A&A 370, 707-714 (2001)
DOI: 10.1051/0004-6361:20010310
N. V. Kryzhevoi^{1,2} - G. V. Efimov^{3} - R. Wehrse^{1,2}
1 - Institut f. Theoretische Astrophysik, Universität
Heidelberg, Tiergartenstr. 15, 69121 Heidelberg, Germany
2 -
Interdisziplinäres Zentrum f. Wissenschaftliches Rechnen,
Universität Heidelberg,
Im Neuenheimer Feld 368, 69120 Heidelberg, Germany
3 - Bogoliubov Laboratory of Theoretical
Physics, Joint Institute for Nuclear Research, 141980 Dubna, Russia
Received 15 November 2000 / Accepted 27 February 2001
Abstract
The analytical solution of the plane-parallel radiative transfer equation is
obtained in the two-stream approximation for a large class of continuous
distributions of the de-excitation coefficient
(constant, linear,
parabolic, with spikes etc.).
We present also the method of the discrete space theory for obtaining
solutions of the transfer equation in the media with strong density
inhomogeneities.
These sets of the analytical solutions can be used for the solution of the
inverse problem. The deduction of the internal distribution of
from observational data is facilitated in the case
of isothermal media, since the characteristic behavior of the solution
refers to the certain behavior of
.
As an example, we find
the corresponding parameters of the constant and linear distributions of
precisely.
Key words: radiative transfer - methods: analytical - methods: statistical
In the present paper we have obtained the analytical solution of the radiative
transfer equation for the different distributions of
in the
plane-parallel media. The description of the problem, the basic equations
and the simplifying assumptions are given in Sect. 2.
In Sect. 3 we present the general solution which requires for the given
only the knowledge of
linearly independent solutions of a second order homogeneous differential
Eq. (2).
In Sect. 4 we give several examples of
and corresponding
linearly independent solutions. In the same section we offer also a method of
the solution of the transfer equation which can be useful for
with spikes.
Having obtained the analytical solutions for the different behaviors of
the solution of the inverse problem becomes possible.
Section 5 is dedicated to the possibility of the diagnostic of
from observational data. In Sect. 6 the case of the medium with a stochastic
distribution of
is considered. The solution is obtained
by the method of the discrete space theory. Section 7 contains a summary.
b^{2} | ||
, - the Airy functions, |
, - the modified Bessel functions, |
- the Kummer confluent hypergeometric function, |
- the Whittaker function, |
Let us suppose that the radiation field can be characterized by a discrete
number of directed streams ("discrete ordinates'') to mimic the true variation
of intensity with
angle. To simplify the problem we consider only two rays in the opposite
directions
that gives already reasonably accurate results.
The Feautrier technique (Mihalas 1978) allows us to transform
Eq. (1) into a second order differential equation
Figure 1: The runs of the mean intensity in the optically thin ( left) and optically thick ( right) isothermal, , slabs with the different distributions of (see insert) | |
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Figure 2: The mean intensity as the function of the optical depth and whose internal distribution can be approximated by a resonance curve. The dependence on the shape and the position of the resonance is shown | |
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After some transformations (see Appendix A) one can get
Some examples of the continuous distributions of and corresponding solutions and (taken from Abramowitz & Stegun 1972; Kamke 1965) have been collected in Table 1. The free parameters must be chosen in such a way to satisfy the condition of the location of in the interval between 0 to 1. In spite of the small variation range of , the solutions obtained for different may have significant difference, especially in optically thick media. So, in Fig. 1 the solutions of Eq. (2) with constant and linear are shown. In optically thin isothermal media (left part) the difference does not exceed 10%. However, it becomes larger with the increasing of the total optical thickness and in the optically thick media can reach at some points 50% (right part).
Although the functions presented in Table 1 are suitable for
the approximation of a large variety of internal distributions of
,
they cannot be applied for the description of media
with strong density condensations. Furthermore, the solution
of the homogeneous Eq. (2) can hardly be found directly with
approximated by a function with spikes.
