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Figure 4.22 Initial mass function for solar neighborhood stars. The dashed line has the same slope as Salpeter's power law.

where C is a normalization constant. It is apparent that £ (M*) is considerably flatter than Salpeter's function (dashed line) below 1.0 M& and approaches it for M* > 10 Me. Both of these general features have been amply confirmed. Our simple power law does not capture the broad maximum near 0.1 MQ seen in the figure, but the true behavior at the lowest masses remains unclear. Their small luminosities make objects in this regime difficult to detect. After extensive searches, a growing population of brown dwarfs, i. e., objects less massive than the hydrogen-burning limit of 0.08 Me, is presently being uncovered. These and other data indicate that £ (M*) is relatively flat near the brown-dwarf limit. Establishing its precise form will require additional effort.

This lingering uncertainty should not obscure the essential message of Figure 4.22. Within any volume undergoing star formation, the number of new stars per unit mass falls rapidly above roughly 0.1 M0. If we make the simplifying assumption that £ (M*) is constant below this value, then equation (4.6) implies that half of all stars are produced with M* > 0.2 Me. Only 12 percent have masses exceeding 1 Me, while the fraction drops to 0.3 percent for stars above 10 M&. Conversely, 70 percent of stars have M* > 0.1 Me. We conclude that the star formation process yields objects with a characteristic mass of a few tenths of M&. Things might have been otherwise. One could imagine stars forming with a pure power-law mass spectrum down to some very low level at the planetary scale. That this is not the case is surely significant. Unfortunately, current theory cannot explain, in any convincing manner, the form of £ (M*). Even the origin of the basic mass scale itself remains uncertain. Needless to say, we shall revisit this central issue in later chapters.

4.5.3 Mass Function in Stellar Groups

It is worth reemphasizing that £ (M*) represents an average over thousands of field stars, i. e., objects outside of known clusters or associations. What is the corresponding function within such groups? In any one system of a few hundred stars or less, statistical fluctuations become significant when addressing this issue. With this caveat in mind, the best candidates for investigation are the relatively unobscured open clusters. Here, the members are no longer accreting molecular gas, while the stellar masses themselves can be read with confidence from the HR diagram, given knowledge of both L* and Teff. On the other hand, the main-sequence turnoffs limit the highest observed masses to modest values, about 15 Mq for a cluster age of 107 yr. One must turn to OB associations to probe the upper end of the distribution. Unfortunately, both the greater distances and the presence of main-sequence turnons make the lowest-mass members difficult to access. Since no one system is ideal, one is forced to sample the mass spectrum within groups in a piecemeal fashion.

Figure 4.23 illustrates the complementary roles of open clusters and OB associations. The left panel shows the number of stars per unit mass, denoted £cluster(M*), for most known members of the Pleiades, along with the curve from equation (4.6). The flattening below M* « 0.15 Mq is similar to that seen in Figure 4.22 for field objects. Note that the luminosities and effective temperatures of many cluster members were obtained here from their Rand I-band magnitudes. The stellar masses then followed by comparison with theoretical pre-main-sequence tracks. In contrast, single measurements at V sufficed for the more massive stars, which are certain to be on the main sequence. It is apparent that the mass distribution in this accessible and populous system also matches the field (i. e., equation (4.6)) between 0.15 and 5 Mq. Other open clusters yield mass functions that are similar but exhibit significant variation. Such deviations from the field-star result do not appear to be correlated with cluster morphology or age, and largely disappear if one adds together the populations of at least a dozen systems.

Turning to OB associations, the most reliable procedure is to focus on rich, spatially compact subgroups. One such subgoup is NGC 6611, a cluster within the Serpens OB1 association, 2.2 kpc distant. The stars here illuminate the Eagle Nebula (M16), long known as a visible HII region traversed by broad lanes of obscuration. Most of the stars in NGC 6611 are still embedded within the local molecular cloud. However, the 150 or so members above 5 Mq are bright enough that they can be placed in the HR diagram through optical measurements. Figure 4.23b shows the masses at this upper end of the distribution. A power-law falloff is evident. Indeed, the best-fit line has r = dlog£/dlog(M*/MQ) = —2.1, close to the Salpeter value of —2.35. Other, less populous clusters within associations have T-values that range from — 1.7 to —3.0 and a mean consistent with the slope of the field-star initial mass function.

We conclude, then, that all groups of sufficient membership display a similar decline in population with mass, at least above the brown-dwarf regime. The steepness of this falloff for stars exceeding several Mq makes it difficult to obtain the complete mass or population of any group by extrapolation from its very brightest components. We stress that the actual form of the initial mass function must currently be viewed as a purely empirical result, one whose proper explanation awaits better understanding of both cluster formation and the termination of protostellar collapse. In particular, the absence of any obvious breaks in the observed distribution does not necessarily imply that a single mechanism is at work in the origin of all stellar masses. On the

Stellar Mass log (MJMq)

Figure 4.23 Empirical mass functions for (a) the Pleiades, and (b) the cluster NGC 6611. The solid curves represent the initial mass function from equation (4.6).

contrary, their distinctive central concentration in bound clusters suggests that the most massive stars form very differently from their more numerous, low-mass counterparts.

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