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

## 3. The corrected luminosity function

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

We can write the solution to the Eq. (2) in the form . To find the inverse of the operator we have used different approximations for the cases , and . Using Eq. (5) we get the true luminosity function, but first we need to choose a probability density for the r.v. , and also determine the extinction . 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 and the extinction . 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

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

where b is a simple function of the scale length ratio

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

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

and for the inhomogeneous model

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 is equivalent (has the same slope) to an homogeneous slab with mean opacity given by the equation

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

### 3.2. The probability density of the random variable

The probability density of the random variable 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 , and the r.v. would be uniformly distributed . 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 is a good approximation in this case. The probability density of the r.v. is obtained by the standard procedure. We get

when i is uniformly distributed, and

when .

Choosing (19) we could obtain slightly major effects for the same value of the mean opacity . 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 in Eq. (7) one gets . The Eq. (2) is then , with given by

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

In Fig. (2) we show the graph of in the case where is a Schechter function. One can prove that , therefore the graph of should cross the graph of the observed luminosity at some luminosity . In this paper we take equal to the Schechter luminosity function, with a cutoff at some small luminosity. The crossing point of the functions , 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 . Substituting (13) into (2) one gets

where is the operator

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

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

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

### 3.5. The case of great opacity

We shall reduce the Eq. (2) to the problem , considered in Sect. (3.3). In this case is close to zero (see Eq. (23)). The norm of the operator 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

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

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

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

with given by Eq. (21).

© European Southern Observatory (ESO) 1998

Online publication: December 16, 1997