Astron. Astrophys. 332, 610-628 (1998)

5. Properties of the irreducible vectors J and S

We discuss in this section the main properties of the six-component vectors and . We investigate in particular the dependence of and on the three parameters defining the magnetic field, , , and . In the results presented below, and are always kept uniform but we discuss one case with a depth-dependent since this feature could not be handled with the FS91 formulation. When is uniform, it is sufficient to calculate the solution with . A simple transformation (see Sect. 5.4) yields the solution for an arbitrary .

All the results shown in this section have been obtained with the atmospheric parameters . For this slab model the product aT is smaller than unity and hence the radiative transfer effects in the line are restricted to the Doppler core. We note also that the thickness T is much smaller than the thermalization length which is around . Hence the slab is effectively thin and the diffuse radiation field is almost independent of . Also because the line is not thermalized, the polarization goes to a constant at mid-slab. However, at depths around unity or less it behaves qualitatively as with the effectively thick slab model used in Sect.  4.

The integral equation (61) for and the relation (47) between and will be used in the analysis of the properties of and .

5.1. Dependence of J on the co-latitude _B

For the computations in this section, we assume and let vary between and . As explained above, we have set the value of to zero. Fig. 5 shows the depth-dependence of the six components , , , for various values of . The upper left panel shows and the other panels , . Figs. 7 to 10 devoted to also show in the upper left panel and , , in the five other panels.

Fig. 5. Symmetries of the irreducible mean intensity vector components with respect to the inclination angle of the magnetic field. The upper left panel shows and the other panels , . Slab model with parameters and . The numbers near the curves refer to the values of . Notice the symmetry/anti-symmetry of the polarized components about , the small sensitivity of to the value of and also the relative magnitudes of and in comparison with

We list below the main properties of . Some of them have already been pointed out in the literature (see e.g. FS91 and the references therein).
is essentially independent of . Actually it is almost independent of the magnetic field as explained below.
depends weakly on and is approximately equal to the resonance polarization value which corresponds to . Near the surface (), it is in absolute value about ten times smaller than .
In absolute value and near the surface, the components and are roughly ten times smaller than .
All the polarization components , , change their sign at roughly the same optical depth .
The components of satisfy the symmetry relations :

These symmetries imply that for . When the components and are identically zero since the radiation field is axisymmetric.

The symmetries of are readily found by examining Eq. (61) with and the coefficients of the matrix given in Eq. (39). A change amounts to change the sign of while keeping unchanged. Hence the coefficients , , , , , , which go to zero when , change sign under this transformation. One can then easily check that it leads to the symmetries presented in Eq. (68).

Consider now the structure of the matrix product . The first column is identical to the first column of . Therefore the components and are coupled to and between themselves but not to . The component is coupled to but this coupling is independent of the magnetic field and fairly weak, as already discussed in Paper I, because it is controlled by the kernel . As for , its dependence on the magnetic field is almost negligible since all the coupling terms with the polarization components, which are anyhow much smaller in magnitude, involve only the kernel .

The optical depth at which and the other polarization components change sign is roughly determined by the optical depth at which the radiation field corresponding to the source function changes its angular dependence from limb darkening (at the surface) to limb brightening (in the interior). For this reason it is essentially independent of the magnetic field.

The values of at the middle of the slab and at the surface can be evaluated with the scaling laws proposed in Frisch (1988) for the Doppler profile. Correcting for a mistake in Eq. (4.6) of that paper, they may be written as

where is a mean value of the primary source term. For the model at hand, . These scaling laws yield good estimates if one chooses to evaluate at the middle of the slab and to evaluate it at the surface. One gets and . The exact numerical values are and .

5.2. Dependence of S on the co-latitude _B

In Fig. 7 we show the dependence of the six components of on . To obtain , it suffices to multiply by the matrix (or in the general case) (see Eq. (47)) and add the primary source term . For the model at hand can be neglected since it contributes only to the intensity component and is at least an order of magnitude smaller than ( as compared to approximately; see Fig. 7). The main properties of the six components of are easy to explain. Because and are much smaller than and is much smaller than , to evaluate the components of it is sufficient to keep the first and second column in the matrix written in Eq. (38). This approximation yields

We show in Fig. 6 the elements of the second column of as function of for different values of . Note that they are of order unity, except for which is identically zero. The approximations (70) to (72) explain why the polarization components of are of the same order of magnitude. Comparing Figs. 6 and 7 we see that the -dependence of the components of follows indeed closely the variation of the . We recall that is almost independent of (see Fig. 5). For example, near the surface, the decrease (in absolute value) of between (resonance polarization) and follows the decrease of the coefficient . Then between and , has a very small rise which also shows up in the variation of .

Fig. 6. Elements of the second column of the matrix as function of for various values of (i.e. of the magnetic field strength B). Solid lines : =0.1, dotted lines : =0.3, short-dashed lines : =0.5, dot-dashed lines : =1, triple-dot-dashed lines : =3, and long-dashed lines : =10. The case of =1 is highlighted for clarity. This figure is useful to understand the and dependence of the polarized source vector components , shown in the Figs. 7 and 9

Fig. 7. Symmetries of the irreducible source vector components with respect to . Same display and same atmospheric and magnetic parameters as in Fig. 5. The numbers near the curves refer to the values of . The Eq. (47) relates the components of source vector to the components of mean intensity , through the matrix. The dominant coupling originates from the elements of the second column , which are plotted in Fig. 6

5.3. A perturbative method based on approximate solutions

The results discussed in this section suggest a perturbative method for solving the transfer problem and an approximation to evaluate the Stokes vector. The perturbation method was used in FS91. The approximation is also to be found in FS91. The idea is to keep only the first two columns in the matrix product . The iterative method and the approximation have a common first step which is the calculation of and . They are obtained by solving a modified resonance polarization problem with the kernel

A simple approximation for can then be set up by using the Eqs. (70) to (72). In comparison with full PALI-H code, the errors on the Stokes parameters for the line centre frequency () are not very large (up to 20 %). In the line wings, they may reach a factor 2. They can be significantly reduced if one uses this first order approximation as starting solution for the iterative process described below. Just five perturbative steps are sufficient to bring them down to a fraction of a percent. In Sect. 6.6 we show the effect of this approximation on a polarization diagram.

