$\begin{figure} \par {\psfig{figure=h2715f15.ps,width=12cm,clip=} }\hfill \end{figure}$

Figure 15: Radial-velocity curve derived from the combined data of this paper and Paper II (top panel). Overdrawn is the Keplerian curve that best fits the nightly averages of the data (solid line), as well as the curve expected if the neutron star has a mass of 1.4 $M_{\odot }$ (dotted line; $K_{\rm opt}=17.5~{\rm km\,s^{-1}}$ ). The residuals to the best-fit are shown in the middle panel. For clarity, the error bars have been omitted. Points taken within one night are connected with lines. In the bottom panel, the residuals averaged in 9 phase bins are shown. The horizontal error bars indicate the size of the phase bins, and the vertical ones the error in the mean. The dotted line indicates the residuals expected for a 1.4 $M_{\odot }$ neutron star.

To obtain the best measure of the radial-velocity amplitude, we combined the radial-velocity measurements of HD 77581 from our blue dataset with those derived from the IUE analysis and those obtained in Paper II from 13 photographic spectra and 40 CCD echelle spectra. We do not use the velocities derived from our red spectra, given their much larger scatter. The resulting radial-velocity curve is shown in Fig. 15. The best-fit radial-velocity amplitude is $K_{\rm opt}=21.7\pm0.8{\rm\,km\,s^{-1}}$ . In Table 3, we compare this with the values inferred in earlier sections from subsets of the data, as well as with those found previously. Clearly, the determinations are all consistent with each other.

7.1 Search for systematic effects with orbital phase

Our approach of obtaining radial velocities over an extended period of time was motivated by the hope that for a given orbital phase the large and highly significant radial-velocity deviations from a Keplerian orbit would average out. In other words, that just by adding measurements with random deviations, the accuracy of the derived radial-velocity amplitude, and thus the mass of the neutron star, would be improved. If there were to be systematic deviations with orbital phase, however, the measured mean radial velocity at a given orbital phase would not necessarily be equal to the Keplerian radial velocity of the star in its orbit.

We can test the above assumption by verifying whether the residuals binned over certain fractions of orbital phase are consistent with zero. In the lower panel of Fig. 15, we show the mean of the residuals determined for 9 phase bins, as well as the associated errors in the mean, calculated from the standard deviation in the phase bins. It is clear that these are not consistent with zero. The root-mean-square deviation of these phase-binned residuals is $2.7~{\rm km\,s^{-1}}$ , while only $1.35~{\rm km\,s^{-1}}$ is expected based on the Monte-Carlo simulations. From those simulations, the likelihood that such a high value occurs by chance, under the assumption of no orbital-phase related systematic effects, is smaller than 0.1%.

Another piece of evidence that systematic, phase-locked effects occur, is that the eccentricity one infers from the optical data is 0.18 (and $\omega=355^\circ$ ). From our simulations, the probability to find a value this much larger than the BATSE value is less than 1%.

Understanding the physical origin of these systematic deviations might allow one to account for them and to further improve the accuracy of the determination of the radial-velocity amplitude. We discuss three possible mechanisms below.

7.2 Wind effects and evidence for the existence of a photoionisation wake

In the orbital phase range 0.45-0.65, strong velocity excursions are observed in the radial-velocity curve obtained for the H $\delta$ line (Sect. 4). In this phase interval, the X-ray source passes through the line of sight of the B supergiant. Inspection of the H $\delta$ line shows that at these orbital phases the absorption profile includes an additional blue-shifted absorption component, similar to what is observed in time series of spectral lines formed in the supergiant's stellar wind (e.g., Kaper et al. 1994). In consequence, the cross-correlation procedure yields a more negative radial velocity for the H $\delta$ line.

