6 Calculation of system parameters

The masses of the two stellar components were calculated according to the method described in Sect. 1. In order to propagate the uncertainties, a Monte Carlo approach was adopted, calculating 10⁶ solutions with input values drawn from a Gaussian distribution within the respective 1$\sigma$ uncertainties of each parameter. The final values were then calculated as the mean result of these 10⁶ solutions, with the 1$\sigma$ uncertainties given by the rms deviation in each case.

The procedure was carried out using $K_{\rm o} = 22.6 \pm 1.5$ km s^-1 from the first fit to means and the results of these calculations are shown in Table 2. We used the value of $K_{\rm x} = 278.1 \pm 0.3$ km s^-1 (Bildsten et al. 1997), to calculate q using Eq. (1) and a value of $33^{\circ}.8\pm1.3^{\circ}$ was adopted for the eclipse half-angle, $\theta_{\rm e}$ from Watson & Griffiths (1977). We note that this value was also used by Wilson & Terrel (1998) for their unified analysis of Vela X-1 and appears to be the most reliable. A value of $0.67\pm0.04$ was adopted for the co-rotation factor, $\Omega$, obtained from comparison of GP Vel spectra with model line profiles by Zuiderwijk (1995). We note that Barziv et al. (2001) also used the $v \sin i$ value derived by Zuiderwijk (1985) and obtained a similar value, namely $\Omega (=f_{\rm co}) = 0.69 \pm 0.08$.

Values for $\beta$ and i cannot be obtained from independent observations. However, since the supergiant is unlikely to be overfilling its Roche lobe, an upper limit to the Roche lobe filling factor is $\beta \leq 1.0$; and an upper limit to the inclination is clearly $i \leq 90^{\circ}$. A lower limit to $\beta$ is found by setting $i=90^{\circ}$, and a lower limit to i is found by setting $\beta = 1$. These limits are indicated in Table 2. We note though that $\beta$ is unlikely to have a fixed value, as the size and shape of the Roche lobes of both components will vary as the separation between the stars varies as they orbit each other eccentrically. Indeed, it has been suggested that GP Vel is likely to fill its Roche lobe at periastron (Zuiderwijk 1995). However the mass ratio implies that Roche lobe overflow would be dynamically unstable and lead to a common-envelope phase, so the limit of $\beta = 1$ is probably implausible. We also note that Barziv et al. (2001) took a somewhat different approach and assumed a filling factor at periastron in the range $\beta = 0.9 {-} 1.0$, thus leading to a minimum value for the inclination of $73^{\circ}$, consistent with our analysis.

Given these two limits on each of $\beta$ and i, Eqs. (2) and (3) then allow us to calculate the masses of the two components for combinations of i and $\beta$ between the two extremes. The limiting values for the masses of the two stars are shown in Table 2. Values between these extremes, as a function of inclination angle, are shown in Fig. 8. With these two limiting situations, we also calculate the semi major axis a of the orbit from Kepler's law, and this is then used to calculate the separation of the stars at periastron according to $a^{\prime} = a (1-e)$. The Roche lobe radius at periastron $R_{\rm L}$ is then determined from the value of $R_{\rm L}/a^{\prime}$ at each extreme, and the radius of the companion star $R_{\rm o}$ is then calculated as $\beta \times R_{\rm L}$ in each case. These values too are shown in Table 2. For each parameter, the actual value may be anywhere within the range encompassed by the two limits.

 

 

Table 2: System parameters for Vela X-1 / GP Vel. The value for $K_{\rm o}$ is that resulting from fitting the phase bin means without a phase shift.

Parameter	Value		Ref.
Observed
$a_{\rm x} \sin i$ / light sec	$113.98 \pm 0.13$		[1]
P / d	$8.964368 \pm 0.000040$		[1]
$T_{\pi/2}$ / MJD	$48895.2186 \pm 0.0012$		[1]
e	$0.0898 \pm 0.0012$		[1]
$\omega$ / deg	$152.59 \pm 0.92$		[1]
$\theta_{\rm e}$ / deg	$33.8 \pm 1.3$		[2]
$\Omega$	$0.67\pm0.04$		[3]
$K_{\rm o}$ / km s-1	$22.6\pm1.5$		[4]
Inferred
$K_{\rm x}$ / km s-1	$278.1 \pm 0.3$
T0 / MJD	$48896.777 \pm 0.009$
q	$0.081 \pm 0.005$
$\beta$	1.000	$0.89 \pm 0.03$
i / deg	$70.1 \pm 2.6$	90.0
$M_{\rm x}$ / $M_{\odot}$	$2.27 \pm 0.17$	$1.88 \pm 0.13$
$M_{\rm o}$ / $M_{\odot}$	$27.9 \pm 1.3$	$23.1 \pm 0.2$
$a^\prime$ at periastron / $R_{\odot}$	$51.4 \pm 0.8$	$48.3 \pm 0.3$
$R_{\rm L}$ at periastron / $R_{\odot}$	$32.1 \pm 0.6$	$30.2 \pm 0.2$
$R_{\rm o}$ / $R_{\odot}$	$32.1 \pm 0.6$	$26.8 \pm 0.9$

[1] Bildsten et al. (1997).
[2] Watson & Griffiths (1977).
[3] Zuiderwijk (1995).
[4] This paper.

Figure 8: The masses of the neutron star (upper panel) and companion star (lower panel) in solar units, as a function of inclination angle. The lower limit to the inclination angle is set by the condition that the companion star cannot over-fill its Roche lobe. Error bars on the masses at the extreme limits are determined as explained in the text.