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Astron. Astrophys. 358, 869-885 (2000)

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2. Converting age distribution into SFH

Assuming that the sample under study is representative of the galactic disk, the star formation rate can be derived from its age distribution, since the number of stars in each age bin is supposed to be correlated with the number of stars initially born at that time.

We use the same 552 stars with which we have derived the AMR (Paper I), after correcting the metallicities of the active stars for the [FORMULA] deficiency (Giménez et al. 1991; Rocha-Pinto & Maciel 1998), which accounts for the influence of the chromospheric activity on the photometric indices. The reader is referred to Paper I for details concerning the sample construction and the derivation of ages, from the chromospheric Ca H and K emission measurements.

The transformation of the chromospheric age distribution into history of the star formation rate comprises three intermediate corrections, namely the volume, evolutionary and scale height corrections. They are explained in what follows.

2.1. Volume correction

Since our sample is not volume-limited, there could be a bias in the relative number of stars in each age bin: stars with different chemical compositions have different magnitudes, thus the volume of space sampled varies from star to star. To correct for this effect, before counting the number of stars in each age bin, we have weighted each star (counting initially as 1) by the same factor [FORMULA] used for the case of the AMR, where d is the maximum distance at which the star would still have apparent magnitude lower than a limit of about 8.3 mag (see Paper I for details).

This correction proves to change significantly the age distribution as can be seen in Fig. 1.

[FIGURE] Fig. 1. Chromospheric age distribution with and without volume correction, which was applied to our sample to allow the derivation of a magnitude-limited SFH.

2.2. Evolutionary corrections

A correction due to stellar evolution is needed when a sample comprises stars with different masses. The more massive stars have a life expectancy lower than the disk age, thus they would be missing in the older age bins. The mass of our stars was calculated from a characteristic mass-magnitude relation for the solar neighbourhood (Scalo 1986). In Fig. 2, the mass distribution is shown. We take the mass range of our sample as 0.8 to 1.4 [FORMULA], which agrees well with the spectral-type range of the sample from nearly F8 V to K1-K2 V. As an example for the necessity of these corrections, the stellar lifetime of a 1.2 [FORMULA] is around 5.5 Gyr (see Fig. 3 below). This means that only the most recent age bins are expected to have stars at the whole mass range of the sample.

[FIGURE] Fig. 2. Mass distribution of the sample. Masses were calculated from a mean mass-magnitude relation given by Scalo (1986). From the figure, we estimate a mass range of 0.8-1.4 [FORMULA] for our sample. Note the substantial absence of massive stars, compared to the left wing of the mass distribution. The evolutionary corrections attempt to alleviate this bias.

[FIGURE] Fig. 3. Stellar Lifetimes from a variety of sources: Bahcall & Piran (1983); VandenBerg (1985), for [FORMULA] and [FORMULA]; Eggleton et al. (1989); Schaller et al. (1992), for [FORMULA] and [FORMULA]; Bressan et al. (1993), for [FORMULA]; Fagotto et al. (1994a), for [FORMULA].

The corrections are given by the following formalism. The number of stars born at time t ago (present time corresponds to [FORMULA]), with mass between 0.8 and 1.4 [FORMULA] is


where [FORMULA] is the initial mass function, assumed constant, and [FORMULA] is the star formation rate in units of [FORMULA] Gyr-1 pc-2. The number of these objects that have already died today is


where [FORMULA] is the mass whose lifetime corresponds to t. From these equations, we can write that the number of still living stars, born at time t, as


Using Eqs. (1) and (2), we have





The number of objects initially born at each age bin can be calculated by using Eq. (6), so that we have to multiply the number of stars presently observed by the [FORMULA] factor. These corrections were independently developed by Tinsley (1974), in a different formalism. RPSMF present another way to express this correction in terms of the stellar lifetime probability function. We stress that all these formalisms yield identical results.

The function [FORMULA] can be calculated by inverting stellar lifetimes relations. Fig. 3 shows stellar lifetimes for a number of studies published in the literature. Note the good agreement between the relations of the Padova group (Bressan et al. 1993; Fagotto et al. 1994a,b) and that by Schaller et al. (1992), as well with Bahcall & Piran (1983)'s lifetimes. The stellar lifetimes for [FORMULA] given by VandenBerg (1985) are underestimated probably due to the old opacity tables used by him. The agreement in the stellar lifetimes shows that the error introduced in the SFH due to the evolutionary corrections is not very large.

The adopted turnoff-mass relation was calculated from the stellar lifetimes by Bressan et al. (1993) and Schaller et al. (1992), for solar metallicity stars:


where t is in yr. This equation is only valid for the mass range [FORMULA].

