The Astrophysical Journal, 575:529–541, 2002 August 10 # 2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.
MAGNETIC FIELDS IN PROMINENCES: INVERSION TECHNIQUES FOR SPECTROPOLARIMETRIC DATA OF THE He i D3 LINE A. Lo´pez Ariste and R. Casini High Altitude Observatory, National Center for Atmospheric Research,1 P.O. Box 3000, Boulder, CO 80307-3000;
[email protected] Received 2002 January 10; accepted 2002 April 17
ABSTRACT We propose the use of principal component analysis (PCA) to invert spectropolarimetric data from prominences. Observation of the full Stokes profiles in prominences is very important for a deeper understanding of magnetic-field topology in these solar structures, and for the testing of theoretical models. The line formation problem, however, is complicated by the special conditions of prominences: anisotropy of light, low magnetic intensities, temperature and density ranges, etc. We created a code to solve this problem in the limit of optically thin plasma and of a collisionless regime, and use it in combination with PCA techniques to invert synthetic data. The results show that inversion is feasible. Subject headings: line: profiles — polarization — Sun: magnetic fields — Sun: prominences
1. INTRODUCTION
The measurement of magnetic fields in the solar photosphere has clearly shown that a reliable diagnostics of the magneticfield vector and of the general properties of the solar atmosphere demands use of spectropolarimetric data. We can expect that measuring magnetic fields in other solar regions or structures should require the same kind of observations. In the particular case of prominences, the work of Leroy and others (Leroy 1977, 1978; Leroy et al. 1977, 1984; Bommier et al. 1994) has shown that this is indeed the case. In that work, only linearly polarized signals were observed with low spatial resolution, and the line ˚ (see Fig. 2 below) polarization was integrated through a relatively large spectral band covering the He i D3 multiplet at 5876 A and other interesting lines. Both theoretical and observational studies of the He i D3 line formation in prominences indicated that the analysis of polarization profiles, including circular polarization, should indeed improve the potential for a diagnostics of the magnetic field topology and of the thermodynamic conditions in these structures (Landi Degl’Innocenti 1982; House & Smartt 1982). However, two different reasons hindered the development of appropriate diagnostic tools. First, the number of instruments capable of providing spectropolarimetric data in prominences was scarce at best. To our knowledge, the Stokes I and II instruments (Baur, House, & Hull 1980; Baur et al. 1981) were the only ones to provide data good enough for use. Second, the full analysis of spectropolarimetric data requires the use of inversion techniques based on forward models, which have started being developed and systematically applied only since the 1990s, limited to the diagnostics of photospheric magnetic fields (Skumanich & Lites 1987; Ruiz Cobo & del Toro Iniesta 1992). While the interest in pursuing extensive and reliable measurements of magnetic fields in prominences is still growing, we believe that some of the past problems have now been overcome. New instruments are becoming available that concentrate on spectropolarimetric capabilities. The French-Italian telescope THEMIS, in particular, has already provided full Stokes profiles of the He i D3 line in prominences (Paletou et al. 2001). In addition, the original, heavy least-squares inversion techniques are being replaced by PCA algorithms, which are much faster and, more importantly, much more stable (Rees et al. 2000; Socas-Navarro, Lo´pez Ariste, & Lites 2001; Lo´pez Ariste 2000). In this work we show how to apply PCA techniques to the problem of inversion of the He i D3 line with full Stokes polarimetry. The tests are done on synthetic data, although some results of fitting real data are presented in x 4. We use a line formation code solving for the statistical equilibrium of a quantum model of the He atom with five atomic terms, and in the presence of magnetic fields. In particular, coherences between fine-structure levels within each atomic term are completely accounted for, allowing for the treatment of both Hanle and Zeeman regimes, including level crossing (incomplete Paschen-Back effect). The main simplification in this code is the absence of radiative transfer. Although the He i D3 line is usually optically thin, this is a constraint that should be relaxed in the future, for a reliable modeling of prominence magnetic fields. We describe this line formation code in the next section. Section 3 is dedicated to the tests performed on the inversion of synthetic profiles, in demonstration of the viability of the PCA approach. In x 4 we apply our code to some real data, although we omit any interpretation of the results. 2. THE LINE FORMATION PROBLEM
The first requirement in the PCA approach to the inversion of Stokes profiles is the completeness of the database against which the observations are to be compared. Although in the present paper we are mainly treating an inversion of prominence data, nonetheless we opt for a reasonably complete coverage of both prominence and filament configurations in the solar atmosphere for our database. 1
The National Center for Atmospheric Research is sponsored by the National Science Foundation.
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Fig. 1.—Geometry of the scattering event at the point O, located at a height h over the solar surface. The direction of y y0 identifies the parallel to the limb, which is assumed as the reference direction for positive Q polarization. The direction of the magnetic field, B, is identified by the angles #B and ’B in the reference frame of the local vertical, and by B and B in the reference frame of the propagation direction, k.
Figure 1 shows a typical geometry of a scattering event. All the different angular parameters involved contribute to the determination of the polarization properties of the scattered radiation. In addition, the strength of the magnetic field and the height of the scattering center at O over the solar limb also play an essential role. The magnetic field strength sets the regime of ‘‘ atomic decoherence,’’ determining whether the Hanle effect (partial decoherence) or the Zeeman effect (complete decoherence) provide the correct description of the scattering event in the magnetized plasma. The height of the scattering center over the photosphere (which is supposed to be the principal layer of the solar atmosphere illuminating the prominence or filament, in the spectral domain of the He i D3 line) determines the dilution and anisotropy factors of the incident radiation field.2 Whereas the dilution factor is usually unimportant for the interpretation of Zeeman-effect observations (unless one is interested in absolute intensity measurement), it plays a fundamental role in the Hanle-effect regime, as it fixes the radiative lifetime of the metastable level 3 S1 of He i due to absorption processes, and therefore the magnetic regime of Hanle depolarization for that level. On the other hand, the anisotropy factor is always an essential ingredient, since it determines the amount of atomic polarization induced by the incident radiation field (in the absence of magnetic field and of depolarizing collisions), and therefore directly affects the amount of linear polarization observed in the scattered radiation. One of the main assumptions is that the incident radiation field is not polarized, and that it possesses cylindrical symmetry around the local solar vertical drawn from the scatterer. This assumption is a good one if there are not bright active regions contributing to the incident radiation field, and if the plasma between the photosphere and the scattering atom is optically thin. However, this restriction will be removed in the near future, in view of a full radiative-transfer implementation of the code. A further important assumption is that the limit of complete frequency redistribution (CRD) in frequency holds, which is equivalent to the requirement that the incident radiation field be spectrally flat over a frequency range comparable to the fine-structure separation of the J levels. We now concentrate on the local problem of the statistical equilibrium (SE) of the He i atoms including atomic polarization. This problem has already been treated in some detail by Landi Degl’Innocenti (1982). However, the hypothesis of CRD allows for some simplifications of the radiative rates with respect to those considered in that paper. For this reason, we re-present here the essential formalism of the SE problem for He i as treated by Landi Degl’Innocenti (1982). We are aware that more sophisticated treatments may be needed for a realistic treatment of this problem, including, for example, collisional coupling between 2 For the inversion tests performed in this work, we neglected limb darkening. Therefore, the anisotropy factor is only determined by the scattering geometry.
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Fig. 2.—Atomic model for the triplet states of He i considered in this work. The infrared transitions between the levels of the third atomic term are also accounted for in the solution of the SE problem, although they are not explicitly drawn in this figure.
the singlet and triplet species and the effect of UV coronal radiation on the ionization balance (see, e.g., Andretta & Jones 1997 and references therein; in these works, however, atomic polarization is consistently neglected). Nonetheless, the simple model we use reproduces the main features observed in the Stokes profiles of the He i D3 line, at least in the optically thin limit, and we therefore consider it sufficient for the purposes of this work. The solution of this problem is rather complicated, because of the atomic structure of He i (see Fig. 2). Even if we assume that L-S coupling is a reasonably good approximation, the smallness of He i fine structure does not allow us to neglect quantum interference between J levels within a given LS term. This ‘‘ multiterm ’’ picture is sensibly more complicated than the ‘‘ multilevel ’’ picture (where J interferences are negligible), and forces us to solve for density-matrix elements of the form LS ðJM; J 0 M 0 Þ. The collisional problem is left aside in this treatment, relying on previous finding that collisions are inessential for the SE of bound-bound transitions within He i (see, e.g., Bommier 1980). With the previous assumptions, the SE equations, written in the frame of reference of the magnetic field (i.e., assuming the magnetic-field direction as the quantization axis of the atomic system), can be written in the form (cf. Landi Degl’Innocenti 1982) d dt
LS K Q ðJ; J 0 Þ
¼
XX
LS K 0 Q0 ðJ 00 ; J 000 Þ
iKM ðLS; KQJJ 0 ; K 0 Q0 J 00 J 000 Þ
K 0 Q0 J 00 J 000
þ RA ðLS; KQJJ 0 ; K 0 Q0 J 00 J 000 Þ þ RE ðLS; KQJJ 0 ; K 0 Q0 J 00 J 000 Þ X X l Ll S K l þ Ql ðJl ; Jl0 ÞTA ðLSKQJJ 0 ; l Ll SKl Ql Jl Jl0 Þ Kl Ql l Ll Jl Jl0
þ
X X
u Lu S Ku Qu ðJu ; Ju0 Þ
TE ðLSKQJJ 0 ; u Lu SKu Qu Ju Ju0 Þ :
ð1Þ
Ku Qu u Lu Ju Ju0
In this equation, the atomic density matrix is expressed in terms of its irreducible spherical tensor components, which are related to the density-matrix elements in the standard representation (of the eigenstates of the atomic Hamiltonian) through the formula (e.g., Landi Degl’Innocenti 1982) LS K Q ðJ; J 0 Þ
¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ð1ÞJM 2K þ 1 M MM 0 X
J0
K
M 0
Q
LS
ðJM; J 0 M 0 Þ :
ð2Þ
The first contribution to the right-hand side of equation (1) is responsible for the atomic depolarization from both J-interferences and the magnetic field. The expression of this ‘‘ magnetic kernel ’’ is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K K 0 1 0 KM ðLS; KQJJ 0 ; K 0 Q0 J 00 J 000 Þ ¼ KK 0 QQ0 JJ 00 J 0 J 000 !LSJ;LSJ 0 þ QQ0 !B ð1ÞJþJ Q ð2K þ 1Þð2K 0 þ 1Þ Q Q 0 " # 0 K K 1 K K0 1 KK 0 000 0 00 JJ 00 ð1Þ LS ðJ ; J Þ 000 ; ð3Þ þ J 0 J 000 LS ðJ; J Þ 00 J J J0 J J J0
