primary eclipse in October of 2009. In March of 2010, I was able to

take several hours worth of data in the .3 phase, which is shortly before the secondary

eclipse. The exposure times for both were 4 seconds and used a V-Bessel filter. This was

the main data utilized to construct the light curve, which will be discussed later in the

processing section.

The telescope at Tzec Maun is located between Cloudcroft and Mayhill, New

Mexico. This location allows for a clear and unobstructed view of the night sky. The

remotely controlled, research grade, internet telescope used a software program called

Astronomy Studio that could be instructed to take either a single or a series of images of the

field containing U Cep. Further information on the telescope can be found through the Tzec

Maun Foundation’s website, which is listed in my references section.


Figure 3: A sample image taken in March 2010 using the Tzec Maun Foundation telescope.

Swarthmore College

Dr. David Cohen and Dr. Eric Jensen took data at Swarthmore College during July of

2010. They used the new RC Optical Systems 24 inch carbon tube telescope equipped with

a CCD camera. Both nights of July 14th and July 29th, Cohen and Jensen took over 450

images of the secondary eclipse. The data of the viewing on July 29th was the data used to

assemble the secondary eclipse for the new light curve constructed for U Cep. The exposure

times for both nights were also 4 seconds and used a V-Bessel filter.

The telescope at Swarthmore College is located in Swarthmore, Pennsylvania near

Philadelphia in the Peter Van De Kamp Observatory that was installed in 2008. This is a

research grade telescope used by professors and their undergraduate students. Further


information about this telescope can be found on the Swarthmore Telescope website that is

listed in the references section.

Kutztown University

Dr. Reed took data with the telescope at Kutztown University in Kutztown, PA,

adding data to the light curve at the .11 phase. This data also helped to check the calibration

on the Kutztown telescope, which has not been utilized much due to field of view issues.

With a slight modification to the CCD camera, the field of view was sufficient to take images

that were similar to those taken by both Swarthmore and Tzec Maun. This provided an

invaluable local resource, verifying that the telescope would be appropriate for research.

The telescope at Kutztown University is a 0.46-meter Cassegrain style reflecting

telescope with a SBIG STL 6303E Research CCD camera. A V-Bessel filter was used and

the exposure time was 4 seconds. (Reed, n.d.) This telescope has recently been refurbished

and this data collection has helped to establish its accuracy at taking research grade

photometric data.

Processing Data

Processing data was probably the most time-consuming part of this process. When

the data was taken, the images recorded by the CCD cameras were only raw data. For the

data to be usable, it first needed to be processed. The first step of this process was retrieving

the files. From all the different sources, the files were downloaded and checked for errors

and then organized so that they could be further reduced in a software program called Mira.

Checking the files for error included several steps. First I checked that the correct

information was located in the header of the image file. Next I needed to make sure that the


image file was clear and usable. If the image had large bright areas or dark areas it was not

considered usable. Although not truly an error, I also needed to check whether the image had

been calibrated using certain reference frames. These reference frames take into account any

thermal noise in the viewing area or errors in the way the CCD camera interprets the data like

when the pixels do not resolve uniformly, that could make the image appear different than

what the sky actually is. Some programs used by observational facilities automatically apply

this filter to the image before rendering to what the user has access. Other facilities provide

this as a separate file and the two files must be merged before the image is reducible.

Once all of the images were prepared as necessary they were loaded individually into

a program called Mira Pro 7 UE. This program is how I converted images into data points,

with each of the 529 images needing to be opened in Mira. After checking for errors, I

would then choose the one or two reference stars used and input the correct magnitudes.

Then I selected U Cep and the program would determine its magnitude by comparing the

chosen star to the reference star(s). In order to select U Cep without selecting any of the

nearby stars, I also had to put the radius information in so that the program would know how

many pixels it should include in its calculations.

