A&A 383, 326-337 (2002)
DOI: 10.1051/0004-6361:20011740
N. A. Silant'ev^{1,2}
1 - Instituto Nacional de Astrofísica, Óptica y
Electrónica, Apartado Postal 51 y 216, Z.P. 72000, Pue. México
2 -
Main Astronomical Observatory of Russian Academy of Sciences,
Pulkovo, 196140 St.-Petersburg, Russia
Received 24 April 2001 / Accepted 25 October 2001
Abstract
We present simple approximate formulae for intensity and the other Stokes
parameters of outgoing radiation for magnetized, plane-parallel optically
thick atmospheres. We assume that the polarization arises as a result of
the light scattering on free electrons in magnetized plasma of an atmosphere.
Our asymptotic formulae take place when the Faraday rotation angle
at
the optical length
is large,
,
and magnetic
field 10^{5} G, when the scattering cross-section has the Thomson value.
The formulae describe the radiation for an arbitrary direction of the
magnetic field. The Milne problem, the atmospheres with homogeneous and
linearly distributed thermal sources, as well as exponential sources are
considered. The superposition of these "standard" sources can represent
practically any real distribution of sources in optically thick accretion
disks. Polarization for some particular models of magnetic fields on the
surface of flat accretion disks are considered (chaotic fields in turbulent
plasma, axially symmetric plane magnetic fields and the field of the magnetic
dipole). The presented results allow us to calculate or estimate the
polarization for a large variety of optically thick magnetized
accretion disks.
Key words: accretion disks - polarization - magnetic fields - stars - quasars - galaxies: nuclei
The existence of accretion disks around very different objects - neutron stars, white dwarfs, young stars, close double systems, quasars and active galactic nuclei, is a commonly accepted paradigm (see Frank et al. 1995). Due to the absence of axial symmetry with respect to our line of sight, the total integrated radiation coming out of accretion disks is polarized. There are numerous observations of light polarization from all the objects mentioned above (see, for example, Stockman et al. 1984; More & Stockman 1984; Bastien & Ménard 1988, 1990; Webb et al. 1993; Korotkar et al. 1995; Yudin 1996; Schmidt et al. 1997; Hutsemékers et al. 1998). The existence of magnetic fields in various cosmic objects is also shown (see the vast review of Vallée 1997). No doubt, many accretion disks are magnetized. Nevertheless, it is difficult to show the existence of a magnetic field in any particular source. Our simple formulae give rise to some characteristic dependence of the polarization spectra on the wave-length and can be used to search for magnetic fields in accretion disks.
The theory of radiative transfer in magnetized atmospheres is developed in a number of papers (Basko 1977; Silant'ev 1979, 1982, 1994; Meszaros et al. 1980; Kaminker et al. 1982). For the case of large magnetic fields, G, when the cross-sections of optical radiation differ from the Thomson value cm^{2}, one can neglect the Faraday rotation of elliptically polarized normal modes and consider the system of coupled equations for the intensities of these modes (Nagel 1980; Kaminker et al. 1982). The different intensities of these normal waves give rise to the net polarization of the radiation. For weaker magnetic fields, B<10^{5} G, the normal waves are right and left circular polarized and have Thomson scattering cross-sections. The anisotropy of the magnetized atmosphere in this case is due to the Faraday rotation only. Such atmospheres were considered in the papers (Silant'ev 1994; Agol & Blaes 1996; Agol et al. 1998). In this paper we consider only this case.
In Silant'ev (1994) the principle of invariance (Chandrasekhar 1950) and the Sobolev technique (Sobolev 1960) were used to obtain the exact solutions of the problems: the Milne problem (the sources are far below the boundary), atmospheres with the power, , and exponential, , sources. Here, is the total optical length, including the scattering and absorption processes. The solutions are expressed in terms of tensor H-functions, which satisfy the nonlinear matrix integral equation. For an oblique magnetic field this equation is very complex.
In Agol & Blaes (1996) and Agol et al. (1998) the Milne problem and the atmosphere with linearly distributed, , thermal sources were considered numerically (Monte Carlo method, the Feautrier technique). The magnetic field was directed along the normal to an atmosphere. We note the very interesting discussion on the influence of the photon true absorbtion in processes of polarization and depolarization (Agol et al. 1998).
