5 Fitting the radial velocity curve

Because Vela X-1 is known to have a significant eccentricity, simply fitting a sinusoidal radial velocity curve to the cross-correlation results as a function of time would not be satisfactory, as an eccentric orbit will show deviations from a pure sinusoid. Instead it may be shown that the observed radial velocity is given by:

$$v = \gamma + \frac{4\pi a_1 \sin i}{P(1-e^{2})^{\frac{1}{2}}} \frac{e\cos \omega + \cos(\nu + \omega)}{2}$$ (10)

where a₁ is the semi-major axis of the orbit of GP Vel, e is the orbital eccentricity, P is the orbital period, $\nu$ is the true anomaly (i.e. the angle between the major axis and a line from the star to the focus of the ellipse), $\omega$ is the periastron angle (i.e. the angle between a line from the centre of the orbit to periastron and a line from the centre of the orbit to the ascending node), i is the angle between the normal to the plane of the orbit and the line-of-sight and $\gamma$ is the radial velocity of the centre of mass of the binary system. The true anomaly is related to the eccentric anomaly, E, by:

$$\tan\left(\frac{\nu}{2}\right)=\left(\frac{1+e}{1-e}\right)^{\frac{1}{2}} \tan \left(\frac{E}{2} \right)$$ (11)

where E is the angle between the major axis of the ellipse and the line joining the position of the object and the centre of the ellipse. In turn, E can be related to M, the mean anomaly, by Kepler's Equation:

$$E-e \sin E = M = \frac{2 \pi}{P}(t-T_0)$$ (12)

where the right hand side of this equation is simply the orbital phase with T₀ the time of periastron passage. This equation can be solved numerically, or using Bessel functions.

The value of T₀ was obtained from the value of $T_{\pi/2}$, the epoch of 90$^{\circ}$ mean orbital longitude, derived by Bildsten et al. (1997) from BATSE pulse timing data. They give $T_{\pi/2} = {\rm MJD}~48~895.2186 \pm 0.0012$, from which

$$T_0 = T_{\pi /2} + \frac{P \left( \omega - \frac{\pi}{2} \right)}{2\pi}$$ (13)

where $\omega = 152.59^{\circ} \pm 0.92^{\circ}$. Hence we calculate the time of periastron passage as $T_0 = {\rm MJD}~48~896.777 \pm 0.009$.

5.1 The first fit

Times, t, were assigned to each radial velocity measurement and Eq. (12) was solved using a numerical grid to calculate the corresponding eccentric anomaly. The true anomaly was then found using Eq. (11), and Fig. 2 shows a plot of radial velocity against true anomaly, with the best fit curve according to Eq. (10). The reduced chi-squared of the fit is $\chi^2_{\rm r} = 3.84$. Scaling the error bars by a factor of 1.96 to reduce $\chi^2_{\rm r}$ to unity, gives the amplitude of the fitted curve as $K_{\rm o} = 21.4 \pm 0.5$ km s^-1. However, we note that, since the use of chi-squared assumes that the errors on all the points are uncorrelated, the uncertainty here is likely to be a gross under-estimate. This will hereafter be referred to as the "first fit''.

Figure 3 shows the residuals to the radial velocity curve in the first fit, plotted against time. It is clear that there are trends apparent in these residuals from night to night and a Fourier analysis shows that the dominant signals are at periods of $9 \pm 1$ d and $2.18 \pm 0.04$ d (Fig. 4). The 9 d signal present in the power spectrum reflects the fact that fixing the zero phase of the radial velocity curve, as implied by the X-ray data of Bildsten et al. (1997) does not provide the best fit to the data. We suggest this effect may be responsible for the "phase-locked'' deviations revealed by Barziv et al. (2001) too. The 2.18 d modulation appears to be relatively stable throughout our two orbits of observations, with an amplitude of around 5 km s^-1, as shown in Fig. 3.

Figure 3: Residuals to the radial velocity curve fit in Fig. 2, plotted against time. (The first fit.) Overlaid is a best-fit sinusoid with a period of 2.18 d.

