4.4 Timing binary pulsars

For binary pulsars, the timing model introduced in Section 4.2 needs to be extended to incorporate the additional motion of the pulsar as it orbits the common centre-of-mass of the binary system. Describing the binary orbit using Kepler’s laws to refer the TOAs to the binary barycentre requires five additional model parameters: the orbital period Pb, projected semi-major orbital axis x, orbital eccentricity e, longitude of periastron ω and the epoch of periastron passage T0. This description, using five “Keplerian parameters”, is identical to that used for spectroscopic binary stars. Analogous to the radial velocity curve in a spectroscopic binary, for binary pulsars the orbit is described by the apparent pulse period against time. An example of this is shown in Panel a of Figure 25View Image. Alternatively, when radial accelerations can be measured, the orbit can also be visualised in a plot of acceleration versus period as shown in Panel b of Figure 25View Image. This method is particularly useful for determining binary pulsar orbits from sparsely sampled data [117Jump To The Next Citation Point].
View Image

Figure 25: Panel a: Keplerian orbital fit to the 669-day binary pulsar J0407+1607 [236Jump To The Next Citation Point]. Panel b: Orbital fit in the period-acceleration plane for the globular cluster pulsar 47 Tuc S [117].

Constraints on the masses of the pulsar mp and the orbiting companion mc can be placed by combining x and Pb to obtain the mass function

4π2 x3 (mc sin i)3 fmass = -----2-= ----------2, (11 ) G Pb (mp + mc )
where G is Newton’s gravitational constant and i is the (initially unknown) angle between the orbital plane and the plane of the sky (i.e. an orbit viewed edge-on corresponds to i = 90 ∘). In the absence of further information, the standard practice is to consider a random distribution of inclination angles. Since the probability that i is less than some value i0 is p(< i0) = 1 − cos(i0), the 90% confidence interval for i is 26∘ < i < 90∘. For an assumed pulsar mass, the 90% confidence interval for mc can be obtained by solving Equation (11View Equation) for i = 26∘ and 90∘.

Although most of the presently known binary pulsar systems can be adequately timed using Kepler’s laws, there are a number which require an additional set of “post-Keplerian” (PK) parameters which have a distinct functional form for a given relativistic theory of gravity [95]. In general relativity (GR) the PK formalism gives the relativistic advance of periastron

( ) −5∕3 ˙ω = 3 Pb- (T⊙M )2∕3(1 − e2)−1, (12 ) 2π
the time dilation and gravitational redshift parameter
( )1 ∕3 Pb- 2∕3 − 4∕3 γ = e 2 π T⊙ M mc (mp + 2mc ), (13 )
the rate of orbital decay due to gravitational radiation
192π ( P ) −5∕3 ( 73 37 ) ( )− 7∕2 P˙b = − ----- --b 1 + --e2 + --e4 1 − e2 T 5⊙∕3mpmcM −1∕3 (14 ) 5 2π 24 96
and the two Shapiro delay parameters
r = T ⊙mc (15 )
( P ) −2∕3 s = x --b T⊙−1∕3M 2∕3m −c1 (16 ) 2π
which describe the delay in the pulses around superior conjunction where the pulsar radiation traverses the gravitational well of its companion. In the above expressions, all masses are in solar units, M ≡ mp + mc, x ≡ apsini∕c, s ≡ sin i and 3 T⊙ ≡ GM ⊙∕c = 4.925490947 μs. Some combinations, or all, of the PK parameters have now been measured for a number of binary pulsar systems. Further PK parameters due to aberration and relativistic deformation [93] are not listed here but may soon be important for the double pulsar [203Jump To The Next Citation Point].
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