The origin of this additional blue-shifted absorption component is likely a so-called photo-ionization wake. The X-ray photons emitted by the compact X-ray source fully ionize the surrounding wind regions, creating an extended Strömgren zone in the stellar wind. The presence of such a zone was predicted by Hatchett & McCray (1977), and can explain the observed strong orbital modulation of the ultraviolet resonance lines of HD 77581 (e.g., Kaper et al. 1993). Due to the high level of ionization of the plasma contained in the Strömgren zone, inside this zone the radiative acceleration of the stellar wind is quenched, leading to low wind velocities within the Strömgren zone. Therefore, at the trailing border of the Strömgren zone a strong shock is formed where the relatively fast ambient flow meets the slow wind that moved through the Strömgren zone. The formation of such a "photo-ionization wake'' is clearly seen in hydrodynamical simulations (Blondin et al. 1990). Additional evidence for the existence of such a structure was found in the X-ray light curve of Vela X-1 (Feldmeier et al. 1996).

On two occassions, in February and May 1996, we obtained spectra of strong optical lines formed in the stellar wind together with some of the (red) spectra that have been used to measure the star's radial velocity. In the period 10-18 February 1996 we monitored the H $\alpha$ line of HD 77581; Fig. 16 shows the dramatic (wind) variability observed in this line. At phase 0.54 a strong, blue-shifted (-200 to $-300~{\rm km\,s^{-1}}$ ) absorption component appears, which at later phases migrates towards more negative velocities. Another blue-shifted absorption component, most probably the remnant of a previous passage of the X-ray source, is seen in the first spectra obtained ( $\phi=0.09$ , 0.20, 0.32). The emission component of the H $\alpha$ profile is variable as well; notice the blue-shifted emission component present at phase 0.3-0.4, most likely caused by emission from the dense photo-ionization wake when the X-ray source is approaching the line of sight. For comparison, Fig. 16 also displays the red spectra obtained within the same nights, where zero velocity corresponds to the rest wavelength (in the heliocentric frame) of the N II line at 5679.56 Å. This line profile clearly demonstrates intrinsic variability which cannot be solely due to orbital motion. However, apart from the enhanced blue-shifted absorption at $\phi=$ 0.54, there does not seem to be an obvious correlation between the variability in the H $\alpha$ and the N II lines. Remarkable is the shallow (incipient emission?) and broad (additional blue-shifted absorption component?) absorption profile observed at phase 0.2.

From May 10 to 16, we monitored the He I 5876 Å line, which is not as strong as the H $\alpha$ line, but is still sensitive to changes in the base of the stellar wind. The systematic changes (Fig. 17) strongly resemble those observed in the H $\beta$ line (Kaper et al. 1994) and are similar to the variations found in H $\alpha$ . Comparison with the red NII spectra shows that at $\phi=0.34$ both the He I and the N II (and Al III) lines are shallower, while at $\phi=0.57$ all lines are deeper. The blue-shifted absorption is enhanced at phases 0.46 and 0.56.

We conclude that the systematic deviation in radial velocity of the H $\delta$ line in the phase interval 0.45-0.65 is caused by the presence of a photo-ionization wake. Weaker spectral lines are much less strongly, though still measurably affected in this phase interval; stronger lines like H $\alpha$ show large deviations in line shape over a much wider range of the orbit. We decided to leave out the radial-velocity measurements based on the H $\delta$ line in our further analysis. In Sect. 7.5, we investigate the impact of the points between phase 0.45 and 0.65 on the determination of the radial-velocity amplitude. It turns out to be small.

7.3 Tidal distortion

Tidal deformation could also cause systematic deviations of the radial-velocity curve, since for a deformed star "center-of-light'' radial-velocity measurements do not necessarily reflect the actual center-of-mass velocity. To investigate the effect this might have on inferred orbital parameters, Van Paradijs et al. (1977b) made model calculations for a star with properties appropriate for HD 77581. For simplicity, they assumed a circular orbit and synchronous rotation of the primary, so that the shape of the star was given by a single Roche equipotential surface at all phases. Using theoretically predicted line profiles, including the Si IV 4089 line, they find that appreciable systematic effects can occur, differing from ion to ion, with the main expected deviations being a positive one around orbital phase 0.1 and a negative one around phase 0.9. As a result, an apparent eccentricity may arise and the radial-velocity amplitude may be overestimated, in unfavourable cases by up to 30%.

The systematic deviations we encounter in the radial-velocity curve behave differently. The larger deviations occur around orbital phases 0.2 and 0.8, and at both phases the deviations are directed towards positive velocities. Thus, it seems the effects of the tidal deformation are not as large as predicted. One should keep in mind, however that the assumptions made by Van Paradijs et al. (1977b) do not correspond to the actual situation in Vela X-1: the orbit is eccentric and HD 77581 rotates sub-synchronously (by a factor of about 2/3, Zuiderwijk 1995). Given the possibly large effects on the inferred parameters, it would be worthwile to reconsider these issues.

7.4 Non-radial pulsations

Apart from distorting the shape of the star, tidal forces can excite (non-radial) pulsations. With a neutron star in a close eccentric orbit (the distance between the neutron star and the non-synchronously rotating surface of HD 77581 is about half a stellar radius) one can expect that tidal waves are excited at the surface of HD 77581 (e.g., Witte & Savonije 1999). The neutron star's orbital frequency may resonate with certain non-radial modes, perhaps phase-locked with the orbit. The properties of these modes depend on the details of the internal structure of the supergiant, which has undergone a complex history of binary evolution, including a phase of mass transfer and a nearby supernova explosion forming the neutron-star companion.

Radial and low-degree non-radial pulsations can change the shape of photospheric absorption line profiles significantly (e.g., Vogt & Penrod 1983). This is due to local temperature variations induced by the pulsations and/or local Doppler shifts related to the 3-dimensional pulsation velocity field.

The question is whether such tidally induced (or self-excited) pulsations, when present in HD 77581, are detectable through inspection of the detailed shape of the line profiles and, if so, could result in systematic deviations in the radial-velocity curve. According to Telting & Schrijvers (1997), low-order pulsation modes (with degree $\ell<$ 3) can lead to detectable radial-velocity variations and changes in the shape of the line-profiles. For instance, moment analyses of the spectral line profiles (Balona 1986; Aerts et al. 1992) have been successfully used to detect low-order modes in single pulsators such as the $\beta$ Cephei stars.

If the presence of non-radial pulsations is the physical cause for the observed deviations in radial velocity, knowledge of the pulsation mode(s) might provide a tool to "correct'' the measured radial velocities and to improve the accuracy of the determination of the Keplerian orbit of HD 77581. However, a mode can only be identified if the time sampling of the data is sufficient to resolve its periodicity (in practice several spectra per hour and continuous - 24 hours per day - coverage). The time sampling of the spectra used in this paper (one spectrum per day) is poor; it is sufficient to cover the change in radial velocity due to orbital motion, but likely insufficient to identify pulsation modes. Another disadvantage of the current dataset is the limited spectral coverage; this excludes the possibility to combine many different spectral lines in order to increase the signal-to-noise ratio and to separate line changes induced by the temperature- and velocity distribution associated with a given pulsation.

A preliminary moment analysis performed on the line profiles comprised in our spectra did not yield a useful result. This is an indication that any putative non-radial pulsation modes are not phase-locked to the orbital motion and/or do not persist for a long period of time. A dataset obtained with better time sampling and spectral coverage is required to further investigate the occurrence of non-radial pulsations in the atmosphere of HD 77581.

