5 Isochrones

   
5.1 Construction of isochrones

From the tracks presented in this paper, we have constructed isochrones adopting the same algorithm of "equivalent evolutionary points'' as used in Bertelli et al. (1990, 1994).

The initial point of each isochrone is the 0.15 $M_{\odot }$ model in the lower main sequence. The terminal stage of the isochrones is either the tip of the TP-AGB for  $M_{\odot }$ (ages of $\sim10^8$ yr), or C-ignition in the core for the remaining stars.

Theoretical luminosities and effective temperatures along the isochrones are translated to magnitudes and colors using extensive tabulations of bolometric corrections and colors, as in Bertelli et al. (1994). The tabulations were obtained from convolving the spectral energy distributions contained in the library of stellar spectra of Kurucz (1992) with the response function of several broad-band filters. The response functions are from Buser & Kurucz (1978) for the UBV pass-bands, from Bessell (1990) for the R and I Cousins, and finally from Bessell & Brett (1988) for the JHK ones.

   
5.2 Description of isochrone tables

In Fig. 5 we present some of the derived isochrones on the HRD, sampled at age intervals of $\Delta\log t=0.3$. They cover the complete age range from about 0.06 to 16 Gyr. Younger isochrones could be constructed only with the aid of evolutionary tracks for stars with initial masses  $M_{\odot }$, which are not presented in this paper. Complete tables with the isochrones can be obtained at the CDS in Strasbourg, or upon request to the authors, or through the WWW site http://pleiadi.pd.astro.it. In this data-base, isochrones are provided at $\Delta\log t=0.05$ intervals; this means that any two consecutive isochrones differ by only 12 percent in their ages.

For each isochrone table, the layout is as follows: A header presents the basic information about the age and metallicity of each isochrone. Column 1 presents the logarithm of the age in yr; Cols. 2 and 3 the initial and actual stellar masses, respectively. We recall that the initial mass is the useful quantity for population synthesis calculations, since together with the initial mass function it determines the relative number of stars in different sections of the isochrones. Then follow the logarithms of surface luminosity (Col. 4), effective temperature (Col. 5), and surface gravity (Col. 6). From Cols. 7 to 15, we have the sequence of absolute magnitudes, starting with the bolometric one and following those in the UBVRIJHK pass-bands. In the last Col.  (16), the indefinite integral over the initial mass M of the initial mass function (IMF) by number, i.e.

$$\mbox{\sc flum} = \int\phi(M) \mbox{d}M$$ (1)

is presented, for the case of the Salpeter IMF, $\phi(M)=AM^{-\alpha}$, with $\alpha=2.35$. When we assume a normalization constant of A=1, FLUM is simply given by FLUM $~= M^{1-\alpha}/(1-\alpha)$. This is a useful quantity since the difference between any two values of FLUM is proportional to the number of stars located in the corresponding mass interval. It is worth remarking that we present FLUM values for the complete mass interval down to 0.15 $M_{\odot }$, always assuming a Salpeter IMF, whereas we know that such an IMF cannot be extended to such low values of the mass. However, the reader can easily derive FLUM relations for alternative choices of the IMF, by using the values of the initial mass we present in the Col. 2 of the isochrone tables.

 

 

Table 3: Sample summary table with the most significant stages along some Z=0.019 isochrones

