Free Access
Volume 577, May 2015
Article Number A55
Number of page(s) 20
Section Stellar structure and evolution
Published online 01 May 2015

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Appendix A: Estimation of the opening angle of the outflow

thumbnail Fig. A.1

Geometry of the hotspot emission. is the normal to the surface of the hotspot and is the normal of the surface , which forms an angle θn with . θflow/ 2 (see text) is equal to θn when the momentum flux integrated in azimuthal angle between 0 and θn equals 68% of the momentum flux integrated over the total hemisphere (between 0 and ).

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The geometry of the escaping stream may be complex, forming a disc-like geometry, and possibly creating a spiral-like structure close to the star (see Sect. 2.2). The geometrical thickness of the disc structure is determined by the opening angle of the outflow. Since the outflow is driven by the radiative flux of momentum pν, we can estimate the geometrical thickness of the outflow by using the dependence of pν on the direction given by θn (angle measured with respect to the direction perpendicular to the hotspot surface; see Fig. A.1).

The total momentum flux, which is normal to an element of surface d that is parallel to the hotspot, is given by (Fig. A.1) (A.1)where Iν is the specific intensity (assumed to be isotropic) originating in the hotspot, is the unit vector normal to the hotspot, c is the speed of light, and dΩ = sinθdθdφ.

To calculate the momentum flux towards an arbitrary angle θn<π/ 2, we introduce the reference frame , where the axis lies in the plane of the hotspot, the axis lies in the plane , and is collinear with , the normal to . We note that θ is measured with respect to . The general expression of Eq. (A.1) for arbitrary angles θn reads (A.2)where is the unit vector pointing in the direction (θ,φ). The absolute value in the second factor of the integrand ensures that radiation traversing the surface from above adds a negative contribution to the total momentum flux. The Heavyside step function accounts for the fact that the hotspot only radiates upwards.

For a fixed angle θn, the vector normal to the hotspot has coordinates (0, − sinθn,cosθn), so that an evaluation of the dot product gives the limits of integration in φ: where the latter expression for φlim,2 results from the symmetry of the problem (the angle φ is 0 above the axis). With these limits, Eq. (A.1) can be expressed as where the first integral (Eq. (A.5)) corresponds to the hemisphere over the surface element in Fig. A.1, and the second integral (Eq. (A.6)) corresponds to the two green parts. The first of the two green parts is subtracted because it lies below the hotspot surface and does not contribute, and the second one makes a negative contribution to the momentum flux because it is below the surface element d. These two regions are identical by symmetry, and they allow us to use the same limits of integration in φ and θ.

Our estimate of the opening angle θflow of the outflow is based on the integral of the momentum flux pν over θn. We define it as the angle that corresponds to a fraction of 0.68 (arbitrarily set; standard 1σ as for a Gaussian distribution) of the total integral, i.e., (A.8)This gives an opening angle of θflow ~ 90°. The “effective” solid angle in which the disc-like outflow spreads is given by: (A.9)where fc is the covering factor that is defined as the fraction of the spherical domain that is covered by gas (Sect. 2.1). This is different from the geometry adopted in Paper I, where the outflow was arbitrarily assumed to be confined in a quarter of a sphere (solid angle equal to π).

Appendix B: Calculation of the effective flux

thumbnail Fig. B.1

Representation of the effective surface of the hotspot over a full orbital period. See text for description.

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In our simulations, the effective radiative flux Feff is given by (B.1)where and are the gainer star and hotspot fluxes, respectively; is the surface area of the star; and is the surface area of the hotspot, i.e. the surface of a calotte with the radius of the base equivalent to the radius of the impacting stream, (B.2)where R and rhs are the star and hotspot radius. Here, is the area of the strip at the equator of the star of height 2rhs (Fig. B.1), (B.3)over which the hotspot flux is averaged. Therefore, the effective flux reads as(B.4)The net flux from the hotspot is thus reduced by the factor (B.5)In our simulations, the source of radiation emits isotropically, so that the total effective luminosity will be (B.6)where .

Appendix C: UV continuum and line variability in W Ser

thumbnail Fig. C.1

Top: continuum in the UV band for W Ser observed with IUE. Black: SWP47439 observation made on 1993-04-07 (20:02:16) at phase Φ = 0.84; red: SWP47457 observation made on 1993-04-10 (21:40:17) at phase Φ = 0.06. Bottom: difference of the two spectra.

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thumbnail Fig. C.2

HST/GHRS spectra of the [Si iv] 1393.76 and 1402.77 Å lines of W Ser at two distinct phases (black: z0lu5108t observation made on 1991-07-12 (05:32:25) at phase Φ = 0.00, primary eclipse; red: z0lu0308t observation made on 1991-07-19 (03:29:13) at phase Φ = 0.5, secondary eclipse).

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W Ser systems are known to show strong line variability in the UV (especially for highly ionised species; see the review by Plavec 1982). In this appendix, we investigate the observed UV spectra for the prototypical system W Ser in the search for variations in the continuum as presented in Sect. 3.2. We also demonstrate the variability of [Si iv] lines for this object.

Figure C.1 shows low-dispersion, large aperture spectra of W Ser in the UV as observed by the International Ultraviolet Explorer (IUE) at two distinct phases. (These phases are estimated thanks to the improved ephemeris of Piirola et al. 2005.) The two spectra have been taken only three days apart, thus limiting the effect of intrinsic variability. The red spectrum is taken between the secondary and the primary eclipses at phase 0.8 and the black one at phase 0.1 during the primary eclipse. We note that at phase 0.1, the absorption in the longer wavelength region of the spectra (λ> 1400 Å) and the Lyman α emission are stronger than during the secondary eclipse, possibly indicating a hot ionised region.

Figure C.2 presents the profiles of the [Si iv] 1393.76 and 1402.77 Å lines as observed with HST/GHRS for W Ser at two different phases: primary eclipse (Φ = 0.0), and secondary eclipse (Φ = 0.5). The same ephemeris has been used. We clearly see an excess of flux in the blue wing at phase Φ = 0.5. The total extent of the wings of both lines reaches 600 km s-1, which is comparable to the escape velocity at the surface of the star (643 km s-1 considering a main sequence gainer star of 1.51 M according to Budding et al. 2004, and following a mass-radius relation for main-sequence stars of the form RM0.8) and with the outflow terminal velocity in our model (850 km s-1). In their attempt to model this line variability, Weiland et al. (1995) found two components, a broad one likely due to the boundary layer of an accretion disc, and a narrow one possibly due to the formation of a hotspot. Further

investigation, especially at different phases, may shed new light on the origin of the [Si iv] emission.

Appendix D: Chemical composition of the outflow

Table D.1 presents the chemical composition of the donor star atmosphere that will be transferred by RLOF onto the accretor’s surface and eventually ejected by the hotspot.

Table D.1

Chemical composition of the outflowing material for the three models computed.

© ESO, 2015

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