Pulsars and GravityR. N. Manchester

Australia Telescope National Facility, CSIRO Sydney Australia

Summary

• Introduction to pulsars and pulsar timing

• Parkes pulsar surveys – the double pulsar

• Tests of gravitational theories using pulsars

• The Parkes Pulsar Timing Array project

Pulsar Origins

• MSPs are very old (~109 years).

• Mostly binary

• They have been ‘recycled’ by accretion from an evolving binary companion.

• This accretion spins up the neutron star to millisecond periods.

• During the accretion phase the system may be detectable as an X-ray binary system.

Normal Pulsars:• Formed in supernova

• Periods between 0.03 and 10 s

• Relatively young (< 107 years)

• Mostly single (non-binary)

Pulsars are believed (by most people) to be rotating neutron stars

Millisecond Pulsars (MSPs):(ESO – VLT)

Spin-Powered Pulsars: A Census

• Number of known pulsars: 1775

• Number of millisecond pulsars: 172

• Number of binary pulsars: 134

• Number of AXPs: 13

• Number of pulsars in globular clusters: 99*

• Number of extragalactic pulsars: 20

Data from ATNF Pulsar Catalogue, V1.32 (www.atnf.csiro.au/research/pulsar/psrcat; Manchester et al. 2005)

* Total known: 137 in 25 clusters (Paulo Freire’s web page)

Pulsars as clocks

• Pulsar periods are incredibly stable and can be measured precisely, e.g. on Jan 16, 1999, PSR J0437-4715 had a period of :

5.757451831072007 0.000000000000008 ms

• Although pulsar periods are stable, they are not constant. Pulsars lose energy and slow down: dP/dt is typically 10-15 for normal pulsars and 10-20 for MSPs

• Precise pulsar timing parameters are measured by comparing observed pulse times of arrival (TOAs) with predicted TOAs based on a model for the pulsar, then using the timing residuals - deviations from the model - to improve the model parameters and to search for unmodelled effects

Sources of Pulsar Timing “Noise” Intrinsic noise

• Period fluctuations, glitches• Pulse shape changes

Perturbations of the pulsar’s motion• Gravitational wave background• Globular cluster accelerations• Orbital perturbations – planets, 1st order Doppler, relativistic effects

Propagation effects• Wind from binary companion• Variations in interstellar dispersion• Scintillation effects

Perturbations of the Earth’s motion• Gravitational wave background• Errors in the Solar-system ephemeris

Clock errors• Timescale errors• Errors in time transfer

Receiver noise

Instrumental errors• Radio-frequency interference and receiver non-linearities• Digitisation artifacts or errors• Calibration errors and signal processing artifacts and errors

Discovered at Arecibo Observatory by Russell Hulse & Joe Taylor in 1975

Pulsar period 59 ms, a recycled pulsar

Doppler shift in observed period due to orbital motion

Orbital period only 7 hr 45 min

Maximum orbital velocity 0.1% of velocity of light

Relativistic effects detectable!

PSR B1913+16: The First Binary Pulsar

Post-Keplerian Parameters: PSR B1913+16

• Periastron advance: 4.226607(7) deg/year M = mp + mc

• Gravitational redshift + Transverse Doppler: 4.294(1) ms

mc(mp + 2mc)M-4/3

• Orbital period decay: -2.4211(14) x 10-12

mp mc M-1/3

Given the Keplerian orbital parameters and assuming general relativity:

First two measurements determine mp and mc. Third measurement checks consistency with adopted theory.

(Weisberg & Taylor 2005)

Mp = 1.4408 0.0003 Msun

Mc = 1.3873 0.0003 Msun

Both neutron stars!

PSR B1913+16 Orbit Decay

• Energy loss to gravitational radiation

• Prediction based on measured Keplerian parameters and Einstein’s general relativity

• Corrected for acceleration in gravitational field of Galaxy

• Pb(obs)/Pb(pred) = 1.0013 0.0021. .

(Weisberg & Taylor 2005)

First observational evidence for gravitational waves!

• First discovery of a binary pulsar

• First accurate determinations of neutron star masses

• First observational evidence for gravitational waves

• Confirmation of General Relativity as an accurate description of strong-field gravity

Nobel Prize for Taylor & Hulse in 1993

The Hulse-Taylor Binary Pulsar

PSR B1913+16

The Parkes radio telescope has found more than twice as many pulsars as the rest of the world’s telescopes put together.

