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# 2. Formal solution of the vector radiative-transfer equation

The VRTE can be written in the form

where is the Stokes vector for the polarized radiation field, is the propagation matrix, is the scalar source function, and is the formal vector . The propagation matrix is then a matrix, whose explicit form will be considered later on.

The optical depth, , is defined such that it increases in the direction opposite to that of the observed radiation. In general, the optical depth is a function of the frequency as well, but in the application of Eq. (1) to the calculation of line profiles it is customary to choose a frequency independent representation of . In particular, if we are considering the case of emission lines in the absence of continuum contribution, then we may define , where is the line absorption coefficient integrated in frequency, and s is the geometrical depth.

In the following, we will adopt such definition of the optical depth. However, since we are parameterizing the solution of Eq. (1) in terms of the optical depth of the medium (instead of the geometrical depth), the explicit dependence of on s, and therefore of on s, is not needed.

Also, in order to keep the complexity of notation at a minimum, we will drop the subscript ``'' from the above defined quantities, implicitly assuming their frequency (or wavelength) dependence.

## 2.1. Formal solution of Eq. (1)

From the general theory of linear ordinary differential equations (see, for instance, Hochstadt 1964), it is known that the formal solution of Eq. (1) can be written in the form

where U is the solution matrix of the adjoint equation of Eq. (1),

Incidentally, we note that U coincides with the evolution operator which was introduced by Landi Degl'Innocenti & Landi Degl'Innocenti (1985).

As we anticipated in the Introduction, we are interested in determining the solution of Eq. (1) in the case of an isolated slab of emitting hydrogen plasma of total optical depth . In the absence of background sources of radiation , so that Eq. (2) becomes

Since we have assumed that the propagation matrix, , is independent of , the formal solution of Eq. (3) is simply given by

apart from an unnecessary multiplicative constant. Thus Eq. (2') becomes

which is the starting point for all the following calculations.

## 2.2. Constant source function

If we let (matching the case of uniform temperature and pressure within the slab), we have, for any optical depth in the interval ,

where is the unit matrix.

We note that, in the limit of optically-thin (emission) lines, the above solution would become

where in the last equivalence we have introduced the emission vector .

In general, we may want to allow the source function to vary with , in order to mimic temperature variations through the slab, and to account for possible departures from LTE. However, we keep the approximation of a constant propagation matrix through the whole slab, so that Eq. (4) is still valid.

Since we are mainly interested in determining the emergent polarized radiation from solar atmospheric structures such as prominences or post-flare loops, we make the additional assumption that the slab has planar symmetry with respect to its median plane. This requirement yields the following constraint on the source function,

which, in particular, gives the two-points boundary condition for the source function

## 2.3. Parabolic source function

The next form of the source function that we consider, after the simplest case of a constant source function which yielded Eq. (5), is the parabolic approximation,

By imposing the two-points boundary condition, Eq. (6b), we easily find

Since the source function can never attain negative values, the parameter must satisfy the constraint

where we also limited to a positive range of values, since we want to describe the realistic physical situation of a slab temperature increasing outward, which is more realistic for the application to prominences and post-flare loops.

Substituting Eq. (7) into Eq. (4) yields, after some tedious calculation,

In the parabolic approximation of the source function, the observed polarization profiles are then given by Eq. (9) for :

## 2.4. Gaussian source function

We next consider a source function with a Gaussian behavior,

where again we limit ourselves to considering only positive values of the parameter , with the additional constraint . Substituting Eq. (11) into Eq. (4), we then obtain

The above integral can be expressed through the error function (e.g., Abramowitz & Stegun 1965), and therefore must in general be calculated through some numerical method.

However, since we are finally interested in determining the outcoming Stokes vector, , we will concentrate on

so to extend the integration limits to the boundary of the slab.

If we then make the assumption that , we can further extend the integration limits to infinity. Such assumption corresponds to imposing the boundary condition or, in other words, it corresponds to the reasonable assumption that the exponential contribution of the source function given by Eq. (11) be negligibly small at the boundary of the slab.

In fact, letting , we note that the exponential contribution to the source function at the boundary of the slab is only about a fraction of the contribution at the center of the slab, if we take . So we will consider the extension of the integration limits to infinity a valid approximation, whenever .

We then observe that we can write, through some simple algebraic transformation,

The last integral can now be evaluated through the formula (see Gradshteyn & Ryzhik 1980)

In fact, it can be shown that the same formula also applies when the parameter q is replaced by any (real) square matrix (like the propagation matrix ). Thus letting and , we easily obtain

so that

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