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

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3. The corrected luminosity function

In this section we shall solve the integral equation (2) for different values of the mean opacity : [FORMULA]. Let us write (2) in the form [FORMULA], where [FORMULA] is the a linear operator defined on the space of the integrable functions [FORMULA]

[EQUATION]

We can write the solution to the Eq. (2) in the form [FORMULA]. To find the inverse of the operator [FORMULA] we have used different approximations for the cases [FORMULA], and [FORMULA]. Using Eq. (5) we get the true luminosity function, but first we need to choose a probability density for the r.v. [FORMULA], and also determine the extinction [FORMULA]. Both depend on the amount and distribution of opaque matter in the galactic disk.

3.1. Distribution of matter in the galactic disk

Let us obtain the function [FORMULA] and the extinction [FORMULA]. We shall consider two models for the distribution of matter. In the first one, luminous and absorbent matter are uniformly distributed on a thin slab. In this case one gets

[EQUATION]

[EQUATION]

In the second model, the optical depth and the emission coefficient are exponentially distributed on a thin disk: [FORMULA]. In this case a simple calculation gives

[EQUATION]

where b is a simple function of the scale length ratio

[EQUATION]

In both models [FORMULA] is a monotone and convex function: [FORMULA], joining the origin to the point [FORMULA] of the [FORMULA] plane. We shall approximate this function by a polygonal

[EQUATION]

where [FORMULA] and [FORMULA] is the slope of the tangent to [FORMULA] at the origin. The values of m for each model are easily calculated. For the homogeneous slab model we obtain

[EQUATION]

and for the inhomogeneous model

[EQUATION]

With the polygonal approximation we are in fact infraestimating the effect of the inclination, but it makes the resolution of Eq. (2) easier. Taking into account expressions (15) and (16) one obtains a useful result: an inhomogeneous model with parameters [FORMULA] is equivalent (has the same slope) to an homogeneous slab with mean opacity [FORMULA] given by the equation

[EQUATION]

For instance, an inhomogeneous model with [FORMULA] and [FORMULA] is equivalent to an homogeneous model with mean optical depth [FORMULA]. This value is suggested by the observations made by Peletier et al (1995). An homogeneous model with [FORMULA], according with the Valentijn's paper (Valentijn 1990), corresponds to an inhomogeneous model with [FORMULA].

3.2. The probability density of the random variable [FORMULA]

The probability density of the random variable [FORMULA] depends on the sample of galaxies used to determine the luminosity function. If the galaxies of the sample have been formed inside a region subtending a very small solid angle, the probability density of the r.v. i will be [FORMULA], and the r.v. [FORMULA] would be uniformly distributed [FORMULA]. But if the galaxies have been formed in a large region of the sky, the r.v. i is more uniformly distributed than in the previous case. We have already shown (Leroy & Portilla 1996) that [FORMULA] is a good approximation in this case. The probability density of the r.v. [FORMULA] is obtained by the standard procedure. We get

[EQUATION]

when i is uniformly distributed, and

[EQUATION]

when [FORMULA].

Choosing (19) we could obtain slightly major effects for the same value of the mean opacity [FORMULA]. With the second option (19) it is possible to reduce the case of infinite opacity to the resolution of an Abel's integral equation, which may be solved analytically. This point is summarized in the next subsection.

3.3. The case of infinite opacity

In the case of infinite opacity the integral Eq. (2) may be solved analytically. Taking the limit [FORMULA] in Eq. (7) one gets [FORMULA]. The Eq. (2) is then [FORMULA], with [FORMULA] given by

[EQUATION]

Substituting the probability density (18) into (20), the equation [FORMULA] is reduced to an Abel's integral equation, which may be solved analytically [FORMULA]:

[EQUATION]

In Fig. (2) we show the graph of [FORMULA] in the case where [FORMULA] is a Schechter function. One can prove that [FORMULA], therefore the graph of [FORMULA] should cross the graph of the observed luminosity [FORMULA] at some luminosity [FORMULA]. In this paper we take [FORMULA] equal to the Schechter luminosity function, with a cutoff at some small luminosity. The crossing point of the functions [FORMULA], [FORMULA] lays outside the interval shown in the figure.

3.4. The case of low opacity

To solve the integral equation (2) for finite values of the mean opacity we shall use the polygonal approximation to the function [FORMULA]. Substituting (13) into (2) one gets

[EQUATION]

where [FORMULA] is the operator

[EQUATION]

Let us denote by [FORMULA] the norm of the operator [FORMULA] in the space of integrable functions. A simple calculation gives

[EQUATION]

Provided that [FORMULA] the Eq. (22) has a solution which may be expressed by a convergent series in powers of [FORMULA] (Kolmogorov& Fomin 1956)

[EQUATION]

The value of [FORMULA] depends on the choice of the probability density [FORMULA]. We have [FORMULA] in the case of (18) and [FORMULA] in the case of (19). The values of [FORMULA] depend on the distribution of opaque matter in the galactic disk. Taking into account (15) and (16) one gets [FORMULA] and [FORMULA] for the homogeneous and inhomogeneous model respectively.

3.5. The case of great opacity

We shall reduce the Eq. (2) to the problem [FORMULA], considered in Sect. (3.3). In this case [FORMULA] is close to zero (see Eq. (23)). The norm of the operator [FORMULA] in Eq. (22) is not smaller than unity and the equation cannot be solved as previously shown (Sect. (3.4). Let us write the Eq. (22) in the form

[EQUATION]

The operator [FORMULA] will be close to A for [FORMULA], so, we change [FORMULA] by [FORMULA] in Eq. (30) , choosing [FORMULA] such that the operators [FORMULA] and [FORMULA] have equal norm. So, we obtain the equation

[EQUATION]

The operator A is invertible, hence, multiplying the previous equation by [FORMULA] we get

[EQUATION]

It can be seen that the norm of the operator [FORMULA] is [FORMULA], therefore, provided that [FORMULA], Eq. (32) may be solved as power series in [FORMULA]

[EQUATION]

with [FORMULA] given by Eq. (21).

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© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997
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