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where !B is the Larmor frequency corresponding to the local magnetic field, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LS ðJ; J Þ ¼ JJ 0 JðJ þ 1Þð2J þ 1Þ þ ð1ÞLþSþJþ1 ð2J þ 1Þð2J 0 þ 1ÞSðS þ 1Þð2S þ 1Þ 0
J S
J0 S
1 L
ð4Þ
is a generalized Lande´ factor, which for J 0 ¼ J reduces to a quantity that is proportional to the ordinary Lande´ factor in the LS-coupling approximation, gLS ðJÞ. The next two contributions to equation (1) describe the ‘‘ relaxation ’’ processes of the atomic LS-term for which the SE equation is being written. These radiative processes correspond to absorption processes toward higher terms (RA ) and emission processes (both spontaneous and stimulated) toward lower terms (RE ). Their expressions are X 0 RA ðLS; KQJJ 0 ; K 0 Q0 J 00 J 000 Þ ¼ ð2L þ 1Þ BðLS ! u Lu SÞð1ÞLu SþJþQ þ1 u Lu
X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L Kr K K 0 Kr K 3ð2K þ 1Þð2K 0 þ 1Þð2Kr þ 1Þ JQrr ð!u Lu S;LS Þ 0 1 1 L Q Q Q u r Kr Qr " pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L Kr K K 0 Kr 1 KþK 0 þKr 0 000 JJ 00 ð1Þ ð2J þ 1Þð2J þ 1Þ 2 J 000 J 0 S J 000 J 0 J pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L Kr K K 0 Kr J 00 J 0 00 þ J 0 J 000 ð1Þ ð2J þ 1Þð2J þ 1Þ ; J 00 J S J 00 J J 0 RE ðLS; KQJJ 0 ; K 0 Q0 J 00 J 000 Þ ¼ KK 0 QQ0 JJ 00 J 0 J 000
X
AðLS ! l Ll SÞ þ ð2L þ 1Þ
l Ll
X pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L L 3ð2K þ 1Þð2K 0 þ 1Þð2Kr þ 1Þ 1 1 Kr Qr " L 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 JJ 00 ð1ÞKþK ð2J 0 þ 1Þð2J 000 þ 1Þ 2 J 000 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L 00 0 þ J 0 J 000 ð1ÞJ J þKr ð2J þ 1Þð2J 00 þ 1Þ J 00
X
ð5Þ
0
BðLS ! l Ll SÞð1ÞLl SþJþQ þ1
l Ll
L
K K 0 Kr J Kr ð!LS;l Ll S Þ Q Q0 Qr Qr K K 0 Kr Kr
J0
S
Kr Ll
L J
Kr S
J 000
K J 00
J0 K0 J
J
Kr J0
:
ð6Þ
In equations (5) and (6), JQK ð!ul Þ are the irreducible spherical tensors of the incident radiation at the frequency !ul between the upper and lower term of an atomic transition. These tensors are related to the Stokes parameters, Si ð!ul ; kÞ (with i ¼ 0, 1, 2, 3 for I, Q, U, and V, respectively), of the incident radiation along the propagation vector k, through the formula I 3 X d k 3 K T ði; kÞ Si ð!ul ; kÞ : ð7Þ JQK ð!ul Þ ¼ 4 i¼0 Q The geometric tensors TQK ði; kÞ are tabulated by, e.g., Bommier (1997), as functions of the angles specifying the direction of propagation, k, and the reference direction for linear polarization measurement on the plane normal to k, in the reference frame of choice (in this case, the one defined by the magnetic field). Since we are assuming that the incident radiation field is unpolarized (Si ¼ 0, for i ¼ 1, 2, 3), only the orders K ¼ 0 and 2 of the radiation field tensors do not vanish, because of the form of the tensor TQK ð0; kÞ. In particular, J00 represents the average intensity of radiation within the radiation cone illuminating the atom, whereas JQ2 represents the anisotropy tensor of the incident radiation in the magnetic reference frame. We note that the magnetic kernel and the relaxation rates relate density-matrix elements pertaining to the same term. In particular, in the absence of radiative processes [JQK ð!ul Þ ¼ 0], the magnetic kernel is responsible for the well-known phenomenon of oscillation of atomic coherences in an isolated atom. In the presence of radiative processes, instead, the relative importance of the magnetic kernel to the relaxation rates determine the depolarization regime of the atomic term. If the average lifetime of the atomic term is very large with respect to the Larmor precession time and/or the inverse of the atomic fine-structure separation frequency, then the atomic coherences for that term are efficiently relaxed. In this asymptotic (or saturated) regime of the Hanle effect, the SE of the atom becomes insensitive to the magnetic-field intensity. In this paper, on the contrary, we are interested in the complementary regime of magnetic fields, such that magnetic decoherence in the atom is only partially achieved (Hanle effect). In reality, the picture presented in the previous paragraph is an oversimplification of the real phenomenon of atomic decoherence. Radiative transitions from upper and lower atomic terms can transfer atomic polarization to the final term of the transition (i.e., the one for which the SE equation is being written). These transfer rates are given by TA and TE in equation (1),
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for absorption and emission processes, respectively. Their expressions are 0
TA ðLSKQJJ 0 ; l Ll SKl Ql Jl Jl0 Þ ¼ ð2Ll þ 1ÞBðl Ll S ! LSÞ ð1ÞKl þQl þJl Jl i1=2 Xh 3ð2J þ 1Þð2J 0 þ 1Þð2Jl þ 1Þð2Jl0 þ 1Þð2K þ 1Þð2Kl þ 1Þð2Kr þ 1Þ K r Qr
L Jl
Ll J
1 S
L Jl0
Ll J0
8 J > 1 < 0 J S > : K
Jl Jl0 Kl
9 1 = > K Kl 1 > ; Q Ql Kr
Kr J Kr ð!LS;l Ll S Þ ; Qr Qr
ð8Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2J þ 1Þð2J 0 þ 1Þð2Ju þ 1Þð2Ju0 þ 1Þ L Lu 1 L Lu 1 Ku þQu þJu0 Ju
TE ðLSKQJJ 0 ; u Lu SKu Qu Ju Ju0 Þ ¼ ð2Lu þ 1Þ ð1Þ "
Ju
J
S
Ju0
J0
J J0 K Ju0 Ju 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð2K þ 1Þð2Ku þ 1Þð2Kr þ 1Þ 0
KKu QQu Aðu Lu S ! LSÞð1ÞJu þJ Qu þ1 þ Bðu Lu S ! LSÞ 8 > : K
X
ð1ÞKr
S
Kr Qr
Ju Ju0 Ku
9 1 > = K 1 > ; Q Kr
Ku Qu
# Kr Kr J ð!u Lu S;LS Þ : Qr Qr
ð9Þ
0 The solution of the SE problem is ultimately determined by imposing the stationary condition ðd=dtÞLS K Q ðJ; J Þ ¼ 0. This 0 Þ. These unknowns, however, are not comðJ; J yields a homogeneous linear system of equations for the unknowns LS K Q pletely independent, because of the additional condition of conservation of the atomic number density, which can be expressed P through the normalization condition J ð2J þ 1Þ1=2 LS 00 ðJ; JÞ ¼ 1. The usual procedure is then to replace one of the SE equations with this normalization condition, in order to transform the original homogeneous linear system with null determinant into a new inhomogeneous linear system with non-null determinant. The solution of the new linear system is finally used in the expression of the Stokes parameters of the scattered radiation, which in the optically thin limit considered here are simply given by the emission coefficients for polarized radiation. These emission coefficients are given by equation (12) of Landi Degl’Innocenti (1982). The complexity of that expression (not reported here) is a consequence of its generality, which is needed to correctly represent the regime of the incomplete Paschen-Back effect, where quantum interferences between distinct J levels become important. This is sometimes referred to as a regime of ‘‘ level crossing,’’ since those J-interferences are mainly due to the interaction of Zeeman sublevels belonging to different J levels, which is obviously stronger when those sublevels are so close in energy that they finally ‘‘ cross each other.’’ The presence of this interaction is apparent from the fact that the magnetic contribution to the atomic Hamiltonian is formally nondiagonal in J. A correct calculation of the scattered radiation implies then the additional complication of the numerical diagonalization of the magnetic Hamiltonian, which is needed to determine the exact frequencies of all Zeeman components of each atomic transition in the model atom. The numerical code used to derive the results presented in this paper takes fully into account the possibility of level crossing in the atomic terms, and can therefore be applied to a wide range of magnetic field strengths.
3. INVERSION WITH PCA TECHNIQUES
Principal component analysis (PCA) is a widely used technique for pattern recognition. Although previously adopted for the analysis of stellar (Bailer-Jones 1997) and galactic spectra (Glazebrook, Offer, & Deeley 1998), the first application to Stokes spectra was done by Rees et al. (2000). The reader is referred to that paper for the fundamentals of this technique. The first real-data application was directed to the inversion of photospheric lines (Socas-Navarro et al. 2001). The problem of measuring magnetic fields in the photosphere had been successfully solved by adopting more usual least-squares fitting techniques (Skumanich & Lites 1987; Ruiz Cobo & del Toro Iniesta 1992). The use of PCA was, in this case, originally motivated by several drawbacks in those least-square fitting techniques, in particular their slowness when compared to the usual data-flow rates of modern instrumentation (Lo´pez Ariste et al. 2001). PCA solved that problem and provided us with an extremely fast inversion technique. At the same time, it brought along another important advantage: whereas the least-squares fitting technique requires a model atmosphere hard-wired in the code, which ultimately affects the efficiency of the inversion algorithm itself, PCA is model-independent in the sense that the algorithm simply performs a comparison of the observed data with a predefined database. The limitations of the model atmosphere adopted therefore only affect the creation of the database, whereas the efficiency of the PCA algorithm is completely independent of them.
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The only problem left is the one of creating a comprehensive database using the algorithm described in the previous section, and of applying the PCA techniques described in the papers cited in the previous paragraph. Our first approach was to launch a random generator of model parameters, in order to create a database of Stokes profiles from which we could extract a complete (in the vector-space sense) set of eigenprofiles. Around 20,000 different parameter configurations were initially included in the database. The varied parameters were (see Fig. 1): the direction of the magnetic field with respect to the solar vertical, ð#B ; ’B Þ, within the full 4 solid angle; the magnetic field strength, B, between 0 and 50 G; the inclination of the line of sight (LOS) from the vertical, #, between 0 and 120 ; and finally, the height of the scatterer, h, between 0 and 0.5 R. The thermodynamic conditions were fixed, assuming a temperature of 10,000 K and no microturbulence. The profiles in the database are used to create a coherency matrix. Singular value decomposition (SVD) of this matrix provides us with an ordered basis of eigenprofiles, the ordering being determined by the relative norm of the associated singular values, so that the first eigenprofile corresponds to the largest (in norm) singular value. Any Stokes profile in the database can be reconstructed as a linear combination of all eigenprofiles. However, we tested that retaining only the first 10 eigenprofiles is generally sufficient for a reconstruction of any profile in the database well beyond the usual noise levels (better than 104 times the intensity in the core of the line), for all the Stokes parameters. We therefore assume that any observed Stokes profile can also be expressed as a linear combination of those 10 eigenprofiles for each Stokes parameter. The resulting 4 10 coefficients of such a linear combination, the eigencoefficients, represent the observed Stokes vector and will be used for the purpose of comparison as required by the PCA inversion techniques. Figure 3 shows the first two eigenprofiles for each of the four Stokes parameters, for the configuration space corresponding to the selected database. The first eigenprofile is always the average of all the profiles in the database, perhaps inverted in sign. (This can easily be seen in Stokes I, where we recognize the negative of the usual emission profile of the He i D3 line with its two visible components.) When comparing Stokes parameters, we remark that the main difference between the average profiles of Stokes I and Q is the different ratio between the two components, indicating a higher polarizability of the red component. It is also worth mentioning that the eigenprofiles for Stokes Q and U are identical, as a consequence of the symmetries involved in the formation of the line. Finally, the first two eigenprofiles for Stokes V can be identified as the contributions from the Hanle effect and the Zeeman effect, respectively. Thus, Stokes V eigenprofiles indicate the presence of a Hanle-induced, asymmetric contribution to the circulation-polarization profile (e.g., Landi Degl’Innocenti 1982), but they also imply that this
Fig. 3.—First two eigenprofiles for each of the four Stokes parameters of the He i D3 multiplet in prominences. Arbitrary units are used for each Stokes parameter. Different eigenprofiles of a particular Stokes parameter share however the same unit scale.
