A&A 472, 1041-1053 (2007)
DOI: 10.1051/0004-6361:20077460
1 - Department of Physics and Astronomy, Institute for
Astronomy, K. U. Leuven, Celestijnenlaan 200 D, 3001 Leuven, Belgium
2 - Sterrenkundig Instituut Anton Pannekoek, University of
Amsterdam, Kruislaan 403, 1098 Amsterdam, The Netherlands
3 -
Department of Astronomy and Space Physics, Uppsala University, Box
515, 751 20 Uppsala, Sweden
Received 12 March 2007 / Accepted 7 June 2007
Abstract
Context. One of the key ingredients in establishing the relation between input signal and output flux from a spectrometer is accurate determination of the spectrophotometric calibration. In the case of spectrometers onboard satellites, the accuracy of this part of the calibration pedigree is ultimately linked to the accuracy of the set of reference spectral energy distributions (SEDs) that the spectrophotometric calibration is built on.
Aims. In this paper, we deal with the spectrophotometric calibration of infrared (IR) spectrometers onboard satellites in the 2 to 200 m wavelength range. We aim at comparing the different reference SEDs used for the IR spectrophotometric calibration. The emphasis is on the reference SEDs of stellar standards with spectral type later than A0, with special focus on the theoretical model atmosphere spectra.
Methods. Using the MARCS model atmosphere code, spectral reference SEDs were constructed for a set of IR stellar standards (A dwarfs, solar analogs, G9-M0 giants). A detailed error analysis was performed to estimate proper uncertainties on the predicted flux values.
Results. It is shown that the uncertainty on the predicted fluxes can be as high as 10%, but in case high-resolution observational optical or near-IR data are available, and IR excess can be excluded, the uncertainty on medium-resolution SEDs can be reduced to 1-2% in the near-IR, to 3% in the mid-IR, and to
5% in the far-IR. Moreover, it is argued that theoretical stellar atmosphere spectra are at the moment the best representations for the IR fluxes of cool stellar standards.
Conclusions. When aiming at a determination of the spectrophotometric calibration of IR spectrometers better than 3%, effort should be put into constructing an appropriate set of stellar reference SEDs based on theoretical atmosphere spectra for some 15 standard stars with spectral types between A0 V and M0 III.
Key words: instrumentation: spectrographs - techniques: spectroscopic - stars: atmospheres - stars: late-type - infrared: stars - techniques: photometric
For (IR) spectrometers, three main steps have to be covered to establish the relation between input signal, flux, and wavelength: spatial, spectral, and photometric calibration. In this paper, we focus on the reference spectral energy distributions (SEDs) used in the spectrophotometric ( = spectral+photometric) calibration process, with main emphasis on the spectral calibration part.
For (IR) spectrometers onboard satellites, the determination of the relative spectral response function (RSRF), which characterises the wavelength-dependent response of a spectrometer, is often a two-step process: (1) first, the RSRF is determined during laboratory tests from measuring a cryogenic blackbody calibration source at differing temperatures; (2) after launch, the RSRF is refined by comparing observations with reference SEDs of various celestial calibration sources, preferably at a spectral resolution comparable to that of the instrument.
Whenever an instrument covers a new spectral window or has a higher
sensitivity than preceding instruments, new reference SEDs have to be
constructed. This was and will be the case for instruments onboard the
ESA-Infrared Space Observatory (ISO), the NASA-Spitzer satellite, the
ESA-Herschel mission, the NASA-James Webb Space Telescope (JWST), and
many others. In this paper, emphasis will be put on the reference SEDs
of calibration sources for instruments covering the 2 to 200 m
wavelength range. When selecting calibration sources, issues like
brightness in the wavelength regime covered by the instrument,
confusion by neighbouring sources, sky visibility, and pointing
accuracy are to be considered. In the 2-200
m wavelength range,
planets, asteroids, and cool standard stars are the ideal calibration
sources to cover the whole dynamic range of present-day instruments
(see Sect. 2). This paper focuses on the
reference SEDs of standard stellar candles, with the aim of
assessing the reliability and achieving accuracy when using IR stellar
standards for spectral calibration purposes. Quite often, calibration
scientists or observers (have to) rely on "quoted'' errors, without
having insight into this part of the calibration pedigree. Since for a
proper interpretation of the output signal one needs to have a grip on
the uncertainties that propagate to the fluxes ultimately calculated
by an instrument, the discussion in this paper is required.
Section 2 gives an overview of the different IR flux calibration sources. The status of stellar reference SEDs in the IR is discussed in Sect. 3, and we focus on the theoretical atmosphere spectra in Sect. 4. Different stellar reference SEDs are compared in Sect. 5, and we end with the conclusions in Sect. 7.
In this section, we discuss different IR calibration sources used in the spectrophotometric calibration process.
For calibration purposes at wavelengths longer than 30
m,
planets are extremely useful since they are among the few astronomical
objects bright enough in the far-IR to allow sufficiently accurate
flux density predictions. At 100
m, the flux values for Neptune
and Uranus range between 100 and 1000 Jy. Uranus is often used as
the primary calibrator, as it is known to be a reliable calibrator
(Griffin & Orton 1993; Sidher et al. 2003), while Neptune
and Mars are used as secondary calibrators.
