A&A 486, 951-970 (2008)
DOI: 10.1051/0004-6361:200809724
B. Gustafsson^{1} - B. Edvardsson^{1} - K. Eriksson^{1} - U. G. Jørgensen^{2} - Å. Nordlund^{2} - B. Plez^{3,1}
1 - Department of Physics and Astronomy, Uppsala Astronomical
Observatory, Box 515, 751 20 Uppsala, Sweden
2 - Niels Bohr Institute for Astronomy, Physics and Geophysics,
Copenhagen University, Blegdamsveg 17, Copenhagen Ø, 2100, Denmark
3 - GRAAL, Université de Montpellier II, 34095 Montpellier Cedex 05, France
Received 5 March 2008 / Accepted 30 April 2008
Abstract
Context. In analyses of stellar spectra and colours, and for the analysis of integrated light from galaxies, a homogeneous grid of model atmospheres of late-type stars and corresponding flux spectra is needed.
Aims. We construct an extensive grid of spherically-symmetric models (supplemented with plane-parallel ones for the highest surface gravities), built on up-to-date atomic and molecular data, and make it available for public use.
Methods. The most recent version of the MARCS program is used.
Results. We present a grid of about 10^{4} model atmospheres for stars with
,
log g = log
(cgs) with various masses and radii,
,
with Fe] = 0.0 and 0.4 and different choices of C and N abundances. This includes ``CN-cycled'' models with C/N = 4.07 (solar), 1.5 and 0.5, C/O ranging from 0.09 to (normally) 5.0 to also represent stars of spectral types R, S and N, and with
km s^{-1}. We also list thermodynamic quantities (T, ,
,
,
partial pressures of molecules, etc.) and provide them on the World Wide Web, as well as calculated fluxes in approximately 108 000 wavelength points. Underlying assumptions in addition to 1D stratification (spherical or plane-parallel) include hydrostatic equilibrium, mixing-length convection and local thermodynamic equilibrium. We discuss a number of general properties of the models, in particular in relation to the effects of changing abundances, of blanketing, and of sphericity. We illustrate positive and negative feedbacks between sphericity and molecular blanketing. We compare the models with those of other available grids and find excellent agreement with plane-parallel models of Castelli & Kurucz (if convection is treated consistently) within the overlapping parameter range. Although there are considerable departures from the spherically-symmetric NextGen models, the agreement with more recent PHOENIX models is gratifying.
Conclusions. The models of the grid show considerable regularities, but some interesting departures from general patterns occur for the coolest models due to the molecular opacities. We have tested a number of approximate ``rules of thumb'' concerning effects of blanketing and sphericity and often found them to be astonishingly accurate. Some interesting new phenomena have been discovered and explored, such as the intricate coupling between blanketing and sphericity, and the strong effects of carbon enhancement on metal-poor models. We give further details of line absorption data for molecules, as well as details of models and comparisons with observations in subsequent papers.
Key words: stars: atmospheres - Sun: abundances - stars: fundamental parameters - stars: general - stars: late-type - stars: supergiants
Since the first grids of line-blanketed, model atmospheres for late-type stars were published (Carbon & Gingerich 1969; Querci et al. 1974; Peytremann 1974; Gustafsson et al. 1975; Kurucz 1979; Johnson et al. 1980), there has been a very impressive improvement in underlying data, in particular, for atomic and molecular absorption. Accurate continuous absorption coefficients for a number of heavy elements have been calculated within the Opacity Project (Seaton et al. 1994), and the Iron Project (Bautista 1997). The Opacity Project also contributed accurate transition probabilities for a wealth of spectral lines for elements where the LS coupling approximation was applicable. Thanks to systematic efforts by Kurucz (see http://kurucz.harvard.edu) line lists with transition probabilities for millions of lines of heavy elements have been calculated. Also, the admirable and systematic work by experimental physicists (e.g. Blackwell et al. 1989; Nave et al. 1994; Hartman et al. 2003; Lawler et al. 2007) has led to the identification and measurements of many more metal transitions, e.g., from Fe I and Fe II, than existed before. The results of these efforts have been made easily accessible in data bases, such as the Vienna Atomic Line Database, VALD (see Piskunov et al. 1995; Stempels et al. 2001; see also http://www.astro.uu.se/~vald). Additional significant progress has been made in the calculation of accurate damping constants for strong atomic lines (Anstee & O'Mara 1995; Barklem et al. 2000a) as well as for hydrogen self broadening (Barklem et al. 2000b).
Impressive progress has also been made in the study of molecular absorption starting from practically nothing 30 years ago (see Jørgensen 1994b, 2005, for reviews). Very extensive line lists have thus been calculated for most diatomic and polyatomic molecules that contribute opacity in stellar atmospheres. The lists are based on laboratory measurements of wavelengths and gf values with simple theoretical extensions, or are the result of more extensive quantum-mechanical ab-initio calculations. Even if the absorption cross-sections are not always of satisfactory quality as yet, the existence of these lists makes it possible to calculate models for, e.g., M and C stars that are realistic enough to compare reasonably well with observations.
Simultaneous with these improvements of basic physics, steps to improve the physical consistency of the models have also been taken. The traditional assumptions of plane-parallel stratification in homogenous layers, of stationary hydrostatic equilibrium, mixing-length convection and local thermodynamic equilibrium (LTE) have stepwise been possible to relax. Grids of spherically-symmetric models for giants and supergiants were made by Plez et al. (1992) and Jørgensen et al. (1992). Dynamic pulsating model atmospheres for cool giants have been developed by Wood (1979) and Bowen (1988), and these were improved with time-dependent dust formation by Fleischer et al. (1992) and Höfner & Dorfi (1997), and further developed to include frequency-dependent radiative transfer by Höfner (1999) and Höfner et al. (2003). Nordlund & Stein and collaborators (Nordlund 1982; Nordlund & Dravins 1990; Stein & Nordlund 1998; Asplund et al. 1999, 2000a) developed 3D simulations with proper hydrodynamics and radiation fields taken into account for solar-type stars, and Freytag (2001) developed full ``star-in-a-box'' models for supergiants. These simulations show a striking agreement with observations of solar granulation and spectral line profiles for solar type stars, and clearly demonstrate the qualitative difference between traditional 1D models and reality: while the temperature structure in the upper layers of the 1D models is determined by radiative cooling and heating, the radiative heating in real late-type stars is balanced to a significant extent also by expansion cooling of upwelling gas.
A consistent treatment of radiative transfer in models for late-type stars without making the assumption of LTE is complicated due to the great number of atomic and molecular species affecting the radiative field; the wealth of levels and transitions in these species; and the lack of basic data for these transitions - not the least of which is the lack of cross sections for atomic and molecular collisions with electrons and hydrogen atoms. From the first attempt to construct a reasonably realistic non-LTE model for a late-type star (the Sun) made by Anderson (1989), the development of algorithms and computers has now made it possible to calculate grids of such models (Hauschildt et al. 2002). The lack of accurate collision cross sections, however, is still a major problem in these efforts.
In the present paper we present an extensive grid of model atmospheres for late-type stars. The models are still classical, in the sense that they are one-dimensional, i.e., spherically symmetric or plane-parallel, with LTE assumed and convection taken into account by using the standard mixing-length theory. In spite of the inadequacy of these underlying physical assumptions, which must be abandoned for detailed analysis of late-type stellar spectra at high accuracy, we still think the present grid will be useful for years to come for various applications extending from studies of individual stars to galactic evolution and populations in external galaxies.