The reference stars that were used were labeled 79 and 102. For the data from Tzec

Maun, the star 79, with a magnitude of 7.925 with a V-Bessel filter. ("Variable Star Plotter

(VSP) | AAVSO," n.d.) This star is a good reference star because its magnitude is constant

over time. By comparing the number of pixels on the CCD image for U Cep and the

reference star, the computer can determine the magnitude of U Cep. The second star used for

reference in the Swarthmore data was star 102, with a magnitude of 10.186. ("Variable Star


Plotter (VSP) | AAVSO," n.d.) Using a second star for reference increased the accuracy of

our calculations and further validated our data points.

The information that Mira placed in spreadsheet form yielded an HJD, magnitude,

and error in magnitude among other things. Each image produced a line with each of these

items that was then compiled into a spreadsheet that looked like Figure 4, below. After

compiling the data into a spreadsheet, I then calculated the intensity from the magnitude and

the phase from the HJD using the equations that follow.

The equation used to calculate the intensity is:

where M0 is the normalization magnitude, 6.8548 in this project, and M is the magnitude of

the phase in question. The calculation is performed for the first line of the data above as a

sample. The equation to used to calculate the phase is:

and the sample calculation of the first line of data from the chart below is:


Magnitude HJD Intensity Phase 6.8496 2455277.6180127 1.00480086439126 0.313398434487709 6.8578 2455277.6238086 0.997240711741549 0.31572322933831 6.8376 2455277.6264302 1.01596793177014 0.316774779963865 6.8336 2455277.6290732 1.01971779922334 0.317834914320898 6.8487 2455277.6315506 1.0056341199161 0.318828624873902 6.8587 2455277.6338544 0.996414410886996 0.3197527027164 6.8889 2455277.6361683 0.969080824194403 0.320680831850034 6.8591 2455277.6386019 0.996047385864622 0.32165697373339 6.8081 2455277.6410216 1.04395072444297 0.322627540188414 6.8254 2455277.643318 1.02744835226935 0.323548649699013 6.8292 2455277.6457495 1.02385864116647 0.324523949384506 6.846 2455277.6482003 1.00813803476751 0.32550699041682

6.8548 2455277.6506114 1 0.326474107390759 6.8537 2455277.6530911 1.00101365083802 0.327468740457689 6.8949 2455277.6554925 0.963740255778264 0.328431966585811 6.8884 2455277.6580643 0.969527205181474 0.329463542048636 6.8735 2455277.6603956 0.982924137287097 0.330398650273715 6.8604 2455277.6631813 0.994855487954515 0.331516023080027 6.8993 2455277.6655231 0.959842553582907 0.332455343228354 6.8822 2455277.6680128 0.975079441717176 0.333453987428392 6.9208 2455277.670577 0.941022483526751 0.33448251424327 6.8691 2455277.6737525 0.9869155685416 0.335756239991063 6.8873 2455277.676497 0.97050996724549 0.336857086928944 6.8642 2455277.6788888 0.991379650183226 0.337816462556056 6.8863 2455277.6818652 0.971404251728767 0.339010327063811 6.8727 2455277.6855387 0.9836486494436 0.340483805559643 6.8722 2455277.6889563 0.984101740707102 0.341854639952921 6.8552 2455277.6918731 0.999631654241084 0.343024598480874 6.9213 2455277.6954821 0.940589226426994 0.344472205308215 6.9059 2455277.6984906 0.954025537236124 0.345678945599957 6.9044 2455277.7018436 0.955344483110094 0.347023868166701

Figure 4: A sample of the data taken on March 22, 2009 used to build the light curve.

with HJD representing the date and time when the data was taken, HJD0 is the time of

primary minimum, and P is the period determined in the ephemeris.

This spreadsheet of data, after adding the data from other observations, was then

utilized to produce a light curve, which can be seen in the U Cephei section. From the light

curve, I was able to determine the temperatures and other parameters of the two stars using a


program called Binary Maker 3.0. After converting the spreadsheet file to a format that

Binary Maker could recognize, I opened the file in the light curve window of the program. I

then used data from a previous study to input the parameters such as radii, mass ratio,

temperatures, inclination, etc. Using these predetermined parameters, I instructed Binary

Maker to produce a light curve based on the previous parameters. After comparing this new

light curve laid over the light curve my data produced, I adjusted the temperatures and

various other parameters to make the two light curves match as closely as possible.

