ASTR 5110, Majewski [FALL 2011]. Lecture Notes
ASTR 511 (Majewski) Lecture Notes
DETECTION LIMITS AND THE ARRAY EQUATION
REFERENCES:
 Howell, Handbook of CCD Astronomy, Section 4.4.
 Newberry 1991, PASP, 103, 122.
 Gullixon 1992, ASPC, 23, 130.


The Array Equation: Simple Photometric Case
In planning and carrying out an observational experiment, it is important to have
an understanding of the needed, expected, and actual final S/N of the experiment.
 The S/N is a measurement of the quality of the experimental result.
 You should have a thorough understanding of how to calculate this
S/N for any instrument you use.
 Section 4.4 of Howell is a must read for a beginning
understanding on noise in CCD arrays.
 The discussion below follows Howell's description closely.
Previously we discussed sources of noise encountered when working with a detector array,
and considered two extreme limits:
 the readoutlimited regime, when the noise is dominated by the readout noise of the detector,
 the skylimited regime, when noise from the sky background dominates.
Now we want to be more explicit and carefully calculate all contributions to the signal
and noise when using a detector array like a CCD.
 Though we are discussing the CCD explicitly here, the equations below generally work for
many other kinds of array detectors as well.
In the case of imaging with an electronic array detector, like a CCD, and a source
extending over n_{pix} pixels, the S/N of the detection of this
source can be estimated in the most basic way as follows:
 It is important to note that the statistics of counting depends
on the number of photons converted to electrons by the detector and
NOT on their ADU representation in the CCD image.
 Thus, all quantities in the above equation are in units of
electrons, which can be obtained from the quantities in ADUs by
knowing the inverse gain (electrons/ADU).
 The other terms in the denominator can be understood in terms
of Poisson statistics (see below).
Clearly, the signal is given by N_{*},
the total photoelectrons obtained from the
source and collected in our "measurement".
 This "measurement" may be from one pixel (if measuring the S/N
per pixel  as is sometimes done in spectroscopy or in measures
of a surface brightness).
 Or the "measurement" may be from multiple pixels, such as all
those contained within a stellar profile or within an aperture.
The random fluctuations in the count, or "noise" N,
from the various contributed
sources are uncorrelated and therefore combine in quadrature.
The total noise is then given by:
where the terms in the square root are, in succession:
 The Poisson noise in the source counts,
(N_{*})^{1/2}.
 The Poisson noise in the sky background counts,
(N_{S})^{1/2}, with equal contributions
(added in quadrature)
from n_{pix} pixels in the "measurement".
 The Poisson noise in the dark current counts,
(N_{D})^{1/2}, with equal contributions
(added in quadrature)
from n_{pix} pixels in the "measurement".
 The contribution of read noise, which is not a Poisson noise
and is usually characterized as N_{R} electrons
per pixel (so, in the quadrature addition, appears as a
squared quantity).
As may be seen, for bright sources, the first source of noise dominates and
we have the classical
For fainter sources, the other noise terms become important (e.g., a
readnoiselimited measurement has the errors dominated by
N_{R}).
The Array Equation: More Exact Case
In some cases (high backgrounds, faint sources, poor spatial sampling,
large inverse gains  see Howell)
a more complex form of the equation is required:
where
 ( 1 + n_{pix} / n_{B} ) is a coefficient
that accounts for noise incurred due to an inaccurate estimation
of the background level on the CCD image (where n_{B}
is the number of pixels used to measure it).
Clearly n_{B} being large is better.
 The term involving the inverse gain, G in electrons/ADU, accounts
for digitization noise within the A/D converter, an error
that becomes considerable as the inverse gain grows.
σ^{2}_{f} is the 1σ error introduced
in the A/D converter of the CCD when it has to decide between two
ADU levels bracketing the electron count divided by G.
The value of this quantity is taken to be
σ^{2}_{f} = 0.289.
Standard Error in Magnitudes
As we have said before, and will explore in more detail soon, the magnitude system
is defined on a logarithmic scale, where m = 2.5 log_{10} F + constant,
where F is the flux.
If one recognizes that the standard error (standard deviation)
of a measurement, σ, is related to the S/N as
then we can derive the error in magnitudes as
where p is
and the factor 1.0857 is the conversion factor one obtains when one calculates
the error in magnitudes (not electrons  derive this yourself!).
Required Integration Times
One of the most important things one needs to calculate in preparation for an
observing run is how long one should expect to integrate to achieve a given
S/N.
 Many observatories have exposure time calculators that
will derive these values for you for a given instrument.
 At minimum, these observatories should be able to tell you
either directly or give you the information needed to find
the count rate expected for a source of a given magnitude
for that instrument.
Then, if instead of thinking of counts, we have the above equations
expressed with count rates per second, we can rewrite the S/N in
terms of integration time, t, as
One then solves this quadratic equation for t to know how long to
integrate to obtain a given S/N within a detection area n_{pix}
(see Howell for an
explicit solution).
The main thing to notice is that in the sourcelimited regime,
S/N ∝ t^{1/2}.
The Array Equation: Spectroscopic Case
In spectroscopic applications, it is common to quote the S/N per pixel or
S/N per resolution element in the continuum.
 In these cases, one must adjust n_{pix}
accordingly, taking into account both the number of pixels in
the dispersion direction (e.g., per resolution element) and
those summed perpendicular to the dispersion direction in the
extraction of the source spectrum.
One can also quote the S/N for a given spectral line or feature.
Note that throughputs of spectrographs are much reduced compared to direct
cameras, amounting generally to a few to 10% or so, even discounting the
possibility of light lost by not making it through the slit.
 Consider the additional amount of optics between the telescope
focal plane and the CCD chip in the case of spectroscopy.
AAO photographs from http://www.aao.gov.au/2df/gallery.html.
All other material copyright © 2002, 2005, 2011, 2015 Steven R. Majewski. All
rights reserved. These notes are intended for the private,
noncommercial use of students enrolled in Astronomy 511 at the
University of Virginia.
