The Effect of Seeing on Star Shapes

[narrowbandpaul]

Hi All

This pdf shows how seeing can really affect your images, and that as you go to higher resolution you end up getting less and less light in each pixel.

I was bored so I decided to try to work out the probability that a photon will enter a given pixel based on the fact that the atmosphere scatters light. I used the fact that for well sampled images stellar profiles look very gaussian in shape. Anyway, the pdf contains the various results I have found, and I think it is quite an enlightening analysis. I have ignored telescope diffraction for the time being....that would be very messy indeed.

Also worked out the size of the star image that contains 80% of the total number of photons collected, a term usually defined as encircled energy, as a function of the seeing and resolution. Also worked out the total fraction of light collected over a 2 pixel radius.

Its a mixture of some nice maths and some nice results.

I couldnt upload a pdf, as the file size was too big. I struggled for hours round this problem but still its too big. I had to convert to word 97-2003, which isnt ideal but I cant upload docx either. To get round this I have put the Word 07 version, which looks much nicer in the attached ZIP folder. If you have Word 07 then have a look at that one.

Questions welcomed

Hope you enjoy

Paul

The effect of seeing on Stellar Profiles_Word 97_2003.doc

The effect of seeing on Stellar Profiles.zip

[FraserClarke]

Very nice Paul. Understanding things like ensquared energy is very useful in understanding instrument sensitivity. One nice thing you could maybe add to your analysis is to show how this ensquared energy plot affects your sensitivity to point-sources in the presence of a sky background (i.e. you should be most sensitive somewhere around/just below) the Nyquist sampling.

I think there is an error in one of your equations; the one where you invoke erf(-x) = -erf(x). This gives you [2*erf(...)]^2, which in turn folds out as 4*[erf(...)]^2, which should cancel the four on the bottom. So I think your probabilities are off by a factor of two... ???

Another cavaet is that the Gaussian approximation is valid where the exposure time is significantly longer than the coherence time of the atmosphere (about 5--10ms typically). Below that you sample the instantaneous state of the atmosphere, and the PSF breaks up into a lot of individual "speckles". These smear out over time, and give you a Gaussian.

[narrowbandpaul]

correct teadwarf!

Cant believe I missed that. Yes one graph will be out: the probability for the centre pixel is off by a factor two! So at nyquist i guess its about 20% or so.

the other graphs should be fine as they were based on the polar co-ord version and no error functions were harmed in the making of those.

yes, the assumption was that a long exposure was used ie t_int>>t_atmosphere

i will think about your sky background idea. Can you provide a bit more info about this idea?

Thanks for pointing out that silly mistake. Much appreciated.

Any other suggestions most welcome.

Paul

[FraserClarke]

Hi Paul,

Your nice derivation gives you the star flux in a given pixel (just treat the central pixel, for example), and shows how that flux varies as a function of the size of that pixel. If you also assume an arbitrary background flux, you can also see how the background flux in a single pixel varies as a function of that pixel size (trivial square of the pixel size). So you can combine those two together into a signal-to-noise, and see how that varies as a function of the pixel size. Of course, to get a real S:N you'd need to include real fluxes for the star and the background, but you should be able to get a relative plot without knowing those. You could also add in read-noise, which will dominate at very small pixel sizes. You'd have three basic regimes;

1) Sky noise dominated (pixels>>FWHM / faint source)

2) Source noise dominated (pixels~FWHM / bright source)

3) Read-noise dominated (pixels<<FWHM)

Not sure if you could come up with a function that works across the whole range, but should be able to derive approximations valid in each one??