INPUT PARAMETERS

To generate tables of Rosseland mean (RM) opacities the user should specify a few input parameters, namely: 1) the grid of the state variables, and 2) the chemical mixture. Please refer to the original paper by Marigo & Aringer (2009, A&A, 508, 1539) for all the details. Below one finds practical indications on how to proceed.

1. GRID OF THE STATE VARIABLES

For a given chemical mixture, one RM opacity table is arranged as a rectangular matrix of dimension $(n_T, n_R)$ , where

• $n_T$ is the number of nodes of $\log_{10}(T)$
• $n_R$ is the number of nodes of $\log_{10}(R)=\log_{10}(\rho/[{\rm gr cm}^{-3}])-3\log_{10}(T) + 18$ .

The lower and upper limits of $\log_{10}(T)$ and $\log_{10}(R)$ and their grid spacing should be defined. The default values are given in Table 1.1.

parameter	min value	max value	grid spacing (dex)
$\color{orange}\log_{10}(T)$	$\color{orange}\log_{10}(T_{\rm min})=$ 3.2	$\color{orange}\log_{10}(T_{\rm max})=$ 4.5	$\color{orange}\Delta_1\log_{10}(T)=$ 0.01 for $3.2 \le \log_{10}(T) \le 3.7$
			$\color{orange}\Delta_2\log_{10}(T)=$ 0.05 for $3.7 < \log_{10}(T) \le 4.5$
$\color{orange}\log_{10}(R)$	$\color{orange}\log_{10}(R_{\rm min})=$ -8.0	$\color{orange}\log_{10}(R_{\rm max})=$ 1.0	$\color{orange}\Delta\log_{10}(R) =$ 0.50 for $-8.0 < \log_{10}(R) \le 1.0$

The user can specify different limits of the state variables, provided that $\log_{10}(T_{\rm min}) \ge 3.2$ and $\log_{10}(T_{\rm max}) \le 4.5$ ; $\log_{10}(R_{\rm min}) \ge -8.0$ and $\log_{10}(R_{\rm max}) \le 1.0$ . Also the grid spacing $\Delta_1\log_{10}(T)$ , $\Delta_2\log_{10}(T)$ , $\Delta\log_{10}(R)$ can be freely defined.

2. CHEMICAL COMPOSITION

It is specified by the user in terms of the following quantities:

2.1 Reference solar mixture

It can be chosen among various options, listed in Table 1. Click on the bibliograpghic reference to view the corresponding compilation of the individual metal abundances and the total metallicity, both in mass fractions and in number fractions.

Table 1: Compilations of the solar chemical composition

Reference	$Z_{\odot}$	(C/O)$_{\odot}$	(C/O) $_{{\rm crit},1}$ 1
Anders & Grevesse 1989 (AG89)	0.0194	0.427	0.958
Grevesse & Noels 1993 (GN93)	0.0173	0.479	0.952
Grevesse & Sauval 1998 (GS98)	0.0170	0.490	0.947
Howeger 2001 (H01)2	0.0149	0.718	0.937
Lodders 2003 (L03)	0.0132	0.501	0.929
Grevesse, Asplund & Sauval 2007 (G07)	0.0122	0.537	0.929
Caffau et al. 2009 (C09)3	0.0155	0.575	0.938

2.2 Reference metallicity

$\color{blue}Z_{\rm ref}$ corresponds to the total abundance (in mass fraction) of metals, i.e. elements with atomic number $Z_i > 3$ , of the reference mixture (see Sect. 2.5). It can be freely chosen as a non-negative quantity, $Z_{\rm ref} \ge 0$ .

2.3 Hydrogen abundance

It is expressed in mass fraction, $\color{blue} X$ , and can be freely chosen in the interval $\color{blue}0 \le X \le 1$ . The user can define a sequence of increasing $X$ values, starting from $\color{orange}X_{\rm min}$ up to $\color{orange}X_{\rm max}$ , with a step $\color{orange}\Delta X$ . This corresponds to compute a set of $\color{blue} N_X={\rm int}[(X_{\rm max}-X_{\rm min})/\Delta X] + 1$ opacity tables. To calculate one table, just set $X_{\rm min}=X_{\rm max}$ .