To set up the iterative process, we must calculate the other harmonic components , . When only the first two columns of are kept, they are given by

with the kernel functions

These components are thus solutions of four scalar transfer equations with known source functions. The source function can then be obtained by applying Eq. (47) with the "full" (or ) matrix. The solution of Eq. (50) combined with Eq. (49) yields a new value for and the process can be iterated.

It must be stressed that keeping only the first and second columns of the matrix may be insufficient when one of the coefficients to becomes close to zero because the self and the harmonic cross-coupling terms which have been neglected may then become the dominant ones. For , the coefficient is close to zero (see Fig. 6), therefore, the component will not be properly evaluated. Similarly, for , it is the component which will not be correctly evaluated. These errors are however of little importance, for polarization diagrams in particular, since they affect only the smallest component of , but they seem to generate convergence problems in the FS91 iterative method of solution. When comparing our PALI-H results with non-perturbative solutions and with the FS91 perturbative ones, we found some discrepancies with the latter solutions actually for the smallest of the components.

5.4. Dependence of S on the azimuthal angle

We now let vary between 0 and , assuming that is independent of the optical depth. As shown below, the dependence on is then very simple.

Eq.  (61) and the factorization (48) suggest to introduce an auxiliary vector

Using , with the () identity matrix, and , it is easy to verify that satisfies the integral equation (61) with replaced by . Hence is independent of and is simply the reduced mean intensity vector for . Eq. (77) yields

and

It is clear that Eqs. (78) and (79) could have been obtained directly by making a Fourier expansion of the Stokes vector in harmonics of () as in FS91.

Fig. 8 shows the dependence on of the six components of for and . First we note that and are independent of . The components are -periodic and change their sign under the transformation . The components are -periodic and change their sign under the transformation . These properties are straightforward consequences of Eq. (79) and of the symmetry properties of which can be read in Eq. (42).

Fig. 8. Dependence of the irreducible source vector components on the magnetic field azimuth . Same display and same atmospheric parameters as in Fig. 5. Magnetic field parameters (, )=(). The numbers near the curves are the values of the azimuthal angle . The components , and do not depend on . The dotted line is drawn only to indicate zero polarization

5.5. Dependence of S on the field strength parameter _B

We now assume that and are fixed, and let vary. Fig. 9 shows the dependence of the 6 components of on for . When , only and are different from zero because of the axial symmetry of the radiation field. For values of , the Hanle effect saturates in the sense that, on further increase in , there is only a small change in the line polarization. Thus the range represents the sensitivity range of the Hanle effect to the changes in magnetic field strength B.

Fig. 9. Dependence of the irreducible source vector components on the magnetic field strength parameter . Same display and same atmospheric parameters as in Fig. 5. Magnetic field parameters (, )=(, ). The numbers near the curves are the values of . The case refers to the non-magnetic resonance scattering polarization. At small optical depths, reach their maxima for , and do so for . Thus a narrow range represents the value of peak sensitivity of a line to the Hanle effect. The effects of on emergent polarization at line centre are shown in Fig. 15

The properties of can be analyzed exactly as in Sect. 5.1 with replaced . The dependence of on is negligible and that of is fairly small (variation of 10% when increases from 0 to 3). According to the approximations (70) to (72) for , the -dependence of is a direct mapping of the -dependence of the elements in the second column of the matrix (see Fig. 6 and Eq. (39)).

We can check on Fig. 9 that the dependence of on is so weak that it cannot be detected on the graph. We see also that, for any , the surface value of monotonically approaches the zero level polarization, following indeed the monotonic decrease of when increases. At the surface decreases by a factor of 3 when increases from 0 to 1. This is the well known Hanle depolarization effect.

For the other components the situation is more complex because of the non-monotonicity of other . In the case of Fig. 9, the first harmonic components exhibit a `peak Hanle sensitivity' around , and the second harmonic components have their peak sensitivity for . By taking the limit or in Eq. (39), one obtains that the source function is dominated by in the former case and by the two components and in the latter case.

5.6. The effect of a depth-dependent azimuth on S

In this section we assume that varies with optical depth according to

where . This profile represents a strong exponential variation in the range of . It gives and . With this model we must solve the transfer equation on the full slab, as there is no symmetry about the mid-plane.

Fig. 10 shows the components of for several values of . This figure should be compared to Fig. 7. The approximation introduced in Sect. 5.3 can also be used here to analyze the results, provided that we replace the elements by the elements ( = , , ). As usual is essentially independent of the magnetic field parameters. Since , the component is almost insensitive to a variation of . For the other components, the most striking feature is the loss of symmetry with respect to their dependence on . In Sect. 6.3 we shall discuss the effect of a depth-dependent azimuthal angle on the emergent polarization.

Fig. 10. The effect of a depth-dependent azimuth on the irreducible source vector components . The depth-dependence of is given by Eq. (80). Same display and same atmospheric parameters as in Fig. 5. Magnetic field strength parameter . The numbers near the curves are the values of . Compare this figure with the corresponding case of depth-independent azimuth shown in Fig. 7. Notice that the symmetries of with respect to are broken by the depth-dependence of .

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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