Figure 4: Power spectra of the residuals to the radial velocity curve fits. The solid curve is the power spectrum of the data in Fig. 3 (the residuals after the first fit) and the dotted curve (offset in the -ve direction by one unit for clarity) is the power spectrum of the data in Fig. 6 (the residuals after the second fit). The highest peaks in the power spectrum of the residuals after the first fit correspond to frequencies of $0.111~{\rm d}^{-1}$ and $0.459~{\rm d}^{-1}$ or periods of 9.0 d and 2.18 d. In each case, window function peaks due to the sampling of the lightcurve have been removed using a 1-dimensional clean (program courtesy H. Lehto).

5.2 The second fit

In order to take account of the correlated residuals from the first fit, another fit to the radial velocity data was performed this time with four free parameters: the $\gamma$ velocity and the $K_{\rm o}$ value as before, plus the amplitude and phase of a sinusoid with a period of 2.18 d. The $K_{\rm o}$part of the fit is sinusoidal as a function of true anomaly, whilst the 2.18 d part of the fit is sinusoidal purely as a function of time. The radial velocity fit according to this prescription, referred to as the second fit, is shown in Fig. 5 and has a reduced chi-squared of 2.40. The improvement over the first fit is therefore clearly apparent. Scaling the error bars by a factor of 1.55 to reduce $\chi^2_{\rm r}$ to unity, gives the amplitude of the fitted curve as $K_{\rm o} = 22.4 \pm 0.5$ km s^-1 and the amplitude of the 2.18 d modulation as $5.4 \pm 0.5$ km s^-1. As with the first fit, we note that the uncertainties here are also likely to be under-estimates. The remaining residuals on the second fit are shown in Fig. 6, and their power spectrum is indicated by the dotted line in Fig. 4. It can be seen that some of the correlated structure in the residuals has been removed by this procedure, although the peak corresponding to a period of $\sim$9 d is still present indicating once again that the zero phase according to the X-ray observations does not provide the best description.

Figure 5: Radial velocity curve for He I, Si III and Si IV lines showing the fitted curve (the second fit) after removal of the 2.18 day signal. Each cluster of radial velocity values represents a single night's data. The fit has a reduced chi-squared of 2.40.

Figure 6: Residuals to the radial velocity curve fit in Fig. 5, plotted against time. (The second fit.)

5.3 Fits to phase binned means

To test whether the deviations from the radial velocity curve displayed by each group of data points are correlated, we subtracted the effect of the 2.18 d period according to the parameters found for it in the second fit, and rebinned the resulting radial velocities into nine phase bins (each $\sim$1 day). The radial velocity fit to these phase-binned means is shown in Fig. 7 and has a reduced chi-squared of 28.3. The errors on the mean data points are calculated as the standard error on the mean from the individual data points that are averaged in each case. The fact that the reduced chi-squared is significantly greater than unity suggests that the deviations are indeed correlated. Scaling the error bars here by a factor of 5.3 to reduce $\chi^2_{\rm r}$ to unity, gives the amplitude of the fitted curve as $K_{\rm o} = 22.6 \pm 1.5$ km s^-1.

Figure 7 clearly shows that a better fit would be obtained if the zero phase of the radial velocity is allowed as a free parameter. This reflects the fact that a $\sim$9 d period is present in the power spectra of the residuals after both the first and second fits (Fig. 4). Allowing the zero phase to vary results in the second fit to the phase binned data shown by the dashed line in Fig. 7. This time the reduced chi-squared is 6.0. Scaling the error bars here by a factor of 2.44 to reduce $\chi^2_{\rm r}$ to unity, gives the amplitude of the fitted curve as $21.2 \pm 0.7$ km s^-1. The shift of the minimum radial velocity with respect to that in the first fit to the pase binned means is a true anomaly of $\Delta \nu = 0.033 \pm 0.007$, corresponding to about 7 hours. We note that this shift in zero phase is significantly greater than the uncertainty implied by extrapolating the X-ray ephemeris of Bildsten et al. (1997) to the epoch of our observations which is only $\sim$0.0006 of a cycle. We conclude that the orbital phase locked deviations remaining in the radial velocities after the first fit to means can therefore mostly be accounted for by a simple phase shift in the radial velocity curve.

Figure 7: Radial velocity curve for He I, Si III and Si IV lines binned into nine phase bins. Each radial velocity value represents two or more night's data. The fit to the phase binned means has a reduced chi-squared of 28.3 and is shown by the solid curve. A second fit to the phase binned means, allowing the zero phase of the radial velocity as a free parameter, is shown by the dashed line and has a reduced chi-squared of 6.0.