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

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3. Accretion disk models

In many persistent LMXB:s the inner disk and compact star have sufficient X-ray luminosity to keep the outer disk completely ionized. (King et al. 1997) This suppresses the thermal accretion disk instability (Tuchman et al. 1990). The Z sources are LMXB:s where the compact star is a neutron star and the accretion rate is probably close to Eddington limit (van der Klis 1989). In these systems the disk temperatures should be high enough to make electron scattering is the dominant opacity source (Frank et al. 1992). SXT:s in outburst have also completely ionized disks.

The vertical structure of accretion disks in LMXB:s is not very well known. It is often assumed that disks are very thin (Frank et al. 1992). However, this is not always a safe assumption (Shaviv et al. 1999). Irradiation may change the vertical disk structure (Dubus et al. 1999). Observations indicate that a thick disk is present at least in some LMXB:s (Mason 1989). Some X-ray (Hellier & Mason 1989) and optical (Hakala et al. 1999) observations are easiest to explain with asymmetric disks.

The standard disk model (see e.g. Frank et al. 1992) has gaussian vertical density profile if the disk is isothermal in vertical direction. The disks of LMXB:s are probably not very similar to standard disk model, as X-ray irradiation from the compact object may affect the disk structure. In my simulations the disk structure is needed only very near the disk surface, where the essential radiative transfer takes place. In contrast, the standard model is used to describe the entire disk and is most relevant at the disk plane, where most of the disk mass is. Disk models used in simulations have exponential vertical density profiles, [FORMULA]. The functional form of the scaleheight H may vary between disk models, but generally [FORMULA], where [FORMULA] is the ratio of scaleheight and radius at the inner disk. As variations in vertical density structure parameters do not produce significant variations in polarization, polarization is probably not very sensitive to the functional form of vertical density distribution.

Effects of the mass flow from the donor and tidal forces can cause deviations from axial symmetry in the disk. These are often observed, as most non-eclipsing LMXB:s show significant orbital variations in their optical brightness (Van Paradijs & McClintock 1995). Asymmetric models require binning of the photons in two dimensions. To obtain statistics comparable to that of axisymmetric simulations, the required processor capacity increases with a factor at least as large as the number of azimuth bins. Small-scale roughness of the disk surface, caused by turbulence, may be present. As the disk rotates with the binary and the gas moves in the disk, the Stokes parameters will be effectively azimuth-averaged in real observations. Phase-resolved polarimetry has generally low signal-to noise ratio, as most LMXB:s have short periods and are very faint. Two simulations of non-axisymmetric disks with small-scale irregularities or warped disks were made. The Stokes parameters produced by simulations were azimuth-averaged to estimate the consequences of these effects.

Brightness temperature at the band in which polarization is measured and electron density are assumed to be proportional to powers of radius, [FORMULA]. The emission of the disk occurs in a layer near the disk surface, between two values of vertical optical depth, [FORMULA] and [FORMULA]. As the polarization of radiation emitted at [FORMULA] is saturated, i.e. does not depend on emission depth, so [FORMULA] is a good upper limit. The dependence of polarization on [FORMULA] is quite weak for values near the saturation depth, as the polarization is close to the saturation value. Any values satisfying [FORMULA] can be used. Further constraints on the values can be derived by considering the X-ray irradiation, which is present in all LMXB:s. The X-rays come from the central source, so they will enter the disk from directions close to the tangent of the disk surface. A part of the X-rays is directly reflected, as others are captured by the gas. The absorption of X-rays is caused by photoelectric absorption and Compton scattering. The relative importance of these effects depends on the ionization states of different elements. Another complication is that the angle between the disk surface and incident X-rays is unknown. For accurate optical emission model, detailed information on the local disk structure and conditions and the X-ray emission geometry is needed. Instead of accurate physical modeling a crude zero-order approximation is used. The values used are [FORMULA] and [FORMULA]. These should not be very far from real values. The vertical optical depth, [FORMULA] at the inner disk edge is set to some value defined by setting [FORMULA] and [FORMULA] in Eq. (4). The five free parameters ([FORMULA]) and the disk scaleheight [FORMULA] define the disk model. As the opacity is produced by electron scattering only, opacity is directly proportional to density. When changing the parameter [FORMULA], also [FORMULA] must be changed to keep the disk optically thick at all radii.

