Issue |
A&A
Volume 517, July 2010
|
|
---|---|---|
Article Number | A57 | |
Number of page(s) | 17 | |
Section | Astronomical instrumentation | |
DOI | https://doi.org/10.1051/0004-6361/201014013 | |
Published online | 05 August 2010 |
Stellar atmosphere parameters with MA
,
a MAssive compression of
for spectral fitting
P. Jofré1 - B. Panter2 - C. J. Hansen3 - A. Weiss1
1 - Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1,
85741 Garching, Germany
2 - Institute for Astronomy, University of Edinburgh, Royal
Observatory, Blackford Hill, Edinburgh EH9 3HJ, UK
3 - European South Observatory (ESO), Karl-Schwarzschild-Str. 2, 85748
Garching, Germany
Received 7 January 2010 / Accepted 19 April 2010
Abstract
MA
is a new tool to estimate parameters from stellar spectra. It is based
on the maximum likelihood method, with the likelihood compressed in a
way that the information stored in the spectral fluxes is conserved.
The compressed data are given by the size of the number of parameters,
rather than by the number of flux points. The optimum speed-up reached
by the compression is the ratio of the data set to the number of
parameters. The method has been tested on a sample of low-resolution
spectra from the Sloan Extension for Galactic Understanding and
Exploration (SEGUE) survey for the estimate of metallicity, effective
temperature and surface gravity, with accuracies of 0.24 dex,
130 K and 0.5 dex, respectively. Our stellar
parameters and those recovered by the SEGUE Stellar Parameter Pipeline
agree reasonably well. A small sample of high-resolution VLT-UVES
spectra is also used to test the method and the results were compared
to a more classical approach. The speed and multi-resolution capability
of MA
combined with its performance compared with other methods indicates
that it will be a useful tool for the analysis of upcoming spectral
surveys.
Key words: techniques: spectroscopic - surveys - stars: fundamental parameters - methods: data analysis - methods: statistical
1 Introduction
The astronomical community has conducted many massive surveys of the Universe, and many more are either ongoing or planned. Recently completed is the Sloan Digital Sky Survey (SDSS, York et al. 2000), notable for assembling in a consistent manner the positions, photometry and spectra of millions of galaxies, quasars and stars, and covering a significant fraction of the sky. The size of such a survey allows us to answer many questions about the structure and evolution of our universe. More locally, massive surveys of stars have been undertaken to reveal their properties, for example the Geneva-Copenhagen Survey for the solar neighborhood (Nordström et al. 2004) and the ELODIE library (Prugniel et al. 2007). These surveys provide a copious amount of stellar photometry and spectra from the solar neighborhood, broadening the range of stellar types studied. The SEGUE catalog - Sloan Extension for Galactic Understanding and Exploration (Yanny et al. 2009) - a component of SDSS, contains additional imaging data at lower galactic latitudes, to better explore the interface between the disk and halo population.
By statistically analyzing the properties of these stars such as chemical abundances, velocities, distances, etc., it has been possible to match the structure and evolution of the Milky Way to the current generation of galaxy formation models (Wyse 2006; Bond et al. 2010; Ivezic et al. 2008; Juric et al. 2008).
Studies of large samples of stars of our galaxy are crucial tests for the theory of the structure and formation of spiral galaxies. Even though the statistics given by SEGUE or the Geneva-Copenhagen survey agree with the models and simulations of galaxy formation, there is some contention over whether the accuracies of measurements adequately support the conclusions. New surveys such as RAVE (Steinmetz et al. 2006) and Gaia (Perryman et al. 2001) will give extremely high accuracies in velocities and positions, LAMOST (Zhao et al. 2006) and part of the SDSS-III project, APOGEE and SEGUE-2 (Rockosi et al. 2009), will give high-quality spectra and therefore stellar parameters and chemical abundances of millions of stars over all the sky. These more accurate properties and larger samples will increase our knowledge of the Milky Way and answer wider questions about the formation and evolution of spiral galaxies.
As data sets grow it becomes of prime importance to create efficient and automatic tools capable of producing robust results in a timely manner. A standard technique for estimating parameters from data is the maximum-likelihood method. In a survey such as SEGUE, which contains more than 240 000 stellar spectra, each with more than 3000 flux measurements, it becomes extremely time-consuming to do spectral fitting with a brute-force search on the multi-dimensional likelihood surface, even more so if one wants to explore the errors on the recovered parameters.
Efforts to reduce the computational burden of characterizing the likelihood surface include Markov-Chain Monte-Carlo methods (Ballot et al. 2006), where a chain is allowed to explore the likelihood surface to determine the global solution. An alternative method is to start at an initial estimate and hope that the likelihood surface is smooth enough that a gradient search will converge on the solution. An example of this approach can be found in Allende Prieto et al. (2006) and Gray et al. (2001), based on the Nelder-Mead downhill simplex method (Nelder & Mead 1965), where the derivatives of the likelihood function give the direction toward the maximum. In both cases, the likelihood is evaluated only partially, leaving large amounts of parameter space untouched. Moreover, the time needed to find the maximum depends on the starting point and the steps used to evaluate the next point.
Another parameter estimation method utilizes neural networks, for example Snider et al. (2001) and Re Fiorentin et al. (2007). While these non-linear regression models can obtain quick accurate results, they are entirely dependent on the quality of the training data, in which many parameters must be previously known. Careful attention must also be paid to the sampling of the model grid which is used to generate the neural network.
An alternative approach to spectral analysis is the Massively
Optimized Parameter Estimation and Data compression method ( MOPED, Heavens
et al. 2000). This novel approach to the maximum
likelihood problem involves compressing both data and models to allow
very rapid evaluation of a set of parameters. The evaluation is fast
enough to do a complete search of the parameter space on a finely
resolved grid of parameters. Using carefully constructed linear
combinations, the data are weighted and the size is reduced from a full
spectrum to only one number per parameter. This number, with certain
caveats (discussed later), contains all the information about the
parameter contained in the full data. The method has been successfully
applied in the fields of CMB analysis (Bond
et al. 1998), medical image registration
and galaxy shape fitting (Tojeiro et al. 2007). A
complete background of the development of the MOPED algorithm can be
found in Tegmark et al. (1997,
hereafter T97), Heavens
et al. (2000, hereafter H00) and Panter et al. (2003).
We present a new derivative of MOPED, MA (MAssive
compression of
),
to analyze stellar spectra. To estimate the metallicity history of a
galaxy for instance, MOPED needs models where the spectra are
the sum of single stellar populations. Here we study one population,
meaning the metallicity has to be estimated from the spectrum of each
single star. It is therefore necessary to develop a specific tool for
this task: MA
.
In Sect. 2
we describe the method, giving a summary of the derivation of the
algorithm. We then test the method in Sect. 3 on the basic
stellar atmosphere parameter estimation of a sample of low-resolution
stars of SEGUE, where we compare our results with the SEGUE Stellar
Parameter Pipeline in Sect. 4 and
finally check the method for a small sample of high-resolution spectra
in Sect. 5,
where we compare our results with a more classical approach for a small
sample of VLT-UVES spectra. A summary of our conclusions is given in
Sect. 6.
2 Method
In this section we describe our method to treat the likelihood surface. We first review the classical maximum-likelihood method, where a parametric model is used to describe a set of data.Then we present the algorithm that compresses the data and hence speeds the likelihood estimation. Finally we show the proof for the lossless nature of the compression procedure.2.1 Maximum-likelihood description
Suppose the data (e.g. the flux of a spectrum) are represented by N
real numbers x1,
x2, ..., xN,
which are arranged in an N-dimensional vector .
