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The interpretation of what we are seeing is straightforward. The portion of a galaxy projecting onto a pixel might have ni stars of luminosity Li (Figure 4).

Figure 4

Figure 4. An HR diagram, showing a contributor to the SBF sum.

Over many samples, Poisson statistics imply that each star contributes a luminosity variance of Li2, so the total luminosity variance is sigma2 = sumni Li2. The mean total luminosity within the pixel is just g = sum niLi, so we are therefore measuring the second moment of the luminosity function divided by the first:

Equation 2.3 (2.3)

bar L varies as a function of color, and depends on stellar population. It is therefore not a perfect "standard candle" (although its total range of variation is much less than most objects which are called standard candles!), but it does vary in a predictable way. We have concentrated on the I band because the original models we calculated using the Revised Yale Isochrones indicated that:

  1. bar I is the brightest M bar in the optical.
  2. Age, metallicity, IMF are degenerate, and bar I is nearly a one-parameter family.
  3. bar I has a very small scatter with respect to its trend with (V - I).
  4. bar I has a very small overall variation with population.

Worthey has subsequently computed better models which confirm items 1-3 above, but we have found that the slope of M barI is steeper than we had thought, with a total change of about 1 mag over the full range of early-type galaxy stellar population. His models are illustrated in Figure 5. Note how the age and metallicity are degenerate in the optical, but become distinct in the IR.

Figure 5

Figure 5. SBF luminosity derived from theoretical models (left) and a blowup of how SBF depends on metallicity and age in the IR.

There is an abrupt change in the behavior of M bar in the IR, which arises because the stars at the tip of the RGB, which have the highest bolometric luminosity, are shrouded in the optical and have a lower flux in the optical than stars lower down the RGB. The IR is potentially better for SBF but

  1. More sensitive to AGB (needs work),
  2. Sky is much brighter (not in outer space, though),
  3. Detectors are worse (but catching up fast),
  4. Metallicity and age become non-degenerate.

To summarize the SBF method then, the measurement is repeatable and perfectible. Given a suitable system and sufficient photons, resolution, and calibration, it is possible to measure m bar to arbitrary accuracy, certainly better than 10 percent. The interpretation also has a solid foundation. Models of stellar populations depend on metallicity, age, IMF, post-RGB evolution, etc. The constraints we can measure include distance independent quantities such as colors, fluctuation colors, and line strengths. If the distance is known, we can add to this list fluctuation absolute magnitudes as well. The models and constraints meet in a library of isochrones which are combined to form a luminosity function and spectral energy distribution. Integration of the SED yields M bar. Both empirical and theoretical work in the I band at least indicates that M barI can be constrained to approx 0.05 mag by measurement of just the (V - I) color.

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