The Initial Mass Function (and Luminosity Function, briefly)
The Initial Mass Function
A Mass Function is simply the distribution of stars by mass.
The present Mass Function of a system
reflects the Initial Mass
Function (IMF) of formed stars plus the effects of stellar
Obviously, evolution acts to remove highest mass stars most quickly.
Let's think about a single starburst first, and just after let
the number of stars formed with masses in range (, + d) be:
Here, N0 is a normalization factor that
depends on the burst strength, while ξ describes the
shape of the IMF.
(Note, in many treatments, the function ξ is
characterized instead by the symbol ψ.)
Regarding N0 -- it is typical to normalize ξ to the
total mass, rather than the total number of new born stars
because there is a large weighting in the total number of
stars towards very low mass stars, which can be very difficult to enumerate.
Using the equation just above, this normalization is done so that:
and N0 then becomes the number of solar masses in the SF burst.
We have no a priori reason to think this, but it appears
that ξ() is a remarkably universal
function that applies to starbursts occurring in a variety of different
circumstances, places, and times in the universe (at least for
Typically assumed to be a simple power law (between limits!):
(Salpeter Mass Function)
x = 1.35 by Salpeter (1955)
Poster from a 2004 meeting showing the Salpeter mass function shape.
Note, mass functions are not typically observable.
Instead, we observe the luminosity function ,
Φ(M), which is the number of stars
visible per absolute magnitude, M. (Note the
different symbol M here for absolute magnitude.)
(Actually, we don't even observe that -- rather
we observe apparent magnitudes and need to get distances too:
a ball of wax I defer for now too!).
We then must convert the LF to the MF with knowledge of
the mass-luminosity relation appropriate to the age of the
population (again, tricky).
How are IMFs derived?
From observations of luminosity functions of course.
Easiest way is to look for very young open cluster and
observe the Initial Luminosity Function,
Convert Φ0(M) to
ξ() with ZAMS mass-luminosity
But, if there has been any evolution, have to account for missing high mass
(high luminosity) and missing low mass (pre-MS; see below) stars.
If we use field stars in the Milky Way, we can, for example,
assume the disk had a constant SFR in the past (not strictly true!).
Measure present LF - Φ(M) of
Main Sequence stars.
Then the Initial LF is:
where t = time since SF started.
Obviously, the proportion of stars whose MS lifetimes
are longer than t is just proportional to what we observe - i.e.,
they are all still around.
But stars evolved away must be accounted for and the t / τMS corrects for the amount of time
that could have contributed to present number of these
stars still around.
(I.e., the only stars of magnitude M we see are those that were formed in the last
(τMS / t ) fraction of the life of the population.)
We will return to the question of the local LF when we talk about starcounts.
Once we have Φ0(M), then:
The relationship between mass and absolute magnitude can be gotten by:
Theoretical stellar atmospheres models coupled to models of main sequence
structure at different masses -- works for most stellar masses but hard for <~ 0.5M☉ because...
Low mass stars are slow to settle onto M.S. -- have to account for
many low mass stars not being on true M.S. yet.
The CMD for a young open cluster, showing stars
at the top of the M.S. evolving away while stars
at the bottom of the M.S. are still getting to the
M.S. From Carroll & Ostlie.
Late type stars have many molecular bands, stellar atmospheres are
hard to calculate.
fully convective - don't have good models for
Observe binary stars, from which masses can be derived using
orbital dynamics (one of the few ways we can measure a stellar mass). But...
Requires a lot of work to get individual orbits monitored
spectroscopically (get spectroscopic parallaxes and radial velocities) and
astrometrically (get trig parallaxes -- if possible -- and the relative
orbits of the two stars around each other, i.e., the relative
masses of the stars to each other from
and the orientation of the orbit on sky and the ellipticity).
With knowledge of the period, P, and the semimajor axis,
a, get (1+
But unless you can determine the a1 and
a2 separately, you cannot portion the relative
mass in each component.
And then there is still the sin i (inclination of the orbit) ambiguity, which, unless
we can resolve (i.e., know i), means we really only at best get
This means we really only get a lower limit on the actual masses (because
sin i < 1 ).
(Note that < sin3i > = 0.59 for randomly oriented
Subject to significant observational uncertainty.
Required data for binary systems including stars
of all masses hard to get - sparse statistics...
An example of a mass-absolute magnitude relationship from binary
star data is given below:
Asteroseismology may be a promising avenue in the future, though currently limited
to brighter stars.
Armed with the mass-absolute magnitude relationship we can proceed to
convert the ILF to the IMF.
But this process is hard because not only do we need M(), but its derivative, because
Data are just not good enough for this.
The actual IMF is more complex than Salpeter with (Scalo 1986)
flatter parts at ends and steeper parts in 1-10 ☉ range.
The peak at the low mass end may be illusory:
A peak in the LF at low luminosities can occur due to a combination
of incompleteness and the fact that near about the H-burning limit,
the luminosity of stars drops precipitously and the stars
within a small range in mass are spread over wide range of magnitudes.
Theory in this part of diagram of mass-magnitude relationship
not good either.
Just hard to work there!
Some various IMF shapes, including more recent papers. From Wikipedia.
For most galaxies, it is safe to assume, to even high mass:
which is an average over the whole galaxy
Note: these high mass stars are somewhat missing from solar
neighborhood, so this is different from what is shown in Fig. 5.12.
However, for a number of reasons, people have debated the possibility that
star formation may be bimodal, with a separate "high mass" mode that occurs
in some extreme cases (place or time; Schwarzschild & Spitzer 1953).
If formation of the very first stars favored very massive stars, then
one could explain the "local dark matter problem" (i.e., that half of the mass
in the solar neighborhood is not accounted for) as residing in the remnants
(black holes and neutron stars) of an initial population of massive stars (Larson
1986, MNRAS, 218, 409).
Note that microlensing experiments, which would be sensitive to the existence
of these "MACHOS" (Massive Compact Halo Objects), have pretty much ruled out that there
are enough to account for the local dark matter.
The observed paucity of low metallicity stars (the so-called "G-dwarf problem")
might be explained by the fact that early star formation underproduced
low mass to high mass stars compared to the present day IMF (Schmidt 1963).
Various versions of bimodal star formation models.
(Left) The bimodal SF model from Larson (1986). (Right)
Schematic of the bimodal IMF predicted by Nakamura & Umemura (2001).
The shadings represent the different outcomes (remnants or enrichment through
supernovae, etc.) for the various mass ranges. From
The colors of the bluest starburst galaxies might be more easily explained if they
are dominated by high mass mode star formation (Charlot et al. 1993).
Observations of metallicity in the intracluster media of some galaxy clusters
indicates a large overabundance of [ α / Fe ], which might come about
from a large number of Type II supernovae which blew their nucleosynthetic
yield into the ICM.
The differences in SF modes may relate to a strong dependence of the
critical mass for cloud fragmentation on the gas temperature.
Note: ~50% of "star systems" contain binary or multiple stars (so ~ 2/3 of all stars are in
(Cecilia Payne-Gaposchkin: "Three out of every two stars is a binary"!)
From solar neighborhood (low mass stars mainly):
(equal mass binaries are preferred and binarity seems lower in M
stars - ~35%).
IMF for halo hard to establish:
Globulars are bad for ξ
() because of dynamical
evolution -- low mass stars lost preferentially. (We will look at this when we
discuss globular clusters.)
Subdwarfs - proper motion selected (biases) ...
dM / d complicated
because of (non-solar) abundance variations.
Overall, a somewhat steeper mass law is suggested: