### 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 , projected semi-major orbital axis , orbital
eccentricity , longitude of periastron and the epoch of periastron passage . 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].
Constraints on the masses of the pulsar and the orbiting companion can be placed by
combining and to obtain the mass function

where is Newton’s gravitational constant and is the (initially unknown) angle between the orbital
plane and the plane of the sky (i.e. an orbit viewed edge-on corresponds to ). In the absence of
further information, the standard practice is to consider a random distribution of inclination angles. Since
the probability that is less than some value is , the 90% confidence interval
for is . For an assumed pulsar mass, the 90% confidence interval for can be
obtained by solving Equation (11) for and .
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, ,
, and . 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].