ASTRONOMY 535 - PROBLEM SET #3 - DUE 02/24/03

1. Imagine you are looking from the Apache Point Observery at a star at a declination of 10 degrees when it crosses the meridian. You are using a V filter for which the extinction coefficient is 0.2. By how much do you need to correct the observed brightness to determine the expected brightness outside the Earth's atmosphere?

   What will the airmass for this object be 3 hours later?

   For star at declination of 10 degree on the meridian at APO, zenith distance is $\sim$ 22 degrees, so airmass is $\sim$ 1.08. Using $m = m_0 + k X$, you need to correct by 0.22 mag, or increase the flux by a factor of 1.22.

2. (ASTR 535 only) Imagine you are going to observe at APO next week and will be doing spectroscopy with DIS. You will need to observe a few spectrophotometric standards in order to calibrate the relative response as a function of wavelength. You can find a list of spectrophotometric standards in Massey et al. 1988 (ApJ 328, 315). From this list, choose one star to observe at the beginning of the night, one in the middle and one at the end of the night. For each star, give the position angle that you need to set the spectrograph slit at (measured from N through E) to eliminate the effect of differential refraction. You may wish to get acquainted with some sort of sky almanac program (e.g., skycalc) to help with this, but it is important that you understand what is happening.

   You want to find stars that are at relatively low airmass (to minimize extinction and differential refraction). Since hour angle = local sidereal time $-$ right ascension, choose stars with RA near the LST at the beginning, middle, and end of the night. To avoid losing light as a function of wavelength, you want to align the slit pointing towards the zenith, in other words, align it along the parallactic angle. You can determine this from the equation given in class, or use, e.g., the skycalc program.

3. The Fourier Transform of the ``rectangle function'' (a function that is 1 between $-1/2 < \rm x < 1/2$ and zero elsewhere) is the sinc function. Imagine that I have several such rectangle functions defined on the x-axis, in total 7 in a row, the first one starting at x= -6.5, the last one ending at x=+6.5 (so f(x) = 1 between x=-6.5 to -5.5, then 0 to x=-4.5, one to x=-3.5, etc). Using the connection between Fourier Transforms and convolutions, in particular the Convolution Theorem, it is easy to write down what the FT of this function will look like. I don't need the answer in terms of one analytic function, just an analytic expression of products and convolutions of particular functions that will give the answer.

   Also make a sketch of what this FT will look like.

   This function can be written as follows:
   

   $$II({x\over 14})(III({x\over2}) \ast II(x))$$
   
   
   The FT of this can be easily derived following the various theorems. The FT of
   
   $$(III({x\over2}) \ast II(x)): 2III(2s)sinc(s)$$
   
   
   and hence the FT of
   
   $$II({x\over 14})(III({x\over2}) \ast II(x)): 14sinc(14s) \ast 2III(2s)sinc(s)$$
   
   
   which is a sampled sinc function convolved with a rapid sinc function.
4. You are going to do some IR imaging. Let's say your observatory has two cameras, one with a HgCdTe device and the other with an InSb device. The HgCdTe chip has a QE of about 40 percent and a readout noise of about 30 electrons; the InSb device has a QE of about 80 percent, but a readout noise of about 100 electrons. You are doing galaxy surface photometry with a broad band filter and want to study the faint outer regions of your galaxies, which have surface brightnesses much fainter than the night sky. Which camera would you choose, and why?

   Your exposures will always be sky noise limited since your object is much fainter than the sky. In this limit, readout noise is unimportant because it is swamped by photon noise from the background. Consequently, you want to choose the device with the higher quantum efficiency, namely, the InSb device.