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

6. The polarization diagrams

Useful tools for extracting magnetic field parameters from the observed Stokes parameters are plots of versus . There are different ways of constructing them (see for instance Bommier et al. 1991; FS91; Stenflo 1994). They have been successfully used for the determination of weak magnetic fields in prominences and in the upper solar atmosphere (references can be found in Stenflo (1994) or Faurobert-Scholl (1996)). Throughout this paper, we show the polarization diagrams for depth , line center (), and selected values of µ. Each diagram is obtained by letting the radiation field azimuth (or the magnetic field azimuth when it is depth-independent) vary in the range . In practice, only Q and U are dependent on . For the resonance scattering problem (), polarization is represented by a point on the axis. For the Hanle effect (), we get closed loops which are Lissajous curves since the azimuthal Fourier expansion of the radiation field is limited to second order terms.

These diagrams are easy to construct since Eqs. (29) and (45) and (46) allow us to explicitly express I, Q and U in terms of the 6 components of the irreducible intensity vector . We thus find

The Stokes parameters depend on W through the factor but also through the components of since W enters in the expression of the kernel of the integral equation for (see Eqs. (50), (51), (56) to (60)). For optically thick lines, there is no simple dependence of U and Q on W. A simple scaling with W holds only for optically thin lines or when using the last scattering approximation as in Stenflo (1982). In these cases the components , =Q, , are proportionnal to .

We note here that we could have chosen the negative root of W when introducing the vectors , (see Eqs. (9) to (13)). The components , =Q, , , would have had opposite signs which would have compensate for the minus sign in front of .

Eq. (81) shows that Stokes I depends on all the six components of . The dominant contribution by far comes from the first term , whatever the line of sight (LOS) and the magnetic field vector. Stokes Q depends on the last five components of . For small values of and a LOS close to the horizontal plane (µ small), the dominant contribution to Q comes from the term proportional to . Stokes U depends only on the last four components of . It is zero when the radiation field is axisymmetric. Equations (81) to (83) hold also for the components , , of the Stokes source vector provided the components of are replaced by the six irreducible source vector components .

The polarization diagrams are generated as follows. We first solve the transfer equation for the axially symmetric irreducible intensity vector with our PALI-H transfer code. We then calculate on a mesh of radiation field azimuths , using Eqs. (81) to (83). The points move in the anti-clockwise direction when is increased from to . When is constant, the polarization diagrams can be constructed by keeping constant and letting vary. Further, for CRD, Stokes Q and Stokes U show similar variation with frequency x : both have a single maxima at line centre and smoothly approach zero, or a constant value at the near wings (), depending on the values of and T (see Faurobert 1987; FS91). Hence, the polarization diagrams using the frequency averaged Stokes parameters exhibit similar shapes as the diagrams presented here for the line centre, except for a proportional decrease in the size of the diagrams, due to averaging. For all the polarization diagrams shown on Figs. 11 - 16, the slab model with the parameters , , and is used. Other parameters are noted in the figure captions, and on the figures.

6.1. Dependence on the radiation field co-latitude

In Fig. 11 we show the polarization diagrams for different values of . The magnetic field parameters are . As varies from the tangential () to the vertical direction (), the amplitude of variation of U decreases and so does the absolute value of Q. This leads to a decrease in the degree of linear polarization defined as . Note also the variation in the shape of the diagrams. It is due to the relative decrease of the first term in Eq. (82) with respect to the third one. In the extreme case of vertical LOS (), the polarization is non-zero, although it is very small as long as because of the contribution from the second harmonic in Q and U. This finite polarization in the vertical direction is known as "Hanle repolarization" (see Bommier et al. 1991) because it is strictly zero when . For ,

We also note that for a given value of , there can be 2 or 4 possible values of , and vice versa. Thus different LOS or different values of may lead to the same degree of linear polarization and also the same angle of rotation of the plane of polarization .

Fig. 11. -dependence of polarization diagrams at line centre. The numbers near the curves refer to the co-latitude of the LOS. The symbols on the curves correspond to different values of . The symbols : plus, asterik, triangle and square correspond respectively to , , and . Atmospheric parameters and magnetic parameters . Notice the expected limb-to-centre decrease in magnitude of the emergent linear polarization

6.2. Dependence on the magnetic field co-latitude _B

In Fig. 12 we show the polarization diagrams for three different values of . The other two magnetic field parameters are held fixed : . The co-latitude of the LOS is . The values of are chosen between and . The diagrams for between and can be obtained by symmetry with respect to the axis . Indeed, when and , Q does not change but U changes its sign. Hence, when , i.e. when the magnetic field vector is horizontal (see Fig. 1), the polarization diagrams are symmetric about the axis. When the LOS is also in the horizontal plane the diagram becomes infinitely thin and looks like an open ended line.

Fig. 12. -dependence of polarization diagrams at the line centre. The numbers near the curves refer to different co-latitudes of the magnetic field. Meaning of symbols and atmospheric parameters as in Fig. 11 ; magnetic field parameters . The co-latitude of the LOS is . Notice the symmetry of the diagram about axis for the particular case

6.3. Polarization diagram for a depth dependent azimuth

In Fig. 13 we compare the polarization diagrams for the cases of constant azimuth (), and a depth dependent azimuth given by Eq. (80). The other magnetic field parameters are . When , the diagrams are symmetric about the axis since we have chosen . This symmetry is broken by a depth dependent . Fig. 13 shows clearly, that the polarization diagrams are almost insensitive to the law used in our model. The sensitivity to the depth variation of strongly depends on the gradient of in the region of formation of the line core. We have noticed that the effects of a variable can become significant with a variation of or more within narrow layers near the surface of the slab (. The diagrams become very asymmetric about the axis, and reduce drastically in size. So unless one has good reasons to suspect a strong variation of within the line core formation region, assuming a uniform is a reasonable hypothesis, in modelling efforts.