Good models for Uranus exist (e.g., Orton & Burgdorf 2003; Sidher et al. 2003; Moreno 1998). Radiative transfer calculations are done in spherical geometry and take the limb into account. The models do not, however, take chemical reactions into account, nor heating and cooling. In the mm and submm-regimes, the models are confirmed within the accuracy of 10-20%. The big advantage of Mars over Uranus is that it is bright. Different models for Mars' surface exist (e.g. Rudy et al. 1987; Hartogh et al. 2005; Lellouch et al. 2000). Using Mars as a primary calibrator, however, would pose different problems: (1) Mars is a planet with a surface and an atmosphere, both contributing to the continuum emission; (2) sandstorms influence the line shapes; (3) the ice caps can influence the continuum if the ice is melting. Neptune could serve almost equally well as an independent (secondary) calibrator. However, many large features are visible in the Voyager IRIS spectra, which may indicate systematic variability with rotational phase (Bishop et al. 1998).
Typical surface temperatures of main-belt asteroids are such that they
emit the bulk of their thermal radiation in the far-IR. At wavelengths
longer than 20
m, the largest asteroids are brighter than
the brightest IR stellar sources: they cover the flux range between
100 and 1000 Jy in the wavelength range between 30 and 45
m, and
hence fill a gap where stellar calibrators are not available
(see Fig. 1 in Müller & Lagerros 1998).
The work of Müller (Müller et al. 2005; Müller & Lagerros 2002,1998,2003) has largely improved the accuracy of the
theoretical model SEDs of asteroids. The most accurate SEDs
at present are based on the thermophysical model (TPM) for describing
the asteroids' thermal emission
(Müller & Lagerros 1998). Due to the asteroids' orbit,
brightness temperatures vary by 17
13%. Müller & Lagerros (2002) quote an accuracy of
5% between 5 and 200
m for Ceres, Pallas, and Vesta, i.e. for the three largest asteroids.
The main limitations for asteroids as IR calibration sources come from the changing background conditions and the flux changes on timescales of hours due to rotation. Moreover, the list of reliable asteroids is still short.
In the near-IR, stars are ideal calibrators, especially since they are
almost point-like and span a range in flux level more than 4 orders
of magnitude between 2 and 50 m, and one can create a database
covering the whole sky. Stellar standards are, however, quite faint in
the far-IR and can henceforth not be used as primary calibrators at
the far-IR wavelength ranges. Good IR calibration sources need to
comply several criteria: be single and non-variable, not have
an IR excess due to a chromosphere, debris disk, or a circumstellar
envelope, and be located in an uncrowded region that can be
observed all or most of the time (Decin et al. 2007). Preferably,
the IR standard is cooler than
10 000 K to provide a good
signal-to-noise ratio over most of the wavelength range.
In the case of spectrophotometric calibration, one extra
requirement should be added to this list regarding the composition of
the sample of used calibrators with regard to the spectral types. For
photometric calibration purposes, the use of only one spectral
type can be justified (Diaz-Miller 2007), but in the case of a
spectrometric calibration, where every single wavelength needs to be
calibrated, none of the available reference SEDs of any spectral type
has high enough accuracy in the full 2 to 200 m wavelength
range (see Sect. 3). Three classes of spectral stellar
standards have been commonly used in the IR spectrometric calibration
pedigree: (i) early A dwarfs, (ii) solar analogs, and
(iii) late-type giants, usually of spectral type G9-M0 III. Each of these groups provides a different challenge (see
below), but by combining them, they (i) reduce the chances for
systematic errors, possibly introduced by the use of only one spectral
type, and (ii) they will increase the total accuracy since the
best SED can be used for each part of the spectrum.
Planck curves represent the simplest way to model stellar far-IR fluxes. A comparison with the predictions of a solar continuum model shows clear deviations, arising to more than 20% in the far-IR (see Fig. 14 in van der Bliek et al. 1996). The reason is the contribution of the H- free-free opacity, shifting the flux-forming region more to outer cooler layers. Neither line or continuous absorption nor back-warming effects are/can be included.
The Engelke function (Engelke 1992) is a more
sophisticated two-parameter analytical approximation to the 2-60 m infrared continuum spectrum for giants and dwarfs with
effective temperature,
Teff, between 3500 and 6000 K. This spectral
function is based on the scaling of a semi-empirical (plane-parallel)
solar atmospheric profile to differing effective temperatures. The
result describes the continuum spectrum expected for stars in
terms of their effective temperature and angular size. The estimated
probable error in absolute flux values is quoted to be
3%
below 10
m, growing to
5% around 25
m and 6% at
60
m.