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signal is, for the typical conditions in prominences, not at all negligible when compared to the better known antisymmetrical Zeeman signal. In fact, for the database used to compute the eigenprofiles of Figure 3, PCA indicates that the Hanle, asymmetric signature occurs more often than the Zeeman, antisymmetric signature. This, of course, is a consequence of our choice for the interval in which B is varied, which excludes relatively strong fields. On the other hand, since this database, although intended to be complete, does not report on the frequency of appearance in prominences of magnetic configurations that would lead to one or the other contribution, we cannot conclude that this predominance is true in general. However, we can assert that both the Hanle effect and the Zeeman effect are important in determining the shape of the Stokes V profile, for the range of magnetic field strengths considered. We determined that the use of random models to build the database is a very inefficient approach, since we inevitably overrepresent some regions of the space spanned by the eigencoefficients (the PCA space) in spite of others. (This is a consequence of the fact that the application from the space of model parameters to the PCA space is not norm-preserving.) We therefore implemented a sieve algorithm, which checks whether the norm in the PCA space (see x 3.1 for a discussion of the definition of this norm) from the proposed random model to the nearest already present in the database is larger than a predetermined threshold. In the affirmative case, the new model is accepted; otherwise it is rejected. In our problem, the threshold was determined after observation of the distance equivalent to a noise of 104 of the intensity of the line core emission. Through the use of this sieve algorithm we created databases of up to 100,000 models uniformly distributed in the PCA space. Another set of random models (no sieve algorithm applied) was eventually used as observations and, after addition of white noise with a standard deviation of, respectively, 103 and 104, in two different experiments, the resulting profiles were inverted using the previously built database. Although the first tests allowed for a high degree of freedom in all the model parameters described in the preceding section, we eventually turned to restricted parameter spaces. The reason for this was practical: although the initial database was built so as to provide eventually the same precision shown in later examples, its high dimensionality implied huge databases, difficult to store and to handle. Tests were done confirming that coarse databases, although not useful for inversion, were still able to put the inverted profile in the correct regions of the parameter space, for example, in the region of low magnetic fields. This result allows one to implement an iterative inversion scheme in which coarse and easy-to-handle databases place a given observed profile in restricted regions for which finer mesh databases have been built. From now on we limit ourselves to those restricted cases, and use them as benchmarks for the test of the PCA technique, without loss of generality. Observations are generally able to fix some of the parameters that we include in our models with a sufficient degree of precision. This fact led us to consider two particular cases for our restricted databases, in which the height and the inclination of the LOS were fixed to values of 1800 3 and 90 (prominence in the plane of the sky [POS]), respectively. Figures 4 and 5 show 3
This value for the height was adopted in order to reproduce the conditions in the recent observations by Paletou et al. (2001).
Fig. 4.—Histograms of the errors in some of the model parameters considered for a fixed-height (1800 above the solar limb) prominence. For each inverted parameter, we compute the values of the maximum errors for 50% and 90%, respectively, of the total amount of models inverted.
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Fig. 5.—Histograms of the errors in some of the model parameters considered for a prominence in the plane of the sky, but variable height
histograms of the errors made in the various parameters involved in the inversion for these two different special cases. Those figures show that the magnetic field vector and the geometry of the LOS are in general determined with satisfactory precision. When looking in detail to the particular distribution of errors in the parameters, we notice several features worth mentioning. It is quite interesting, for example, to note the tail of the error distribution of the inclination #B of the magnetic field with respect to the local solar vertical. That tail is due to well-known ambiguities in the polarization by resonance scattering (see, e.g., Bommier 1980; Casini 2002), involving a loss of information in the parameter #B . The validity of the previous statement can be checked by looking at the error distribution of the corresponding angular parameter in the observer’s reference frame, B , showing similar errors but without the extended tail. We note that only plots of absolute errors are shown. In fact, relative errors are only meaningful when the responsiveness of the Stokes profiles to changes in the model parameters are much larger than both noise and the error introduced by the discrete character of the database. For instance, for the range of magnetic strengths considered in the database, one can expect a higher responsiveness to B of the linear-polarization Stokes profiles, Q and U, between approximately 0.1 and 20 G, where the Hanle depolarization of the upper term of He i D3 is fastest. However, from Figure 4 we see that an absolute error of 2 G must be accounted for in the inversion, which is essentially due to the coarseness of a database with ‘‘ only ’’ 100,000 points. For the case of the prominence in the POS (Fig. 5), we retrieve the height of the scatterer center as well. The observed errors are numerically acceptable; however, they still allow for many cases in which the error of the inversion is much worse than what can be obtained by a simple measurement of the position of the telescope above the solar limb and from a rough knowledge of the topology of the prominence. This is an indication that observational constraints should be introduced in the analysis. 3.1. Distances in the Eigenfeature Space Inversion with our PCA algorithm is performed by selecting the model in the database whose distance to the observed point is a minimum. The database is searched globally, thus avoiding the problem of local minima, so common and troublesome with other inversion techniques (least-squares fitting and related methods for example). The only ambiguity left in this search for the minimum distance is the actual definition of distance. Each of the four Stokes parameters is treated independently. Sets of eigenprofiles are calculated for each one, and the mathematical theorem for singular value decomposition ensures (see, e.g., Golub & Van Loan 1993) that the eigenprofiles thus obtained form an orthonormal basis of the space spanned by the Stokes profile. We should therefore make use of the Euclidean norm inside each of the four spaces. How should the four spaces be combined, however, into a meaningful distance? Any such distance between two models should take into account the position of the models in the four eigencoefficient spaces. Since the intensity profile shows much larger values than any of the other Stokes profiles (at least 10 times larger), so do the corre-
