The conversion from Luminosity Functions to Mass ... - CiteSeerX

The conversion from Luminosity Functions to Mass Functions Francesca D'Antona Osservatorio Astronomico di Roma, I{00040 Monte Porzio, Italy

Abstract. The aim of this talk is to review the methodology required

to translate the observed luminosity function of a stellar sample into the mass function of the stars constituting the sample. Emphasis is given to the stellar models which necessarily underlie this operation, as the mass function is proportional to the derivative of the mass{luminosity relation, and this latter quantity can not be derived from observations with a sucient precision. On the other hand, when adopting a theoretical mass{luminosity relation, some useful rules must be kept in mind to obtain a physically meaningful result. The methodology is applied to the following speci c elds: 1. derivation of the initial mass function (IMF) of nearby low mass stars and of disk stars; 2. derivation of the IMF for the low masses in globular clusters; 3. derivation of the IMF in young stellar populations. In all cases, the stellar physics which enters into the determination of the mass{luminosity relation, and its uncertainties, is shortly discussed.

1. Introduction This review naturally begins with a series of very nave statements. Nevertheless, some of the astronomy of the Initial Mass Function (IMF) of the latest 20 years has been built up while ignoring this navete, so that it becomes mandatory to remember them in a review devoted to the mass{luminosity relation (MLR). The IMF of stars, along with its dependence on the metallicity, is a fundamental ingredient in any study of the mass content of galaxies and in the understanding of their chemical evolution. In this respect, both the functional form of the IMF at high mass, at low, and at very low masses is important. In fact, the masses in the upper mass range evolve rapidly and contribute to the enrichment in heavy elements of the galactic gas, but the low mass stars have a double role: once matter is locked into low mass stars, which do not evolve in a Hubble time, it is \lost" for chemical evolution (apart from the possible long term eect of low rate winds); second, the functional form of the IMF at the end of the main sequence may give us hints about whether brown dwarfs exist in numbers so large that they contribute to barionic dark matter, either in the disk or in the halo of galaxies. Throughout this paper we will generally refer to the \mass function" dN=dM, that is the number of stars per unit mass, and not its logarithmic form dN=d log M, 1

which is more useful in describing the mass budget. The mass function is often described by a power law: dN=dM / M , where = ?2:35 for Salpeter's (1955) IMF. The role of the MLR is crucial for a correct understanding of the IMF, especially for low mass stars. To write down the expression of the IMF in terms of the luminosity function, for a system system of age t0 , we call Mband the magnitude in a given band (e.g. MI is the absolute magnitude in the Johnson I band). Theoretical models in general provide log L=L , and the bolometric magnitude is de ned as:

Mbol = ?2:5 log L=L + MV + BC (1) where MV = 4:83  0:03. For a given band: Mband = Mbol ? BCband (2) so that the complete de nition of mass function from a luminosity function (dN=dMband ) given in a particular band reads as: dN dN dM = dMband

, "

#

dM    dMbol  dMbol t0 dMband

(3)

which can be reduced simply to: dN = dN   dM  (4) dM dMband dMband t0 remembering that there is an implicit consideration here of the bolometric magnitude to band magnitude relation. The term [dM=dMband ]t0 is the derivative of the MLR. What matters for a correct derivation of the IMF is thus not the MLR itself, but its derivative. Semi-empirical MLRs, based on the very few binary system mass determinations (Popper 1980, Andersen 1991, see also Malkov et al. 1997) are the best way to derive the masses of main sequence stars, knowing their magnitude, chemical composition and distances. Even a mass derivation through the colors of a main sequence star might not be too bad, once we have calibrated the main sequence masses through the binaries. Empirical MLRs are often given in the form of a power law, so that they have (dM=dMband ) = constant. More complicated empirical MLRs have been given recently, guided by theoretical stellar models, e.g. by Kroupa, Tout and Gilmore (1993) and Henry and McCarthy (1993)|see also Reid (1998). When we need information on the IMF, a power law approximation to the MLR becomes too poor, and even misleading, as [dM=dMband ] is not constant in reality. Also the more complicated empirical laws may have derivatives which do not resemble the \true" hidden derivatives that underlie them, which only correct stellar models can provide. A typical relation for this derivative is shown in Figure 1: it shows peaks and dips. The combination with even a very smooth IMF will result in an LF which ampli es these features. 2

