Appendix B: Normalisation and continuum variability

Comparison of column densities towards the primary as derived from line absorption requires the continuum to be normalised to eliminate continuum variability of the primary - which is mainly due to tidal deformation of the primary, with a possible contribution of X-ray heating of the primary by the X-ray source. It must be realised that normalising the continuum is in principle incorrect when studying line emission: at a certain orbital phase, the emission is due to scattered light originating from parts of the primary that are not necessarily the part from which the observed continuum originates.

For each object, three wavelength intervals were chosen that show no intrinsic variability other than a possible continuum variability. These are listed in Table B.1. To normalise the flux scales of the spectra relative to eachother, each spectrum k was divided by a normalisation factor f_k. This factor was constructed from the spectral points $\{ x_{ki} \}_{k=1}^{ N}$ in the three intervals $\{(A_{j},B_{j})\}_{j=1}^{ 3}$, with interval j having n_j spectral points:

$$f_{k} = \sum_{j=1}^{3} \left[ \left( \frac{ n_{j} }{ \sum_{j=... ...}{ \sum_{k=1}^{N} \sum_{i=A_{j}}^{B_{j}} x_{ki} } \right]\cdot$$ (B.1)

In this way the spectra are corrected for continuum variability as long as the continuum slope remains constant.

Figure B.1: Lightcurves of the UV continuum flux and the UV colours as defined in the text.

The normalisation constants may in principle be used to construct UV continuum lightcurves, after correcting for the change in sensitivity of the detector according to the correction table of Bohlin & Grillmair (1988). The correction procedure does not discriminate between the two resolution modes (Cassatella et al. 1994). Not all spectra can be used to construct a lightcurve. For the HMXBs in low resolution all spectra available are used except for SWP3989 (HDE 226868/Cyg X-1) and SWP1458 and 1459 (Sk-Ph/LMC X-4) that had a large (factor of 3 to 5) offset in flux level. For the HMXBs in high resolution all spectra of HD 77581/Vela X-1 are used, but SWP1476, 1714 and 1972 through 5180 (HD 153919/4U1700-37) are omitted because they were taken through the small aperture causing the loss of an unknown fraction of the total light.

 

 

Table B.1: Intervals (A,B), in Å, used for constructing the continuum lightcurves.

HMXB	Band 1	Band 2	Band 3
Cyg X-1	(1265, 1295)	(1759, 1765)	(1820, 1895)
LMC X-4	(1650, 1670)	(1690, 1710)	(1820, 1895)
SMC X-1	(1605, 1635)	(1746, 1762)	(1820, 1895)
Vela X-1	(1272, 1278)	(1492, 1498)	(1696, 1702)
4U1700-37	(1465, 1470)	(1515, 1518)	(1772, 1777)

To study variability in the spectrum tilt, we define UV colours in the following way:

$$U_{\rm far} = L_1 - L_2 \hspace{10mm} \& \hspace{10mm} U_{\rm mid} = L_2 - L_3$$ (B.2)

where L₁, L₂ and L₃ are the fluxes (normalised to unity) for the wavelength bands 1, 2 and 3 in order of increasing wavelength. Because of the difficulty to find suitable wavelength bands of the best quality for each HMXB, the bands - and therefore the colours - are defined for each HMXB individually.

The UV continuum lightcurves are displayed in Fig. B.1. The $L_{\rm ave}$lightcurve is an average of all points within the 3 wavelength bands. It is not straightforward to determine the errors on the individual points in the lightcurve. The photometric accuracy of IUE is reported to be $\sim$6% at a 95% confidence level (Bohlin et al. 1980). This is probably a conservative estimate, judging from the small degree of scatter. For the error on the colour measurement we adopt the standard deviation of the distribution function of the measured $U_{\rm mid}$ points, which is a very conservative error estimate, and the same value is assigned to the error of the $U_{\rm far}$ colour.

B.1 HDE 226868/Cyg X-1

The lightcurve of HDE 226868/Cyg X-1 (Fig. B.1) has a minimum to maximum amplitude of $\sim$17% (the deviating point at $\phi=0.281$ probably does not contain all the flux of the source). This is consistent with the scatter of similar amplitude in the UV photometry around 1500 and 1800 Å obtained by Wu et al. (1982) with the Astronomical Netherlands Satellite (ANS), but considerably larger than the 8% found by Treves et al. (1980) who analysed only 7 spectra by integrating the entire spectrum between 1250 and 1900 Å. The absence of X-ray eclipses limits the inclination to $i\sim60\hbox{$^\circ$ }$ (Bolton 1975). Hence the deprojected amplitude is even larger, implying a severe deformation of HDE 226868.

The $U_{\rm far}$ colour shows clear orbital modulation similar to the average flux: when the HMXB becomes fainter in the UV, the spectrum becomes redder. This may reflect a lower photospheric temperature of HDE 226868 at the side that faces to and away from Cyg X-1, probably a result of the tidal deformation of the primary (Hutchings 1974). HDE 226868 is possibly filling its tidal lobe during peri-astron of the slightly eccentric orbit ($e\sim0.05$: Bolton 1975). No heating of the photosphere of the primary by the X-ray companion is observed, in agreement with Strömgren photometry by Hilditch & Hill (1974).

B.2 Sk-Ph/LMC X-4

The lightcurve of Sk-Ph/LMC X-4 (Fig. B.1) has a minimum to maximum amplitude of $\sim$19%, and is an improvement over the lightcurve derived from only 15 spectra by van der Klis et al. (1982) (see also Vrtilek et al. 1997). The variable depth of the $\phi =0.5$ minimum was explained by a precessing accretion disk (van der Klis et al. 1982). The precession causes variations in the amount of surface of Sk-Ph that is exposed to X-ray heating, as well as variations in the fraction of Sk-Ph that is eclipsed by the disk. Heemskerk & van Paradijs (1989) confirmed the presence of an accretion disk that precesses with a period of $30.36\,\pm\,0.02$ days, whilst X-ray observations had already revealed a $30.42\,\pm\,0.03$ days period (Pakull et al. 1985); we adopt $P=30.38\,\pm\,0.03$ days. Phase ${\phi}_{\rm prec}=0$ corresponds to the phase when the accretion disk is seen edge-on.

Figure B.2: Continuum lightcurves of Sk-Ph/LMC X-4 for orbital phase bins as indicated in each of the nine frames. The cycle parameters are those of the precessing disk: $\phi _0 \equiv$ JD 2443392.52(2); $P=30{\hbox{$.\!\!^{\rm d}$ }}38$(3). The dotted cosine waves are fits-by-eye to the data (see text).

Lightcurves are created according to the cycle parameters of the precession of the accretion disk, for bins around orbital phases ${\phi}_{\rm orb}=0$, 0.25 & 0.75, and 0.5 (Fig. B.2). For orbital phases ${\phi}_{\rm orb}\sim0$ no clear variability with the disk precession period is seen. For ${\phi}_{\rm orb}\sim0.5$ the luminosity is lowest when the accretion disk eclipses the largest part of Sk-Ph ( ${\phi}_{\rm prec}=0.5$), and highest when the accretion disk is seen edge-on ( ${\phi}_{\rm prec}=0$). This is especially clear when narrower orbital phase bins are taken. At ${\phi}_{\rm orb}\sim0.25$ & 0.75 the luminosity varies in anti-phase with the variation at ${\phi}_{\rm orb}=0.5$. Our results agree well with those of Heemskerk & van Paradijs (1989) from optical data. The amplitude of variability of the UV luminosity at ${\phi}_{\rm orb}=0.5$ is 10% - the same as in the optical.

The $U_{\rm far}$ colour (Fig. B.1) is modulated with the orbital period. The colour behaviour around orbital phase ${\phi}_{\rm orb}\sim0.5$ may be dominated by X-ray heating, causing the UV spectrum to become hotter close to ${\phi}_{\rm orb}=0.5$. The scatter in the $U_{\rm far}$ colour very close to ${\phi}_{\rm orb}=0.5$ may be due to variable obscuration by the precessing disk.

B.3 Sk 160/SMC X-1

The lightcurve of Sk 160/SMC X-1 (Fig. B.1) has a minimum to maximum amplitude of $\sim$17%. In optical lightcurves the $\phi =0.5$ minimum is observed to be weaker than the $\phi =0$ minimum, explained by X-ray heating by the bright X-ray source (Hutchings 1974). Van Paradijs & Zuiderwijk (1977) and Howarth (1982), however, found that X-ray heating is not sufficient and that an emitting accretion disk is required, in agreement with the suggestion that the mass transfer is dominated by Roche-lobe overflow (Hutchings et al. 1977). Despite using more spectra than van der Klis et al. (1982), the orbital phase coverage of our UV lightcurve around $\phi =0.5$ is too poor to answer the question of X-ray heating and accretion disk. An $\sim\!\!60$ day periodicity in the X-ray characteristics of SMC X-1 has been attributed to a precessing accretion disk (e.g. Wojdowski et al. 1998).

B.4 HD 77581/Vela X-1

The lightcurve of HD 77581/Vela X-1 (Fig. B.1) has a minimum to maximum amplitude of $\sim$20% and clear orbital modulation, although Dupree et al. (1980) could not see continuum variability in the 5 low resolution spectra they used. The $\phi =0.5$ minimum is significantly deeper than the $\phi =0$minimum, much alike the visual lightcurve (Zuiderwijk et al. 1977) that has a somewhat smaller amplitude of $\sim$15%. Zuiderwijk et al. argue that HD 77581 is nearly filling its Roche lobe.

The $U_{\rm far}$ colour is modulated with the orbital period, becoming redder around $\phi =0.5$. This may be explained by temperature gradients over the photosphere of HD 77581. The scatter of the $U_{\rm far}$ colour near $\phi =0.5$ may result from variable X-ray heating of the part of the photosphere of HD 77581 that faces Vela X-1, due to the strongly variable X-ray flux.

B.5 HD 153919/4U1700-37

The lightcurve of HD 153919/4U1700-37 (Fig. B.1) is flat within $\sim$4 to 5%, possibly with a marginable maximum at $\phi =0.5$. This is less than the UV amplitude around 1500 and 1800 Å of 8% found by Hammerschlag-Hensberge & Wu (1977) using the ANS. Optical amplitudes are $\sim$4 to 8% (Hammerschlag-Hensberge & Zuiderwijk 1977; van Paradijs et al. 1978). The small amplitude may be due to a large mass ratio of the system and therefore small distortion of HD 153919. The optical lightcurve cannot be described by tidal distortion of HD 153919 alone (van Paradijs et al. 1978). In particular the deepest minimum does not occur at $\phi=0.50$ but at $\phi=0.56$. Possible explanations include extra absorption originating from the mass flow in the system. The UV lightcurves presented here also suggest a minimum near $\phi=0.6$. Hints for the signature of a photo-ionization wake in Strömgren photometry are presented in Appendix C. The $U_{\rm far}$ colour is bluest at orbital phase $\phi =0.5$, which suggests X-ray heating of the part of the photosphere of HD 153919 facing 4U1700-37.