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Astron. Astrophys. 353, 646-654 (2000)

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3. Results

3.1. A distance estimate

In all low resolution spectra the TiO absorption bands from the secondary are easily recognized. We therefore use the calibration method described in Beuermann & Weichhold (1999) to determine the surface flux from the flux difference between 7165 and 7500 Å. In July and Dec. 95 RX J1313.2-3259 was in a low state, so we used the flux differences as observed. For the Mar. 97 data we first estimated the gradient of the cyclotron emission between 7165 Å and 7500 Å by subtracting suitably scaled spectra of the M-dwarf Gl 207.1 and corrected the measured flux differences for that. We also tried to fit spectra of other M-dwarfs (e.g. Gl 205, Gl 352) but with less satisfactory results. From these 35 spectra we obtained a mean value of [FORMULA][FORMULA] erg cm-2 s- 1 Å-1 for the flux difference. This value is reduced by 2% if we exclude all spectra with phases between 0.25 and 0.75, i.e. looking at the unilluminated backside of the secondary. In the following we use a value of [FORMULA][FORMULA] erg cm-2 s- 1 Å-1.

Filling its critical Roche lobe the secondary must have a mean density of [FORMULA] for a system period of 4.19 h. This value depends slightly on the mass ratio but the distance changes by no more than 2% for white dwarf masses between 0.5 and 1.2 [FORMULA]. Assuming the secondary to be a main-sequence star the stellar models of Baraffe et al. (1998) suggest a mass of 0.51 [FORMULA] and a radius of 0.486 [FORMULA], with a slight dependence on mass ratio. However, the secondary, due to its mass-loss history, may be out of thermal equilibrium and therefore have a lower mean density than a main-sequence star of the same mass. Comparing the low-state spectra to those of several M-stars we estimate the spectral type to be M2.5 [FORMULA]. Since the spectral type does not change drastically for stars out of thermal equilibrium (Kolb & Baraffe 1999), we obtain an estimate for the mass of the secondary from a comparison with models of Baraffe et al. (1998): [FORMULA]. Using the relation between spectral type and the surface brightness [FORMULA] by Beuermann & Weichhold (1999) we arrive at a distance of

[EQUATION]

3.2. System parameters

The orbital period of RX J1313.2-3259 was obtained from a sinusoidal fit to the radial velocity measurements of the absorption lines Na I [FORMULA]8183,8195 and K I [FORMULA]7699, obtained in Aug. 92, Feb. 93, and Mar. 97. Possible aliases have been checked against other radial velocity measurements obtained at many different epochs (see Table 1). The zero point for the resulting ephemeris is defined as the blue-to-red zero-crossing of the radial velocity curves. The fits to the narrow emission line components of the Mar. 97 data give a slightly earlier zero point than the absorption line data (difference in phase: [FORMULA]) and we have used the mean of both data sets. The resulting ephemeris is (errors of the last digits are given in brackets):

[EQUATION]

The relatively large error in the zero point reflects the difference in determining the blue-to-red zero-crossing from both the narrow emission and the absorption line measurements. The total time span between our first and last observation amounts to 2667 days or 15276 cycles. Thus the maximum error in the phasing calculated from the error for the period is 0.013, which excludes any errors in the cycle count.

The measurements shown in the two lower panels of Fig. 8 clearly demonstrate that the narrow emission line flux is stronger and the absorption line flux weaker than average when the illuminated side of the secondary is in view of the observer. Such a behavior can be understood with the irradiation model of Beuermann & Thomas (1990). Using the amplitude of [FORMULA] km s-1 measured for the absorption lines the model determines the inclination angle as a function of mass ratio. We have assumed that the contribution to the absorption line flux from the illuminated side is negligible. For primary masses of 0.5 and 1.1 [FORMULA] we obtain inclinations of 52o and 31o, respectively, taking 0.45 [FORMULA] for the mass of the secondary. The corresponding velocities for the emission lines are then 51 and 98 km s-1. Looking at the measured values (Table 2) this would favor the low inclination and so a high primary mass. A low inclination also roughly fits the observed line fluxes, an example is shown in Fig. 8 (middle panel, dotted line). For the inclination of 66o derived below (Sect. 3.3) this is not the case. But if we assume that a substantial contribution to the line flux comes from the unilluminated side of the secondary the situation changes. To reproduce the observed average amplitude of 102.6 km s-1 for the narrow emission lines, [FORMULA]% of the line flux (averaged over the surface of the secondary) must be generated by illumination. We then obtain the line flux variation displayed in Fig. 8 (middle panel, solid line), which fits the observed variation best, except for those measurements where the radial velocity curves of broad and narrow component cross each other and therefore a separation into the two components is difficult to achieve. While this result depends only weakly on the assumed inclination it shows that the observed line flux and velocity amplitude for the narrow emission are consistent with a high inclination if part of the emission line flux is generated on the unilluminated side of the secondary.

