2. General conditions for the solvability of the Hanle diagnostic problem
The Hanle depolarization due to a magnetic field can be expressed in terms of the factor , which represents the ratio between the observed polarization amplitude in the selected spectral line and the corresponding amplitude in the absence of a magnetic field. Thus
The in the upper index position of refers to the -multipole and the circumstance that it is the multipole with (the electric quadrupole) that is responsible for atomic alignment and the linear polarization (while (magnetic dipole, atomic orientation) governs the scattering of the circular polarization).
For the fields that we will be dealing with here the Zeeman splitting is much smaller than the Doppler width. In this case the Hanle depolarization factor for a turbulent magnetic field of strength with an isotropic distribution of field vectors is according to Stenflo (1982) given by
where m and e are the mass and charge of the electron, is the Landé factor of the excited level, and is the natural decay rate of the excited state (due to spontaneous emission). For a field of strength the Larmor precession frequency is comparable to the decay rate (or inverse life time) .
where is the collisional broadening rate, and represents the rate at which the -multipole is collisionally destroyed (Stenflo 1994; Faurobert-Scholl et al. 1995). thus represents the effective characteristic field strength of the Hanle effect that results when we compare the Larmor frequency with the total destruction rate of the atomic polarization of the excited state due to the combined effect of spontaneous decay and collisions.
If we could somehow determine the Hanle depolarization factor from observational data, we could find from Eq. (2) and from Eq. (3), provided that the effective characteristic field strength is known. Assuming that we can determine the line polarization from the observations (which turns out to be a complex task due to the continuum polarization, see below), still cannot be found directly from Eq. (1), since the amplitude in the absence of magnetic fields is in general an unknown quantity. Furthermore, while the field strength parameter can sometimes be found from atomic physics data (if the natural life time and Landé factor of the excited state are known), the collisional branching ratio depends both on the atmospheric densities and temperatures and on the atomic physics. This general problem was solved by Faurobert-Scholl (1993) and Faurobert-Scholl et al. (1995) by doing sophisticated radiative-transfer modelling to calculate and for the Sr I 4607 Å line.
If we want to avoid such radiative-transfer modelling, then we are stuck with three unknowns, , , and , while there is only one observable, (assuming that we are able to separate the line contribution from the observed total polarization of both continuum and line). The problem may thus appear unsolvable. This is however so only if we consider a single spectral line. The number of observables and unknowns multiply at different rates when we observe several spectral lines in several different solar regions. If the numbers of lines and regions are sufficiently large, one passes a cross-over, beyond which the number of observables is larger than the number of unknowns and the problem becomes solvable. Let us now make the book-keeping to see when this cross-over takes place.
Let us assume that we have an observational data set from which we are able to extract the empirical line polarizations for different spectral lines in different regions on the Sun, which all have the same limb distance but different field strengths . There are two unknowns ( and ) and one observable () per spectral line, while there is one new unknown () for each solar region. The number of unknowns is thus , while the number of observables is . The necessary (although not necessarily sufficient) condition for the problem to be solvable is then
A solution is thus possible only if we can observe the lines in at least 3 different solar regions with different turbulent field strengths . If , then we must have .
The problem can be extended further by adding more free parameters, but there will always be a cross-over such that with a sufficient number of spectral lines and solar regions observed, the number of observables will exceed the number of unknowns. For instance, there is always some uncertainty in the precise limb distance of a given observation, and since the polarization amplitude varies steeply with limb distance, one could apply a scaling factor as a free parameter to all the lines that are covered within the spectral field of view. This would compensate for an error in the limb distance (assuming that all the covered lines have a center-to-limb dependence with the same relative steepness). One could also introduce an extended parametrization to model the turbulent magnetic field and let it be described not just by a single field strength, but by a parametrized distribution of field strengths. The collisional depolarization factor could in principle also be allowed to vary from one solar region to the next.
The circumstance that one can always assemble a data set for which the number of observables exceeds the number of unknowns does however by no means guarantee that the problem can be solved, i.e., that the inversion will have a unique solution. For the inversion problem to be well conditioned it is necessary that the selected spectral lines have a large spread in their sensitivities to the Hanle effect (i.e., in their values), and that the selected solar regions have a large spread in their values (within the Hanle sensitivity range of the selected set of lines). Furthermore the empirical line polarization must be a well defined quantity. Surprisingly enough, this last point turns out to be the biggest problem.
If there were no continuum radiation, then in Eq. (1) would be the directly observed polarization amplitude in the line. In reality, however, the spectral lines are superposed on a background continuum that is polarized. For most lines near the solar limb (like at ) the continuum polarization is of the same order of magnitude as the line polarization, with the exception of a few particularly strongly polarizing lines, like Ca I 4227 Å , Sr I 4607 Å , and Na I D2. Unfortunately the line polarization cannot simply be measured from the continuum level, since for large Hanle depolarizations the observations show (see below) that the polarization in the line can lie far below the continuum level. The Hanle depolarization cannot lead to a change in sign of the polarization, only to a reduction of the line polarization amplitude with a certain, magnetic-field dependent factor. All values of the observed line polarization in Eq. (1) have to be positive to be "physical" and for the inversion procedure to work. To extract from the observations we have to define some reference level from which the observed polarization should be measured to represent the line polarization. However, due to the mixing of the polarizing line and continuum opacities this reference level is neither the polarization zero level nor the level of the continuum polarization beside the line, but something in between.
This problem is further aggravated by the circumstance that Stokes cross talk (due to instrumental polarization) makes the zero level of the polarization scale unknown when working with telescopes that are not polarization free (Stenflo & Keller 1997).
In spite of these difficulties we have carried out an observing project to measure a large number of spectral lines in as many solar regions as possible that have the same limb distance () but have clearly different Hanle depolarizations, which can be verified by direct inspection of the qualitative appearance of the polarized spectra. A heuristic way of defining a reference level for the line polarization is found after some experimentation with the data, and the inversion problem is solved after careful conditioning of the data set to avoid the condition that the problem is ill posed. The inversion finds turbulent field strengths in the range 4-40 G for the various solar regions. One main aim of this exercise is to expose the various problems that have to be addressed and solved before the Hanle effect can be considered to be a reliable and well established diagnostic tool for exploring magnetic fields on the solar disk.
© European Southern Observatory (ESO) 1998
Online publication: November 24, 1997