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Figure 1:
Ratio between the continuum flux predicted
from a theoretical plane-parallel stellar atmosphere with
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The main limitation of the Engelke function is neglect of the
influence of the surface gravity. In Fig. 1 the
difference between the Engelke function and the continuum flux of a
plane-parallel theoretical stellar atmosphere (see Sect. 4)
with an effective temperature,
Teff, of 3500 K for different values
of the gravity is shown. For values of the logarithm of the gravity,
log g, lower than 3.00, the absolute deviations may be as high as
25%. At a temperature of 6000 K, the absolute deviations
approximate the errors quoted by
Engelke (1992). Important, however, for the
discussion in this paper of the RSRF determination are the
systematic differences between the Engelke continuum and both
(1) the continua calculated by ab-initio model atmosphere calculations
(see Fig. 1) and (2) real continua deduced from
observations (see Engelke et al. 2006). This systematic
discrepancy occurs both for solar analogs and the cooler K-M giants:
the same type of curves as displayed for the lower gravity values in
Fig. 1 (M-giants) also occur in an analogous plot
for G dwarfs, but with a smaller amplitude of 3%. Note
also that for almost all values of log g, the ratio as displayed in
Fig. 1 is systematically higher than one, implying that
when either the effective temperature or the angular diameter (
) is
kept fixed, the other parameter will be overestimated.
Another issue not taken into account are sphericity effects, important when dealing with the extended atmospheres of giants. Figure 2 compares a theoretical atmosphere spectrum (see Sect. 4) calculated using spherical geometry with the spectrum obtained in plane-parallel geometry for the case of an M0 III giant. Compared to a plane-parallel atmosphere, the radiation field of a spherical model is diluted in the upper photospheric layers, causing the temperature (and hence the source function) there to be lower. The lower surface flux is, however, compensated by the fact that the IR flux arises from higher layers. The net result is an infrared excess of the spherical model relative to the plane-parallel model. Important for this paper is the rising continuum in the bottom panel of Fig. 2, implying that the use of the Engelke function or a plane-parallel geometry to represent the SED of giants may introduce uncertainties on the broad-band RSRF characterisation of a few percent.
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Figure 2:
Upper panel: model atmosphere spectrum in spherical
geometry (black) compared to the theoretical spectrum calculated in
plane-parallel geometry (grey) representing the M0 III giant
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In a series of papers on spectral irradiance calibration in the
infrared, Cohen and colleagues have presented a self-consistent
network of absolutely calibrated reference spectra between 1.2 and
35 m for over 600 stars spanning spectral types from A0 to M4 and
distributed over the entire sky (Cohen et al. 1999,2003,1996a,1992b,a,1995, often referred to as the CWW
network). The CWW network
is the result of a tremendous and consistent work done during the past
15 years. Four main steps can be distinguished in the network. (i)
The approach is based on a pair of absolutely calibrated models of the
two A-type dwarfs Vega and Sirius calculated by
Kurucz (1993)
(Cohen et al. 1992a). (ii) The next layer in the network
is a set of composites of 13 secondary calibration
stars. Composites are spectra measured by various ground-based and
airborne telescopes (KAO, IRAS LRS, etc.) that are averaged and/or
spliced together to form a continuous spectrum between 1.2 and
35
m. This set compromises one G2 dwarf -
Cen (G2 V); seven K giants -
Tau (K5 III),
Boo (K2 IIIp),
Gem(K0 III),
Hya (K3 II-III),
TrA (K2 III),
Car (K3 III),
Dra (K5 III); and
5 M giants -
Peg (M2.5 II-III),
And (M0 III),
Cet (M1.5 III),
Cru
(M3.4 III),
UMa (M0 III)
(Cohen et al. 1992b,1996a,b,1995). The KAO was no
longer operational at the time the composites of
Dra,
Cet,
Cru and
UMa were constructed. Spectral regions that are opaque from the
ground were therefore replaced by spectra of other stars with the same
spectral type. Note that the "composite'' of
Cen is
a Kurucz model-atmosphere spectrum, which was then used as a reference
to create the composite spectra of
TrA and
Car. For wavelengths longer than
22
m,
the Engelke function was used for twelve composites to represent the
stellar spectrum, the exception being
Tau. (iii) These 13
composite spectra are the basis for a set of absolutely calibrated
spectral templates for 602 stars with spectral types between
G9.5-M0.5 III (Cohen et al. 1999,2003). For each star a smoothed composite spectrum
is chosen as its spectral template according to the spectral
class. The template is corrected for reddening and normalised using
available optical photometry. For purpose of the NASA-Spitzer Infrared
Array Camera (IRAC) calibration an extra set of absolutely calibrated
0.275-35
m spectra of 33 optical standard stars with spectral
types between A0-A5 V and K0-M0 III, and flux levels down to V
11-12, was prepared (Cohen et al. 2003). (iv) In
support of the Infrared Space Observatory (ISO) PHT and LWS
instruments, CWW have provided extrapolated continuum spectra of few
composites till 300
m (Cohen et al. 1996b). The
continuum spectra have been interpolated in a grid of four
plane-parallel model-atmosphere spectra (see also
Sect. 3.2) with solar abundances. To obtain the
temperature-versus-continuum optical depth relation
(
), a scaling based on the ratio
Teff(desired)/
Teff(grid model) was performed. Differences in gravity
were not accounted for. An uncertainty of 5.67% was assessed to
these extrapolations.
It should be emphasised that the CWW network has some major advantages, the main ones being (1) the common calibration pedigree; and (2) the good sky-coverage (see Fig. 2 in Cohen et al. 1999). For that reason, the CWW network was used to support the calibration of many spaceborne, airborne, and ground-based instruments, as e.g. instruments onboard ISO, Spitzer, AKARI, MSX, etc.
