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Astron. Astrophys. 320, 428-439 (1997)

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4. Solar neighbourhood SFR and IMF

In the following, we briefly discuss the uncertainties that affect the SFR history, as obtained from the study of solar neighbourhood stars. Then, we assess the accuracy of the IMF derived from the solar neighbourhood LF.

4.1. Uncertainties in the SFR

The history of the stellar birthrate in the solar neighbourhood has been derived through several techniques over the last 20 years. A general feeling from many different studies is that the birthrate in the galactic disc remained more or less constant since the formation of the disc. However, it is difficult to assess what "constant" means exactly because quite different assumptions have been made regarding the disc dynamical evolution and the adopted scale heights.

For instance, the SFR found by Twarog (1980) (irrespective of uncertainties due to stellar evolution theory and main sequence datation) hinges on the IMF slope in the mass range of 1 to 3 solar masses, and scale height corrections. The IMF in this mass range is usually assumed Salpeter-like (x=1.35) or Scalo like (x=1.7) for which Twarog's SFR has a maximum equal to about three, 9 Gyrs ago, with a subsequent slow decrease. However, there is no evidence that the IMF slope in the mass range 1-3  [FORMULA] is around x =1.3-1.7. Quoting Twarog (1980) study for a constant SFR, often used in galactic disc models, would implies a corresponding very steep IMF in the intermediate mass range, in the order of x =3. The decreasing SFR envisaged by Twarog (1980), corresponded to a flat slope for the IMF (x =-1), according to his data. Scalo (1986) has favored a bimodal IMF on the basis of his LF study, also implying a decreasing SFR.

As an illustration of the SFR uncertainties originating in the age- [FORMULA] relation, we have calculated the scale height of a given isothermal population of stars embedded in a disc made of several components using the formula of Talbot & Arnett (1975):


The relevant quantity here is the ratio [FORMULA], which we have calculated in the case of the relation ([FORMULA]) and ([FORMULA]). Applying these scale height ratios to the data of Twarog (his Table 3, column 4, 6), the SFR variations we obtain are displayed in Fig. 6, for IMF x =-1 and x =3. As can be seen from the figure, the SFR still covers a wide range of possible variations, from decreasing SFR by a factor of 6, to first increasing and then decreasing SFR, by a factor of 2.5. In 1977, Mayor & Martinet had reached a similar conclusion, and were not able to constraint SFR variation to better than a factor of 7.

[FIGURE] Fig. 6. The SFRs from Twarog (1980) for x =-1 (dashed-dotted curves) using scale height corrections from age- [FORMULA] relation ([FORMULA]) (lower curve) and ([FORMULA]) (upper curve). Continuous curves are for x =3, and same age- [FORMULA] relations.

Other methods have made use of the white dwarf LF and the chromospheric activity of red dwarfs, the uncertainties and advantage of each being well described in the recent papers of Wood (1992), Yuan (1993) and Soderblom et al. (1991). The accuracy on the SFR determination using the white dwarf LF is also limited to within a factor of 7.

4.2. Uncertainties in the IMF

A widely used method to estimate changes between past and present SFR is the study of the shape of the LF. A comprehensive review on this issue was given by Scalo (1986). The LF gives a measure of the variation between the mean SFR in the galactic disc since its formation and the present-day SFR. The fast change of star lifetimes along the main sequence implies that most stars seen at [FORMULA] [FORMULA] 1 are young, and therefore witness the SFR at the present epoch whereas stars at [FORMULA] [FORMULA] 4 are long-lived stars ([FORMULA] greater than the age of the galactic disc), and their number gives the mean intensity of the SFR since the beginning of star formation in the disc. As emphasized by Scalo (1986), the ratio [FORMULA] gives a hint to the overall variation of the SFR provided the present SFR is not special in any respect. For instance, if star formation in the Galaxy proceeds as a succession of burst, as has been suggested (Scalo, 1988), then this ratio is barely a meaningful measurement of the overall SFR variation.

