National Optical Astronomy Observatory, 950 North Cherry Avenue, Tucson, AZ 85719, Email: mighell@noao.edu

Using a simple 2-D Gaussian Point Spread Function (PSF) on a
constant (flat) sky background, I derive a theoretical photometric and
astrometric performance model for analytical and digital PSF-fitting stellar
photometry. The theoretical model makes excellent predictions for the
photometric and astrometric performance of over-sampled and under-sampled CCD
stellar observations even with cameras with pixels that have large
* intra*-pixel quantum efficiency variations. The performance model
accurately predicts the photometric and astrometric performance
of realistic space-based observations from segmented-mirror telescope concepts
like the Next Generation Space Telescope with the MATPHOT algorithm for
digital PSF CCD stellar photometry which I presented last year at ADASS XI.
The key PSF-based parameter of the theoretical performance model is the
effective background area which is defined to be the reciprocal of the volume
integral of the square of the (normalized) PSF; a
critically-sampled PSF has an effective background area of
() pixels. A bright star with a million photons can
theoretically simultaneously achieve a signal-to-noise ratio of 1000 with
a (relative) astrometric error of a * milli*pixel. The photometric
performance is maximized when either the effective background area or the
effective-background-level measurement error is minimized.
Real-world considerations, like the use of poor CCD flat fields to calibrate
the observations, can and do cause many existing space-based and
ground-based CCD imagers to fail to live up to their theoretical performance
limits. Future optical and infrared imaging instruments can be designed and
operated to avoid the limitations of some existing space-based and ground-based
cameras. This work is supported by grants from the Office of Space Science
of the National Aeronautics and Space Administration (NASA).

Let us assume that the variance of the noise associated with the
pixel of an observation of a bright star is due only to
stellar photon noise,

where is the normalized

as expected from photon statistics with a normalized unsampled PSF ().

Let us assume that we can replace the measurement error associated with the
pixel of an observation of a faint star
with an average * constant* rms value of

where is the constant background level in electrons per pixel ( ) and is the square of the rms readout noise ( ). Using this approximation, we find that the variance of the stellar intensity measurement of faint

where the constant is the ``effective background area'' defined as the

A simple performance model for photometry can be created by combining
the bright and faint star limits developed above.
The total variance of the stellar intensity measurement of
over-sampled stars is thus

The term in brackets in the last equation can physically be thought of as the ``effective background level''.

An important, but frequently ignored, noise source
is the uncertainty of the measurement of the effective background level
(
).
If the ``sky'' background is assumed to be flat, then the lower limit for
measurement error of the effective background level is

In order to have a more realistic performance model for photometry, this noise source must be added as the square of because it is a systematic error:

Photometric performance will be maximized when either the effective background area ( ) or the effective-background-level measurement error () is minimized.

We now have the basis for a simple, yet realistic, photometric performance model for PSF-fitting algorithms. An upper limit for the theoretical signal-to-noise ratio of a PSF-fitting algorithm is

Let us again assume
that the variance of the noise associated with the
pixel of an observation of a bright star is due only to
stellar photon noise.
The variance of the stellar position measurement, , of bright
* over-sampled* stars with a
normalized * unsampled* Gaussian PSF at the pixel is

is

where is the effective background area as defined above. By symmetry, the variance of the stellar position measurement of bright over-sampled stars is the same.

Let us again assume that noise contribution from the star
is negligibly small and that can replace with
with an average * constant* rms value of
.
Using this approximation, we find that the
variance of the stellar position measurement of faint * over-sampled*
stars with a normalized * unsampled* Gaussian PSF is

By symmetry, the variance of the stellar position measurement of faint over-sampled stars is the same.

We can now create
a simple performance model for astrometry
by combining the bright and faint star limits developed above.
The expected lower limit of the rms measurement error of the
stellar position
of a PSF-fitting algorithm is

King, I. R. 1983, PASP, 95, 163

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