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GALACTIC STELLAR AND SUBSTELLAR INITIAL MASS
FUNCTION
Gilles Chabrier
Ecole Normale Supe´rieure de Lyon,
Centre de Recherche Astrophysique de Lyon (UMR CNRS 5574), 69364 Lyon Cedex 07,
France
(chabrier@ens-lyon.fr)
Short title: THE GALACTIC MASS FUNCTION

– 2 –
ABSTRACT
We review recent determinations of the present day mass function (PDMF) and initial
mass functions (IMF) in various components of the Galaxy, disk, spheroid, young and globu-
lar clusters and in conditions characteristic of early star formation. As a general feature, the
IMF is found to depend weakly on the environment and to be well described by a power-law
form for m & 1M⊙ and a lognormal form below, except possibly for early star formation
conditions. The disk IMF for single objects has a characteristic mass around mc ∼ 0.08M⊙
and a variance in logarithmic mass σ ∼ 0.7, whereas the IMF for multiple systems has
mc ∼ 0.2M⊙ and σ ∼ 0.6. The extension of the single MF into the brown dwarf regime is in
good agreement with present estimates of L- and T-dwarf densities and yields a disk brown
dwarf number density comparable to the stellar one nBD ∼ n⋆ ∼ 0.1 pc−3. The IMF of young
clusters is found to be consistent with the disk field IMF, providing the same correction for
unresolved binaries, confirming the fact that young star clusters and disk field stars represent
the same stellar population. Dynamical effects, yielding depletion of the lowest-mass objects,
are found to become consequential for ages & 130 Myr. The spheroid IMF relies on much
less robust grounds. The large metallicity spread in the local subdwarf photometric sample,
in particular, remains puzzling. Recent observations suggest that there is a continuous kine-
matic shear between the thick-disk population, present in local samples, and the genuine
spheroid one. This enables us to derive only an upper limit for the spheroid mass density
and IMF. Within all the uncertainties, this latter is found to be similar to the one derived
for globular clusters, and is well represented also by a lognormal form with a characteristic
mass slightly larger than for the disk, mc ∼ 0.2-0.3 M⊙, excluding a significant population of
brown dwarfs in globular clusters and in the spheroid. The IMF characteristic of early star
formation at large redshift remains undetermined, but different observational constraints
suggest that it does not extend below ∼ 1M⊙. These results suggest a characteristic mass
for star formation which decreases with time, from conditions prevailing at large redshift to
conditions characteristic of the spheroid (or thick-disk), to present-day conditions. These
conclusions, however, remain speculative, given the large uncertainties in the spheroid and
early star IMF determinations.
These IMFs allow a reasonably robust determination of the Galactic present-day and
initial stellar and brown dwarf contents. They also have important galactic implications
beyond the Milky Way in yielding more accurate mass-to-light ratio determinations. The
mass-to-light ratios obtained with the disk and the spheroid IMF yield values 1.8 to 1.4
smaller than for a Salpeter IMF, respectively, in agreement with various recent dynamical
determinations. This general IMF determination is examined in the context of star forma-
tion theory. None of the theories based on a Jeans-type mechanism, where fragmentation is
due only to gravity, can fulfill all the observational constraints on star formation and predict

– 3 –
a large number of substellar objects. On the other hand, recent numerical simulations of
compressible turbulence, in particular in super-Alfve´nic conditions, seem to reproduce both
qualitatively and quantitatively the stellar and substellar IMF, and thus provide an appeal-
ing theoretical foundation. In this picture, star formation is induced by the dissipation of
large scale turbulence to smaller scales, through radiative MHD shocks, producing filamen-
tary structures. These shocks produce local, non-equilibrium structures with large density
contrasts, which collapse eventually in gravitationally bound objects under the combined
influence of turbulence and gravity. The concept of a single Jeans mass is replaced by a dis-
tribution of local Jeans masses, representative of the lognormal probability density function
of the turbulent gas. Objects below the mean thermal Jeans mass still have a possibility to
collapse, although with a decreasing probability.

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1 Introduction
Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Mass-magnitude relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 The Galactic disk mass function
2.1 The field mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 The disk stellar luminosity function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 The disk stellar mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Correction for binaries. The disk system mass function . . . . . . . . . . . . . . . 17
2.1.4 The disk brown dwarf mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 The young cluster mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 The planetary mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 The Galactic spheroid mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4 The globular cluster mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 The dark halo and early star mass function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6 Galactic mass budget. Mass-to-light ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
7 The initial mass function theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
8 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

– 5 –
1. INTRODUCTION
1.1. Historical Perspective
Since the pioneering paper of Salpeter (1955), several fundamental reviews on the Galac-
tic stellar mass function (MF) have been written by, in particular, Schmidt (1959), Miller
& Scalo (1979, hereafter MS79), Scalo (1986). A shorter, more recent discussion is given by
Kroupa (2002). The determination of the stellar MF is a cornerstone in astrophysics, for the
stellar mass distribution determines the evolution, surface brightness, chemical enrichment,
and baryonic content of galaxies. Determinating whether this MF has been constant along
the evolution of the universe, or varies with redshift, bears crucial consequences on the so-
called cosmic star formation, i.e. on the universe light and matter evolution. Furthermore,
the knowledge of the MF in our Galaxy yields the complete census of its stellar and substel-
lar population, and provides an essential diagnostic to understand the formation of star-like
objects. As emphasized by Scalo (1986), the stellar and substellar mass distribution is the
link between stellar and galactic evolution.
As too rarely stressed, there is no direct observational determination of the MF. What
is observed is the individual or integrated light of objects, i.e. the luminosity function (LF)
or the surface brightness. Transformation of this observable quantity into the MF thus relies
on theories of stellar evolution, and more precisely, on the relationship between mass, age
and light, i.e. mass-age-luminosity relations.
Until recently, only the LFs of giants and sun-like stars, i.e. objects with mass m & 1M⊙,
were observed with enough precision to derive stellar MFs. These latter were presented
as power law approximations, dN/dm ∝ m−α, as initially suggested by Salpeter (1955),
with an exponent close to the Salpeter value α = 2.35. A departure from this monotonic
behaviour, with a flattening of the MF below∼ 1M⊙, was first proposed by MS79, suggesting
a lognormal form. The tremendous progress realized within the past few years from the
observational side, from both ground-based and space-based surveys, now probes the M-
dwarf stellar distribution down to the bottom of the main sequence (MS). Moreover, over
a hundred brown dwarfs have now been discovered, both in the Galactic field and in young
clusters, down to a few Jupiter masses, providing important constraints on the census of
substellar objects in the Galaxy. Not mentioning the ongoing detection of planets orbiting
stars outside our solar system. All these recent discoveries show unambiguously that the
stellar MF extends well below the hydrogen- and probably the deuterium-burning limit, and
urge a revised determination of the stellar and substellar census in the Galaxy, and thus of
its MF. In the meantime, the general theory of low-mass star and brown dwarf evolution has
now reached a mature state, allowing a reasonably robust description of the mechanical and

– 6 –
thermal properties of these complex objects, and of their observational signatures.
It is the purpose of this review to summarize these recent discoveries, to examine which
lessons from the Milky Way can be applied to a more general galactic and cosmological
context, and to determine our present knowledge of the Galactic MF and our present un-
derstanding of star formation. Detailed discussions on the MF of massive stars (m & 1M⊙)
have been developed in the remarkable reviews of MS79 and Scalo (1986) and we orient the
reader to these papers for these objects. The present review will focus on the low-mass part
of the MF in various regions of the Galaxy, and its extension into the substellar regime. Low-
mass stars (hereafter stars with mass m < 1M⊙) have effective temperatures Teff . 6000 K,
which implies eventually formation of molecules in their atmosphere. Below Teff ∼ 4000 K,
their spectral energy distribution strongly departs from a blackbody distribution, and peaks
generally in the visible or near-infrared, with yellow to red colors (see Chabrier & Baraffe
2000 for a recent review). These objects live for a Hubble time, or longer, and provide the
overwhelming majority of the galactic stellar contents.
The organization of the paper is as follows: in §1, we summarize the various defini-
tions used in the present review and we briefly summarize our present knowledge, and the
remaining uncertainties, of the mass-magnitude relationship. Sections 2-5 are devoted re-
spectively to the determination of the Galactic field and young clusters, Galactic spheroid,
globular cluster and dark halo and early star MFs. The stellar and substellar Galactic mass
budget and the cosmological implications are presented in §6. Examination of our present
understanding of star formation is discussed in §7, while §8 is devoted to the conclusion.
1.2. Definitions
1.2.1. Mass function
The MF was originally defined by Salpeter (1995) as the number of stars N in a volume
of space V observed at a time t per logarithmic mass interval d logm:
ξ(logm) =
d(N/V )
d logm
=
dn
d logm
(1)
where n = N/V is the stellar number-density, in pc−3 in the following.