Once this was achieved, I was able to take note of the new temperatures and, by

these, determine the stage of life the stars were now in. This also allowed me to discover that

the secondary had filled its Roche lobe, verifying that U Cep is a semi-detached binary

system. This means that the mass has spread out far enough in the red giant secondary to

escape the pull of the stars gravity and flow over the inner Lagrangian point, L1, as seen in

the diagram below. This inner Lagrangian point is lower than the surrounding gravitational

potentials represented by the 3-D image at the top of the figure. It can be thought of like a

topographical map; if water fills the left hand well, the first place it will spill over into is the

right hand well through the saddle point at L1.

Summary

Many people and methods were employed in obtaining the data used to create the

new light curve for U Cep. It was a painstaking process at times, but the outcome is

incredibly useful. This information can be used not only for further study of U Cep, but also

for study of binary stars and stellar evolution in general. The process of taking and

processing the data had to be handled carefully since the results may be used in further


experiments, and any contamination in the data upstream will cause a more dramatic error in

the results downstream.

Figure 5: A three-dimensional representation of a binary with a mass ratio of 2.0, showing the Lagrangian point at L1. ("Information on astronomy: Formation and evolution of compact binaries," n.d.)

U Cephei

Introduction

Now that the purpose of this project and some background information has been

defined, this review of literature will help the reader to understand what work has been done

previously with U Cep and from what data I performed my calculations. One paper was

relied upon heavily for the original parameters when I began to model the light curve I


obtained in Binary Maker; this was the paper by Burnett. (1993) Several other papers

contributed various specifications or background information for this study, including papers

by Khan and Budding (1986) and McCluskey (1988).

Following are the charts and figures produced by this project. The first is the light

curve mentioned previously that was constructed using the data collected through

observations. This data was first collected in a spreadsheet and binned by fives. This is a

process of collecting five data points in the same data set and averaging them to produce a

single data point. This helps to account for any minor fluctuations or other error that could

skew the data in an individual data point, but is lost when the data is averaged together.

Charts and Figures

Figure 6: Light curve for U Cep with binary model data superimposed in red.

The observation of the primary eclipse, along with some data collected from

American Association of Variable Star Observers (AAVSO), was used to supplement


Figure 7: Images produced in Binary Maker demonstrating U Cep in phases 0, 0.25, 0.50, and 0.75.


previous data producing an Observed minus calculated chart, as seen in Figure 7. This chart

shows a change in period over several thousand cycles. The E on the x-axis represents the

epoch, or one complete cycle. The numbers show the most recent data at 0 and the oldest

data at just over -4000 E, or -4000 times the period of approximately 2.5 days and divided by

the number of days in a year, 365.25. This shows the oldest data was taken approximately 27

years ago. “This period change indicates a rather high rate of mass transfer that is consistent

with an actively interacting Algol-type binary system.” (Reed & Lauria, April 24, 2010)

Figure 8: Observed minus calculated (O-C) curve for U Cep. (Reed & Lauria, April 24, 2010)


Calculations

In order to determine the mass transfer rate, the O-C curve must be utilized. There is

an equation associated with the line of best fit on the O-C curve that is shown below. Before

any calculations take place, it must first be assumed that all the mass transfer in the system

takes place between the two stars, with no mass being lost to the surroundings. This is called

conservative mass transfer and is, of course, improbable; however, the calculations would be

entirely too complex if this were not the case.

It can be seen that the mass transfer rate is changing due to the curve of the line.

Previous studies had determined that the mass transfer rate was constant due to, what is

believed to be, incomplete data. If the line between the data points is straight, there would be

no E2 term and, thus, the mass transfer rate would be a linear term. In Figure 6, however, it

can be seen that the line does in fact have a quadratic term and, hence, a curved shape.

