Scuola nazionale de Astrofisica Radio Pulsars 2: Timing and ISM

1. Scuola nazionale de Astrofisica Radio Pulsars 2: Timing and ISM Outline • Timing methods • Glitches and timing noise • Binary pulsar timing • Post-Keplerian effects, PSR B1913+16 • Dispersion, pulsar distances • Faraday Rotation – Galactic magnetic field • Scintillation: DISS, RISS

2. 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 • Young pulsars suffer period irregularities and glitches (DP/P <~ 10-6) but these are weak or absent in MSPs

3. Techniques of Pulsar Timing • Need telescope, receiver, spectrometer (filterbank, digital correlator, digital filterbank or baseband system), data acquisition system • Start observation at known time and synchronously average 1000 or more pulses (typically 5 - 10 minutes), dedisperse and sum orthogonal polarisations to get mean total intensity (Stokes I) 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 (tobs) over days – weeks – months – years • TOA rms uncertainty: • Correct observed TOA to infinite frequency at Solar System Barycentre (SSB) tclk: Observatory clock correction to TAI (= UTC + leap sec), via GPS D: dispersion constant (D = DM/(2.41x10-16) s R: propagation (Roemer) delay to SSB (Uses SS Ephemeris, e.g. DE405) S: Solar-system Shapiro delay E: Einstein delay at Earth

4. Timing Techniques (continued) • Have series of TOAs corrected to SSB: ti • Model pulsar frequency by Taylor series, integrate to get pulse phase ( = 1 => P) • Choose t = 0 to be first TOA, t0 • Form residual ri = i - ni, where ni is nearest integer to i • If pulsar model is accurate, then ri << 1 • Corrections to model parameters obtained by making least-squares fit to trends in ri • Timing program (e.g. TEMPO or TEMPO2) does SSB correction, computes ri and improved model parameters • Can solve for pulsar position from error in SSB correction • For binary pulsar, there are additional terms representing Roemer and other (relativistic) delays in binary system

5. Sources of Timing “Noise” • Intrinsic noise • Period fluctuations, glitches • Pulse shape changes • Perturbations of pulsar 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

6. Spin Evolution • For magnetic dipole radiation, braking torque ~ 3 • Generalised braking law defines braking index n • n = 3 for dipole magnetic field • Measured for ~8 pulsars • Crab: n = 2.515 • PSR B1509-58: n = 2.839 • Can differentiate again to give second braking index m, expected value mo • Secular decrease in n observed for Crab and PSR B1509-58 • For PSR B1509-58, mo = 13.26, m = 18.3  2.9 • Implies growing magnetic field (Livingston et al. 2005)

7. Derived Parameters • Actual age of pulsar is function of initial frequency or period and braking index (assumed constant) • For P0 << P, n = 3, have “characteristic age” • If know true age, can compute initial period • From braking equation, can derive B0, magnetic field at NS surface, R = NS radius. Gives value at NS equator; value at pole 2B0 • Numerical value assumes R = 10 km, I = 1045 gm cm2, n = 3 • For dipole field, can derive magnetic field at light cylinder • Especially for MSPs, these values significantly modified by “Shklovskii term” due to transverse motion, e.g. for PSR J0437-4715, 65% of observed P is due to Shklovskii term .

8. Pulsar Glitches First Vela glitch (Wang et al. 2000) (Radhakrishnan & Manchester 1969) Probably due to sudden unpinning of vortices in superfluid core of the neutron star transferring angular momentum to the solid crust. Quasi-exponential recovery to equilibrium slowdown rate.

9. Intrinsic Timing Noise • Quasi-random fluctuations in pulsar periods • Noise typically has a very ‘red’ spectrum • Often well represented by a cubic term in timing residuals Stability 8 measured with data span of 108 s ~ 3 years used as a noise parameter

10. Binary pulsars • Some pulsars are in orbit around another star. Orbital periods range from 1.6 hours to several years • Only a few percent of normal pulsars, but more than half of all millisecond pulsars, are binary. • Pulsar companion stars range from very low-mass white dwarfs (~0.01 solar masses) to heavy normal stars (10 - 15 solar masses). • Five or six pulsars have neutron-starcompanions. • One pulsar has three planets in orbit around it.

11. Keplerian parameters: • Pb: Orbital period • x = ap sin i: Projected semi-major axis • : Longitude of periastron • e: Eccentricity of orbit • T0: Time of periastron Kepler’s Third Law: (Lorimer & Kramer 2005) PSR B1913+16 From first-order (non-relativistic) timing, can’t determine inclination or masses. Mass function: For minimum mass, i = 90o For median mass, i = 60o

12. PSR B1257+12 – First detection of extra-solar planets A: 3.4 Earth masses, 66.5-day orbit B: 2.8 Earth masses, 98.2-day orbit C: ~ 1 Moon mass, 25.3-day orbit Wolszczan & Frail (1992); Wolszczan et al. (2000)

13. Post-Keplerian Parameters Expressions for post-Keplerian parameters depend on theory of gravity. For general relativity: . : Periastron precession : Time dilation and grav. redshift r: Shapiro delay “range” s: Shapiro delay “shape” Pb: Orbit decay due to GW emission geod: Frequency of geodetic precession resulting from spin-orbit coupling . PSR B1913+16: , , Pb measured PSR J0737-3039A/B , , r, s, Pb measured . . . .

14. Shapiro Delay - PSR J1909-3744 • P = 2.947 ms • Pb = 1.533 d • Parkes timing with CPSR2 • Rms residuals: • 10-min: 230 ns • Daily (~2 hr): 74 ns • From Shapiro delay: • i = 86.58  0.1 deg • mc = 0.204  0.002 Msun • From mass function: • mp = 1.438  0.024 Msun (Jacoby et al. 2005)

15. Post-Keplerian Parameters: PSR B1913+16 Given the Keplerian orbital parameters and assuming general relativity: • 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 First two measurements determine mp and mc. Third measurement checks consistency with adopted theory. Mp = 1.4408  0.0003 Msun Mc = 1.3873  0.0003 Msun Both neutron stars! (Weisberg & Taylor 2005)

16. 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 . . First observational evidence for gravitational waves! (Weisberg & Taylor 2005)

17. PSR B1913+16 The Hulse-Taylor Binary Pulsar • 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

18. Interstellar Dispersion Ionised gas in the interstellar medium causes lower radio frequencies to arrive at the Earth with a small delay compared to higher frequencies. Given a model for the distribution of ionised gas in the Galaxy, the amount of delay can be used to estimate the distance to the pulsar.

19. Dispersion & Pulsar Distances • For pulsars with independent distances (parallax, SNR association, HI absorption) can detemine mean ne along path. Typical values ~ 0.03 cm-3 • From many such measurements can develop model for Galactic ne distribution, e.g. NE2001 model (Cordes & Lazio 2002) • Can then use model to determine distances to other pulsars

20. (Han et al. 2005) Faraday Rotation & Galactic Magnetic Field

21. Interstellar Scintillation • Small-scale irregularities in the IS electron density deflect and distort the wavefront from the pulsar • Rays from different directions interfere resulting in modulation in space and frequency - diffractive ISS • Motion of the pulsar moves the pattern across the Earth • Larger-scale irregularities cause focussing/defocussing of wavefront - refractive ISS

22. Dynamic Spectra resulting from DISS (Bhat et al., 1999)

23. DISS Secondary Spectrum • Take 2-D Fourier transform of dynamic spectra • Sec spectrum shows remarkable parabolic structures • Not fully understood but main structure results from interference between core and outer rays (Stinebring 2006)

24. ISM Fluctuation Spectrum • Spectrum of interstellar electron density fluctuations • Follows Kolmogorov power-law spectrum over 12 orders of magnitude in scale size (from 10-4 AU to 100 pc) • Mostly based on pulsar observations (Armstrong et al. 1995)

25. End of Part 2

26. First detection of pulsar proper motion PSR B1133+16 Derived proper motion: 375 mas yr-1 Manchester et al. (1974)