The grid extends from models for A-type stars to M, S, and C star models, from dwarfs to supergiants in luminosity, from 10^{-5} times solar to 10 times solar in metallicity, and with various choices of parameters like radius and mass, abundances of C, N, and O, of `` elements'' (Ne, Mg, Si, S, Ar, Ca, Ti) relative to Fe, and of microturbulence. We also provide model spectral energy distributions (SEDs) sampled in about 10^{5} frequency points. With respect to methods and underlying data used, the model grid is homogeneous.
The grid is presented in a number of papers. In this initial paper, we (Sect. 2) sketch the development of our computer code MARCS, from which models produced at different earlier stages are in wide use, give a general outline of the methods now used (Sects. 3 and 4), and underlying data (Sect. 5), as well as properties of the grid (Sect. 6). Models are also compared with models from other contemporary grids (Sect. 7). In subsequent papers, we shall discuss models for A-G stars (Edvardsson et al., Paper II), K and R stars (Eriksson et al., Paper III), M stars (Plez et al., Paper IV), S stars (Plez et al., Paper V), C stars (Jørgensen et al., Paper VI), and models for very metal-poor stars (Paper VII). In these subsequent papers, the most important new opacity data are described; the model structures are described and analysed; and the model SEDs, fluxes and colours are compared to some observational data to explore and illustrate the applicability of the models. The total number of models is about 10^{4}. The details of these models, including their calculated fluxes at different wavelengths, are available via http://marcs.astro.uu.se.
Since the early 1970ies, we have developed and used the code MARCS for constructing late-type model atmospheres. Spectral-line blanketing was first considered using opacity distribution functions (ODFs, Gustafsson et al. 1975) and this technique and its underlying ODF approximation was proven to be fully adequate for F, G, and K stars. Thus, models could be calculated with just a few hundred frequency points and extensive grids for G and K giants (Bell et al. 1976) as well as R stars (Olander 1981) were issued on the basis of the ODF approximation. The ODFs were constructed with a program called SSG, originally devised for synthetic colour calculations by Bell (see, e.g., Bell 1971). The line list used by this program was based on laboratory wavelengths and oscillator strengths, as well as ``astrophysical oscillator strengths'' derived from the solar spectrum. The total list contained about 50 000 lines. The SSG program was used next to calculate model spectra and colours, which were systematically compared with observations (e.g. Gustafsson & Bell 1979; Bell & Gustafsson 1989). From these comparisons, a generally good agreement was found, which gave us the confidence to apply the models in a number of studies, e.g., of abundances in globular cluster stars. However, there was a mismatch in certain wavelength regions, in particular in the ultraviolet and violet, where the models were obviously too bright. We tentatively ascribed this discrepancy to the absence in the line lists and the models of many weak lines, which together affect the stellar spectra.
There were also more fundamental problems with the ODF method. One was that the ODFs had to be recalculated when the chemical composition was changed, which made it inflexible for stars with special and peculiar abundances, such as carbon stars. To circumvent this, a special method to add ODFs for individual molecules was invented (Saxner & Gustafsson 1984). However, for N-type carbon star models, we also found that the effects of molecules in the upper atmospheric layers, contributing strong opacity at wavelengths different from the atomic absorption at greater depths, made the ODF approximation unsatisfactory (cf., Ekberg et al. 1986). To provide satisfactory analysis for a study of the chemical composition of carbon stars (Lambert et al. 1986), we consequently changed the scheme to opacity sampling, in which the adequate absorption at each monochromatic wavelength point is treated in full detail (yet, assuming LTE in our case), following Peytremann (1974) and Sneden et al. (1976). In order to obtain an accurate representation of the total radiation contribution to the heat balance and the radiative pressure force, this required an increase of the number of frequency points by one to two orders of magnitude. A number of model grids were calculated with this new program, for carbon stars (cf., Lambert et al. 1986; Jørgensen et al. 1992), for M giants (Plez et al. 1992) and M dwarfs (Brett & Plez 1993) and applied to studies of individual stars. We gradually extended the underlying line list by calculating and adopting more complete data for diatomic molecules, such as TiO (Jørgensen 1994a; Plez et al. 1992; Plez 1998), as well as polyatomic molecules like HCN (Jørgensen et al. 1985; Jørgensen 1990), C_{2}H_{2} (Jørgensen thesis; cited in Jørgensen 1989) and H_{2}O (Plez et al. 1992; Alvarez & Plez 1998; Jørgensen et al. 2001; Decin et al. 2000).
In order to provide models for a major effort to study the build-up of chemical elements in the Galactic disk using solar-type stars, we further extended the line list, adding the great number of metal lines from Kurucz's calculations (Plez et al. 1992; Edvardsson et al. 1993). This version of the code was also used in calculating a grid for hydrogen-poor carbon stars (R Cr B stars, Asplund et al. 1997), and applied in an abundance study of such stars (Asplund et al. 2000b). Somewhat different versions of the code were used to compare models to ISO spectra and other IR spectra, partly for calibration purposes of the satellite instrumentation, of F, G, and K stars (van der Bliek et al. 1996, and subsequent papers; Decin et al. 2003, and references therein) as well as of M stars (Alvarez et al. 2000; Fluks et al. 1994) and carbon stars (Loidl et al. 2001). We also made comparisons of model colours with observed ones (Bessell et al. 1998), and applied colours and SEDs for establishing temperature scales (see, e.g., Massey et al. 2007, and references therein).
The development of MARCS has always been driven by our own (and collaborators') needs for models for studies of particular stars, mainly for abundance determination. It has been a flexible ``laboratory set-up'', rather than a ``common-user instrument'', and we have been reluctant to calculate models for stars of types, which we cannot explore ourselves to understand the limitations of our models. Until now, less than 10^{3} models have been published, which is a small number compared to the number actually calculated and used. Also, several different versions of MARCS have appeared, which has led to some confusion because of inhomogeneities among the grids produced. It was, therefore, judged important to develop a common updated version, and to use it to construct and publish an extensive homogeneous grid. The results of this effort is presented in this and subsequent papers.
The basic assumptions have been listed above. We shall not discuss their adequacy further here (for a review, see, e.g., Gustafsson & Jørgensen 1994), but comment on them one at a time, and in connection with that list some of the corresponding fundamental equations, for further reference below.
Assuming spherical symmetry we may write the equation of hydrostatic equilibrium:
Thus, a model with a turbulent velocity may be represented by a model with a reduced gravity and , according to this recipe. Similarly, effects from the radiation force may be mimicked by changes in g or M. However, the Planck mean may vary strongly with depth and with stellar fundamental parameters, and one should therefore be careful in the use of Eq. (7) for exploring the radiative effects on the atmospheric structure and, e.g., the possible effects on mass-loss rates. For a more detailed study for red supergiants and asymptotic-branch stars, see Gustafsson & Plez (1992) and Jørgensen & Johnson (1992), respectively. We have tested the use of Eq. (7) to simulate the effects of turbulent pressure for a number of models at various points in the grid and find that it leads to very small errors in the temperature structure (less than 5 K in the temperature throughout the model for a depth independent in the interval 0 to 10 km s^{-1}). We, therefore, have chosen to set for all grid models, and advise those who would have liked a different choice to use models with a different mass or g, according to the recipe given in Eq. (7). It should be noted that the mixing-length treatment of convection adopted here (see Sect. 2.2 below) leads to a rapidly varying formal convective velocity, in particular close to the boundary where Schwarzschild stability sets in. If this variation were included the term would get a major contribution from . However, more realistic simulations of convection show that varies much less with depth than , in accordance with what was assumed in the derivation of Eq. (7) above.