Below is the equation determined by this graph with a c2 term of 2.74 x 10-9.

This c2 term is related to the change in period, P⋅ , as shown below.

Using this relationship, the rate of change in the period can be determined by

knowing the c2 term from the O-C equation above and the period. This result means that the

period of U Cep changes by 2.2 x 10-9 Heliocentric Julian days each Heliocentric Julian day.

This may not seem like much at first, but compounded over centuries this can be quite a

dramatic shift.


Once this rate of change in the period is known, it can be used to determine the rate of mass

transfer, m⋅ , shown below. M⨀ represents solar mass, which is equivalent to 1.981 ✕ 1030

kilograms.

The results of the change in mass transfer rate will be discussed further in the

following chapter.

Summary

Much data was compiled during this project and this data in table alone is often

difficult to interpret. In order that the data can be correctly understood, it must first be turned

into readable charts that allow us to compare and contrast the data with other factors that are

known about the system. Charts and tables allow the data to be clearly compared and

contrasted and also allow researchers to average the data to determine equations that best

represent the physical changes that are taking place. Therefore, these figures are an integral

part of any scientific study.

New Model of U Cephei

Introduction

Like all experiments, the original intent of this project was to calculate real, usable

statistics that could tell astronomers something new about how our universe works. Above I


have shown the process that was followed to obtain, reduce, and interpret the data. Below

can be found the actual numerical results determined by this research project. Nothing here

is earth shattering, but it does show that even an amateur can produce usable data that

contributes to the scientific community at large.

New Results

The first major result from this experiment was the determination of a new ephemeris.

Using just the data from this project, the new time of primary minimum was determined to be

HJD 2455144.70344, which corresponds to Monday 2009 November 09 04:52:57.2

Universal Time (UT). This was the data taken by Dr. Reed, which is also represented by the

primary minimum on the light curve.

The mass transfer rate, as calculated previously in the Calculations section, is

equivalent to

which may not seem easily discernable, but when converted into units that mean something

in non-scientific terms it is equivalent to:

This information, and the information obtained from the use of the program Binary

Maker has also determined that the classifications of the two stars are as follows. The

primary star is approximately 11,600 K and is a blue star on the main sequence, type B. The


secondary star is a red giant at approximately 5125 K and is a K-type star. This is a minor

change from previous reported parameters, which can most likely be attributed to the mass

transfer occurring within the system.

A reflection coefficient was also utilized in the model system, which is equivalent to

a heating of the near side of the secondary star of about 25% by the primary star. This

reflection coefficient accounts for the heat of the primary being reflected back to the observer

during observations of a secondary eclipse. It is similar to the idea of how the sun heats up

the near side of the Earth, reflecting some of it’s own heat back toward itself.

Comparison and Error Analysis

A comparison to previously published papers shows that the period has stayed the

same, up to four significant figures, since Burnett (1993) published his findings. The

parameters determined in this study can be compared to Burnett’s (1993) findings in the chart

below.

Burnett (1993) Current Results

Primary Temperature 11 250 K 11 600 K

Primary Radius (back) .186 .185

Secondary Temperature 4 980 K 5 125 K

Secondary Radius (back) .316 Roche lobe filled

Figure 9: A comparison table of old and new parameters.

Some differences can be attributed to the fact that this study is only photometric and

therefore, only a few parameters could be determined with the given data. Another

difference is that this experiment had strong data for both the primary and secondary eclipse,


whereas Burnett (1993) used mostly primary data to fit his model. This could lead to several

factors such as reflection being overlooked. As compared to data from a century ago, the

differences can mostly be attributed to an advance in the technology used to produce these

results.

The only error that can be calculated in this experiment is in the magnitudes that were

produced by Mira. Luckily, the hard part is done by Mira, which automatically generates a

margin of error when it produces its line of data for each image. The average error of the

magnitude of U Cep as calculated by the images used in this experiment is less than ±0.0009.

