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Astron. Astrophys. 342, 201-211 (1999)

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5. An approximation for a general atmosphere

Model atmospheres are usually not so perfect as to be considered in the cases treated in the last section. In the absence of a general analytical solution for the evolution operator, we must manage those general models numerically. Our strategy is to integrate everything we can and to linearize the rest. For this purpose we borrow a technique from other successful numerical integrators as the well-known DELO (Rees et al., 1989). In what follows we develop this idea.

We want to integrate the transfer equation

[EQUATION]

or alternatively

[EQUATION]

The integration is to be made in the interval [FORMULA] and the atmosphere in the two extreme points [FORMULA] and [FORMULA] is specified, that is, we know [FORMULA] and [FORMULA] and also [FORMULA] and [FORMULA]. The incoming light [FORMULA] is also given. We want to obtain the polarized light at [FORMULA]: [FORMULA]. After the last section we already know how to integrate the transfer equation if the atmosphere is characterized by the prescriptions given there. Note that, given the atmosphere at the two points [FORMULA] and [FORMULA], one can always calculate the angles [FORMULA] and [FORMULA] and the matrix [FORMULA] at those two levels, and look for a suitable integrable atmosphere in agreement with expression (27) which satisfy the data at [FORMULA] and [FORMULA] as far as possible. So let us approximate our atmosphere between these two levels by this integrable atmosphere, represented by a matrix [FORMULA] and an emission vector [FORMULA]. We can obtain a solution [FORMULA] for the equation

[EQUATION]

taking as initial condition [FORMULA]. Substraction of Eq. (65) from Eq. (64) results in

[EQUATION]

and upon formal integration:

[EQUATION]

where [FORMULA] and [FORMULA] given as solution to Eq. (65). This equation reflects the error made under the previous approximation. Following our strategy, once we have solved for the integrable part we linearize the rest. So that we now assume that the right hand side of equation (66) is small and can be linearized in the interval [FORMULA]. We define

[EQUATION]

and upon linearization we write

[EQUATION]

where

[EQUATION]

At [FORMULA], we have [FORMULA], and we obtain

[EQUATION]

To solve Eq. (67) we write

[EQUATION]

And by means of the linearization the last integral becomes

[EQUATION]

so that, substituting a and b by its complete expressions

[EQUATION]

In this expression everything is already known except for [FORMULA] that is precisely what we want to calculate.

A convenient choice of the overlined parameters may render equation (73) simpler. For illustration, let us choose

[EQUATION]

and [FORMULA], [FORMULA]. We then obtain

[EQUATION]

a solution for [FORMULA]. This solution is not exact, its precision depends on how good the linear approximation is. In the limit, we can made the integration interval [FORMULA] as small as we want but at the cost of increasing the number of layers. A compromise will be necessary between speed and required precision.

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© European Southern Observatory (ESO) 1999

Online publication: December 22, 1998
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