The acceleration of gravity, g, for our spherical models is a depth-varying quantity, according
to Eq. (8). Also the stellar energy flux F(r) and
are varying with depth. Thus, we have
We have used the version of the mixing-length ``theory'' as presented by
Henyey et al. (1965).
The convective energy flux is given by
and
is the local pressure scale height,
All number densities of all atoms and molecules are assumed to follow from the corresponding laws for thermal equilibrium; the Saha equation; and the corresponding equation of chemical equilibrium for molecules (Gibson & Heitler 1928; see also Russell 1934). Similarly, all excitation equilibria and all partition functions are calculated adopting the Boltzmann distribution, with higher terms in the atomic partition functions cut according to the method of Irwin (1981).
The radiation source function is assumed to be
The volume of physical data needed in the calculation of model atmospheres of late-type stars is considerable. Data are needed for the calculation of the ionisation equilibrium of atoms and the dissociation equilibrium of molecules. These data include chemical composition data, ionisation energies and dissociation energies, as well as partition functions. Moreover, continuous absorption and scattering coefficients are needed. The, by far, most extensive and most demanding need, however, is the various data that are necessary for the proper calculation of the line absorption. Here, we shall briefly present the data used to calculate ionisation-dissociation equilibria as well as continuous absorption and scattering. Some more details concerning the line-absorption data are given in Papers II-VII.
The basic chemical composition adopted is that of the Sun, as listed by Grevesse et al. (2007). There is still some dispute on the C, N, and O abundances adopted there (C = 8.39, N = 7.78, and O = 8.66) and, therefore, we have taken the data of Grevesse & Sauval (1998) with CNO abundances higher by about 0.2 dex for an alternative solar-metallicity sub-grid. The effects of these differences are explored in Sect. 6.3. In varying the overall metallicity of the models ([Me/H]), we have as a standard scaled the abundances of all elements heavier than He in unison. The abundances of so-called elements O, Ne, Mg, Si, S, Ar, Ca, and Ti tend to vary somewhat differently for galactic stars (see, e.g., Edvardsson et al. 1993; Reddy et al. 2003; Ryde & Lambert 2004; Nissen et al. 2004; Cayrel et al. 2004). The variation, however, is not necessarily the same for all stellar populations in the Galaxy (see, e.g., Fuhrmann 1998; and Bensby et al. 2005) or in other galaxies (see Venn et al. 2004). Therefore, we offer models with two different sets of abundances: a set with a uniform scaling for each element with [Me/H]; and a set where the elements have been scaled as follows: [/Fe] = 0.4 for ; [/Fe] = for ; and [/Fe] = 0.0 for [Me/H] 0.0. For the giant stars, the results of the first dredge up of CNO processed material is known to lead to a reduced carbon abundance and a correspondingly increased N abundance (cf. e.g., Boothroyd & Sackmann 1999). We therefore also offer models with revised C and N such that C/N = 1.5 and 0.5 by number, respectively, as alternatives to the solar value of C/N = 4.07, though keeping C+N constant (cf. Sect. 6.3). For the corresponding giant stars, the changed C/N ratio is usually accompanied by a decrease in the isotopic ratio ^{12}C/^{13}C. For the ``CN processed'' models we have therefore changed this ratio from solar to 20 and 4, respectively. Finally, we also present models with increased C abundances for the M stars and carbon stars such that C/O ranges from 0.54 to 0.99 (M and S star models) to above 1.0 (models for C stars).
We use the atomic partition functions and ionisation energies from Irwin (1981) with some modifications. The molecular equilibria were calculated using the partition functions and dissociation energies of Sauval (private communication), which is an update of Sauval & Tatum (1984) for diatomic molecules, and Irwin (1988, and private communication) for polyatomic molecules. Altogether 519 molecules were included in the equilibrium calculations. A comparison with equilibria calculated by Piskunov (private communication) shows a very good agreement.
The continuous absorption from H I, H^{-}, H_{2}^{-}, H_{2}^{+}, He I, He^{-}, C I, C II, C^{-}, N I, N II, N^{-}, O I, O II, O^{-}, Mg I, Mg II, Al I, Al II, Si I, Si II, Ca I, Ca II, Fe I, and Fe II, as well as CH, OH, CO^{-}, and H_{2}O^{-} were calculated from sources, according to Table 1.
Table 1: Sources of data for continuous opacities.
Corrections were made to the data for C I, Mg I, Al I, Si I to account for the fact that TOP base photon cross sections are published with theoretical energy levels derived from the model atom, which depart significantly from the much better known data from laboratory work. For these four species, corrections were considered necessary since they significantly affect the UV fluxes of the models. We identified the individual TOP base energy levels for these species and shifted them to the empirical energies given by the National Institute of Standards and Technology (NIST) data base (http://physics.nist.gov/PhysRefData/ASD/index.html). The TOP base gives the photon absorption cross section (in Mbarn) as a function of the energy of the ejected electron. This electron energy was converted to the energy of the absorbed photon by adding the ionisation energy from the specific electronic level.
The cross-section data for all levels was summed up, assuming LTE for a number of temperatures, and plotted at the full original wavelength resolution at different temperatures. Also collision-induced absorption of H I+H I, H I+He I, H_{2} + H I, H_{2} + H_{2}, and He I was included. H and H CIA data from Borysow et al., referred to in Table 1, are available from http:www.stella.nbi.dk/pub/scan. More extensive data sets for the CIA are publicly accessible from http://www.astro.ku.dk/~aborysow. Continuous electron scattering as well as Rayleigh scattering from H I, H_{2} and He I was included and assumed to be isotropic.
Line absorption data for atoms and first ions were originally adopted from VALD-1 (Piskunov et al. 1995), but recently data for the ten most important species were modified according to VALD-2 (Stempels et al. 2001). The updated species are Si I, Ca I, Ca II, Ti II, Cr I, Cr II, Mn II, Fe I, Fe II, and Ni II. The VALD-data adopted include the very numerous line data calculated by Kurucz & Bell (1995, see also http://kurucz.harvard.edu), including lines between predicted energy levels that are not yet experimentally verified. The gf values were, however, modified for 948 lines from VALD-1 and 1276 lines from VALD-2 in the wavelength range 3783 to 8968 Å, based on fits of synthetic solar intensity spectra. These model spectra were calculated with the Holweger & Müller (1974) model and compared with the observed solar disk-centre FTS spectrum of Brault & Neckel (1987), see Paper II. The model spectrum was required to match the observed equivalent widths to within 0.1 dexin log gf. The resulting gf values are listed at http://marcs.astro.uu.se. Line absorption data for diatomic and polyatomic molecules were considered for species according to Table 2. Most line lists of Jørgensen and collaborators are described by Jørgensen (1997) and may be obtained by anonymous ftp via www.stella.nbi.dk/pub/scan. Line lists for HCN, C_{2}H_{2} and C_{3} will be added to this data base later in connection with the paper on carbon-enriched stars in the present series. The line lists of Plez referred to in Table 2 as ``unpublished'' are available via http://marcs.astro.uu.se. Some of these lists were especially tailored for the present grid; in such cases a more detailed description of the line list for the particular species is to be found in one of the subsequent papers in the present series. For the OH A-X system we did not use the data of Gillis et al. (2001) since the Kurucz (1995b) list includes higher vibration and rotation states. For MgH we did not include the new data of Skory et al. (2003) for ^{24}MgH since we were anxious to be consistent with the MgH lines for other Mg isotopes.
Hydrogen line and bound-free opacity, and their merging, were modelled using a code by Barklem (http://www.astro.uu.se/~barklem/hlinop.html), which is based on the occupation probability formalism of Däppen et al. (1987). The details of the description of this line opacity are given by Barklem & Piskunov (2003). For the atomic lines of metals, the damping wings were also calculated, using the best available data (Anstee & O'Mara 1995; Barklem et al. 2000a, and references therein, 2000b; Barklem & Aspelund-Johansson 2005; and Barklem, private communication). For weak lines where such data were missing we just adopted the Unsöld recipe (cf., Unsöld 1955, his Eq. (76.43)), with a constant enhancement factor set to 2.0 for Na I, 1.3 for Si I, 1.8 for Ca I, 1.4 for Fe I and 2.5 for all other species. (The damping constants given in the line list of Kurucz, which were obtained by perturbation theory, might have been preferred to the use of Unsöld values.) For the molecules, damping was not taken into account - the lack of proper damping parameters makes any such attempt questionable. Also, the huge density of molecular lines often makes the damping wings of a line less important since the intensity of the Doppler cores of neighbouring lines dominate.
In general, it should be noted that although the present atomic and molecular data are very extensive, and much more complete and accurate than, e.g., a decade ago, they are far from perfect. Considerable improvements are still needed, both by adding many more faint lines and by determining line strengths more accurately. The risk that present models are still underblanketed must be appreciated.
Table 2: Sources of data for molecular line opacities.
Equations (1)-(3), (13), (20) or (30), (27), and (28) with relevant boundary conditions form a closed system of equations for determining the dependent variables , T, , , , , as well as quantities like , , or alternatively . In addition to these quantities, we also need to calculate quantities characteristic (in LTE at least) of the local temperature and pressure: , , , Q, and . The system of equations is solved on a scale, with calculated by integration over all 10^{5} wavelength points. This choice of depth scale leads to temperature structures that are only moderately affected by changes in the fundamental stellar parameters. Basically, a standard multi-dimensional Newton-Raphson method is applied to solve the highly non-linear system. All equations are first discretised, in the variables and . Next, all the resulting equations are linearized in the dependent variables listed above. The input variables at the calculation of thermodynamic quantities and absorption coefficients are T and , which is a very appropriate choice as long as H^{-} is a dominating opacity source. (For the coolest models, e.g., or could have been more advantageous.) As a result of the linearisation, a set of linear equations in the variations of the dependent variables is obtained. The coefficients of this system contain derivatives of a great number of quantities relative to the dependent variables. First, a starting model is adopted to make a first calculation of the coefficients possible. Next, the system of linear equations is solved numerically, the resulting variations are applied to the dependent variables, new coefficients are calculated, and a new solution is obtained for the variations. The elimination scheme is basically that of Rybicki (1971), also applied by Gustafsson & Nissen (1972) and Gustafsson et al. (1975), with an elimination work that scales linearly with the number of wavelength points, .
The total radiative flux,
,
and
are non-local functionals of the
model structure and can be calculated from a solution of the transfer
equation, Eq. (25).
For the model structure obtained after each iteration, we need an accurate
solution of the equation with the source function
given by Eqs. (21) and (23), which then enables us to
calculate
and
for that structure.
These quantities are used to calculate the corrections (right hand sides) in the
next Newton-Raphson iteration.
We also need a linearisation of
and
in T and
to calculate factors, symbolically written as, e.g.,
,
or
,
to be used in the
coefficient matrix in the forth-coming iteration.
Here,
and
are any two radial optical depths in the stellar
atmosphere. I.e., we need to estimate how the radiation
contribution to the energy balance at each point in the atmosphere is affected
by variations in temperature or pressure anywhere else. Both these tasks are
accomplished with the iterative method of Nordlund (1984), which was applied to
model atmosphere calculations by Plez et al. (1992) for M giants
and by Jørgensen et al. (1992) for carbon stars. Here, we shall
only give a brief summary for reference.
Figure 1: Sketch of the geometric representation used for solving the spherically-symmetric radiative transfer problem. The full radius mesh used for representing the variables as functions of radius (about 7 points per decade in ) is indicated along the radius vector by a set of concentric circles. The distance of rays from the centre of symmetry is measured by the impact parameter p. About 15 rays with impact parameters , the core radius, are used and are represented here by two dashed lines. The set of rays that do not hit the core, typically about 50 rays but dependent of wavelength, are here represented by five parallel rays. The mesh of points used for solving the equation of radiative transfer are the crossing points between the circles and the horizontal lines. Distances along the rays are measured by where . | |
Open with DEXTER |
The equation of radiative transfer is solved along a set of parallel rays, which cover a suitable range of impact parameters, chosen individually for each wavelength (cf. Fig. 1). In practice, about 15 rays equidistant in (the cosine of the angle relative to the normal) are chosen for impact parameters <, where is the radius for the stellar non-transparent ``core''. This core is defined by a radial , to allow a great enough thermalization depth for scattered light. Rays that never hit this core, but only pass through the transparent ``atmospheric'' region, are all chosen to go through pre-selected points in the vertical scale along a line through the stellar centre and perpendicular to the ray. These rays are distributed with about 5 rays per decade in along this line.
Scattering is handled by an iterative technique where the radiative transfer
equation along a particular single ray is used to correct residual errors.
This technique uses a ``form factor'' ,
The representative ray is chosen in the following way: at the radius in the discretisation of the scale where , measured radially is about the set of rays hits the sphere at a set of angles relative to the normal. We take the ray where the cosine for this angle is close to , a natural choice in view of the result of the Eddington approximation that the outgoing intensity from a gray atmosphere with a linear source function in this direction has a characteristic depth of formation which can be taken as an average depth of formation for all disk rays.
In order to obtain a sufficient numerical accuracy we linearise the radiative flux directly (Eq. (20)) only for optical depths , while for smaller depths we linearise the divergence of the flux (i.e. Eq. (30)). This avoids the loss of accuracy associated with the nearly constant monochromatic fluxes at most wavelengths in the optically thin layers. In the discretisation in we use 56 points, distributed between and .
The integrals over are calculated using the trapezian rule. In the discretisation, we use 10 000 points during the first iterations. In the final iterations there are 108 000 points, set to produce a sampling density , with Å for models warmer than 3500 K and 1300 Å for cooler models, and . We have made a series of numerical experiments to investigate the errors resulting from the sampling of the spectrum with a resolution smaller than what corresponds to the characteristic line widths. We did this investigation by making a number of models, each with 30 000 wavelength points. The models were identical, except for the particular wavelength points which were sampled differently. We then found the temperature errors for to be less than K with the maximum errors in the upper layers of the model (above ). The corresponding errors in the flux integrals and flux derivative integrals (i.e., the integrals in Eqs. (28) and (30)) by the sampling procedure are less than 1.5%. For the grid models with 108 000 wavelength points, the sampling errors are correspondingly smaller (by approximately a factor of (108 000/30 000)^{-1/2}). They then lead to temperature errors of about K or less. For the M and C star models, we have found the corresponding errors in fluxes and temperatures to be reduced by a factor of three or more. This is a result of the overlapping molecular-line absorption, which reduces the dominance of a few strong metal lines in the flux derivative integrals in the upper model layers, lines which otherwise demand a high density of sampling points if no special measures are taken to cover them properly by quadrature points.