– 7 –
This definition was used also by MS791 and Scalo (1986). Since the formation of star-
like objects is now observed to take place over five orders of magnitude in mass, from about
100 to 10−3 M⊙, such a logarithmic definition of the MF seems to be the most satisfactory
representation of the mass distribution in the Galaxy. Conversely, Scalo (1986) defines the
mass spectrum as the number density distribution per mass interval dn/dm with the obvious
relation:
ξ(m) =
dn
dm
=
1
m (Ln 10)
ξ(logm) (2)
With these definitions, if the MF is approximated as a power law, the exponents are
usually denoted respectively x and α, with ξ(logm) ∝ m−x and ξ(m) ∝ m−α, x = α − 1.
The original Salpeter value is x = 1.35, α = 2.35.
Stars eventually evolve off the main sequence (MS) after a certain age, so that the
present-day MF (PDMF) of MS stars, which can be determined from the observed present-
day LF, differs from the so-called initial mass function (IMF), i.e. the number of stars which
were originally created per mass-interval in the Galaxy. Indeed, stars with masses above
the minimum so-called ”turn-off” mass will have evolved as red giants and white dwarfs, or
neutron stars or black holes as the end product of type-II supernovae explosion, depending
on the initial MS mass. Here the minimum turn-off mass is defined as the mass for which
the age at which the star starts evolving off the MS on the giant branch equals the age of the
Galaxy (or the age of a given cluster). For an age τD ≈ 10 Gyr, about the age of the Galactic
disk, this corresponds to a mass mTO ≈ 0.9M⊙ for solar metallicity. The determination of
the PDMF thus involves the stellar formation rate (SFR) b(t), i.e. the number of stars
(more generically star-like objects) formed per time interval along galactic evolution. For
this reason, the quantity to be considered to link the PDMF and IMF is the so-called stellar
creation function, as introduced by MS79.
1.2.2. Creation function
The creation function C(log m, t) is defined as the number of stars per unit volume
formed in the mass range [logm, logm + d logm] during the time interval [t, t + dt]. Given
1Note that Miller & Scalo use the stellar surface-density, in pc−2, in their definition of the MF, to be
divided by the respective Galactic scale heights of the various stellar populations to get the volume density
in the solar neighborhood.

– 8 –
this definition, the total number of star-like objects per unit volume ever formed in the
Galaxy reads:
ntot =
∫ log(msup)
log(minf )
∫ τG
0
C(logm, t) d logmdt (3)
where minf and msup denote respectively the minimum and maximum mass for the formation
of star-like objects, and τG denotes the age of the Galaxy2.
The creation function is related to the total birthrate B(t), i.e. the total number-density
of star-like objects ever formed per unit time, as:
B(t) =
∫ log(msup)
log(minf )
C(logm, t) d logm (4)
Following MS79, we refer to the star-like formation rate (SFR), as the ratio of the
absolute birthrate at time t over the average birthrate:
b(t) =
B(t)
1
τG
∫ τG
0 B(t) dt
(5)
so that
∫ τG
0 b(t) dt = τG.
It is generally admitted that the creation function can be separated into the product of
a function of mass - the mass function - and a function of time - the formation rate. The
underlying physical hypothesis is that the MF, the issue of the physical process which drives
star formation per mass interval, does not depend on time. In fact, as will be illustrated
later on, time may play a role in this mechanism in determining some characteristic mass,
but without affecting the generic form of ξ(m). Under such a condition of separability of
mass and time, the creation function C(m, t) can be rewritten:
C(logm, t) = ξ(logm)× B(t)∫ τG
0 B(t) dt
= ξ(logm)× b(t)
τG
(6)
2Or the age of a given cluster if one wants to determine a cluster stellar content.

– 9 –
The IMF, i.e. the total number-density of star-like objects ever formed per unit log
mass, thus reads:
ξ(logm) =
∫ τG
0
C(logm, t) dt (7)
From the definitions (3) and (5), the IMF and the SFR are related to the total number-
density of star-like objects ever formed in the Galaxy by:
ntot(t = τG) =
1
τG
∫ τG
0
b(t)dt×
∫ log(msup)
log(minf )
ξ(logm)d logm =
∫ log(msup)
log(minf )
ξ(logm)d logm (8)
As noted by MS79, all stars with MS lifetimes greater than the age of the Galaxy are
still on the MS. In that case the PDMF and the IMF are equivalent. This holds for brown
dwarfs (BD) too. Brown dwarfs have unlimited lifetimes so that all BDs ever formed in the
Galaxy still exist today, regardless on when they were formed, and the BD PDMF is the BD
IMF. For stars with MS lifetimes τMS less than the age of the Galaxy, only those within the
last τMS are observed today as MS stars. In that case the PDMF φMS(log m) and the IMF
ξ(logm) are different, and - using the separability condition for the creation function - obey
the condition (MS79):
φMS(log m) =
ξ(logm)
τG
×
∫ τG
τG−τMS
b(t) dt τMS < τG (9)
1.2.3. Functional forms
The most widely used functional form for the MF is the power law, as suggested origi-
nally by Salpeter (1955):
ξ(log m) = Am−x (10)
This form is believed to adequately describe the IMF of massive stars in our Galaxy,
m & 1M⊙, with an exponent x ≃ 1.7 (Scalo 1986, Table VII), for a standard fraction of

– 11 –
∫ msup
minf
p(m) dm = 1 (14)
with the probability for a star to have a mass ∈ [minf , m]:
P (m) =
∫ m
minf
p(x) dx =
1
ntot
∫ m
minf
ξ(x) dx (15)
1.3. Mass-magnitude relationships
The only possible direct determination of a stellar mass is by use of Kepler’s third law
in a binary system, providing a long enough time basis to get the appropriate dynamical
information. As shown below, the statistics of such a sample is largely insufficient to allow a
reasonable estimate of the MF, but it certainly provides stringent constraints for the models.
Therefore, as mentioned earlier, the only possible way to determine a PDMF is by transfor-
mation of an observed LF Φ = dN/dM , i.e. the number of stars N per absolute magnitude
interval dM , into a MF. This involves the derivative of a mass-luminosity relationship, for a
given age τ , or preferentially of a mass-magnitude relationship (MMR) which applies directly
in the observed magnitude and avoids the use of often ill-determined bolometric corrections:
dn
dm
(m)τ = (
dn
dMλ(m)
)× ( dm
dMλ(m)
)−1τ (16)
where Mλ denotes the absolute magnitude in a given bandpass. Another way to proceed is to
attribute a mass to each star of the sample, which avoids involving explicitly the derivative
of the MMR. In practice, both methods should yield similar results.
A first compilation of mass-luminosity data in the M-dwarf domain was published by
Popper (1980), and was subsequently extended by Henry & McCarthy (1993), who used
speckle interferometry to obtain MMRs in the V, J, H and K-bands. The determination of
the V-magnitude was improved subsequently with the HST (Henry et al. 1999), reducing
appreciably the uncertainty in the m-MV relation. This sample has been improved signifi-
cantly recently by Delfosse et al. (2000) and Se´gransan et al. (2003b). Combining adaptative
optic images and accurate radial velocities, these authors determined the MMR of about 20
objects between ∼ 0.6 and ∼ 0.09 M⊙ in the aforementioned bands with mass accuracies of
0.2 to 5%.

– 14 –
(MV < 9.5), which defines the Gliese Catalogue of Nearby Stars, a few parsecs for the faint
end. For the faintest M-dwarfs, the estimated completeness distance is rcompl ≈ 5 pc (Henry
et al. 1997). This yields the so-called nearby LF Φnear. The main caveat of the nearby
LF is that, given the limited distance, it covers only a limited volume and thus a limited
sample of objects. This yields important statistical undeterminations at faint magnitudes
(MV & 12). On the other hand, a fundamental advantage of the nearby LF, besides the
reduced error on the distance, and thus on the magnitude, is the accurate identification of
binary systems. A V-band nearby LF can be derived by combining Hipparcos parallax data
(ESA 1997), which is essentially complete for MV < 12 at r=10 pc, and the sample of nearby
stars with ground-based parallaxes for MV > 12 to a completeness distance r=5.2 pc (Dahn
et al. 1986). Henry & MCarthy (1990) used speckle interferometry to resolve companions of
every known M-dwarf within 5 pc and obtained the complete M-dwarf LF Φnear in the H and
K bands. Their sample recovers the Dahn et al. (1986) one, plus one previously unresolved
companion (GL 866B). Reid and Gizis (1997) and more recently Reid, Gizis and Hawley
(2002) extended this determination to a larger volume and determined a nearby LF based
on a volume sample within about 8 pc. It turns out that, down to the limit of completeness
claimed for this sample, MV ∼ 14, the two LFs agree reasonably well.
Other determinations of the disk LF are based on photographic surveys, which extend
to d ≈ 100-200 pc from the Sun, and thus encompass a significantly larger amount of stars.
However, photometric LFs Φphot suffer in general from significant Malmquist bias and, as
mentioned above, the low spatial resolution of photographic surveys does not allow the
resolution of binaries at faint magnitudes. An extensive analysis of the different nearby and
photometric LFs has been conducted by Kroupa (1995). As shown by this author, most of
the discrepancy between photometric and nearby LFs for MV > 12 results from Malmquist
bias and unresolved binary systems in the low-spatial resolution photographic surveys (see
also Reid & Gizis (1997) for an alternative point of view). A recent determination of Φphot has
been obtained with the HST (Gould, Bahcall & Flynn 1997, hereafter GBF97), which extends
to an apparent magnitude I . 24. The Malmquist bias is negligible because all stars down to
∼ 0.1M⊙ are seen through to the edge of the thick disk. A major caveat of any photometric
LF, however, is that the determination of the distance relies on a photometric determination
from a color-magnitude diagram. The former analysis of the HST data (GBF97) used for the
entire sample a color-magnitude transformation characteristic of stars with solar-abundances.
A significant fraction of the sample probed by the HST, however, lies at galactic scale height
‖z‖ & 1 kpc (Zheng et al. 2001, Fig. 2) and thus belongs to the thick-disk population, and
is likely to have metal-depleted abundances −0.5 . [M/H] . 0. Assuming solar metallicity
for the entire sample results in an underestimate of the absolute magnitude for a given color
(the lower the metallicity, the fainter the absolute magnitude for a given color), and thus

