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Scuola nazionale de Astrofisica
Radio Pulsars 2: Timing and ISM
• 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
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
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
DR: propagation (Roemer) delay to SSB (Uses SS Ephemeris, e.g. DE405)
DS: Solar-system Shapiro delay
DE: Einstein delay at Earth
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
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
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)
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,
• For dipole field, can derive magnetic field at light
• Especially for MSPs, these values significantly
modified by “Shklovskii term” due to transverse
e.g. for PSR J0437-4715, 65% of observed P is due to
Shklovskii term
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.
Intrinsic Timing Noise
• Quasi-random fluctuations in
pulsar periods
• Noise typically has a very ‘red’
• Often well represented by a cubic
term in timing residuals
Stability D8
measured with
data span of 108
s ~ 3 years
used as a noise
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-star companions.
• One pulsar has three planets in orbit around it.
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
PSR B1257+12 – First detection of extra-solar planets
A: 3.4 Earth
masses, 66.5-day
B: 2.8 Earth
masses, 98.2-day
C: ~ 1 Moon
mass, 25.3-day
Wolszczan & Frail (1992); Wolszczan et al. (2000)
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
PSR B1913+16:
. .
, , Pb measured
PSR J0737-3039A/B
, , r, s, Pb measured
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)
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)
PSR B1913+16 Orbit Decay
• Energy loss to gravitational
• 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)
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
Interstellar Dispersion
Ionised gas in the interstellar
medium causes lower radio
frequencies to arrive at the
Earth with a small delay
compared to higher
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.
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
Faraday Rotation & Galactic Magnetic Field
(Han et al. 2005)
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
Dynamic Spectra resulting from DISS
(Bhat et al., 1999)
• Take 2-D Fourier
transform of dynamic
• Sec spectrum shows
remarkable parabolic
• Not fully understood
but main structure
results from
interference between
core and outer rays
(Stinebring 2006)
ISM Fluctuation
• 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
• Mostly based on pulsar
(Armstrong et al. 1995)
End of Part 2
First detection of pulsar proper motion
Derived proper motion: 375 mas yr-1
Manchester et al. (1974)