Figure 1. The bottom gure shows the mass versus luminosity, and the top gure shows the derivative of the same relation from models which describe a Population II coeval population of age 12 Gyr. The chemical composition is Y = 0:23 and Z = 3  10?4 (Silvestri 1997). I will give the following golden rule for ANY derivation of an IMF from an LF: if an LF presents features like peaks or dips, they should not be automatically attributed to features of the IMF, before one has excluded the possibility that they correspond (even if not exactly) to expected features of the MLR.1 Another warning is necessary here: blind use of a theoretical MLR does not necessarily provide a correct solution, and might even worsen the problem. The theoretical results may be aected by systematic errors (e.g. in the opacities, or simply in the choice of the chemical composition; for the low and very low masses, the equation of state may in uence the MLR too) and the features of the theoretical MLR might not correspond perfectly to the features of the LF. In this latter case, combination of the LF and the MLR may give rise to a series of spurious peaks in the IMF. This result is not so bad if one takes into account the errors, but it may also lead to over-interpretations and to interesting (although meaningless) consequences such as the \multimodal stellar fragmentation" sometimes found in the literature. A recent example is discussed in Section 4.2. The outline of this review is as follows: in Section 2 we show examples of the derivation of the IMF of low mass disk stars, and the importance of the MLR; Section 3 deals with the low mass stellar models and their present uncertainties; Section 4 shows critically how the IMF of globular cluster stars depends on the 1

This rule is obviously a trivial version of the \Occam's razor" rule of economy: non sunt multiplicanda entia praeter necessitatem.

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distance modulus assumed for the cluster and on the age through the MLR; Section 5 deals with the problems of young stellar populations.

2. An example: application|or not|of the golden rule The rst application of the simple statement given in the introduction as golden rule was published some 25 years ago by Mazzitelli (1972), who derived and analyzed the LF of the nearby stars in Gliese's (1969) catalogue: he showed that the LF presented a dip at magnitude MV  7. To obtain the mass function (MF), he used the direct relation: MF = dN=dM = (dN=dMV )=(dM=dMV ) (5) multiplying each bin of the observed LF (dN=dMV ) by the derivative of the MLR (dM=dMV ) obtained from the Copeland et al. (1970) models of main sequence stars. He then approximated with a smooth and monotonic MF the points obtained, and rederived the LF: (LF )new = dN=dM  dM=dMV (6) where he again used (dM=dMV ) from Copeland et al. models; in this way, from a monotonic MF, he obtained an LF with a gap at MV ' 7. The gap was due to an in ection point in the derivative of the MLR of stellar models around MV = 7. This feature of the LF has taken the name of \Wielen dip", following Wielen's (1974) analysis of the LF of nearby stars presented at the IAU, and sometimes it was interpreted in terms of a discontinuity in the IMF. Mazzitelli's analysis was often forgotten, although it had already provided a very simple physical explanation for the gap. Wielen (1974) showed that the nearby stars LF presented a maximum around MV = 12{13, and declined at lower luminosity. This maximum was con rmed in the analysis by Wielen et al. (1983), from the updated catalogue of nearby stars. Stellar models show that, going down along the main sequence, around M ' 0:25 M the MLR has another change of slope and its derivative shows a maximum (see Figure 1). Here, in fact, the LF will show a maximum, even for a steeply increasing IMF. This characteristic behaviour is present in all stellar populations; the location of the maximum occurs at a magnitude increasing with metallicity, and it is now well con rmed observationally (e.g. von Hippel et al. 1996). This feature of stellar models is much more important than the in ection at M  0:7 M , as it will provide the shape of the IMF towards the low masses, whose mass budget is important for our understanding of Galaxy evolution. However, it was only recognized in 1983, as modeling of low mass stars is not straightforward (D'Antona and Mazzitelli 1983), and it was not considered either in the IMF derivation by Miller and Scalo (1979) or in its update (Scalo 1986). For the low mass unevolved stars, Scalo (1986) used a power law IMF, coupled with an LF whose shape was taken to be that for the nearby stars, so the IMF re ected the presence of the Mazzitelli{Wielen gap at MV = 7, and the maximum at MV = 13. The result is seen in the famous Figure 16 of Scalo (1986), in which, below 1 M , one can recognize the features of the LF translated 4

as if they were real features of the IMF.2 The errors plotted in Scalo's gure are very large, so that the Wielen dip is certainly not considered to be true, and also the decline at M <  0:2 M is generally not considered to be a real feature of the IMF. Nevertheless, use of the MLR of stellar models, in place of an MLR having a constant derivative, provides a continuously increasing IMF in place of Scalo's IMF showing a decline at very low masses. In terms of a power law of index , D'Antona and Mazzitelli (1986) obtain = ?1:68  0:1 in the whole range from 0.1 to 1 M , and = ?2:0  0:4 in the range from 0.1 to 0.3 M . The update result by Mera et al. 1996, with their recent stellar models, provides

= ?1:3 in the range from 0.25 to 0.6 M and = ?2:5 for M  0:25 M . The 8 pc sample, which is relatively complete provides  ?1 (Reid 1998) in the low mass range, but it is obtained via an empirical MLR. The errors on these numbers are very large; it is mandatory to extend the LF of nearby stars to a larger volume of space. For the 20 pc sample, the number of stars which is predicted at MV = 13 is 597 (Wielen et al. 1983), but this number is based on only 7 actually detected stars. Interestingly enough, for our knowledge of the IMF of the low mass stars it would be more important to reduce the error bars in the range 12 < MV < 15 than to search for very faint stars at the end of the main sequence. The photometrically based LFs, although exploring a deeper fraction of the disk in limited areas, are at variance with the LFs of the nearby stars, showing a very pronounced peak at MV ' 12 (e.g. Stobie et al. 1989, Kroupa et al. 1990). A long literature has tried to solve this problem. Apart from the consideration of unresolved binaries (Kroupa 1995), recently Reid and Gizis (1997) have shown that the conversion from color to absolute magnitude is dramatically in uenced by the adopted color{MV relation, and we are probably close to a solution of this problem. The global result from the analysis of Population I stars is that the IMF of low mass stars derived by combination of the LF with stellar models, in the mass range below 1 M , has a certainly smaller than Salpeter's (?2.35), but there is no clear sign of decline in the index towards the end of the main sequence, contrary to the result obtained by using a constant (dM=dMband ).