It should be noted that the fluxes, even if they are monochromatically precise within the model constraints, may be far from representative for the regions surrounding each wavelength point in the spectrum. Thus, model fluxes integrated to represent narrow spectral regions, wide, will have relative errors of typically . E.g., for 100 Å wide bands at 5000 Å, the sampling errors will correspond to standard deviations of typically 1.6%. Obviously, synthetic spectra more detailed than the model fluxes must be calculated for theoretical calibrations of, e.g., narrow-band photometry. We have included the surface fluxes in all 108 000 wavelength points into the model files published on http://marcs.astro.uu.se - more detailed high-resolution synthetic spectra will also be gradually added.
For the integration of the moments of the radiative intensity (in practice ) we use the points defined by the angles in the crossing points between the different parallel rays with impact parameter p_{i} and the concentric spheres around the stellar centre with radii r_{k} corresponding to the discretisation of the vertical scale, cf. Fig. 1. I.e., , directions for which is directly available. Considering the character of the radiation field in a spherical atmosphere, this is an adequate set of points. This, despite the fact that they are different for different radii and different wavelengths. The typical number of points is 6 in the plane-parallel case and typically 40-60 in the spherical models. The quadrature in is performed using piece-wise cubic splines, as described by Nordlund (1984).
The line opacities have been pretabulated in the following way: for each
individual species (e.g. H I, C I, C II, CH, C_{2}, etc.) and the full
set of wavelength
points, arrays of line absorption were calculated for a number of T-P_{6} points
(or T points for the molecules).
The value of P_{6} is given by
In the calculation of opacity data tables, we added the relevant absorption for each spectral line at each T and P_{6} value to the table for wavelengths points progressing to the blue and the red side of the central line wavelength as far as the line opacity exceeded a predefined cross section. Also, for very weak lines, we then included each line at a minimum of two wavelength points. To limit the total computing time needed and still guarantee that no significant opacity was lost, these limiting cross sections were empirically determined for groups of atoms and ions.
The total computing time needed for the calculation of a model
atmosphere (the pre-calculation of the absorption coefficient tables
excluded) scales approximately as follows: (1) for the radiative transfer
part of the problem the time is proportional to
;
(2) for the calculation of ionisation equilibria and molecular equilibria the time scales as
;
and (3) for the calculation of absorption coefficients as
.
In addition, (4) some initiation time is needed. Setting
,
and
and noting that we usually set
in
the plane-parallel case and that
in the spherical case, we find
empirically that the time
needed per iteration is, in seconds,
In multi-dimensional Newton-Raphson schemes like the present one, convergence is rapid, provided that a starting solution, which is close enough to the final solution, has been chosen. When model calculations are started from scratch with the MARCS program, usually a gray starting model is chosen for the radiative zone. When the calculated flux in the convectively unstable zone of the first starting model exceeds the total flux, the temperature gradient is automatically and directly reduced, beginning at the onset of the convective instability until the convective flux is smaller than the prescribed total flux. In the present grid, we usually start from a nearby model in the fundamental-parameter space. If the starting model has a different effective temperature, a simple scaling of the temperature structure may be applied, but this is not necessary if steps of only a few hundred K are taken in .
Figure 2: The temperature structures for a set of model atmospheres with different , log g = 3 and different metallicities. | |
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The route towards convergence is often rapid, but not always quadratical. The pronounced non-linearities, not the least in the temperature dependencies of the molecular equilibria with strong effects on the opacities may slow down the convergence if one is not very close to the final solution. Also, the changing presence and depth of the convective zone with temperature and metallicity affect the convergence. Usually, convergence to temperature corrections of less than a few K, and logarithmic pressure corrections smaller than 0.01 dex, is obtained all through the models after 4 to 10 iterations. For models with K, the convergence is close to quadratic with corrections reduced by almost one order of magnitude from one iteration to the next. For the cooler models the convergence is slower but still fast. For some parameter choices, however, converged models were not obtained. In particular, this is the case for models with a strong radiative pressure gradient (i.e., close to the effective Eddington limit, see Gustafsson & Plez 1992) which is close to upsetting the hydrostatic equilibrium condition, i.e., models with high temperatures and low gravities. Another difficulty occurs for models in a small band in the diagram, extending from about , log g = 5, to , log g = 3, where convergence is not achieved in the deepest layers of the models because the convectively unstable regions are swapping between thin convective zones and zones extending to depths below the bottom of the model. This difficulty may be circumvented by extending the depth scale of the model to deeper layers. Some of the models, though not fully converged in the deepest layers, are nevertheless presented in the grid since the spectrum forming regions (above ) are not affected at all. Convergence problems occurred for some of the coolest models with H_{2} convection zones at the surface as well as for models where the radiative pressure totally dominates the gas pressure.
The effects of spectral lines on stellar atmospheres (``blanketing'') have been explored thoroughly since it became possible to calculate models with lines included in the 1960s and 1970s. Here, we shall make a short summary of these results as a background for some more specific comments on the blanketing effects in the new grid.
Figure 3: The blocking fraction X in percent for models in the grid with two different metallicities. The dwarf models all have log g=4.5 while the giant models have log g values increasing with temperature, from log g=0.0 at K to log g=3.0 at K. | |
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In general, the blocking of
radiation leads to heating of the deeper layers of the atmospheres (``back-warming''). Here and subsequently we measure the
back-warming effect at a given continuum optical depth, in practice at
nm (i.e., not
,
which for the cooler models is strongly
affected by line absorption). The back-warming is in fact
easily estimated directly from the definition of the effective temperature
by assuming that a model with effective temperature
is heated such that the deeper layers
correspond to an unblanketed model with an effective temperature,
Figure 4: The effects of switching off line absorption on the temperature structure of a sequence of models with log g=3.0 and solar metallicity. Note that . It is seen that the blanketing effects are fairly independent of effective temperature for models with | |
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We also calculated the total blocking fraction X for the grid models and find the remarkable behaviour shown in Fig. 3. Thus, X stays nearly constant for the temperature interval K for a given metallicity, and is closely identical for dwarfs and giants, while below 4000 K it increases as temperature goes down, due to the heavy TiO and H_{2}O blocking. Here, the metallicity sensitivity of X decreases while the gravity sensitivity increases. The decreasing metallicity sensitivity reflects the fact that the molecular lines fill in the continuum regions in the spectrum and define the scale ( being rather robust against metallicity changes).
For the upper layers of the atmospheres the situation is more complex,
as was discussed, e.g., by Gustafsson et al. (1975);
Gustafsson & Olander (1979); and Carbon (1979); and as is illustrated
in Fig. 4. In short, for LTE the spectral-line absorption may cause a cooling or a heating, depending on whether the absorption is located
on the long-wavelength side of the local Planck function (cooling, Case 1) or on the short-wavelength side. In the latter case,
heating (2a) or cooling (2b) may result, depending on whether the absorption is concentrated to the upper layers (2a) or
whether it extends to the deeper atmosphere (2b). Examples of Case 1 are the cooling by the IR VR lines of CO for models of late G stars
(cf. Johnson 1973; Gustafsson et al. 1975) and by
H_{2}O in the IR for later M stars.
A clear example of Case 2a is the heating by
the electronic bands of TiO in early M-star models (Krupp et al. 1978), while a Case 2b example is provided by the metal-line
absorption in the UV and blue for F and G stars. We note in passing that these surface effects are all dependent on the LTE assumption - in
particular, if the spectral lines are partially formed in scattering processes the effects may be significantly smaller due to the weaker coupling
of the radiation to the temperature of the local gas. The back-warming effects, on the other hand, are not strongly dependent on the
assumed line-formation mechanism as long as the amount of blocking is unchanged. Missing or erroneous
line absorption data may, however, still be of significance as a source of systematic errors in the back-warming effects.
Figure 5: Structural effects on the surface (black triangles) and depth (open circles) of model atmospheres by different groups of spectral lines. The solid lines show surface-heating and bottom-cooling as defined in Eq. (37) when all spectral lines are removed in the calculation of solar-metallicity dwarf models ( ). The dotted, dashed, and dash-dotted lines show, respectively, the effects of removing only the metal lines, the molecular lines, or the hydrogen lines. | |
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The general effects of blanketing are illustrated for the grid models by a test where all line absorption was switched off. All models with and solar metallicity show a similar response: the temperature is increased by about in the outermost layers if the line opacity is neglected, an increase that gradually diminishes to zero at and then changes to a lowering of the temperatures at greater depth since back-warming is inhibited (cf., Fig. 4). For the coolest giants the effects become considerably greater.
We shall now give some more specific comments on effects of blanketing in the new grid, with remarks on the effects of various opacity sources and on the effects of changing abundances and microturbulence. Furthermore, we shall comment on the effects of sphericity and explore some interesting coupling between these latter effects and blanketing.
We see from Fig. 5 that for a sequence of main-sequence models with solar metallicity the total blanketing
effect, both at the surface and in the deeper layers, is rather temperature independent. The structural effects of spectral lines
are surprisingly similar
for all solar-type stars. Thus, the mean surface cooling
amounts to about
of
and the mean heating of the layers below
is about
over most of the effective-temperature interval. I.e., the somewhat decreasing
importance of metal lines when proceeding towards the hot end of our effective-temperature interval is rather well compensated for
by the increasing hydrogen-line blanketing. The diminishing effects of metals for the cooler models are compensated for by
the molecular blanketing. These compensation effects explain why relatively old differential abundance analyses, made by
primitive model atmospheres such as scaled solar models, are often found to agree rather well with more recent results
based on models with much more complete atomic and molecular data.
Figure 6: Similar to Fig. 5. The structural effects of molecular lines separated for different species to show the results of removing the line opacity of TiO, water, and CO, respectively. | |
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Figure 7: The effects of dredge-up of CN-cycled material to the surfaces of Pop I giant stars as illustrated by model atmospheres with different C and N abundances. In the top panel, carbon has been converted to nitrogen such that the C/N abundance ratio has been lowered from the solar ratio of 4.07 to 1.5, typical for the first dredge-up in low-mass stars. In the lower panel, the ratio has been further lowered to 0.5 and there also the effects of changing the ^{12}C/^{13}C ratio are displayed. The stronger back-warming of the 3000 K models is due to the increased importance of water as more oxygen gets available when the CO abundance is decreased by the reduction of carbon while the heating effect at the surface merely reflects the inadequacy of for this heavy line-blocked model - for a plot of the temperature vs. , an increased cooling at the surface appears. In the 5000 K model, the decreased CO cooling leads to a hotter surface. | |
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The recent considerable downward corrections of solar CNO abundances by Asplund and collaborators (Grevesse et al. 2007, and references therein) are found to lead to very minor effects for most regions in the parameter space spanned by the present grid. Thus, for the temperatures at given , the effects are less than for all depths and luminosities for giants at solar metallicity. The most pronounced effects in the surface layers appear in models with K and models with K where the reduced cooling due to H_{2}O and CO, respectively, is visible.
The CNO abundances in stars are modified through the first dredge-up along the subgiant-giant branch, mostly converting C to N (see, e.g., Boothroyd & Sackmann 1999; Charbonnel 1994). The effects of this on the model atmospheres are, however, small or moderate, as illustrated in Fig. 7. The two most important effects are again the enhanced H_{2}O surface cooling for the coolest models (since more O is available if C and thus CO is depleted), and a diminished CO cooling for the somewhat hotter models. The changes due to effects by CN are marginal. The accompanying changes of ratios are obviously only of minor significance.
Figure 8: A sequence of MARCS spherical model atmospheres with K, log g = 0.0, . The models have different abundance ratios, C/O, marked in the figure, but otherwise solar metal abundances. Points on the temperature-pressure relations with log , -4, -3,-2, -1, 1, and 2 are marked with circles, which are filled for the carbon-rich models ( ). Points where log are marked with . The figure demonstrates the strongly increasing pressures at given temperature and optical depth as the C/O ratio increases, until C/O reaches 1.0 when the situation is drastically reversed as the opacities increase and the pressures decrease again. The models with show M-type model spectra, those in the interval S-type spectra, while those with greater C/O show N-star spectra. | |
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Figure 9: The effects of non-solar CNO abundance ratios on extreme Pop II star models with K, log g=1.0, [Me/H] = -3.0. In the top panel, the carbon and nitrogen abundances have been increased by factors of 10, 100, and 300 above the solar ratio to iron, oxygen and all other elements. The heating in the middle layers, primarily due to C_{2}, is strong, and even more so at depth, which is due to the increased electron pressures and continuous opacities. In the lower panel, oxygen is also enhanced. The increased electron pressures in the more CN(O)-rich models lowers the density and convection efficiency at a certain optical depth which increases the temperature gradient. The major surface cooling agent in the outer surface layers is CO. | |
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The most severe atmospheric effects of CNO abundance changes occur, however, when the carbon abundance is raised by dredge-up on the asymptotic-giant branch and, as the carbon abundance approaches or exceeds that of oxygen, the star becomes an S or a C star, respectively, with drastic changes in radiation fields and thus temperature-pressure structure. This transition has been explored using MARCS models (for references, see above), and will be further discussed in Papers V and VI in the current series. As an illustration here, however, we present a sequence of models, with C/O ratios ranging from 0.5 to 2.4 in Fig. 8.
The most metal-poor stars known until now show severe departures from ``standard'' CNO abundances: they seem to be comparatively very rich in CNO elements (cf., Christlieb et al. 2004; Frebel et al. 2005; Norris et al. 2007, and references therein). In general, there is a group of low-metallicity stars with considerably enriched carbon abundances, so-called carbon-enriched metal-poor (CEMP) stars, (see, e.g., Aoki et al. 2007), which may show carbon enrichments ranging in the interval between a factor of 10 and 1000 relative to a solar C/Fe, as well as similarly considerable N enrichments and possibly also O enrichments. The effects on the atmospheric structures from these enrichments are illustrated in Fig. 9 for one representative CEMP giant model.
It is seen that the enhanced C and N abundances heat the layers considerably for these model parameters (a result which was also found by Hill et al. 2000). Also, the deep layers get warmer, which is due to the increased electron contributions from the enhanced carbon. This increases the H^{-} opacity, which lowers the density at a given optical depth and therefore reduces the convective energy flux, which increases the inner temperature gradient. These effects change if O is also enhanced correspondingly. Then the CO cooling takes over in the surface layers, and much of the C_{2} and CN heating absorption vanishes since most of the carbon is bound in the CO molecules.
Figure 10: The effects of an increase in the -element abundances by +0.4 dex on the temperature structure of models of dwarfs and giants with [Fe/H] = -1.0. The comparison model in each case is a corresponding model with no -element enhancement ([/Fe] = 0.0). The increased electron pressure lowers the density and convection efficiency at a certain Rosseland optical depth, which increases the temperature gradient in the deeper parts of a model. | |
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The effects of the observed increase of Fe] for galactic, though seemingly not all, metal-poor stars can also be explored using the grid models. Although these changes are easily seen in spectra of the models, e.g., in the strengths of the Mg and Ca lines, the effects on the temperature-pressure structures are only minor. As is seen in Fig. 10, they are, however, of some significance for models with where the higher Ti abundance increases the surface heating as well as the back-warming of the TiO absorption. Simultaneously, the electron contributions from Mg and Ca increase and raise the H^{-} opacity, which partly reduces the effects of the Ti abundance change. As discussed above for the CEMP star models, the higher electron pressures also reduce the densities at a given optical depth, which diminishes the convective flux and thus admits a stronger temperature gradient in the deep atmosphere.
