David Thomas Hill PhD Thesis - Research@StAndrews:FullText ...

Chapter 1. Introduction: The luminosity distribution and total luminosity density of
galaxies
1/Vmax method
The Volume corrected distribution method (1/Vmax) is the simplest, and oldest (Schmidt,
1968) 1 , method of calculating galaxy luminosity distributions. It works by weighting each
galaxy by the volume that it could reside within and still be included within the sample.
Galaxies are then allocated into bins based upon their luminosity. The value for each
bin is the sum of the galaxy weights for all sources within it. For example, Equation 1.5
shows the formulae required to calculate the luminosity density within a bin that covers
dM centred on Ms:
φ(Ms)dM =
Ng �
i=1
N(Ms − dM
2 ≤ Mi ≤ Ms + dM
2 )
Vmax(Mi)
(1.5)
where Ng is the number of galaxies within the sample, N(x) is 1 if the statement x is true,
and 0 otherwise, and Vmax(x) gives the volume a xmagnitude galaxy would be visible
within.
Unfortunately this method is dependent on the size of bins used; different sizes of bins
can lead to different results for the same dataset. It has the advantage of making no
assumptions about the shape of the LF, and has no dependence on the environment (Bell
et al., 2003). A recent approach has been to calculate 1/Vmax for each source, and sum
the parameters provided by the entire population to obtain the luminosity distribution.
SWML method
The SWML method, which is described in detail in Efstathiou et al. (1988), is a maximal-
likelihood method of calculating binned luminosity distributions in a non-parametric man-
ner.
Maximum-likelihood approaches attempt to find the distribution that has the highest
probability of generating the observed sample. In the SWML case, this is computed by
maximising L with respect to a discretised luminosity distribution, where L is the product
of the probabilities of observing each galaxy within the sample’s absolute magnitude limits
(given the redshift of the galaxy). The resulting luminosity distribution can then be fit
via χ 2 -minimisation to determine the Schechter function parameters.
1 This method was first proposed by Kafka (1967), in an unpublished preprint. Schmidt cites this, but has
since been generally credited with its creation. (Felten, 1976)
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