Neutron Star (Mostly Pulsar) MassesIngrid Stairs

UBC VancouverCAWONAPS

TRIUMFDec. 9, 2010

Pulsars are one manifestation of neutron stars:spin periods ~1.4 ms to 8.5 seconds,magnetic fields ~108 to ~5 x1013 G. Low fluxes mean large radio telescopesare needed to detect them.

Modern observing systems usethe coherent dedispersiontechnique over hundreds ofMHz of bandwidth.

Result: sharp profile featuresare resolved, leading to moreprecise pulse Times of Arrival(TOAs). The wide bandwidthsincrease the signal-to-noiseratio and therefore alsoTOA precision.

Build up a “standard profile”over hours/days of observing.

Standardprofile

Observedprofile

Measure offset

Pulsar Timing in a Nutshell

Record the start time of the observation with the observatory clock (typically a maser). The offset gives us a Time of Arrival (TOA) for the observed pulse. Then transform to Solar System Barycentre, enumerate pulses and fit the timing model.

Timing Residuals: Actual TOAs – Predicted

Gaussian residuals imply a good ephemeris.

PSR J1751-2857 – Stairs et al. (2005)

The Pulsar Population

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τ =P

2 ˙ P

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B = 3.2 ×1019 P ˙ P G

Binary Pulsars

Doppler shift in period is quickly obvious. All systems: fit basicKeplerian parameters: Pb, ecc, asini = x, ω and T0.

The 5 Keplerian parameters leave the two massesand the orbital inclination angle i unknown.These are related by the mass function:

... but there are still two unknowns. €

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2πPb

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

x 3

T0

=m2 sin i( )

3

m1 + m2( )2

Mass function:

So measuring masses is difficult!

It's only possible when we can measure “post-Keplerian” (PK) parameters: advance of periastron, time dilation/redshift, orbital period decay and/or Shapiro delay.

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2πPb

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

x 3

T0

=m2 sin i( )

3

m1 + m2( )2

Assuming general relativity (GR) is the correcttheory of gravity, we can get the masses fromthe PK parameters:

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˙ ω = 3Pb

2π ⎛ ⎝ ⎜

⎞ ⎠ ⎟−5 / 3

T0M( )2 / 3

1− e2( )−1

γ = ePb

2π ⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/ 3

T02 / 3M −4 / 3m2 m1 + 2m2( )

˙ P b =−192

5Pb

2π ⎛ ⎝ ⎜

⎞ ⎠ ⎟−5 / 3

1+7324

e2 +3796

e4 ⎛ ⎝ ⎜

⎞ ⎠ ⎟1− e2( )

−7 / 2T0

5 / 3m1m2M −1/ 3

r = T0m2

s = xPb

2π ⎛ ⎝ ⎜

⎞ ⎠ ⎟−2 / 3

T0−1/ 3M −2 / 3m2

−1

M = m1 + m2

T0 = 4.925490947μs

The mass constraints are easily understoodvia a mass-mass diagram.

Nice, Stairs & Kasian,“40 Years of Pulsars”proceedings (2007)

J0621+1002:advance of periastronmeasured; limit onShapiro delay

Pulsar mass: 1.70+0.10-0.17 Mʘ

Shapiro Delay

Timing signature can bepartly absorbed intoKeplerian parametersfor circular orbits-- unless the orbit isnearly edge-on to theline of sight.

1802-2124: WD: 0.78 ± 0.04MʘNS: 1.24 ± 0.11 Mʘ

Ferdman et al. 2010

Shapiro delay in PSR J1713+0747

Also other constraintson the inclination anglevia geometric effects.

Splaver et al., ApJ620, 405 (2005).

PSR J0751+1807: Shapiro delay and orbital decayThis was “the 2.1 Mʘ pulsar.” Early data was folded with a bad ephemeris, distorting the profiles, and this turns out to have corrupted the measurement.

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˙ P b

Nice, Stairs & Kasian, “40 Years of Pulsars” proceedings (2007).

PSR J0751+1807: Shapiro delay and orbital decayThis was “the 2.1 Mʘ pulsar.” Early data was folded with a bad ephemeris, distorting the profiles, and this turns out to have corrupted the measurement.

Pulsar mass is now 1.26 ± 0.14 Mʘ at 68% confidence.€

˙ P b

More PK params in double-NS binaries

PSR B1534+12(Stairs et al, unpublished)

5 PK parameters includingShapiro delay ⇒masses precisely measured

PLUS self-consistencytests of GR or othertheories of strong-fieldgravity.

The double pulsar J0737-3039A/B

5 PK params plusthe mass ratio!(Therefore even moretests of strong-fieldgravity.)

Kramer et al., 2006.

Well-measuredmasses for X-ray binaries can be found in Lattimer 2010 review in Ap&SS. Here are masses for radio pulsars.

Are the differences significant? due to evolution?

There are also several eccentric systems in globular clustersfor which we have measured .

M28 and Ter5numbersstillpreliminary-- do notquote!

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˙ ω

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˙ ω About half of these are pointing to highish pulsar masses – if is purely due to GR with no mass-quadrupole effects. Not sure what is going on with these! Note also several low-mass pulsars.

Future Prospects for Mass Measurements J1614-2230 and J1903+0327: starting to put useful constraints on the NS equation of state.

J0737-3039: advance of periastron is so precisely measured that we may be able to measure the contribution from spin-orbit coupling and get the moment of inertia of pulsar A. Requires measuring s and very precisely.

Hope to clear up masses in those eccentric globular-cluster systems, by measuring time dilation (γ) in some cases.

Wide-band observing systems should lead to new results for more pulsars.

Long-term: stay tuned for results from the SKA!

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˙ P b