We have also considered the effects of the metallicity-dependent lifetimes on the turnoff mass. To account for this dependence, we have adopted the stellar lifetimes for different chemical compositions, as given by Bressan et al. (1993) and Fagotto et al. (1994a,b). Equations similar to Eq. (7) were derived for each set of isochrones and the metallicity dependence of the coefficients was calculated. We arrive at the following equation:


where [FORMULA], [FORMULA], [FORMULA]. Since [Fe/H] depends on time we use a third-order polynomial fitted to the AMR derived in Paper I. In that work, we have also shown that the AMR is very affected at older ages, due to the errors in the chromospheric bins. The real AMR must be probably steeper, and the disk initial metallicity around -0.70 dex. The effect of this in the SFH is small. The use of a steeper AMR increases the turnoff mass at older ages, decreasing the stellar evolutionary correction factors (Eq. 6). As a result, the SFH features at young and intermediate age bins (ages lower than 8 Gyr) increases slightly related to the older features, in units of relative birthrate which is the kind of plot we will work in the next sections.

Note that Eq. (8) does not reduce to Eq. (7) when [FORMULA]. The former was calculated from an average between two solar-metallicity stellar evolutionary models, while the latter uses the results of the same model with varying composition. The difference in the turnoff mass from these equations amount 12-15% from 0.4 to 15 Gyr.

The initial mass function (IMF) also enters in the formalism of the [FORMULA] factor. For the mass range under consideration, the IMF depends on the SFH, more specifically on the present star formation rate. It could be derived from open clusters, but they are probably severely affected by mass segregation, unresolved binaries and so on (Scalo 1998). We have adopted the IMF by Miller & Scalo (1979), for a constant SFH, which gives an average value for the mass range under study. Power-law IMFs were also used to see the effect on the results.

In Fig. 4 we show how this factor varies with age. The curves represent Eqs. (7; dashed curve) and (8; solid curve) using the Miller-Scalo's IMF. A third curve (shown by dots) gives the results using a Salpeter IMF with the turnoff-mass given by Eq. (7). The [FORMULA] factor does not vary very much when we use a different IMF. Being flatter than Salpeter IMF, the correction factors given by the Miller-Scalo IMF are higher. However, the effects of neglecting the metallicity-dependence of the stellar lifetimes are much more important in the calculation of this correcting factor. Since low-metallicity stars live less than their richer counterparts, the turnoff-masses at older ages are highly affected. In the following section, we will use the [FORMULA] factors calculated for metallicity-dependent lifetimes.

[FIGURE] Fig. 4. Stellar evolution correction factors. The curves stand for Eqs. (7; dashed line) and (8; solid line) and Miller-Scalo's IMF. A third curve (dotted line) gives the results of using a Salpeter IMF with the turnoff-mass given by Eq. (7).

2.3. Scale height correction

Another depopulation mechanism, affecting samples limited to the galactic plane, is the heating of the stellar orbits which increases the scale heights of the older objects. To correct for this we use the following equations. Assuming that the scale heights in the disk are exponential, the transformation of the observed age distribution, [FORMULA], into the function [FORMULA] giving the total number of stars born at time t is


where [FORMULA] is the average scale height as a function of the stellar age. A problem arises since scale heights are always given as a function of absolute magnitude or mass. To solve for this, we use an average stellar age corresponding to a given mass, following the iterative procedure outlined in Noh & Scalo (1990). This average age, [FORMULA], can be obtained by


where [FORMULA] is the lifetime of stars having mass m, and [FORMULA] is the star formation rate. Since [FORMULA] depends on the star formation rate, which on the other hand depends on the average ages through the definition of [FORMULA], Eqs. (9) and (10) can only be solved by iteration. We use the chromospheric age distribution as the first guess [FORMULA], and calculate the average ages [FORMULA]. These are used to convert [FORMULA] to [FORMULA], and the star formation history is found by Eq. (9), giving [FORMULA]. This quantity is used to calculate [FORMULA] and a new star formation rate, [FORMULA]. Note that, in Eq. (9), the quantity that varies in each iteration is [FORMULA], not the chromospheric age distribution [FORMULA]. Our calculations have shown that convergence is attained rapidly, generally after the second iteration.

Great uncertainties are still present in the scale heights for disk stars. Few works have addressed them since Scalo (1986)'s review (see e.g., Haywood, Robin & Crezé 1997). We will be working with two different scale heights: Scalo (1986) and Holmberg & Flynn (2000), that are shown in Fig. 5. Haywood et al.'s scale heights are just in the middle of these, so they set the limits on the effects in the derivation of the SFH.

[FIGURE] Fig. 5. Scale heights given by Scalo (1986, solid line) and Holmberg & Flynn (2000, dotted line).

The major effect of the scale heights is to increase the contribution of the older stars in the SFH. Better scale heights would not change significantly the results, so that we limit our discussion to these two derivations.

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Online publication: June 20, 2000