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Astron. Astrophys. 364, 587-596 (2000)

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4. Results

The number of free parameters needed to define a disk model is large enough to make extensive mapping of the parameter space practically impossible. Some of these parameters can not be deduced from observations or constrained theoretically. Fortunately, most of the parameters do not influence the polarization levels when physical values are used. Simulated disk models are listed in Table 1. The errors of all axisymmetric simulations are smaller than [FORMULA] for inclinations above [FORMULA] and smaller than [FORMULA] for inclinations below [FORMULA]. Error bars have been left out of the figures for clarity.


Table 1. Parameters of simulated disk models

The parameters of the disks modeled in Fig. 1 are [FORMULA], corresponding to standard viscously heated disk, [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. The disks are thin, and radiation is generated not very near the disk surface. [FORMULA] parameter (vertical optical depth at the inner disk) varies between 10 000 and 2000. As the vertical optical depth decreases outward, the lowest value corresponds to a disk which is optically thin in the outer parts. As the emissivity of the disk is proportional to [FORMULA], the contribution of the outer disk to the radiation is negligible. Consequently, the outer disk radius has no effect in polarization. The results shown in Fig. 1 clearly show that the vertical optical depth of the disk has no large effects on polarization. Therefore disk mass or accretion rate can not be estimated from polarization observations. The polarization levels are easily observable for brightest LMXBs, if their inclination is above [FORMULA]. The disk rim is not modelled correctly, as the disk is only cut off at a sufficiently large radius. Therefore our simulations have only marginal significance for eclipsing systems, where a large fraction of the observed optical flux comes from the outer disk. The simulations are more relevant to medium and low inclination systems ([FORMULA]), where the inner disk produces most of the observed flux.

[FIGURE] Fig. 1. Polarization from a thin, viscous disk. Vertical optical depth [FORMULA] at the inner disk from the left: 10000, 5000, 3000 and 2000. Solid line: only single scattering, dashed line: double scattering included, dotted line: multiple scattering included.

Fig. 2 shows results for a viscous disk where the disk scaleheight is varied. The fixed disk parametres are [FORMULA] and [FORMULA]. The disk scaleheight varies from [FORMULA] to [FORMULA]. The effect of increasing thickness is observable at high inclinations, where the disk rim modelling should be more detailed to produce reliable results. No effect in polarization levels is seen at lower inclinations. In Fig. 3 and Fig. 4 results for irradiated (X-ray heated) disks, are shown. These disks are similar to the viscous disks in Fig. 2, but the radiation is generated closer to the disk surface. The radial emissivity profile of disks in Fig. 3 is the same as those in Fig. 2 ([FORMULA]), whereas for Fig. 4 the emissivity profile is less steep ([FORMULA]). The hybrid models of Fig. 3 with radial temperature distribution of a viscous disk and vertical emissivity distribution of an irradiated disk were simulated to allow separation between the effects of both radial and vertical emissivity variations on polarization. The optical luminosity of a real disk is a sum of the released gravitational potential energy and the energy of the compact object X-rays absorbed in the disk surface layers, so the real radial dependence of emissivity is roughly [FORMULA], so the value of [FORMULA] is between -1/2 and -3/4, and [FORMULA]. The somewhat arbitrary division between different emissivity profiles is motivated by the fact Stokes parameters are additive. Therefore results of the simulations are also additive, if parameters [FORMULA] and [FORMULA] are changed and other parameters, which describe the distribution of the electrons, are kept constant. Specially, simulations of different emission mechanisms, such as those presented in Fig. 2, Fig 3 and Fig. 4, can therefore be combined to give a more realistic polarization model.

[FIGURE] Fig. 2. Polarization from a viscous disk. Disk parametres are [FORMULA] and [FORMULA]. The disk scaleheights are from the left [FORMULA], [FORMULA] [FORMULA] and [FORMULA] Solid line: only single scattering, dashed line: double scattering included, dotted line: multiple scattering included.

[FIGURE] Fig. 3. Polarization from a lightly irradiated disk. Disk parametres are [FORMULA] and [FORMULA]. The disk scaleheight H is from the left: [FORMULA], [FORMULA], [FORMULA], and [FORMULA]. Solid line: only single scattering, dashed line: double scattering included, dotted line: multiple scattering included.