3. Physics of the mass{luminosity relation: open problems The examples given above should convince the reader that the LFs actually amplify features of the MLR through Equation 4. We should then ask why the MLR shows so much structure, and what are the uncertainties in this relation. It is easy to recognize that also the HR diagram presents quite a complex morphology. Figure 2 shows a collection of observational data, guided by the model interpretation. The right part of the diagram corresponds to the population I main sequence. It is described observationally by Monet et al. (1992) data and by the Pleiades sequence (Stauer, private communication). The left part is bounded by the metal poor globular cluster NGC 6397 (data from King et 2

Scalo (1986) plots dN=d log M, so that the at part of the function at M <  0:5 M (? = means that the IMF dN=dM is increasing with slope = ?1 towards low masses.

+ 1 = 0)

5

Figure 2. Composite HR diagram of Population I and II stars in the (V ? I ,MI ) plane. Symbols are as follows: black dots: stars listed in Monet et al. 1992; open triangles: other parallax stars from Naval Observatory, as given in Monet et al. (1992); crosses: Pleiades stars from a list of Stauer (private communication) plotted assuming distance modulus (m ? M )0 = 5:6 and reddening E (B ? V ) = 0:04. Threepointed stars are the data from King et al. (1997) for the globular cluster NGC 6397, shifted by (m ? M )0 = 12:25 and dereddened by E (B ? V ) = 0:15. Notice the white dwarfs at the left corner. The models shown are: open squares: a sequence having metal mass fraction Z = 3  10?4 , helium mass fraction Y = 0:23 and age 1010 yr, transformed into the observational plane adopting the correlations from Allard and Hauschildt (1995) and from Kurucz (1993); full squares: models with Z = 0:019, Y = 0:28, age 109 yr, by D'Antona and Mazzitelli 1994. Below 0.15 M the colors have been adjusted to reproduce the \second kink" region. Also the other present day lowest mass models (e.g. Barae et al. 1995) lie below the sequence shown, possibly because grains are not included in the model atmosphere computations.

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al. 1997), shifted at the cluster distance and dereddened. The structure points correspond to models by D'Antona and Mazzitelli (1994) of age 1 Gyr for the population I and to unpublished models by myself for Z = 3  10?4 , age 10 Gyr, for the globular cluster. The HR diagrams of other more metallic globular clusters, as well as some of Monet's subdwarfs, lie in the gap between NGC 6397 and the Population I stars, in agreement with theoretical expectations (e.g. Barae et al. 1997, whose models would also provide a very good t of the data shown for NGC 6397). Figure 2 emphasizes one important result of recent years: the exploration of the HR diagrams of several Globular Clusters (GCs) down to very low luminosities, and, at least in the case shown, down to the lowest possible masses in Population II: the four lowest open squares in the 10 Gyr Population II isochrone correspond to masses 0.12, 0.11, 0.10 and 0.096 M . The last two points then include masses distributed over an interval of only 0.004 M ! No doubt, because the IMF of this cluster is quite at, we do not expect to nd many stars there. The observations of NGC 6397 also show the interesting \double kink" shape (D'Antona 1995) predicted by theoretical models. A description of the physics behind this morphology is given in D'Antona (1995) and D'Antona and Mazzitelli (1996). It is easily traced, in the observational data, due to the homogeneity of the globular cluster chemical composition, and to the fact that the sample is located at a unique distance. The Population I data also show a double kink morphology in the HR diagram, more dicult to recognize, both because of uncertainties in the distances and because of the spread in color due to the dierent metallicities of disk stars. Let me consider separately the regions which form the double kink structure. 1) The pre- rst-kink models: these have masses above  0:5{0.6 M for Population I, and above  0:45 M for Population II. These models are certainly those for which the input physics is best known. The HR diagram location depends on the details of the convection model adopted, as convection is superadiabatic in the outer layers, but the luminosity does not depend on the atmospheric boundary conditions or on convection. The main sequence models burn hydrogen through the proton-proton chain, the nuclear reactions involved must include both 3 He + 3 He and the beryllium-7 chain at least above 0.7M . The opacities are well known, grey boundary conditions are safe enough to model the atmosphere, and the main requirement of modern stellar models is to achieve adequate relations between Te , colors and bolometric corrections. 2) The models in between the rst and second kinks: the mass range is 0:5  M=M  0:15 for Population I, and 0:4  M=M  0:12 for Population II. The models become more complicated: boundary conditions play a key role (Barae et al. 1995 and 1997). Although the main eect in the change of slope of the theoretical main sequence is to be attributed to the eect of dissociation of H2 molecules in the envelope, and to the consequent lowering of the adiabatic gradient (Copeland et al. 1970), at the same time molecules must be considered in the atmospheric integration which becomes more dicult, and uncertainties begin to weigh on the relation Te {colors. This is particularly true for the Population I models, but also Population II models are probably aected at a level that is dicult to quantify. Progress is possible, thanks to the eorts of the model atmosphere researchers (Allard and Hauschildt 1995; Tsuji et al. 7

