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luminosity functionThe relative numbers of objects of different luminosities in a standard volume of space (usually a cubic parsec or cubic megaparsec). The luminosity may be in the optical band, the radio band, or at any other defined waveband. Observationally the luminosity function is determined as a histogram; theoreticians fit this with an algebraic expression.
In optical astronomy the luminosity function is given the symbol Φ(M) and is the number of stars per cubic parsec or the number of galaxies per cubic megaparsec with absolute magnitudes M ± ½. The luminosity function for stars within 10 parsecs of the Sun shows a peak at an absolute magnitude of about +15: there is thus a predominance of intrinsically faint stars in the solar neighborhood. The luminosity function for any given magnitude interval is not the same everywhere in the Galaxy: for instance, the luminosity function of a globular cluster is different from that of an open cluster. Stars of all spectral types contribute to the value Φ(M ). The specific luminosity function Φ(M,S) considers one spectral type at a time, each showing a peak distribution at a different mean magnitude.
The luminosity function of galaxies (or quasars) is usually expressed as the number of galaxies in a given luminosity interval per cubic megaparsec and has a characteristic shape that is almost flat at faint magnitudes and falls off sharply at the bright end. The behavior of the bright and faint end of the luminosity function is divided by the ‘typical’ galaxy with an absolute magnitude of around –21.5, known as the L* galaxy , at the ‘knee’ of the luminosity function.
The evolution of the quasar luminosity function is usually expressed as one of two forms of behavior: either the luminosity of all quasars is assumed to increase uniformly with redshift, or the number of quasars increases with redshift. In practice, present quasar luminosity functions fit a combination of the two models. See also luminosity–volume test.
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.
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