[FIGURE] Fig. 4. Polarization from a heavily irradiated disk. Disk parameters are [FORMULA] and [FORMULA]. The disk scaleheight H is from the left: [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. Solid line: only single scattering, dashed line: double scattering included, dotted line: multiple scattering included.

The fixed disk parameters are [FORMULA] and [FORMULA] for both figures. The lower values of [FORMULA] and [FORMULA] are chosen to model the irradiated disk, as the energy absorbed from X-rays will be emitted closer to the disk surface. The parameter [FORMULA] is set to [FORMULA] for Fig. 3 and to [FORMULA] for Fig. 4. The value [FORMULA] corresponds to a disk where irradiation is the dominant source of energy, and the emissivity scales as [FORMULA]. The radiation is now generated between approximately five and six scaleheights from the disk plane. At [FORMULA], the a radiating area is at [FORMULA]. The geometry deviates so much from thin disk, that this is clearly seen in polarization. This sets a limit for geometrically thin accretion disk model. This kind of bulged disk may be present in soft X-ray transients at the onset of the transient outburst, when most of the disk mass is falling inwards very rapidly. In thick disks the structure of the outer disk may have larger effects on polarization, as larger fraction of inner disk radiation is processed in the outer disk. The results shown in Fig. 2, Fig. 3 and Fig. 4 indicate that disk optical thickness and scaleheight of medium-inclination systems can not be derived from polarization observations. Conversely, no information of disk thickness or scaleheight is needed when inclination estimates are derived from polarization observations. However, it is challenging to distinguish effects caused by inclination from those related to the vertical structure.

The results of Fig. 2, Fig. 3 and Fig. 4 show clearly, that polarization levels are quite sensitive to the vertical temperature profile. Radial temperature profile has a smaller effect on polarization, and other parameters do not produce observable effects, if the disk is thin. This has two important observational consequences: As the color temperatures corresponding to different wavelengths have variable radial dependence, they should also have different polarization levels. This effect could perhaps be seen if polarization measurements were carried out at different wavebands, and it may be responsible for the lower S/N ratio in the I band compared to UBVR in polarimetric observations of Her X-1 (Egonsson & Hakala 1991). The depths where of different emission lines are formed could perhaps be deduced from spectropolarimetric observations, but this is not possible with present instruments and telescopes.

The low viscosity of gas in accretion disks probably causes turbulence (Frank et al. 1992). In turbulent disks, small-scale irregularities may be present. Irregularities of emission and scattering regions cause reduction in the polarization, as directions of polarization vectors have larger variations, and therefore larger fraction of polarization is cancelled. Small-scale irregularities were modelled by varying the disk scaleheight with azimuth. Azimuth-integrated results for these disks are presented in Fig. 5, where all disks have parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. The disk in the first panel on the left has [FORMULA], corresponding to a regular disk. In the two following panels, [FORMULA] and [FORMULA]. These disks have a little lower polarization levels, as expected. The irregularities in these models have small size, so time resolution of polarization observations is not sufficient to observe real variations of polarization. Only cancellation of polarization due to more random polarization directions is observed. The rightmost panel has [FORMULA], approximating a warped disk. Warped disks are a possible explanation for long-term photometric variations seen in some X-ray binaries (Pringle 1996).

[FIGURE] Fig. 5. Polarization from irregular disks. Disk parameters are [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. First panel on the left, a regular disk ([FORMULA]) for comparison, Second panel: ([FORMULA]), Third panel: ([FORMULA]), Fourth panel: Warped disk, ([FORMULA]) Solid line: only single scattering, dashed line: double scattering included, dotted line: multiple scattering included.

Some simulations of disks with gaussian vertical density profile and with disk models having different radial density profiles, [FORMULA] or [FORMULA] were made. The results were almost identical to those shown in previous figures, so [FORMULA] and vertical profile do not have large influence on polarization. The evaluation of integrals with gaussian profiles is more complicated that of exponentials. This increases the required computing time with almost one order of magnitude.

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Online publication: January 29, 2001