1 Introduction
Abstract

We study the standard spectral line radiative transfer equation for media consisting of resonant atoms and non-resonant components (the dust grains and the atoms without considered spectral transition). Our goal is to study the intensity and polarization of the resonance radiation emerging from semi-infinite atmosphere. Using the known technique of resolvent matrices, we obtain the exact solution of vectorial radiative transfer equation for various sources of non-polarized radiation in semi-infinite atmosphere. Homogeneous, linear increasing and exponentially decreasing sources are considered. Recall, that’ with the homogeneous sources correspond to the isothermal atmosphere and the exponential ones correspond to the incident radiation with different angles of inclination.

Keywords: Radiative transfer, spectral lines, scattering, gas-dusty atmosphere

Spectral line intensity and polarization in gas-dusty medium

N. A. Silant’evthanks: E-mail: nsilant@bk.ru  , G. A. Alekseeva,  V. V. Novikov

Central Astronomical Observatory at Pulkovo of Russian Academy of Sciences,

196140, Saint-Petersburg, Pulkovskoe shosse 65, Russia

E-mail: nsilant@bk.ru

1 Introduction

The observation of spectral lines is the most important technique of investigation in astrophysics. First of all the intensity of spectral line was considered. The angular distribution of scattered radiation was assumed isotropic. The problems of the scalar spectral line transfer were presented in known monographs by Ivanov (1973) and Mihalas (1982).

The problems with non-isotropic and polarized radiation are more difficult to consider (see, for example, Stenflo 1976, Rees & Saliba 1982; Faurobert 1988, Faurobert-Sholl & Frisch 1989; Faurobert-Sholl et al. 1997). It should be also noted the papers by Ivanov 1996, Ivanov et al. 1997, and Dementyev 2008, 2010. Simple quantum mechanical study of the resonance line problems is given in the standard text-book by Landau & Lifshitz vol. 4, part 1 (1968) .

The polarization of spectral line depends also on the magnetic field. Most exhaustive radiative transfer problems in magnetized atmosphere are presented in the books by Fluri (2003) and Landi degl’Innocenti & Landolfi (2004).

We present here the transfer equation for polarized spectral line radiation in non-magnetized media considering three types of particles - atoms with the resonant level, atoms without considered resonant level and the dust grains.

A detailed quantum-mechanical theory (see Landau & Lifshitz 1968) gives rise to the following formulas for the scattering cross-section and the absorption one ( describing the process of collisional deexitation, or destruction, see Hummer & Rybicki 1971):

 σ(s)res(ν)=σ(s)res(ν−ν0)2+γ2,
 σ(a)res(ν)∼Constγ(ν−ν0)2+γ2. (1)

These cross-sections have the Lorentz shape . Here is the central frequency of resonance line, the value is the line width. The absorption coefficient is proportional to small parameter . Note that the line width includes all types of scattering - both elastic and non-elastic. Formulas (1) correspond to the rest frame of the resonance atom.

The allowance for Doppler shift of frequencies in the thermal motions of atoms gives rise to the final form of cross-sections (see Ivanov 1973):

 σ(s)res(ν)=σ(s)resφV(ν),
 σ(a)res(ν)=σ(a)resφV(ν), (2)

where is known Voigt normalized profile.

Further we will use the total cross-section notation:

 σ(t)≡σ(s)+σ(a). (3)

Very important part of our study is the choice of redistribution matrix . We chose the most simple form used in many papers, in particular, in Ivanov et al. 1997. This is complete frequency redistribution matrix (see Eqs.(4) and (11)). The matrix describes the probability to scatter the initial resonance radiation taking the frequency and direction into radiation with frequency and direction , as a result, of the scattering on a resonant atom.

The complete frequency redistribution matrix has the form:

 ^R(ν,n;ν′,n′)=(1/4π)φ(ν)φ(ν′)^P(n,n′). (4)

Here is normalized profile of scattering and absorption cross-sections; the matrix for a number of cases is given in the book by Chandrasekhar (1960).

In radiative transfer equation without introduction of dimensionless optical depth, the integral term, descibing the scattering of radiation, contains redistribution matrix with the term . Here is the number density of resonant atoms, is frequency averaged cross-section of the resonance scattering. Note that the detailed formulas for and are given in books by Ivanov (1973) and by Mihalas (1982).

Below we consider the axially symmetric problems, where the Stokes parameter . In this case and the transforms to form, where , are the cosines of angles between , and the normal to the surface of the atmosphere.

