1 Introduction

Binary systems containing an X-ray pulsar are very important astrophysically as they can offer a direct measurement of the neutron star mass. The mass ratio of the system, q, is simply given by the ratio of the radial velocity semi-amplitudes for each component:

$$q = \frac{M_{\rm x}}{M_{\rm o}}=\frac{K_{\rm o}}{K_{\rm x}}$$ (1)

where $M_{\rm x}$ is the neutron star mass, $M_{\rm o}$ is the mass of the optical component, $K_{\rm o}$ is the semi-amplitude of the radial velocity of the optical component, and $K_{\rm x}$ is the semi-amplitude of the radial velocity of the neutron star. In addition, for an elliptical orbit, it can be shown that,

$$M_{\rm o} = \frac{K_{\rm x}^{3}P \left( 1-e^2 \right)^{\frac32}} {2\pi G \sin^{3} i} \left( 1+q \right)^{2}$$ (2)

and similarly,

$$M_{\rm x} = \frac{K_{\rm o}^{3}P \left( 1-e^2 \right)^{\frac32}} {2\pi G \sin^{3} i} \left( 1+\frac{1}{q}\right)^{2}$$ (3)

where i is the inclination of the orbital plane to the line of sight, e is the eccentricity and P is the period of the orbit.

We therefore have a means of calculating the mass of the neutron star if the orbits of the two components and the inclination of the system are known. Such a combination is possible in an eclipsing X-ray binary system, in which the neutron star is a pulsar. X-ray pulse timing delays around the neutron star orbit yield the value of $K_{\rm x}$, and conventional radial velocity measurements from optical spectra yield $K_{\rm o}$.

A value for i can be obtained from the following approximations:

$$\sin i \approx \frac{\left(1 - \beta^{2}\left(\frac{R_{\rm L}}{a^{\prime}}\right)^{2}\right)^{\frac{1}{2}}} {\cos \theta_{\rm e}}$$ (4)

$$\frac{R_{\rm L}}{a^{\prime}} \approx A + B \log q + C \log^{2} q$$ (5)

where $R_{\rm L}$ is the radius of the optical companion's Roche lobe, $\beta = R_{\rm o}/R_{\rm L}$ is the ratio of the radius of the optical companion to that of its Roche lobe, $a^\prime$ is the separation between the centres of masses of the two components, and $\theta_{\rm e}$ is the eclipse half-angle. A, B, and C have been determined by Rappaport & Joss (1983) to be:

$$\approx 0.398 - 0.026 \Omega^{2} + 0.004 \Omega^{3}$$                              A (6)

$$\approx -0.264 + 0.052 \Omega^{2} - 0.015 \Omega^{3}$$ B (7)

$$\approx -0.023 - 0.005 \Omega^{2}$$ C (8)

where $\Omega$ is the ratio of the rotational period of the companion star to the orbital period of the system. Note that, whilst $R_{\rm L}/a^{\prime}$ is expected to be constant for a given system, $R_{\rm L}$, $a^\prime$and $\beta$ will vary around the orbit for a system which has an appreciable eccentricity.

Only seven eclipsing X-ray binary pulsars are known (namely Her X-1, Cen X-3, Vela X-1, SMC X-1, LMC X-4, QV Nor and OAO1657-415). Orbital parameters for the first six of these are still relatively poorly known, whilst the counterpart to OAO1657-415 has only recently been identified (Chakrabarty et al. 2002) and no optical radial velocity curve has been measured. In addition, the mass of the neutron star in a seventh eclipsing X-ray binary (4U1700-37) has recently been determined by Clark et al. (2002). This system does not contain an X-ray pulsar though, and the mass determination is based on a Monte-Carlo modelling method which relies on the spectral type of the companion and is thus highly uncertain. Vela X-1 is the only one of these systems to have an eccentric orbit, and apart from OAO1657-415 is the one with the longest orbital period. For these reasons, and others discussed below, an accurate determination of the mass of the neutron star in Vela X-1 has always been difficult to obtain.