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 of the point source detection (typically 5 or greater). This limit is approximately
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.