2 Radiative transfer equation for Stokes parameters I(τ,ν,μ) and Q(τ,ν,μ) in the case of spectral line

We will consider the vectorial radiative transfer equation for the intensity and the Stokes parameter , which describe axially symmetric problems. Usually one uses the optical depth:

 dτ=Nres(σ(s)res+σ(a)res)dz≡Nresσ(t)resdz, (5)

independent of the frequency , where is the number density of resonant atoms, and are the frequency averaged corresponding cross-sections. Recall, that the cross-sections:

 σ(s,a)res(ν)=σ(s,a)resφ(ν), (6)

where normalized function describes the shape of the scattering and absorption cross-sections. As the - function we take the function, which describes the Doppler shape of spectral line. This shape arises from the Voigt function for very small value of parameter :

 φD(ν)=1√πΔνDexp[−(ν−ν0ΔνD)2],
 ∞∫−∞dνφD(ν)=1. (7)

Here is Doppler’s width:

, where the thermal velocity along the line of sight is , and the analogous turbulent velocity is the mean value of chaotic motions . Parameter is the speed of light.

The first term in radiative transfer equation describes the extinction of the radiation. In our case of the three - component medium the extiction factor has the form:

 χ(ν)=Nresσ(t)resφ(ν)+Ngrainσ(t)grain+Natomσ(t)atom≡
 Nresσ(t)resα(ν), (8)

where dimensionless extinction factor is:

 α(ν)=φ(ν)+β,
 β=Ngrainσ(t)grain+Natomσ(t)atomNresσ(t)res≡βgrain+βatom. (9)

Parameter descibes the extinction by dust grains and non-resonant atoms, i.e. by particles which do not have the considered resonant level. and are corresponding number densities.

According to usual procedure of derivation of radiative transfer equation with the optical depth , we obtain the following equation:

 μdI(τ,ν,μ)dτ=α(ν)I(τ,ν,μ)−
 (1−ϵ)1∫−1dμ′∞∫−∞dν′^R(ν,μ;ν′,μ′)I(τ,ν′,μ′)−
 φ(ν)s(τ,ν)(10). (10)

Here we introduced the (column) vector I with the components (), where and [erg/cm Hz s sr] are the intensity and the Q- Stokes parameter, respectively. The value is the frequency of light, . The angle is the angle between the line of sight and the normal to the plane-parallel atmosphere . The parameter is the probability of scattering on the resonant atom. Parameter is the destruction probability (see Frisch & Frisch 1977). The source term is the scattered non-polarized isotropic Plank’s radiation.

The cross-section of the resonance line scattering in optical range is of the order cm (see, for example, Gasiorovich 1996). The cross-section of the scattering for non-resonant radiation is much lesser cm. Remind, that the Thomson cross-section of scattering on free electrons is cm, larger than atom’s cross-section. The cross-section , where is the radius of a grain. It appears that .

It is also seen from Eq.(9) that the extinction factor is the sum of the part, depending on the dust grains, and part, depending on non-resonant atoms. The case may correspond to . We will investigate the dependence of the radiation intensity and polarization on the parameter which consists of the sum of these parameters. Note that only the particular models can present the values and separately.

The matrix in general has very complex form

(see McKenna 1985: Landi degl’Innocenti & Landolfi 2004).

This is the reason why one uses the model of complete (full) redistributed matrix:

 ^R(ν,μ;ν′,μ′)=12φ(ν)φ(ν′)^P(μ,μ′), (11)

It appears that formula (11) corresponds to the case when during the lifetime of atomic level many impacts hold.

The matrix has the form:

 ^P(μ,μ′)=^A(μ2)^AT(μ′2). (12)

The matrix is (see Ivanov et al. 1997):

 ^A(μ2)=⎛⎜ ⎜⎝1,√W8(1−3μ2)0,3√W8(1−μ2)⎞⎟ ⎟⎠. (13)

Here parameter depends on quantum numbers of transition atomic levels. For simplest case of dipole transition . Our calculations in Ch. 5 corresponds to this case. Note that the superscript T will be used for matrix transpose. Note also that there exists the equality:

 (10)≡^A(μ2)(10). (14)

Using the dimensionless frequencies and identity (14) , the transfer equation (10) can be written in the form:

 μdI(τ,x,μ)dτ=α(x)I(τ,x,μ)−φ(x)^A(μ2)S(τ), (15)

where the vector is:

 S(τ)=s(τ)(10)+
 1−ϵ21∫−1dμ∞∫−∞dxφ(x)^AT(μ2)I(τ,x,μ). (16)

It is easy check that for the case , and there exists the conservation law for the total flux of radiation:

 dF(τ)dτ=0,F(τ)=∫1−1dμ∫∞−∞dxI(τ,x,μ). (17)

Using known formal solution of Eq.(15) (see Chandraserhar 1960, Silant’ev et al. 2015), we derive the integral equation for :

 S(τ)=g(τ)+∞∫0dτ′^L(|τ−τ′|)S(τ′). (18)

The free term has the form:

 g(τ)=s(τ)(10). (19)

The matrix kernel of integral equation (18) is the following:

 ^L(|τ−τ′|)=1∫0dμμ∞∫−∞dxφ2(x)×
 exp(−α(x)|τ−τ′|μ)^Ψ(μ2), (20)

where

 ^Ψ(μ2)=1−ϵ2^AT(μ2)^A(μ2). (21)

Note that the matrix is symmetric: . This property gives rise to symmetry of kernel . The explicit form of matrix is the following:

 ^Ψ(μ2)=
 (22)

Below we follow to general theory of resolvent matrices, which is given in Silant’ev et al. (2015).

3 Solution of integral equation for S(τ) using resolvent matrix

According to the standard theory of integral equations (see, for example, Smirnov 1964), the solution of Eq. (18) can be presented in the form:

 S(τ)=g(τ)+∞∫0dτ′^R(τ,τ′)g(τ′), (23)

where the resolvent matrix obeys the integral equation

 ^R(τ,τ′)=^L(|τ−τ′|)+∞∫0dτ′′^L(|τ−τ′′|)^R(τ′′,τ′). (24)

It has the property . We see that the equation for follows from Eq.(24):

 ^R(τ,0)=^L(τ)+∞∫0dτ′^L(|τ−τ′|)^R(τ′,0). (25)

The general theory (see Sobolev (1969) and Silant’ev et al. 2015) demonstrates that the resolvent can be calculated, if we know the martices and . This is seen directly from the expression for double Laplace transform of with the parameters and :

 ~~^R(a,b)=1a+b[~^R(a,0)+~^R(0,b)+~^R(a,0)~^R(0,b)]. (26)

Taking the Laplace transform of and using the relation (26), we can derive non-linear equation for -matrix:

 ^H(z)=^E+~^R(1z,0), (27)

where is the Laplace transform of with parameter . is the unit matrix. This equation has the form:

 ^H(z)=^E+^H(z)∫∞−∞dx′φ2(x′)∫10dμ′μ′×
 11/z+α(x′)/μ′^HT(μ′α(x′))^Ψ(μ′2), (28)

The -matrix can be calculated, if we know , which obeys the following non-linear equation:

 ^H(μα(x))=^E+^H(μα(x))∫∞−∞dx′∫10dμ′μ′×
 φ2(x′)α(x)/μ+α(x′)/μ′^HT(μ′α(x′))^Ψ(μ′2). (29)

The effective methods of numerical calculation of -matrix are presented in papers Krease & Siewert (1971) , Rooij et al.(1989) and Dementyev (2008).

According to Eq.(15), the vector , describing the outgoing radiation, has the form:

 I(0,x,μ)≡I(x,μ)=φ(x)^A(μ2)×
 ∞∫0dτμexp(−α(x)τμ)S(τ), (30)

i.e. this expression is proportional to the Laplace transform of over variable . The presence of function in Eq.(30) guarantees that tends to zero for . The vector is presented in Eq.(23). The Laplace transform of this vector can be written as:

 ~S(α(x)μ)=
 ∞∫0dτ[^Eexp(−α(x)τμ)+~^R(α(x)μ,τ)]g(τ). (31)

Thus, expression (30) acquires the form:

 ∞∫0dτ~^R(α(x)μ,τ)g(τ)⎤⎥⎦. (32)

Eq.(32) is the general expression for outgoing radiation, where the term characterizes the distribution of sources of non-polarized radiation in an atmosphere. Explicit form of can be obtained from the model of an atmosphere (see Eq.(19)). It appears, the distribution of (i.e. the model of an atmosphere) can be approximated in the form:

 s(τ)≃∑nsnhexp(−hnτ)+s0+s1τ+s2τ2+... (33)

For source function the expression for acquires comparatively simple form. In this case the expression for depends on and , i.e. it does not depend on the total matrix .