Fig. 13. Polarization diagrams at line center, for a uniform (= ) and a depth dependent given by Eq. (80). Atmospheric parameters as in Fig. 11 and magnetic field parameters . Constant azimuth : full lines with bigger symbols. Depth dependent azimuth : dotted lines with smaller symbols. The numbers near the curves refer to co-latitude of the LOS. Notice the loss of symmetry about the axis when the magnetic field azimuth is depth dependent

6.4. Dependence on the magnetic field strength parameter _B

In Fig. 14 we present the polarization diagrams for different values of , the strength parameter of the magnetic field. The field direction is fixed at . The models are the same as those used for Fig. 9. The co-latitude of the LOS is . The non-magnetic resonance scattering polarization () yields the point %. For small values of (), the component is the dominant one (see Fig. 9), hence U is small and the diagrams are quite flat. When , the last five components are more or less of the same order, which explains the butterfly shape of the diagrams. Fig. 14 also shows clearly that for large values of (), the Hanle effect becomes negligible.

Fig. 14. The effect of magnetic field strength on the polarization diagrams at line center and of the LOS. Meaning of symbols and atmospheric parameters as in Fig. 11 ; magnetic field parameters . The different values of are indicated near the diagrams. Notice the strong increase in between and and the saturation of the depolarizing efficiency for large values of

6.5. The two-parameter polarization diagrams

As suggested in Bommier et al. (1991), the determination of magnetic field parameters from observational data can be attempted with the help of two-parameter diagrams showing a network of iso-strength and iso-azimuth curves. For a given LOS, determined by the values of and , one chooses a value of and vary the two other parameters of the magnetic field, and . Then in the plots of versus , one draws not only the iso-strength curves as in Fig. 14 but also the iso-azimuth curves. Fig. 15 shows such a diagram for a LOS with and and a magnetic field with . To draw this figure we have used the slab model () and varied in the range 0-100. Similar two-parameter diagrams for different choices of and are shown in Bommier et al. (1991).

Fig. 15. The two-parameter polarization diagrams for Hanle effect. Same atmospheric parameters as in Fig. 11. Magnetic field parameters , in the range (0-100) and in the range (0- ). The iso-azimuth curves (solid lines) are drawn by fixing and varying . The iso-strength curves (dotted lines) are drawn by fixing and varying . All the curves in the figure are symmetric about the axis. The LOS is fixed at . The results are presented for the line centre. The iso-strength curves show the Hanle depolarization and saturation effects clearly. The iso-azimuth curves show the effect of rotation of the plane of polarization when varies

When both the LOS and the magnetic field are lying in the horizontal plane, as in Fig. 15, the iso-strength loops become infinitely thin and look like open ended curves. Also it is sufficient to let vary in the range 0- to cover a whole iso-strength curve (dotted line). When is small the iso-strength curves are almost horizontal for the reason given in Sect. 6.4. When is large the two prongs of the curve become almost vertical. When , the ordinate of the lowest point reaches a limiting value given by .

The iso-azimuth curves (solid lines) are open ended lines which start at and end at . All the iso-azimuth curves merge at the point where and . The iso-azimuth curves for (and ) are straight lines which coincide with the axis.

When an observational data point falls within an approximate interval defined by , we get upper and lower limits on the possible values of and , which may further be used for modelling. This approach is reasonable when an independent estimate of is available. It is worthwhile to note that in order to construct a series of such two-parameter polarization diagrams corresponding to different values of the fixed magnetic field parameter, a fast method for the computation of emergent Stokes parameters is required. The approximation introduced in Sect. 5.3 can serve that purpose.

6.6. A fast method of generating the polarization diagrams

The method is very simple. One first solves a two-component polarized transfer problem with the kernel given in Eq. (73) to calculate and . The approximations (70)-(72) then yield all the components of . The only remaining task is the solution of six scalar transfer equations with known source functions.

When , the coefficients in (70)-(72) should be replaced by the coefficients . This amounts to making the changes,

where the and , are defined in Eqs. (43) and (44).

In Fig. 16 we compare polarization diagrams obtained with this approximation and with a full PALI-H iterative method. It is clear that the differences fall within the error bars of a standard measurement. The calculation of the approximate solution is a factor of 10 faster than the full PALI-H code. It is so fast that one can think of using it as part of an inversion code to set up estimates of the vector magnetic field.

Fig. 16. Polarization diagrams showing the relative accuracy of the approximation defined in Sect. 5.3. Same atmospheric model as in Fig. 11 with magnetic field parameters and different values of . The diagrams refer to line centre and the LOS co-latitude . The solid lines show the results of full PALI-H computations, and the dotted lines the approximate solutions. The relative errors are 20 %, or less

For the purpose of estimating the magnetic field, it is possible to use an even cruder version of the above approximation, already suggested in FS91. The components and and the components of are calculated as above. The surface polarization is then estimated with the Eddington-Barbier approximation, , with . For and negligible compared to unity, this approximation yields for the surface value of the Stokes parameters :

An approximation for the three components of the Stokes source vector constructed with the same method is given in FS91 (it contains however some misprints).

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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