# test data - Excel spreadsheet

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Lymm Observatory

optimal photometry

how it works

test data 1

test data 2

real data

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Let's put the following star into the spreadsheet:

sky = 40

amplitude = 40

fwhm = 15

If you think of pixels 1 unit wide, then the fwhm is 15 pixels, and the central pixel has 40 photons from the sky and about 40 from the star. Alternatively, you could think of this as a sky background of 4000 photons per pixel, with a star fwhm of 1.5 pixels (although the image will then be processed in 0.1-pixel-width rings, which we can't do in practice, of course). The spreadsheet tells you the total flux from this star is 10208 photons, regardless of actual pixel size. Ignore the two comparison stars for now.

The extraction PSF should have an FWHM equal to the actual star PSF. The camera gain is assumed to be 1 (1 count = photon) and the errors are computed based on photon statistics and pixel sampling.

Notice the S/N achieved with the different methods. The most precise is obtained with well sampled optimal photometry, followed by 'poorly sampled' optimal photometry (reducing the image to 10 unit wide annuli), and finally aperture photometry. At best, optimal photometry gives about 10% better S/N than aperture photometry, although there is a suggestion optimal extraction suffers with a poorly sampled image.

Some comments are in order about aperture photometry. First, in this case, the highest S/N is obtained with an aperture that samples about 80% of the total light from the star. There's a simple reason for this - the remainder of the light is thinly dispersed against the sky background, and inclusion of this part of the image actually reduces the overall S/N. Second, in practice, aperture photometry is difficult to do with a small FWHM. I'll explore this further later (when discussing results form synthetic images).

Now try varying PSF size, whilst keeping the total flux constant, with the following parameters,

amplitude = 20   fwhm = 20.213

amplitude = 14.4   fwhm = 25

amplitude = 10   fwhm = 30.

Notice how S/N drops as the image is made larger and the starlight diluted against a greater area of noisy sky. There is also a suggestion optimal extraction performs closer to the theoretical ideal with a wider (better sampled) PSF.

Now, let's consider the two comparison stars. Enter some values for the flux (but brighter than the target star).

One important issue is what happens if the we can't model the star's PSF exactly - which is why I provide the option to use a different extraction PSF. Try changing the width of the extraction PSF slightly - and notice the following:

(1) The extracted flux changes (rather similar to changing the aperture size in aperture photometry). (2) The effect is the same for target and comparison stars, so flux ratios (differential magnitudes) are still correct (again like aperture photometry).

It is useful that the differential magnitudes do not depend critically upon the extraction PSF, since we are unlikely to be able to precisely model the actual PSF anyway!

Finally, there is the question of which star to use to generate the weight mask. Again there is an analogy with aperture photometry - fainter stars yield highest S/N with smaller apertures than brigher stars, but we must use the same aperture for all stars. With optimal photometry, we use the faintest star of interest to generate the weight mask. You can demonstrate this with the spreadsheet. For example, let the target (and weighting star) have an amplitude of 40, and the comparison stars fluxes of 400 to 4000 units. The spreadsheet calculates the error in the flux ratio (differential magnitude) between the two comparison stars. Now increase the flux of the target star - this error remains more or less constant until the flux reaches that of the fainter comparison star.