The sources of types are related with exponential source by the simple formula:

 sn(τ)=(−1)nsndndhnexp(−hτ)|h=0. (34)

It means that the sources of type (33) can be considered on the base of the exponential source.

Let us consider this case in detail. Taking in Eq.(31), we obtain:

 Ih(x,μ)=shφ(x)^A(μ2)×
 ^H(μα(x))^HT(1h)α(x)+μh(10). (35)

Homogeneous source corresponds to . Physically this case corresponds to homogeneous isothermal atmosphere. In this case we obtain from Eq. (35):

 I0(x,μ)=s0φ(x)α(x)^A(μ2)×
 (36)

The emerging radiation for the source can be calculated following the formulas in Silant’ev et al. (2015).

5 The results of calculations

It is easy to check (see, for example, Ivanov 1973) that Eq.(29) for matrix can be presented as the equation for variable . The emerging radiation (see Eq. (35)) depends on the product . We prefer to derive the equation for . In this case we are not deal with the complex problem of approximation of from the value . From Eqs.(21) and (29) we derive the following equation for :

 ^H0(x,μ)=^A(μ2)+(1−ϵ)^H0(x,μ)μ∫∞0dx′∫10dμ′×
 φ2(x′)α(x)μ′+α(x′)μ^HT0(x′,μ′)^A(μ′2). (37)

Here we take into account that depends on . For this reason we take the - integration in the interval . It is clearly that Eq.(37) depends on parameters and . Recall, that ().

The solution of Eq.(37) can be carry out by iteration method, analogous to known Chandrasekhar’s (1960) method (see also Dementyev (2008)). For values of parameter we chose .

Note that our calculations correspond to the case . Recall, that in astrophysical conditions parameter is small 10 (see Ivanov 1973, Frisch & Frisch 1977).

5.1 The shapes of resonance line

The shapes of resonance line

are given in Fig.1 (for homogeneous and linearly increasing sources), and in Fig.2 ( for exponentially decreasing sources and . In calculations we assume that all coefficients and are equal to unity. Here the function is the flux of intensity :

 FI(x)=∫10dμμI(x,μ). (38)

The value presents the intensity flux as the function of the dimensionless frequency . Recall, that the flux one observes from the distant spherical stars (really one observes the value

, where is the distance to a star and is the radius of a star).

The linear polarization of radiation from the spherical star is equal to zero due to axial symmetry of the problem. The Figures 1 and 2 - demonstrate the dependence of spectral line shape on the parameter .

The homogeneous sources with small parameters and give rise to the minimum of outgoing radiation at . In these cases the resonance line near looks like an absorption line. For the maximum value of is and corresponds to . For the maximum of the line is considerably less and corresponds to . For and the shape of the line looks like usual line profile with the maximum at .

The linearly growing sources are result in qualitatively the same behavior, but the maxima of the lines are more profound ( at for ) and ( at for .)

The presence of in Eq.(30) means that the intensity of the resonance line tends to zero for . Why does the peak of the resonance line arise for small values of absorption parameter ? Let us bear in mind that the radiation density increases due to diffusion of radiation with the growing . For small the emerging radiation goes from the regions far from the surface, where radiation density is much larger than that near the surface.

The exponential sources are close to the boundary of a medium and the diffusion of radiation does not give rise to large radiation density far from the boundary. As a result, the peaks do not arise. Fig.2 shows this.

The absolute values of intensity flux can be obtained from the function if we know the flux and the constants :

 FI(x)=snFI(0)J(x). (39)

The values we present in Tables 1 and 2 for a number of parameters . The value presents the flux in the centre of a spectral line. In these Tables we also present the integrals:

 Φint=∫40dxFI(x). (40)

It is seen, that the values and decrease with the increasing of parameter .

5.2 Integral angular distribution and polarization from optically thick accretion discs

If we observe by a telescope the intensity and polarization of radiation from the distant optically thick accretion disc, we observe the radiation fluxes:

 F(Tel)I(x,μ)=SR2μI(x,μ)
 F(Tel)Q(x,μ)=SR2μQ(x,μ). (41)

Here is the distance to a disc, is the angle between the line of sight and the normal to a disc ; , S - is the observed surface of homogeneous disc.