7.5 Orbital-phase intervals excluded

$\begin{figure} \par {\psfig{figure=h2715f18.ps,width=8.5cm,clip=} }% \end{figure}$

Figure 18: Results for fits to the radial-velocity curve with phase intervals of width 1/9 (left) and 2/9 (right) excluded. The points in the top panel show the best-fit radial-velocity amplitudes found when the phase range indicated by the horizontal error bars is excluded. The short-dashed contours indicate the 95% and 99% confidence range of the variation expected from our simulations (for which it is assumed that the variability of the star is independent of orbital phase.) The long-dashed lines indicate the best-fit amplitude to all data and the associated 95% confidence uncertainties. The panels in the second row show the root-mean-square deviation of the phase-binned residuals. The short-dashed lines indicate the levels below which 50%, 95%, and 99% of the simulated data are contained. In the bottom two rows of panels, the best-fit values for the eccentricity and periastron angle are shown. The long-dashed lines indicate the accurate values inferred from the BATSE measurements (Table 4), the short-dashed contours the 95% and 99% ranges expected from the simulations.

In the H $\delta$ profile, we found clear evidence for an ionisation wake in the phase interval between 0.45 and 0.65. It may be that the photoionisation wake has a (less obvious) effect on the other spectral lines as well. We can verify to what extent the exclusion of this phase interval would affect the amplitude of the radial-velocity curve (and thereby the measurement of the neutron-star mass). Similarly, in previous studies it was found that the largest deviations occurred near velocity minimum (Paper II and references therein). From Fig. 15, one sees that the phase bin at velocity minimum again has one of the largest and most significant deviations from zero.

In order to see what is the effect of these phase intervals on the solution, we decided to exclude, like in Paper II, all possible one ninth and two ninth cycle wide orbital-phase intervals (i.e., approximately one and two-day wide) from the fit to the radial-velocity curves, starting from orbital phase zero until one with a step of one ninth. Subsequently, we fitted the remaining points and derived the amplitude of the radial-velocity curve, as well as the uncertainty estimated using our Monte-Carlo simulations. The resulting amplitude is shown as a function of the excluded phase intervals in Fig. 18. Also shown is the rms of the phase-binned residuals. If a specific phase interval were responsible for virtually all of the phase-locked deviation, one would expect that the binned residuals would become consistent with zero when that phase bin is excluded. From the figure, one sees that excluding the bins near inferior conjunction of the neutron star, when the ionisation wake is most important, and near velocity minimum have the largest effect on the rms. Excluding inferior conjunction, the inferred radial-velocity amplitude is not affected much, since the velocities are at a zero crossing, while excluding velocity minimum, a much larger velocity amplitude is inferred; indeed, excluding this bin leads to the only change in radial-velocity amplitude whose probability of occurring by chance, as inferred from the simulations, is less than 1%.

7.6 Fitting eccentricity and periastron angle

In another attempt to determine whether any specific bin is responsible for the phase-locked systematic effects, we repeated the above simulations but with this time the eccentricity and the periastron angle as free parameters in the fitting procedure. In principle, of course, one should reproduce the values found from the X-ray analysis. The bottom panels of Fig. 18 show the resulting eccentricity and periastron angle. We find that when early phase intervals are excluded, the inferred eccentricity remains far larger than the true value, measured by Bildsten et al. (1997; Table 4). When phase intervals around velocity minimum are excluded, however, the inferred eccentricity becomes much closer to the true value.

7.7 The uncertainty in the radial-velocity amplitude

The above results suggest that the main systematic effects on the radial-velocity curve occur near inferior conjunction and at velocity minimum. If so, the radial-velocity amplitude inferred from all the data likely is an underestimate of the true value. However, while we believe the velocity deviation at inferior conjunction can be understood - in terms of a photo-ionisation wake - we do not have a good idea for the cause of the other systematic deviations. Lacking this, it is difficult to estimate the uncertainty that is introduced. Probably best is to assume there is no bias to high or low radial-velocity amplitude, and base the estimate on the increase in the uncertainties required to match the observed root-mean-square deviations of the phase-binned residuals. This is about a factor two, and corresponds to a similar increase in the uncertainty on the final radial-velocity amplitude. In consequence, our final estimate is not much more precise than values given previously, in which the influence of systematic effects with orbital phase was not taken into account. We conclude that our best present estimate is $K_{\rm {}opt}=21.7\pm1.6~{\rm km\,s^{-1}}$ .