Luminosity Function

luminosity function [‚lü·mə′näs·əd·ē ‚fəŋk·shən]

(astronomy)

The functional relationship between stellar magnitude and the number and distribution of stars of each magnitude interval. Also known as relative luminosity factor.

(optics)

A standard measure of the response of an eye to monochromatic light at various wavelengths; the function is normalized to unity at its maximum value. Also known as luminosity curve; spectral luminous efficiency; visibility function.

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Luminosity Function 

an empirical relation characterizing distributions of stars by luminosity or by absolute stellar magnitude. The luminosity function ϕ(M) makes it possible to determine the fraction N of stars located in some volume of space that have absolute stellar magnitudes lying in the range from M to M + dM. Sometimes, the function Ф(M) = D(r)ϕ(M), which makes it possible to calculate the absolute number of stars of a given stellar magnitude occurring in a unit volume (usually 10³ parsec³), is called the luminosity function. Here, D(r) is the density of stars in space. In some cases, the luminosity function is considered for stars of different spectral classes.

Various methods of determining the luminosity function have been worked out; in any method, the main difficulty is to introduce corrections that allow for the incompleteness of the data used. The function ϕ(M) can be determined by isolating a number of stars up to some apparent stellar magnitude and determining by some method the absolute stellar magnitude M for each star. It is necessary to remember that stars of different luminosity are located at different distances from the observer and thus exist in different volumes of space. If all the known stars within some distance are selected for use in determining ϕ(M), then such a selection will not affect the computed result as much; however, this method does not permit the determination of the density of stars of high luminosity, since the probability that they will fall within a small volume (less than 10 par-secs in cross section) is small, and only within such a distance from the sun may all stars be assumed to be known.

An indirect method of determining the luminosity function is based on the statistical relations between the parallaxes, proper motions, and apparent stellar magnitudes. This method was first used by J. Kapteyn in 1902 and was later used repeatedly by other investigators.

The luminosity function for the neighborhood of the sun is shown in Figure 1. The function exhibits appreciable asymmetry. Initially, as the luminosity decreases, the function increases, reaching a maximum at M ≈ + 15; it then begins to decrease rapidly. This decrease, however, is apparently due to the incompleteness of our knowledge of stars of low luminosity.

Figure 1. Luminosity function of stars in the neighborhood of the sun

The type of luminosity function depends on the composition of the stellar population and is different for different parts of the Galaxy. Given the luminosity function, it is possible to estimate the total mass of stars in the Galaxy on the basis of the mass-luminosity relation and to determine the stellar density by solving the integral equations of stellar statistics.

E. D. PAVLOVSKAIA