Parkes Multibeam Pulsar Survey

Principal papers:

• Covers strip along Galactic plane, -100o < l < 50o, |b| < 5o

• Central frequency 1374 MHz, bandwidth 288 MHz, 96 channels/poln/beam• Sampling interval 250 s, time/pointing 35 min, 3080 pointings• Survey observations commenced 1997, completed 2003• Processed on work-station clusters at ATNF, JBO and McGill• 740 pulsars discovered, 1015 detected• At least 18 months of timing data obtained for each pulsar

I: Manchester et al., MNRAS, 328, 17 (2001)System and survey description, 100 pulsars

II: Morris et al., MNRAS, 335, 275 (2002)120 pulsars, preliminary population statistics

III: Kramer et al., MNRAS, 342, 1299 (2003)200 pulsars, young pulsars and -ray sources

IV: Hobbs et al., MNRAS, 352, 1439 (2004)180 pulsars, 281 previously known pulsars

V: Faulkner et al., MNRAS, 355, 147 (2004)Reprocessing methods, 17 binary/MSPs

VI: Lorimer et al., MNRAS, 372, 777 (2006) 142 pulsars, Galactic population and evolution

Parkes Multibeam Surveys: P vs P

.J1119-6127

• New sample of young, high-B, long-period pulsars

• Large increase in sample of mildly recycled binary pulsars

• Three new double-neutron-star systems and one double pulsar!

J0737-3039

The first double pulsar!

Discovered at Parkes in 2003

One of top ten science break-throughs of 2004 - Science

PA = 22 ms, PB = 2.7 s

Orbital period 2.4 hours!

Periastron advance 16.9 deg/yr!(Burgay et al., 2003; Lyne et al. 2004)

Highly relativistic binary system!

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are needed to see this picture.

PSR J0730-3039A/B

PSR J0737-3039B

(Lyne et al., Science, 303, 1153, 2004)

• “Double-line binary” gives the mass ratio for the two stars – strong constraint on gravity theories

0.2 pulse periods

Orb

ital p

erio

d

• MSP blows away most of B magnetosphere - dramatic effect on pulse emission

(Spitkovsky & Arons 2005)

Binary pulsars and Gravity

Tests of Equivalence Principles

Limits on Parameterised Post-Newtonian (PPN) parameters

Dipolar gravitational radiation – dPb/dt

Variation of gravitational constant G – dP/dt, dPb/dt

Orbit ‘polarisation’ due to external field – orbit circularity

Binary pulsars give limits comparable to or better than Solar-system tests, but in strong-field conditions (GM/Rc2 ~ 0.1 compared to 10-5 for Solar-system tests)

PSR J1853+1303 and Nordvedt Effect• Long-period binary MSP discovered in Parkes Multibeam Survey

• P = 4.09 ms, Pb = 115 d, Ecc = 0.00002369(9), Min Mcomp= 0.24 Msun

• White dwarf companion

• Test of Strong Equivalence Principle: Differential acceleration in Galactic gravitational field leads to “forced” eccentricity (Damour & Schaefer 1991)

• Bayesian analysis with 20 other known low-mass wide binary pulsars

• || < 5 x 10-3 (95% confidence) Comparable to LLR limit but in strong field regime.

(Stairs et al. 2005)

Constraints on Gravitational Theories from PSR J0737-3039A/B

• Mass functions: sin i < 1 for A and B

• Mass ratio R = MA/MB Measured value: 1.0714 0.0011

Independent of theory to 1PN order. Strong constraint!

• Periastron advance : 16.8995 0.0007 deg/yr

Already gives masses of two stars (assuming GR):

MA = 1.3381 0.0007 Msun

MB = 1.2489 0.0007 Msun

Star B is a very low-mass NS!Mass Function A

Mass function B

.

(Kramer et al. Science, 314, 97, 2006)

GR value Measured value Improves as

Periast. adv. (deg/yr) - 16.8995 0.0007 T1.5

Grav. Redshift (ms) 0.3842 0.386 0.003 T1.5

Pb Orbit decay -1.248 x 10-12 (-1.252 0.017) x 10-12 T2.5

r Shapiro range (s) 6.15 6.2 0.3 T0.5

s Shapiro sin i 0.99987 0.99974 T0.5

Measured Post-Keplerian Parameters for PSR J0737-3039A/B

.