3.3. Cyclotron emission

For the data taken during a low state in Dec. 95, we represent the spectral flux in the 14 individual low-resolution spectra by the sum of a Rayleigh-Jeans spectrum, an M-star spectrum, and a cyclotron spectrum. This was done by adjusting the Rayleigh-Jeans to the flux between 4040 and 4080 Å and the M-star template to the flux difference between 7165 and 7500 Å. (As discussed in Sect. 3.1 the best M-star template is that of the M3 star Gl 207.1). Subtracting these two contributions we are left with the cyclotron spectra shown in Fig. 10. The variation of the Rayleigh-Jeans component in the V-band is roughly sinusoidal and can be fitted with a mean of [FORMULA] and an amplitude of [FORMULA], both in units of [FORMULA] erg cm-2 s- 1 Å-1, the maximum occuring at phase [FORMULA]. Such a variation indicates heating of an area around the accretion spot (see Sect. 3.3). The flux of the M-star displays maxima at phases near quadrature. The origin of the these variations must be due to ellipsoidal modulation from the M-star. The flux ratio between phases 0.5 (primary minimum) and 0.25/0.75 (maxima) amounts to [FORMULA].

[FIGURE] Fig. 10. Cyclotron flux extracted from spectra taken in Dec 95. The model fits are discussed in Sect. 3.3, orbital phases are given on the right-hand side.

To model the Dec. 95 variation of cyclotron flux with orbital phase we adapted the emission of an isothermal homogeneous plasma slab (Barrett & Chanmugam 1985) to the data. The input parameters were the magnetic field strength B, the plasma temperature T, the inclination i of the system, the angle [FORMULA] between the rotation axis and the field direction, the angles [FORMULA] and [FORMULA] describing the location of the accretion region (relative to the rotation axis and the line connecting the two stars), and the dimensionless thickness [FORMULA] and the area [FORMULA] of the accretion region. From the phasing of the broad emission lines we took the azimuthal angle between field direction and direction to the companion [FORMULA] to be 17o. The fit shown in Fig. 10 required [FORMULA], [FORMULA], [FORMULA], [FORMULA] = 7o, [FORMULA] = 24o, [FORMULA] = 18o, [FORMULA] = 26, and [FORMULA] cm2 (for a distance of 200 pc). It was obtained in a iterative procedure starting with a coarse grid of cyclotron spectra which allowed to fix B, T, i, and [FORMULA] and then refining the grid for the other parameters. The poor fit at phase 0.915 may be the result of absorption by the accretion stream which is in front of the accretion area around phase 0.95. The rather high inclination can only be made consistent with the results from the irradiation model for a white dwarf mass of 0.39 [FORMULA] (see Sect. 3.5). The values of [FORMULA] and [FORMULA] are in conflict with the assumption of a pure dipolar field configuration because that requires [FORMULA]. Also this simple model does not reproduce the spectral shape at short wavelengths, probably due to a more complicated accretion geometry than used here.

Using the parameters derived above we computed the gravity darkening in the Roche geometry. For the ratio of primary minimum to maximum we obtained a value between 0.87 (without limb-darkening) and 0.83 (50% limb-darkening), in agreement with the value for the ellipsoidal variation deduced above.

For the values of i, [FORMULA], and [FORMULA] the sinusoidal variation of the Rayleigh-Jeans component can be explained assuming a constant contribution from the white dwarf photosphere plus a varying contribution from the accretion area. The fluxes in the V-band then amount to 2.8 and 0.85, respectively, both in units of [FORMULA] erg cm-2 s- 1 Å-1. At the derived distance the constant flux gives an absolute magnitude of [FORMULA] = [FORMULA], corresponding to a photospheric temperature of [FORMULA] 14 000 K. The flux ratio of the variable to constant component in an accretion area occupying a fraction f on the white dwarfs surface requires a temperature which is a factor [FORMULA] higher than the temperature of the white dwarf, in reasonable aggreement with the results of Gänsicke et al. (1999).

We also tried to fit the spectropolarimetric data from May 98, which do not suffer from the uncertainties introduced by the subtraction of other flux contributions. But because of the incomplete orbital coverage we can only state that magnetic field strength, inclination, and field direction are similar, while the thickness [FORMULA] must be somewhat larger to produce the high circularly polarized flux at 3950 Å around phase 0.4 (see Fig. 9). We plan to repeat these observations, which will then allow us to check the system parameters derived above. Cyclotron spectra extracted from the Mar. 97 data show similar variations with phase as the data of Dec. 95, but the removal of other flux contributions introduced too large an uncertainty, so we did not try to fit cyclotron spectra to that dataset.

