For a fixed amount of gas turned into stars, different IMFs obviously
imply different proportions of low mass and high mass stars. This is
illustrated in Figure 8.1 showing three
different IMFs, all with the same slope below 0.5
*M*_{}, that
is *s* = 1 + *x* = 1.35, and three different slopes above:

(1) |

where the factor 0.5^{1.3-s} ensures the continuity of
the IMF at *M* = 0.5
*M*_{}. The
normalization of the three IMFs corresponds to
a fixed amount of gas
turned into stars, that is, for fixed

(2) |

Here the case *s* = 2.35 corresponds to the Salpeter-diet IMF already
encountered in previous chapters. Thick lines in
Figure 8.1b show the
cumulative distributions, defined as the number of stars with mass less
than *M*, *N*(*M'* < *M*) =
∫_{0.1}^{M}
(*M*')
*dM*', while thin lines show
the fraction of mass in stars less massive than *M*.
In a Salpeter-diet IMF ~ 0.6% of all stars are more massive than
8 *M*_{},
while for *s* = 3.35 and 1.5 these fractions are 0.03%
and 9 % respectively. The mass in stars heavier than 8
*M*_{} is
20% for a Salpeter-diet IMF; it drops to 1% for *s* =
3.35, and is boosted to 77 % for *s* = 1.5, a *top-heavy*
IMF. Figure 8.1 wants to convey the message that IMF variations have
a drastic effect on stellar demography, and therefore on several key
properties of stellar populations. Suffice it to say that most of the
nucleosynthesis comes from *M* > 10
*M*_{} stars,
whereas the light of
an old population (say, *t* > 10 Gyr) comes from stars with
*M*
1 *M*_{}, and
therefore is proportional to
(*M*
= *M*_{}).

Figure 8.2 shows the variation of scale factor
*A*
(cf. Chapter 2) as a function of the IMF slope, again for a fixed amount
of gas turned into
stars. The scale factor *A* has a maximum for *s*
2.75, pretty close to
the Salpeter's slope. Since by construction *A* =
(*M*
= 1 *M*_{})
and the luminosity of a > ~ 10 Gyr old population is proportional to
(*M* =
1 *M*_{}), an IMF
with the Salpeter's slope has the remarkable property of almost
maximizing the light output of an old population, for fixed mass
turned into stars. A flat IMF (*s* = 1.35) is much less efficient
in this respect, indeed by a factor of ~ 8 compared to the Salpeter's
slope, as shown by Figure 8.2. This figure
also shows the mass-to-number conversion factor
*K*_{}, giving the number of stars *N*_{T}
formed out of a unit amount of gas turned into stars, that is,
*N*_{T} = *K*_{}
/
*M*_{}. Thus,
for a Salpeter-diet IMF *K*_{}
1.5, saying that ~ 150 stars are formed out of 100
*M*_{} of
gas turned into stars.

An empirically motivated, broken-line IMF such as that shown in Figure 8.1 is widely adopted in current astrophysical applications, yet Nature is unlike to make such a cuspy IMF. Perhaps a more elegant rendition of basically the same empirical data is represented by a Salpeter+lognormal distribution in which a lognormal IMF at low masses joins smoothly to a Salpeter IMF at higher masses, that is:

(3) |

where *A*_{1} = 0.159, *M*_{c} = 0.079,
= 0.69,
*A*_{2} = 0.0443
and *x* = 1.3. Thus, this *Chabrier* IMF is almost identical
to the Salpeter IMF above 1
*M*_{}, and
smoothly flattens below, being
almost indistinguishable from the Salpeter-diet IMF.

Explorations of variable IMFs can be made by either changing its
slope, or by moving to higher/lower masses the break of the IMF slope
with respect to Equation (8.1), or allowing the
characteristic mass *M*_{c} in Equation (8.3) to vary.
Figure 8.3 shows examples of such evolving IMFs.
The two slope IMF with *M*_{break} = 0.5
*M*_{} and
the Chabrier IMF with *M*_{c} = 0.079
*M*_{} (lines
*a* and *c* in Figure 8.3) fit each
other extremely well and both provide a good fit to the local
empirical IMF. By moving the break/characteristic mass to higher
values one can explore the effects of such evolving IMF, for example
mimicking a systematic trend with redshift.
The cases with *M*_{c}
*M*_{break}
4
*M*_{} are
shown in Figure 8.3 (lines *b* and
*d*). Having normalized all IMFs to the same value of
(*M*
= 1 *M*_{}),
Figure 8.3 allows one to immediately gauge the
relative importance of massive stars compared to solar mass stars,
with the latter ones providing the bulk of the light from old (> ~
10 Gyr) populations.