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2.1.3. Mass-to-light Ratio of the Spheroid Component

The mass-to-light ratio of stars and their remnants in spheroid populations can be inferred from star population synthesis calculations, assuming the stars are close to coeval (so that the integrated B luminosity is not much affected by recent episodes of star formation) and the spheroid stars in our neighborhood are representative. Within these assumptions the most serious uncertainty in the calculation is the lower cut off of the initial mass function (the IMF). In the past the cutoff was somewhat arbitrary, between 0.1 Msun and 0.2 Msun, and the choice seriously affected the estimate of M/L. Thus for the Salpeter IMF M differs by 24% between these two choices, while L is virtually unaffected. This uncertainty for the local IMF has been removed by the recent advance in observation of M subdwarf star counts with the Hubble Space Telescope: it has been shown that the IMF of the local Galactic stars has a turnover at about 0.4 Msun (Gould, Bahcall & Flynn 1996; hereafter GBF), and the integral over the GBF mass function converges without an arbitrary cut off. For M > Msun the GBF mass function agrees well with the Salpeter function. Using the GBF mass function for M < Msun, instead of the Salpeter mass function with x = 1.35 (where x is the index of the IMF dn/dM propto M-[1+x]), the mass integral is 0.70 times that obtained with the Salpeter mass function cut off at 0.15 Msun. The light in the blue-visual bands contributed by stars less massive than one solar mass is small: from Table 4 of Tinsley & Gunn (1976), we estimate that the V light integral decreases only 2% with the use of the GBF mass function. Therefore, the mass-to-light ratio with the GBF mass function is (M/LV)GBF = 0.72 (M/LV)Salpeter, where the latter ratio is computed for the cutoff at 0.15 Msun. Charlot, Worthey, & Bressan (1996) have given a compilation of mass-to-light ratios from various stellar population synthesis calculations. They show that M/LV is reasonably consistent among authors when the lower mass cutoff is fixed. From the Charlot et al. (1996) compilation of population synthesis calculations, the GBF IMF, and the conversion formula discussed above, we obtain (M/LV)GBF = (4.0 ± 0.3) + 0.38 (tG - 10 Gyr), where tG is the age of the spheroid and the error represents the model-dependent uncertainty. Using the Charlot et al. (1996) synthetic calculation of the B - V color, we obtain M/LB = (5.4 ± 0.3) + 0.7 (tG - 10 Gyr). For tG = 12 ± 2 Gyr we have M/LV = 4.0-5.9, or

Equation 5 (5)

The M/L ratio in equation (5) may be compared to the values inferred from the kinematics of the nuclear regions of elliptical galaxies. van der Marel (1991) finds M/LR = (6.64 ± 0.28)h (Johnson R), which translates to M/LV = (7.72 ± 0.33)h or

Equation 6 (6)

using < V - RJ > = 0.68, corresponding to the average color < B - V > = 0.92 for the 37 elliptical galaxies he used (< B - V >sun = 0.65 and < V - RJ >sun = 0.52 are also used). There is excellent consistency between equations (5) and (6) if the Hubble constant is in the range h = 0.6-0.8 and spheroid populations dominate the the mass of these nuclear regions.

Dynamical measures show M/L depends on the luminosity in elliptical galaxies (Kormendy 1986): van der Marel (1991) finds M/L = (L/L*)0.35 ± 0.05. This manifestation of the color-magnitude relation of early-type galaxies probably reflects the effect of metallicity and perhaps also of dark matter halos. In the former case we estimate this dependence may reduce the effective mean value of M/L weighted by the contribution to the luminosity density by 17%, and accordingly reduce the lower end of the allowed range by this amount.

We conclude that a good estimate of the mass-to-light ratio of stars and their remnants in spheroids is

Equation 7 (7)

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