In the restricted range of magnitude [FORMULA] [FORMULA], the LF depends on both the IMF and SFR history. Scalo (1986) has investigated the possible shape of the IMF assuming different SFR histories. He favoured a bimodal IMF (see his Fig. 17) assuming two modes of star formation, dominant at masses m=0.3 and m=1.1-1.3  [FORMULA]. We discuss below the different conversion factors that are required to obtain the IMF from the LF, and the accuracy of the new IMF we obtain. A second method, utilized in Sect. 4.3, consist of calculating theoretical luminosity functions, as described in Haywood (1994), and compare them with the observed one. An interesting consistency check which we also discuss in Sect. 4.3 is to use the (IMF,SFR) derived with the first method as input parameters to the second.

Information about the present-day mass function (PDMF) is usually obtained by applying the following transformations to the LF (Miller & Scalo, 1979):


which include the derivative of the absolute magnitude to mass relation, the correction for the increase of the scale heights of the stars with absolute magnitude, H([FORMULA]), and the corrections for the percentage of evolved stars at a given absolute magnitude, [FORMULA] ([FORMULA]).

These last two factors depend on the history of the galactic disc. For instance, the ratio of the evolved to main sequence stars ought to be high if the SFR was greater at remote times. Similarly, it is expected that main sequence stars at a given magnitude are a mixture of ages with different characteristics scale heights, oldest stars having the largest scale heights. The final scale height at a given magnitude will therefore depend on the relative weight of each of these age groups, and hence on the star formation history. These may be second order effects compared to the other uncertainties (particularly the uncertainty in the mass-absolute magnitude relation), but it is nonetheless necessary to quantify them.

4.2.1. Scale heights correction

For the conversion from volume density to surface density, the method described in Sect. 4.2 uses a mean absolute visual magnitude-scale height correction. We may calculate, using our model, the [FORMULA] - [FORMULA] relation one would obtain, adopting for example a constant SFR and the age- [FORMULA] relation [FORMULA]. This relation is illustrated in Fig. 7, where we have plotted the exponential density laws as given by Scalo (1986), and the density profiles we obtained with our model. Only the integral under each curve is important, not the detailed density variation. We may calculate an equivalent exponential scale height to our model curves, by dividing the surface density with the volume density. The scale height corresponding to unevolved dwarfs (triangles) is approximately 260 pc if the age- [FORMULA] relation [FORMULA] is the correct relation. The scale height one would obtain with age- [FORMULA] relation [FORMULA] would be [FORMULA]  345pc, more akin to the one adopted by Scalo (1986). In the case for decreasing SFR, and with this last age- [FORMULA] relation, the equivalent scale height would be [FORMULA]  380pc. Fig. 7 also shows that the relation of Scalo systematically underestimate the scale heights of stars with [FORMULA] =2 and 3, therefore underestimating their surface density.

[FIGURE] Fig. 7. Comparison of the exponential profiles used by Scalo (1986) (empty symbols) and the one deduced from our model (filled symbols), normalized to 1 at z=0, and for different absolute magnitudes. The thickening of the disc in our model is described through age- [FORMULA] relation [FORMULA], and the age distribution of stars along the main sequence was calculated assuming a constant SFR. Squares are for [FORMULA] =0, diamonds for [FORMULA] =2, circles [FORMULA] =3, triangles for [FORMULA] =4 and 5.

4.2.2. Correction for evolved stars

To illustrate how this factor depends on the star formation history in the disc, we have plotted in Fig. 8 the correction for evolved stars that one has to apply to the LF, as it comes out of our model (i.e assuming the evolutionary tracks mentioned in Sect. 2.2), and for different SFRs. The thick curve is the one adopted by Scalo (1986). The continuous and dotted curves correspond to constant and increasing SFR respectively. They differ significantly from the curve adopted by Scalo, which decreases to 0.6 at [FORMULA] =1 while our model indicates more than 0.8 if the SFR has been constant or decreasing. In this region, the stars belong mainly to the main sequence and to the subgiant branch. Main sequence lifetimes vary from 10 Gyrs to 1 Gyrs according to evolutionary tracks, while subgiant branch time scale is approximately 10 time smaller.

