Astron. Astrophys. 355, 759-768 (2000)
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
, the continuum optical depth, as
![[EQUATION]](img48.gif)
The radiative transfer equation can be solved analytically in this
case to obtain the emergent intensity normal to the surface,
![[EQUATION]](img49.gif)
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:
, the line to continuum opacity ratio
at line centre; and which controls
the line broadening.
First we fix the line broadening parameter at
, and generate a database of 19
profiles using the opacity ratio values
, i.e. from 1 to 10 in steps of 0.5.
The profiles computed at 61 wavelengths
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
. 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 . 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
.
![[FIGURE]](img56.gif) |
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 and
for 81 wavelengths
. Profiles for
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]](img65.gif) |
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 and are shown respectively as dashed and full lines.
|
3.2. H flux profiles
The efficacy of PCA analysis is not restricted to such simple
models. To emphasise this fact we consider the strong line
H 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 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 (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 and
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 (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 and
H 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 line centre were used in the PCA
analysis. Examples are shown in Fig. 4a for
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]](img73.gif) |
Fig. 4a-h. H 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.
|
© European Southern Observatory (ESO) 2000
Online publication: March 9, 2000
helpdesk.link@springer.de  |