It should finally be noted that systematic errors in the models, due to missing opacities, departures from LTE or convectively generated inhomogeneities and errors in calculated convective fluxes, may well vary systematically with chemical composition. Thus, the trends discussed in the present section may have to be revised when more realistic models are available.
In the present grid, there are models calculated with different microturbulence parameters , but with otherwise identical input data. This makes it possible to systematically explore the dependence of the blanketing effects on microturbulence. Before discussing the results, we will present some considerations of what one could expect.
An increased microturbulence enhances the blocking of the spectrum by isolated
saturated spectral lines, in proportion to
.
So, if all lines were saturated, one might estimate from
Eq. (35) that the back-warming
would increase by about
.
However, this is an upper limit since a
considerable fraction of the spectral lines (though not necessarily those that contribute most significantly
to the blocking) will not be saturated - their equivalent widths will be unaffected by a changing
and thus the total blocking
is not changed. Also, for more metal-rich stars the overlapping of strong lines reduces the differential
effects of a
increase as does the fact that many of these lines have strong damping wings. We thus
expect the effects of microturbulence changes on the back-warming to be small for the most metal-poor stars where
even the strongest lines are unsaturated,
then increase but finally level off for the most metal-rich and coolest models, and never reach the
estimated upper limit. These expectations are verified by the models, as is
illustrated in Fig. 11. The differential back-warming caused by an increased microturbulence parameter,
e.g., by a factor of 2, is of about the same magnitude as the result of doubling the metallicity. This suggests that the total blocking
contribution of lines on the flat part of the curve-of-growth (mainly sensitive
to
)
is of similar significance
as that of weak spectral lines (proportional in strength to the metallicity); this picture is complicated
by both the effects of lines on the damping-part of the curve-of-growth (with only a square-root dependence on abundance)
and the strong abundance sensitivity of some diatomic molecules.
Figure 11: for models with different microturbulence parameters , minus for a model with km s^{-1}, for solar composition, log g=2.0 and different effective temperatures. Thick lines denote models with km s^{-1}, thin lines with km s^{-1}. | |
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Changes in the macroturbulent parameter (classically representing velocity shifts on geometrical scales corresponding to at least one optical depth unit in the continuum) are not able to change the model structures through radiative field changes. See Sect. 3 above concerning effects of changes in the turbulent pressure, which may be related to both macro- and microturbulent motions.
Real stellar atmospheres are not believed to be well represented by the concepts of micro- and macroturbulence. Velocity gradients on intermediate scales are of vital importance, as has been shown by convective simulations (Nordlund & Dravins 1990; Asplund et al. 2000a), and global gradients caused by pulsations as demonstrated in Lederer et al. (2006) and references therein. With respect to the effects on the radiative field, these gradients allow for more heating in spectral lines since the gas high up in the atmosphere may absorb hot radiation from lower layers that was not absorbed below; similarly, the gas may cool more efficiently through spectral lines since the shifts make the gas more transparent in overlying layers. However, even more important are the hydrodynamical effects due to gas expansion and compression on the energy balance. Altogether, this makes the representation of the energy balance in real stars by classical models quite problematic; also, there is no hope to gain further insight into the physics of these systems by fine-tuning the turbulence parameters.
The properties of spherically-symmetric model atmospheres for late-type stars have been discussed or commented upon in numerous studies. The area was first pioneered by Schmid-Burgk & Scholz (1975) and Schmid-Burgk et al. (1981, and papers cited therein); and later explored by Plez (1990), Plez et al. (1992); Jørgensen et al. (1992); Hauschildt et al. (1999); and Heiter & Eriksson (2006). The papers by Plez and collaborators, by Jørgensen and collaborators, and by Heiter & Eriksson, all build on earlier versions of the MARCS code. These papers demonstrate that the effects of sphericity are in general important for the temperature structures, causing relative temperature effects on the order of 1% or more, for extensions of the atmosphere normalised on the stellar radius greater than about 2%. The relative extensions scale roughly as g^{-1/2}; the sphericity effects are thus mainly important for low-gravity models. These papers do not, however, systematically explore the effects on the models and fluxes of changing from plane-parallel to spherical symmetric geometry. Here, some further comments will be made on this topic on the basis of the new model grid.
The extension z_{0} of the stellar atmosphere, measured as the geometrical thickness of the
atmosphere (for instance, defined as the layers with
)
is a key
factor in the discussions.
It is easy to estimate the extension from the hydrostatic-equilibrium
equation, the definition of optical depth and the gas law for an ideal gas. The relative thickness of the
atmosphere from optical depth
up toward the surface at
is thus:
Figure 12: The value of C as calculated from Eq. (40) with model values of z_{0}/R_{1} for a set of models with ranging from 3000 K to 8000 K (with bigger symbols the higher the effective temperature) and metallicities [Me/H] = 0.0, except for two models with [Me/H] = -1.0, which are attached with thin lines to corresponding [Me/H] = 0.0 models. was chosen = 1. The value of C when based on the value of the extension z_{0}/R_{1} estimated from Eq. (39) is 1.0 and shown by the horizontal dotted line. | |
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Figure 13: The difference in the surface temperature (at log ) between spherical and plane-parallel models with ranging from 3000 K to 7000 K (with bigger symbols the higher log g) and metallicities [Me/H] = 0.0, except for two models with , which are attached with thin lines to corresponding [Me/H] = 0.0 models (the lower symbols in the pairs representing the more metal-poor models). Three models at K have also been calculated with TiO absorption neglected and are denoted by crosses - it is seen that the positions of these models depart very significantly from the corresponding standard models. The relation according to Eq. (41) is shown as a straight line. | |
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According to our analytical estimate of the relative extension in Eq. (39),
the quantity
We now turn to the sphericity effects on the temperature structure of the models. The temperature in radiative
equilibrium is set by the radiative field, following Eq. (30). The basic effects due to sphericity at a
point high up in a stellar atmosphere come
from the general dilution of the radiative field, partially due to the
absence of incoming radiation from angles
where
gradually increases the further from the
optically-thick layers that the point is situated. Basically, this leads to less radiative heating of the
upper layers of the atmosphere and thus leads to cooler temperatures there. One may prove that this reduction of the mean intensity
for small extensions will scale with
(z_{0}/R_{1})^{1/2}.