The angle of the Faraday rotation
may be written in the form
(Gnedin & Silant'ev 1997):
Photons escape the optically thick atmosphere mainly from the boundary layer with . If the Faraday rotation angle corresponding to this optical length is about unity, then the outgoing radiation will be depolarized as a result of summation of radiation fluxes with very different angles of Faraday's rotation. Only for directions of the light propagation with the Faraday rotation angle is small, or absent, and depolarization does not occur. The diffusion of light in the inner parts of an atmosphere depolarizes the radiation even in the absence of a magnetic field, as a result of multiple photon scattering. For this reason, far from the boundary, radiation is unpolarized. The existence of the Faraday rotation only increases the depolarization process. Thus, the polarization of outgoing radiation acquires the peak-like angular dependence with its maximum for . The sharpness of the peak increases with the increasing of magnetic field values and is of the order .
As is known, the polarization terms (the Stokes parameters Q and U) weakly influence the intensity of radiation I. So, the solution of the transfer equation for the intensity I only (with the Rayleigh phase function) gives for the Milne problem with q=0. The same problem accounting for the polarization terms gives the value 3.06. Here, is the angle between the outward normal to the surface of the atmosphere and the direction of the light propagation ( ). Furthermore, the peak-like polarization terms for the case of a magnetized atmosphere are practically insignificant when determining the intensity I. This means that the intensity of radiation in the atmospheres with sufficient Faraday's rotation ( ) can be calculated from the separate transfer equation with the Rayleigh phase function. Thus, for such atmospheres, the intensity does not depend on the magnetic field value.
The contribution of the polarization terms Q and U by calculation of the polarization itself is rather large for atmospheres without a magnetic field. So, the calculation of the polarization using the known intensity of radiation for the Milne problem gives the value , instead of the known value of . It means that the difference, ( of total polarization), is due to the contribution of the polarization terms in the transfer equation. In the presence of the Faraday rotation, contribution of the polarization terms is much smaller and tends to zero with the increase of the parameter . This contribution can be estimated as . Thus, for we can neglect this contribution and calculate the linear polarization using the known intensity of radiation. Such an approach gives us the opportunity to easily calculate the polarization of light for an arbitrary direction of magnetic field and gives rise to simple analytical formulae.
The aim of this paper is to give simple formulae for parameters Q and Ufor an arbitrary magnetic field inclination. The only restriction is . We give the solutions of the following problems: the Milne problem in the presence of absorbtion (); a plane-parallel atmosphere with the sources of thermal radiation of the forms: , , and ( h=0.5, 1, 2, 5). The linear superposition of these standard sources can approximate practically all forms of real sources in accretion disks.
We use the general theory (Silant'ev 1994) which acquires a very simple form for the large Faraday rotation case, considered in this paper. We present here only the final simplified formulae of this theory. The advantage of our approach, using the principle of invariance and Sobolev's technique, is that we can obtain the azimuthal dependence of polarization for an oblique magnetic field in an analytical form. In this aproach it is not required to solve the radiative transfer equation for particular values of the azimuthal angle. Finally, the observed intensity and polarization of outgoing radiation are expressed in terms of one scalar H-function, which is tabulated for a number of values of the absorption degree q. The exactness of our formulae can be estimated by the comparison with the results of the numerical calculations of Agol et al. (1998) for the Milne problem. The comparison shows that when our asymptotic formulae coincide with the exact numerical calculations. For they are accurate to 10%. For the limiting case (atmosphere without magnetic field) our asymptotic formulae only qualitatively correspond to the exact values of polarization presented in the papers Silant'ev (1980) and Loskutov & Sobolev (1981). So, for q=0 first, at , asymptotic value is less by 20% the real polarization, and then, when and polarization tends to zero, our asymptotic polarization is twice the exact value. For q=0.1-0.5the asymptotic values are less to 40% than exact polarization.
As was discussed, the intensity
in the case of large Faraday
rotation obeys the separate transfer equation with Rayleigh's phase function
.
We consider only the
case of thermal sources when this equation acquires the form:
The principle of invariance (Chandrasekhar 1950), applied to Eq. (2), gives
rise to the system of coupled nonlinear equations for two H-functions,
and .
Sobolev (1972) showed that these functions
can be expressed in terms of one H-function. This function satisfies the
usual nonlinear equation for H-functions:
q=0 | 0.05 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | ||||||||||||||
A | B | A | B | A | B | A | B | A | B | A | B | A | B | |||||||
0 | 0.25 | 0.5 | 0.25 | 0.5 | 0.25 | 0.5 | 0.25 | 0.5 | 0.25 | 0.5 | 0.25 | 0.5 | 0.25 | 0.5 | ||||||
0.05 | 0.2940 | 0.5559 | 0.2856 | 0.5473 | 0.2817 | 0.5427 | 0.2760 | 0.5354 | 0.2715 | 0.5293 | 0.2676 | 0.5239 | 0.2642 | 0.5190 | ||||||
0.10 | 0.3322 | 0.5938 | 0.3146 | 0.5765 | 0.3069 | 0.5678 | 0.2961 | 0.5546 | 0.2878 | 0.5440 | 0.2809 | 0.5349 | 0.2749 | 0.5268 | ||||||
0.15 | 0.3698 | 0.6245 | 0.3421 | 0.5978 | 0.3306 | 0.5851 | 0.3147 | 0.5665 | 0.3029 | 0.5520 | 0.2934 | 0.5398 | 0.2852 | 0.5291 | ||||||
0.20 | 0.4076 | 0.6499 | 0.3689 | 0.6133 | 0.3535 | 0.5967 | 0.3328 | 0.5730 | 0.3178 | 0.5550 | 0.3057 | 0.5401 | 0.2956 | 0.5273 | ||||||
0.25 | 0.4459 | 0.6709 | 0.3956 | 0.6238 | 0.3763 | 0.6034 | 0.3508 | 0.5749 | 0.3326 | 0.5537 | 0.3183 | 0.5364 | 0.3064 | 0.5217 | ||||||
0.30 | 0.4850 | 0.6879 | 0.4223 | 0.6299 | 0.3990 | 0.6057 | 0.3689 | 0.5726 | 0.3478 | 0.5485 | 0.3314 | 0.5290 | 0.3179 | 0.5126 | ||||||
0.35 | 0.5249 | 0.7013 | 0.4493 | 0.6319 | 0.4220 | 0.6040 | 0.3873 | 0.5665 | 0.3635 | 0.5396 | 0.3451 | 0.5182 | 0.3301 | 0.5004 | ||||||
0.40 | 0.5657 | 0.7113 | 0.4766 | 0.6300 | 0.4453 | 0.5984 | 0.4062 | 0.5567 | 0.3797 | 0.5273 | 0.3595 | 0.5042 | 0.3432 | 0.4851 | ||||||
0.45 | 0.6075 | 0.7180 | 0.5043 | 0.6243 | 0.4691 | 0.5891 | 0.4256 | 0.5435 | 0.3966 | 0.5117 | 0.3748 | 0.4870 | 0.3572 | 0.4667 | ||||||
0.50 | 0.6503 | 0.7214 | 0.5325 | 0.6151 | 0.4933 | 0.5763 | 0.4457 | 0.5269 | 0.4143 | 0.4929 | 0.3908 | 0.4668 | 0.3721 | 0.4455 | ||||||
0.55 | 0.6942 | 0.7217 | 0.5612 | 0.6024 | 0.5181 | 0.5601 | 0.4665 | 0.5070 | 0.4328 | 0.4710 | 0.4078 | 0.4436 | 0.3880 | 0.4213 | ||||||
0.60 | 0.7391 | 0.7189 | 0.5905 | 0.5862 | 0.5436 | 0.5405 | 0.4880 | 0.4839 | 0.4522 | 0.4461 | 0.4258 | 0.4174 | 0.4050 | 0.3944 | ||||||
0.65 | 0.7851 | 0.7131 | 0.6205 | 0.5667 | 0.5698 | 0.5176 | 0.5104 | 0.4577 | 0.4725 | 0.4181 | 0.4448 | 0.3884 | 0.4230 | 0.3646 | ||||||
0.70 | 0.8322 | 0.7043 | 0.6512 | 0.5438 | 0.5967 | 0.4915 | 0.5336 | 0.4284 | 0.4937 | 0.3872 | 0.4648 | 0.3565 | 0.4421 | 0.3321 | ||||||
0.75 | 0.8804 | 0.6925 | 0.6825 | 0.5178 | 0.6243 | 0.4622 | 0.5576 | 0.3961 | 0.5160 | 0.3534 | 0.4858 | 0.3218 | 0.4623 | 0.2968 | ||||||
0.80 | 0.9297 | 0.6778 | 0.7146 | 0.4884 | 0.6527 | 0.4297 | 0.5826 | 0.3608 | 0.5392 | 0.3167 | 0.5079 | 0.2843 | 0.4836 | 0.2588 | ||||||
0.85 | 0.9800 | 0.6602 | 0.7475 | 0.4559 | 0.6820 | 0.3942 | 0.6086 | 0.3225 | 0.5634 | 0.2771 | 0.5311 | 0.2440 | 0.5061 | 0.2181 | ||||||
0.90 | 1.0316 | 0.6396 | 0.7812 | 0.4203 | 0.7121 | 0.3556 | 0.6355 | 0.2813 | 0.5887 | 0.2347 | 0.5554 | 0.2010 | 0.5297 | 0.1747 | ||||||
0.95 | 1.0842 | 0.6162 | 0.8156 | 0.3815 | 0.7432 | 0.3139 | 0.6634 | 0.2371 | 0.6151 | 0.1895 | 0.5808 | 0.1552 | 0.5545 | 0.1286 | ||||||
1 | 1.1380 | 0.5898 | 0.8509 | 0.3396 | 0.7751 | 0.2692 | 0.6923 | 0.1901 | 0.6425 | 0.1415 | 0.6074 | 0.1067 | 0.5804 | 0.0799 |
The Milne problem deals with the case when sources are in very deep
layers of an atmosphere (). According to the general theory
(Silant'ev 1994) we have for intensity (erg cm^{-2} s^{-1} Hz^{-1} sterad^{-1}) and the Stokes parameters
and
the following expressions:
q=0 | 0.05 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | ||||||||
J | J | J | J | J | J | J | ||||||||
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |||||||
0.05 | 1.1467 | 1.1427 | 1.1386 | 1.1301 | 1.1213 | 1.1122 | 1.1030 | |||||||
0.10 | 1.2647 | 1.2591 | 1.2532 | 1.2407 | 1.2274 | 1.2133 | 1.1986 | |||||||
0.15 | 1.3746 | 1.3691 | 1.3631 | 1.3496 | 1.3342 | 1.3173 | 1.2992 | |||||||
0.20 | 1.4801 | 1.4766 | 1.4721 | 1.4606 | 1.4459 | 1.4284 | 1.4087 | |||||||
0.25 | 1.5829 | 1.5832 | 1.5822 | 1.5761 | 1.5651 | 1.5495 | 1.5303 | |||||||
0.30 | 1.6836 | 1.6901 | 1.6946 | 1.6981 | 1.6942 | 1.6836 | 1.6674 | |||||||
0.35 | 1.7829 | 1.7979 | 1.8105 | 1.8282 | 1.8359 | 1.8340 | 1.8239 | |||||||
0.40 | 1.8811 | 1.9074 | 1.9308 | 1.9684 | 1.9930 | 2.0047 | 2.0048 | |||||||
0.45 | 1.9784 | 2.0190 | 2.0565 | 2.1208 | 2.1692 | 2.2009 | 2.2166 | |||||||
0.50 | 2.0750 | 2.1332 | 2.1884 | 2.2878 | 2.3689 | 2.4290 | 2.4680 | |||||||
0.55 | 2.1710 | 2.2504 | 2.3277 | 2.4723 | 2.5976 | 2.6981 | 2.7709 | |||||||
0.60 | 2.2666 | 2.3711 | 2.4754 | 2.6778 | 2.8629 | 3.0202 | 3.1428 | |||||||
0.65 | 2.3617 | 2.4958 | 2.6328 | 2.9088 | 3.1745 | 3.4128 | 3.6092 | |||||||
0.70 | 2.4564 | 2.6250 | 2.8014 | 3.1709 | 3.5462 | 3.9017 | 4.2104 | |||||||
0.75 | 2.5509 | 2.7592 | 2.9828 | 3.4712 | 3.9974 | 4.5267 | 5.0128 | |||||||
0.80 | 2.6451 | 2.8989 | 3.1789 | 3.8193 | 4.5569 | 5.3529 | 6.1348 | |||||||
0.85 | 2.7390 | 3.0447 | 3.3919 | 4.2281 | 5.2688 | 6.4945 | 7.8096 | |||||||
0.90 | 2.8328 | 3.1972 | 3.6246 | 4.7151 | 6.2049 | 8.1718 | 10.5709 | |||||||
0.95 | 2.9264 | 3.3573 | 3.8802 | 5.3059 | 7.4904 | 10.8723 | 15.9632 | |||||||
1 | 3.0198 | 3.5257 | 4.1626 | 6.0376 | 9.3641 | 15.9318 | 31.0925 | |||||||
k | 0 | 0.378659 | 0.523200 | 0.704828 | 0.819984 | 0.896901 | 0.947380 | |||||||
g | 0.83255 | 0.80264 | 0.77129 | 0.70405 | 0.63055 | 0.55063 | 0.46437 | |||||||
J_{1} | 1.19402 | 1.29277 | 1.40518 | 1.68238 | 2.05696 | 2.58190 | 3.35377 |
The degree of polarization
p=(Q^{2}+U^{2})^{1/2}/I has the form:
Figure 1: Polarization for the Milne problem. Solid lines present maximum polarization for . The numbers denote the values of the absorption degree q. Dotted lines demonstrate the peak-like polarization for and . The dashed line presents Chandrasekhar's polarization for a non-magnetized and non-absorbing atmosphere. | |
Open with DEXTER |
For maximum polarization ( ) one has and , i.e. the electric field oscillations are perpendicular to the plane . Because the degree of absorption depends on the wavelength, the known dependence exists only in nonabsorbing atmospheres with q=0 (for more detailed discussion, see Agol et al. 1998).
If the sources of radiation are distributed homogeneously,
,
the formulae for the intensity and polarization of outgoing radiation acquire
the form:
q | 0.01 | 0.05 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 0.99 |
10.41642 | 4.59871 | 3.22726 | 2.26350 | 1.83976 | 1.58872 | 1.41832 | 1.29312 | 1.19622 | 1.11841 | 1.05417 | 1.00504 | |
9.11007 | 4.20755 | 3.02812 | 2.18017 | 1.79738 | 1.56586 | 1.40596 | 1.28673 | 1.19323 | 1.11728 | 1.05393 | 1.00504 | |
53.71482 | 9.08645 | 3.95636 | 1.58043 | 0.85654 | 0.52000 | 0.33067 | 0.21171 | 0.13137 | 0.07425 | 0.03207 | 0.00285 | |
44.55025 | 7.43478 | 3.18844 | 1.23946 | 0.65542 | 0.38894 | 0.24208 | 0.15188 | 0.09243 | 0.05128 | 0.02176 | 0.00190 | |
3.51342 | 2.62789 | 2.20445 | 1.79188 | 1.56578 | 1.41507 | 1.30452 | 1.21860 | 1.14916 | 1.09142 | 1.04238 | 1.00398 | |
3.34751 | 2.57104 | 2.18634 | 1.79904 | 1.57984 | 1.43029 | 1.31855 | 1.23037 | 1.15818 | 1.09747 | 1.04539 | 1.00428 | |
2.48587 | 2.07557 | 1.84303 | 1.58814 | 1.43472 | 1.32641 | 1.24366 | 1.17731 | 1.12234 | 1.07569 | 1.03537 | 1.00334 | |
2.46209 | 2.09256 | 1.87347 | 1.62382 | 1.46810 | 1.35539 | 1.26762 | 1.19615 | 1.13616 | 1.08468 | 1.03974 | 1.00377 | |
1.83887 | 1.65720 | 1.54014 | 1.39788 | 1.30425 | 1.23413 | 1.17815 | 1.13167 | 1.09204 | 1.05758 | 1.02716 | 1.00259 | |
1.88210 | 1.71258 | 1.59693 | 1.44977 | 1.34897 | 1.27144 | 1.20828 | 1.15501 | 1.10899 | 1.06853 | 1.03247 | 1.00310 | |
1.38258 | 1.31945 | 1.27401 | 1.21302 | 1.16898 | 1.13375 | 1.10412 | 1.07845 | 1.05576 | 1.03540 | 1.01692 | 1.00163 | |
1.44277 | 1.38058 | 1.33241 | 1.26403 | 1.21237 | 1.16982 | 1.13328 | 1.10110 | 1.07226 | 1.04610 | 1.02213 | 1.00214 |
Figure 2: Polarization for an atmosphere with homogeneous sources. The notations are the same as in Fig. 1. The solid lines present maximum polarization for . The upper and low dotted lines present the polarization for and 50, respectively. | |
Open with DEXTER |
Now we give the formulae for sources of the form
:
In contrast to homogeneous sources, the angular distribution acquires a sharper form with the increase of the absorption coefficient q, as happens for the Milne problem. For the coefficients and tend to zero such that . This means that tends to , in contrast to the Milne problem, where this polarization tends to . Another difference with the Milne problem is the peak-like form of for . Note that for the Milne problem both the intensity and the Stokes parameters and include the characteristic denominator , i.e. they, even for , are very sharp functions of for . But the polarization degree does not depend on this value and is proportional to for large values of q. On the contrary, the Q^{(1)} and U^{(1)} at are not sharp compared with I^{(1)}. This explains the sharpness of the angular distribution of the maximum polarization . These characteristic features are seen in Fig. 3. Of course, the Faraday rotation additionally increases the peak-like form of the polarization curves.
Figure 3: Polarization for an atmosphere with linearly distributed sources. The notations are the same as in Fig. 1. The solid lines present maximum polarization for . | |
Open with DEXTER |
These sources can arise if the surface of an atmosphere is bombarded by
fluxes of particles and radiation. Also, they can be used to