1996; Brett 1995). Below  0:4 M the stars become fully convective, so that the luminosity output is directly constrained by the conditions in the radiative optical photosphere. Therefore, not only the color, but also the luminosity is aected by an incorrect treatment of the atmospheric boundary conditions. 3) Models below the second kink: in Population II the second kink is due to the reaching of degeneracy in the stellar interior. Here the models become more and more uncertain. In particular the equation of state problems are not yet solved. For Population II, in fact, the models below ' 0:12 M enter in the partial dissociation and ionization region of a real gas. Saumon, Chabrier and Van Horn 1995 have provided the most modern computation of EOS for the density{temperature region covered by low mass stars. This is employed e.g. in the models for Populations I and II by the Lyon group. The Saumon et al. EOS is computed for pure hydrogen and pure helium composition only. The neglect of metals might not be particularly signi cant for the low metallicity stars. On the other hand, the physical quantities for intermediate hydrogen plus helium mixtures are obtained by the additive volume law, which can not be used in the real{gas partial dissociation and ionization regime (Saumon et al. 1995). In fact, the ionization of hydrogen has a direct in uence on the helium ionization, and this cannot be taken into account by interpolation between pure elemental equations of state. While only a small mass fraction of the very low mass stars lies in this regime, an incorrect value for the adiabatic gradient here aects the central temperature and therefore the determination of the end of the main sequence and of the mass luminosity relation. The \best" equation of state for this region remains that from Magni and Mazzitelli (1979), who considered the intermediate helium compositions directly in the EOS computation; but a modern recomputation, taking advantage of present day computing power, would be important. Notice in addition that the bolometric corrections, too, become uncertain in this region, so that it is still dicult to try to constraint the IMF at the very low mass end in globular clusters. The \after the second kink" region in Population I (which is, by the way, related to the discrepancy noted by Reid and Gizis [1997] between the linear ts of the main sequence and its location at V ? I  3) is more ambiguous. It apparently begins at M  0:15 M , well above the minimum hydrogen burning mass, so that its presence could be more easily attributed to an atmospheric problem (e.g. the formation of grains in the atmosphere|Tsuji et al. 1996; Allard 1998) than to a structure problem. Present day models are not able to t it. An inspection of the problem shows that the radius of the very low mass structure close to degeneracy is too small.

4. The IMF of globular clusters Figure 3 shows the comparison between the MLR from the recent Barae et al. (1997) models for [M=H] = ?1:5 and the equivalent models from D'Antona and Mazzitelli (1996, DM96) (referring to Z = 10?3 ). We see that the dierences are most relevant at masses  0:3 M , where in fact the stars are fully convective and the dierent equation of state and boundary conditions play the maximum role. Dierences amount to  0:2 mag. There are dierences also at the low 8

Figure 3. The MLR from models with Y = 0:23 and Z = 1  10?3 from D'Antona and Mazzitelli (1996) (full line) and the corresponding relation by Barae et al. (1997), for [Fe=H] = ?1:5 (dotted line). Remember that [Fe=H] = log(Z=Z ) and Z = 0:018.

Figure 4. The derivative of the MLR from models with Y = 0:23 and Z = 3  10?4 , for dierent ages. It clearly shows the signature at MV  7:5 which in Population I is the cause of the Mazzitelli{Wielen dip. 9

mass end, which is sharper in the DM96 models. This can be attributed either to dierences in the opacities, or again to dierences in the equation of state. As we have noticed, it is not at all clear that the Barae et al. (1997) models are better in this region. Notice however that the two sets of models dier noticeably ( 0:1 mag) for masses above 0.6 M , where the uncertainties should be minimal or non-existing: this could be due, e.g., to the fact that Barae et al. (1997) do not include the 3 He + 4 He reaction in their network, and demonstrates the necessity of having accurate models in all of the necessary parameter space, as we will see later on. The dierences in color V ? I are more striking than the dierences in magnitudes, but this has always been a problem in the globular cluster tting of isochrones: researchers generally try to t the color within a few hundredths of a magnitude, while in any case the distance modulus is uncertain by  0:3 mag between the so called \short" (Carney et al. 1992) and \long" (Sandage 1993, Walker 1994) distance scales (see the discussion below).