However, aiming for a (spectrophotometric) calibration better than a few percent, some steps used in the construction of the CWW network deserve some comments.
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Figure 3:
Observed (red - diamonds) versus synthetic (green -
stars) magnitude difference between Vega and Sirius
(Mag
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Figure 4:
UKIRT CGS3 15-22 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Table 1:
Summary of uncertainties attributed to the standard star spectra in
the CWW network in the 2-200 m range.
In order to remove the "molecular feature'' bias that can enter into the template construction, the computation of ab-initio theoretical model atmosphere spectra may offer a solution (see next section). Moreover, theoretical atmosphere spectra can be computed at any spectral resolution (comparable to that of the instrument) and in that way compete with the template spectra, which have quite low spectral resolution.
Different groups have given a lot of effort into developing trustworthy
hydrostatic 1-dimensional (1D) model atmosphere codes, the
most famous ones in the field of cool stellar atmospheres (
Teff
10 000 K) being the ATLAS atmosphere code
(Castelli & Kurucz 2004; Kurucz 1996,1970,1993), the
MARCS code (originally presented in
Gustafsson et al. 1975, with the current grid being
described in Gustafsson et al. 2003), and the
PHOENIX code (Hauschildt et al. 1999a,b). Also in the case of late-type (variable)
asymptotic giant branch (AGB) stars, the 1D hydrostatic models are
still useful in providing us with physical information, due to the
great detail now included in their treatment of
opacities. Hydrodynamic models in 1D
(e.g. Höfner et al. 2003; Winters et al. 2000; Bessell et al. 1996) include detailed physics, like
time-dependent dust formation and the coupling between gas, dust, and
radiation. Few dynamical models are now able to simultaneously solve
the equations of hydrodynamcics and frequency-dependent radiative
transfer leading to consistent dynamical density-temperature
structures (see, e.g., Höfner et al. 2003). Using these
kinds of models, one can already quantitatively reproduce the variation
in line profiles due to the influence of gas velocities (see,
e.g., Nowotny et al. 2005). Recently, 3D
hydrodynamical simulations of stellar surface convection have become
feasible thanks to advances in computer technology and efficient
numerical algorithms (Collet et al. 2006). The 3D models
can shed light on the coupling between convection and pulsation, as
well as be employed in element abundance analysis. These 3D model
computations are, however, still very computer-time intensive, and it is
not known how well they represent the surface layers of the stars, so
spectrum synthesis of a whole sample of stellar spectra is still
beyond the scope.
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Figure 5:
Upper panels: Template spectra of
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For the scope of this project of developing accurate SEDs for IR stellar standards, we therefore rely on a 1D hydrostatic model atmosphere code. We here opt to use the M ARCS code (see next section).
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Figure 6:
Upper panel: comparison between the ATLAS,
PHOENIX, and M ARCS model spectra. The PHOENIX
model is shifted downwards with a factor 10, and the M ARCS
model with a factor 100. Middle panel: comparison between the
ATLAS, PHOENIX and M ARCS model spectra at a
resolution of 100 for
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The M ARCS model atmosphere and spectrum synthesis code was
developed in Uppsala
(Gustafsson et al. 2003).
This local thermodynamic equilibrium (LTE) model atmosphere code is
built on the assumptions of spherical or plane-parallel stratification
in homogeneous stationary layers and hydrostatic equilibrium. Energy
conservation is required for radiative and convective flux, where the
energy transport due to convection was treated through a local
mixing-length theory. For a discussion of the method for solving the
radiative transfer equations in the atmospheric models and spectrum
synthesis, we refer to Nordlund (1984) and
Plez et al. (1992).
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Figure 7:
Contribution of CO (blue), SiO (orange), OH (green), CN
(pink), H2O (red), atoms (purple) to the 2-200 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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In the framework of the release of a grid of M ARCS models,
colleagues from the University of Uppsala are comparing the model
structures of the M ARCS, ATLAS, and PHOENIX
codes. An example typical of a cool K giant is shown in
Fig. 6. The plot is made in the
F
-space to clearly distinguish the differences. The continuum
level between the M ARCS and PHOENIX models do agree
within 0.5%. A larger difference of
5% is seen between
the M ARCS and ATLAS models for the continuum at
> 10
m. The difference in the molecular features is
4%
between M ARCS and PHOENIX and
8% between
MARCS and ATLAS at a resolution of 100. The reason for the
difference is the use of a plane-parallel geometry in the case of the
ATLAS models, whereas the PHOENIX and M ARCS models
use a spherically symmetric geometry. Slightly different surface
temperature and geometrical extension explain the difference between
M ARCS and PHOENIX model fluxes. Moreover, the
MARCS models are based on the Asplund et al. (2004)
solar abundances, while the PHOENIX models follow the results
of Grevesse et al. (1996). Note also that the
MARCS model uses a more complete OH and SiO line list at
> 10
m (see also Fig. 7).
