When not understanding something the best strategy to continue is sometimes simply to 'forget' the problem and continue along the path of least resistance. Thus, while the problem ^near = ^phot, My > 12 could not be explained immediately, it turned out to be constructive to ascertain first which LF shape must be the correct one by using entirely different arguments.
In Equation (24.5) the slope of the stellar mass-luminosity relation of stars enters, giving a clue. Figure 2 in Kroupa (2002) shows the mass-luminosity data of binary stars with Kepler orbits, and demonstrates that there exists a non-linearity near 0.33 MQ such that a pronounced peak in -dm/dMy appears at My ~ 11.5, with an amplitude and width essentially identical to the maximum seen in the photometric LF at this luminosity (Figure 24.1). This agreement of
(a) the location of the maximum and
(b) the amplitude and
(c) the width of the extremum convincingly suggests stellar astrophysics to be the origin of the peak in the LF, rather than the MF. The Wielen dip (Figure 24.1) similarly results from subtle structure in the mass-luminosity relation. Thus, simply by counting stars on the sky we are able to direct our gaze within their interiors. It is the internal constitution of stars which changes with changing main-sequence mass and this is what drives the structure in the mass-luminosity relation.
Having thus established that the peak in the LF must, in fact be there where it is also found as a result of fundamental astrophysical processes, we can test this result using star clusters that constitute single-age, single-metallicity and equal-distance stellar samples. Figure 24.2 does exactly this, and a very pronounced peak is indeed evident at exactly the right location and with the right width and height (Kroupa 2002, Figure 24.1). We can therefore trust the peak in ^phot. In Figure 24.1 it can be seen that ^near also shows evidence for this peak, by noting that the peak is smeared apart in the local stellar sample because the stars have a wide spread in metallicities. The metallicity-dependence of the peak has been shown to be in agreement with the LFs of star clusters over a large range of metallicity (von Hippel et al. 1996; Kroupa & Tout 1997).
Kroupa et al. (1993) then performed a trick to get at the correct mass-luminosity relation without having to resort to theoretical or purely empirical relations, which are very uncertain in their derivatives (fig. 2 in Kroupa 2002): The Malmquist-corrected ^phot is used to define the amplitude, width and location of the extremum in the single-age, single-metallicity average dm /dMy, and integrat-
10 12 14 16 18
10 12 14 16 18
Figure 24.2. I -band LFs of stellar systems in four star clusters: globular cluster (GC) M15 (de Marchi & Paresce (1995a), distance modulus Am = m - M = 15.25 mag); GC NGC 6397 (Paresce etal. (1995), Am = 12.2); the young Galactic cluster Pleiades (Hambly et al. (1991), Am = 5.48); and GC 47 Tuc (de Marchi etal. (1995b), Am = 13.35).
ing the resulting constraint leads to a semi-empirical m (MV) relation. It is a semi-empirical relation because we used theoretical stellar models to place, so to speak, zeroth-order constraints on this relation, i.e. to prove the existence of the extremum and to estimate its location, width and amplitude. With this theoretical knowledge in hand we then used the LF to constrain the detailed run of m (MV). The result is in amazing agreement even with the most-recent high-quality binary-star constraints published by Delfosse etal. (2000). With the so-obtained m (MV) relation, whichhas the correct derivative, it is now possible to take another step towards constraining the MF.
But first the problem ^near = ^phot, MV > 12, needs to be addressed.
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