4.1. The very low mass end

By looking at Figure 3 we see that general shape of the MLR in the two sets of models is in practice very similar. When we wish to understand the IMF at the end of the main sequence, however, even the subtle dierences in the shape at masses below 0.12 M , where the EOS problems are unsolved with the present generation of EOS, will play a role. Here I wish to stress that results from such analyses should not be taken at face value, considering both the EOS problems and the uncertainties in the \opacity" to be attributed to the comparison between data and models. The opacity uncertainty has several aspects: 1) the global metallicity of globular cluster stars is a problem in itself not completely settled (see e.g. the Carretta and Gratton 1997 scale versus the Zinn and West 1984 scale); 2) the -enhancement problem is far from being settled within a factor of about 0.2 dex; 3) recent generation Rosseland mean opacities (Alexander and Ferguson 1994) represent a great improvement over the past but present theoretical models generally employ solar scaled mixtures to simulate {enhanced mixtures. These reasons, connected to the fact that we wish to explore a region where, intrinsically, the dM=dMV factor becomes close to zero, make the derivation of the IMF close to the globular cluster minimum mass for hydrogen burning very uncertain. Therefore, claims about the IMF \falling down"|but also on the IMF steeply climbing|must be regarded mainly as educated guesses than as rmly established results. Nevertheless, based on the golden rule, I think we must be suspicious about conclusions on a change of shape in the IMF just very close to the minimum mass for hydrogen burning, where the models are most uncertain and dM=dMV becomes very small in any case. The simplest hypothesis is to infer that the IMF remains smooth, and see whether the uncertainty in the models may accomodate such a hypothesis. After all, nobody would try to derive the IMF of globular cluster stars based on the red giant branch luminosity function, given that the dierence in mass between the turn-o stars and the red giant tip stars is  0:03 M .

4.2. The role of the distance modulus

Recently, the attention or researchers has been focused so much on the low mass tail of the IMF, that the way today's literature on the IMF of globular clusters 10

from HST data estimate the distance scale to be adopted passes in silence, whereas it is a very relevant issue, which has to do with our golden rule. Figure 4 shows the derivative of the MLR for models with Z = 3  10?4 at dierent ages. It shows a prominent dip at MV ' 7, which is the Population II analog of the in ection in the Population I MLR, which produces the Mazzitelli{Wielen dip. My models are shown, but a similar dip, at about the same location, is obtained by examining, e.g., the Barae et al. 1997 models. In addition, the location of the dip is independent of metallicity. Two important points must be noticed:  According to the chosen distance modulus, the dM=dMV value corresponding to each apparent magnitude bin is dierent. As there are ranges in which this derivative is not monotonic, we are exactly in the situation described in the introduction: a bad match of LF and MLR will produce peaks and gaps in the derived IMF.  Only for magnitudes MV  8 (masses well below 0.4 M ) can one adopt an MLR independent of age. If we wish to derive the IMF for a wider range of masses, we must take into account the age eect, and choose an age corresponding to the distance modulus. Looking at the recent literature it is easy to see that these precautions have not been taken. For some reasons, the \short" scale distances have always been adopted for the metal poor globular clusters, but models corresponding to an age of 10 Gyr have been used, in place of more appropriate models corresponding to an age of  16 Gyr. As a result of the rst choice, all the derived IMFs for low metallicity clusters show a prominent dip at log M  ?0:4, in spite of dierent authors and dierent adopted models (e.g. Piotto et al. 1997; Ferraro et al. 1997; Elson et al. 1995; Chabrier and Mera 1997). Notice that practically all the LFs of low Z clusters show a attening in the LF just above the turno, which should correspond to the dip at MV ' 7:5 in Figure 4. In practice, the relation dN=dMV = (dN=dM)  (dM=dMV ) (7) tells us that the dM=dMV is re ected in the LF, modulated through the IMF shape. The choice of the \short" distance scale produces an IMF with structure. If we adopt a larger distance modulus the result is a smooth power law IMF: in fact the features of the LF and the features of the dM=dMV relation become well matched. Now, if we wish to minimize our hypotheses, a smooth IMF is more credible than an IMF with structure at  0:4 M ; we come to the conclusion that the MLR of low mass stars gives us information about the distance of globular clusters, that is, about the age of the Universe. This result is exempli ed in Figure 5, where the IMF for the globular cluster M30 (NGC 7099) from the data of Piotto et al. (1997) is shown for two choices of the distance modulus. We also mention that the LF of NGC 6397 presents a sharper dip below the turn-o (Piotto et al. 1997, King et al. 1997): this is probably due to the fact that the IMF of this cluster is atter than average, so that in Equation 7 the IMF does not act to smear out the dip. This result is preliminary, and it is given only as an example. Remember however that the \pre- rst-kink" models should be the most reliable, so that we must consider in detail the possibility of obtaining a smooth IMF in this region by adjusting models and distance modulus. 11