For larger distances from the star, a simple and approximative estimate for the temperature
effect, was presented by Gustafsson et al. (1975; see also Böhm-Vitense 1972) who assumed that the radiative
flux should be roughly proportional to radius r^{-2} and to temperature T^{4} as one may estimate from
Stefan-Boltzmann's law. This leads to the estimate
Figure 14: The difference between temperature structures of spherically symmetric and corresponding plane-parallel models with log g=1.0, , [Me/H] = 0.0, and ranging from 3000 K to 6000 K. The complex variation of the sphericity effects in the upper layers when TiO sets in as a heating agent at K is illustrated. | |
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Figure 15: The difference between temperature structures of spherically-symmetric and corresponding plane-parallel models with K and different log g. and [Me/H] = 0.0, and with the TiO absorption alternatively switched on and off (dashed lines, the gravities of these can be traced via the models to which they adhere at depth). The complex variation of the sphericity effects in the upper layers due to TiO is illustrated. | |
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Now turning to the detailed model results, the sphericity effects on are shown in Fig. 14 as a function of for a sequence of supergiant models with different effective temperatures. Obviously, the effects shift with . First being relatively small for the hottest models, they considerably increase for , then diminish again to reach a minimum at about 3800 K and finally again increase strongly as goes towards 3000 K. This behaviour may be interpreted as a coupling between the sphericity effects and the effects of molecular blanketing. The coupling may show up as positive or negative feedback. Sphericity cools the upper layers, relative to a plane-parallel model, and the molecular formation increases. Around 5000 K this leads to a strong surface cooling by CO. For cooler models TiO forms, which conversely heats the upper atmosphere. Finally, H_{2}O takes over as a dominating opacity source, and that again cools the model surface. That this explains the behaviour can be tested by blocking out different molecular opacities from the model calculations. In Fig. 15 we thus show a sequence of models at with different log g and with and without TiO. It is clear that the differential sphericity effects above are changed significantly by the TiO absorption. From this figure we also see that the sphericity effects in the temperature structure are significantly greater than 1% for models with log . However, if turbulent pressures are taken into account this latter value may be higher, according to Eq. (7).
Figure 16: MARCS model atmospheres (solid) for dwarf stars with log g=4.5, [Me/H] = 0.0 and corresponding models from the Castelli & Kurucz ODFNEW grid (dashed). The curves are labelled with relevant values of . | |
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Figure 17: MARCS model atmospheres for giants and supergiants with [Me/H] = 0.0 (solid) and corresponding models from the recent Castelli & Kurucz grid (dashed). The curves are labelled with relevant values of and log g. | |
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Figure 18: MARCS model atmospheres with [Me/H] = -2.0 (solid) and corresponding models from the Castelli & Kurucz ODFNEW grid (dashed). The curves are labelled with relevant values of and log g. | |
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The most extensive and widely-used grid of models for most of our parameter space are the plane-parallel models of Kurucz and collaborators (available on http://kurucz.harvard.edu). Among the different sets of models published there, it is particularly relevant to compare our models with those calculated by Castelli & Kurucz (2003) with a standard mixing-length theory (without ``convective overshoot''), which also seem to produce more consistent model parameters when applied to real data (Castelli et al. 1997). Some sample comparisons of model structures are shown in Figs. 16-18. It is seen that the agreement in the temperature structures is almost perfect for the models of solar-type dwarf stars of different metallicities. This is also true for models of early M dwarfs. For the models of giants and supergiants, the agreement is again good, although the MARCS models tend to be some 10-80 K cooler in the surface layers where . The agreement is also very satisfactory if pressures or densities are intercompared. In view of the fact that these two grids of models are made with two totally independent numerical methods and computer codes, with independent choices of basic data (although Kurucz's extensive lists of atomic line transitions are key data underlying both grids), this overall agreement is both satisfactory and gratifying.
Figure 19: Model atmospheres with solar abundances, from the MARCS, NextGen, and PHOENIX-ACES-2008 grids and (for K) the ODFNEW Castelli & Kurucz grid. The log g parameter is 3.0 (5000 K) and 0.0 (3000 K), respectively. The MARCS and NextGen models are spherically symmetric with while the Kurucz model is plane-parallel. Along the MARCS curve every decade in is marked by a sign, and by a . The general impression of a better agreement between MARCS and Castelli & Kurucz models than between MARCS and NextGen models is also shown if the temperature is plotted relative to optical depth. | |
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Figure 20: Model atmospheres with solar abundances, from the present MARCS grid compared with corresponding MARCS models from Bell et al. (1976, BGEN, dashed) and with one model from Plez (1992). The log g parameter is 3.0 and the geometry is plane-parallel for the 6000 K, 5000 K and 4000 K models; the 3000 K models are spherically symmetric with M=5 and log g=0.0. | |
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A number of comparisons with models made by previous versions of MARCS have also been performed. In Fig. 20, we thus compare with models of G and K giants (Bell et al. 1976) calculated with the first version of MARCS. It is seen that the present temperature structures agree quite well with the older ones in spite of the much more incomplete line absorption data of those models. (An exception from this is seen in the outer layers of the 4000 K model, where the TiO absorption, missing in the BGEN models, contributes a surface heating of the more recent model). The pressures in the present models are, however, significantly lower, which reflects their more heavy opacities. The good agreement in the temperature structures must thus, at least partially, be fortuitous, e.g., reflecting that the line absorption missing in the 1976 models is not heavily biassed towards the ultraviolet or the red spectral regions relative to the absorption that is now included. In Fig. 20, we also show one example where a spherically-symmetric model for a late-type M supergiant is compared with a previous spherically-symmetric MARCS model of Plez et al. (1992). The considerable differences in the upper photosphere are due to the new and improved molecular data used now and its more detailed representation, in particular, for the H_{2}O absorption.
It is useful to be able to interpolate the model structures to a set of parameters , , and [Me/H] different from those tabulated in the grid. Such an interpolation is not completely straightforward, as strong non-linearities appear in the behaviour of some of the thermodynamic variables when stellar parameters are varied. One program was developed by Masseron (2006) and extensively tested by him on a previous grid of MARCS models encompassing the following range of parameters: 3800 K, , . It interpolates , and the geometrical depth as a function of . The interpolated model must lie inside a complete cube of existing models in the parameter space ( , , [Me/H]), and the interpolation is optimized to account for non-linearities in the grid. With the actual grid parameter steps, maximum errors in the interpolated quantities should remain below 0.25% for T and a few % for and , whereas and the column density (rhox) are more difficult to interpolate. The program, with a detailed user manual, is available on the MARCS web site: http://marcs.astro.uu.se. An interpolation routine, applied to structures, as well as fluxes and colours of models by Kurucz and collaborators, has also been described by Nendwich et al. (2004), see also Valenti & Fischer (2005).
Further presentations and discussions of particular properties of these models will appear in subsequent papers in this series. Hopefully, these will also serve as starting points for further studies of the adequacy and shortcomings of the standard models, and as a stimulus for endeavours to set higher standards in the art of modelling stellar atmospheres in the future.
Acknowledgements
We thank the Swedish Science Research Council and the Centre National de la Recherche Scientifique for long-standing support during the years. B.P. thanks Sveneric Johansson for his hospitality for several years at the Department for Physics at Lund. Previous collaborators in the development of the MARCS code are also thanked: Roger A Bell, Martin Asplund, John Brett, Leen Decin, Ulf Ekberg, Göran Hammarbäck, Dan Kiselman, Michelle Mizuno-Wiedner, Olle Morell, Nils Olander, Mikael Saxner and Nicole van der Bliek, as well as contributors and critics of important atomic and molecular data: Jan Almlöf, Bernhard Aringer, Paul Barklem, Manuel Bautista, Aleksandra Borysow, Alan Irwin, Per Jensen, Sveneric Johansson, Alain Jorissen, Bob Kurucz, David Lambert, Mats Larsson, Bo Lindgren, Nik Piskunov, Francois and Monique Querci, Jaques Sauval, Per Siegbahn and Georg Ole Sørensen. Peter Hauschildt is thanked for providing PHOENIX models for comparison and Thomas Masseron for providing his interpolation routine for MARCS models. Martin Asplund, Ulrike Heiter, Bob Kurucz, Jeff Linsky and Amy Mednick are thanked for valuable comments on the manuscript.