They are the flux measurements at N wavelength
points. Assume that each data point xi
has a signal part
and a noise contribution
![]() |
(1) |
If the noise has zero mean,




with the iron abundance
![${\rm [Fe/H]}$](/articles/aa/full_html/2010/09/aa14013-10/img25.png)





![]() |
(3) |
gives the probability for the parameters, where

The position of its maximum estimates the set of parameters






2.2 Data compression
In practice not all data points give information about the parameters, because either they are noisy or not sensitive to the parameter under study. The MOPEDalgorithm uses this knowledge to construct weighting vectors which neglect some data without losing information. A way to do this is by forming linear combinations of the data.
Let us compress the information of a given parameter
from our spectrum. The idea is to capture as much information as
possible about this particular parameter. We define our weighting
vector
as
where




Figure 1
shows the example of the weighting vectors for the parameters of the
set (2) using a
fiducial model with ,
K and
.
The upper panel corresponds to the synthetic spectrum at the SDSS
resolving power (R = 2000) and the panels B, C and D
to the weighting vector for metallicity, effective temperature and
surface gravity, respectively. This figure is a graphical
representation of Eq. (5),
and helps to better visualize the weight in relevant regions of the
spectrum.
For metallicity (panel B), the major weight is concentrated in the two peaks at 3933 and 3962 Å, corresponding to the CaII K and H lines seen in the synthetic spectrum in panel A. The second important region with weights is close to 5180 Å, corresponding to the MgIb triplet. A minor peak is seen at 4300 Å, which is the G-band of the CH molecule. These three features are metallic lines, therefore show a larger dependency on metallicity than the rest of the vector, which is dominated by a zero main weight.
For the temperature (panel C) the greatest weight is focused on the hydrogen Balmer lines at 4101, 4340 and 4861 Å. Peaks at the previous metallic lines are also seen, but with a minor amplitude. Finally, the surface gravity (panel D) presents the strongest dependence on the Mg I triplet. The wings of the Ca II and Balmer lines also present small dependence. As in the case for metallicity, the rest of the continuum is weighted by a mean close to zero.
![]() |
Figure 1:
(A) Fiducial model with parameters |
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The weighting is done by multiplying these vectors with the spectrum,
where the information kept from the spectrum is given by the peaks of
the weighting vector. We define this procedure as ``compression'',
where the information about the parameter
is expressed as
with







In order to perform a compression for all the parameters
simultaneously, we require that yk
is uncorrelated with yj,
with .
This means that the
-vectors
must be orthogonal, i.e.
.
Following the procedure of H00 we find the other
by the Gram-Schmidt ortogonalization
![]() |
(7) |
For m = 3 we have the numbers







Goodness of fit
The definition of ,
which is the sum of the differences of the data and the model,
motivates us to define our goodness of fit. The ``compressed''
is the sum of the differences of the compressed data and model
![]() |
(8) |
which gives the probability for the parameter

![]() |
(9) |
Because the yj numbers are by construction uncorrelated, the ``compressed'' likelihood of the parameters is obtained by multiplication of the likelihood of each single parameter

![]() |
(10) |
where
is the compressed

As in the classical approach, the peak of the compressed likelihood
estimates the parameters
that generate the model
,
which reproduces best the data
.
The advantage of using the compressed likelihood is that it is fast,
because the calculation of
needs only m iterations and not N
as in the usual
computation. The search of the maximum point in the likelihood is
therefore done in an m-dimensional grid of y-numbers
and not of N-length fluxes.
![]() |
Figure 2: Correlations between the Fisher matrix values obtained from the full and compressed likelihoods for metallicity (upper panel) and effective temperature (lower panel) for 75 randomly selected stars of SEGUE. The line corresponds to the one-to-one relation. |
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2.3 No loss of information by the compression
The Fisher Information Matrix describes the behavior of the likelihood close to the maximum. Here we use it only to show the lossless compression offered by the MOPED method, but a more extensive study is given in T97 and H00.
To understand how well the compressed data constrain a
solution we consider the behavior of the logarithm of the likelihood L
near the peak. Bellow we denote a generic L that
can correspond to the compress likelihood
or the full one
,
because the procedure is the same. In the Taylor expansion the first
derivatives
vanish at the peak and the behavior around it is dominated by the
quadratic terms
The likelihood function is approximately a Gaussian near the peak and the second derivatives, which are the components of the inverse of the Fisher Matrix

![]() |
(13) |
In Fig. 2 we plotted the values
![$\mathcal{F}_{{\rm [Fe/H]}}$](/articles/aa/full_html/2010/09/aa14013-10/img71.png)

Figure 3 shows full and compressed likelihood surfaces of metallicity and temperature for four randomly selected stars, where eight equally spaced contour levels have been plotted in each case. This is another way to visualize the statement that the Fisher Matrices correlate. The curvature of both likelihoods is the same close to the peak, as predicted. In Fig. 3 we also over-plotted the maximum point of the compressed likelihood as a triangle and the maximum point of the full one as a diamond. It can be seen that the maximum points of the compressed and full data set lie inside the first contour level of the likelihoods. This shows that the compressed data set gives the same solution as the full one - even after the dramatic compression.
![]() |
Figure 3: Likelihoods of the compress data set ( left) and full data set ( right) in the parameter space of effective temperature and metallicity for four randomly selected SEGUE stars. In each panel eight equally spaced contour levels are plotted. The triangle and diamond correspond to the maximum point of the compressed and the full data set, respectively. Both maxima lie in the first confidence contour level, meaning that both data sets reach the same solution for these two parameters. |
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In the bellow parts the numerical values of
correspond to the classical reduced
.
3 Implementation on low-resolution spectra
The sample of stars used to test the method are F-G dwarf stars. Based on the metallicity given by the SEGUE Stellar Parameter Pipeline, we chose metal-poor stars as will be explained below. This choice allows us to avoid the problem of the saturation of metallic lines such as Ca II K (Beers et al. 1999), which is a very strong spectral feature that serves as a metallicity indicator in the low-resolution spectra from SEGUE. These stars fall in the temperature range where Balmer lines are sensitive to temperature and the spectral lines are not affected by molecules (Gray 1992). With these considerations it is correct to assume that the spectra will behave similar under changes of metallicity, temperature and gravity. This allows the choice of a random fiducial model from our grid of models for the creation of the weighting vectors, which will represent well the dependence of the parameters in all the stars.
3.1 Data
We used a sample of SEGUE stellar spectra (Yanny et al. 2009), part of the Seventh Data Release (DR7, Abazajian et al. 2009) of SDSS. The survey was performed with the 2.5-m telescope at the Apache Point Observatory in southern New Mexico and contains spectra of
![$-999 < {\rm [Fe/H]} < -0.5$](/articles/aa/full_html/2010/09/aa14013-10/img75.png)
Our final sample contains spectra of 17 274 stars
with a resolving power of
and a signal-to-noise up to 10. The wavelength range is
and the spectra are with absolute flux.