approximate the real distribution of sources in an atmosphere. For such
sources,
,
we give:
If the thickness of such layer is small (the cases h=2 and 5), the angular distribution of outgoing radiation is extended for directions near the surface, compared to the direction along the normal . For the first directions we collect the photons from the line of sight, where the sources exist ad infinitum. On the other hand, for the direction along the normal, the density of radiation decreases rapidly beyond the optical depth 1/h.
For thicker layers (h=0.5 and 1) the situation is more complex. First, for small absorption the angular distribution is extended along the normal . So, we have J^{(1/2)}(1)=1.955 and J^{(1)}(1)=1.446 for q=0. Then, with the increase of q, the values J(1) diminish and the maximum of tends to ( for h=0.5 and for h=1). After that ( and , respectively) the J(1) - values became less than unity. For large q( and ) these angular distributions monotonically decrease from the standard value 1 at .
It is natural that polarization depends on the particular form of the angular distribution of photons near the surface. Figures 4 and 5 reflect this dependence. As usual, we present the maximum degree of polarization , corresponding to .
Figure 4: The Stokes parameter for an atmosphere with exponential sources (h=0.5) and for . The numbers denote the values of the absorption degree q. Descending polarization curves for q=0.8 and 0.9 are depicted as the dotted curves. | |
Open with DEXTER |
First, consider Fig. 4 for the case h=0.5. Because for small q the angular distribution of photons is extended along the normal , the scattered radiation is polarized perpendicular to the plane and the Stokes parameter Q^{(1/2)} < 0. At , when the angular distribution is practically isotropic, the polarization does not arise at all, . For q>0.158, the angular distribution looks as directed parallel to the surface and creates a "negative" polarization with the electric field oscillations in the scattering plane . Such polarization by multiple scattering was first discovered by Nagirner (1962) and explained by Gnedin & Silant'ev (1978). Up to , this negative polarization grows by ( ) and then tends monotonically to zero for , when the scattering of photons disappears. It is of interest to note that numerical calculations of Bastien & Ménard (1988, 1990) have also demonstrated the existence of negative polarization for the optically thick accretion disks consisting of Mie dust spheres.
Figure 5: Polarization for an atmosphere with exponential sources (h=5) and for . The numbers denote the values of the absorption degree q. The curves with q=0.01 and 0.1for clarity are depicted as the dashed curves. | |
Open with DEXTER |
Now consider Fig. 5, corresponding to a very thin source layer with h=5. For this case the angular distribution is ``flat'', mostly parallel to atmosphere's surface, and the polarization is negative for all values of q. Here the limiting case q=0 does not correspond to maximum polarization, as in the case of h=0.5. We have . Up to q=0.05 the polarization grows slightly ( ) and then decreases up to zero for .
