4.3 Binary pulsars 

For binary pulsars, the timing model described in § 4.2 needs to be extended to incorporate the additional radial accelerations 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, orbital eccentricity, longitude of periastron and the epoch of periastron passage. Light travel times over binary pulsar systems are typically of the order of several seconds. Since this corresponds to a large number of pulse periods, obtaining a phase coherent timing solution thus requires accurate determination of the orbital parameters from a densely sampled initial set of observations.

The Keplerian formularism is analogous to spectroscopic binary stars where 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 familiar mass function:

 

where G is the universal gravitational constant. Assuming a pulsar mass of , the mass of the orbiting companion can be estimated as a function of the (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 < 26 is only 10%; evaluating the companion mass for this inclination angle constrains the mass range between and (in a probabilistic sense) at the 90% confidence level. The mass distribution of the companions in the present sample discussed in § 2.4 indicates that the companions are white dwarfs, other neutron stars or main sequence stars.

An example of a purely Keplerian orbital solution is shown in Fig.  12 by the fit to a set of pulse period measurements for the binary pulsar J1012+5307. The peak-to-peak variations in period correspond to a change in radial velocity during the 14.5 hour binary orbit of about 46 km s . The eccentricity of the orbit dictates the shape of the orbital curve. In this case and the Doppler shifts are indistinguishable from a sinusoid. For the higher-mass binary systems with e > 0.3 the curve becomes increasingly like a saw-tooth in appearance.

Although many of the presently known binary pulsar systems can be adequately described by a Keplerian orbit, there are a number of systems, including the original binary pulsar B1913+16, which exhibit a number of relativistic effects requiring an additional set of ``post-Keplerian'' parameters [36, 72, 51, 152]. Taylor & Weisberg [151] measured two such parameters from an analysis of timing data for B1913+16 between 1974 and 1981. The first of these parameters is advance of the longitude of periastron, analogous to the general relativistic perihelion advance of Mercury. For PSR B1913+16 this amounts to about 4.2 degrees per year [151], some 4.5 orders of magnitude larger than for Mercury. The second post-Keplerian parameter is related to the gravitational redshift and transverse Doppler shifts in the orbit. Together with the five Keplerian parameters, these two post-Keplerian parameters allow an unambiguous description of the masses of the binary components, together with the inclination angle of the orbit.

  

Figure 12: The variation in apparent pulse period for the binary pulsar J1012+5307 as it moves about the centre of mass every 14.5 hours. Points denote observations of the pulse period, the smooth curve is a Keplerian fit to the data.

The latest published measurements for PSR B1913+16 [152] determine the mass of the pulsar and its (unseen) companion to be and respectively, demonstrating that the companion is most likely another neutron star (§ 2.4). Both these post-Keplerian parameters have now been measured for the two other double neutron star binary systems that will merge within a Hubble time: B1534+12 [166] and B2127+11C [53]. These measurements provide similar, but slightly less precise, determinations of neutron star masses. For a number of other systems, the advance of the longitude of periastron has been measured yielding the total system mass and thus allowing interesting constraints to be placed on the component masses [154, 120].

An important general relativistic prediction for eccentric double neutron star systems is the orbital decay due to the emission of gravitational radiation. Taylor & Weisberg were able to measure this for B1913+16 and found it to be in excellent agreement with the predicted value.

  

Figure 13: Orbital decay in the binary pulsar B1913+16 system demonstrated as an increasing orbital phase shift for periastron passages with time. The general relativistic prediction due entirely to the emission of gravitational radiation is shown by the parabola. (After Taylor & Weisberg 1982).

The orbital decay, which corresponds to a shrinkage of about 1.5 cm per orbit, is seen most dramatically as the gradually increasing shift in orbital phase for periastron passages with respect to a non-decaying orbit. This is reproduced from Taylor & Weisberg's classic paper [151] in Fig.  13 . Further observations of this system demonstrate that the agreement with General Relativity is at the level of about 1% [152]. Hulse and Taylor were awarded the Nobel prize in Physics in 1993 [1] in recognition of the discovery of this remarkable laboratory for testing General Relativity.

For those binary systems which are oriented nearly edge-on to the line-of-sight, a significant delay is expected for orbital phases around superior conjunction where the pulsar radiation is bent in the gravitational potential well of the companion star. This effect, analogous to the solar system Shapiro delay, has so far been measured for two neutron star-white dwarf binary systems: B1855+09 and J1713+0747 [134, 87, 41].

Binary and Millisecond Pulsars D. R. Lorimer (dunc@mpifr-bonn.mpg.de) http://www.livingreviews.org/lrr-1998-10 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de