– 15 –
an overestimate of the distance and an underestimate of the number density, in particular
near the faint end of the LF. An extension and a reanalysis of the HST sample, taking into
account a statistically weighted metallicity gradient along the Galactic scale height, and a
related color-magnitude-metallicity relationship, yields a revised ΦHST (Zheng et al. 2001),
with indeed a larger number of M-dwarfs at faint absolute magnitude. However, because of
its limited angular resolution (< 0.1′′), the HST misses all the binaries, and the LF must be
corrected from this caveat to yield a single LF (§2.1.3 below).
In practice, the determination of the MF from the LF implies the knowledge of each
star chemical composition, since the colors and the magnitude depend on the metallicity.
This metallicity spread translates into a spread in the LF and in the MF. Analysis of the
Hipparcos color-magnitude diagram, however, indicates that ∼ 90% of the thin-disk stars
have abundances [M/H] = 0 ± 0.2 (Reid 1999), so the spread of metallicity in the solar
neighborhood should not affect significantly the derivation of the MF through the MMR.
The magnitude of an object, however, varies with age, so the determination of its mass
through a theoretical MMR necessitates the knowledge of its age. The luminosity of MS
stars above m ∼ 0.7M⊙ starts increasing substantially after ∼ 10 Gyr, about the age of
the Galactic disk. On the other hand, objects below m ∼ 0.13M⊙ (for a solar composition)
take more than 5×108 years to reach the main sequence (see Table 1 of Chabrier & Baraffe
2000). Therefore, for a constant SFR and a age of the disk τD = 10 Gyr, at most ∼ 5%
of the nearby stars in the mass range 0.13 ≤ m/M⊙ ≤ 0.7 might still be contracting on
the pre-main sequence, an uncertainty well within the statistical observational ones. Within
this mass-range, the position of the star is fixed in the mass-luminosity diagram. Below 0.1
M⊙, and in particular in the BD domain, age variations must be taken into account, within
a given SFR, for a proper determination of the MF from the observed star or BD counts
(Chabrier 2002).
2.1.2. The disk stellar mass function
Recently, Chabrier (2001) has determined the Galactic disk M-dwarf MF from the 5-pc
and 8-pc Φnear. He has shown that, although still rising down to the H-burning limit, the
IMF ξ(log m) starts shallowing with respect to the Scalo or Salpeter value below ∼1 M⊙
and flattens off below ∼0.3 M⊙, as noted previously by MS79 and Kroupa et al. (1993).
Combining the M-dwarf MF with the Scalo (1986) power law for masses above 1 M⊙, and
fulfilling the so-called continuity condition for stars with τTO ≈ τG (MS79), i.e. m ≃ 0.9M⊙,
Chabrier (2001) showed that the MF is well described over the entire stellar mass range,
from about 100 M⊙ to 0.1 M⊙, by any of the functional forms mentioned in §1.2.3, i.e. a

– 16 –
two-segment power law, a lognormal form or an exponential (Rosin-Rammler) form. This
analysis has been completed by Chabrier (2003), who has calculated the MF from the nearby
LF Φnear obtained both in the V-band (Dahn et al. 1986) and in the K-band (Henry &
McCarthy 1990). Figure 1 displays such a comparison. The conversion of the V-band LF
into an MF was done using the Delfosse et al. (2000) m-MV relation, which fits the observed
data, whereas the BCAH98 m-MK relation was used to convert the K-band LF. We note the
very good agreement between the two determinations, which establishes the consistency of
the two observed samples and the validity of the mass-magnitude relationships. The ∼1.5-σ
difference in the mass range logm ∼ -0.5 to -0.6, i.e. MV ∼ 12-13, reflects the remaining
uncertainties either in the MMR or in the LF Φnear. The MF derived from the new V-band
LF or Reid et al. (2002), not displayed in the figure, closely resembles the one derived from
the Henry & McCarthy (1990) K-band LF. The solid line displays an analytic form which
gives a fairly good representation of the results. The uncertainties in the MF are illustrated
by the surrounding dashed lines. This analytic form for the disk MF for single objects below
1 M⊙, within these uncertainties, is given by the following lognormal form (Chabrier 2003):
ξ(log m)m≤1 = 0.158+0.051−0.046 × exp
{
−(log m − log 0.079
−0.016
+0.021)
2
2× (0.69−0.01+0.05)2
}
(logM⊙)−1 pc−3 (17)
The derivation of this MF from the Hipparcos and local sample provides the normaliza-
tion at 0.7 M⊙: ( dndm)0.7 = 3.8 × 10−2 M−1⊙ pc−3, with at most a 5% uncertainty. Age effects
above 0.7 M⊙ are illustrated in the figure by the empty circles and empty squares which
display the MF obtained with the MMR for t = 10 Gyr and 1 Gyr, respectively, whereas
the solid circles correspond to t = 5 Gyr, the average age for the Galactic thin-disk3. As
mentioned previously, age effects become negligible below m = 0.7 M⊙. The dotted line
displays part of the 4-segment power-law MF derived by Kroupa (2002). This MF slightly
overestimates the M-dwarf density.
Note that eq.[17] yields the Scalo (1986) normalization for 5 Gyr at 1 M⊙, ( dndm)1.0 =
1.9 × 10−2 M−1⊙ pc−3, which corresponds to a condition MV = 4.72 for m = 1.0M⊙ at 5 Gyr.
As shown by Scalo (1986) and illustrated in Figure 1, 1 M⊙ is about the limit for which the
disk PDMF and IMF start to differ appreciably, so that only the m > 1M⊙ power-law part
of the MF will differ, depending whether the IMF (x = 1.3± 0.3) or the PDMF (x given by
eq.[11]) is considered. This yields the global disk PDMF and IMF, as summarized in Table 1.
As mentioned earlier, substantial uncertainty remains in the value of x at large masses for the
3The vast majority of stars in the Galactic mid-plane belong to the old-disk (h ∼ 300 pc, τ ∼ 5 Gyr),
and about 20% to the young disk (h ∼ 100 pc, τ ∼ 1 Gyr) (see e.g. Gilmore & Reid 1983).

– 18 –
al. 2001) HST LF. This author has shown that the MF derived from the revised HST LF (i)
is very similar to the local so-called system MF, i.e. the MF derived from the local LF once
the companions of all identified multiple systems have been merged into unresolved systems,
(ii) is consistent with the single MF (eq.[17]) providing a binary fraction X ≈ 50% among
M-dwarfs, with the mass of both the single objects and the companions originating from
the same single MF (eq.[17]). This multiplicity rate implies that about ∼ 30% of M-dwarfs
have a stellar (M-dwarf) companion, whereas about ∼ 20% have a substellar (BD) compan-
ion (Chabrier 2003), a result in agreement with present-day determinations of the M-dwarf
binary fraction in the solar neighborhood (Marchal et al. 2003) and of BD companions of
M-dwarfs (Gizis et al. 2001, Close et al. 2003). This system MF can be parametrized by
the same type of lognormal form as the single MF (eq.[17]), with the same normalization at
1 M⊙, with the coefficients (Chabrier 2003):
ξ(log m)m≤1 = 0.086× exp
{
−(log m − log 0.22)
2
2× 0.572
}
(logM⊙)−1 pc−3 (18)
and is displayed by the long-dash line in Figure (2).
These calculations show that the disk stellar IMF determined from either the nearby
geometric (parallax) LF or the HST photometric LF are consistent, and that the previous
source of disagreement was due to two effects, namely (i) incorrect color-magnitude deter-
mined parallaxes, due to the fact that a substantial fraction of the HST M-dwarf sample
belongs to the metal-depleted, thick-disk population and (ii) unresolved binaries. As dis-
cussed in Chabrier (2003), these results yield a reinterpretation of the so-called ”brown
dwarf desert”. This latter expresses the lack of BD companions to solar-type stars (G-K),
as compared with stellar or planetary companions, at separation of less than 5 AU (Marcy,
Cochran & Mayor 2000). Indeed, proper motion data from Hipparcos have revealed that
a significant fraction of low-mass companions in the substellar regime have low-inclination
and thus larger, possibly stellar masses (Marcy et al. 2000, Halbwachs et al. 2000). Cor-
rection for this inclination yields a deficit of small-separation BD companions, the so-called
BD desert, suggesting that the MF of substellar companions to solar-type stars, at least at
separations less than 5 AU, differs significantly from the one determined for the field. The
present calculations, however, show that this ”desert” should be reinterpreted as a lack of
high-mass ratio (q = m2/m1 . 0.1) systems, and does not preclude a substantial fraction of
BDs as companions of M-dwarfs or other BDs, as suggested by recent analysis (Marchal et
al. 2003, Burgasser et al. 2003, Close et al. 2003).

– 19 –
2.1.4. Disk brown dwarf mass function
As shown in §2.1.2, the IMF (eq.[17]) gives a good representation of the stellar regime
in the disk down to log m ∼ −0.9 (m ∼ 0.12M⊙), where all objects have reached the
MS. This IMF, which closely resembles the IMF2 derived in Chabrier (2001), gives also
a good description of the star counts in the deep field of the ESO Imaging Survey (EIS)
(Groenewegen et al. 2002), better than the power-law forms of Kroupa (2001) or IMF1 of
Chabrier (2001). It has been shown also to agree fairly well with the L-dwarf and T-dwarf
BD detections of various field surveys (Chabrier 2002). Figure 3 displays the predicted BD
luminosity functions (BDLF) in the K-magnitude and in terms of fundamental parameters
(Teff , L/L⊙) from the bottom of the MS over the entire BD domain. These BDLFs were
obtained fromMonte Carlo simulations, with mass, age and distance probability distributions
as described in Chabrier (2002). Only the case of a constant SFR has been considered
presently. The various dashed and dotted lines display the relative contributions of BDs (m ≤
0.072M⊙ for solar metallicity, Chabrier & Baraffe 1997), T-dwarfs, identified as faint objects
with (J-H)< 0.5, (H-K)< 0.5 (Kirkpatrick et al. 2000) and objects below the deuterium-
burning limit (m ≤ 0.012M⊙). The predictions are compared with various available data,
namely the nearby K-band LF (Henry & McCarthy 1990), converted into a bolometric LF
on the bottom panel with the M-dwarf bolometric corrections of Tinney et al. (1993) and
Leggett et al. (1996), the L-dwarf density estimate of Gizis et al. (2000), and the L-
dwarf and T-dwarf estimated densities of Kirkpatrick et al. (1999, 2000) and Burgasser
(2001). It is important to mention that the Vmax and thus the explored volume and density
determinations for BD surveys are a very delicate task, affected by numerous uncertainties
(see Burgasser 2001). A ∼ 1 mag uncertainty in the maximum limit of detection translates
into a factor of∼ 4 in the Vmax and thus in the estimated density Φ = ΣV −1max. Not mentioning
difficult completeness corrections for such surveys. Furthermore, the observational Teff
and bolometric correction determinations remain presently ill-determined for BDs. On the
other hand, theoretical models of BD cooling, although now in a mature state, are still far
from including all complex processes such as dust sedimentation, cloud diffusion, or non-
equilibrium chemistry. For all these reasons, the present results should be considered with
caution. The BDLFs calculated with the IMF (eq.[17]) yield a very good agreement with
the determinations of Kirkpatrick et al. (1999, 2000) but seem to overestimate by a factor of
about 3 the density of L-dwarfs obtained by Gizis et al. (2000) and the number of bright T-
dwarfs observed by Burgasser (2001). The decreasing number of L-dwarfs in the Kirkpatrick
et al. (1999) survey at bright magnitudes is due to their color selection (J-K > 1.3). Given
all the aforementioned uncertainties, the comparison between the observed and predicted
LFs can be considered as satisfactory. This assesses the validity of the present disk IMF
determination in the BD regime.