This error is so small as to be insignificant when the types of calculations performed are

taken into account.

Summary

The results of this experiment although not substantial enough to support the

publication in a scientific journal alone, are a strong base from which to proceed. Future

researchers, including others or myself, can use this data to produce results that are worthy of

publication in a nationally recognized journal. These results are sufficient on their own,

however, to be published as data that can be utilized by professionals around the world

through the AAVSO website, which maintains a publicly accessible bank of data related to

variable stars of all kinds.

Conclusion

Introduction

The results obtained from this research project are by no means earth shattering.

They are, however, as part of a collection of data pertaining to U Cep in specific and stellar


evolution in general, a key component to furthering mankind’s knowledge about stars. By

providing more current data to the information previously gathered about U Cep, we can

determine the path of evolution the two stars in the system. This data can be an invaluable

resource when astronomers are looking for information to compare to new results that might

change our entire perspective on the evolution of astronomical objects.

Directions for Future Research

The opportunities that have presented themselves at the conclusion of this research

are innumerable. In addition to continuing research on U Cep itself, proceeding in the same

vein as this project has, other types of studies could be performed.

Another area that opens itself up is to study U Cep spectroscopically. This would

entail using different types of equipment to view the system and analyzing the types of

elements that are present in the stars. This will give me a better idea of the stage of the life

cycle and chemical composition of the system. I can use this data in addition to the

information produced by this project to verify my findings.

The final area that I can continue research into is another binary star. Now that I have

learned the basic process of a photometric study of U Cep, I can transfer this knowledge to

the study of other binary systems. There are hundreds of known binary stars that have been

studied over the last century or so. Many of these studies were incomplete due to lack of

equipment to do a proper job. Now that computers can be utilized to process the data more

accurately and model the systems, study of these stars can reveal more accurate and complete

details.


Summary

In short, the physical data determined by this project, although valuable, pales in

comparison to the invaluable experience of actually exploring a binary star in depth and

learning the mechanics behind the process. In addition to learning about binary stars and

astronomy, I have also learned about the research process and the writing of scientific

research papers. Although this paper was written to summarize a project performed as an

undergraduate and geared towards a non-scientific audience and is therefore not written as

scientifically or as concisely as would be in a journal, the methods of researching can be

reproduced in a more technical setting that could lead to publication in a nationally

recognized scientific journal.


Reference List

Burnett, B. J., Etzel, P. B., & Olson, E. C. (1993). New Six-Color Intermediate-Band

Photometry and Photometric Solutions for U-Cephei. The Astronomical Journal,

106(4), 1627-1638. Retrieved from NASA Astrophysics Data System.

Carroll, B. W., & Ostlie, D. A. (2007). An Introduction to Modern Astrophysics. San

Francisco, CA: Pearson Addison-Wesley.

Chandra :: Educational Materials :: Stellar Evolution. (2008, September 24). The Chandra X-

ray Observatory Center :: Gateway to the Universe of X-ray Astronomy! Retrieved

from http://chandra.harvard.edu/edu/formal/stellar_ev/Dictionary and Thesaurus -

Merriam-Webster Online. (n.d.). Retrieved from

http://mw3.merriamwebster.com/dictionary/

Gimenez, A., Guinan, E. F., & Gonzalez-Riestra, R. (1993). UV and X-ray Emission in the

Interacting Binary U Cephei. Astronomy & Astrophysics Supplement Series, 97th ser.,

261-263. Retrieved from NASA Astrophysics Data System.

Information on astronomy: Formation and evolution of compact binaries. (n.d.). The Stars for

the Netherlands and Belgium. Retrieved from

http://hemel.waarnemen.com/Informatie/Sterren/hoofdstuk6.html

Julian Date Converter. (n.d.). Astronomical Applications Department. Retrieved April 17,

2011, from http://aa.usno.navy.mil/data/docs/JulianDate.php

Khan, M. A., & Budding, E. (1986). Photometry and Discussion of the Classical Algol

Systems U Sge and U Cep. Astrophysics and Space Science, 125th ser., 219-242.