3.2 Grid of models
The synthetic spectra were created with the synthesis code SPECTRUM (Gray & Corbally 1994),
which uses the new fully blanketed stellar atmosphere models of Kurucz (1992) and computes
the emergent stellar spectrum under the assumption of local thermal
equilibrium (LTE). The stellar atmosphere models assume the solar
abundances of Grevesse & Sauval
(1998) and a plane parallel line-blanketed model structure in
one dimension. For the creation of the synthetic spectra, we set a
microturbulence velocity of 2 km s-1,
based on the atmosphere model value. The line-list file and atomic data
were taken directly from the SPECTRUM webpage.
In these files, the lines were taken from the NIST Atomic Spectra
Database
and the Kurucz webside
. No molecular opacity was
considered in the model generation.
We created an initial three dimensional grid of synthetic
spectra starting from the ATLAS9 Grid of stellar atmosphere models of Castelli & Kurucz (2003) by
varying the parameters
of Eq. (2).
They cover a wavelength range from 3800 to 7000
in steps of
,
based on the wavelength range of SDSS spectra together with our
line-list. This wavelength range is broad enough for a proper continuum
subtraction, as described in Sect. 3.3. The
spectra have an absolute flux and were finally smoothed to a resolving
power of R = 2000, according to SDSS resolution.
In order to have a finer grid of models, we linearly
interpolated the fluxes created for the initial grid. It has models
with
with
dex,
with
K and
with
dex.
The scaling factor for the normalization varies linearly from 0.85 to
1.15 in steps of 0.01 (see below). Our final grid has
models. We set a linearly varying
of
for stars in the metallicity range of
and
for stars with
,
to reproduce the abundance of
elements in the Milky Way, as in Lee
et al. (2008a). These varying
abundances were also calculated with interpolation of fluxes created
from solar and
-enhanced
([
/Fe] = 0.4)
stellar atmosphere models. We did not include the
abundances as a free parameter, because we aim to estimate the
metallicity from lines of
elements (Ca, Mg), meaning we would obtain a degeneracy in both
parameters. We prefered to create a grid where the
abundances were already ``known''.
Table 1: Strongest lines in F-G dwarf stars.
![]() |
Figure 4: Radial velocities found using minimum fluxes of strong lines given by Table 1 (MAX) compared with those of the SEGUE database. The difference of the radial velocities given by the SEGUE database from those obtained by our method is indicated in the legend as offset, with its standard deviation as scatter. |
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The grid steps between the parameters are smaller than the accuracies expected from the low resolution SEGUE data. The extra time required to calculate the compressed likelihood in this finer grid is not significant, and retaining the larger size allows us to demonstrate the suitability of the method for future more accurate data capable of using the full grid.
3.3 Matching models to data
Data and model needed to be prepared for the analysis. First of all, the data needed to be corrected from the vacuum wavelength frame of the observations to the air wavelength frame of the laboratory. Secondly, we needed to correct for the Doppler effect zc. This was done by using the flux minimum of the lines indicated in Table 1 except for the Mg Ib triplet, because this feature is not seen clearly enough in every spectrum, driving in our automatic zc calculation to unrealistic values. A comparison of the radial velocity found with this method with the value given by the SEGUE database f is shown in Fig. 4. The difference of the radial velocities (SEGUE - MAX) has a mean (offset) of -20.15 km s-1 with standard deviation (scatter) of 7.85 km s-1. The effect of this difference on the final parameter estimation is discussed in Sect.4. Once our data were corrected for the Doppler effect, they had to be interpolated to the wavelength points of the models to get the same data points.
For automatic fitting of an extensive sample of stars with a
large grid of models, we decided to normalize the flux to a fitted
continuum. In this way the dependence of the parameters is this is not
past, it is a fact that happens now too) concentrated mainly on the
line profiles. The choice of the normalization method is a difficult
task, because none of them is perfect. It becomes especially difficult
in regions where the spectrum shows too many strong absorption
features, as the Ca II lines of Table 1. Using the same
method for models and data, difficult regions behave similar in both
cases, which allowed us a final fitting in these complicated regions as
well. We adopted the normalization method of Allende
Prieto et al. (2006), because it works well for the
extended spectral range of SDSS spectra. It is based on an iteratively
polynomial fitting of the pseudo-continuum, considering only the points
that lie inside the range of 4
above and 0.5
below. Then we divided the absolute flux by the final pseudo-continumm.
Due to noise, the continuum of the final flux may not necessary be at
unity. We included a new degree of freedom in the analysis that scaled
this subtracted continuum. This scaling factor for the normalization is
then another free parameter, which means that we have finally four
parameters to estimate - three stellar atmosphere parameters and the
scaling factor. The final step is to choose the spectral range for the
analysis.
3.4 Compression procedure
We chose the fiducial model with parameters K,
and
,
to calculate the weighting vectors,
,
,
and
,
corresponding to metallicity, effective temperature, gravity and
scaling factor for normalization, respectively. Then, we calculated the
set of y-numbers using Eq. (6) for each point in the
grid of synthetic spectra by projecting the
-vectors onto the spectrum,
resulting in a four dimensional y-grid, with every
point a single number. With the respective
-vectors we calculated the y-numbers
for the observed spectra, the expression (11) for every point
in the grid and finally we found the minimum value in the grid which
corresponds to the maximum point of the compressed likelihood.
To refine our solution we found the ``real'' minimum
with a quadratic interpolation. For the errors, we looked at the models
within the confidence contour of
with
corresponding to the 1
error in a likelihood with four free parameters (for further details
see chapter 14 of Press 1993).
4 Application to SEGUE spectra
We chose to use the wavelength range of [3850, 5200] Å as information about metallicity, temperature and gravity, which is available in the lines listed in Table 1. These spectral lines are strong in F-G stars, meaning they can be identified at low resolution without difficulty. The wings of Balmer lines are sensitive to temperature and the wings of strong Mg lines are sensitive to gravity (Fuhrmann 1998). Beacuse iron lines are not strong enough to be distinguished from the noise of our spectra, the Ca II K and Mg Ib lines are our indicators of metallicity (Beers et al. 2000; Allende Prieto et al. 2006; Lee et al. 2008a).
It is important to discuss the carbon feature known as the
G-band at 4304 Å in our spectral window. The fraction of
carbon-enhanced metal-poor (CEMP) stars is expected to increase with
decreasing metallicity (Beers
et al. 1992) to about 20% at below
(Lucatello et al. 2006),
possibly implying a strong carbon feature in the observed spectrum. As
discussed by Marsteller
et al. (2009), a strong G-band may also affect the
measurement of the continuum at the Ca II lines, which could result in
an underestimation of the stellar metallicity. We commented in
Sect. 2
on the influence of the lines in the weighting vectors used for the
compression and we saw that the G-band also displayes minor peaks (see
Fig. 1).
The role of this minor dependence compared with those from the Ca II ,
Balmer and Mg I lines can be studied by comparing the final
parameter estimation when using the whole spectral range or only those
regions with the lines of Table 1, where the
G-band is not considered. A further motivation for performing an
analysis of an entire range against spectral windows is also explained
below. The implications of this tests in terms of fraction of CEMP
stars is discussed below.
4.1 Whole domain vs. spectral windows
The classical
fitting procedure uses every datapoint; therefore, a straight-forward
method to speed up the analysis would be to mask those parts of the
spectrum which do not contain information about the parameters,
essentially those that are merely continuum. Certainly, by considering
only the Ca II , Balmer and Mg Ib lines, the number of operations
becomes smaller, increasing the processing speed of
calculations. This is clumsy however: an empirical decision must be
made about the relevance of pixels, and no extra weighting is
considered - and the time taken for the parameters estimation is still
long if one decides to do it for many spectra. The use of the
-vectors in
the MA
method means that very little weight is placed on pixels which do not
significantly change with the parameter under study, automatically
removing the sectors without lines.