For spectrum synthesis, data on the absorption by atomic and molecular
species are collected from different databases. For a discussion of
various available atomic and molecular line lists, we refer to
Decin (2000). For the purpose of this work, the following infrared
spectroscopic line lists were used: CO line list computed by
Goorvitch & Chackerian (1994), SiO by Langhoff & Bauschlicher (1993), CN
by Plez (priv. comm.), OH by Goldman et al. (1998), H2O by
Partridge & Schwenke (1997), NO, HF, NH, HCl, and CH by Sauval (priv. comm.), and atomic line lists by
van Hoof (1998), by Hirata & Horaguchi (1995), by Sauval
(priv. comm.), and of VALD (Kupka et al. 1999; Piskunov et al. 1995; Ryabchikova et al. 1997). Using the
model photosphere as input, synthetic spectra are calculated for a
typical resolution of
/
300 000, even
though the final instrumental resolution is often lower. With a
typical microturbulence of 2 km s-1, this means we are certain to sample
all lines in the atomic and molecular database. We note that, for the
purpose of calculating theoretical spectra in the mid to far-IR, some
line lists are still far from complete or accurate (see
Sect. 4.5). An typical example of a K2 giant is shown in
Fig. 7. At a medium resolution of
/
= 1500, the depression of flux due to the line veiling is
3% at
wavelengths longer than
30
m.
For the purpose of using model atmosphere spectra to represent the SED of standard stars, an appropriate analysis of different sources of uncertainties contributing to total uncertainty in the spectrum predictions is in place. The following sections discuss the effects of sources of error as a) the input stellar parameters, b) uncertainties in the model assumptions, c) the possible presence of a chromosphere, ionised wind or circumstellar dust shell, d) the adopted continuum opacity, and e) the used line lists.
Model atmospheres are defined by the fundamental parameters effective
temperature, gravity, and metallicity (and stellar mass or radius in
case of a spherical geometry). As demonstrated in
Decin et al. (2000), the influence of the stellar mass on
the synthetic spectrum is small. Varying the other fundamental
parameters shows the effect of errors in the determination of the
fundamental parameters on the synthetic flux distribution. Although
this effect is dependent on the full set of fundamental stellar
parameters, we may summarise that, for
> 50
m (1.), varying the effective temperature by
200 K for stars
with spectral type between G and K, roughly corresponds to a change of
4% in the continuum flux in the IR, (2.) a change in
the logarithm of the surface gravity of 0.20 dex introduces
uncertainties in the continuous flux distribution of about 0.5%,
and (3.) an uncertainty in the metallicity of about 0.20 dex
corresponds to an uncertainty in the IR continuum flux of
0.1%. The uncertainty in the near-IR and on molecular
absorption features is, however, much greater, as can easily be seen in
Fig. 8, arising to 7% (at a resolution of 100).
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Figure 8:
Ratio between synthetic spectra with differing stellar
parameters at a resolution of
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The same order of magnitude on the uncertainties also applies to the continuum predictions of the hotter A-type stars for the same relative changes in stellar parameters. On one hand, one may argue that the absence of broad molecular absorption bands makes these hotter dwarfs more eligible as standards. On the other hand, exactly the presence of molecular features in the cooler giants can be used as a strong diagnostic tool to estimate the stellar parameters, and hence to provide a fiducial prediction of the whole IR spectrum. Especially in spectral regions with strong hydrogen lines, for which the computation of the self-broadening remains problematic (Barklem et al. 2000; Decin et al. 2003b), the use of cooler giant as stellar standards will improve the accuracy of the spectrophotometric calibration.
Model atmosphere calculations are based on a number of assumptions,
one of them being radiative and convective flux conservation also in
the outermost layers of the photosphere. This simplification may be
the reason that the, otherwise almost perfect, match between the
FTS-Kitt Peak Spectrum of
Boo and theoretical
predictions based on M ARCS atmospheres (see Figs. 16-17
in Decin et al. 2003b) shows small deviations at the 1-2% level at a
resolution of
/
60 000 in the
low-excitation CO and OH lines.
Not only for K giants (as
Boo), but also for the
most studied star, the Sun, there is still quite some debate
concerning the assumed mean thermal profile in the outer layers of the
photosphere. Both for G and K-type stars, indications are found from,
e.g., the Ca II H and K lines that the temperature structure has a
minimum before segueing into the mechanically heated chromosphere
(e.g. Ayres & Linsky 1975). Controversially, the
analysis of CO lines indicates a cooler brightness temperature at the
same altitude (e.g. Wiedemann et al. 1994; Ayres & Testerman 1981; Ayres et al. 2006).
As done by van der Bliek et al. (1996) and
Vanhollebeke (2003), one can simulate uncertainties in the
temperature structure T() to mimic different effects: a)
flatten or steepen T(
) for a Rosseland optical depth
< 0.01 to study the effect of less, respectively
more, line blanketing; b) steepen T(
) around
= 0.1 to study the effect of convective overshoot
in the outer layers of the model photosphere; or c) steepen
T(
) for
> 1 to study the effect of convection
in the deeper layers of the atmosphere. From Table 3 in
van der Bliek et al. (1996) and the results of
Vanhollebeke (2003), we can conclude that this type of error
in the temperature distribution gives rise to uncertainties in the
predicted continuum flux at 100
m lower than 3.5%. This
uncertainty can, however, be strongly reduced in case high-resolution
spectra are available for atomic or molecular lines with different
strengths and excitation energies. From a proper analysis of the line
shapes, one can pin down T(
) to
50 K in the outer layers,
reducing the uncertainties in IR continuum flux predictions to
1-2% (van der Bliek et al. 1996).
