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Neutron Star (Mostly Pulsar) Masses. Ingrid Stairs UBC Vancouver CAWONAPS TRIUMF Dec. 9, 2010. Pulsars are one manifestation of neutron stars: spin periods ~ 1.4 ms to 8.5 seconds, magnetic fields ~10 8 to ~5 x10 13 G. Low fluxes mean large radio telescopes - PowerPoint PPT Presentation
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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