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9.3 Uncertainties

Fluctuation fluxes are subject to the usual photometric calibration errors. Fundamentally, we need to transfer the known magnitude of a standard star to the power spectrum of the psf that we fit to a data power spectrum, since zero-wavenumber of a power spectrum is just the squared integral of the flux of the psf. Applying the same photometric reduction to the standard star and the psf star accomplishes this transfer with an accuracy of roughly 1-2%. Uncertainties in atmospheric extinction, galactic extinction, and extinction within the galaxy vary, but can also contribute an error of 1-2% (or more if there is considerable intervening dust).

There is a substantial systematic error in fluctuation measurements if the fluctuation component of the power spectrum is buried by the white noise component. The reason for this can be seen by referring to Figure 19. The variance (power) in fluctuations is equal to the total number of photons collected per pixel times the number of photons collected per barm. The variance due to photon statistics is just the total number of photons collected. In Figure 19, where the variance has been divided by the mean number of photons per pixel from the galaxy, the photon statistics level is constant with exposure time at 1 plus the ratio of sky to galaxy brightness. The fluctuation signal, however, rises proportional to time. Until the fluctuation signal has cleanly emerged from the photon statistics background at wavenumbers which are uncontaminated by low frequency noise, it is not possible to make an accurate measurement of the fluctuation amplitude. In practice this contributes an error which is roughly 30% of (P1 / P0), or about 3% if there are 10 photons detected per barm.

Depending on the density of point sources and the depth of the photometry, Pr may be a large or small fraction of P0. The fundamental limitation in any photometry is the noise found within a region of area w2, where w is the FWHM (measured in pixels) of the psf. psf fitting photometry programs such as DoPHOT (Mateo and Schechter 1989) and DAOPHOT (Stetson 1987) approach this limit. The fundamental limitation of observations of point sources seen against a galaxy background is in the fluctuations themselves, because, unlike photon statistics, their amplitude increases proportional to time. When we have reached the point that P0 emerges from P1, we obtain no further advantage from longer observations in measuring point sources. These considerations lead to a fundamental photometry limit at a magnitude which depends on the galaxy surface brightness s (magnitude per pixel), the fluctuation apparent magnitude barm, and the signal-to-noise limit eta of the point source detection (typically 5 or greater). This limit is approximately

Equation 21 (21)

It is a coincidence that, regardless of the distance of a galaxy, the density of globular clusters at the peak of the GC luminosity function turns out to be comparable to that of background galaxies of similar brightness, since the (magnitude) luminosity function of galaxies is roughly a power law of slope -1. Thus we simultaneously have to remove GCs and background galaxies, but we can expect not to be overwhelmed by background galaxies as we push to greater distances. (In fact in the I band the relative numbers of galaxies to clusters decreases slightly with distance.)

An accurate assessment of Pr must be carried out for any region that we analyze, but a good rule of thumb is that the Pr variance per pixel is the square of the flux corresponding to mlim multiplied by the density of point sources nlim (number per pixel per magnitude) at mlim. At the distance of Virgo, in 1 arcsec seeing, with a specific frequency of globular clusters of 6 (number per MV = -15), the residual point sources contribute an error of about 6% in the V and 3% in the I band. Equation (21) shows that, given sufficient observing time to reach the photometry limit, the limiting magnitude depends on the product of distance and seeing. The errors above go roughly as the square of that product.

A source of error which is difficult to assess is the error in P0 that leaks in from low wavenumbers. Sources of this error include poor flattening, uneven sky flux from fringing in the CCD, poor fitting of the mean galaxy profile, or mottling from uneven dust absorption. At the distance of Virgo, the ratio of fluctuation rms to galaxy brightness is about 1% within a 1 arcsec area, and these effects are all small relative to 1%, provided that the detector is good and the seeing is 1 arcsec or better. With worse seeing or at greater distance these effects become appreciable, and they are certainly a concern at the 0.2% level, encountered at distances 5 times that of Virgo. We routinely estimate this contribution by performing the fluctuation analysis on blank sky and looking for excess variance above that expected from residual point sources.

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