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Astron. Astrophys. 333, 1130-1142 (1998)

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3. The general solution

The system described by Eq. (A1) is a linear, homogeneous system of equations. A normalization condition on the populations

[EQUATION]

determines the solution.

We now specify the model atom used for our calculations. This is composed by the levels having [FORMULA], and 3. The allowed transitions and Einstein coefficients for this model atom are given in Table 1.


[TABLE]

Table 1. Values of the Einstein A coefficients.


As the tensor [FORMULA] contains only the multipolar components [FORMULA] and [FORMULA], only [FORMULA] 's with K even have to be considered. We have to solve a linear system of 38 equations plus an extra equation (the normalization condition).

Solving numerically the system for the populations, one finds that the population of the ground level [FORMULA] is nearly equal to [FORMULA]. This is the value obtained in Eq. (10) setting all populations to zero except for the population of the ground level.

Obviously, there is a numerical solution of the system for each value of the atomic velocity, a parameter on which an average has to be performed. To circumvent this problem we prefer to obtain an approximate analytical solution for the [FORMULA].

3.1. The analytical solution for [FORMULA]

We introduce the following notations

[EQUATION]

where [FORMULA] stand for the frequencies of H [FORMULA], Ly [FORMULA], Ly [FORMULA], respectively. The density of radiation is

[EQUATION]

Using the data available for the Sun, Delbouille et al. (1973 ) and Allen (1973 ) for the H [FORMULA] line and the results of Gouttebroze et al. (1978 ) for the Ly [FORMULA] and Ly [FORMULA] lines, we summarize the results in Table 2. Note that [FORMULA], [FORMULA], [FORMULA] are dimensionless quantities proportional to the number photons at frequency [FORMULA]. The values of the intensites [FORMULA] of the Ly [FORMULA] and Ly [FORMULA] lines are taken at the center of the lines and at disk center, while the intensity [FORMULA] of the H [FORMULA] line is taken in the continuum surrounding the line at disk center.


[TABLE]

Table 2. Values for the f quantities.


We will not write down explicitly the Kramer system [FORMULA] where [FORMULA] is the ([FORMULA]) matrix of coefficients. But when all the coefficients are calculated (which implies a heavy use of Racah algebra), it is readily seen that one can neglect some couplings between the elements and thus reduce the whole system to a set of subsystems of maximum dimension 3. Another simplification is made by writing

[EQUATION]

for the population of the ground level.

For the populations we get the results contained in Table 3, where the last column is obtained by solving numerically the Kramer system, for a zero velocity.


[TABLE]

Table 3. Analytical expressions and numerical values of the populations of the levels involved in all transitions.


The results refer to the value

[EQUATION]

valid for an atom at a height [FORMULA] km. The results for the higher tensorial components are given in Table 4.


[TABLE]

Table 4. Analytical expressions and numerical values of the higher tensorial components.


The agreement between the analytical approximation and the exact numerical results is very good.

The quantity [FORMULA] appearing in Table 4 is defined by

[EQUATION]

Moreover,

[EQUATION]

We can observe that all the tensorial components of the d states [FORMULA] and [FORMULA] are proportional to the product [FORMULA]. This is indeed a process of two step excitation, [FORMULA] from the ground level.

3.2. The density matrix of the reemitted photons

The density matrix of the reemitted photons in H [FORMULA] is given by the expression (Bommier 1977 , Eq. (III-45); see also Bommier & Sahal-Bréchot 1978 , Eq. (38))

[EQUATION]

where the sum has to be performed over all initial states with [FORMULA] and all final states with [FORMULA].

Using the useful relations between Einstein A coefficients

[EQUATION]

we find after some calculations

[EQUATION]

[EQUATION]

and, replacing the [FORMULA] with their analytical approximation,

[EQUATION]

[EQUATION]

where the [FORMULA] is defined in Eq. (3).

3.3. The Stokes parameters of the H [FORMULA] line

We now have to transform the matrix [FORMULA] to the frame AXYZ of the observer (see Fig. 2.1), which is obtained after a rotation of the frame Axyz by the Euler angles (Sahal-Bréchot et al. 1998 )

[EQUATION]

The general transformation of the tensors, under a rotation characterized by the Euler angles [FORMULA], is

[EQUATION]

where [FORMULA] are the rotation operators as defined in Messiah (1959 ). For our particular rotation, we have

[EQUATION]

For the matrix [FORMULA] expressed in the observer's frame we thus obtain (Sahal-Bréchot et al. 1998 )

[EQUATION]

[EQUATION]

[EQUATION]

The Stokes parameters of the H [FORMULA] line are then given by (Sahal-Bréchot et al. 1998 )

[EQUATION]

[EQUATION]

[EQUATION]

The distribution of velocities used here is the shifted Maxwellian of Eq. (6), where [FORMULA] or [FORMULA] is the velocity of the stream of particles in the spicule.

We simplify the final expression by setting [FORMULA], that is by supposing the atom is in the plane of the sky. We thus obtain

[EQUATION]

[EQUATION]

[EQUATION]

We now substitute Eqs. (15), and, in order to simplify the resulting formulae, we intoduce some notations. Putting

[EQUATION]

the density matrix elements of the H [FORMULA] incoming radiation can be written as (Sahal-Bréchot & Choucq-Bruston 1994 , Sahal-Bréchot et al. 1998 )

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Introducing the quantities (depending on frequencies and direction)

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

one obtains

[EQUATION]

[EQUATION]

[EQUATION]

Here [FORMULA], [FORMULA], [FORMULA] stand, respectively, for [FORMULA], [FORMULA], [FORMULA]. Note that we neglected the terms proportional to

[EQUATION]

These terms are indeed negligible due to the fact that

[EQUATION]

and that

[EQUATION]

3.4. Integration over the velocity distribution

From Eq. (21) we see that we can split the calculation of the integral into two parts: part I coming from the first term (the continuum), and part II coming from the the second term (containing the profile of the H [FORMULA] line). Part I is easily evaluated since the velocity distribution is normalized, so that

[EQUATION]

We obtain

[EQUATION]

[EQUATION]

[EQUATION]

For part II we shall not give here the details of the proof of the following formula, which will be published elsewhere for any [FORMULA] value (Sahal-Bréchot et al. 1998 ). Introducing the quantities

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

and introducing the formal operator

[EQUATION]

we can write the Stokes parameters as

[EQUATION]

[EQUATION]

[EQUATION]

Integrating [FORMULA], [FORMULA], [FORMULA] over frequencies, we obtain the simpler formula

[EQUATION]

where

[EQUATION]

In particular, for [FORMULA], we have

[EQUATION]

and, for [FORMULA]

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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