$\log({\rm age/yr})$	$M_{\rm ini}$	$M_{\rm act}$	$\log(L/L_{\odot})$	$\log T\mbox{$_{\rm eff}$ }$	$\log G$	MV	U-B	B-V	V-I	stage
7.80	5.6503	5.644	3.147	4.206	3.81	-1.725	-0.643	-0.174	-0.161	TO
7.80	6.3176	6.305	3.474	4.164	3.37	-2.789	-0.586	-0.156	-0.135	Te-m
7.80	6.3592	6.346	3.572	4.223	3.51	-2.690	-0.688	-0.187	-0.167	Te-M
7.80	6.3625	6.349	3.576	4.067	2.88	-3.584	-0.394	-0.113	-0.068	L-M
7.80	6.3666	6.353	3.314	3.656	1.50	-3.026	1.170	1.222	1.155	RGBb
7.80	6.3733	6.356	3.898	3.588	0.64	-3.845	1.806	1.542	1.657	RGBt
7.80	6.4397	6.397	3.491	3.635	1.24	-3.291	1.458	1.348	1.282	BHeb
7.80	6.4899	6.440	3.686	3.819	1.78	-4.523	0.230	0.363	0.431	Te-M
7.80	6.6046	6.530	3.703	3.612	0.94	-3.599	1.697	1.473	1.438	EHeb
7.80	6.6152	6.526	4.317	3.553	0.09	-4.262	1.813	1.581	2.230	Cb
$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$
9.00	1.7933	1.791	1.160	3.871	3.96	1.826	0.025	0.270	0.308	TO
9.00	2.0510	2.047	1.443	3.825	3.55	1.148	0.004	0.414	0.495	Te-m
9.00	2.0663	2.062	1.609	3.890	3.65	0.693	0.107	0.181	0.205	Te-M
9.00	2.0671	2.063	1.560	3.827	3.45	0.851	0.014	0.406	0.486	L-M
9.00	2.0728	2.068	1.292	3.713	3.26	1.747	0.494	0.883	0.906	RGBb
9.00	2.0866	2.080	2.353	3.635	1.89	-0.454	1.293	1.283	1.280	RGBt
9.00	2.0913	2.084	1.528	3.691	2.94	1.253	0.683	0.991	0.996	BHeb
9.00	2.1686	2.159	1.620	3.692	2.87	1.016	0.677	0.987	0.989	Te-M
9.00	2.2986	2.283	2.113	3.664	2.28	-0.056	0.978	1.135	1.124	EHeb
9.00	2.3080	2.279	3.326	3.564	0.67	-2.051	1.818	1.570	2.000	1TP
9.00	2.3096	0.672	4.069	3.473	-0.97	-1.287	1.377	1.617	3.764	AGBt
$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$	$\vdots$
10.20	0.8938	0.892	0.090	3.742	4.21	4.665	0.269	0.748	0.791	TO
10.20	0.9190	0.917	0.240	3.702	3.91	4.436	0.557	0.917	0.955	RGBb
10.20	0.9321	0.927	1.350	3.653	2.61	1.928	1.021	1.162	1.184	L-M
10.20	0.9324	0.926	1.289	3.656	2.69	2.057	0.984	1.143	1.167	L-m
10.20	0.9342	0.743	3.370	3.485	-0.18	0.191	1.399	1.615	3.577	RGBt
10.20	0.9342	0.743	1.574	3.666	2.35	1.276	0.948	1.121	1.112	BHeb
10.20	0.9343	0.742	1.625	3.661	2.27	1.180	0.999	1.146	1.135	Te-m
10.20	0.9354	0.738	1.586	3.670	2.35	1.220	0.916	1.104	1.093	Te-M
10.20	0.9363	0.735	2.019	3.631	1.76	0.415	1.360	1.311	1.303	EHeb
10.20	0.9363	0.734	2.267	3.609	1.42	0.017	1.647	1.450	1.454	L-M
10.20	0.9363	0.733	2.142	3.620	1.59	0.210	1.502	1.377	1.373	L-m
10.20	0.9365	0.674	3.314	3.487	-0.15	0.275	1.404	1.614	3.537	1TP
10.20	0.9365	0.529	3.499	3.465	-0.53	0.315	1.362	1.618	3.886	AGBt

We also provide summary tables containing basic information for the most significant stages along the isochrones. A sample table of this kind is presented in Table 3 below, for three Z=0.019 isochrones, with age values $\log({\rm age/yr})=7.8$, 9.0 and 10.2. The evolutionary stages are listed in the last column, and are, in sequence:

• TO: the turn-off point, i.e. the point of highest $T\mbox{$_{\rm eff}$ }$ during the core-H burning phase;
• If present, Te-m and Te-M signal the coldest and hottest points, respectively, of stars in the overall contraction phase at the end of core-H burning; in this case Te-M roughly corresponds to the stars in the stage of core H-exhaustion. Occasionally, this stage is followed by a local maximum of luminosity, L-M, of stars which are crossing Hertzsprung gap;
• RGBb: the base of the RGB;
• if present, L-M and L-m limit the luminosity interval of RGB stars which are crossing the discontinuity in chemical profile left by the first dredge-up event; so, this interval corresponds to the bump in the luminosity function along the RGB;
• RGBt: the tip of the RGB;
• BHeb: the beginning of the CHeB phase. It is defined as the point of lowest luminosity for CHeB stars;
• If present, Te-m and Te-M signal the coldest and hottest points, respectively, for CHeB stars. For the youngest isochrones, Te-M represents the maximum extension of the Cepheid loop;
• EHeb: the end of the CHeB phase;
• In the oldest isochrones, L-M and L-m limit the luminosity range of early-AGB stars; this interval corresponds to the clump of early-AGB stars in colour-magnitude diagrams;
• 1TP: the beginning of the thermally pulsing AGB phase;
• AGBt: the end of the AGB phase;
• Cb: the stage of C-ignition in the cases the AGB phase does not occur.

Similar tables are presented in the data-base, for other values of age and metallicity.

In addition, we provide tables with the integrated broad-band colours of single-burst stellar populations. Such tables are derived by integrating the stellar luminosities, weighted by the predicted number of stars in each bin, along the isochrones.