Consider this very common case: you are measuring the brightness of a star by imaging it onto a CCD detector. We focus the image as well as possible then make an image of duration T (seconds). When we read out the detector we can clearly see the image of the star, and the rest of the detector seems "empty." In fact the act of reading-out the CCD places some noise on each pixel, in addition to which the CCD will have registered photons from the night sky. So the chip is not "empty" anywhere. How do we best measure the star's brightness and estimate the error in this measurement?
One way is to define a box around the star and to compare the counts in this box with the counts in a similar but "empty" box. The difference should give the stellar flux.
The error in the determination of the "empty" is the poisson noise over the area of the box, which is approximately the square root of the total counts from the sky. The read noise contribution is usually expressed as a number, say rn which is the square root of the variance in the electronic background level. So this is roughly the same as another sky background contribution except that it does not depend on T. Total noise per pixel is thus sqrt(bkg + rn2) and the total noise within the box is sqrt(N(bkg + rn2)) where N is the area of the box.
For the box containing the star the error can be calculated in the same way except that the variance in the flux of the star itself (S) must be included. This is, again, just S, so the noise is sqrt(S). Because of the poissonian nature of photon arrival times the counts accumulated in time T is subject to an uncertainty which approaches the square root of the number of counts detected.
Since the flux estimate requires that one box be subtracted from the other, the final noise estimate will be the sqrt of the sum of the squares of the individual errors.
This is a robust and simple method which does not require one to know the PSF (point spread function) of the stellar image. You only need to make the boxes large enough to include effectively all of the stellar light, which can be checked by experiment.
Simulated data:
sky background (mean = 1600 per pixel)
2 stars at S/N=16 (4000 counts)
2 stars at S/N=5 (1200 counts)
same image, gaussian smoothed (sigma=2.5)
A second way to measure brightness is to establish the PSF (perhaps by inspecting many stars on a different image) and then to fit the PSF to the stellar image. The background level is then fit to the zero level of the PSF and you do not need to have an "empty" box.
As an exercise, we calculated the magnitude of a star which could just be detected in a 1-hour observation with a given telescope. I have written out this calculation in the form of an iraf script which you can modify as you wish and then run under iraf (instructions are in the file itself). Note that we are only attempting to detect the line, so it is really just Signal v Noise. In the second part of the exercise we want 99% photometric accuracy - since photometry requires you to subtract the sky background from the image the sky noise figures in twice. Details of the calculation are also included in the program.
General definition of a magnitude scale:
m = -2.5 log(flux) + constant
Stellar magnitudes are established by comparison with other stars, especially "standard" stars. But here are some numbers which allow magnitudes to be calculated directly, or allow expected counts to be calculates for a star of known magnitude:
Material:
Kitchin fig 3.1.1 (UBV bands), tables of extended photometric bands (R...Q) etc.
Bradt Table 8.2 (photometric bands), Fig 8.2 UBV curves, Lena's book has similar information.
The process of combining monochrome images taken with various filters to create
a colour image is mostly taken for aesthetic or promotional reasons. However
doing this right does require some photometric (and other) expertise.
For example before being combined
the images have to be aligned and smoothed so that the final
stellar images have the same PSF.
Once this is ok, a quick method is to use the IRAF color package.
A better method is described in a
tutorial
by astronomer Tom Statler, who describes how to scale and convert the data using
the IRAF export package. In particular this can provide better
translation of the (U)BVR band data to RGB values.
There are two main effects which degrade your data and which can be
improved by flat-fielding.
Non-uniform illumination: the telescope does not have the same "throughput"
of light all over the field of view - usually, there is some "vignetting" which
causes the image to be a little fainter the further you are from the centre. You
might notice this especially towards the corners. Even if you do not notice it,
it is probably present and since your object is generally not in the same place
in the image as the standard star, it will affect the quality of the photometry.
Also, you will sometimes notice strange patterns or "defects" which also affect the
photometry - they are often doughnut-shaped and are usually caused by
out-of-focus images dust on a lens or mirror. To correct these effects we take
a one or more sky flats. A sky flat is an image taken with the telescope
pointed at the sky just after sunset or before sunrise. The sky should be featureless,
so the flat field will only differ from a constant value because of vignetting
and defects. As the vignetting and defects are assumed to be a constant
property of the telecope, they can be corrected
by dividing the images by the flat field. One may combine many sky flats to get higher
S/N, and it may be smoothed in order to take out any residual "high frequency" noise
(see below). Finally
one usually normalises the sky flat before using it to correct the image.
Pixel-to-pixel variation: the pixels of the CCD have themselves different
sensitivities, but to correct these without actually adding poisson noise you need
to use "flats" which have a very large number of counts. It is hard to do this
with sky flats because the twilight is quite a brief period. Instead, you can use
dome flats
which are taken during the day when the telescope is otherwise idle. The telescope is
pointed literally at the inside of the dome, which is illuminated by a strong lamp
(e.g. a halogen lamp). Such "flats" usually have some general structure, which is
usually removed by dividing the summed dome flat by a smoothed version of itself. The
"high frequency" information, i.e. the pixel-to-pixel variation, is retained, and this
can be used to correct your images.
Summary of data-reduction steps