The remarkable result of the MA method is that it
is possible to determine the maximum of the compressed likelihood in
10 ms with a present day standard desktop PC. This is at least
300 times faster than the same procedure when doing an efficient
evaluation of the uncompressed data.
The 1
confidence contours indicate errors in the parameters of
0.24 dex in metallicity on average, 130 K in
temperature and 0.5 dex in gravity. Examples of fits are shown
in Fig. 5.
The upper plot shows the fit between a randomly selected SEGUE spectrum
and the best model - the crosses correspond to the observed spectrum
and the dashed red line to the model, with stellar atmosphere
parameters in the legend, as well as the resulting reduced
of the fit. The plot in the bottom is the fit of the same star, but
considering only the data points within the line regions identified in
Table 1,
as discussed above. Again, the red dashed line indicates the model with
parameters in the legend. The
of the fit is also given.
Because the lines contain the most information about
metallicity, temperature and gravity, we expect to obtain the same
results whether we use all the data points or only those corresponding
to the lines. The legends in Fig. 5 show the parameters
estimated in both analyses. The small differences between them are
within the 1
errors. We plotted in the upper panel of Fig. 6 the
comparison for metallicity (left panel), temperature (middle) and
gravity (right) of our sample of stars when using the whole spectral
range (``whole'') or only spectral windows with the lines (``win'').
The offsets and scatters of the distribution as well as the one-to-one
relation are indicated in the legend of each plot. Here we randomly
selected 150 stars to plot in Fig. 6 to
visualize better the correlations of the results. Offsets and scatters
are calculated with the entire star sample. Metallicity has excellent
agreements, with a small scatter of 0.094 dex, as shown in the
legend of the plot. Temperature has a negligible offset of
38 K and a scatter of 69 K, which is also less than
the accuracies obtained in the temperature estimation. Gravity has the
largest scatter offset of 0.1 dex and a scatter of
0.34 dex, but this is still within the errors.
![]() |
Figure 5:
Example of the fit between a randomly selected SEGUE star (crosses) and
a synthetic spectrum (red dashed line). The legend indicates the
stellar atmosphere parameters of the model and the value of the reduced
|
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![]() |
Figure 6:
Upper panel: metallicity ( left),
effective temperature ( middle) and surface gravity (
right) obtained using MA |
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The negligible offsets obtained when using the entire spectral range against the limited regions is an encouraging result in terms of the effect of the G-band on the spectrum. As pointed out above, the dependence on this feature in the parameters under study is not as strong as the rest of the lines of Table 1. This is translated into less weight for the compression, as seen in Fig. 1, meaning the G-band does not play a crucial role in our compressed data set for the parameter recovery.
Let us remark that this does not mean a lack of CEMP stars in
our sample, we simply do not see them in the compressed space. A
spectrum with a strong observed CH molecule gives a similar compressed
as one with a weak one, because the compression does not consider
variations in carbon abundance. The full
,
on the other hand, will certainly be larger for the spectrum with a
strong G-band, because our models do not have different carbon
abundances.
In any case, the fraction of CEMP stars is high and it is
certainly interesting to locate them with the MA method in the
future. This implies that we need to create another dimension in the
grid of synthetic spectra - models with varying carbon abundances - and
increase the number of parameters to analyze to five. One more
dimension certainly means a heavier grid of models, but in terms of
parameter recovery, there is no big difference to use four or five y-numbers
for the compressed
calculation. This study goes beyond the scope of this paper, however,
where we only introduce the MA
method.
4.2 Effect on radial velocities
As mentioned in Sect. 3.3
and seen in Fig. 4,
radial velocities given by the SEGUE database have a mean difference of
20 km s-1 compared to ours. To
study the effect of this difference in the final parameter estimation,
we analyzed a sub-sample of randomly selected 2670 stars of
our sample using SEGUE radial velocities. The comparison of the
metallicity, effective temperature and surface gravity obtained when
using our radial velocities (``rv MAX'') and the SEGUE ones (``rv
SEGUE'') can be seen in the lower panel of Fig. 6. In this
plot, also a small sample of randomly selected 150 stars was used to
better visualize the results, but offsets and scatters were calculated
considering the 2670 stars. A difference in
20 km s-1 produces offsets and
scatters in the three parameters, with dex
for metallicity,
K for temperature and
dex
for gravity. These offsets are negligible when compared with the 1
errors obtained the parameter estimation.
![]() |
Figure 7: Upper plot: comparison of the results from the SEGUE Stellar Parameter Pipeline (SSPP) with different methods: this work (MAX, black), (Lee et al. 2008a, NGS1, blue), (Re Fiorentin et al. 2007, ANNSR, red) and (Allende Prieto et al. 2006, k24, green). Lower part: comparisons of individual methods for a randomly selected subsample of stars. Each plot has a line with slope of unity. The different offsets and scatters between the results are indicated in Table 2. In each plot, a selection of 150 random stars has been used. |
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4.3 Comparison with the SEGUE Stellar Parameter Pipeline
The SEGUE Stellar Parameter Pipeline (SSPP, Allende Prieto et al. 2008; Lee et al. 2008b,a) is a combination of different techniques to estimate the stellar parameters of SEGUE. Some of them are to fit the data to a grid of synthetic spectra like the k24 (Allende Prieto et al. 2006), and the k13, NGS1 and NGS2 ones (Lee et al. 2008a). Another method are the ANNSR and ANNRR of Re Fiorentin et al. (2007), which are an artificial neural network (ANN) trained to use a grid of synthetic (S) or real (R) spectra to estimate the parameters of real spectra (R). Other options are line indices (Wilhelm et al. 1999) and the Ca II K line index (Beers et al. 1999).
We compare our results with the final adopted SSPP value (ADOP), and those of the k24, NGS1 and ANNSR grids of synthetic spectra. Each of these results can be found in the SSPP tables of the SDSS database. The grids of models are created using Kurucz stellar atmosphere models, but k24 and NGS1 cover the wavelength range of [4400, 5500] Å. The ANNSR grid includes the Ca II triplet close to 8600 Å.
Figure 7
shows the correlations between the results of these methods and our
own. As in Fig. 6,
we plotted a random selection of 150 stars to better visualize the
correlations, but values of offsets and scatters were made using the
entire sample of 17 274 stars. The left panels
correspond to the metallicity, the middle ones to temperature and the
right ones to the surface gravity. The top row of plots shows a general
comparison of all the different methods, plotted with different colors.
The y-axis indicates the ADOP results and the x-axis
the methods MA
(black), NGS1 (blue), ANNSR (red) and k24 (green). The one-to-one line
is also plotted in the figures. For each distribution a histogram of (
)
with
was fitted with a Gaussian to obtain estimations for offset (mean
)
and scatter (standard deviation
). The results are summarized
in Table 2.
In the lower panel of Fig. 7 we plotted the
comparisons of individual methods for 150 randomly selected stars.
These panels help to better visualize how the single methods of the
pipeline and our own correlate with each other. Figure 7 and
Table 2
show that our results reasonably agree with the pipeline, and that the
scatter is of the same order on magnitude as that between the
individual SSPP methods.
Considering individual parameters, our metallicities have a
general tendency to be 0.3 dex
lower than adopted metallicity of SSPP (ADOP). A similarly large offset
exists between ANNSR and NGS1, with -0.22 dex. This could be
due to the consideration of the Ca II lines in the fit, which are not
included in the NGS1 and k24 ones. ANNSR also includes Ca II lines, the
offset of -0.17 dex to our results is smaller than 1
error. If the same lines are used in the fitting, no offset should be
seen, as will be shown in Sect. 5.
