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Mighell, K. J. 2003, in ASP Conf. Ser., Vol. 295 Astronomical Data Analysis Software and Systems XII, eds. H. E. Payne, R. I. Jedrzejewski, & R. N.
Hook (San Francisco: ASP), 395
A Theoretical Photometric and Astrometric Performance Model for
Point Spread Function CCD Stellar Photometry
Kenneth J. Mighell
National Optical Astronomy Observatory,
950 North Cherry Avenue,
Tucson, AZ 85719, Email: mighell@noao.edu
Abstract:
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
![$4\pi$](img1.gif)
(
![$\approx 12.57$](img2.gif)
) 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
millipixel. 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 sampled Point Spread Function
(PSF).
All other noise sources
(for example, the background sky, instrumental readout noise, etc.)
are assumed, in this case, to be negligibly small.
The variance of the stellar intensity measurement of bright
over-sampled stars is thus
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 over-sampled
stars is
where the constant
is the
``effective background area'' defined as the
reciprocal of the volume integral of the
square of the normalized unsampled PSF (
).
The effective background area
for a normalized Gaussian PSF with a standard deviation of
px is
px
;
the value for a critically-sampled normalized Gaussian is,
by definition,
(
12.57) px
.
King (1983) identifies
as the ``equivalent-noise area''
and notes that numerical integration of a realistic ground-based
stellar profile gives an equivalent area of
px
instead of the value of
px
for a
Gaussian profile.
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
By symmetry,
the expected lower limit of the variance of the
measured
coordinate of the stellar position
of a PSF-fitting algorithm is the same.
Acknowledgments
This work is supported by
grants awarded by the Office of Space Science of the
National Aeronautics and Space Administration (NASA) from the
Applied Information Systems Research Program (AISRP) and the
Long-Term Space Astrophysics (LTSA) program.
References
King, I. R. 1983, PASP, 95, 163
© Copyright 2003 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA
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