To avoid these difficulties we suggest the following procedure:
if
can be represented as
= | (9) |
For example, the choice of
Figure 3: The mean intensity at the boundary as a function of the optical thickness . The curves correspond to the different runs of (see insert): ( dashed), ( dashed-dotted), ( dotted) and ( solid) | |
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The prediction of the internal structure of a medium from observational
data is one of the most important task in astrophysics.
The only observable quantity of the problem is the emergent intensity
that is a function of wavelength .
The total optical thickness of a
layer also depends on .
The knowledge of these functions allows us
to plot
and therefore makes the prediction
of
possible.
In the general case when the solution depends both on the temperature
and on
the diagnostic of
is hardly
possible. However, in the isothermal media the features of the solutions
associate only with the definite behavior of
and therefore
the derivation of the corresponding parameters of such behavior seems not to be
so hopeless. In order to confirm that we consider a slab with B=1,
constant and linear
.
In these cases the integration in
(7) gives
Figure 4: The mean intensity at the boundary of a medium with a strong density condensation | |
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The derivation of the corresponding parameters can be done much easier if we
take into account the behavior of these curves at the limit of large and small
.
At the limit of small
these functions are proportional to
whereas at large
they saturate (Fig. 3) in
accordance with the following
(13) |
As mentioned above the presence of some peculiarities can point to the definite behavior of and, thus, simplify its diagnostic. For example, the maximum of the function may indicate to the linearly decreasing (Fig. 3), although other distributions of the de-excitation coefficient may also result in such feature.
(14) |
We do not use the formalism of Peraiah (1984) or that of
Schmidt & Wehrse (1987) based on the interaction
principle which relates the incident and emergent intensities in a layer. Instead
of these we propose another method which relates the mean intensity and the flux
at one boundary of a layer with the same functions at another one. So, a vector
= | (15) |
A successive application of Eq. (16) - with the corresponding boundary
conditions - allows us to study the evolution of the mean intensity in the
medium (see Appendix B)
= | (19) | ||
= |
In our example we divide the slab into 200 layers.
We require that values of
lie in interval [0,1] and
probability of appearance of small
be higher.
As an example of such
we take the following
(20) |
One realization of
is shown in Fig. 5.
In Fig. 6 one can see the set of solutions for 50 realizations
of
as well as the solution for
in a
medium with B=1. Their statistical distributions at
different
are shown in Fig. 7.
Figure 5: An example of the stochastic distribution of | |
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Figure 6: Solutions obtained for 50 different realizations of . The bold curve was obtained for | |
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Figure 7: Statistical distribution of from Fig. 6. Top left: The mean value is , the standard deviation , the value of the bold curve (see Fig. 6) at this point is . Top right: , , . Bottom left: , . . Bottom right: , , | |
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However, in very inhomogeneous media whose properties can only be treated statistically this method no longer works. Therefore we had to use another technique, namely, the method of the discrete space theory. The solution obtained is written as a sequence of the products of -matrices and two-components vectors that is very easy to be implemented. It is no longer necessary to solve the system of 2(N-1) linear equations (N - number of layers) (Wehrse 1981) or to use the method of the forward elimination and back substitution (Peraiah 1984) for the determination of the internal distribution of J. The problems related to the finding of the corresponding inverse matrix and the keeping of many coefficients are thus avoided, so that the numerical calculations are speeded up.
The presence of the analytical solutions enables us to solve the inverse problem. The more accurate diagnostic of can be done in the isothermal media, since there the features of the solutions refer to the definite behavior of only. By using the characteristic behaviors of the solutions in the limit of the large and small the exact derivation of the corresponding parameters is possible.
As mentioned in the Introduction, there are many classes of objects for which a spectral analysis requires the consideration of complicated depth dependencies of (including stochastic ones) and their consequences for the system parameters. We plan to apply the algorithm developed here to the radiation fields of accretion disks.
Figure 8: The variation range vs. the number of layers taken at | |
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Acknowledgements
We gratefully acknowledge support by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 359/C2) and the Graduiertenkolleg at the Interdisciplinary Center for Scientific Computing at the University of Heidelberg.
C_{1} | = | (A.1) | |
C_{2} | = | (A.2) |
= | |||
+ |
Let us now introduce new variables
and
Taking into account the following identity
with
and
we get the expression of A(N,j,1)