### 1. HOW SBF IS MEASURED

The idea behind the measurement and interpretation of surface brightness fluctuations (SBF) is simple. Figure 1 shows an image of a globular cluster (M3) and a Virgo galaxy. As far as mean surface brightness goes, the two images are nearly identical. However the globular cluster image is much more mottled.

 Figure 1. Images of the GC M3 and the galaxy N4472.

Setting aside the technical difficulties, the measurement of SBF is straightforward:

1. Pick a patch of galaxy.
2. Measure the pixel-to-pixel rms: .
3. Measure the average flux: g.

The flux that we see in a pixel is coming from a finite number N of stars of average flux , and so we expect that g = N, and = N1/2 , so we can solve for this average flux as = 2 / g. For example, at a radius of 1 arcmin in the bulge of M31 we might see an I surface brightness of g = 16 mag sec-2. Assuming we're in the limit of no readout noise and perfect seeing so that adjacent pixels are negligibly blurred with each other (see below how to deal with the true situation), we might find that the irreduceable mottling in 1" pixels gives = 19.5 mag, i.e. about 4 percent of the mean brightness. We can then immediately solve for the magnitude corresponding to as I = 2×19.5 - 16.0 = 23.0 mag.

There are, of course, technical difficulties. The first is that seeing blurs pixels together, reducing the from what it would be at the top of the atmosphere, with only Poisson statistics operating. Since we know accurately what the point spread function (psf) is from stars in the field, we can compensate for this effect and compute what , would be in the absence of seeing. Our method is to do this in the Fourier domain, where convolution by the psf becomes multiplication by the transform of the psf.

A second challenge is that there are other sources of noise. We have some detector readout noise, and there is noise from the Poisson statistics of the photons collected as well as the Poisson noise from the finite number of stars. In our method we exploit the fact that these sources of noise are not blurred by the psf and therefore have a white noise spatial power spectrum. Figure 2 illustrates schematically what the power spectrum looks like.

 Figure 2. A schematic drawing of the SBF power spectrum.

Galaxies can also be bumpy if the stars are correlated with each other, either through their own gravity, or more likely from the gravity perturbations from a large-scale mode. Spiral arms are an example of this, and the SBF method is not applicable in such situations. However, in "hot" stellar systems, elliptical galaxies and the bulges of S0 and spiral galaxies, the scale length for such correlations is many kpc, and they produce negligible power at the pixel sizes of 20 pc. Another problem is the mottling produced by dust. We have done experiments in the bulges of M31 and M81, where we identify the dust in the B band and then measure SBF in the I band, in regions where the dust appears to be negligible as well as regions where the dust is prominent. We find that the SBF measurement is quite robust in the presence of these weak (< 10 percent) dust features, both because the dust absorption is much less in the I band, and also because the dust has a spatial power spectrum which is mostly at lower frequencies than where we fit for .

 Figure 3. A schematic cut through a galaxy image, showing noise and point sources (left), and the resulting luminosity function of the point sources (right).

The most serious problem is the removal of "point sources", i.e. foreground stars, globular clusters, and background galaxies which appear in an image. We accomplish this by using DoPhot to identify them to some completeness limit, fitting the resulting luminosity function, deleting the sources brighter than some completeness limit, and estimating the residual variance left in the image by integrating an extrapolation of our luminosity function beyond the completeness limit. This is the fundamental distance limitation to SBF, since the delectability of point sources drops radically with distance at a fixed seeing, and this compensation for residual variance is relatively uncertain.

The bottom line is that the minimal requirements for an SBF measurement are:

 Object: clean (low dust) E / S0 / spiral bulge Signal: 2 > 5(gxy + sky + readnoise), i.e. > 5 photon / . Seeing: psf < 1.0" (d / (2000 km s-1)) (I band)

Such minimal exposure and psf should give a reliable with and error of about 0.2 mag. If you improve both by a factor of two, the error will drop to about 0.1 mag, but further improvement is slowed by systematics of flattening, color terms, etc.