.

GR is OK! Consistent at the 0.05% level!

(Kramer et al. 2006)Non-radiative test - distinct from PSR B1913+16

+16 -39

PSR J0737-3039A/B Post-Keplerian Effects

R: Mass ratio

: periastron advance

: gravitational redshift

r & s: Shapiro delay

Pb: orbit decay

(Kramer et al. 2006)

.

.

• Six measured parameters

• Four independent tests

• Fully consistent with general relativity (0.05%)

Orbit Decay - PSR J0737-3039A/B

• Measured Pb = (-1.252 0.017) x 10-12 in 2.5 years

• Will improve at least as T2.5

• Not limited by Galactic acceleration!

System is much closer to Sun - uncertainty in Pb,Gal ~ 10-16

• Main uncertainty is in Shklovskii term due to uncertainty in transverse velocity and distance

Scintillation gives Vperp = 66 15 km s-1

Timing gives Vperp ~10 km s-1 -- correction at 0.02% level

VLBI measurements should give improved distance

.

.

Will surpass PSR B1913+16 in ~5 years and improve rapidly!

PSR J0737-3039: More Post-Keplerian Parameters!

• Relativistic orbit deformation: er = e (1 + r)

e = e (1 + ) ~ T2.5

Should be measurable in a few years

• Spin orbit coupling:

Geodetic precession - precession of spin axis about total angular momentum

Changes in pulse profile should give misalignment angle

Periastron precession - higher order terms

Can give measurement of NS moment of inertia

• Aberration: xobs = a1 sin i = (1 +A)xint

Will change due to geodetic precession

(Damour & Deruelle 1985)

Geodetic Precession of Spin Axis• For J0737-3039A, precession period ~ 75 yr (Damour & Ruffini 1974, Barker & O’Connell 1975)

• Expect changes in pulse profile as line-of-sight cut moves across beam (observed in PSR B1913+16, B1534+12, J1141-6545)

Difference profiles over 1.5 years

(Manchester et al. 2005)

(Kramer 1998,2003; Weisberg & Taylor 2002)

Not observed in PSR J0737-3039A! Small misalignment angle? Small natal kick? Light NS, low velocity, small eccentricity Different NS formation mechanism?

(Piran & Shaviv 2005)

PSR B1913+16

Detection of Gravitational Waves

• Prediction of general relativity and other theories of gravity

• Generated by acceleration of massive object(s)

(K. Thorne, T. Carnahan, LISA Gallery)

• Astrophysical sources:

Inflation era

Cosmic strings

SN, BH formation in early Universe

Binary black holes in galaxies

Coalescing neutron-star binaries

Compact X-ray binaries

(NASA GSFC)

Detection of Gravitational Waves• Huge efforts over more than four decades to detect gravitational waves

• Initial efforts used bar detectors pioneered by Weber

• More recent efforts use laser interferometer systems, e.g., LIGO, VIRGO, LISA

• Two sites in USA• Perpendicular 4-km arms• Spectral range 10 – 500 Hz• Initial phase now operating• Advanced LIGO ~ 2011

LISALIGO• Orbits Sun, 20o behind the Earth• Three spacecraft in triangle• Arm length 5 million km• Spectral range 10-4 – 10-1 Hz• Planned launch ~2017

Double-Neutron-Star Binary Mergers

• Around one thousand systems similar to PSR J0737-3039A/B in Galaxy

• Galactic merger rate between 80 and 370 per Myr (1 )

• Detection rate for initial LIGO between one per 8 years and one per 35 years.

• Factor of seven increase over rates estimated from PSR B1913+16

(Kalogera et al. 2004)

• Prime candidate source for ground laser-interferometer systems

• Predicted detection rate dominated by double pulsar!