The temperature, on the other hand, shows a small offset of -61 K with respect to the ADOP value of the pipeline. It is encouraging to see that we obtain the best agreement, except for ANNSR, which has no offset. A large mean difference is found between k24 and NGS1, where the offset is 244 K. The scatter in the various comparisons varies from 100 K (NGS1 v/s ADOP) to 200K (k24 v/s ANNSR); our scatter of 112 K with respect to ADOP is one of the lowest values.
Finally,
shows the largest offset. We derived gravity values 0.51 dex
higher than the ADOP ones. The worst case is the comparison between our
method and ANNSR with 0.63 dex. Between the methods of the
pipeline, the largest difference (0.27 dex) is found between
ANNSR and NGS1. The best agreement we found for gravities are with
NGS1, with an offset of 0.38 dex. The scatter for gravities
varies from 0.23 dex (NGS1 v/s ADOP) to 0.48 dex
(ANNSR v/s k24 and MA
v/s
k24). A possible approach to correct our gravities would be to shift
the zero point by -0.38 dex to agree with those of NGS1. We
prefer to accept that gravity is our least constrained parameter, given
the lack of sensitive features except for the wings of the Mg Ib lines,
and the noise in the spectra does not restrict
effectively. In order to deal with this problem, k24 and NGS1 smooth
the spectra to half of the resolution, gaining signal-to-noise by this
procedure. We remark that the k24 grid includes
colors, also to constrain the temperature. This automatically leads to
different gravity values. A more extensive discussion of this aspect
will be given in Sect. 5.6.
The right panels of Fig. 7
shows that all
estimates are rather uncorrelated with each other with respect to the
other ones.
![]() |
Figure 8:
Histograms of the differences of the results obtained using 8 different
fiducial models for the compression. Left:
metallicity, middle: temperature and right:
gravity. The parameters of the fiducial model are indicated in the
right side of each plot, labeled as [[Fe/H], |
Open with DEXTER |
Table 2:
Offsets ()
and scatter (
)
of the differences (raw - column) between the methods: MA
(this work).
4.4 Systematic errors: choice of fiducial model
The -vectors
are basically derivatives with respect to the parameters.With the
assumption that the responses of the model to these derivatives are
continuous, the choice of the fiducial model (FM) used to build the
-vectors is
free. This assumption, however, is only correct for certain regions of
the parameter space.
For example, Balmer lines are good temperature indicators for stars no
hotter than 8000 K (Gray 1992),
and for cold stars spectral lines are affected by molecular lines.
Limiting our candidate stars to the range of 5000 K to
8000 K we know that we have a clean spectra with Balmer lines
as effective temperature indicators. This allows us to choose a FM with
any temperature in that range to construct our weighting vector
,
which will represent well the dependence of the model spectrum on
temperature. Similar assumptions are made for the other two model
parameters.
In order to test this assumption we studied the effect of
using different fiducial models in the final results of the parameter
estimation. In Fig. 8
we plot the results of metallicity (left), temperature (middle) and
gravity (right) for 150 randomly selected SEGUE stars. There are four
sets of different fiducial models [A,B] used for the
vector calculation. The parameters of the FM are indicated at the right
side in each set. From top to bottom the first one compares the results
of a cold (
K)
and a hot (
K)
fiducial model of our grid of synthetic spectra, the second one a
metal-poor (
)
and a metal-rich (
)
one. The third one considers two different gravities (
and
). For these three sets, all
other parameters are identically calculated. Finally, the last one uses
models which vary all three parameters. The histograms of each plot
correspond to the difference of the results obtained with the fiducial
model with a set of the parameters A and B, respectively. The Gaussian
fit of the histogram is overplotted and its mean (
)
and standard deviation (
)
are indicated in the legend at the top.
Metallicity shows a well defined behavior under different -vectors.
There are negligible offsets when varying only one parameter in the FM,
except for the case where the FM is totally different. But even in that
case the offset of -0.15 dex is less than the 1
errors of 0.25 dex obtained in the metallicity estimation. We
conclude that this parameter does not show real systematic offsets due
to the choice of fiducial model. The scatter obtained in the
metallicity is also less than the 1
errors of the results. Only in the last case (bottom left) it becomes
comparable with the errors. Because gravity is a poorly constrained
parameter (see discussions above) and the y-numbers
are uncorrelated, the effect when using different gravities in the FM
should be completely negligible in the result of the other parameters.
This can be seen in the third panel from top to bottom of Fig. 8, where
metallicity and temperature show extremely small offsets and scatters.
The derivation of temperature using different fiducial models
shows a similar behavior. The discrepancies between results when using
different FM are on the order of 50 K, except when the metallicity is
varied. The mean difference of the final temperature estimates when
using a metal-poor fiducial model and a metal-rich one is of 141 K,
which is smaller than some differences seen in Table 2 between the
methods of the SEGUE Stellar Parameter Pipeline. This probably happens
because the spectrum of a metal-rich star presents more lines than a
metal-poor one, which is translated into the -vector as
temperature-dependent regions that do not exist. In the case of SEGUE
data, many of the weak lines seen in a metal-rich synthetic spectrum
and(or) the
-vector
of a metal-rich fiducial model are hidden by the noise in the observed
spectra and the assumption of a temperature dependence due to these
weak lines may not be correct. For the analysis of low signal-to-noise
spectra, it is preferable to choose a FM with a rather low metallicity.
But even when using a high metallicity for the FM, the offset is
comparable with the 1
error of 150 K for the temperature estimations. We conclude
that temperature does not show a significant systematic offset due to
choice of fiducial model either.
Finally, gravity shows larger discrepancies in the final
results when varying the FM. The differences can be as large as
0.4 dex in the worst case when using different metallicities
in the FM. This could also be because a metal-rich FM contains more
lines and therefore greater sensitivities to gravity, which in our low
signal-to-noise spectra is not the case. These sensitivities are
confused with the noise in the observed spectrum. Scatters of
0.5 dex are in general on the order of the
errors.
5 Application to high-resolution spectra
As a check, MA
also was tested with a smaller sample of 28 high-resolution spectra.
For consistency with the low resolution
implementation, again metal-poor dwarf stars were selected.
5.1 Data
The spectra were obtained with UVES, the Ultraviolet Echelle
Spectrograph (Dekker et al. 2000)
at the ESO VLT 8 m Kueyen telescope
in Chile. The resolving power is
and the signal-to-noise
is typically above 300 in our spectra. Because MA
is an automatic
fitting tool
for synthetic spectra, we first had to be sure that the observed
spectra could be fitted by our grid of synthetic spectra. This means
that we
had to have easily distinguishable unblended spectral lines and a
clear continuum. For these reasons we selected a part of the red
setting of the UVES spectra, which covers the wavelength range
580 nm
- 680 nm, where there are many unblended lines.
5.2 Grid of high-resolution models
When modeling spectra in high resolution, there are some differences to the low-resolution case and the following considerations have to be made:- A value for microturbulence must be set, because the shape
of strong lines sensitively depends on the value chosen for vt.