– 20 –
The factor ∼ 3 overestimate of the predicted LF, if confirmed, might stem from various
plausible explanations. First, this might indicate too high a normalization of the IMF near
the bottom of the MS, due to the presence of hot BDs, misidentified as MS very-low-mass
stars, in the faintest bins of the nearby LF. However, as seen from the top panel of Figure
3, the contribution of young BDs to the local LF is zero for MK ≤ 9, which corresponds to
a ∼0.12 M⊙ MS M dwarf, and thus does not affect the MF normalization at this mass. An
alternative, similar explanation would be the presence of a statistically significant number of
very-low-mass stars younger than 108 yr, still contracting on the PMS, in the local sample.
This implies a small scale height for these objects. Indeed, for a constant SFR and a young-
disk age τ ≃ 1 Gyr, the probability to find an object with t < 108 yr is ∼10%, for a
homogeneous sample. Only redoing the same observations in a few hundred million years
could help resolving this issue, a rather challenging task ! A second possible explanation could
be unresolved BD binaries. The dotted line in the bottom panel of Figure 3 displays the LF
obtained with the IMF (eq. [18]), illustrating the effect due to BD unresolved systems. The
effect of unresolved binaries on the BDLF is much more dramatic than on the stellar LF. This
stems from the much steeper mass-magnitude relationship at a given age in the BD regime.
At 1 Gyr, a factor of 2 in mass corresponds to about ∼ 2 mag difference in the stellar regime,
against ∼ 4 mag or more in the BD regime. On the other hand, remember that the difference
between the single (eq.[17]) and system (eq.[18]) MFs assumes a binary correction X ≈ 50%
among stellar objects. Extrapolating these corrections into the BD domain assumes that
the binary rate in star formation does not depend on the mass of the primary. If the
present discrepancy between theory and observation is confirmed, it might indicate that this
frequency is significantly smaller in the BD regime (with X . 20%) due for example to
the fact that very-low-mass systems cannot form with large mass ratio (q = m2/m1 << 1)
or with large orbital separations (see e.g. Burgasser et al. 20034). Again, long time basis
observations are mandatory to nail down this issue. A third, appealing explanation for the
factor 3 discrepancy might be substantial incompleteness of the present BD surveys resulting
from selection effects. Salim et al. (2003) estimate that some 40% of bright L-dwarfs are
missed because they lie close to the Galactic plane, a region avoided by most searches. This
correction factor would bring the present theoretical predictions in perfect agreement with
the observational BD census. Finally, the present factor ∼ 3 disagreement between the
predicted and observationally-derived counts might just reflect the remaining imperfections
in BD cooling models.
It is interesting to note the bimodal form of the stellar+BD LFs. The stellar LF peaks
4Note that the BD binaries observed by Burgasser et al. (2003) have an orbital separation > 1 A.U., and
do not include BD systems with smaller separation such as PPL15

– 22 –
clusters. Second of all, extinction and differential reddening caused by the surrounding dust
in star forming regions modifies both the intrinsic magnitude and the colors of each individ-
ual object, preventing direct mass-magnitude-color determinations and making photometric
determinations very uncertain, not mentioning the amount of accretion which varies signif-
icantly from an object to an other (see e.g. Comero´n et al. 2003). Moreover, the near-IR
excess of embedded young clusters associated with hot circumstellar dusty disk complicates
significantly the interpretation of near-IR luminosity into stellar luminosity functions. Third
of all, some dynamical evaporation may have taken place, rejecting low-mass objects to the
periphery of the cluster, where contamination from field stars become important. This holds
even for very young clusters (< 1 Myr) which contain O stars, like e.g. the Orion Nebula
cluster (Kroupa, Aarseth & Hurley 2001). Finally, there is presently no appropriate effective
temperature calibration for gravities characteristic of PMS M-dwarfs, yielding people to rely
on empirical Teff -spectral type (Sp) determinations, as discussed below.
From the theoretical point of view, accurate models must include gravity effects, which,
for young objects, affect both the spectrum and the evolution (Baraffe et al. 2002). As shown
by these authors, no theoretical model is presently reliable for ages younger than ∼ 106 yr.
At such young ages, the evolution is severely affected by several uncertainties, like e.g. the
unknown convection efficiency (and thus mixing length parameter), the accretion rate, the
deuterium abundance, not mentioning the fact that at these ages the models are affected
by the (arbitrary) initial conditions. As shown by Baraffe et al. (2002), the evolution along
the contracting PMS phase for t . 106 yr depends not only on the (unknown) efficiency
of convection but also, for the coolest objects, on the formation of molecular hydrogen H2
in the atmosphere. Both effects affect significantly the evolution. Therefore, assuming a
constant Teff evolution for a given mass in a HR diagram for young, very-low-mass objects,
as done sometimes in the literature, may lead to inaccurate mass determinations and the
inferred IMFs must be considered with great caution. In fact, 3D calculations are necessary
to determine accurately the entropy profile of objects in the initial accreting, gravitational
contracting phase, for 1D collapse calculations yield erroneous results (Hartmann et al.
1997, Hartmann 2003, Baraffe et al. 2003). No such consistent calculation and thus no
reliable temperature and mass calibration exists today for low-mass PMS stars. Only for
ages t & 106 yr, do these uncertainties disappear, or at least become less important, and can
reasonably reliable PMS models be calculated (Baraffe et al. 2002). Finally, as pointed out
by Luhman et al. (2000), what is really observed in star forming clusters is not the IMF
but the creation function (see §1.2.2). The same underlying IMF convolved with different
age distributions will yield different LFs, a result which can be misinterpreted as originating
from different IMFs. Conversely, assuming a single, median age for objects in star-forming
regions, where the typical age spread can reach a few Myr, yields an IMF of limited validity,

– 29 –
1980) survey of southern (δ < 30) stars with µ > 0.7
′′
yr−1, might be underestimated by
∼ 30%, due to incompleteness of Eggen’s sample in the Galactic Plane.
Differences between the HST and nearby LFs might be due to the HST small field of
view. On the other hand, it is generally admitted that the spheroid is substantially flattened,
with q ∼ 0.7, so that most of the local subdwarfs would reside close to the disk, and this
population would not be included in the HST sample (see e.g. Digby et al. 2003). Sommer-
Larsen & Zhen (1990) estimate this subdwarf fraction to be about 40%. For this reason the
local normalization of the spheroid subdwarf density from the number-density observed at
large distances from the plane, as done with the HST (GFB98), is a very uncertain task.
We note also some difference, at the ∼ 2σ level, between the Dahn et al. (1995, 2002) LF
and the NLTT one (Gould 2003), this latter rising more steeply and peaking at a ∼ 1 mag
brighter magnitude. The reason for such a difference is unclear. It might stem from the
limited statistics in the Dahn et al. survey (∼ 10 to 30 stars per bin in the MV =9-12
range) or from the simple color-magnitude relations adopted by Gould (2003). On the other
hand, Dahn et al. used a purely kinematic criterion to select halo objects in their sample.
As acknowledged by these authors themselves, this undoubtedly rejects bona fide spheroid
subdwarfs due to their directional locations in the sky. Such an even small correction might
be consequential in the last bins. Uncompleteness of the LHS Catalogue at faint magnitude
would also affect the faint part of the LF. All these uncertainties must be kept in mind when
considering the present results.
The spheroid population can also be identified photometrically, which strongly correlates
with metallicity. Figure 7 displays the 114 spheroid stars identified in the Dahn et al. (1995)
survey in a MV -(V -I) color-magnitude diagram as well as the observed thin-disk M-dwarf
sequence (Monet et al. 1992, small dots) and superimposed to the observations five 10 Gyr
isochrones with metallicities [M/H] =-2.0, -1.5, -1.3, -1.0 and -0.5, respectively. Recall that
these isochrones reproduce accurately the observed sequences of various globular clusters of
comparable metallicity, except for the more metal-rich ones ([M/H] & −1.0) (see BCAH97
and §1.3). This figure clearly shows that the kinematically-identified spheroid subdwarf
population covers a wide range of metallicities, from ∼1% solar to near-solar, with an average
value 〈[M/H]〉 ≃ -1.0 to -1.3, i.e. [Fe/H] ≃ -1.7 to -1.49 (see also Fuchs, Jahreiss & Wielen
1999). Such a large dispersion remains unexplained and is at odd with a burst of star
formation in the spheroid ∼ 10 to 12 Gyr ago. Accretion during the star orbital motion across
the disk is unlikely. A Bondi-Hoyle accretion rate, most likely an upper limit except possibly
during the early stages of evolution, yields m˙acc ≈ 2pi(Gm)2nmH/v3 ≈ 2.6×10−19(m/M⊙)2×
9For metal depleted objects, a metallicity [M/H] corresponds to an iron to hydrogen abundance [Fe/H ] ≃
[M/H]− 0.35, due to the α-element enrichment (see BCAH97).

