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 B
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 J had a period of :
 ms
Although pulsar periods are stable, they are not constant. Pulsars lose energy and slow down: dP/dt is typically for normal pulsars and 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 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 B : n = 2.839
Can differentiate again to give second braking index m, expected value mo
Secular decrease in n observed for Crab and PSR B
For PSR B , 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 J , 65% of observed P is due to Shklovskii term .

8. Pulsar Glitches First Vela glitch
(Radhakrishnan & Manchester 1969)
(Wang et al. 2000)
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 ( solar masses).
Five or six pulsars have neutron-star companions.
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 B
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 B : , , Pb measured
PSR J A/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 =  0.1 deg
mc =  Msun
From mass function:
mp =  Msun
(Jacoby et al. 2005)

15. Post-Keplerian Parameters: PSR B1913+16
Given the Keplerian orbital parameters and assuming general relativity:
Periastron advance: (7) deg/year
M = mp + mc
Gravitational redshift + Transverse Doppler: 4.294(1) ms
mc(mp + 2mc)M-4/3
Orbital period decay: (14) x 10-12
mp mc M-1/3
First two measurements determine mp and mc. Third measurement checks consistency with adopted theory.
Mp =  Msun
Mc =  Msun
Both neutron stars!
(Weisberg & Taylor 2005)

16. PSR B 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) =  . .
First observational evidence for gravitational waves!
(Weisberg & Taylor 2005)

17. PSR B1913+16 Nobel Prize for Taylor & Hulse in 1993
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. Faraday Rotation & Galactic Magnetic Field
(Han et al. 2005)

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 B

Derived proper motion: 375 mas yr-1
Manchester et al. (1974)