The Milky Way and Beyond (6 page)

A further difference between the two populations became clear as the study of stellar evolution advanced. It was found that Population II was exclusively made up of stars that are very old. Estimates of the age of Population II stars have varied over the years, depending on the degree of sophistication of the calculated models and the manner in which observations for globular clusters are fitted to these models. They have ranged from 10
⁹
years up to 2 × 10
¹⁰
years. Recent comparisons of these data suggest that the halo globular clusters have ages of approximately 1.1–1.3 × 10
¹⁰
years. The work of American astronomer Allan Sandage and his collaborators proved without a doubt that the range in age for globular clusters was relatively small and that the detailed characteristics of the giant branches of their colour-magnitude diagrams were correlated with age and small differences in chemical abundances.

On the other hand, stars of Population I were found to have a wide range of ages. Stellar associations and galactic clusters with bright blue main-sequence stars
have ages of a few million years (stars are still in the process of forming in some of them) to a few hundred million years. Studies of the stars nearest the Sun indicate a mixture of ages with a considerable number of stars of great age—on the order of 10
⁹
years. Careful searches, however, have shown that there are no stars in the solar neighbourhood and no galactic clusters whatsoever that are older than the globular clusters. This is an indication that globular clusters, and thus Population II objects, formed first in the Galaxy and that Population I stars have been forming since.

In short, as the understanding of stellar populations grew, the division into Population I and Population II became understood in terms of three parameters: age, chemical composition, and kinematics. (A fourth parameter, spatial distribution, appeared to be clearly another manifestation of kinematics.) The correlations among these three parameters were not perfect but seemed to be reasonably good for the Galaxy, even though it was not yet known whether these correlations were applicable to other galaxies. As various types of galaxies were explored more completely, it became clear that the mix of populations in galaxies was correlated with their Hubble type, which separates galaxies into ellipticals, barred spirals, and spirals without bars. Spiral galaxies such as the Milky Way Galaxy have Population I concentrated in the spiral disk and Population II spread out in a thick disk and/or a spherical halo. Elliptical galaxies are nearly pure Population II, while irregular galaxies are dominated by a thick disk of Population I, with only a small number of Population II stars. Furthermore, the populations vary with galaxy mass; while the Milky Way Galaxy, a massive example of a spiral galaxy, contains no stars of young age and a low heavy-metal abundance, low-mass galaxies, such as the dwarf irregulars, contain young, low heavy-element stars, as the buildup of heavy elements in stars has not proceeded far in such small galaxies.

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 10
⁶
years. Calculations of stellar evolution indicate that in an additional 10
⁹
or 10
¹⁰
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 10
¹⁰
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 10
¹⁰
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 has been stated 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.

The density distribution of stars near the Sun can be used to calculate the mass density of material (in the form of stars) at the Sun's distance within the Galaxy. It is therefore of interest not only from the point of view of stellar statistics but also in relation to galactic dynamics. In principle, the density distribution can be calculated by integrating the stellar luminosity function. In practice, because of uncertainties in the luminosity function at the faint end and because of variations at the bright end, the local density distribution is not simply derived nor is there agreement between different studies in the final result.

In the vicinity of the Sun, stellar density can be determined from the various surveys of nearby stars and from estimates of their completeness. For example, Wilhelm Gliese's catalog of nearby stars, a commonly used resource, contains 1,049 stars in a volume within a radius of 65 light-years. This is a density of about 0.001 stars per cubic light-year. However, even this catalog is incomplete, and its incompleteness is probably attributable to the fact that it is difficult to detect the faintest stars and faint companions, especially extremely faint stars such as brown dwarfs.

In short, the true density of stars in the solar neighbourhood is difficult to establish. The value most commonly quoted is 0.003 stars per cubic light-year, a value obtained by integrating the van Rhijn luminosity function with a cutoff taken
M
= 14.3. This is, however, distinctly smaller than the true density as calculated for the most complete sampling volume discussed above and is therefore an underestimate. Gliese has estimated that when incompleteness of the catalogs is taken into account, the true stellar density is on the order of 0.004 stars per cubic light-year, which includes the probable number of unseen companions of multiple systems.