As [FORMULA], H and [FORMULA] determine the density distribution, they represent total disk mass, [FORMULA] and density at inner disk radius, [FORMULA]. [FORMULA] can be derived from Eq. (4) and definition of [FORMULA].


where [FORMULA] is density scaleheight at inner disk in meters, and exponential vertical profile is assumed The density can be directly integrated to get disk mass:


where [FORMULA] is the ratio of inner and outer disk radii, and [FORMULA] is inner disk radius in kilometers. If [FORMULA], the fraction should be replaced with [FORMULA]. Setting [FORMULA] and [FORMULA] gives [FORMULA]. LMXB disk masses are not well known, but a rough estimate for the upper limit can be obtained assuming Eddington-limited accretion [FORMULA] and accretion timescale of [FORMULA], giving [FORMULA]. The vertical optical depth at the outer disk edge is [FORMULA], and this should be larger than one to keep the disk optically thick even at the outer radii. This can be combined with the requirement of optically thick disk at all radii to yield [FORMULA].

In most persistent LMXB:s the companion star has much smaller luminosity than the accretion disk. This is also true for SXT:s during the outburst. In some cases the secondary star is heated by X-rays from the compact star, which is observed as low-amplitude (generally below 0.2 magnitudes) ellipsoidal variations in the lightcurve. The electron scattering of the radiation from the secondary star may produce an additional polarization component. This effect can be significant in CV:s, but in persistent LMXB:s and SXT:s in outburst this should be a very small effect. In quiescent transients this effect is probably responsible for most of the polarization (Dolan & Tapia 1989). Electron scattering of the disk radiation by the secondary star should be insignificant, as the secondary stars in LMXB:s have low surface temperatures and therefore low free electron densities.

As the accuracy of polarization measurements does not allow accurate determination of the disk model, I used very simple disk models. All disk models used are axisymmetric or have azimuth-averaged Stokes parameters, and magnetic effects are ignored. As the polarization levels produced by electron scattering are quite small, other sources of polarization must also be considered.

The effects of magnetic fields, such as cyclotron or synchrotron radiation and disk disruption by magnetic field, are observed in some LMXB:s, e.g Her X-1. Many LMXB:s exhibit X-ray bursts indicating neutron star primaries with low magnetic field. Even in these systems the effect of the magnetic field can not be directly neglected. The magnetic field can cause the ionized gas emit elliptically polarized cyclotron/synchrotron radiation. Electron scattering in non-magnetic medium causes only linear polarization, but scattering in magnetic field produces elliptically polarized radiation (Whitney 1991), with larger effects in circular polarization. Magnetic field effects on scattering are most important when the frequency of the radiation is close to electron cyclotron frequency, [FORMULA]. For optical light, this would require magnetic fields of order [FORMULA] T. The field strengths of neutron stars in disk-accreting LMXB:s can not be directly estimated, but propable LMXB endproducts, millisecond pulsars, have surface field strenghts of the order [FORMULA] T (Bhattacharya 1995). It is also possible that some neutron stars in LMXB:s have even lower fields, as no magnetic effects are observed in some neutron star LMXB:s. For dipole field, [FORMULA], so magnetic effects are restricted to the innermost disk. The field could be temporarily screened by the accreted matter, reducing the magnetic effects further or suppressing them completely (Bhattacharya & Srinivasan 1995and references therein). Measurements of circular polarization should be used to estimate the amount of linear polarization with magnetic origin in neutron star LMXB:s.

Differential saturation (Calamai et al. 1975) is an effect that produces pure linear polarization, when saturated absorption lines are Zeeman-splitted. In a typical LMXB spectrum most lines are in emission, so differential saturation from the disk can be safely neglected. In VY Scl stars, also known as anti-dwarf novae, most of the accretion disk is also ionized, and only the outermost parts of the disk may be neutral (Leach et al. 1999). In these systems the accretion rate shows long-term variations. Two explanations for this have been presented: shielding of the secondary by the accretion disk rim (Wu et al. 1995) and a magnetic mechanism, where starspots of the secondary close to the inner Lagrange point reduce the accretion flow (Livio & Pringle 1994). The latter scenario assumes that the secondary has a high surface magnetic field. CV secondaries are late-type stars with many absorption lines, and the secondary contribution to the optical luminosity may be significant. Therefore differential saturation may cause linear polarization in VY Scl stars. In some cases, the polarization produced by differential saturation may cancel some of the polarization produced by Thomson scattering (Huovelin 1990). If my results are applied to these systems, the polarization produced by differential saturation should be carefully estimated.

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Online publication: January 29, 2001