– 30 –
(n/1 cm−3)(v/220 km.s−1)−3 M⊙yr−1, i.e. macc . 10−9 M⊙, for subdwarf masses, in 10 Gyr
(here n is the density of the ISM and v the velocity of the star). An alternative possibility
is a metallicity and velocity gradient along the spheroid vertical structure above the disk.
In that case, the subdwarfs discovered with the HST should be more metal-depleted than
the one in the local sample. Recent observations (Gilmore et al. 2002) have detected a
substantial population of stars a few kpc above the Galactic disk with kinematic properties
(rotational velocity and velocity dispersion) intermediate between the canonical thick disk
and the spheroid. These authors interpret this ”vertical shear” as an extension of the thick
disk, caused by the ancient merging of a nearby galaxy. This interpretation confirms the
previous analysis of Fuchs et al. (1999) and is supported by the recent analysis of Fuhrmann
(2002) who finds that the majority of subdwarfs within 25 pc from the Sun with large space
velocities ((U2 + V 2 + W 2)1/2 & 100 km.s−1) have a chemical composition characteristic of
the thick disk ([Fe/H] . −0.5, [Fe/Mg] ≈ −0.5). If this interpretation is confirmed, this
implies a substantial revision of the thick-disk and spheroid models. In that case, the local
subdwarf sample and the one observed with the HST probe two different stellar populations.
In particular, as discussed by BC86, the inclusion of a few stars with high-velocity belonging
to this extended thick-disk population in the local genuine spheroid subdwarf sample will
yield a severe overestimate of the supposed spheroid density. Until this issue is solved, we
will assume that the NLTT (Gould 2003) or LHS (Dahn et al. 1995, 2002) samples are
representative of the spheroid one10, keeping in mind that these samples may include a
fraction of thick-disk stars with high dispersion velocities. Such an assumption yields the
maximum mass contribution and local normalization of the Galactic spheroidal component.
A correct analysis of the subdwarf metallicity would require a statistical approach but
the metallicity probability distribution for these stars is presently unknown, and the deriva-
tion of such a distribution from a two-color criterion only is of weak significance. For this
reason, we have converted the observed LFs of Figure 6 into MFs, based on the BCAH97
mass-MV relationships, assuming that all stars have a given metallicity. In order to estimate
the uncertainty on the MF due to possible metallicity variations, we have used m-MV re-
lationships for [M/H] = -1.5, -1.0 and -0.5, respectively. Figure 8 displays the MF derived
from the NLTT LF for these three metallicities. To illustrate the uncertainty due to the
different LFs, the MF derived from the Dahn et al. (1995, 2002) LF is also shown, for
[M/H] =-1.0 (dot-line). Interestingly enough, the differences between the MFs derived for
the three different metallicities remain modest, a consequence of the limited effect of metal-
10Note that the NLTT LF of Gould (2003) is in agreement with the one derived recently from a detailed
reduced proper motion analysis of the Sloan and SuperCosmos surveys (Digby et al. 2003), presumably
probing the genuine spheroid subdwarf population.

– 31 –
licity, in the aforementioned range, on the slope dMV /dm of the MMR (BCAH97). The
effect is dominant at the low-mass end of the MF: the lower the metallicity the steeper the
MF. The wiggly behaviour of the MF derived from the Dahn et al. LF, with a peak around
logm = −0.2 followed by a dip, stems from the flattening behaviour of both their LF and
the m-MV relation in the MV ≈ 8-10 range.
The MF is reasonably well described by the following lognormal form below 0.7 M⊙,
illustrated by the solid line in Figure (8):
ξ(log m) = 3.6× 10−4 exp{− [log m − log(0.22± 0.05)]
2
2× 0.332 }, m ≤ 0.7M⊙ (20)
Although an IMF similar to the disk one (eq.[17] ) can not be totally excluded, in
particular if the Dahn et al. (1995, 2002) LF happens to be more correct than the Gould
(2003) one, equation [20] gives a better representation of the data. For comparison, the
IMF derived from the HST LF decreases as a straight line ξ(log m) ∝ m0.25 below 0.7 M⊙
(GFB98), a consequence of the much smaller number of faint subdwarfs detected by the
HST, as noted previously (Figure 6).
For an age t > 10 Gyr, a lower limit for the spheroid, stellar evolution either on or off
the MS affects objects with mass m & 0.7M⊙, i.e. MV . 6. Objects brighter than this
magnitude must then be ignored for the IMF determination and normalization. The shape
of the IMF above the turn-off mass (. 0.9M⊙ for t > 10 Gyr), is undetermined. Various
analysis of the high-mass part of the IMF in the LMC, SMC and in various spheroidal
galaxies (Massey 1998 and references therein) seem to be consistent with a Salpeter slope,
for all these metal-depleted environments. We thus elected to prolongate the IMF (eq.[20])
by such a power-law, with a common normalization at 0.7 M⊙, yielding the global spheroid
IMF given in Table 2.
Equation [20] yields a normalization at 0.70 M⊙ : ξ(logm)0.7 = (1.13±0.5)×10−4 (logM⊙)−1 pc−3.
This yields a spheroid main-sequence star number-density nMS ≃ (2.4±0.1)×10−4 pc−3 and
mass density ρMS ≃ (6.6±0.7)×10−5 M⊙ pc−3. Note that this value is more than twice larger
than the determination of GFB98, due to the different IMFs, as mentioned above. Note that
we assumed that the power-law form extends to 0.7 M⊙. Given the unknown slope of the IMF
for spheroid stars in this mass range, we could as well have chosen 0.9 M⊙ for the limit of the
power-law part of the IMF and have extended the lognormal form to this limit. The difference
in the derived densities, however, is small and largely within the present uncertainties of the
IMF. Integration of this IMF in the substellar regime yields a negligible BD number-density
nBD . 3× 10−5 pc−3 and mass-density contribution ρBDsph . 2× 10−6 M⊙ pc−3.

– 35 –
the twelve deep LFs on the cluster dynamical history, in spite of the very different cluster
conditions. With the noticeable exception of NGC6712 (De Marchi et al. 1999) which is
probably close to complete disruption. These results are consistent with the fact that the
12 clusters examined by Paresce and De Marchi (2000) are located well inside the survival
boundaries of the vital diagrams obtained from numerical simulations (Gnedin & Ostriker
1997). This suggests that the clusters probably remained undisturbed in their internal parts.
Therefore the MF measured near the half-light radius for these clusters should resemble very
closely the IMF.
The most striking conclusion of the study of Paresce and De Marchi (2000) is that
(i) a single power-law MF can not reproduce both the bright part and the faint part of
the observed LFs, (ii) the PDMFs derived for all the clusters are consistent with the same
lognormal form peaked at mc = 0.33± 0.03M⊙, with a standard deviation σ = 0.34± 0.04,
the error bars illustrating the variations between all clusters:
ξ(log m) ∝ exp{−(log m − log (0.33± 0.3))
2
2× (0.34± 0.04)2 }, m ≤ 0.9M⊙ (23)
The limit ∼ 0.8-0.9 M⊙ corresponds to the turn-off mass for an age t ≈ 10 Gyr for metal-
depleted environments.
This MF is displayed by the dash-line in Figure 9, superimposed to the MFs derived in
the present paper with the BCAH97 MMRs of appropriate metallicity from the 12 cluster LFs
observed both in optical (WFPC) and infrared (NICMOS) colors with the HST by Paresce
and De Marchi (2000). These observations are in excellent agreement with observations of
the same clusters by other groups. As cautiously stressed by Paresce and De Marchi (2000),
however, only 4 cluster sequences extend significantly beyond the peak of the LF, which
corresponds to a mass m ≈ 0.3M⊙. Future large, deep field surveys, for example with the
HST Advanced Camera (ACS), are necessary to make sure that all the cluster IMFs are
adequately reproduced by the aforementioned IMF. Since this latter, however, adequately
reproduces the bright part of the LF, i.e. the upper part of the IMF, there seems to be
no reason why significant departures would occur at the lower part. Except if - contrarily
to what seems to have been established from either Michie-King models or Fokker-Planck
calculations - mass segregation or tidal shocks affect significantly the shape of the IMF even
near the half-light radius, yielding a deficiency of low-mass stars (see e.g. Baumgardt &
Makino 2003). Or, similarly, if the half-mass radius does not correspond to the observed
half-light radius. The dotted lines on Figure 9 illustrate the IMF derived in the previous
section for the spheroid population (eq.[20]), with the characteristic mass shifted by 1 to
2 σ’s (i.e. mc = 0.22 to 0.32 M⊙). The agreement between the globular cluster and the

