Timing binary pulsars

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 25

. 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 25

. This method is particularly useful for determining binary pulsar orbits from sparsely sampled data [117

Figure 25: Panel a: Keplerian orbital fit to the 669-day binary pulsar J0407+1607 [236

]. 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

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 (11

) 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

the time dilation and gravitational redshift parameter

the rate of orbital decay due to gravitational radiation

and the two Shapiro delay parameters

and

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 [203