This is not the case in low resolution, so that there is no need to
fine-tune this parameter. In high resolution the set of basic stellar
atmosphere parameters is
. To do a proper parameter estimation we would need to create a four-dimensional grid of synthetic spectra, varying all the parameters indicated above. This goes further than the purposes of this paper, where we aim to check the applicability of our method. Therefore we fixed the microturbulence parameter to a typical value, guided by the results of a standard ``classical'' spectral analysis (see below).
- To compute synthetic spectra, a list of lines with their
wavelength and atomic data must be included. The atomic data consist of
oscillator strength
and excitation potentials, which are an important source of differences in the final shape of a given line. For the comparison consistent line lists are needed.
- In order to make a proper fit to a broad wavelength range, the models should be perfect. Our models were computed under the simplifying assumption of LTE and with plane-parallel atmosphere layers; the atomic data have errors, too. There are too many features to fit, therefore a global fit becomes difficult, almost impossible. For this reason we selected windows of a limited spectral range and fitted the observed spectra only in these.
We determined the three parameters temperature, metallicity and gravity
using
neutral and ionized iron lines. Neutral lines are sensitive to
temperature and single ionized ones to gravity (Fuhrmann
1998). The metallicity was
effectively obtained from both Fe I and II, for a given temperature and
gravity. In
the wavelength range of 580 nm-680 nm, we have six Fe
II lines and 20 Fe I ones that are unblended and strong enough to be
present in most
metal-poor stars of our sample. The lines are indicated in
Table 3,
where wavelength and
values are
listed. The
values and excitation potentials are taken from
Nissen et al. (2002)
and the VALD database (Kupka
et al. 2000).
Our grid of high resolution models covers a range in
metallicity of ,
in effective temperature of
,
in surface gravity of
and in scaling factor for normalization from 0.85 to 1.15. The grid
steps are the same
as in Sect. 3.2.
The models have
and vt =
1.2 km s-1, according to the
values obtained for our stars with the ``classical'' method (see
below).
![]() |
Figure 9:
Fit of the spectrum of the star HD 195633 (points) with the
best model (red dashed line). The parameters of the synthetic spectrum
are [Fe/H] = -0.5, |
Open with DEXTER |
Table 3: Fe I and II lines used for the high-resolution fits.
5.3 Preparing the observed data for synthetic spectral fitting
The reduction of the data was carried out in the same way as in
Sect. 3.3,
with the following difference in the normalization
procedure. At this resolution, neither the polynomial-fitting approach
by
Allende Prieto et al. (2006)
nor IRAF were able to automatically
find a good
pseudo-continuum. Hence, the observed spectra were
normalized interactively using Midas (Crane
& Banse 1982), and the synthetic spectra were
computed
with normalized flux. The compression procedure was the same as
in Sect. 3.4.
Compressing the grid of high-resolution
spectra takes more time than for the low-resolution ones, but this has
to be
carried out only once to obtain a new grid of
y-numbers.
5.4 Results of the high-resolution analysis using MA
Figure 9
shows an example fit of HD 195633
(points) with the best MA
model (red dashed line) and the model with
parameters found with the ``classical'' method (blue dashed line; see
below). Each panel
represents a spectral window with the lines of Table 3
used for the parameter estimation. All three parameters were
determined simultaneously and the best fit corresponds
to the model with parameters of [Fe/H] = -0.5,
K and
= 4.61. The final parameters of all stars in
this sample are given in Table 4. The first column
of the table
indicates the name of the star and the six next ones are the
parameters found with two different MA
analyses. The first set
(``free'') corresponds to the standard approach of determining all
parameters simultaneously from the spectrum only.
In the second variant (``restricted'') the gravity is determined
independently using Eq. (14), which will be
introduced below. The last four columns are the
parameters obtained from the ``classical'' approach (below).
Table 4:
Parameters of the stars (Col. 1) obtained with MA
for both types of analysis discussed in the text.
5.5 Results of a ``classical'' analysis
In order to compare our MA
results, we analyzed the spectra
with a classical procedure, determining the stellar parameters through
an iterative process. The method is basically the same as in, e.g.,
Nissen et al. (2002):
For the effective temperature it relies on the infrared flux method, which provides the coefficients to convert colors to effective temperatures. We used the (V-K) color and the calibration of Alonso et al. (1996), after converting the K 2MASS filter (Skrutskie et al. 1997) to the Johnson filter (Bessell 2005) and de-reddening the color. The extinction was taken from Schlegel's dust maps (Schlegel et al. 1998) in the few cases where we could not find the values in Nissen et al. (2004,2002).
Surface gravities were determined using the basic parallax
relation
where
is the V magnitude corrected for interstellar
absorption, BC the bolometric correction and
the parallax
in arcsec. We adopted a different mass value for each star based on
those of Nissen et al. (2002),
which are between 0.7 and 1.1
.
The bolometric correction BC was calculated with
the solar calibration of
as in Nissen et al. (1997).
The equivalent widths of neutral and ionized iron lines were
used to
determine the metallicity [Fe/H]. It was
obtained using Fitline (François
et al. 2007), which fits Gaussians to the line
profiles. The
equivalent widths computed from these Gaussians fits were converted to
abundances by running MOOG
(Sneden 1973; Sobeck, priv.
comm.) with the plane-parallel LTE
MARCS model atmospheres of (Gustafsson
et al. 2008), which differ from those
used for MA.
The line list contains the lines of
Table 3
and more taken from bluer wavelengths in the range
300-580 nm (Hansen, priv. comm.).
Finally, the value for the microturbulence was found by requiring that all the equivalent widths of neutral lines should give the same Fe abundance as the ionized lines.
In order to obtain the four parameters in this way, we started with an initial guess for each of the interdependent parameters. After determining them with the above steps all values will change due to the interdependence, hence we had to iteratively determine the parameters until their values showed a negligible change.
We are aware that our metallicity based on Fe I lines is
ignoring any non-LTE
effects (Asplund
2005; Collet
et al. 2005). But because neither the
classically nor the MAanalysis
are taking this into account, the results
obtained by both methods can well be compared.
![]() |
Figure 10:
Upper panels: Correlations between the
results obtained with the automatic fitting (MAX) and the classical
method (EW) for metallicity ( left), temperature (
middle) and gravity ( right). In
each panel, the one-to-one line is overplotted and a legend
containing the mean difference (in the sense ``EW-MA |
Open with DEXTER |
The upper panels of Fig. 10
show the results of metallicity (left),
temperature (middle) and gravity (right) for the entire sample of
stars. The x-axis shows the results of the
automatic fitting
MA
and the y-axis the ``classical'' parameter
estimation (EW).
The one-to-one line is overplotted in each figure and the legend
indicates the mean of the differences (MA
- EW) and its standard
deviation, denoted as offset and scatter, respectively. Each point
corresponds to a star of Table 4 with parameters
obtained by
the ``free'' method.
Gravity shows a scatter of dex
and a negligible offset. The determination of the gravity from
FeII lines has always been a problematic task: Fuhrmann
(1998) in an
extensive study of parameters of nearby stars showed that surface
gravities of F-type stars located at the turn-off point can easily
differ by up to
0.4 dex
if derived from either LTE iron
ionization equilibrium or parallaxes. The amount of ionized lines
(gravity dependent) present in the
spectra is usually smaller than that of neutral ones (temperature
dependent). By performing an
automatic fitting of weighted spectra that contain only six Fe II lines
compared to 20 Fe I lines, a scatter of 0.46 dex is
reasonable.
The result for the effective temperature shows a very small
offset of
only -10 K, but a quite
large scatter of K.
Given that two
different methods were employed to determine it, this still appears to
be acceptable. The scatter may also be
affected by our fixed value for the microturbulence, which was taken
from the average of the values found with the EW
approach (see
Table 4),
but which in individual cases may differ severely.
One can decrease the scatter in the temperature difference by
removing
the three most discrepant objects from the sample. The star
HD 63598 (triangle) has an
offset of 398 K. The Schlegel maps give a de-reddening for
this star that is unrealistically large, so instead we set it to zero.
This led to a temperature that was too low. Jonsell
et al. (2005) have found a temperature of
5845 K for this star, reducing the difference to
233 K with respect to our result.
The star G005-040
(asterisk) is the second case, where the offset is 462 K. For
this
star, the continuum subtraction was not perfect in every spectral
window, resulting in a fit where the parameters were quite
unreliable. Finally, the weak lines of the observed spectrum of
HD 19445
pushed MA
to the border of the grid. The parameters in this case
were undetermined.
By removing these three stars from our sample, the
scatter for temperature is reduced to 197 K.
Metallicity, on the
other hand, shows a very good agreement, with a negligible offset of
0.02 dex and a scatter of 0.16 dex (upper left panel
of Fig. 10).
The
lines used for the automatic fitting (MA)
and for the classical
analysis (EW) are in most cases identical, except for the lowest
metallicities,
where some of them are hardly visible. In these cases the classical
method also resorted to other lines outside the MA
wavelength range.
The atomic data are identical, but the value for microturbulence are
not, as
mentioned before. In view of all this, the very good correlation of
metallicity
is encouraging.
5.6 ``Restricted'' parameter recovery
An example for the individual fit of the best MA




It is also interesting to notice from the upper panels of
Fig. 10
that stars
with large discrepancies in gravity
(for example those with special symbols) also give large discrepancies
in temperature, as expected. They show
a quite unsatisfying fit for the Fe II lines. Motivated by this, we
did an additional test by restricting MA
to the determination
of the parameters with input values for
obtained
from the EW method, i.e. we found a local maximum
point of the likelihood in a restricted area. To do this we chose the
three closest gravity values from our grid of synthetic spectra to the
classical one found with Eq. (14)
- see Col. 9 of Table 4 - and we searched
for the maximum point within this range. May be that for
there is no local maximum in this range, and the final estimation will
go to the border of the grid, as the case of CD-3018140, where
.
The general tendency is anyway a local maximum close to the input
EW-value.
Now the agreement
with the classical method for the effective temperatures became
excellent, with a negligible offset of only
7.42 K and a scatter of 128 K, as seen in the lower panels of
Fig. 10.
Gravity was also better constrained, with a small scatter of
0.14 dex
and a negligible offset. Note that we did not necessarily obtain
identical values, because the MA
gravity is obtained by using the value from Eq. (14) only as input, and
we were looking for a final solution close to this value. The behavior
of the metallicity
does not change with respect to the ``free'' case, demonstrating the
robustness of the determination of this parameter. The results
obtained when using parallaxes for the initial guess for
are
summarized in Table 4
in the columns under the heading
``MA
,
restr.''.
It is instructive to discuss the implication of this
comparison.
For the ``classical'' EW method we determined the
parameters making use of the best information available and of the
freedom to adopt the method to each star individually. The
iterative process allowed us to decide where to stop the iteration, or,
if no satisfying convergence could be reached, to draw on the options
to
move to another spectral window, to use other lines, or disregard
problematic lines. Moreover, the continuum could be
separately subtracted for each line, thus creating locally
perfect normalized flux levels. For the effective temperature, the
more reliable infrared flux method employing photometric data could be
used
instead of only relying on spectra. The advantage of determining
independently was
already demonstrated.
On the other hand we were
attempting an estimate of the parameters only from the spectral
information without fine-tuning the models or fit procedure for each
star for our automatic MA
method. Given this, the comparison with the full interactive method is
surprisingly good. We demonstrated that by using additional information
for one parameter (here
)
it becomes even better for the
remaining two quantities. Table 4 shows our final
parameter
estimate for our stars, when we make use of the parallaxes as
additional
information.
The result of this test is that we demonstrated our ability to estimate the basic stellar atmosphere parameters quickly and accurately enough for a substantial sample of stars. While individually severe outliers may occur, the method is accurate overall for a whole population of stars. For this purpose the EW method would be much too slow and tedious. Our method may also serve as a quick and rough estimate for a more detailed follow-up analysis.
6 Summary and conclusions
We described MA,
a new derivative of MOPEDfor the estimation of parameters from stellar
spectra. In this case the parameters were the metallicity, effective
temperature and surface gravity. The method reduces the data to a
compressed data set of three numbers, one per parameter. Assuming that
the noise is independent of the parameters, the compressed data contain
as much information about the parameters of the spectrum as the entire
data itself. As a result, the likelihood surface around the peak is
locally identical for the entire and compressed data sets, and the
compression is ``lossless''. This massive compression, with the degree
of compression given by the ratio of the size of the data to the number
of parameters, allows the cost function calculation for a parameter set
to be sped up by the same factor. For SDSS data, the spectra are of the
order of 1000 datapoints, each with a corresponding error, which must
be compared to a similarly sized model. The speed up factor in this
case is then at least 1000/3,
333x.
This extremely fast multiple parameter estimate make the MA
method a powerful tool for the analysis of large samples of stellar
spectra. We have applied it to a sample of 17 274 metal-poor
dwarf stars with low-resolution spectra from SEGUE, using a grid of
synthetic spectra with the parameter range of
[-2.5, -0.5] dex in metallicity, [5000, 8000] K in
effective temperature and [3.5, 5] dex in surface gravity,
covering a wavelength range of [3850, 5200] Å.
From the Ca II , Balmer and Mg Ib lines, which are the
strongest absorption features identified in SDSS spectra, we estimated
the metallicity with averaged accuracies of 0.24 dex, the
temperature with 130 K and
with 0.5 dex, corresponding to the 1
errors. Surface gravity is a poorly constrained parameter using these
data, mainly due to the lack of sensitive features of this parameter
(apart from some degree of sensitivity of the wings of the Mg Ib
triplet) and the considerable noise in the spectra when compared to
high-resolution spectra. Additional information to the spectra, such as
photometry, would help to constrain the gravity parameter more.
MA
has the option to simultaneously fit different spectral windows. We
have compared estimates of the parameters using the whole spectrum and
only those data ranges where known lines exist. Both analyses take
approximately the same time and agree excellent in recovered
parameters. This suggests that for these low-resolution spectra there
is no need to carefully and laboriously select the spectral windows to
be analyzed with MA
,
the method calculates this automatically as part of its weighting
procedure.
With the assumption that the spectra behave similarly under
changes of the parameters, the choice of the fiducial model for the
compression is free. We have created eight compressed grids using
different fiducial models for the -vector calculation and obtain
agreement between the recovered parameters. Given the low
signal-to-noise of our data, it is better to use a more metal-poor
fiducial model, as the dependence on the parameters will be focussed on
the strong lines.
We have comprehensively investigated the correlations of our
results with those obtained for the SEGUE Stellar Parameter Pipeline (Lee et al. 2008a), which
reports results from a number of different methods. The results from MA
agree well with those of the various pipelines, and any differences are
consistent with those between the various accepted approaches. We are
aware of the MATISSE (Recio-Blanco
et al. 2006) method of parameter estimate, which
uses a different combination of weighting data but is closer to the MA
approach than the standard pipeline methods. We look forward to
comparing our results with those of MATISSE when they become available.
