View Full Version : Resolution and Pixel Size

2016-Oct-23, 10:20 PM
I am doing an Introduction to Astrophysics course. We have just covered the Airy Disc and Diffraction problems.

I have a tutorial question which I don't understand. I have done the first part:

The EELT will have a mirror of diameter 39m. Calculate the best possible resolution if it observes at 500nm.

I get that to be 1.564 x 10^-8 radians which converts to 3.23 x 10^-3 arcseconds using the equation for the angular size of the Airy Disc:
Theta = 1.22 Lamda/Aperture

OK so far I think. Starightforward.

But then I am asked "Assume that the EELT uses a detector with square pixels, and the size of each pixel matches the resolution of the telescope, calculate the Flux received from Alpha Centauri by one pixel. I am told that Alpha Centauri B has a radius 0.86 of the Sun and is 1.3 pc away.

I can calculate the Flux at Earth from Alpha Centauri. What I don't know is how to relate the size of the pixel to the resolution of the telescope which I calculated to be 1.564 x 10^-8 radians. I am assuming that the pixel will have the dimensions of the Airy Disc. But how would I calculate that?

We haven't done focal lengths yet. I am thinking that maybe there is a geometrical solution where if I have the Mirror Diameter Size, and I have an angle, so if there's a focal length implied by the mirror size then that would allow me to calculate the Airy DIsc size and hence the pixel size and hence the Fkux received by each pixel?

Can anyone help?

Thank you

2016-Oct-24, 12:15 AM
Do you need the flux per unit area on the surface of the pixel, or just the total flux on the pixel?

2016-Oct-24, 12:30 AM
The problem is poorly phrased. You should politely complain to the person who wrote the tutorial. Or, if you tell me, I will :-)

I suspect that the problem wants to say this:

The optics and detector yield a plate scale in which the side length of each pixel is equal to the Full-Width at Half-Maximum of the Point-Spread Function. Assume that the PSF has the form of a gaussian. Assume further that the star is positioned so that its central falls directly on the central region of one pixel.

Now, how many photons per second strike this pixel?

That's a pretty standard sort of question to ask. It requires the student to figure out the fraction of all the light from a gaussian PSF which falls within some particular distance of the center, which means doing a relatively simple integral.

If my guess is correct, you'll want to look up properties of the "erf", or "error function."

Good luck!

(no, really, who did write this question?)

2016-Oct-24, 07:33 AM
Do you need the flux per unit area on the surface of the pixel, or just the total flux on the pixel?

The question asks for the "Flux Received from Alpha Centauri B by one pixel".

I have a lecture in half an hour, and then a further one tomorow, by which time I should in theory know some more.

I'll have a look at the suggestion above and of course when I get the answer I'll let you know more. Thanks for the replies!

2016-Oct-24, 12:23 PM
One piece of your puzzle is to estimate how many photons are being generated by Alpha Centauri B per second, and then figure out what fraction of the whole does the mirror take up. Then what fraction is lost because the pixel doesn't catch what's outside it in the Airy disk.