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sponding eigencoefficients. It is then obvious that a given absolute Euclidean distance between two models in Stokes I might translate into a small difference in the actual intensity profile, which can be safely neglected, while the same absolute distance in the Stokes V space would produce completely different profiles. We conclude that distances in any of the spaces should be weighed in a ratio inversely proportional to the typical signals of the corresponding Stokes profiles. For instance, if it is observed that Stokes I is usually about 100 times larger than Stokes Q and U, and about 1000 times larger than Stokes V, then one could use weights 1 : 100 : 100 : 1000 for the distances in I, Q, U, and V, respectively. On the other hand, we have observed that whenever Stokes V is above the noise level, a good fitting of it ensures smaller errors in the vector magnetic field, even when Stokes Q and U are not fitted just as well. Therefore, we decided to adopt a local weight equal to the inverse of the amplitude of the profile, whenever it happens to be above the noise level, and equal to the noise level otherwise. In this approach, the weaker profiles become the most relevant for inversion purposes. We have not found other justification for this weighting of the four eigencoefficient spaces than our intuition regarding the importance of the different Stokes parameters and their usual observed amplitudes, as well as the fact that by using it the statistics improves greatly. More work is necessary in this direction. 3.2. The Physical Meaning of the Eigenprofiles Skumanich & Lo´pez Ariste (2002) have shown that the eigencoefficients obtained from the analysis of photospheric lines formed in the magnetic regime of the Zeeman effect contain direct physical information. For example, the first eigencoefficient of Stokes V appears to be proportional to the longitudinal magnetic flux. The question arises whether a direct physical interpretation can also be given of the eigencoefficients of resonance-scattering Stokes profiles formed in the magnetic regime of the Hanle effect. As a guideline for what kind of dependence we should seek between the eigencoefficients and the physical parameters of the model atmosphere, we relied on the Hanle diagrams that were first proposed by Bommier & Sahal-Bre´chot (1978) as a diagnostic tool of vector magnetic fields in prominences. We considered the particular example illustrated by Landi Degl’Innocenti (1982) of a prominence in the POS observed at a height of 7000 above the solar limb, and with the magnetic field perpendicular to the local vertical. For this particular geometry, we calculated a series of models with magnetic field intensities varying from 0 to 60 G, and azimuths varying from 0 to 180 . From the resulting Stokes profiles of the He i D3 line we can select some spectral band centered on each of the visible components of the line, and compute qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Q 2 þ U 2; p¼ Q ¼ arctan ; ð10Þ 2 U and U are the integrals of the Q and U Stokes profiles in the selected spectral bands. Plotting p against , we would where Q then obtain the Hanle diagrams of Figures 5 and 6 of Landi Degl’Innocenti (1982; see also our Fig. 7). This procedure is analogous to the one adopted to create Hanle diagrams from real spectropolarimetric data. From the standpoint of PCA, we can imagine a procedure of taking the first eigencoefficients of the Q and U Stokes profiles, c0Q and c0U , and build the two following quantities: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c0Q 1 p0 ¼ ðc0Q Þ2 þ ðc0U Þ2 ; 0 ¼ arctan 0 ; ð11Þ 2 cU and plot them analogously to the Hanle diagram. The result is shown in Figure 6. The similarities between this diagram and
Fig. 6.—Hanle-like diagram for the zeroth eigenprofile (p0max =c0I ¼ 0:0550). Solid lines follow isoinclination models. Dashed lines link models with equal magnetic field strength.
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Fig. 7.—Hanle-like diagrams for the zeroth (p0max =c0I ¼ 0:0534; left panel) and first (p0max =c0I ¼ 0:0129; right panel) eigenprofiles, after rotation to isolate the red and blue lines (see text). See Fig. 6 for line identification. These diagrams are identical in shape to the ones obtained by measuring Stokes Q and U amplitudes in the blue and red components, respectively.
the one in Figure 5 of Landi Degl’Innocenti (1982) are striking. The combinations (eq. [9]) of the first eigencoefficients of Stokes Q and U appear to determine patterns similar to those of the linear polarization of the blue component of the He i D3 line. Nonetheless, there are some differences between the two figures. The reason for this is that the usual Hanle diagrams are built independently for the two visible components of the line, through the procedure outlined above, whereas the eigencoefficients take into account the full profile, spanning both components. In order to make a closer comparison with the Hanle diagrams, the eigenprofiles should be restricted to one or the other of the two components. Looking back at the eigenprofiles for Stokes Q and U shown in Figure 3, we see that the two visible components have the same sign in the first eigenprofile, but opposite signs in the second eigenprofile. As shown by Skumanich & Lo´pez Ariste (2002), this is an indication that the spectropolarimetric information contents of the two components are, in some sense, independent. In this case, from Skumanich & Lo´pez Ariste (2002) we see that it is possible to isolate the spectropolarimetric information of one or the other of the two components through an adequate combination of the first two eigenprofiles of Q and U. In order to preserve the orthonormality of the set of eigenprofiles, we require that this combination be a rotation. An angle of 15 in the case of Q, and of 12 in the case of U, succeeds in isolating the two components. Therefore, in the case of Q, 0 q0 q0 cos 15 sin 15 ¼ ; ð12Þ q01 q1 sin 15 cos 15 and analogously for U. The new zeroth-order eigenprofiles, q00 ðÞ and u00 ðÞ, contain exclusively the blue component, and computation of the quantities p0 and 0 with the corresponding eigencoefficient results in the diagram shown in the left panel of Figure 7. This time the diagram is identical in all of its details to the one shown in Figure 5 of Landi Degl’Innocenti (1982). Analogously, it is found that the eigencoefficients of the new first-order eigenprofiles, q01 ðÞ and u01 ðÞ, represent exclusively the red component, and the resulting 0 -p0 diagram is identical to the one shown in Figure 6 of Landi Degl’Innocenti (1982) (right panel of our Fig. 7). We can conclude that the first two eigencoefficients of Stokes Q and U contain the same physical information that can be retrieved by polarization filtergrams of both components of the He i D3 line. In other words, a PCA inversion based on only these two eigencoefficients allows the same diagnostics done in the past on the vector magnetic field of prominences (see, e.g., Querfeld et al. 1985). Furthermore, our inversion scheme not only includes Stokes V, but also extends from 2 to 10 the number of eigencoefficients adopted in the diagnostic procedure. Therefore, we should expect a sensible improvement of the inversion results. Figure 8 shows scatter plots for the magnetic field strength, B, and the inclination in the local reference system, #B , in the case of a prominence located at a height of 1800 . The dispersion in the results obtained when only the first two eigencoefficients of Stokes Q and U are taken into account (Fig. 8, right panel) is very large, compared to the PCA inversion using 10 eigencoefficients for each Stokes parameter (left panel), supporting our previous conclusion.
