Scuola nazionale de AstrofisicaRadio 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

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 (P/P <~ 10-6) but these are weak or absent in MSPs

• 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)

Techniques of Pulsar Timing

tclk: Observatory clock correction to TAI (= UTC + leap sec), via GPSD: 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

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 = 2.515PSR 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, 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

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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’ 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

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

PSR B1913+16From first-order (non-relativistic) timing, can’t determine inclination or masses.

Mass function:

For minimum mass, i = 90o

For median mass, i = 60o

Kepler’s Third Law:

(Lorimer & Kramer 2005)

PSR B1257+12 – First detection of extra-solar planets

Wolszczan & Frail (1992); Wolszczan et al. (2000)

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

Post-Keplerian ParametersExpressions 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

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Shapiro Delay - PSR J1909-3744

(Jacoby et al. 2005)

• 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

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

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.

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)

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)

ISM Fluctuation Spectrum

(Armstrong et al. 1995)

• 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

End of Part 2

First detection of pulsar proper motion

PSR B1133+16

Manchester et al. (1974)

Derived proper motion: 375 mas yr-1