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Astron. Astrophys. 329, 840-844 (1998)

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2. The observed and the true luminosity function

If the galactic disk is considered as a mixture of luminous and absorbent matter one has a relation between the apparent luminosity l and the inclination i of the galactic plane

[EQUATION]

where [FORMULA], and [FORMULA] is the apparent luminosity when the galaxy is seen face on, with [FORMULA]. The function [FORMULA] depends on how luminous and absorbent matter are distributed throughout the disk. This is considered in Sect. (3.1). Absolute luminosity is determined from the apparent luminosity and the distance to the galaxy, therefore, a similar relation stands for the absolute luminosity [FORMULA]. If we knew the inclination of the galactic disc we could use this to determine the face on absolute luminosity [FORMULA] from that obtained directly from the apparent luminosity. As this is not the case, we can proceed alternatively considering [FORMULA] and [FORMULA] as random variables with probability densities [FORMULA] respectively. The first one will be identified with the observed luminosity function, the second one is the face-on luminosity function, the probability of the random variable [FORMULA] will be considered below). These probability functions are related by an integral equation (Papoulis 1965)

[EQUATION]

We shall take two functions for representing the observed LF (see the Introduction). One is a gaussian with mean of [FORMULA] and a dispersion of [FORMULA] mag, and the other one a Schechter function

[EQUATION]

with parameters : [FORMULA]. Or, in terms of absolute magnitudes

[EQUATION]

The function [FORMULA] is knwon in a range of values of L. A cutoff (or an inflection point) for low luminosities is necessary to make the integral of [FORMULA] finite. Once the integral Eq. (2) has been solved, and the face-on LF [FORMULA] has been determined, one must correct for face on extinction, [FORMULA], to determine the true absolute magnitude [FORMULA]. So, the probability density function of the true absolute magnitude will be

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997
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