Detecting Gravitational Waves with Pulsars• Observed pulse periods affected by presence of gravitational waves in Galaxy

• With observations of <10 pulsars, can only put limit on strength of stochastic GW background

• Best limits are obtained for GW frequencies ~ 1/T where T is length of data span

• Analysis of 8-year sequence of Arecibo observations of PSR B1855+09 gives g = GW/c < 10-7 (Kaspi et al. 1994, McHugh et al.1996)

• Extended 17-year data set gives better limit, but non-uniformity makes quantitative analysis difficult (Lommen 2001, Damour & Vilenkin 2004)

Timing residuals for PSR B1855+09

A Pulsar Timing Array• With observations of many pulsars widely distributed on the sky can in principle detect a stochastic gravitational wave background

• Gravitational waves passing over the pulsars are uncorrelated

• Gravitational waves passing over Earth produce a correlated signal in the TOA residuals for all pulsars

• Requires observations of ~20 MSPs over 5 – 10 years; could give the first direct detection of gravitational waves!

• A timing array can detect instabilities in terrestrial time standards – establish a pulsar timescale

• Can improve knowledge of Solar system properties, e.g. masses and orbits of outer planets and asteroids

Idea first discussed by Hellings & Downs (1983), Romani (1989) and Foster & Backer (1990)

Clock errors

All pulsars have the same TOA variations: monopole signature

Solar-System ephemeris errors

Dipole signature

Gravitational waves

Quadrupole signature

Can separate these effects provided there is a sufficient number of widely distributed pulsars

Detecting a Stochastic GW Background

Simulation using Parkes Pulsar Timing Array (PPTA) pulsars with GW background from binary black holes in galaxies

(Hobbs et al., 2008)

The Parkes Pulsar Timing Array ProjectCollaborators:

Australia Telescope National Facility, CSIRO, SydneyDick Manchester, George Hobbs, David Champion, John Sarkissian, John Reynolds, Mike Kesteven, Grant Hampson, Andrew Brown, David Smith, Jonathan Khoo, (Russell Edwards)

Swinburne University of Technology, MelbourneMatthew Bailes, Ramesh Bhat, Willem van Straten, Joris Verbiest, Sarah Burke, Andrew Jameson

University of Texas, BrownsvilleRick Jenet

University of Sydney, SydneyDaniel Yardley

National Observatories of China, BeijingJohnny Wen

Peking University, BeijingKejia Lee

Southwest University, ChongqingXiaopeng You

Curtin University, PerthAidan Hotan

To detect gravitational waves of astrophysical origin

To establish a pulsar-based timescale and to investigate irregularities in terrestrial timescales

To improve on the Solar System ephemeris used for barycentric correction

The PPTA Project: Goals

To achieve these goals we need ~weekly observations of ~20 MSPs over at least five years with TOA precisions of

~100 ns for ~10 pulsars and < 1 s for rest

• Modelling and detection algorithms for GW signals

• Measurement and correction for interstellar and Solar System propagation effects

• Implementation of radio-frequency interference mitigation techniques

Sky Distribution of Millisecond PulsarsP < 20 ms and not in globular clusters

PPTA Pulsars: Recent Results using PDFB2

• 20 MSPs - all in Galactic disk except J1824-2452 (B1821-24) in M28

• ~200 days of timing data at 2 -3 week intervals at 10cm and 20cm

• Uncorrected for DM variations

• Two pulsars with rms timing residuals < 100 ns, seven < 500 ns, eleven < 1 s, all < 2.6 s

• Best results on J0437-4715 (52 ns) and J1909-3744 (97 ns)

Highest precision timing results ever obtained!

Name Period (ms)

DM (cm-3 pc)

Orbital period (d)

Rms Residual ( s)

J0437-4715 5.757 2.65 5.74 0.05

J0613-0200 3.062 38.78 1.20 0.69

J0711-6830 5.491 18.41 - 1.26

J1022+1001 16.453 10.25 7.81 1.03

J1024-0719 5.162 6.49 - 1.01

J1045-4509 7.474 58.15 4.08 1.51

J1600-3053 3.598 52.19 14.34 0.41

J1603-7202 14.842 38.05 6.31 0.92

J1643-1224 4.622 62.41 147.02 1.00

J1713+0747 4.570 15.99 67.83 0.16

J1730-2304 8.123 9.61 - 1.34

J1732-5049 5.313 56.84 5.26 1.34

J1744-1134 4.075 3.14 - 0.36

J1824-2452 3.054 119.86 - 1.66

J1857+0943 5.362 13.31 12.33 0.72

J1909-3744 2.947 10.39 1.53 0.10

J1939+2134 1.558 71.04 - 0.13

J2124-3358 4.931 4.62 - 2.66

J2129-5721 3.726 31.85 6.63 1.57

J2145-0750 16.052 9.00 6.84 0.46

Still not quite good enough though!!