Figure 5. The IMF derived for the globular cluster M30 by assuming two dierent distance moduli. The MLR is from models with Y = 0:23 and Z = 3  10?4 (from Silvestri 1997). The \long" distance scale which we are inclined to choose for M30 provides an age of  12 Gyr for the cluster, and is consistent both with recent models for the horizontal branch stars (Mazzitelli et al. 1995, Caloi et al. 1997) and with the recent redetermination of the subdwarf parallaxes from Hipparcos (Reid 1997, Gratton et al. 1997). I note nally that also the slope of the IMF derived depends on the distance modulus: Figure 5 shows that the larger distance modulus provides a smaller

, which is reduced from ?2 (a value reasonably close to Salpeter's slope) to about ?1.5. Consequently, it is mandatory to know the distance modulus if we wish to understand whether there is or is not the possibility of splitting o a signi cant mass budget as brown dwarfs. My conclusion at this point is that present day observations, together with the most plausible (long scale) distance moduli, provide 's much smaller than ?2 for the low Z globular clusters, and therefore the globular cluster stars do not provide any evidence in favour of the diversion of a substantial mass budget into brown dwarfs. No evidence exists, however, that the IMF is declining in globular clusters, so that a substantial number of brown dwarfs is probably fragmented.

5. The young stellar populations Recent results, mainly from near IR observations, have opened up a new window on our knowledge of low mass stars. The youngest stellar systems are the place were we may in principle determine the IMF of brown dwarfs (BD) from observations, rather than from extrapolation of the stellar IMF, due to the relatively large intrinsic luminosity of BDs in these phases. 12

The problems encountered here are twofold: rst, one must in any case recognize BDs in a dicult dusty environment, and it is not yet clear whether this will bring reliable results; second, we do not know much about the very early phases of evolution of very low mass stars and BDs, simply because we do not know how they form, and on the contrary, it is just from these observations that we hope to get information on the formation mechanisms. Also in the case of young stellar populations the LFs are used to derive information about the IMF. The problems, however, are very dierent from those encountered when dealing with old stellar systems. In particular, the systems are generally not coeval: the spread in the star formation ages is generally of the order of the age itself, so that Equation 4 turns out to be a poor approximation. Another tool used to derive the IMF is simply to derive the mass of each star by locating it on the theoretical isochrones. In particular Hillenbrand (1997), for the Orion region, has separated the stars according to spectral type (Te ) and not in intervals of luminosity. This can be useful, as stars of very dierent mass may reside at the same luminosity, due to the age spread. This eld is rapidly evolving observationally, so that it is important to understand a few tools for the theoretical interpretation, and its limitations. A critical review of the theory of pre{MS evolution and of the uncertainties in the models is given in D'Antona and Mazzitelli 1997.

5.1. Coeval luminosity functions: importance of very low mass stars

In recent years, the set of pre{MS models by D'Antona and Mazzitelli (1994) have been widely used to derive information on the young star forming regions. In particular, they have been the basis for constructing LFs for coeval (and non-coeval) young stellar populations (e.g. Lada 1998). Apart from the eect of Deuterium (D) burning, the hydrostatic pre-mainsequence (pre-MS) models simply represent the gravothermal contraction on the Kelvin{Helmholtz timescale of the star of xed mass; that is, they assume that the protostellar accretion phase is already nished. D'Antona and Mazzitelli (1997) discuss the zero point of evolutionary times. This is found to be consistent with the zero point of the above evolutionary models, at least for the masses smaller than  0:5 M which do burn D during the pre-MS stage. D-burning is certainly an important phase for the low mass structure: its duration will depend on how much of it has been burned during the protostellar phases, and on how large the initial abundance is. In the hydrostatic evolution, the star contracts till the central temperature reaches the  106 K necessary for D ignition. The energy released from the D+ p reaction is large enough to practically stabilize the luminosity of burning. The smaller the stellar mass, the lower is the luminosity of burning, in a way such that the time spent in this phase increases when decreasing the mass (left side of Figure 6). Due to this characteristic of the D?burning phase, at a given age the MLR shows a attening (right side of Figure 6). In a coeval stellar population, this will produce a peak in the resulting LF, as rst noticed by Zinnecker et al. (1993). The shape and location of the peak will depend on the initial D-abundance in the star. The absence of strong peaks in the LFs of young stellar populations may indicate that they are not strictly coeval (as expected), and not that the D-burning does not play a role. Actually, from an examination of the binaries in the Hartigan et al. (1994) list, 13

