Author: Michael Gutzwiller Date: 970630 Revision: #0 970630 Key Words: photometry, techniques, computation
This Technical Note describes the optimized aperture used for photometry in the latest version of Michael Gutzwiller's "star" program for generating star lists.
The size and shape of the aperture used for aperture photometry will affect both the signal measured within the aperture and the noise level associated with it. Using the PSF the signal within an aperture is:
(1) S(n) = A*sum(p(i))
where:
S(n) is the signal of the star in electrons on the CCD
for an aperture with n pixels,
A is the total amplitude of the star in electrons on the CCD and
p(i) is the PSF at each pixel within the aperture.
The noise associated with the signal is:
(2) N(n) = sqrt(A*sum(p(i)) + n*sigmaB*sigmaB + n*n*sigmaB*sigmaB/k)
where:
N is the noise in electrons for an aperture with n pixels, n is the number of pixels in the aperture, sigmaB is the CCD readout and sky background noise and k is the number of pixels used to measure the background
For the purposes of our discussion we will assume that k >> n so that equation (2) becomes:
(3) N(n) = sqrt(A*sum(p(i)) + n*sigmaB*sigmaB)
Now suppose we want to optimize our aperture to get the highest signal to noise ratio. We can accomplish this by starting with the smallest possible aperture, a single pixel at the peak of the PSF, and adding more pixels to the aperture until the S/N starts to go down instead of up. This relation is shown in equation (4)
S(n+1) S(n)
(4) ------ < ----
N(n+1) N(n)
Lets first take the case of faint stars. For these stars we can assume:
(5) A*sum(p(i)) << n*sigmaB*sigmaB
Actually for TASS images faint by this category isn't very faint. For a typical image from Tom the first term is bigger than the second only for stars brighter than mag 8! Using equation (5) equation (3) can be further simplified to:
(6) N(n) = sqrt(n)*sigmaB
and equation (4) becomes:
A*p(n+1) + A*sum(p(i)) A*sum(p(i))
(7) ---------------------- < --------------
sqrt(n+1)*sigmaB sqrt(n)*sigmaB
taking out the common terms we have:
p(n+1) + sum(p(i)) sum(p(i))
(8) ------------------ < ---------
sqrt(n+1) sqrt(n)
For reasonably large n we can approximate
(9) sqrt(n+1) ~ sqrt(n)*(1 + (1/(2*n)))
Substituting (9) into (7) and dropping the common sqrt(n) we have:
p(n+1) + sum(p(i))
(10) ------------------ < sum(p(i))
(1 + (1/(2*n)))
Solving for p(n+1) we get:
sum(p(i))
(11) p(n+1) < ---------
2*n
So for faint stars we should stop adding pixels to the aperture when the next pixel's psf value is less than the sum of the psf pixels we have so far divided by two times the number of pixels in the aperture.
Brighter stars are more of a problem. If we take an approach similar to equation (5) but with the ineqality reversed we find that adding more pixels to the aperture never lowers the S/N. Thus brighter stars want more pixels in the aperture. To accommodate the brighter stars and also account for jitter in the determination of the location of the peak for the star candidate, the "star" program "relaxes" equation (11) empirically to the following:
sum(p(i))
(12) p(n+1) < ---------
4*n
This allows more pixels for the brighter stars and doesn't degrade the measurement to fainter stars unduly.