4. APPLICATION TO REAL DATA AND DISCUSSION
The passage from synthetic data to real observations in the development of a PCA-based diagnostics of vector magnetic field in prominences is hampered by a severe scarcity of spectropolarimetric data on these solar structures. To our knowledge, observations of the He i D3 line with full spectropolarimetry are limited to the work of House & Smartt (1982) (see also
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Fig. 8.—Scatter plots showing inversion results vs. model values for the magnetic field strength, B (top row), and the inclination in the local reference system, #B (bottom row), for a prominence located at a height of 1800 . The left column shows the results of PCA inversion with 10 eigencoefficients for each Stokes parameter. In the right column, only the first two eigencoefficients of Stokes Q and U were considered for inversion. The latter is equivalent to independent filtergram observations in the two visible components of the line.
Querfeld et al. 1985; Landi Degl’Innocenti 1982) using HAO Stokes I and II polarimeters at Sacramento Peak Observatory. Recently, Paletou et al. (2001) obtained data using the THEMIS telescope in the Canary Islands. Although the data are still very crude and instrumental errors are ubiquitous, in this section we use them as a benchmark for our code. The purpose of this test is to illustrate PCA application to real prominence data, and as a code testing, rather than to advance any conclusions on prominence physics. The observational setup is described by Paletou et al. (2001). Fifty exposures of a prominence were taken with fixed slit placed at approximately 1500 above the limb and parallel to it. Following Paletou et al. (2001), we decided to add those 50 exposures in one single image containing 100 points along the prominence, in order to increase the S/N and smooth some of the instrumental errors. The four Stokes profiles were scaled down by the peak intensity, the same normalization being performed on the database. After checking that the averaged image did not show important differences in velocity from point to point, we introduced a fixed line shift to the profiles in the database. Therefore, the velocity is considered to be zero in this approximation. After these adjustments, the inversion was performed according to the procedure illustrated in the previous section. For all the inverted points, our code was able to extract from the database synthetic profiles reproducing the main features of the observed profiles. The most frequent error was a mismatched profile amplitude between the two visible components. We have selected two particular examples of profiles that can be considered as generally good fits (Figs. 9 and 10), showing some of the errors. The simple model used to represent the line formation problem is partially responsible for those errors. In particular, the absence of radiative transfer fixes the ratio of the intensity amplitudes of the blue and red components to its theoretical value of 8, which therefore is a constant feature of the synthetic profiles in the database. Radiative transfer is expected to introduce variations of this ratio, and its effect can be seen in the observed profiles, although in general the optically thin approximation proves to be a reasonable assumption (see Fig. 11). Another possible effect of the absence of radiative transfer is the fact that the synthetic profiles for the intensity look very Gaussian, so they cannot reproduce the extended wings of the observed profiles, as seen in Figures 9 and 10. However, collisions could also affect the shape of the observed profile wings. In general, the inversion of the THEMIS data gives a magnetic field vector oriented almost horizontally (i.e., parallel to the solar surface), and approximately along the LOS, with strengths around 40 G. This result agrees with the preliminary analysis performed by Paletou et al. (2001), and is confirmed by the evident Zeeman-like Stokes V profiles observed. However, we insist that such results should be considered very cautiously, because of the quality of the data and the simple model adopted in creating the database. The main conclusion of this test is that the PCA inversion scheme is just as able to deal with real data as with synthetic data, and that models within the proposed database are found that fit the observed data with an acceptable degree of confidence. 5. CONCLUSION
To understand the physics of prominences, good radiative signatures of the magnetic field and its evolution in these solar structures is required. Accurate diagnosis cannot be pursued other than through a detailed analysis of spectropolarimetric
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Fig. 9.—Example of inversion (solid line) of observed Stokes profiles (dotted line)
observations of appropriate spectral lines. As first shown by Bommier & Sahal-Bre´chot (1978), the He i D3 line is one such line, for which we can be confident that we understand the main aspects of its formation in all four Stokes parameters (Landi Degl’Innocenti 1982). While we still are in urgent need of full Stokes profiles observations of these solar structures, recent work by Paletou et al. (2001) seems to indicate that such data will soon be available. Besides the uncertainty of the existing models, the only other missing ingredient up to now was a reliable tool to relate the observed Stokes profiles with those calculated through forward modeling. This tool should meet the basic requirements of being statistically reliable and stable, and able to incorporate in a transparent way increasingly sophisticated models of prominences. Least-square fitting (and related) techniques have been used in the past to handle spectropolarimetric data from photospheric lines formed in the magnetic regime of the Zeeman effect. Previous experience with this techniques shows that a considerable effort would be spent to ensure the stability of the algorithm. In addition, in this kind of inversion techniques, the line
Fig. 10.—Example of inversion (solid line) of observed Stokes profiles (dotted line)
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Fig. 11.—Histogram of the ratio of intensity amplitudes between the blue and the red components of He i D3 multiplet in the observed data. The quantum theoretical ratio in the optically thin limit is 8. Smaller ratios likely indicate the presence of radiative transfer effects. The main peak is for a ratio of 6.8, in good agreement with previous results from House & Smartt (1982).