[FIGURE] Fig. 8. The ratio of main sequence stars to (main sequence + evolved) stars as a function of [FORMULA]. Three models are shown on the figure, dotted curve is for increasing SFR, continuous thin curve is for constant SFR, while dashed-dotted is for decreasing SFR. The thick line is the relation adopted by Scalo.

The curve given for a decreasing SFR (factor 7) bears some ressemblance with the one adopted by Scalo. The decrease in main sequence/evolved stars in the range [FORMULA] =1-4 is similar to Scalo's, while it seems to shows less evolved stars.

4.2.3. The mass- [FORMULA] relation

Scalo (1986) quoted in his review that mass- [FORMULA] is probably the most important source of uncertainty in the derivation of the IMF. We want to point out here its importance for the IMF at masses less than solar. Scalo advocated that the IMF is bimodal, because the shape of the observed LF at [FORMULA] =5-9, shows a flattening or a dip that he attributed to an equivalent feature in the mass function. He demonstrated that this dip was followed by a rise in the IMF with a maximum at 1.3  [FORMULA], if a constant or decreasing SFR was assumed. Scalo favoured the case of a decreasing SFR and an age for the disc of 15 Gyrs, therefore implying strong bimodality. At the same time, D'Antona & Mazzitelli (1986) showed that the mass- [FORMULA] relation could be responsible for the flattening in the LF if the non-linear variation of the luminosity with mass was taken into account, a result further demonstrated by KTG (1990, 1991, 1993) and Haywood (1994).

Although these studies have weakened the argument for a dip in the mass function at 0.5-1.0  [FORMULA], there remain surprising differences between the value derived by D'Antonna & Mazzitelli and Haywood (1994) (x [FORMULA] 0.7) on the one side, and KTG (1993) (x=1.2) on the other side. Such a difference is not expected, because the LF is known to have satisfactory accuracy in the magnitude range [FORMULA] =5-10. Because the LF used in all three studies is the same, the only difference is the mass- [FORMULA] relation. In DM and KTG, the link between the mass and the magnitude is a one-to-one relation, while in Haywood (1994), the LF was computed through the evolution of stars on tracks over a period of 10 Gyr. The resulting area spanned by the stars over the mass- [FORMULA] plane is shown in Fig. 9. The thin continuous curve on the figure is the relation adopted by KTG, which, as can be seen agrees well with our own relation. However, adopting a mean mass- [FORMULA] relation becomes a rough approximation when its derivative has to be taken for the conversion to mass. As an illustrative example and in order to explain the difference between our result and the one of KTG, we have considered two other mass- [FORMULA] relations, which are shown in Fig. 9 as thick lines, and are polynomial fits that bracket the mass- [FORMULA] relation obtained from evolutionary tracks. We then simply calculate their derivatives, and the corresponding IMF.

[FIGURE] Fig. 9. The Mass- [FORMULA] relation as deduced from our model (dotted area) and the two continuous thick lines which are polynomial fits limitting this relation. The thin continuous line is the mean relation adopted by KTG (1993). Most observational points are from Popper (1980).

If we use the derivative given by KTG (their figure 2), we found an IMF which is adequately fitted by a straight line. A minimum [FORMULA] fit shows that its slope is x =1.15, a value that is (not surprisingly) in agreement with the one given by KTG (1993). However, the slopes obtained with our two polynomial fits are 1.15 and 0.43, which is the range of uncertainty that results from this method. Hence we cannot decide what is the best slope for the IMF within this range; the two slopes derived here are equally representative of the local IMF derived by this method, but none of them tell us what is the true IMF. A correct measurement of the IMF at M [FORMULA] 0.5  [FORMULA] must take into account the distribution in magnitude at a given mass. This can only be done by taking into account the luminosity evolution of the stars over their main sequence life. In Haywood (1994), such a calculation was done, and we found that the IMF was correctly represented using an index of 0.7 at M [FORMULA][FORMULA], which is a more exact value of the IMF slope. It is the value adopted hereafter.

4.2.4. The solar neighbourhood IMF revisited

Evaluating the uncertainty in the IMF due to assuming the derivative of a mean mass- [FORMULA] relation at masses higher than solar becomes a difficult and unproductive task. Although the computation of theoretical LF depends on stellar evolution theory, it does not rely on fragile approximations (mean mass- [FORMULA] relation, hypothetical evolved to main sequence ratio etc..). It is a much more straighforward procedure, and one that offers the advantage of comparisons that can also be made on the evolved stars contained in the LF. This will be done in the next section. We want to illustrate now what are the possible IMF when taking into account all the uncertainties discussed above.

Having eliminated the possibility for a dip in the IMF at M=0.7-0.8 [FORMULA], we return to a problem mentioned by Miller & Scalo (1979): if one assumes a constant or decreasing SFR, the corrections for evolved stars are important, and give rise to a change of behaviour in the IMF at M=1  [FORMULA], which could be assigned to bimodality. As noted by Miller & Scalo (1979), it is quite problematic that the rise in the IMF appears just at 1 [FORMULA], which designates the mass of our sun as being special. We now derive new IMFs, assuming an age for the galactic disc of 10 Gyrs (therefore minimizing the correction for SFR history), to see how this feature is present in our calculations. These IMFs are plotted in Fig. 10 and given in Table 2. The IMF at M [FORMULA][FORMULA] is the one obtained for x =0.7. At M [FORMULA][FORMULA] we adopt the mass- [FORMULA] relation of Scalo (1986).

[FIGURE] Fig. 10. The IMF in the solar neighbourhood derived using the new factors in Table 2. The upper IMF corresponds to the decreasing SFR, calculated with scale height corrections from Scalo. The two others curves are for constant and increasing IMF (lower curve), calculated assuming a scale height of 250 pc at [FORMULA] 4, and Scalo corrections otherwise. Dotted vertical lines give the limit for 1 and 2 solar masses, while the indexes indicated in the box are for power-law IMFs with dN/dM [FORMULA]  M [FORMULA].


Table 2. IMF given different assumptions on the SFR and scale height of the disc. Columns (1) to (5) give the absolute visual magnitude, the mass, the observed luminosity function, the derivative of the mass- [FORMULA] relation, the scale height corrections. Column (6) gives the IMF assuming a constant SFR and for scale heights corrections of column (5). Column (7) gives the IMF obtained with an exponentially increasing SFR with a characteristic time scale of 5 Gyrs, and with the same scale height corrections. Column (8) gives the scale height corrections of Scalo (1986), and column (9) the IMF when an exponentially decreasing SFR with a characteristic time scale of 5 Gyrs and scale height corrections of column (8) are assumed. The IMF at M [FORMULA][FORMULA] is calculated assuming x =0.7, starting from m=0.962  [FORMULA].

Fig. 10 illustrates the very great uncertainties that remain in the IMF, due mostly to scale height corrections, and to our ignorance of the SFR history. Because of scale height corrections, the IMF at masses less than solar remains uncertain up to at least 20 to 30 %. At masses greater than solar mass, the conjugated effects of the scale heights and SFR, the number of stars that formed within this mass range is uncertain to a factor of 5-10. In the case of the constant and increasing SFR, the IMF in Fig. 10 is compatible with a smooth IMF, and we think that although the IMF shows a flattening at 1.0-1.1, no bimodality is apparent, given the uncertainties that still remain, in particular due to the adopt of a mean mass- [FORMULA] relation. At masses greater than 1.1  [FORMULA], the IMF corrected for a constant SFR may be represented with a slope [FORMULA] 2.0-2.2, depending on the considered upper mass (log M=0.27 or 0.4). In the case of the increasing SFR, the IMF is near x =2.6-2.9 over the same mass range.

On the contrary, if a decreasing SFR is assumed, the change in the IMF at 1 [FORMULA] leans in favour of a bimodal IMF, and the problem raised by Miller & Scalo is relevant. As mentioned above, either we should considered that the mean mass- [FORMULA] relation is responsible for the problem, or that the history of the SFR is not one which has decreased over the last 10 Gyrs.

4.3. Synthetic luminosity function

One way to check how the results found by Scalo (1986) or hereabove are consistent with the observed LF is to use the IMF derived through the present-day mass function and the corresponding SFR as input parameters to compute theoretical LF using synthetic HR diagrams. Fig. 11 presents a number of theoretical LF that are compared with the observed LF. Two observed LFs are shown: the LF from Scalo (1986), which was used to obtain the IMF in the last section, and the LF from Wielen et al. (1983). The characteristics of the model used for each curve are given in Table 3.

[FIGURE] Fig. 11a-c. LFs obtained for different SFRs and IMFs. The observed LF in Scalo (1986) (dots, [FORMULA] [FORMULA] 4), and Wielen et al. (1983) (squares). Plot a is for decreasing SFR, b is for constant SFR, and c increasing SFR. See text for detail.


Table 3. Characteristics of each model used to calculate the LFs of Fig. 11. The first column gives the number of the curve on each of the plot of Fig. 11. The 3 power law IMF slopes are given over the mass ranges M [FORMULA] 1.0  [FORMULA], 1-3  [FORMULA], and M [FORMULA] 3.0  [FORMULA]. T0 is the disc age.

In Fig. 11a, the curve labelled 1 is the LF obtained for an exponentially decreasing SFR exp(-t/ [FORMULA]) ([FORMULA] =12 Gyrs) and an age for the disc of 12 Gyrs, while the other two curves were obtained for an age of 10 Gyrs. The IMF is the one given by Scalo (1986) for the same parameters (see his figure 17). The curve labelled 2 was obtained for the IMF of Sect. 4.2.4 that corresponds to a decreasing SFR (see Fig. 10). Finally, the curve labelled 3 is for an IMF with a slope 0.7/1.5/2 on mass intervals 0.1-1.0, 1.0-3.0 and [FORMULA][FORMULA], and the same SFR.

It is striking to see how the LF (1) in Fig. 11a, calculated with the IMF given by Scalo (see his Fig. 17), contradicts the result obtained by Scalo. Clearly such a combination of IMF and SFR does not yield an acceptable fit with the observed LF. This calculated LF tells us that if a decreasing SFR is the true SFR, then necessarily something was wrong in the derivation of the IMF by Scalo (1986), most probably his mean mass- [FORMULA] relation. This point of view is confirmed by the curve (3) in Fig. 11a: since the fit to the observed LF is correct in this case, according to our synthetic LF, a decreasing SFR implies an IMF with a slope of x =1.5, in contradiction with the strongly bimodal IMF found by Scalo. The same effect is present in the case of LF (2), while the difference with the observed LF is much smaller, due to the fact that we considered the change of slope in the IMF at 1.0  [FORMULA] (and not at 0.7  [FORMULA]). The bump at [FORMULA] =4 corresponds to the maximum at M=1.17  [FORMULA] in the mass function (see Fig. 10).

In Fig. 11b, LF (1) was obtained for a disc age of 12 Gyrs, with IMF slopes 0.7/0.0/2, chosen to represent the similar IMF Scalo derived assuming a constant SFR (see his Fig. 17). Here again, his IMF overestimates the number of stars in the range 0.7 to 1.17, and the LF appears systematically higher than the observed one. In view of the uncertainties in the data, the LF (2, 3, 4) all look acceptable IMF.

In Fig. 11c, the increasing SFR with a IMF slope x =1.5 yields an LF which systematically overestimates the observed LF at [FORMULA] [FORMULA] 3. IMF with x =2.5 falls well within the range of observed LFs. IMF slope x =3.5 gives the upper limit of acceptable IMFs.

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Online publication: June 30, 1998