Figure 6. The left gure shows the pause due to D-burning in the luminosity evolution of stars of masses from 0.02M to 0.10M (mass step 0.01M ), 0.15M and from 0.20 to 1.0M (mass step 0.1M ). The right gure shows the in uence of dierent D-abundances on the mass{J -magnitude relation Censori and D'Antona (1998) have shown that it is easier to attribute to the components a unique age (as it would seem reasonable) if D-burning is included than if it is neglected. An important point to be noticed here is that the masses M < 0:1 M must be considered when constructing the LFs (Censori and D'Antona 1998). In fact, in the rst release of the D'Antona and Mazzitelli (1994) pre-MS evolution, the tracks below 0.1 M were not included, and theoretical LFs built up from those tracks did not consider the low masses, which should provide a noticeable contribution at MJ  7{8 for ages from 106 to 107 yr. This is shown in Figure 7, where the LFs built up by Strom et al. (1993) from the DM94 tracks for masses above 0.1 M are compared with the LFs which include the lower masses. If the observational LFs do not show the counts predicted at these deep magnitudes, this can have a number of possible explanations. Censori and D'Antona (1998) suggest that the LFs of the region L1641 in Taurus should be interpreted with models including quite a large D-abundance, about twice the \standard" one XD = 2  10?5 . It is also possible that the D-burning in these low mass models is not correctly treated, as the mixing timescale becomes comparable to the timescale of burning (Chabrier and Barae 1997), so that the luminosity of Dburning is smaller than given in these models. In this case, however, the time spent in the D-burning phase will be even longer, so that, if brown dwarfs are formed in any (even minor) fraction, they must be ultimately discovered in star forming regions. 14

Figure 7. Comparison between the Strom et al. (1993) LFs (dotted) and the LFs including the lowest mass stars (full lines) for coeval systems of the labelled age (from Censori and D'Antona 1998).

5.2. Mass calibration through location in the HR diagram

The IMF in a young star forming region can be derived by attributing a mass to each star from its location in the HR diagram. In this context, we must remember that this procedure requires caution. In fact, while the temporal luminosity evolution of contracting stars is well known, the Te of the tracks depends considerably on the convection model adopted. An inecient convection (e.g. mixing length theory with  = l=Hp = 1) will provide, for the same mass, cooler tracks than those provided by ecient (e.g.  = 2) convection. The DM94 set of models \Alexander CM", employing the Canuto and Mazzitelli (1991) model for superadiabatic convection, provide hotter tracks than the MLT based models, especially for the low masses. The derivation of the IMF through these models therefore provides a somewhat steeper IMF than with MLT based models. An exhaustive discussion of this point is given by D'Antona and Mazzitelli (1997).

5.3. The Orion IMF

An IMF derivation in the Orion region was made by Hillenbrand (1997) by counting the stars in stripes of Te , that is, adopting in practice a relation of the type dN=dM = (dN=dTe )  (dTe =dM); (8) where the term (dTe =dM) is taken from models, in particular from DM94. The resulting IMF presents a sharp peak at M ' 0:2 M . We have recently computed new sets of models (D'Antona and Mazzitelli 1997), which at low masses dier from DM94 due to a particular choice for the treatment of external 15

Figure 8. Mass{Te relation for an age of 106 yr, from the models by D'Antona and Mazzitelli 1994 (DM 94, full line) and from the new models by D'Antona and Mazzitelli (1997; DM 97, dotted line). The new models show a change in the slope of the relation at M  0:2M . Can this be the reason why there is a peak in the IMF derived from the Hillenbrand (1997) study?

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convection (D'Antona and Mazzitelli 1998). The new models dier from the old ones around 0.2 M : Figure 8 shows that the Te {mass relation becomes steeper at such a mass; therefore, again, if we take into account in Equation 8 the decrease in the term (dTe =dM), the resulting IMF might not show a peak if interpreted with these models. It is not clear whether the DM97 models are more reliable than the DM94 models for these masses, but this possibility should be kept in mind. Acknowledgments. The golden rule was devised in the course of a discussion with Hans Zinnecker, who is gratefully thanked for insisting on the necessity of clarifying this issue. M. Meyer put forward the problem of a possible change in the slope of the mass{Te relation, discussed in Section 5.3. Ivan King and collaborators are thanked for making available their recent data for NGC 6397 before publication. Ivan King also took the trouble of correcting the English and the readibility of the paper. I wish to dedicate this review to Italo Mazzitelli, after|or in spite of|our 25 year long marriage and scienti c cooperation.

References Alexander, D. R., & Ferguson, J. W. 1994, ApJ, 437, 879 Allard, F. 1998, in \Brown Dwarfs and Extrasolar Planets", eds. R.Rebolo, E.L. Martin, & M.R. Zapatero Osorio, A.S.P. Conf. Series, in press Allard, F., & Hauschildt, P.H. 1995, ApJ, 445, 433 Andersen, J. 1991, A&A Rev., 3, 91 Barae, I., Chabrier, G., Allard, F., & Hauschildt, P.H. 1995, ApJ, 446, L35 Barae, I., Chabrier, G., Allard, F., & Hauschildt, P.H. 1997, A&A, in press Bessell, M.S. 1991, AJ, 101, 662 Brett, J.M. 1995, A&A, 295, 736 Caloi, V., D'Antona, F., & Mazzitelli, I. 1997, A&A, 320, 823 Canuto, V.M., & Mazzitelli, I. 1991, ApJ 370, 295 Carney, B.W., Storm J., & Jones, R.W. 1992, ApJ, 386, 663 Carretta, E., & Gratton, R.G. 1997, A&AS, 121, 95 Censori, C. & D'Antona, F. 1998, in \Brown Dwarfs and Extrasolar Planets", eds. R.Rebolo, E.L. Martin, & M.R. Zapatero Osorio, A.S.P. Conf. Series, in press Chabrier, G., & Mera, D. 1997, A&A, in press Chabrier, G., & Barae, I. 1997, A&A, in press Copeland, H., Jensen, J.O., & Jorgensen, H.E. 1970, A&A, 5,12 D'Antona F., 1995, in \The bottom of the Main Sequence | and beyond", ESO Astrophysics Symposia (Springer), ed. C. Tinney, p. 13 D'Antona, F., & Mazzitelli, I. 1983, A&A, 127, 149 D'Antona, F., & Mazzitelli, I. 1986, A&A, 162, 80 D'Antona, F., & Mazzitelli, I. 1994, ApJS, 90, 467 (DM94) D'Antona, F., & Mazzitelli, I. 1996, ApJ, 456, 329 (DM96) 17