formation model becomes an essential ingredient of the inversion algorithm, to the point that changing the model almost amounts to rewriting the inversion algorithm altogether. We cannot expect any improvement of this situation when applying least-square inversion techniques to prominence spectropolarimetric data. In fact, one should rather expect the problem to become even more difficult, due to the additional complication introduced by the non-LTE line formation problem associated with the polarization of resonance-scattering radiation. A new and completely different approach was needed. In this paper, we present the application of pattern-recognition techniques to the analysis of spectropolarimetric data from prominences. Pattern recognition was originally introduced in the analysis of spectropolarimetric data with the main goal of increasing speed and reliability in the inversion of vector magnetic fields in the solar photosphere, a goal that has been met with great success (Rees et al. 2000; Socas-Navarro et al. 2001). With this paper, we prove that the very same technique is also able to fit spectropolarimetric data in prominences, and to retrieve information on the topology of the magnetic field in those structures of the solar atmosphere. Future work should allow for more sophisticated and realistic models. In particular, realistic MHD structures of prominences and radiative transfer (three-dimensional) should be included in the formation problem of Stokes profiles (see, e.g., Heinzel & Anzer 2001). In analogy with the diagnostics of photospheric fields, we should also consider more spectral lines that can be ˚ seems a particularly suitable candidate. First, it has the observed simultaneously in the same region. The line He i 10830 A advantage of being part of the same SE problem as the D3 line; second, it is usually optically thick in prominence plasma (Trujillo Bueno et al. 2002), which should help in understanding the radiative transfer effects on both lines; finally, this line is also observed in filaments on the disk, under situations that overcome some of the problems of polarized resonance scattering off the limb (e.g., the well-known 180 ambiguity of linear-polarization measurements). In conclusion, this work shows that (1) PCA techniques can be used to infer the magnetic field topology in prominences, and (2) full spectropolarimetry is needed in order to achieve accurate fits. In particular, we reaffirm the necessity of observing Stokes V in spite of its much smaller amplitude (although in many cases comparable to Stokes U), because it dramatically reduces the inversion error on the vector magnetic field (see Fig. 8). REFERENCES Leroy, J. L., Bommier, V., & Sahal-Brechot, S. 1984, A&A, 131, 33 Andretta, V., & Jones, H. P. 1997, ApJ, 489, 375 Bailer-Jones, C. A. L. 1997, PASP, 109, 932 Leroy, J. L., Ratier, G., & Bommier, V. 1977, A&A, 54, 811 Baur, T. G., Elmore, D. E., Lee, R. H., Querfeld, C. W., & Rogers, S. R. Lo´pez Ariste, A. 2000, in ASP Conf. Ser. 236, Advanced Solar Polarimetry: 1981, Sol. Phys., 70, 395 Theory, Observation and Instrumentation, ed. M. Sigwarth (San Baur, T. G., House, L. L., & Hull, H. K. 1980, Sol. Phys., 65, 111 Francisco: ASP), 521 Bommier, V. 1980, A&A, 87, 109 Lo´pez Ariste, A., Rees, D. E., Socas-Navarro, H., & Lites, B. W. 2001, ———. 1997, A&A, 328, 706 Proc. SPIE, 4477, 96 Bommier, V., Landi Degl’Innocenti, E., Leroy, J. L., & Sahal-Bre´chot, S. Paletou, F., Lo´pez Ariste, A., Bommier, V., & Semel, M. 2001, A&A, 375, 1994, Sol. Phys., 154, 231 L39 Bommier, V., & Sahal-Bre´chot, S. 1978, A&A, 69, 57 Querfeld, C. W., Smartt, R. N., Bommier, V., Landi Degl’Innocenti, E., & Casini, R. 2002, ApJ, 568, 1056 House, L. L. 1985, Sol. Phys., 96, 277 Glazebrook, K., Offer, A. R., & Deeley, K. 1998, ApJ, 492, 98 Rees, D. E., Lo´pez Ariste, A., Thatcher, J., & Semel, M. 2000, A&A, 355, Golub, G. H., & Van Loan, C. F. 1993, Matrix Computations (Baltimore: 759 Johns Hopkins) Ruiz Cobo, B., & del Toro Iniesta, J. 1992, ApJ, 398, 375 Heinzel, P., & Anzer, U. 2001, A&A, 375, 1082 Skumanich, A., & Lites, B. W. 1987, ApJ, 322, 473 House, L. L., & Smartt, R. N. 1982, Sol. Phys., 80, 53 Skumanich, A., & Lo´pez Ariste, A. 2002, ApJ, 570, 379 Landi Degl’Innocenti, E. 1982, Sol. Phys., 79, 291 Socas-Navarro, H., Lo´pez Ariste, A., & Lites, B. W. 2001, ApJ, 553, 949 Leroy, J. L. 1977, A&A, 60, 79 Trujilo Bueno, J., Landi Degl’Innocenti, E., Collados, M., Merenda, L., & ———. 1978, A&A, 64, 247 Maso Saintz, R. 2002, Nature, 415, 403