These considerations also allow us to qualitatively understand the behavior of the V-flux variations (Fig. 6). In a low state both the cyclotron flux and the ellipsoidal variations of the secondary cause a light curve with two maxima during one orbit, while in a high state the contribution of the secondary is negligible and the cyclotron flux contribution most likely comes from a more extended accretion area. The minimum around phase 0.5 which resulted from a viewing angle close to 90o will then be filled in because of contributions from other parts of the accretion area at lower viewing angles.

3.4. X-ray emission

We now turn to the X-ray light curves and spectra obtained during the RASS and the two ROSAT PSPC and HRI pointed observations. Since the inclination is larger than the field line direction it is expected that at least part of the X-ray flux is absorbed by the stream of matter towards the white dwarf. This explains the low flux around phase 0.95 both in the RASS and to some degree in the HRI pointing. The reduction in the count rate by a factor 50 between the RASS and the PSPC pointing and the corresponding change in the hardness ratio [FORMULA] from -0.96 to 0.28 could partially be the result of a lower blackbody temperature. Although the best fit temperature for the pointing is 50 eV ([FORMULA]/d.o.f. 22/10), with a reduction of the blackbody flux by a factor 230 compared to the RASS, the fit allows for much higher blackbody fluxes if the temperature would be lower: at 15 eV, the total blackbody flux decreases by a factor 14 only compared to the RASS ([FORMULA]/d.o.f. 27/11). This is of the same order as the reduction in cyclotron flux between the observations in Feb. 93 and Dec. 95 (factor 7.4). The contribution of the bremsstrahlung flux to the PSPC pointed data is reduced by a factor 2.3 compared to the RASS.

3.5. Masses of the components

The masses determined above are in conflict with present understanding of stable mass transfer. For donor stars with masses below [FORMULA] [FORMULA] mass transfer is dynamically unstable for mass ratios M2/M1 [FORMULA] 0.7 (Webbink 1985). The lower limit of 0.38 [FORMULA] for the mass of the secondary (Sect. 3.1) also does not fulfil the criterion for stability. Even if we assume an evolved secondary as in Kolb & Baraffe (1999) we arrive at masses of 0.31 [FORMULA] for the white dwarf and 0.25 [FORMULA] for the M star, again violating the stability criterion. The mass ratio is determined from the observed absorption line velocity and the inclination. So we turn the question around and ask, for which inclination would the stability criterion be satisfied? Taking the derived mass of the secondary we then need a primary mass of 0.64 [FORMULA] and an inclination of 45.3o. That implies a variation of the viewing angle around this value, which in turn causes cyclotron emission lines to be shifted to the blue as compared to our data in Fig. 10. Especially above 8000 Å the spectral flux should strongly increase with wavelength for all phases due to the presence of the next lower harmonic, which in Fig. 10 only marginally shows up around phase zero. Therefore with our data we can see no remedy to the present situation. So the stability problem makes RX J1313.2-3259 an interesting system for future observations in four aspects: a) the contribution of the secondary star to the spectrum, best observable during a low state, b) the absorption line flux variation relevant for modelling the illumination, c) the degree of circular polarization as a function of orbital phase, and d) the white dwarf contribution to the spectrum, best observable in the ultraviolet regime.

Finally, we note that constraining the mass of the white dwarf is important for our understanding of the formation of cataclysmic variables, because with the period determined above models of common envelope evolution predict that the mass of the white dwarf should exceed 0.6 [FORMULA] (de Kool 1992).

3.6. Mass accretion rates

For an estimate of the mass accretion rate one has to sum up all flux contributions from the accretion area. Although the optical and X-ray observations are not simultaneous we take the optical observations in Feb. 93 and the X-ray observations during the RASS as representative of the high state and those of Dec. 95 in the optical and July 92 in X-rays of the low state. To extend the observations to the whole frequency range we used the model fits for cyclotron emission, blackbody radiation, and thermal bremsstrahlung, respectively. The main uncertainty in these values is caused by the uncertainty in the blackbody temperature. Therefore we took the [FORMULA] upper and lower limits of 68 and 41 eV (see Fig. 3) for the RASS data and assumed temperature limits of 15 and 50 eV for the PSPC data (see Sect. 3.4), to provide a range for the derived fluxes. The results are summarized in Table 3. From the total fluxes we computed the mass accretion rates for a distance of 200 pc, a white dwarf mass of 0.4 [FORMULA], and a white dwarf radius of [FORMULA] cm.


[TABLE]

Table 3. Integrated fluxes from the accretion area and mass accretion rates. The units are [FORMULA] erg cm- 2 s-1 for the fluxes and [FORMULA] [FORMULA] yr-1 for the mass accretion rates.


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© European Southern Observatory (ESO) 2000

Online publication: December 17, 1999
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