Figure 6: The boundary between negative (Q>0) and positive (Q<0) polarization for atmospheres with exponential sources of the type . | |
Open with DEXTER |
In Fig. 6 we give the curve dividing the plane of parameters q and hinto regions with , corresponding to the usual polarization with the electric oscillations perpendicular to the plane , and the vast region of "negative" polarization. The curve itself corresponds to the absence of any polarization. For h>1.65 polarization is always negative. The absence of polarization for h=0.5 and 1 occurs at q=0.158and q=0.0254, respectively. We note that our approximation gives rise to a change of sign of the Stokes parameter for all values of . This is in contrast to the usual non-magnetized atmosphere, where for the polarization is always "positive" (see Loskutov & Sobolev 1981; Silant'ev 1980). Loskutov & Sobolev (1981) give the polarization degree for the Milne problem and homogeneous and linearly distributed sources. Silant'ev (1980) also presents the case of exponential sources and the angular distributions for all considered atmospheres. Note that our calculations of angular distributions in a magnetized atmosphere are very close to those without any magnetic field. The degree of polarization qualitatively corresponds to the case of a non-magnetized atmosphere. The existence of the Faraday rotation gives rise to the peak-like form of polarization near , in accordance with the discussion in the Introduction.
If one has a model of the distribution of sources
of thermal
radiation inside the accretion disk, we can approximate this distribution
as a linear superposition of our "standard" sources.
To obtain the fluxes of radiation F_{I}, F_{Q} and F_{U}(erg cm^{-2} s^{-1} Hz^{-1}), observed in a telescope, we
integrate the values ,
and
over all the observed surface of an accretion disk. In the general case
all values - ,
F, q, I, Q and U depend on their position
on a surface. Usually one supposes that the disk is flat, i.e. the outward
normal
to the surface is independent of the position. In this case
we can directly integrate parameters
and
,
which refer to the system with its x-axis in the plane
.
If an accretion disk is not flat, the
- plane depends on
its position on the curved surface. In this case we take into
account the transformation law of the parameters Q and U. Thus, what we
observe with the telescope is:
Consider the light polarization for some model distributions of magnetic fields in flat accretion disks. These models may be used for crude estimates by analysis of the observed polarization.
If an accretion plasma is turbulent and the mean magnetic field is
much smaller than the chaotic part of the field, we can average our formulae,
considering that all directions of the magnetic field are equally probable. As a
result of such averaging, the parameter
disappears because direct
and inverse magnetic fields are equally present in an atmosphere. In this
situation the plane
is physically distinguished
and, hence, the parameter
.
After averaging, we have:
It should be noted that such an average has sense if the characteristic length of turbulence is greater than the free path of photons. In the opposite case of small-scale turbulence, the mean value of the Faraday rotation angle during the photon free path propagation is equal to zero, and the averaged inclination of the wave oscillation plane maintains its initial value. For small-scale turbulences only mean large-scale magnetic fields depolarize the linear polarization.
If a magnetic field is axially symmetric and its force lines lie on
the surface of an accretion disk (B_{z}=0,
and
), then
We see that all parameters F_{Q} do not depend on the relative relation between radial and azimuthal components of the magnetic field. What is important is the radial dependence of the total magnetic energy which is described by the parameter . Thus, for example, the radial magnetic field ( ) and the azimuthal field ( ) give the same net polarization.
The interaction of the magnetic field of a central object with an accretion disk is a very complex problem (see, for example, Pringle & Rees 1962; Lipunov & Shakura 1980; Lai 1999; Bisnovatyi-Kogan 1999; Bisnovatyi-Kogan & Lovelace 2000). Usually one assumes that the central object has a magnetic dipole field and studies the interaction of this field with the plasma of an accretion disk. For this reason, the model with a magnetic dipole field is of importance. Frequently one assumes that a magnetic dipole is perpendicular to the surface of a disk. In this case we have on the surface and the azimuthal angle integration in (27)-(28) gives the factor , for axially symmetric models. The Stokes parameters for oblique inclination of magnetic dipole are more difficult to calculate. Our simple analytic formulae for and allow us to obtain the result of the azimuthal angle integration in an analytic form.
