and Cosmology

3. The World of Galaxies
118
should be noted here explicitly that both definitions of
the luminosity function are denoted by the same symbol,
although they represent different mathematical functions,
i.e., they describe different functional relations. It
is therefore important (and in most cases not difficult)
to deduce from the context which of these two functions
is being referred to.
Problems in Determining the Luminosity Function.
At first sight, the task of determining the luminosity
function of galaxies does not seem very difficult. The
history of this topic shows, however, that we encounter
a number of problems in practice. As a first step, the
determination of galaxy luminosities is required, for
which, besides measuring the flux, distance estimates
are also necessary. For very distant galaxies redshift is
a sufficiently reliable measure of distance, whereas for
nearby galaxies the methods discussed in Sect. 3.6 have
to be applied.
Another problem occurs for nearby galaxies, namely
the large-scale structure of the galaxy distribution. To
obtain a representative sample of galaxies, a sufficiently
large volume has to be surveyed because the
galaxy distribution is heavily structured on scales of
∼ 100 h −1 Mpc. On the other hand, galaxies of particularly
low luminosity can only be observed locally, so
the determination of Φ(L) for small L always needs
to refer to local galaxies. Finally, one has to deal with
the so-called Malmquist bias; in a flux-limited sample
luminous galaxies will always be overrepresented because
they are visible at larger distances (and therefore
are selected from a larger volume). A correction for this
effect is always necessary.
3.7.1 The Schechter Luminosity Function
The global galaxy distribution is well approximated by
the Schechter luminosity function
( Φ
∗
)( ) L α
Φ(L) =
L ∗ L ∗ exp ( −L/L ∗) , (3.38)
where L ∗ is a characteristic luminosity above which the
distribution decreases exponentially, α is the slope of
the luminosity function for small L, andΦ ∗ specifies
the normalization of the distribution. A schematic plot
of this function is shown in Fig. 3.31.
Expressed in magnitudes, this function appears much
more complicated. Considering that an interval dL in
luminosity corresponds to an interval dM in absolute
magnitude, with dL/L =−0.4 ln10dM, and using
Φ(L) dL = Φ(M) dM, i.e., the number of sources in
these intervals are of course the same, we obtain
Φ(M) = Φ(L)
dL
∣dM
∣ = Φ(L) 0.4 ln10L (3.39)
= (0.4 ln10)Φ ∗ 10 0.4(α+1)(M∗ −M)
(
)
× exp −10 0.4(M∗ −M)
. (3.40)
As mentioned above, the determination of the parameters
entering the Schechter function is difficult; a set of
parameters in the blue band is
Φ ∗ = 1.6 × 10 −2 h 3 Mpc −3 ,
M ∗ B =−19.7 + 5 log h , or
L ∗ B = 1.2 × 1010 h −2 L ⊙ , (3.41)
α =−1.07 .
While the blue light of galaxies is strongly affected by
star formation, the luminosity function in the red bands
measures the typical stellar distribution. In the K-band,
we have
Φ ∗ = 1.6 × 10 −2 h 3 Mpc −3 ,
MK ∗ =−23.1 + 5 log h , (3.42)
α =−0.9 .
The total number density of galaxies is formally infinite
if α ≤−1, but the validity of the Schechter function
does of course not extend to arbitrarily small L. The
luminosity density
l tot =
∫ ∞
0
dL LΦ(L) = Φ ∗ L ∗ Γ(2 + α) (3.43)
is finite for α ≥−2. 6 The integral in (3.43), for α ∼−1,
is dominated by L ∼ L ∗ ,andn = Φ ∗ is thus a good
estimate for the mean density of L ∗ -galaxies.
6 Γ(x) is the Gamma function, defined by
∫ ∞
Γ(x) = dy y (x−1) e −y . (3.44)
0
For positive integers, Γ(n +1) = n!.WehaveΓ(0.7) ≈ 1.30, Γ(1) = 1,
Γ(1.3) ≈ 0.90. Since these values are all close to unity, l tot ∼ Φ ∗ L ∗
is a good approximation for the luminosity density.