## 4. Solar neighbourhood SFR and IMFIn 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 SFRThe 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 As an illustration of the SFR uncertainties originating in the age-
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 ,
which we have calculated in the case of the relation
() and (). 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
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 IMFA 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 1 are young, and therefore witness the SFR at the present epoch whereas stars at 4 are long-lived stars ( 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 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 , 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 . 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(), and the corrections for the percentage of evolved stars at a given absolute magnitude, (). 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 correctionFor 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 - relation one would obtain, adopting for example a constant SFR and the age- relation . 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- relation is the correct relation. The scale height one would obtain with age- relation would be 345pc, more akin to the one adopted by Scalo (1986). In the case for decreasing SFR, and with this last age- relation, the equivalent scale height would be 380pc. Fig. 7 also shows that the relation of Scalo systematically underestimate the scale heights of stars with =2 and 3, therefore underestimating their surface density.
## 4.2.2. Correction for evolved starsTo 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 =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.
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 =1-4 is similar to Scalo's, while it seems to shows less evolved stars. ## 4.2.3. The mass- relationScalo (1986) quoted in his review that mass- 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 =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 , 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- 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 , there remain surprising differences between the value derived by D'Antonna & Mazzitelli and Haywood (1994) (x 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 =5-10. Because the LF used in all three studies is the same, the only difference is the mass- 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- 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- 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- relations, which are shown in Fig. 9 as thick lines, and are polynomial fits that bracket the mass- relation obtained from evolutionary tracks. We then simply calculate their derivatives, and the corresponding IMF.
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
fit shows that its slope is ## 4.2.4. The solar neighbourhood IMF revisitedEvaluating the uncertainty in the IMF due to assuming the derivative of a mean mass- 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- 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
, 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 , 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 , 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 1 is the
one obtained for
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- relation. At masses greater than
1.1 , the IMF corrected for a constant SFR
may be represented with a slope 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 On the contrary, if a decreasing SFR is assumed, the change in the IMF at 1 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- 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 functionOne 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.
In Fig. 11a, the curve labelled 1 is the LF obtained for an exponentially decreasing SFR exp(-t/ ) ( =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 3 , 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- 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 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 © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |