8 Discussion

Our final value for $K_{\rm o}$ of $22.6\pm1.5$ km s^-1 is slightly higher than, but comparable to, those quoted by other recent authors, noted in Sect. 2. Our result for the neutron star mass is compatible with van Kerkwijk et al.'s (1995) mass value of $M_{\rm x}=1.9^{+0.7}_{-0.5}~M_{\odot}$, and that of Barziv et al. (2001), namely $M_{\rm x}=1.86 \pm 0.32~M_{\odot}$. We therefore support the view that the neutron star in Vela X-1 is more massive than the canonical value of 1.4 $M_{\odot}$. However, there are a number of potential sources of systematic error, as noted in our earlier paper concerning the mass of the neutron star in Cen X-3 (Ash et al. 1999), some of which will be examined below.

8.1 Stellar wind

As GP Vel is an early-type star, it has a significant stellar wind, and a strong and variable stellar wind can have disastrous consequences when making radial velocity measurements, especially over an extended period of time. Barziv et al. (2001) made a detailed study of the effects of winds on their observations and showed that deviations in the Balmer lines in particular indicated the presence of a photo-ionization wake. We note that the supergiant GP Vel is of later type (B0) than that of Cen X-3 (O6-7), and thus has a weaker wind. Additionally, as our observations were made over a relatively short time period, the potential effect of long-term wind variation will be less than if the observations were spread over a number of years, as in the case of our Cen X-3 campaign, or over a year, as in the case of Barziv et al's Vela X-1 campaign. In other words, although the effect of the supergiant's stellar wind cannot be discounted, its effect should be less than its effect on our Cen X-3 results and less than on the results of Barziv et al. (2001).

The apparent increase in amplitude of the radial velocity curve seen with increasing order in the Balmer series lines mentioned earlier, may also be an indication of a stellar wind. Conceivably, the different order lines have different contributions from the wind material as it moves further out, and this could produce the correlation seen. Since we have not used the Balmer lines in our analysis however, this effect does not influence our final result.

8.2 X-ray heating

The effect of X-ray heating of the surface of GP Vel may be compared with the corresponding effect in Cen X-3. The ratio of intrinsic to irradiated fluxes may be estimated for each system as $(L_{\rm o} / 4\pi R_{\rm o}^2) / (L_{\rm x} / 4\pi (a - R_{\rm o})^2)$. The RXTE ASM lightcurves show that both sources have similar X-ray fluxes, but Cen X-3 is at over three times the distance of Vela X-1 (Sadakane et al. 1985; Humphreys & Whelan 1975) so its X-ray luminosity is at least a factor of ten higher. The luminosities of the supergiant stars in both systems are similar, but the radius of the star in Cen X-3 and the separation of the supergiant and neutron star are both around three times smaller in Cen X-3 (Ash et al. 1999) than in Vela X-1. Hence the effect of X-ray heating in Vela X-1 is of order ten times less than in Cen X-3. We were able to rule out significant X-ray heating in the case of Cen X-3 (Ash et al. 1999), and the effect should therefore be even less in this case.

8.3 Deviations from Keplerian radial velocity curve

Clearly, another source of uncertainty in the case of GP Vel arises from the observed deviations from a pure Keplerian radial velocity curve. van Kerkwijk et al. (1995) suggested that these deviations do not last longer than a single night, although our results (Fig. 3) suggest that they may last for up to two system orbits. They suggested that the deviations are the result of short-lived, high-order pulsations of the photosphere and we confirm that the frequencies appear to be harmonically related to the orbital frequency. It is possible therefore that they do represent tidally induced oscillations. As well as introducing another source of error into our attempts to measure the radial velocity of the centre of mass of the supergiant, the "wobbles'' cause another problem. If the shape of the star is constantly changing, this will affect the extent to which the star fills its Roche lobe. Additionally, the size and shape of the Roche lobe itself will vary with phase, as the two components orbit each other in eccentric orbits. We believe that we have partially accounted for this effect by removing the 2.18 d modulation shown in Fig. 3.

However, as Fig. 7 demonstrates, there is clearly a remaining effect which may be characterised by an additional radial velocity variation at the orbital period. This manifests itself as a phase shift in the radial velocity curve with respect to the X-ray orbital ephemeris. The measured amplitude of the radial velocity curve is therefore a combination of the orbital motion of GP Vel and an additional component that may be due to a non-radial pulsation induced by the eccentric orbit of the neutron star. There may even be another component of this non-radial pulsation at the orbital period that is in phase with the orbital motion and so undetectable. Our $K_{\rm o}$value, and presumably all previous measurements too, are therefore subject to an unknown systematic error, and the stellar masses calculated from them are also uncertain because of this.

Another possibility for the origin of the extra velocity component at the orbital period is large-scale motion required to keep the tidal bulge pointing in the direction of the neutron star given the asynchronous stellar rotation ( $\Omega = 0.67$). However, such a tidal bulge would extend in both directions - towards and away from the neutron star - and so the effect may cancel out.

As a further result, our identification of these departures from the radial velocity curve in GP Vel suggest that it may be the third recent candidate to exhibit tidally induced non-radial oscillations, after HD 177863 (De Cat 2001; Willems & Aerts 2002), and HD 209295 (Handler et al. 2002).

8.4 Neutron star mass

The minimum neutron star mass consistent with our results is around 1.74 $M_{\odot}$. A recent review of neutron star equations of state by Lattimer & Prakash (2000) shows that such a mass would exclude the schematic potential models of Prakash, Ainsworth and Latimer, and the various models based on a field theoretical approach by Glendenning & Moszkowski, by Glendenning and Schaffner-Bielich, and by Prakash et al. Hence the "softer'' equations of state that incorporate components such as hyperons, kaons and quark matter are ruled out.

In order to be consistent with a neutron star mass of $\sim$1.4 $~M_{\odot}$, the radial velocity amplitude of GP Vel would have to be $K_{\rm o} \sim 18$ km s^-1, implying that the additional velocity component due to the non-radial pulsation has an amplitude of around $\sim$4.5 km s^-1.

We conclude by noting that there are two large sources of uncertainty in deriving the masses of the stellar components. The first is that of the value $\beta$ (the ratio of the radius of the companion star to its Roche lobe radius). This in turn leads to a large uncertainty in $\sin i$ (and hence the masses) and is unlikely to be reduced even with optical spectroscopic observations of greater signal-to-noise. The second source of uncertainty is the additional radial velocity component at the orbital period that we have discovered and represented by a phase shift. Removing the effect of such a velocity component is likely to require detailed modelling of tidally excited oscillation modes in GP Vel which in turn requires an accurate knowledge of the stellar parameters. Consequently, the mass measurement presented here may represent the best that is achievable.

Acknowledgements

We thank the Mount Stromlo Observatory staff for maintaining the 74-inch telescope. The data analysis reported here was carried out using facilities provided by PPARC, Starlink and the OU research committee. HQ was employed under PPARC grant number GR/L64621 during the course of this work. The authors would also like to thank Deepto Chakrabarty for his help, and the referee, Marten van Kerkwijk, for his many detailed and constructive suggestions which have helped to significantly improve the paper.