– 37 –
lack of detection of very metal-depleted ([M/H] ≪ −4) stars in the halo population might be
an artefact due to too faint detection limits. The faintest giants in the HES survey extend
up to 20 kpc or more. Further analysis of the survey should tell us whether or not it reveals
the tip of the red giant branch of a Pop III stellar population.
(iii) The existence of a significant remnant population in the dark halo is not a com-
pletely settled issue yet. As mentioned in the previous section, the maximum contribution
from spheroid and/or dark halo WDs predicted by the IMF (eq.[20]) represents at most
∼ 0.5% of the dark matter density, i.e. ρWD ≈ 4× 10−5 M⊙ pc−3, so that the unambiguous
detection of a genuine halo WD population with a significantly larger density would imply
that the halo IMF differs significantly from this form and peaks in the 1-10 M⊙ mass range.
Microlensing observations of dark matter baryonic candidates in the halo, however, remain
controversial. The MACHO observations (Alcock et al. 2000) yield a microlensing optical
depth, based on 13 to 17 events, τ = 1.2+0.4−0.3×10−7, with a total mass in the objects within 50
kpc M50 = 9+4−3×1010 M⊙, i.e. . 20% of the dynamical mass. For a standard isothermal halo
model with a velocity dispersion v⊥ = 220 km.s−1, the event time distribution corresponds
to a peak in the range ∼ 0.5 ± 0.4M⊙. Since M-dwarf stars are excluded as a significant
dark halo population (point (i) above), this implies halo WDs. The EROS project, exploring
a larger field around the disks of the LMC and SMC, derived an upper limit contribution
to the dark matter of 25% for objects in the mass range 2 × 10−7 ≤ m/M⊙ ≤ 1 at the
95% confidence level (Afonso et al. 2003). Interestingly enough, the only events detected
today towards the SMC have been shown to belong to the SMC population. One thus can
not exclude that events detected towards the LMC are mainly due to self-lensing events,
as pointed out originally by Sahu (1994), or that some events such as supernovae or very
long-period variables have been misidentified as microlensing events.
Another important constraint on the dark halo population stems from the abundances
of helium and heavy elements, which point to a primordial WD mass fraction in the halo
ρWD . 0.1 × ρdyn . 10−4 M⊙ pc−3 (Gibson & Mould 1997, Fields et al. 2000). This is
confirmed by recent nucleosynthesis calculations of zero or near-zero metallicity low-mass and
intermediate-mass stars which show that helium burning and CNO-cycle processes material
(in particular C and O) to the surface (Fujimoto et al. 2000, Siess et al. 2002).
Several detections of faint, cool, high-velocity WDs in the solar neighborhood, based
on either spectroscopic, kinematic or photometic identifications, have been claimed recently
(Oppenheimer et al. 2001). These detections, however, remain controversial, and are based
on a limited number of objects. Indeed, if the extended thick-disk suggested by Gilmore
et al. (2002) and Fuhrmann (2002) is confirmed, with kinematic properties intermediate
between the standard thick-disk and the spheroid ones, a substantial fraction of the high-

– 38 –
proper motion WDs discovered by Oppenheimer et al. (2001) might indeed belong to this
population. As shown by Chabrier (1999), one needs large field (> 1 sq.deg.) surveys at
faint magnitude (V,R, I > 20) to really nail down this issue and derive a reasonably robust
estimate of the halo WD density.
It seems thus clear that the present dark halo contains only a negligible fraction of the
Galactic baryonic mass.
5.2. The early star mass function
This brings us to the hypothetical determination of the IMF of primordial stars, formed
at large redshift. Only indirect information on such early star formation processes can be
inferred from various observational constraints and from galactic evolution (see e.g. Larson
1998).
(i) The modest increase of metallicity along Galactic history, from [M/H] ≈ −2, charac-
teristic of the spheroid, to [M/H] = 0, but the scarcity of very-metal-depleted [M/H] << −4.0
stars in the Milky Way as well as in other galaxies, the so-called G-dwarf problem, or con-
versely the similarity of the massive (oxygen producing) stars over low-mass star ratio be-
tween spheroid and disk, imply that relatively few LMS were formed when the metallicity
was very low at early times.
(ii) Observations of young galaxies at z > 1 in the submillimetre and far-IR domains
rule out a Salpeter IMF extending down to the H-burning minimum mass and suggest a
top-heavy IMF, with a cut-off near ∼ 0.7M⊙, to produce massive stars without producing
low-mass stars of which light would remain visible to the present time (Dwek et al. 1998,
Blain et al. 1999).
(iii) The observed abundances of heavy elements in clusters of galaxies require an in-
crease by a factor of ∼ 3 in the total mass of heavy elements than predicted by a Salpeter
IMF. Thus a comparable increase in the ratio of heavy elements per solar-mass produced by
high-mass stars relative to the number of low-mass stars formed.
(iv) A top-heavy IMF at early times of galactic evolution increases the number of SNII
per visible stars, providing more excess thermal energy and thus a larger amount of hot
gas and heavy elements ejected in the IGM from the bound clusters of galaxies. This is
consistent with the fact that most of the heavy elements in clusters are in the IGM rather
than in galaxies.
(v) Increasing M/L ratio and Mg/H and Mg/Fe abundances with mass are observed

– 39 –
in early-type galaxies (Worthey, Faber & Gonza´lez 1992). This points to a larger relative
contribution from massive stars, i.e. a dominant high-mass mode formation, and more mass
locked in remnants.
(vi) Recent observations of the large scale polarization of the cosmic microwave back-
ground measured by the WMAP satellite require a mean optical depth to Thomson scatter-
ing τe ∼ 0.17, suggesting that reionization of the universe must have begun at large redshift
(z & 10). A possible (but not unique) solution is a top heavy IMF for primordial, nearly
metal-free stars (Ciardi, Ferrara & White 2003, Cen 2003).
Although certainly not conclusive, all these independent constraints (to be considered
with caution) point to an early-type IMF with a minimum low-mass cut-off & 1M⊙. On
the other hand, [α-element/Fe] ratios measured in the intergalactic hot gas seem to be only
slightly oversolar, which implies a significant contribution from type Ia SN, suggesting a
constant Salpeter-like slope of the high-mass tail (m & 1M⊙) of the IMF. Indeed, an IMF
with a Scalo slope (ξ(logm) ∝ m−1.7) seems to underestimate the fraction of very massive
stars to solar-type stars in high-z field galaxies, producing too much long-wavelength light
by the present epoch (Lilly et al. 1996, Madau, Pozzetti & Dickinson 1998)11.
Indeed, the thermal Jeans mass strongly depends on the temperature (∝ T 3/2) and
more weakly on the pressure (∝ P−1/2). Although there is no reason for this latter to have
changed significantly during the universe evolution, the temperature did evolve significantly.
As pointed out by Larson (1998), the very minimum ambient temperature of the medium
is given by the cosmic background radiation 2.73(1 + z) K so that the thermal Jeans mass,
i.e. the minimum mass for gravitationally-bound objects, increases with redshift. Whether
this mass is the very characteristic mass in star formation, or whether a distribution of Jeans
masses is more relevant, will be examined in §7. It is also important to note that, in the
absence of a significant fraction of metals, the cooling, and thus fragmentation, of the cloud
proceeds via collisional excitation and radiative deexcitation of H2, which can not cool below
85 K (first rotational level of H2) (see e.g. Abel et al. 2000, Nakamura & Umemura 2002,
Bromm et al. 2002).
Given all this general context, it is interesting to examine the signature of a primordial
IMF biased towards large masses (> 1M⊙), and more specifically towards WD progenitors,
i.e. with a characteristic mass in the AGB-mass range, like the following form :
11Note, however, that these results depend on the correction due to dust extinction and should be consid-
ered with due caution.

– 45 –
line widths for . 0.1 pc, with a non-thermal to thermal velocity dispersion ratio of H2
σNT /σT ∼ 0.7 (Belloche et al. 2001).
Star formation theories must now be confronted to the general results (i)-(v). In a
canonical theory for isolated star formation, low-mass stars form from the collapse of ini-
tially hydrostatic but unstable dense cloud cores which have reached a ρ(r) ∝ r−2 density
distribution of a singular isothermal spheroid (Shu, Adams & Lizano 1987). In this scenario,
deuterium-burning, which occurs at pre-main-sequence ages, is a key ingredient to trigger
star formation. The onset of D-burning induces convective instability in the interior. Com-
bined with the rapid rotation resulting from the accretion of angular momentum with mass,
this convection is believed to generate a strong magnetic field through dynamo process. The
field will drive a magnetocentrifugal wind that ultimately sweeps away the surrounding ac-
creting material, and determines the mass of the nescent star. Within this picture, objects
below the deuterium-burning limit can not reverse the infal and thus can not form gravita-
tionally bound objects. Such a scenario can now be reasonably excluded on several grounds.
First of all, substellar objects are fully convective, with or without deuterium-burning, ex-
cept for the oldest ones which develop a conductive core at late ages (Chabrier & Baraffe
2000, Chabrier et al. 2000a). In fact, the numerous detections of free-floating objects at
the limit and below the deuterium-burning minimum mass, and the rising mass spectrum
down to this limit in several young clusters (see Figure 5 and Najita et al. 2000, Be´jar et al.
2001, Mart´ın et al 2001, Lucas et al. 2001) seem to exclude deuterium burning as a peculiar
process in star formation. This should close the ongoing debate in the literature arguing that
the deuterium-burning minimum mass distinguishes BDs from planets, since such a distinc-
tion does not appear to be supported by physical considerations. Second, as mentioned above,
star formation in young clusters appear to form over a timescale significantly shorter than
the ambipolar diffusion timescale & 10 Myr, indicating that, if magnetic field plays some
role in star formation, it is unlikely to be a dominant process. In the ambipolar diffusion
scenario, the cloud must survive long enough, in near equilibrium between magnetic and
gravitational pressure. This is not consistent with observations of rapid star formation and
cloud dissipation, and with the observed turbulent nature of clouds. Indeed, equipartition
between kinetic, gravitational and magnetic energy fails to reproduce the observed properties
of molecular clouds, which are dominated by super-Alfve´nic and supersonic motions, where
kinetic energy dominates magnetic energy, with a decay timescale approximately equal to
a dynamical timescale (see e.g. Padoan & Nordlund 1999 and references therein). In fact,
ambipolar diffusion models require large static magnetic field strengths (∼30-100 µG) ex-
ceeding Zeeman estimates for low-mass dense cores (. 10 µG) (Crutcher & Troland 2000,
Padoan & Nordlund 1999, Padoan et al. 2001a,b, 2001b Bourke et al. 2001).
Another version of the wind-limited accretion model relates the gas cloud properties,