Figure 7: The flat accretion disk (the x,y-plane) in the magnetic dipole field of a central source. | |
Open with DEXTER |
We consider the magnetic dipole
of a central object (neutron star,
early type star etc.) characterized by the longitudinal and azimuthal
angles
and ,
respectively (see Fig. 7). As is well
known, the magnetic field of the magnetic dipole has the form:
According to the general expression (29), we have:
The parameter
in (41) is equal to (1) with B=B_{0}.
We also introduce the auxiliary values:
After the azimuthal angle integration, formulae (27)-(28) acquire the form:
For diamagnetic (perfectly conducting) plasma the normal z-component of
magnetic field does not penetrate inside the disk (see Lai 1999). Practically
this occurs if the time of molecular diffusion, depending on a conductivity,
is much shorter than other dynamical characteristic times (period of
cyclotron rotation, time of turbulent mixing etc.). For this case we
can omit the factor
in
the -coefficient in (41). Considering that the most important
dependence on
occurs due to the factor
in ,
and taking out the integrands all other quantities in the mean value of
,
we obtain the following formula for the degree of polarization:
The simple formula (54) was derived under the assumption that the parameter
It is interesting that for very large Faraday's rotation ( ) the polarization degree is proportional to for all problems. This dependence has a simple explanation. In a non-magnetized atmosphere the outgoing radiation is mostly collected from the total optical depth . For the intensity I this is true also for an atmosphere with Faraday rotation. For large Faraday's rotation the values of the Stokes parameters Q and U are determined by the contribution of a thin surface layer, where the angle of rotation (see Eq. (1)). The optical depth of this layer . Thus, the Q and U-parameters are proportional to . If , the Faraday rotation does not affect the polarization of the outgoing radiation.
Using the assumption of a large Faraday rotation (the Faraday rotation angle at the optical length ), we obtained simple approximate formulae for the intensity and the Stokes parameters of outgoing radiation for a number of "standard" problems (the Milne problem, the atmospheres with homogeneous and linear distribution of thermal sources as well as exponentially decreasing distribution of sources in a magnetized, plane-parallel, optically thick atmosphere). The polarization arises as a result of the light scattering on free electrons in a magnetized plasma of an atmosphere. The magnetic field is assumed to be <10^{5} G, when the scattering cross-sections are equal to the Thomson value. Practically any real distribution of sources in such atmospheres can be approximated as a superposition of these standard sources. Thus, our formulae allow us to obtain the polarization for various real atmospheres. Of course, the distribution of sources in disks depends also on the illumination from the central objects (star, quasar etc.). Sometimes, direct radiation from these central objects must be also taken into account and the corresponding intensity and the Stokes parameters are to be added to our formulae. The most suitable objects to apply our simple formulae are hot accretion disks around quasars and active galactic nuclei (see Wills et al. 1992 and references therein). The characteristic spectra of polarization, corresponding to atmospheres with Faraday rotation, can be used to confirm the existence of magnetic fields in these disks and to give some estimates of its values.
It was shown that the Faraday rotation very effectively depolarizes the radiation inside the optically thick atmosphere and to obtain the polarization of outgoing radiation, one can consider only the last scattering of non-polarized radiation before the escape from an atmosphere. In this approximation the intensity of radiation satisfies the separate transfer equation with Rayleigh's phase function.
The classic methods of Chandrasekhar and Sobolev allowed us to express all values as depending on one scalar H-function. This function can be easily calculated from the known Ambartzumian - Chandrasekhar nonlinear equation. We numerically calculated all required values and present the results in tables and figures.
We also present the calculation of polarization for some simple models of the magnetic field distribution in an accretion disk (chaotic field in turbulent plasma, axially symmetric plane magnetic field and the field of magnetic dipole). For a diamagnetic (perfectly conducting) accretion disk in the dipole field of a central object we obtained very simple asymptotic formulae for polarization, depending on the longitudinal and azimuthal angles of the magnetic dipole position and the angle of observation.
These results allow us to calculate and estimate the polarization for a large variety of models of optically thick, magnetized accretion disks.