One of the largest uncertainties in model spectrum predictions is the
(unknown) presence of a chromosphere, an ionised wind, or a
circumstellar dust envelope. These structures may easily yield a flux
excess in the order of 10% at mid to far-infrared wavelengths
(van der Bliek et al. 1996). Recent studies by
Dehaes et al. (2007) show that all of the six studied cool standard
stars, used for the calibration of the Short-Wavelength Spectrometer
(SWS, 2.38-45.2 m) onboard ISO, with spectral types between
K2 III and M0 III show a flux excess due to an ionized wind. Only for
two of them, the flux excess starts at wavelengths shorter than 1 mm,
but still longer than
200
m. In the case of the Spitzer-IRS
(5.3-18
m) calibration, the IRS team had to reject a
considerable fraction of candidate standards as calibrators due to IR
excess emission (Sloan 2005, priv. comm.).
The continuous opacity of solar-type and cooler stars is dominated by
free-free absorption of the negative hydrogen ion. The M ARCS
model atmosphere calculations make use of the free-free absorption
coefficients as calculated by Bell & Berrington (1987). A
quantitative assessment of the reliability of H
-ffabsorption coefficients by John (1994) shows that
the absorption coefficients tabulated by
Bell & Berrington (1987) are accurate to about 1% for
wavelengths greater than 0.5 m over the temperature range
between 1400 and 10 080 K. Stilley & Callaway (1970)
neglected the adiabatic exchange potential term yielding absorption
coefficients that may be off by
4.5% at temperatures lower
than
2500 K. Using these coefficients, one may introduce errors
in the order of 4% at wavelengths <10
m, diminishing to
less than 0.3% for longer wavelengths (see Fig. 9). The
prominent feature, seen around 8
m in Fig. 9, arises
from the response of SiO on the slightly different temperature structure.
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Figure 9:
Ratio between the synthetic spectrum calculated with the
H
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Table 2:
Summary of
uncertainties attributed to the theoretical atmosphere spectra in the
2-200 m range at a resolution
/
100.
Atomic and molecular data bases (including very different numbers of molecular lines and based on computations spanning a wide range in quality) are used in the construction of model atmospheres. Knowing the limitations on the accuracy and completeness of the used line list is of key importance when calculating theoretical spectra. The M ARCS code, with the input data used, is particularly tuned for computations of late-type stars where molecular opacities play an important role. As said in the introduction of Sect. 4, we here rely on the study done by Decin (2000). In this study, it has been shown that for molecules such as CO, SiO, and OH, different molecular line lists reach a high level of accuracy and completeness. In the case of CN, the dissociation energy is, however, still a matter of debate (see, e.g., Lambert 1994). Most problematic is the situation for the H2O line lists. The most often used theoretical H2O data bases are the SCAN list (Jørgensen et al. 2001), the list of Partridge & Schwenke (1997), and the very recent list of Barber et al. (2006). These line lists are likely to be satisfactory for opacity calculations and probably for comparing observed and synthetic spectra at low and medium resolution, but at higher resolution line positions based on laboratory measurements (e.g., from Polyansky et al. 1997) should be used if possible.
Focusing on the wavelength range between 60 and 210 m (covered
by the ESA PACS instrument onboard Herschel, launch foreseen in 2008),
one should realise that the atomic
VALD
database
presently only contains 1380 lines with
< 122
m, the
NIST
database
only 90 atomic lines, while the atomic line list of
van Hoof (1998) tabulates 13 527 lines, of which
the oscillator strengths are only known for 1660.
M ARCS model atmospheres were used by, among others,
Decin et al. (2000) and Ryde & Eriksson (2002)
to predict IR spectra of cool stars. In their comparison between the
high-resolution (
/
60 000) Fourier
Transform Spectrometer (FTS) spectrum of
Boo and
theoretical M ARCS predictions, Decin et al. (2003b) show that,
in the case of this well-known giant, the model predictions in the
0.9-5.3
m range only differ from the observational data by
1-2%! In general, IR model spectra predictions in the wavelength
range <25
m for A-G dwarfs and for non-pulsating K-M0
giants may be as accurate as 3% at medium resolution
(
/
1500) and as
5% at high
resolution (
/
100 000), as judged from
the available observational data (Decin et al. 2003a,b). An
exception may be the hydrogen-line predictions due to the
problematic computation of the self-broadening
(Barklem et al. 2000). At longer wavelengths, the
accuracy and resolution of today's modern instruments remain too
poor to constrain the model atmosphere spectra at a few percent
level. Luckily, the depression of flux due to the line veiling in
the FIR is estimated to be
3% at a resolution of 1500 (see
Fig. 7), rendering the representation of the SED of
stellar standards by theoretical model atmosphere spectra still
useful. In order to avoid the propagation of inaccurate predictions
of molecular/atomic lines to the RSRF determination, different
spectral classes should be used. At temperatures lower than
3500 K, H2O becomes a dominant opacity source, excluding giant
stars with a cooler spectral type than M0 III as standard calibrators.
