The Initial Mass Function

Any attempt to understand the origin of stellar groups must address the issue of their internal mass distribution. It is not obvious, of course, that any single function will adequately describe all existing systems. In principle, the natural variation in such environmental factors as the ambient magnetic field or the molecular cloud temperature prior to cluster formation could yield a wide variety of distributions. However, we have already seen from numerous examples that massive stars are intrinsically rarer than their low-mass counterparts. We now seek to quantify this notion. As a practical matter, the masses of embedded stars are difficult to obtain empirically, so we first look to field stars in the solar neighborhood. We will then show that the mass distribution found here also appears to hold, at least approximately, for discrete clusters and associations. This important finding bolsters the view that all stars are born within such groups.

4.5.1 Luminosities Past and Present

Even for an unobscured field star at a known distance, it is the luminosity within a certain wavelength range, rather than the mass, that is directly observable. A fundamental statistical property of field stars is thus the general luminosity function, $ (MV). This function is defined so that $ (MV) AMV is the number of stars per cubic parsec in the solar neighborhood with absolute visual magnitude between MV - AMV/2 and MV + AMV /2. Obtaining the general luminosity function is no trivial matter. One must derive distances to large numbers of stars and make proper extrapolations for the even larger numbers whose distances are unavailable directly. Other, more subtle complications abound. For example, the Galactic scale heights of stars vary inversely with mass, the brightest stars hovering close to the midplane during their relatively short lifetimes. These same stars can be seen out to distances much greater than their scale heights. Thus, they appear to occupy flattened disks, whose volumes must be accurately assessed when obtaining densities. Beginning with the pioneering efforts of P. J. van Rhijn in

the 1920s, such difficulties have gradually been overcome, and a modern result is displayed as the solid curve in Figure 4.21.

Over most of the magnitude range shown, the general luminosity function rises with increasing MV, i. e., there are more dim stars than bright ones in any fixed magnitude interval. This trend continues up to MV ~ +12, after which there is a steady decline. The details of this slow falloff remain somewhat uncertain, since many of the observed "stars" in this regime are actually binaries that have not been spatially resolved. On the other hand, there is no doubt that the slope of §(MV) is considerably steeper for the most luminous stars. This initial sharp rise levels off rather abruptly at about My = +5. Reference to Table 1.1 reveals that a star at this transitional magnitude has a main-sequence lifetime tMS slightly greater than 1010 yr. This time is close to the age of the Galactic disk, currently estimated at tgal = 1 x 1010 yr. If we recall that any star's post-main-sequence lifetime is brief compared to tMS, the origin of the change in slope becomes clear. Relatively dim, low-mass stars with MV > My have been accumulating steadily over the lifetime of the Galaxy, while only a fraction of the brighter, short-lived stars with MV < My have survived.

It is apparent, then, that $(MV) itself does not accurately reflect the relative production rate of stars with various MV-values. However, the foregoing argument can readily be quantified to make the necessary modification. Following the notation of Chapter 1, let my(t) be the total Galactic star formation rate per square parsec near the solar position. Notice that we use the rate integrated over the disk thickness, in order to account for the diffusion of stars from the midplane during their evolution. To a fair approximation, in fact, the stellar volume density falls off exponentially away from the plane, with a scale height H that is a function of MV. We further define the initial luminosity function MV) to be the relative frequency with which stars of a given MV first appear. This function is normalized to unity: / ^(Mv) dMV = 1. Resetting My to that magnitude for which tMS = tgal precisely, we can write the general luminosity function as an integral over time, where the integration limits depend on MV:

f fy . dtm*(t)V(Mv) [2H(Mv)]-1 if Mv < My $(Mv) = J v (4.4)

[ /0igal dtm y(t) $(Mv) [2 H (Mv )]-1 if Mv > My .

In writing equation (4.4), we have ignored any possible time-dependence in ^(Mv) or H(Mv). Furthermore, we are actually interested in the appearance of main-sequence stars with various magnitudes, while $(Mv) encompasses bright field stars that are giants and super-giants. Accordingly, the luminosity function on the left side of equation (4.4) must be diminished at the lowest values of Mv, corresponding to the brightest stars.

After making this correction, knowledge of the main-sequence lifetimes tMs (Mv) allows us to invert equation (4.4) and obtain ^(Mv), provided we know the rate my(t). Neither theory nor observation is of much help in this regard, beyond the general statement that my(t) should diminish with time. Fortunately, the final result is rather insensitive to the prescription adopted here, so we follow the standard expedient of ignoring the time dependence and adopting a fixed rate. With ^(Mv) in hand, it is a straightforward matter to apply bolometric corrections and obtain the relative birthrates of stars as a function of Lbol rather than Mv. This is ^(Lbol), the form of the initial luminosity function already shown in Figure 4.13. As anticipated, the curve here is smoother than $(Mv), since it lacks the age-dependent falloff for the brightest stars.

4.5.2 Character of the Mass Distribution

Our true goal, however, is the distribution at birth of various stellar masses. We accordingly define £ (My) to be the initial mass function (IMF), the relative number of stars produced per unit mass interval. Again normalizing this function to unity, we have simply m*) = HMv)d-^-. (4.5)

The derivative on the righthand side refers to variations along the main sequence and can be obtained numerically from Table 1.1 or its equivalent.3

Historically, E. E. Salpeter proceeded in the manner we have outlined to find that £ (My) varies as My, with 7 = -2.35. This simple power law is still frequently employed to obtain approximate results, but has long since been supplanted by other investigations using more extensive data. Figure 4.22 displays the results of a later study. For convenience, we may approximate the mass function as a sequence of power laws:

£ (My) = <( C (My/Mo)-2'7 1.0 < My/MQ < 10 (4.6)

3 Many authors define the IMF as the relative number of stars per logarithmic mass interval, i. e., as Mt £(Mt). The reader should check carefully in each case.

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