Milky Way Galaxy The stellar luminosity functionastronomy

The stellar luminosity function is a description of the relative number of stars of different absolute luminosities. It is often used to describe the stellar content of various parts of the Galaxy or other groups of stars, but it most commonly refers to the absolute number of stars of different absolute magnitudes in the solar neighbourhood. In this form it is usually called the van Rhijn function, named after the Dutch astronomer Pieter J. van Rhijn. The van Rhijn function is a basic datum for the local portion of the Galaxy, but it is not necessarily representative for an area larger than the immediate solar neighbourhood. Investigators have found that elsewhere in the Galaxy, and in the external galaxies (as well as in star clusters), the form of the luminosity function differs in various respects from the van Rhijn function.

The detailed determination of the luminosity function of the solar neighbourhood is an extremely complicated process. Difficulties arise because of (1) the incompleteness of existing surveys of stars of all luminosities in any sample of space and (2) the uncertainties in the basic data (distances and magnitudes). In determining the van Rhijn function, it is normally preferable to specify exactly what volume of space is being sampled and to state explicitly the way in which problems of incompleteness and data uncertainties are handled.

In general there are four different methods for determining the local luminosity function. Most commonly, trigonometric parallaxes are employed as the basic sample. Alternative but somewhat less certain methods include the use of spectroscopic parallaxes, which can involve much larger volumes of space. A third method entails the use of mean parallaxes of a star of a given proper motion and apparent magnitude; this yields a statistical sample of stars of approximately known and uniform distance. The fourth method involves examining the distribution of proper motions and tangential velocities (the speeds at which stellar objects move at right angles to the line of sight) of stars near the Sun.

Because the solar neighbourhood is a mixture of stars of various ages and different types, it is difficult to interpret the van Rhijn function in physical terms without recourse to other sources of information, such as the study of star clusters of various types, ages, and dynamical families. Globular clusters are the best samples to use for determining the luminosity function of old stars having a low abundance of heavy elements (Population II stars).

Globular-cluster luminosity functions show a conspicuous peak at absolute magnitude M_V = 0.5, and this is clearly due to the enrichment of stars at that magnitude from the horizontal branch of the cluster. The height of this peak in the data is related to the richness of the horizontal branch, which is in turn related to the age and chemical composition of the stars in the cluster. A comparison of the observed M3 luminosity function with the van Rhijn function shows a depletion of stars, relative to fainter stars, for absolute magnitudes brighter than roughly M_V = 3.5. This discrepancy is important in the discussion of the physical significance of the van Rhijn function and luminosity functions for clusters of different ages and so will be dealt with more fully below.

Many studies of the component stars of open clusters have shown that the luminosity functions of these objects vary widely. The two most conspicuous differences are the overabundance of stars of brighter absolute luminosities and the underabundance or absence of stars of faint absolute luminosities. The overabundance at the bright end is clearly related to the age of the cluster (as determined from the main-sequence turnoff point) in the sense that younger star clusters have more of the highly luminous stars. This is completely understandable in terms of the evolution of the clusters and can be accounted for in detail by calculations of the rate of evolution of stars of different absolute magnitudes and mass. For example, the luminosity function for the young clusters h and χ Persei, when compared with the van Rhijn function, clearly shows a large overabundance of bright stars due to the extremely young age of the cluster, which is on the order of 10⁶ years. Calculations of stellar evolution indicate that in an additional 10⁹ or 10¹⁰ years all of these stars will have evolved away and disappeared from the bright end of the luminosity function.

In 1955 the first detailed attempt to interpret the shape of the general van Rhijn luminosity function was made by the Austrian-born American astronomer Edwin E. Salpeter, who pointed out that the change in slope of this function near M_V = +3.5 is most likely the result of the depletion of the stars brighter than this limit. Salpeter noted that this particular absolute luminosity is very close to the turnoff point of the main sequence for stars of an age equal to the oldest in the solar neighbourhood—approximately 10¹⁰ years. Thus, all stars of the luminosity function with fainter absolute magnitudes have not suffered depletion of their numbers because of stellar evolution, as there has not been enough time for them to have evolved from the main sequence. On the other hand, the ranks of stars of brighter absolute luminosity have been variously depleted by evolution, and so the form of the luminosity function in this range is a composite curve contributed by stars of ages ranging from 0 to 10¹⁰ years. Salpeter hypothesized that there might exist a time-independent function, the so-called formation function, which would describe the general initial distribution of luminosities, taking into account all stars at the time of formation. Then, by assuming that the rate of star formation in the solar neighbourhood has been uniform since the beginning of this process and by using available calculations of the rate of evolution of stars of different masses and luminosities, he showed that it is possible to apply a correction to the van Rhijn function in order to obtain the form of the initial luminosity function. Comparisons of open clusters of various ages have shown that these clusters agree much more closely with the initial formation function than with the van Rhijn function; this is especially true for the very young clusters. Consequently, investigators believe that the formation function, as derived by Salpeter, is a reasonable representation of the distribution of star luminosities at the time of formation, even though they are not certain that the assumption of a uniform rate of formation of stars can be precisely true or that the rate is uniform throughout a galaxy.

It was stated above that open-cluster luminosity functions show two discrepancies when compared with the van Rhijn function. The first is due to the evolution of stars from the bright end of the luminosity function such that young clusters have too many stars of high luminosity, as compared with the solar neighbourhood. The second discrepancy is that very old clusters such as the globular clusters have too few high-luminosity stars, as compared with the van Rhijn function, and this is clearly the result of stellar evolution away from the main sequence. Stars do not, however, disappear completely from the luminosity function; most become white dwarfs and reappear at the faint end. In his early comparisons of formation functions with luminosity functions of galactic clusters, Sandage calculated the number of white dwarfs expected in various clusters; present searches for these objects in a few of the clusters (e.g., the Hyades) have supported his conclusions.

Open clusters also disagree with the van Rhijn function at the faint end—i.e., for absolute magnitudes fainter than approximately M_V = +6. In all likelihood this is mainly due to a depletion of another sort, the result of dynamical effects on the clusters that arise because of internal and external forces. Stars of low mass in such clusters escape from the system under certain common conditions. The formation functions for these clusters may be different from the Salpeter function and may exclude faint stars. A further effect is the result of the finite amount of time it takes for stars to condense; very young clusters have few faint stars partly because there has not been sufficient time for them to have reached their main-sequence luminosity.