In Figs. we give the angular distributions and degrees of polarization for all considered types of sources, when the telescope observes the radiation in the -interval (0,4), as a whole:

 F(Tel)I(μ)=SR2μIint(μ),
 F(Tel)Q(μ)=SR2μQint(μ), (42)

where and are:

 Iint(μ)=∫40dxI(x,μ),
 Qint(μ)=∫40dxQ(x,μ). (43)

The angular distribution of integral radiation has the following definition:

 Jint(μ)=Iint(μ)Iint(0). (44)

The degree of polarization is determined by the ratio:

 pint(μ)=Qint(μ)Iint(μ). (45)

Recall, that negative polarization denotes that the electric field of electromagnetic wave oscillates perpendicular to the plane , i.e. parallel to the disc’s plane. Such oscillations take place at multiple scattering of radiation on free electrons (the Milne problem, see Chandrasekhar 1960).

The integral values (44) and (45) we observe by the telescope which does not resolve particular frequencies inside the spectral line. Recall, that frequencies are blanketed by continuum radiation and by the wings of neighboring spectral lines. The values and are given in Figures .

In Table 3 we give the -values for a number of considered types of sources. It is seen that all values decrease with the increasing of the parameter .

For homogeneous sources the angular distribution increases monotomically with . For we have . For and the increase is less - and , respectively. For larger values of the angular distribution tends to be isotropic. Such behavior of angular distribution of a spectral line is opposite to angular distribution of continuum radiation for the Milne problem, where for we have and for one has (see Chandrasekhar 1960; Silant’ev 1980). It appears, this behavior is due to the presence of radiation peaks for small values of (the less is parameter , the greater is the intensity of outgoing radiation).

The total picture of angular distribution depends on two reasons - the presence of peaks and the usual dependence of emerging radiation . The latter gives rise to greater intensity at compared to . Recall, that the outgoing radiation of the resonance line arises at different , depending on the frequency .

The polarization of outgoing radiation mostly arises by last scattering before escape from an atmosphere. If the incident radiation is parallel to , then degree of polarization and the wave electric field oscillates perpendicular to scattering plane . If the incident radiation is perpendicular to , then the oscillations of integral radiation corresponds to oscillations in the plane . Of course, the total polarization also depends on angle . In general, when the intensity of incident radiation along is much greater than that in perpendicular direction, the total polarization of outgoing radiation has oscillations perpendicular to the scattering plane , i.e. are parallel to the accretion disc. More detail consideration is presented in Dolginov et al. (1995). Just such behavior we observe in our case. For resonance line at the radiation is directed mostly along and, as a result, we have large degree of negative polarization . For and , when the incident radiation is mostly perpendicular to the plane , the wave electric field oscillates in the plane . However, this polarization is not high and acquires the maximum value for the case .

For linearly increasing source the angular distribution has more complicated behavior than that in the case of homogeneous source. Every curve monotonically increases with . However, the - dependence differs from that for homogeneous . Firstly the value decreases monotonically with the growth of from the value at to at . Then value begins increase up to value at . It appears, both mechanisms, mentioned above, work more intensive because the density of radiation increases more rapidly with the optical depth than in the case of homogeneous sources. We see that all angular distributions have very elongated form. As a result, the polarization is negative for every value , i.e. the wave electric field oscillations are perpendicular to the plane . The polarization acquires maximum value at . These values are within the limits (- .)

For exponential sources and

the density of radiation is located near the surface of the medium. That is why the angular distributions have the maximum values at . The incident radiation falls on the resonant atoms mostly perpendicular to the normal . As a result, the polarization of emerging radiation is positive, i.e. the wave electric field oscillates in the plane . The degrees of polarization have the maximum values at (). The angular dependence and polarization for sources and are close with one another. This is due to diffusion of the radiadion which gives rise to the smoothing of the radiation density from initial sources located near the surface of the medium.

It is of interest that for homogeneous source at the value is independent of the frequency . This follows from Eq.(37), if we recall that and .

Tables 4 and 5 present , for homogeneous, linearly increased and exponential (h=1) sources at for and , correspondingly. These tables can be useful for estimates of polarization of spectral line radiation emitted from optically thick accretion discs.

For the same aim we give the Tables 6 and 7, which characterize the intensity and polarization in the centre of resonance line . In these Tables the value characterizes the angular distribution of emerging radiation and is the polarization degree of this radiation.