1. Plate Scale and CCDs (15 points)

I'm
working on a project to get photometry for stars throughout the
globular cluster M13, using the prime focus camera on the Paul Harding
Telescope at Kitt Peak (where typical seeing is about 1 arcsec). When I
turn up for my observing run, I am asked what CCD I wish to use. I
don't want to let on that I have not read the observers manual, so a
quick look at the telescope tells me the distance from the primary to
the focal plane is 40 feet. The choice is between a CCD A, which has
3000x3000 pixels with each pixel 9x9 microns, and CCD B which has
2048x2048 pixels with each pixel 24x24 microns. Which CCD would you
pick and why?

2. Filters (15 points for ASTR 306; 30 points for ASTR 406)

Here is the filter tracing (transmission as a
function of wavelength) for one of the Kitt Peak R band filters.
From this tracing, calculate the filter's central wavelength and
the width of the "equivalent square filter." Plot the filter transmission function and overplot the
transmission function for the equivalent square filter.

**ASTR406****:** If an object has a spectrum given by
I(lambda) = I_{0} * (lambda/5450A)^n, where I_{0}
= 3.63x10^{-13} erg/s/cm^{2}/A. For this filter,
and for the equivalent square filter, calculate m_{R} in
the STMAG system for n=-2 (a blue spectrum) and n=+2 (a red
spectrum). Comment on the differences between the numbers for
the two filters with the different values of n -- that is, why do
you get different values depending on whether you use the real
filter tracing or the equivalent square filter, and why do the
differences depend on the spectrum of the object?

3. Calculating a curve of growth for stellar photometry (20 points)

For this problem, use
a gaussian profile as our model for the point spread function
(PSF) of an image.

- First, calculate the relationship between sigma and FWHM for a gaussian profile.

- Now download this SDSS image of the globular cluster M13.
Characterize the data as follows: what is the FWHM of the
stars in the image (in both pixels and arcseconds)? What is
the background level and its dispersion (in counts)? You should check a few
stars and a few sky areas to make sure you are getting
consistent values.

- Assuming a gaussian profile, calculate the total enclosed light as a function of radius (also known as a "curve of growth"). That is, given sigma (or FWHM), how what fraction of the total light from a star do you expect inside a circular aperture of size r? What is the radius which contains half the light? 80% of the light (a common "spec" by which optical designs are rated)? 90% of the light? 99% of the light? Make a plot of enclosed light as a function of radius, using the value of sigma you got from the image. It should go from 0% at r=0 to 100% at r=big.

Helpful
tip for this part: normalize radius by the gaussian sigma. In other
words, make your radius measure r/sigma instead of r. In the integral,
do the substitution x=r/sigma and calculate (and plot) enclosed light
as a function of x. Then when overplotting your curve of growth from
the data (the last step of this problem), do the same thing for your
measured data, using the sigma you infer from your measurement of FWHM
for stars on the image.

- Now measure a "curve of growth" for three good stars. That is,
change the parameters in imexam for the apertures ("epar
rimexam"), and see how the total flux of a star changes as a
function of aperture size. Make sure you are choosing sensible
values for the photometry parameters "radius", "buffer", and
"width" -- describe what you set them to and why. And also
describe which stars you picked to do this analysis, and why
you picked them. Make sure you have set the rimexam parameters
center=yes, iterati=1!

- Finally, compare (ie overplot) your measured curve of growth to that of the gaussian model. How good is the model? Comment on what might be causing the differences you see.

Charge Transfer Efficiency (CTE) describes the fraction of electrons successfully moved during a single pixel shift during CCD readout. That is, a CTE of 0.99 means that 1 out of every 100 electrons is left behind during a shift. If you have a 4096x4096 pixel CCD and want to make sure 99% (or more) of the electrons held in each pixel are moved successfully into the serial readout register (meaning they don't get left behind in a trailing pixel), what kind of CTE do you need for your CCD?

5. Sky Background (15 points)

You are wanting to take spectra
of stars using a fiber fed spectrograph. The typical seeing is not great, about
1.5" FWHM, and the night sky has a surface brightness of mu_{R}
= 20.8 mag/arcsec^{2}. Plot the fraction of total light in the
aperture that comes from a star of magnitude m_{R}=15 as
a function of aperture diameter, with aperture diameters going
from d=0-5 arcsec. Overplot similar curves for stars of magnitude m_{R}=16, 17, and 18. Why would big fibers be good if you want
to look at faint stars? Why would they be bad? (Don't worry about spectral resolution for this question.)