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Astron. Astrophys. 355, 759-768 (2000)

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3. PCA of unpolarised spectra

In this section we illustrate these ideas using unpolarised intensity and flux profiles formed in models of increasing complexity.

3.1. Intensity profiles

Consider a spectral line formed in static Milne-Eddington model of a plane-parallel atmosphere with source function equal to the Planck function written as a linear function of [FORMULA], the continuum optical depth, as


The radiative transfer equation can be solved analytically in this case to obtain the emergent intensity normal to the surface,


where x is wavelength measured from line centre in units of an arbitrary Doppler width, and we have assumed purely Doppler broadening. This model has only two adjustable parameters: [FORMULA], the line to continuum opacity ratio at line centre; and [FORMULA] which controls the line broadening.

First we fix the line broadening parameter at [FORMULA], and generate a database of 19 profiles using the opacity ratio values [FORMULA], i.e. from 1 to 10 in steps of 0.5. The profiles computed at 61 wavelengths [FORMULA] are shown in Fig. 2a. Only the first two PCA eigenvalues in Fig. 2b are significant; note that in this and later figures we plot the normalised eigenvalues [FORMULA]. Thus the profiles can be reconstructed to high accuracy using the mean profile and the two eigenprofiles in Fig. 2c. The absolute percentage errors between the reconstructed and the original data for all wavelengths are plotted in Fig. 2d. Note that the eigenfeatures in Fig. 2e are smooth functions of [FORMULA]. Clearly eigenfeatures at parameter values not used originally can be estimated accurately by interpolation. This is a recurring theme in this paper. As shown in Fig. 2f the model manifold in two-dimensional eigenfeature space is a smooth curve parameterised by [FORMULA].

[FIGURE] Fig. 2a-f. One-parameter Milne-Eddington model: a Intensity profiles; b normalised eigenvalues; c eigenprofiles; d reconstruction errors; e eigenfeatures; f model manifold.

Now we allow both parameters to vary, generating a database of 209 profiles with [FORMULA] and [FORMULA] for 81 wavelengths [FORMULA]. Profiles for [FORMULA] and 10 are shown in Fig. 3a. Eigenvalues and eigenprofiles are shown in Fig. 3b and Fig. 3c respectively. The reconstruction error (Fig. 3d) using only 5 eigenprofiles is less than 2% for the entire database. As we cannot visualise the model manifold in 5 dimensions, we consider instead its projection on the 3-dimensional space spanned by the first 3 eigenprofiles. The first 3 eigenfeatures shown in Figs. 3d-f are again smooth functions of the model parameters, and the model manifold in Fig. 3g is a smooth surface.

[FIGURE] Fig. 3a-h. Two-parameter Milne-Eddington model: a intensity profiles; b normalised eigenvalues; c eigenprofiles; d reconstruction errors; e eigenfeatures; f model manifold. On the model manifold the isocontours for the parameters [FORMULA] and [FORMULA] are shown respectively as dashed and full lines.

3.2. H[FORMULA] flux profiles

The efficacy of PCA analysis is not restricted to such simple models. To emphasise this fact we consider the strong line H[FORMULA] formed in non-LTE in the chromosphere of a cool star. The stellar atmosphere model adopted here is typical of that for a K0 dwarf star with log g = 4.478 (eg. Kelch 1978, Simon et al. 1980), and follows the solar paradigm, having a temperature distribution consisting of a negative photospheric temperature gradient with height, a temperature minimum region, a gradual positive gradient (the chromospheric temperature rise), followed by a temperature plateau and a steep transition zone. We use a model with only two parameters: T, the temperature of the plateau; and m, the logarithm of the column mass at the onset of the transition zone.

The H[FORMULA] line exhibits strong variations. As a photo-ionisation controlled line in quiet atmospheres, the line is in absorption, but as atmospheric parameters are changed, source function control moves to collisional, resulting in core filling and wing emissions which are readily modelled. It is well known that varying the parameter m controls the core depth of H[FORMULA] (since the core is formed at the base of the transition zone), and that T controls the wing emission (Thatcher, 1994). A database of 147 flux profiles was generated with [FORMULA] and [FORMULA] K, which are typical values of these parameters in models of moderately to highly active K dwarf stars such as 70 Oph, Epsilon Eri and EQ Vir (Thatcher, 1994).

For our computations, we used the non-LTE computer code "Multi" (Carlsson, 1986) to solve the coupled statistical equilibrium and radiative transfer equations in a semi-infinite, plane parallel atmosphere under the constraint of hydrostatic equilibrium. The adopted hydrogen atom is the five bound level plus continuum model of Lites et al. (1987). All bound-bound transitions and the Lyman, Balmer, Paschen and Brackett continua are calculated in detail. Scattering in frequency is modelled by complete redistribution which is a good approximation for H[FORMULA] (Thomas, 1957; Shine et al., 1975). To approximate the strong partial redistribution effects in the hydrogen Lyman lines, we restricted the radiative transfer calculations in these lines to the Doppler core (Milkey and Mihalas, 1973; Lites et al., 1987). This has a significant effect on the results, causing as much as a 5% variation in H[FORMULA] and H[FORMULA] core depths and a factor of 10 change in the departure coefficients when compared with results where the transfer calculations are not restricted to the core. All computations and atomic models are discussed in more detail in Thatcher (1994) and Thatcher et al. (1991).

Flux profiles normalised to the background continuum for 81 wavelengths -4(0.1)4 Å measured from H[FORMULA] line centre were used in the PCA analysis. Examples are shown in Fig. 4a for [FORMULA] and 8500, and all values of m. Eigenvalues and eigenprofiles are shown in Fig. 4b and Fig. 4c respectively. Only 4 eigenprofiles are needed to reconstruct the entire database with less than 4% error, and the error is less than 1% for most of the database (see Fig. 4d). Despite the complexity of the line formation model, the basic properties of PCA analysis prevail: the eigenfeatures (Figs. 4e-g) are smooth functions of the model parameters; and the model manifold projected into 3 dimensions (Fig. 4h) is a smooth surface. Clearly resampling by spline interpolation is far preferable to detailed non-LTE computations to estimate eigenfeatures at other model parameter values. This approach could lead to considerable time saving in modelling of stellar (and solar) chromospheres, especially when one tries to fit spectra from many different atomic species, as done by Thatcher et al. (1991).

[FIGURE] Fig. 4a-h. H[FORMULA] formed in two-parameter chromospheric model: a flux profiles; b normalised eigenvalues; c eigenprofiles; d reconstruction errors; e -g eigenfeatures; h model manifold. On the model manifold isocontours of parameters T and m are shown respectively as dashed and full lines.

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Online publication: March 9, 2000