## 4.4 Binary pulsars and Kepler's laws

For binary pulsars, the simple timing model introduced in §  4.2 needs to be extended to incorporate the additional radial acceleration of the pulsar as it orbits the common centre-of-mass of the binary system. Treating 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 (, see below), orbital eccentricity (e), longitude of periastron () and the epoch of periastron passage (). This description, using five ``Keplerian parameters'', is identical to that used for spectroscopic binary stars.

For spectroscopic binaries the orbital velocity curve shows the radial component of the star's velocity as a function of time. The analogous plot for pulsars is the apparent pulse period against time. Two examples are given in Fig.  22 .

Figure 22: Orbital velocity curves for two binary pulsars. Left: PSR J1012+5307, a 5.25-ms pulsar in a 14.5-hour circular orbit around a low-mass white dwarf companion [177, 256, 125]. Right: PSR J1811-1736, a 104-ms pulsar in a highly eccentric 18.8-day orbit around a massive companion (probably another neutron star) [147].

Constraints on the mass of the orbiting companion can be placed by combining the projected semi-major axis and the orbital period to obtain the mass function:

where G is the universal gravitational constant. Assuming a pulsar mass of (see below), the mass of the orbiting companion can be estimated as a function of the (initially unknown) angle i between the orbital plane and the plane of the sky. The minimum companion mass occurs when the orbit is assumed edge-on (). For a random distribution of orbital inclination angles, the probability of observing a binary system at an angle less than some value is . This implies that the chances of observing a binary system inclined at an angle is only 10%; evaluating the companion mass for this inclination angle constrains the mass range between and at the 90% confidence level.

 Binary and Millisecond Pulsars at the New Millennium Duncan R. Lorimer http://www.livingreviews.org/lrr-2001-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de