Retrieved from NASA Astrophysics Data System.


McCluskey, Jr., G. E., Kondo, Y., & Olson, E. C. (1988). IUE Spectroscopy of U Cephei

During the Mass Flow Outburst of 1986 June. The Astrophysical Journal, 332nd ser.,

1019-1029. Retrieved from NASA Astrophysics Data System.

Reed, P. (n.d.). Kutztown University Observatory. Retrieved April 19, 2011, from

http://faculty.kutztown.edu/preed/observatory.html

Reed, P., & Lauria, K. N. (2010, April 24). New Times of Minimum Light of the Interacting

Binary Star U Cephei [Poster session presented at the 30th Annual Central

Pennsylvania Consortium Astronomer's Meeting, Gettysburg, PA].

Reed, P. A. (2008). Ultraviolet Spectroscopy of R Arae: An Interactive Binary Star (Doctoral

dissertation, Lehigh University, 2008). Bethlehem: Lehigh University.

Swarthmore Telescope. (n.d.). Swarthmore College :: ITS :: Staff Directory. Retrieved April

18, 2011, from http://wikis.swarthmore.edu/telescope/index.php/Main_Page

Tzec Maun Foundation | Free Access to Internet Telescopes for Students and Researchers.

(n.d.). Retrieved from http://www.tzecmaun.org/

Variable Star Plotter (VSP) | AAVSO. (n.d.). AAVSO | American Association of Variable

Star Observers. Retrieved from http://www.aavso.org/vsp/chart

Wilson, R. E. (1992). Documentation of Eclipsing Binary Computer Model. Privately

Distributed.


Glossary of Astronomical Terms

Algol – a class of binary stars where the larger member completely eclipses the smaller, brighter star which will cause a cyclical variation in brightness apparent magnitude – the brightness of a star based on its observation from Earth, normalized to the value it would take without the atmosphere, with a lower value assigned to brighter objects binary star – a system containing two stars which orbit each other due to the gravitational attraction between the two objects around their center of mass. black hole – an object in space that has such a strong gravitational field that nothing, not even light, can escape the pull of it’s gravity eclipsing binary – a binary star with a orbital plane that lies in the line of sight of the observer, allowing the observer to calculate it’s period by production of a light curve generated by two eclipses ephemeris – an equation that allows for the calculation of any part of a period of a cyclical object epoch – a part of the ephemeris that determines which part of the period is calculated intensity – the flux times the surface area divided by the distance squared interacting binary – a binary system undergoing mass transfer interstellar medium – the matter that occupies space between stars Julian Date – the interval of time in days and fractions of a day since January 1, 4713 BC Greenwich noon Kelvin - the base unit of temperature in the International System of Units that has its zero point at -273 Centigrade light curve - a graph showing the variation in intensity of a binary star over a period of time limb darkening – the effect of the decrease in brightness as one travels from the center of a star to its edge due to a gradient of temperature with the core being the hottest main-sequence – 90% of known stars make up the group that are in some stage of the path a normal star would follow during it’s lifetime neutron star – a small but massive object that is the result of a much larger, massive star collapsing


nuclear fusion – the fusing of two nuclei which results in a larger nucleus and a large amount of energy phase – the stage during a period of cyclical motion as referenced by an arbitrary zero point photometry – a branch of science that deals with measurement of the intensity of light protostar – part of the interstellar medium that is believed to develop into a star solar mass - the mass of the sun used as a unit for the expression of the masses of other objects and equal to about 2 × 1030 kilograms spectroscopy – the use of light or particle emission from a distant object, such as a star, that provide useful information about the object such as its chemical composition supernova – the death of a star that can be brighter than one billion times the sun’s luminosity white dwarf – a small, dense, white star with mass approximately that of our sun, which is hot, but does not give off much light

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