– 47 –
i.e. in a highly non-homogeneous manner. Furthermore, turbulence is a highly non-linear
process, opposite to the basic assumption of gravitational instability model.
Alternative models suggested that stars grow from protostar collisions and/or coales-
cence between gas clumps until the bound fragment becomes optically thick (Nakano 1966,
Nakano, Hasegawa & Norman 1995). The aforementioned rapid timescale for star forma-
tion, however, is much shorter than the typical collision time between multiple protostellar
clumps, and thus seems to exclude this scenario, at least for the low-mass stars. It is also dif-
ficult to reconcile this star formation mechanism, involving kind of feedback effects, with the
universal form of the IMF, suggesting that this latter reflects the initial conditions imposed
in the cloud. Not mentioning the difficulty to reconcile coalescence process with supersonic
turbulence. A recent extension of this type of scenario, where the IMF is determined by the
competitive accretion between the various stellar cores, and a combination of mass accretion
and stellar mergers, has been proposed by Bonnel, Bate and collaborators (Bonnel et al.
2001a, 2001b, Bonnell & Bate 2002), based on hydrodynamical simulations of gas accretion
onto a pre-existing cluster of 1000 stars. Based on 15 stars with m & 5M⊙ at the end of
the calculations, the high-mass tail of the IMF, obtained with such an accretion and merger
scenario, seems to reproduce a Salpeter slope, although the reason for such a result is not
clear, whereas the lower-mass part of the IMF yields a shallower power-law with x ≈ 0.5.
Although, as mentioned earlier, the accretion process certainly plays some role in shaping
the final stellar mass, the present scenario relies on some assumptions for the initial condi-
tions, e.g. an ensemble of already formed stars of equal mass as nucleation centers and a gas
reservoir apparently not supersonic, which appear rather unrealistic.
Finally, some models suggest that star formation is not due to a dominant process,
but is rather the byproduct of several independent processes of comparable importance.
The product of a large number of statistically-independent processes naturally points to
the central limit theorem, as initially suggested by Larson (1973), Zinnecker (1984) and
Elmegreen (1983), and later on by Adams & Fatuzzo (1996). The final product of the
central limit theorem is a gaussian distribution, i.e. a lognormal form in a logarithmic
plane. In this type of theory, however, the statistical aspect of star formation still arises
from the hierarchical structure produced by fragmentation, and thus is linked back to the
original concept of Hoyle. Moreover, this theory is frustrating from the physics point of view,
since it relies on a purely statistical mechanism and prevents understanding star formation
from identified physical processes. Not mentioning delicate applications of the central limit
theorem concept, which strictly speaking implies an infinity of statistically independent
variables, in real nature ! In fact, no current theory of the IMF is consistent with all the
aforementioned constraints (i)-(v) and predicts in particular the formation of free-floating
objects in significant numbers at very low mass.

– 49 –
similar to the Salpeter value.
In these simulations, the typical core mass formed in shocked gas reads :
m(L) ∼ ρl3 ∼ ρ0L3/MA(L)2 ∼ L3−2α ∼ L2.2 (31)
In this turbulent picture of fragmentation, the distribution of cores arises essentially
from internal cloud turbulent dissipation. The collapse of these cores into protostars is
then determined by the dynamical timescale of supersonic MHD turbulence, τdyn = L/σ(L),
rather than by the local gravitational timescale (Gρ)−1/2. Sufficiently massive cores continue
to collapse under selfgravity, so for large m the distribution of cores is directly reflected in the
distribution of stellar masses. At smaller masses, only cores with sufficient density are able to
collapse further, which reduces the number of stars formed out of a given distribution of cores
with mass m, and causes the IMF to deviate from the large m behavior. Thus, in a generic
sense the roll-over of the IMF happens when gravity is no longer able to cause the collapse of
most cores of a given mass. At this stage, different factors such as cooling functions, equations
of state, additional fragmentation during collapse etc., become important. Quantitative
predictions then require detailed numerical simulations, such as those of Bate et al (2002,
2003) or Klessen (2001). If, for example, because of gravitational fragmentation during the
collapse, each core gives rise to a distribution of stars, the IMF will be shifted to smaller
masses. This will not influence the power law shape on the high mass side, but will shift the
maximum mass of the IMF to a smaller value, affecting the expected number of low-mass
stars and brown dwarfs.
Qualitatively, however, the roll-over of the IMF is displayed already for idealized as-
sumptions with isothermal conditions. A universal behaviour of turbulent fragmentation for
an isothermal gas is that it produces a lognormal probability distribution function (PDF) of
gas density in unit of mean density x = n/n0:
p(x) d lnx ∝ exp
{
−(ln x− 〈lnx〉)2/2σ2
}
d ln x (32)
where n0 is the mean density, and σ2 ≈ ln (1 + 0.25M2) (Padoan et al. 1997, Padoan &
Nordlund 1999, Ostriker et al. 1999). In the present context of star formation, this yields no
longer a unique Jeans mass but a distribution of local Jeans masses p(mJ ), obtained from the
PDF of gas density, assuming that the distribution of average density of clumps of a given
mass has also a lognormal distribution (Padoan et al. 1997, Padoan & Nordlund 2002):

– 50 –
p(mJ) d ln mJ =
2√
2piσ
m−2J exp
{
−(ln m
2
J − |〈ln x〉|)2
2 σ2
}
d ln mJ (33)
where mJ is written in units of the thermal Jeans mass at mean density n0:
MJ = 1.2M⊙ (
T
10K
)3/2 (
n0
104 cm−3
)−1/2 = 1.2M⊙ (
T
10K
)2 (
P0
105 cm−3 K
)−1/2 (34)
which thus ranges from ∼ 1.2 to ∼ 0.12M⊙ for characteristic low-pressure to high-pressure
clumps13. The Jeans mass is approximately the same for spheres and for filaments (Larson
1985).
The fraction of small cores of mass m < MJ to collapse to gravitationally-bound struc-
tures is thus given by the probability distribution P (m) =
∫ m
0 p(mJ )dmJ , and the mass
distribution of collapsing cores reads (from eq. [30]):
N(m)d lnm ∝ m− 34−β P (m) d lnm (35)
Therefore, although the average star mass is similar to the average thermal Jeans mass
of the medium, the global mass distribution extends well below this limit, with decreasing
probability. Within this picture, star formation proceeds as follows (see Nordlund & Padoan
2002):
(i) Supersonic turbulence in the ISM, produced by large amounts of kinetic energy
at large scales, dissipates in fragmenting molecular clouds (preventing a global collapse of
the cloud) into highly anisotropic filaments, due to the random convergence of the velocity
field. These filaments form dense cores, with large density contrasts (much larger than the
maximum value ∼ 14 for a self-gravitating, pressure-bounded Bonnor-Ebert sphere) via the
action of radiative MHD shocks and thus determine the fragmentation length scale over
which collapse is possible. Cooling becomes more efficient as density increases in these
dense cores, of typical dimensions ∼ 0.01-0.1 pc, which become self-gravitating and begin to
collapse. During this stage, the star formation process itself, during which gas is converted
13Note the incorrect density scaling factor, 103 cm−3 in Padoan & Nordlund (2002, Eq.(21)) (Nordlund,
private communication). Using the same expression for the Jeans mass as Bonnell et al. (2001a, Eq. 1), the
scaling constant changes from 1.2 to 1.9 M⊙

– 54 –
IMF or similarly from a more or less uniform mass ratio distribution. The extension of
the single MF into the BD regime is in good agreement with present estimates of L- and
T-dwarfs densities, when considering all the uncertainties in these estimates. This yields
a disk BD number density comparable to the stellar one, namely ∼ 0.1 pc−3. The IMF
of several young clusters is found to be consistent with this same field IMF, providing a
similar correction for unresolved binaries, confirming the fact that young star clusters and
disk field stars represent the same stellar population. Dynamical effects, yielding depletion
of the lowest-mass objects, are found to become consequential for ages slightly older than
the age of the Pleiades, i.e. & 130 Myr.
The spheroid IMF relies on much less robust grounds. The large metallicity spread in
the photometric local sample, in particular, remains puzzling. Recent observations suggest
that there is a continuous kinematic shear between the thick-disk population, present in the
local samples, and the spheroid one, observed with the HST. This enables us to derive only
an upper limit for the spheroid mass contribution and IMF. This latter is found to be similar
to the one derived for globular clusters, and is well described also by a lognormal form, but
with a characteristic mass slightly larger than for the disk, around ∼ 0.2-0.3 M⊙. Such
an IMF excludes a significant population of BDs in globular clusters and in the spheroid,
i.e. in metal-depleted environments. These results, however, remain hampered by large
uncertainties such as the exact amount of dynamical evolution near the half-mass radius
of a globular cluster, the exact identification of the genuine spheroid population, the exact
fraction of binaries in globular cluster and spheroid populations.
The early star IMF, representative of stellar populations formed at large redshift (z & 5),
remains undetermined, but different observational constraints suggest that it does not extend
below ∼ 1M⊙. Whether it extends down to this mass range, implying the existence of a
primordial white dwarf population, or whether the cutoff for this primordial IMF occurs at
much larger masses remains unsettled. In any case, the baryonic content of the dark halo
represents very likely at most a few percents of the Galactic dark matter.
These determinations point to a characteristic mass for star formation which decreases
with time, from early star formation conditions of temperature and metallicity to conditions
characteristic of the spheroid or thick-disk environments, to present-day conditions. These
results, however, remain more suggestive than conclusive. These IMFs allow a reasonably
robust determination of the Galactic stellar and brown dwarf content. They have also im-
portant galactic implications beyond the Milky Way in yielding more accurate mass-to-light
ratio determinations. The IMFs determined for the disk and the spheroid yield mass-to-light
ratios a factor of 1.8 to 1.4 smaller than for a Salpeter IMF, respectively, in agreement with
various recent dynamical determinations.

– 55 –
This IMF determination is examined in the context of star formation theory. Theories
based on a pure Jeans-type mechanism, where fragmentation is due only to gravity, appear
to have difficulties explaining the determined IMF and various observational constraints
on star formation. On the other hand, recent numerical simulations of compressible tur-
bulence, in particular in super-Alfve´nic conditions, reproduce qualitatively and reasonably
quantitatively the determined IMF, and thus provide an appealing solution. In this pic-
ture, star formation is induced by the dissipation of large scale turbulence to smaller scales,
through radiative shocks, producing filamentary structures. These shock produce local, non-
equilibrium structures with large density contrasts. Some of these dense cores then collapse
eventually in gravitationally bound objects, under the combined action of turbulence and
gravity. The concept of a single Jeans mass, however, is replaced by a distribution of local
Jeans masses, representative of the lognormal probability density function of the turbulent
gas. Cores exceeding the average Jeans mass (& 1M⊙) naturally collapse into stars under
the action of gravity whereas objects below this limit still have a possibility to collapse, but
with a decreasing probability, as gravity selects only the densest cores in a certain mass
range (the ones such that the mass exceeds the local Jeans mass mJ ). This picture, com-
bining turbulence, as the initial mechanism for fragmentation, and gravity thus provides a
natural explanation for a scale free, power-law IMF at large scales and a broad lognormal
form below about 1 M⊙. Additional mechanisms, such as accretion, subfragmentation of
the cores, multiplicity will not affect significantly the high-mass, power-law part of the mass
spectrum, but can modify the extension of its low-mass part. The initial level of turbulence
in the cloud, and its initial density, can also affect the low-mass part of the IMF.
Future improvements, both on the theoretical and observational sides, should confirm
(or refute) this general scenario and help quantifying the details of the interaction between
turbulence and gravity, but it is encouraging to see that we are now reaching a reasonable
paradigm in our understanding of the Galactic mass function over 5 orders of magnitudes,
from very massive stars to Jupiter-like objects, of the census of baryonic objects in the
Galaxy, which can be applied to external galaxies, and of the dominant physical mechanisms
underlying the process of star formation.
Acknowledgments:
It is a great pleasure for me to thank many colleagues who contributed to this review.
A special thank to D. Barrado y Navascues, V. Be´jar, J. Bouvier, A. Burgasser, F. Comero´n,
C. Dahn, P. Dobbie, N. Hambly, H. Harris, D. Kirkpatrick, K. Luhman, E. Moraux, for
sending their data and in some cases sharing unpublished results. My profound gratitude
to S. Charlot for calculating the mass-to-light ratios in §6. Finally, I am deeply indebted
to I. Baraffe, J. Bouvier, G. DeMarchi, L. Hartmann, P. Kroupa, R. Larson, A˚. Nordlund,

– 56 –
for numerous discussions, highly valuable comments and for reading a preliminary version of
this review. Finally a special thank to Anne Cowley and David Hartwick, editors of PASP,
for their astronomical patience before they received the present manuscript.

– 57 –
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Table 1: Disk initial mass function (IMF) and present-day mass function (PDMF) for single
objects. For unresolved binary systems, the coefficients are given by eq.[18].The normaliza-
tion coefficient A is in pc−3(log M⊙)−1.
m ≤ 1.0M⊙ m > 1.0M⊙
ξ(log m) = A exp{− (log m − log mc)22σ2 } ξ(log m) = Am−x
IMF A=0.158+0.051−0.046 A=4.43×10−2
mc = 0.079−0.016+0.021 x=1.3 ±0.3
σ = 0.69−0.01+0.05
PDMF A=0.158+0.051−0.046 0 ≤ log m ≤ 0.54 : A = 4.4× 10−2 , x=4.37
mc = 0.079−0.016+0.021 0.54 ≤ log m ≤ 1.26 : A = 1.5× 10−2 , x=3.53
σ = 0.69−0.01+0.05 1.26 ≤ log m ≤ 1.80 : A = 2.5× 10−4 , x=2.11

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Table 3: Present day stellar and brown dwarf Galactic budget. The number densities n are
in [pc−3], the mass densities ρ are in [M⊙ pc−3].
Disk Spheroid Dark halo
nBD 0.13±0.06 . 3.5× 10−5
ρBD (0.4± 0.2)× 10−2 . 2.3× 10−6
n⋆(≤ 1M⊙) 0.13± 0.02 ≤ (2.4± 0.1)× 10−4
ρ⋆(≤ 1M⊙) (3.5± 0.3)× 10−2 ≤ (6.6± 0.7)× 10−5 ≪ 10−5
n⋆(> 1M⊙) 0.4× 10−2 0
ρ⋆(> 1M⊙) 0.6× 10−2 0
nrem (0.7± 0.1)× 10−2 ≤ (2.7± 1.2)× 10−5 ?
ρrem (0.6± 0.1)× 10−2 ≤ (1.8± 0.8)× 10−5
ntot 0.27± 0.06 ≤ 3.0× 10−4
ρtot (5.1± 0.3)× 10−2 ≤ (9.4± 1.0)× 10−5 ≪ 10−5
N (BD); M(BD) 0.48; 0.08 0.10; 0.03
N (LMS); M(LMS) 0.48; 0.68 0.80; 0.77
N (IMS); M(IMS) 0.015; 0.11 0.; 0.
N (HMS); M(HMS) 0; 0 0.; 0.
N (rem.); M(rem.) 0.025; 0.13 0.10; 0.20 ?

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Table 4: Initial stellar and brown dwarf Galactic budget (from Table 2).
Disk Spheroid
nBD 0.13±0.06 ∼ 3.5× 10−5
ρBD (0.4± 0.2)× 10−2 ∼ 2.3× 10−6
n⋆(≤ 1M⊙) 0.13± 0.02 ≤ (2.4± 0.1)× 10−4
ρ⋆(≤ 1M⊙) (3.5± 0.3)× 10−2 ≤ (6.6± 0.7)× 10−5
n⋆(> 1M⊙) 1.5× 10−2 ≤ 2.3× 10−5
ρ⋆(> 1M⊙) 4.7× 10−2 ≤ 1.0× 10−4
ntot 0.27 ≤ 3.0× 10−4
ρtot (8.6± 0.3)× 10−2 ≤ 2.0× 10−4
N (BD); M(BD) 0.48; 0.04 0.10; 0
N (LMS); M(LMS) 0.48; 0.41 0.80; 0.48
N (IMS); M(IMS) 0.04; 0.35 0.09; 0.34
N (HMS); M(HMS) 0; 0.20 0; 0.18

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Figure Legends
Fig. 1.— Disk mass function derived from the local V-band LF (circles and solid line)
and K-band LF (squares and dash-line). The solid line and the two surrounding dash-lines
display the lognormal form given by eq.[17], whereas the dotted line illustrates the 4-segment
power-law form of Kroupa (2002). The empty circles and squares for log m ≥ −0.15 display
the MF obtained for t = 10 Gyr and 1 Gyr, respectively, illustrating the age uncertainty on
the MF for m > 0.7M⊙. The empty triangles and dotted error bars display the MF obtained
from the bulge LF (see text).
Fig. 2.— Disk mass function derived from the system K-band LF (solid squares and solid
line) and the HST corrected MF (triangles and short-dash-line) from Zheng et al. (2001).
The solid line and surrounding dotted lines display the lognormal form given by eq.[17] for
single objects, as in Figure 1, whereas the dash-line illustrates the lognormal form given by
eq.[18].
Fig. 3.— Luminosity functions for the Galactic disk predicted with the IMF (eq.[17]) and a
constant SFR. Solid: stars+BDs; long-dash: BDs only (m ≤ 0.072M⊙); long dash-dot: T-
dwarfs only (J-H<0.5 and H-K<0.5); short dash-dot: objects below the D-burning minimum
mass (m ≤ 0.012M⊙). The short-dash lines illustrate the range of uncertainty in the IMF
(eq.[17]). The dotted line in the middle panel displays the result obtained with a power-law
IMF with x = 0 (ξ(log m) = constant) with the same normalisation at 0.1 M⊙ as IMF
(eq.[17]). The dotted line in the bottom panel displays the results obtained with the system
IMF (eq.[18]). The histogram displays the nearby LF (Henry & McCarthy 1990). Empty
and filled squares are estimated L-dwarf densities by Gizis et al. (2000) and Kirkpatrick
(1999, 2000)+Burgasser (2001), respectively. Triangles are estimated T-dwarf densities from
Burgasser (2001).
Fig. 4.— Pleiades Mass Function calculated with the Baraffe et al. (1998) and Chabrier et
al. (2000) MMRs, from various observations : squares : Hambly et al. (1999); triangles :
Dobbie et al. (2002b); circles : Moraux et al. (2003). The short-dash and long-dash lines
display the single (eq.[17]) and system (eq.[18]) field MFs, respectively, arbitrarily normalized
to the present data.

– 72 –
Fig. 10.— Mass-to-light ratios in various passbands, in units of stellar mass per solar lu-
minosity in the considered band, calculated wiht the Salpeter IMF (dotted line), the disk
IMF (eq.[17]) (solid line), the spheroid IMF (eq.[20]) (long-dash line) and the top-heavy
IMF (eq.[24]) (short-dash line). The calculations correspond to Simple Stellar Populations
(SSP), i.e. a stellar birthrate parameter b = 0 (see text). All IMFs are normalized to
∫ 100
0.01 m (dN/dm) dm = 1. Courtesy of S. Charlot.
Fig. 11.— Hertzsprung-Russell diagram for young objects in Chameleon (circles), Lupus
(triangles), IC348 (squares) from Comero´n et al. (2003) and in Taurus (dots) from Bricen˜o
et al. (2002). Superposed are various isochrones of Baraffe et al. (2002), for τ = 106, 2 ×
106, 3× 106, 5× 106, 107 yr from top to bottom, for different masses, as indicated.

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