A Pulsar Timescale• Terrestrial time defined by a weighted average of caesium clocks at time centres around the world

• Comparison of TAI with TT(BIPM03) shows variations of amplitude ~1 s even after trend removed

• Revisions of TT(BIPM) show variations of ~50 ns

(Petit 2004)• Pulsar timescale is not absolute, but can reveal irregularities in TAI and other terrestrial timescales

• Current best pulsars give a 10-year stability (z) comparable to TT(NIST) - TT(PTB)

• Full PPTA will define a pulsar timescale with precision of ~50 ns or better at 2-weekly intervals and model long-term trends to 5 ns or better

Current and Future Limits on the Stochastic GW Background

(Jenet et al. 2006)

10 s

Timing Residuals• Arecibo data for PSR B1855+09 (Kaspi et al. 1994) and recent PPTA data

• Monte Carlo methods used to determine detection limit for stochastic background described by hc = A(f/1yr) (where = -2/3 for SMBH, ~ -1 for relic radiation, ~ -7/6 for cosmic strings)

Current limit: gw(1/8 yr) ~ 2 10-8

For full PPTA (100ns, 5 yr): ~ 10-10

• Currently consistent with all SMBH evolutionary models (e.g., Jaffe & Backer 2003; Wyithe & Loeb 2003, Enoki et al. 2004)

• If no detection with full PPTA, all current models ruled out

• Already limiting EOS of matter in epoch of inflation (w = p/ > -1.3) and tension in cosmic strings (Grishchuk 2005; Damour & Vilenkin 2005)

Future ProspectsSingle source detection

Stochastic GW BackgroundPPTA

SKA

Range of predicted amplitudes(Jaffe & Backer 2003; Wyithe & Loeb 2003)

5 years, 100 ns

Difficult to get sufficient observations with PPTA alone - international collaborations important!

Predicted merger rates for 5 x 108 M binaries (Wen & Jenet 2008)

PPTA can’t detect individual binary systems - but SKA will!

The Gravitational Wave Spectrum

Pulsars are extraordinarily good clocks and provide highly sensitive probes of a range of gravitational effects

Parkes multibeam pulsar surveys have been extremely successful, more than doubling the number of known pulsars

First-known double-pulsar system detected! Makes possible additional independent tests of relativistic gravity

Direct detection of gravitational waves (GW) is a major goal of current astrophysics - it will open a new window on the Universe

A pulsar timing array can detect GW from astrophysical sources (or rule out most current models)

Parkes Pulsar Timing Array (PPTA) timing 20 MSPs since mid-2004. Goal is ~100 ns rms residuals on at least half of sample; currently have two with rms residuals < 100 ns and seven less 500 ns

A pulsar-based timescale will have better long-term stability than current best terrestrial timescales

SKA will herald a new era in the study of gravitation using pulsars!

Summary

Pulsar Model

(Bennet Link)

• Rotating neutron star

• Light cylinder RLC = c/= 5 x 104 P(s) km

•Charge flow along open field lines

• Radio beam from magnetic pole (in most cases)

• High-energy emission from outer magnetosphere

• Rotation braked by reaction to magnetic-dipole radiation and/or charge acceleration:

= -K -3

• Characteristic age: c = P/(2P)

• Surface dipole magnetic field: Bs ~ (PP)1/2

..

.

Neutron Stars• Formed in Type II supernova explosion - core collapse of massive star

• Diameter 20 - 30 km

• Mass ~ 1.4 Msun

(Stairs 2004)

(MT77)

(Lattimer & Prakash 2004)

Geodetic Precession in PSR J1141-6545• Relatively young 394-ms pulsar in 4.7-h binary orbit (Kaspi et al. 2000)

• From timing, companion mass 0.99 +/- 0.02 Msun: white dwarf (Bailes et al. 2003)

• Relativistic precession of periastron observed• Expected rate of geodetic precession 1.36 deg/yr - precession period 265 yr• Dramatic evolution of pulse profile observed! (7-yr data span)

(Manchester et al. (2007)

PSR J1141-6545 Geodetic Precession: Polarisation evolution and beam model

Model timing residuals

• Period: P = 5 x 10-16 s

• dP/dt: P = 4 x 10-23

• Position: = 1 mas

• Proper motion: = 5 mas/yr

• Parallax: = 10 mas

.

Maximum Spin Frequency • In LMXB systems, long evolution time allows spin-up to > 1 kHz

• Most neutron-star EOSs allow spin at > 1 kHz

• X-ray observations and recent radio observations have little or no observational selection against sub-ms pulsars

• But, maximum observed spin frequency ~ 700 Hz

Mass asymmetry due to accretion (I/I ~ 10-7) results in GW emission (e.g., Bildsten 1998)

r-mode instability in NS leads to viscous damping & GW emission (e.g., Ho & Lai 2000)

MSPs and Gravity:

(Arras 2004)

Neutron-star masses: PSR 1913+16

(Diagram from C.M. Will, 2001)

• Periastron advance

• Grav. Redshift

• Orbit decay

Mp = 1.4408 0.0003 Msun

Mc = 1.3873 0.0003 Msun

Both neutron stars!

(Weisberg & Taylor 2005)

Correlated scintillation in A and B: implies i = 90o.26 +/- 0o.13

Measured Shapiro delay implies i = 87o.8 +/- 1o.2

(Kramer et al. 2005)

(Coles et al. 2005)

Inconsistent!?

Geodetic Precession - PSR J0737-3039A• Precession period ~ 75 yr

• Expect changes in pulse profile as line-of-sight cut moves across beam (observed in PSR B1913+16, B1534+12, J1141-6545)

Difference profiles over 1.5 years

(Manchester et al. 2005)

(Kramer 1998,2003; Weisberg & Taylor 2002)

Not observed in PSR J0737-3039A! Small misalignment angle? Small natal kick? Light NS, low velocity, small eccentricity Different NS formation mechanism?

(Piran & Shaviv 2005)

PSR B1913+16

Orbital Modulation of PSR J0737-3039B

Secular changes are observed! Mechanism for orbital modulation not fully understood

Can’t separate effects of periastron precession and geodetic precession

(Burgay et al. 2005)

• Start observation at known time and average 1000 or more pulses to get mean pulse profile.

• Cross-correlate this with a standard template to give the arrival time at the telescope of a fiducial point on profile, usually the pulse peak – the pulse time-of-arrival (TOA).

• Measure a series of TOAs over days – weeks – months – years.

• Compare observed TOAs with predicted values from a model for pulsar using TEMPO - differences are called timing residuals.

• Fit the observed residuals with functions representing errors in the model parameters (pulsar position, period, binary period etc.).

• Remaining residuals may be noise – or may be science!

Measurement of pulsar periods

Periastron Precession - higher order terms

1PN 2PN

Spin A Spin B

Geometry NS Structure• If geometry can be understood, measurement of variation in rate of periastron precession can be used to estimate NS moment of inertia (Damour & Schaefer 1988)

1PN: 16.9 o/yr

2PN: 0.0004 o/yr

Spin: 0.0002 o/yr

Current: 0.0007 o/yr• Since mass known, strong limit

on EOS!(Morrison et al. 2004; Lattimer & Schutz 2005)

Not Easy!

S/N ~ T1.5

The PPTA Project: Methods• Using the Parkes 64-m telescope at three frequencies (680, 1400 and 3100 MHz)

• Digital filterbank system, 256 MHz bandwidth (1 GHz early 2007), 8-bit sampling, polyphase filter

• CPSR2 baseband system 2 x 64 MHz bandwidth, 2-bit sampling, coherent de-dispersion

• Developing APSR with 512 MHz bandwidth and 8-bit sampling

• Implementing real-time RFI mitigation for 50-cm band• TEMPO2: New timing analysis program, systematic errors in TOA corrections < 1 ns, graphical interfaces, predictions and simulations (Hobbs et al. 2006, Edwards et al. 2006)

• Observing 20 MSPs at 2 - 3 week intervals since mid-2004

• International collaboration and co-operation to obtain improved data sampling including pulsars at northern declinations

Dispersion Measure Variations

• DM from 10/50cm or 20/50cm observation pairs

• Variations observed in most of PPTA pulsars

• DM typically a few x 10-3 cm-3 pc

• Weak correlation of d(DM)/dt with DM, closer to linear rather than DM1/2

• Effect of Solar wind observed in pulsars with low ecliptic latitude

(You et al., in prep.)