D'Antona, F., & Mazzitelli, I. 1997, in \Cool stars in Clusters and Associations", eds. R. Pallavicini & G. Micela, Mem.S.A.It., 68, n.4 D'Antona, F., & Mazzitelli, I. 1998, in \Brown Dwarfs and Extrasolar Planets", eds. R.Rebolo, E.L. Martin, & M.R. Zapatero Osorio, A.S.P. Conf. Series, in press Elson, R.A.W., Gilmore, G.F., Santiago, B.X., & Casertano, S. 1995, AJ, 110, 682 Ferraro, F.R., Carretta, E., Bragaglia, A., Renzini, A., & Ortolani, S. 1997, MNRAS, in press Gliese, W. 1969, Catalogue of Nearby Stars, Edition 1969 Veroentlichungen des Astronomischen Rechen{Instituts, Heidelberg, No. 22 Gratton, R.G., et al. 1997, ApJ, in press Hartigan, P., Strom, K. M., & Strom, S.E. 1994, ApJ. 427, 961 Henry, T.J., & McCarthy, D.W. 1993, AJ, 106, 773 Hillenbrand, L.A. 1997, AJ, 113, 1733 von Hippel, T., Gilmore, G., Tanvir, N., Robinson, D., & Jones, D.H.P. 1996, AJ, 112, 192 King, I.R., Anderson, J., Cool, A.M. & Piotto, G. 1997, ApJ, in press Kroupa, P. 1995, ApJ, 453, 358 Kroupa, P., Tout, C., & Gilmore, G. 1990, MNRAS, 244, 76 Kroupa, P., Tout, C., & Gilmore, G. 1993, MNRAS, 262, 545 Kurucz, R.L. 1993, ATLAS9 Stellar Atmosphere Programs and 2Km s?1 Grid (Kurucz CD ROM No. 13) Lada, E. 1998, in this volume Malkov, O.Yu., Piskunov, A.E., & Shpil'kina, D.A. 1997, A&A, 320, 79 Magni, G., & Mazzitelli, I. 1979, A&A, 72, 134 Mazzitelli, I., D'Antona, F. & Caloi, V. 1995, A&A, 302, 382 Mazzitelli, I. 1972, Astrophys. Space Sci., 17, 378 Mera, D., Chabrier, G., & Barae, I. 1996, ApJ, 459, L87 Miller, G.E., & Scalo, J.M. 1979, ApJS, 41, 513 Monet, D. G., Dahn, C. C., Vrba, F. J., Harris, H. C., Pier, J. R., Luginbuhl, C. B., & Ables, H. D. 1992, AJ, 103, 638 Piotto, G., Cool, A., & King, I.R. 1997, AJ, 113, 1345 Popper, D.M. 1980, ARA&A, 18, 115 Reid, I.N. 1997, AJ, 114, 161 Reid, I.N. 1998, in this volume Reid, I.N., & Gizis, J.E. 1997, AJ, 113, 2246 Sandage, A. 1993, AJ, 106, 703 Saumon, D., Chabrier, G., van Horn, H. M. 1995, ApJS, 99, 713 Salpeter, E.E. 1955, ApJ, 121, 161 Scalo, J.M. 1986, Fundam. Cosmic Phys., 11, 1 Silvestri, F. 1997, Thesis of Laurea in Physics, University of Roma \La Sapienza" 18

Stobie, R.S., Ishida, K., & Peacock, J.A. 1989, MNRAS, 238, 709 Strom, K.M., Strom, S.E. & Merrill, K.M. 1993, ApJ, 412, 233 Tsuji, T., Ohnaka, K., & Aoki, W. 1996, A&A, 305, L1 Walker, A.R. 1994, AJ, 108, 555 Wielen, R. 1974, Highlights of Astronomy, 3, 395 Wielen, R., Jahreiss, H., & Kruger, R. 1983, in IAU Coll. 76, 163 Zinn, R., & West, M.J. 1984, ApJS, 55, 45 Zinnecker, H., McCaughrean, M. J., & Wilking, B.A., 1993 in \Protostars & Planets III, eds. E.H. Levy, J.I. Lunine, & M.S. Matthews (Tucson, AZ: University of Arizona), p. 429

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