![]() |
Figure 10:
Comparison between the various reference SEDs discussed in
this paper in the 2 to 200 ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
In last few sub-sections many uncertainties on the calculations of
model atmospheres are summarised (see
Table 2 for an overview), and
the total uncertainty is not (as often done) estimated as the
root-sum-square of all above-mentioned uncertainties since they are
often mutually dependent. The total uncertainty on the stellar
reference SEDs depends on the spectral type of the target, the
wavelength range under study and the instrumental resolution. In case
the standard star has been properly studied using high-resolution
optical and/or near-IR data and the presence of an IR excess can be
excluded, one can constrain
Teff and the temperature distribution to
within 50 K, yielding near-IR (continuum + line)
medium-resolution predictions better than 1-2%, mid-IR predictions
better than
3% and far-IR predictions better than
5%
for stars with earlier spectral type than M0. Each of the discussed
spectral classes (early A dwarfs, solar analogs, and G9-M0 giants) has
its own drawbacks. Consequently, with the aim of constraining the RSRF
at a few percent level, the selected stellar standard candles should
cover a wide diversity in spectral types to avoid biases to spectral
features typical of different spectral classes.
It is instructive to compare the different IR stellar reference SEDs
presented in this paper. We therefore have chosen to compare the
proper blackbody, Engelke function, and model atmosphere spectrum with
the spectra of two composites (as explained in
Sect. 3.3) (
Boo (K2 IIIp) and
And (M0 III)) and two templates (
Dra (K2 III) and
Tuc (K3 III)) from the sample of
CWW (see Fig. 10). Figure 10 also plots the
IRAS LRS and PSC data and different photometric data points collected
from the ISO Ground Based Preparatory Programme
(GBPP)
as obtained by Hammersley et al. (1998) and
Hammersley & Jourdain de Muizon (2003).
For the M ARCS atmosphere spectra, stellar parameters as
determined by Decin et al. (2003a) were used as input parameters. Main
parameters for the absolute flux,
Teff, and
, are listed in
Table 3. For the computation of the flux values of the
Engelke function for both composites,
Teff and
, as given
in the headers by CWW, are used. Note that for
And,
the re-scaled (by CWW) angular diameter of 13.71 mas is used: based
on the InfraRed Flux Method (IRFM),
Blackwell et al. (1991) determined a lower angular
diameter of 13.219 mas corresponding to the stellar temperature of
3839 K. As noted in Sect. 3.2, the Engelke function needs a
higher angular diameter (or
Teff) for low-gravity, low-temperature
giants in order to attain the correct flux level. For both templates, same values as for the M ARCS spectrum are used. The
input for the blackbody calculation is the same as for the Engelke
function.
Concentrating on the two composites,
Boo and
And, of which a large part of the CWW spectrum
consists of observational data, it is immediately clear that the
Engelke function does a much better job than the blackbody function of
representing the observed continuum stellar SED. The simple blackbody
function can not even be used to estimate broad-band features in the
RSRF. Between
23 and
35
m, the CWW spectrum of both
composites is given by their Engelke function. As discussed in
Figs. 1 and 2, the shape of the
continuum flux differs between a proper M ARCS model spectrum and
the flux values given by the Engelke function, the reason being the
neglect of the influence of the gravity and sphericity effects, which
are important for giant stellar atmospheres. Between 20 and
200
m, the maximum difference between the Engelke function used
by CWW and the M ARCS model atmosphere for
Boo
is near 60
m, where it attains
2.5%, rising to
7.5% when the same
Teff and
are used as for the
M ARCS model spectrum (Fig. 11).
Table 3:
Stellar parameters used to for the
calculation of (1.) the M ARCS model atmosphere spectra
(Cols. 2-5) and (2.) the Engelke function and blackbody
(BB) (Cols. 6, 7) in Fig. 10. Effective
temperature,
T, is given in Kelvin, gravity, g, in cm/s2, and
angular diameter,
, in mas.
The large absolute difference between the IRAS LRS spectrum of
Boo and both the M ARCS model atmosphere spectrum and
CWW composite (
35%, see
also Van Malderen et al. 2004) remains unexplained. The LRS
raw data are extracted from the Groningen IRAS database and calibrated
with the LRSCAL routine in the GIPSY package. Although
the IRAS LRS data were originally not meant to be absoluted
calibrated, absolute calibration factors were determined by
Volk & Cohen (1989) and Cohen et al. (1992b).
The same procedure was used by Cohen et al. (1996b), who
however only needed a factor of 0.95 to splice the LRS data to the
CGS3 data between 7.5 and 13
m.