More specifically, temperature agree excellently with the
averaged temperature of SSPP, with a negligible offset of
-61 K and low scatter of 112 K. Our metallicities
show a tendency to be -0.32 dex lower than SSPP averages. The
small scatter of 0.23 dex suggests that the different spectral
features used in the analysis (mainly Ca II lines) could shift the zero
point of the metallicity. The most pronounced discrepancy is found in
surface gravity, where Ma
reports values 0.51 dex (
= 0.39) higher than the averaged value of the pipeline. We saw that
this is consistent with the discrepancies seen between other methods.
We have also tested MA
on a sample of 28 high resolution spectra from VLT-UVES, where no
offset in metallicity is seen. In this case we have carefully chosen
the models and spectral range for comparing our parameter estimation
against a ``classical'' approach: temperature from photometry, surface
gravity from parallaxes and metallicity from equivalent widths of
neutral and ionized iron lines. We have calculated the parameters
ourselves to avoid additional systematic offsets that various different
methods would introduce in stellar parameter scales. These results were
compared with the automatic fitting of 20 Fe I and 6 Fe II
lines (that coincide in most cases with the equivalent widths
calculations) made with MA
.
We obtained large scatter in gravity and temperature (0.44 dex
and 220 K, respectively), but no systematic offset. The normalization
of the continuum in some cases was not perfectly done, making it
difficult to fit every line properly: especially for the Fe II lines.
They produce a scatter in gravity, which drives a scatter in
temperature.
We have seen in our fits that our best model does not differ very much from the model with parameters found by the ``classical'' approach. Motivated by this we decreased the scatter in gravity by using additional information to the spectrum, i.e. using the gravities determined from parallaxes as input value. This forced our method to find a local maximum point of the likelihood close to this input gravity value. Fixing the gravity gave an improved agreement in temperature, now with a scatter of 128 K. Metallicity does not change when forcing gravities, illustrating the robustness of determination for this parameter.
MA
is an extremely rapid fitting technique and as such is independent of
model and data used. It will work for any star for which an appropriate
grid of synthetic spectra can be calculated. Although the grid
calculation is time consuming, it only needs to be performed once
allowing an extremely rapid processing of individual stars.
MA
is one of the fastest approaches to parameter determination, and its
accuracy is comparable with other methods. To develop MA
further in preparation for the next generation of surveys (e.g. Gaia,
APOGEE or LAMOST), we plan to expand our range of model grid and
integrate photometric data to improve the determination of the
parameters, especially surface gravity. We must be aware that in a grid
of synthetic spectra with a broader parameter range a set of
-vectors
from one single fiducial model will not represent well the dependence
on the parameters in the entire sample. To cover the entire sample with
well represented dependences, we plan to compute different compressed
grids, each of them with
-vectors calculated from
different fiducial models. The parameter estimate in this case will be
made with the different compressed grids using a final convergence
test. After convergence, the final parameters should be estimated from
the compressed grid with the fiducial model with parameters close to
the final result.
This work is part of the PhD Thesis of Paula Jofré and is funded by an IMPRS fellowship. We thank Richard Gray, Martin Asplund, Francesca Primas and Jennifer Sobeck for their helpful comments in interpreting our results, Alan Heavens for algorithmic advice and especially Carlos Allende Prieto, for all the great advice and for sharing with the authors the continuum subtraction routine. P. Jofré is thankful to Timo Anguita and Manuel Aravena for programing tips and Thomas Mädler for his careful reading of the manuscript and the support to take this paper out. Finally, the authors gratefully acknowledge the many and detailed positive contributions made by the referee. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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Footnotes
- ... MOPED
- MOPED is protected by US Patent 6,433,710, owned by The University Court of the University of Edinburgh (GB)
- ... registration
- http://www.blackfordanalysis.com
- ... webpage
- http://www1.appstate.edu/dept/physics/spectrum/spectrum.html
- ... Database
- http://physics.nist.gov/PhysRefData/ASD/index.html
- ... webside
- http://kurucz.harvard.edu/linelists.html
- ... IRAF
- IRAF is distributed by the National Optical Observatory, which is operated by the Association of Universities of Research in Astronomy, Inc., under contract with the National Science Foundation.
All Tables
Table 1: Strongest lines in F-G dwarf stars.
Table 2:
Offsets ()
and scatter (
)
of the differences (raw - column) between the methods: MA
(this work).
Table 3: Fe I and II lines used for the high-resolution fits.
Table 4:
Parameters of the stars (Col. 1) obtained with MA
for both types of analysis discussed in the text.
All Figures
![]() |
Figure 1:
(A) Fiducial model with parameters |
Open with DEXTER | |
In the text |
![]() |
Figure 2: Correlations between the Fisher matrix values obtained from the full and compressed likelihoods for metallicity (upper panel) and effective temperature (lower panel) for 75 randomly selected stars of SEGUE. The line corresponds to the one-to-one relation. |
Open with DEXTER | |
In the text |
![]() |
Figure 3: Likelihoods of the compress data set ( left) and full data set ( right) in the parameter space of effective temperature and metallicity for four randomly selected SEGUE stars. In each panel eight equally spaced contour levels are plotted. The triangle and diamond correspond to the maximum point of the compressed and the full data set, respectively. Both maxima lie in the first confidence contour level, meaning that both data sets reach the same solution for these two parameters. |
Open with DEXTER | |
In the text |
![]() |
Figure 4: Radial velocities found using minimum fluxes of strong lines given by Table 1 (MAX) compared with those of the SEGUE database. The difference of the radial velocities given by the SEGUE database from those obtained by our method is indicated in the legend as offset, with its standard deviation as scatter. |
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Example of the fit between a randomly selected SEGUE star (crosses) and
a synthetic spectrum (red dashed line). The legend indicates the
stellar atmosphere parameters of the model and the value of the reduced
|
Open with DEXTER | |
In the text |
![]() |
Figure 6:
Upper panel: metallicity ( left),
effective temperature ( middle) and surface gravity (
right) obtained using MA |
Open with DEXTER | |
In the text |
![]() |
Figure 7: Upper plot: comparison of the results from the SEGUE Stellar Parameter Pipeline (SSPP) with different methods: this work (MAX, black), (Lee et al. 2008a, NGS1, blue), (Re Fiorentin et al. 2007, ANNSR, red) and (Allende Prieto et al. 2006, k24, green). Lower part: comparisons of individual methods for a randomly selected subsample of stars. Each plot has a line with slope of unity. The different offsets and scatters between the results are indicated in Table 2. In each plot, a selection of 150 random stars has been used. |
Open with DEXTER | |
In the text |
![]() |
Figure 8:
Histograms of the differences of the results obtained using 8 different
fiducial models for the compression. Left:
metallicity, middle: temperature and right:
gravity. The parameters of the fiducial model are indicated in the
right side of each plot, labeled as [[Fe/H], |
Open with DEXTER | |
In the text |
![]() |
Figure 9:
Fit of the spectrum of the star HD 195633 (points) with the
best model (red dashed line). The parameters of the synthetic spectrum
are [Fe/H] = -0.5, |
Open with DEXTER | |
In the text |
![]() |
Figure 10:
Upper panels: Correlations between the
results obtained with the automatic fitting (MAX) and the classical
method (EW) for metallicity ( left), temperature (
middle) and gravity ( right). In
each panel, the one-to-one line is overplotted and a legend
containing the mean difference (in the sense ``EW-MA |
Open with DEXTER | |
In the text |
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