2.4 Normalization of the abundances

The elemental abundances can be expressed either in mass fractions, $X_i$ , or in number fractions, $\varepsilon_i$ , according to:

$$\color{blue} {\color{blue}X_i} = \displaystyle \frac{A_i N_i}{\su... ...}{N_{\rm a}}=\displaystyle \frac{N_i}{\sum_{j=1}^{\mathcal{N}_{\rm el}} N_j} ,$$ (1)

where $\mathcal{N}_{\rm el}$ the number of elements, $N_i$ is the number density of nuclei of type $i$ with atomic mass $A_i$ , and $N_{\rm a}$ is the total number density of all atomic species (hydrogen, helium and metals). In both cases the normalization condition must hold, i.e. $\sum_{i=1}^{\mathcal{N}_{\rm el}}X_i=1$ and $\sum_{i=1}^{\mathcal{N}_{\rm el}}\varepsilon_i=1$ .

The user should choose the preferred formalism, and express the abundance variation factors of metals consistently (see Sects 2.5 and 2.6). For instance, if one wants to double the abundance of carbon with respects to its reference value, it is necessary that ÆSOPUS knows whether the user's adopted abundance is in mass fraction, i.e. $X_{\rm C} = 2 X_{\rm C, ref}$ or in number fraction, $\varepsilon_{\rm C} = 2 \varepsilon_{\rm C, ref}$ .

2.5 Primordial mixture

For those ones interested in metal-free mixtures (with $Z_{\rm ref}=0$ ), having a primordial chemical composition produced by the Big-Bang nucleosynthesis, it is possible to specify the abundance of lithium, expressed by the ratio Li/H.

The default configuration assumes $\color{blue}\varepsilon_{\rm Li}/\color{blue} \varepsilon_{\rm H}=4.15 10^{-10}$ as predicted by the standard Big-Bang nucleosynthesis in accordance with the WMAP results (Coc et al. 2004).

2.6 Reference mixture

Please refer to Sects. 3.1 and 4.3 in Marigo & Aringer (2009, A&A submitted) for more details and applications to $\color{black}\alpha$ -enhanced mixtures.
By construction the reference mixture has a metallicity $\color{orange}Z=Z_{\rm ref}$ , previously selected by the user.
The reference abundances of the metals, $X_{i,{\rm ref}}$ or $\varepsilon_{i,{\rm ref}}$ are defined by the ratios (in dex):

$$\color{blue}[A_i/{\rm Fe}] = \log \left(\frac{X_{i, {\rm ref}}}{X_{\rm Fe, ref}}\right) -\log \left(\frac{X_{i,\odot}}{X_{\rm Fe,\odot}}\right)$$ (2)

or

$$\color{blue}[A_i/{\rm Fe}] = \log \left(\frac{\varepsilon_{i, {\r... ...t) - \log \left(\frac{\varepsilon_{i,\odot}}{\varepsilon_{\rm Fe,\odot}}\right)$$ (3)

which should be specified in the interactive mask. For each selected species the ratio can be set either positive (corresponding to supersolar ratios), or negative (corresponding to subsolar ratios).

The default configuration assumes $\color{orange}[A_i/{\rm Fe}]=0$ for all metals, i.e. the reference mixture consists of scaled-solar abundances.

Once specified the $\color{orange}[A_i/{\rm Fe}]$ ratios for the selected elements, the user should decide how the reference mixture is constructed in order to preserve the reference metallicity, namely:

• Mixture $\color{black}\color{orange} A$ The total abundance variation of the selected metals is balanced by the abundance variation of all other metals.
• Mixture $\color{black}\color{orange} B$ The total abundance variation of the selected metals is balanced by the abundance variation of iron-group elements.

Frequent applications of this setting section likely deal with $\color{orange}\alpha$ -enhanced mixtures, i.e. with supersolar ratios $[\alpha/{\rm Fe}]$ of $\color{black}\alpha$ -elements (O, Ne, Mg, Si, S, Ca, and Ti).

2.7 Additional chemical pattern

Finally, superimposed to the reference chemical mixture it is also possible to specify an additional chemical pattern, defined by the abundance enhancement/depression factor $\color{orange}f_i$ or $\color{orange}g_i$ of each metal species (heavier than helium), with respect to its reference abundance.

$$X_i=f_i X_{i,{\rm ref}} {\rm and} \varepsilon_i = g_i \varepsilon_{i,{\rm ref}} .$$ (4)

In the web-mask the user should specify the decimal logarithm of the variation factors: $\color{orange}\log_{10}(f_i)$ or $\color{orange}\log_{10}(g_i)$

The default configuration assumes $\color{orange}\log_{10}(f_i)=0$ or $\color{orange}\log_{10}(g_i)=0$ for all metals, i.e. the final mixture coincides with the reference mixture.

Where the variations factors of the selected elements are set $\color{orange}\log_{10}(f_i)\ne 0$ (or $\color{orange}\log_{10}(g_i)\ne 0$ ), then the actual metallicity $\color{orange}{Z \ne Z_{\rm ref}}$ , i.e. the enhancement/depression factors $f_i$ of the selected elements produce a net increase/depletion of total metal content $Z$ relative to the reference metallicity $Z_{\rm ref}$ .
In this case all $\mathcal{N}_Z$ variation factors $f_i$ can be freely specified without any additional constrain.

2.8 C/O, C, and N variation grids

These optional settings should be of particular interest to researchers dealing with the atmospheres of TP-AGB stars, as their C/O ratio, as well as the absolute C, N, and O abundances, may be significantly altered by the occurrence of the third dregdge-up and hot-bottom burning.

For the three composition parameters, the user can define a few values (in dex), each sequence corresponding to a grid of opacity tables. For instance, let us suppose that the selected normalization of the abundances is in number fraction, then one can choose the variation factors defined as:

$${\color{blue}\log_{10}(g_{\rm CO})_i} = {\color{blue}\log_{10}\left(\frac{\varepsilon_{\rm C}}{\varepsilo... ...m C, ref}}{\varepsilon_{\rm O,ref}}\right)} \quad i=1,\cdots N_{\rm CO}$$ (5)

$${\color{blue}\log_{10}(g_{\rm C})_i} = {\color{blue}\log_{10}(\varepsilon_{\rm C})-(\varepsilon_{\rm C,ref})} \quad\quad\quad\quad\quad\quad i=1,\cdots N_{\rm C}$$ (6)

$${\color{blue}\log_{10}(g_{\rm N})_i} = {\color{blue}\log_{10}(\varepsilon_{\rm N})-(\varepsilon_{\rm N,ref})} \quad\quad\quad\quad\quad\quad i=1,\cdots N_{\rm N}$$ (7)

Note that, by construction, the variation factors for O are given by ${\color{blue}\log_{10}(g_{\rm O})=\log_{10}(g_{\rm C})-\log_{10}(g_{\rm CO})}$ .

Filling in these fields will supersede the variations factors for C, N, and O set in the previous mask (see Sect. 2.6).

The maximum allowed numbers of values are: $\color{black}\color{orange}N_{\rm CO}^{\rm max}=10$ ; $\color{black}\color{orange}N_{\rm C}^{\rm max}=6$ ; $\color{black}\color{orange}N_{\rm N}^{\rm max}=6$

ÆSOPUS computes RM opacity tables for all combinations of $\log_{10}(g_{\rm CO})$ , $\log_{10}(g_{\rm C})$ , and $\log_{10}(g_{\rm N})$ .

It follows that the total number of tables (for given $Z_{\rm ref}$ ) is $\color{blue}N_{\rm tables}= N_X \times N_{\rm CO}\times N_{\rm C} \times N_{\rm N}$ .
Pay attention that the resulting number of tables may become quite high!