![]() |
Figure 11:
Upper panel: comparison between the M ARCS model
spectrum of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
Inspecting the molecular absorption features in both templates,
Dra and
Tuc, a clear difference is
visible between the template spectrum (red) and the M ARCS model
atmospheres (black). As discussed in Decin et al. (2000)
and in Sect 4.1, differences in abundance pattern (here
mainly the C and O abundance), temperature, gravity, and metallicity
result in substantial differences in strength in molecular
absorption. In the case of
Dra, the composite of
Boo was used to construct the template, the
composite of
Hya (K2 II-III) was used for
Tuc. Inspecting the literature study of
Boo and
Dra presented in Appendix D in
Decin et al. (2003a) clearly shows that independent studies
incorporating both objects in general yield an effective temperature
higher by
120 K, a logarithm of the gravity higher by
1 dex, and a metallicity [Fe/H] higher by
0.50 dex for
Dra with respect to
Boo.
Figure 12 compares the M ARCS flux
predictions of
Boo and
Dra. Especially in the regions
of molecular absorption, the (relative) difference between both
spectra may be 10% or even larger, explaining why the difference
between the CWW spectrum and the M ARCS model spectrum is larger
than the uncertainties quoted by CWW. Consequently, the use of the
composite spectrum of the metal-deficient K2-gaint
Boo as
a template to represent a whole class of K2 giants in the CWW network
will introduce additional uncertainties, which may propagate through
the RSRF determination.
![]() |
Figure 12:
Upper panel: comparison between the M ARCS model
spectrum of ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Open with DEXTER |
For both composites, the far-IR extrapolations constructed by
Cohen et al. (1996b) differ by a few percent from
the M ARCS atmospheric predictions, since the M ARCS models
are computed in spherical geometry, while the CWW predictions are
interpolations in a grid of plane-parallel models. For the cooler,
more-extended giant calibrators, sphericity effects may yield an infrared flux
excess in the order of 5% (see Fig. 2).
The four reference SEDs constructed from the M ARCS atmosphere code presented in Fig. 10 are available via the electronic version of the article. Other reference SEDs constructed in the framework of the calibration of e.g. ISO, Spitzer, MIRI etc. are available upon request. If one needs extra reference SEDs for additional stellar calibration sources, one may always contact the authors, who are willing to provide you with the appropriate theoretical model atmosphere spectrum in the wavelength range and at the resolution requested.
In this paper, various sets of reference SEDs used for
determining the spectrophotometric calibration of IR
spectrometers (onboard satellites) have been discussed. Our main emphasis
was on the stellar reference SEDs, with special focus on model
atmosphere spectra. It is shown that the predicted medium-resolution
IR model atmosphere spectra are accurate within 1-2% in the
near-IR, 3% in the mid-IR, and
5% in the far-IR for
stars with spectral types earlier than M0. From the four types of
stellar reference SEDs discussed in this study (blackbody, Engelke
function, templates in the CWW network, and theoretical atmosphere
spectra), it is believed that theoretical atmosphere spectra make up
the best representations nowadays for the stellar SED, especially in
case one wants to calibrate instruments with a spectral resolution
500. We note, however, that at
> 20
m, stellar
calibrators might be too faint to be used in the spectrophotometric
calibration pedigree.
Since the ultimate goal of the calibration system is often to be
capable of deriving spectral flux values that are trustworthy to 3%
or better on both absolute and relative scales, one should aim at
building a highly accurate system of stellar reference SEDs. A good
sky coverage by the calibrators is an important ingredient in terms of
a time-efficient determination of the (spectrophotometric)
calibration. In that sense, the IR network constructed by CWW is a
good starting point. However, the CWW network has its limitations in
terms of accuracy at representing the (molecular) spectral features
and the SED at wavelengths longer than 20
m. For a list of
candidate calibrators, one therefore should put effort into obtaining
ancillary observational data to both (1) constrain the reliability of
the candidates as standard stellar sources and (2) estimate the
stellar parameters to compute a set of highly reliable theoretical
model atmosphere SEDs. This set should compromise standard stellar
candles with different spectral types ranging between A0 V and
M0 III. Experiences with the Short-Wavelength Spectrometer (SWS)
onboard ISO and the InfraRed Spectrometer (IRS) onboard Spitzer have
shown that this set should consist of some ten to fifteen stellar
calibrators. In the framework of the calibration plans for PACS
onboard the ESA-Herschel satellite and MIRI onboard the NASA-JWST
satellite, a set compromising theoretical spectra of
15 stellar
calibrators is in preparation (Decin et al. 2007). The input
stellar parameters for each set of model spectra have to be estimated
from a proper analysis of high-resolution optical or near-IR data. A
systematic cross-calibration with planets and asteroids is the last
step in the spectrophotometric calibration pedigree. Only then can a
highly accurate calibration system be developed.
Acknowledgements
L.D. acknowledges financial support from the Fund for Scientific Research - Flanders (Belgium) and KE from the Swedish Research Council. We are grateful to B. Gustafsson, B. Edvardsson, and B. Plez for their ongoing support when using the M ARCS model atmosphere code developed at the Uppsala University. L.D. thanks C. Waelkens, B. Vandenbussche, T. Verhoelst, J. Blommaert, and colleagues from the ISO, Spitzer, Herschel, and MIRI calibration teams for many fruitful discussions on the use of the most appropriate SEDs to represent the spectrum of standard stars used in the spectrophotometric calibration process of these infrared instruments. T. Verhoelst is thanked for Fig. 3.