ASTRONOMY 535 - PROBLEM SET #2 - DUE 9/7/01

1. Consider observations of two stars which give 10 and 100 photons respectively, with two choices of background, 0 and 100 photons (four combinations altogether). Determine the signal-to-noise, the instrumental stellar magnitudes:
   
   $$m = -2.5\log(\mathrm{total photons - background photons}),$$
   
   
   and the estimated uncertainties in these magnitudes, assuming that the background is measured exactly.

   Use the following
   

   $$S/N = {S\over \sqrt{S+B}}$$
   
   
   
   
   $$m = -2.5 \log (\mathrm{tot-B})$$
   
   
   $$\begin{eqnarray*} \sigma_m^2 &=& \sigma_{tot}^2 \left({dm\over d\mathrm{tot}}\ri... ...\\ &=& \sigma_{tot}^2 \left({-2.5 \log e\over S}XS\right)^2\\ \end{eqnarray*}$$
   
   
   
   
   
   $$\begin{eqnarray*} \sigma_m &=& 1.086 {\sigma_{tot} \over S}\\ &=& {1.086 \sqrt{S+B} \over S} \end{eqnarray*}$$
   
   
   
   
   
   This gives

   S	B	S/N	m	$\sigma_m$
   10	0	3.16	-2.5	0.34
   10	100	0.95	-2.5	1.14
   100	0	10	-5	0.1
   100	100	7.07	-5	0.15

2. You are doing photometry on a field of stars with a range of brightnesses, so you observe the field at several different exposure times of 100, 600, and 1800 seconds. For one particular star you measure 960, 6050, and 17900 photons in the 3 different observations. Imagine that you are using a CCD with a readout noise which corresponds to 10 photons/pixel, and that you are summing over 15 pixels to get the total counts. What is your best estimate of the mean count rate from the star?

   We wish to get the mean count rate $r$:
   

   $$r \equiv {c\over t}$$
   
   
   
   
   $$\sigma_r = {\sigma_c \over t}$$
   
   
   However, we can only estimate $\sigma_c$, because it depends on $r$,
   
   $$\sigma_c = \sqrt{r t + n\sigma_{rn}^2}$$
   
   
   where $n$ is the number of pixels over which we are summing. Clearly, the 3 different measurements have different S/N, so we want to use a weighted mean to get the best estimate of the count rate. To avoid biases, we want to use the correct relative weights, at least. To do so, we make some estimate of the mean count rate, then use this estimate to get the correct relative weights. It doesn't really matter how we get our initial estimate of $r$; once the relative weights are correct, the resulting bias will be negligible. So let's just take the rate from the highest S/N exposure as our estimate $r_{est} = 17900/1800 = 9.94 photons/sec$. Then, we have:
   
   $$r = {\sum {r_i\over \sigma_i^2} \over \sum {1\over \sigma_i^2}}$$
   
   
   where
   
   $$\sigma_i^2 = r_{est} t + n \sigma_{rn}^2$$
   
   
   Using $n=15$, $\sigma_rn = 10$, we get:
   
   $$r = { {{960 \over 100} \over {((9.94) (100) + 1500)\over 100... ...(9.94) (1800) + 1500)\over 1800^2}} }= 9.968 \textrm{photons/s}$$
   
   

3. What does it mean for an object to have B-V = 1.0 ? B and V are defined as:
   
   $$mag\equiv -2.5 \log {\int F_\lambda(star) d\lambda\over \int F_\lambda (Vega) d\lambda}$$
   
   
   So, $B-V=1.$ means that the ratio of B band light to V band light is 2.512 times larger in the star than in Vega, i.e. the star is redder. The key point here is that it is the color relative to Vega.
4. Estimate the number of photons/second you would receive for a star with V=22 with the NMSU 1m assuming that the combined detection efficiency of the telescope and detector is 50 percent. You can assume that the V bandpass is rectangular with a central wavelength of 5500 Å, a full width of 1000 Å, with an in-band transmission of 80 percent.
   
   $$S=T \int d\lambda F_\lambda {\lambda\over h c} q f$$
   
   
   where $q$ gives the efficiency of telescope and detector (0.5) and $f$ is the filter transmission. For a rectangular bandpass,
   
   $$S=T \Delta\lambda F_\lambda {\lambda\over h c} q f = \pi (50... ...\times 10^{-9}) ({5500\times 10^{-8} \over hc})= 4.98photons/s$$
   
   

5. Imagine you are planning an observation with the NMSU 1m. In the V bandpass, let's say a star with V=22 gives 5 photons/second at the 1m detector. Let's say the background in V is 21 mag/square arcsecond and that the typical image size is 2 arcseconds FWHM. How long would you need to expose to reach a S/N of 10 for a star with V=24?

6. How long if the image quality were to improve to 0.8 arcsec?

   Signal from star is given by $S=5\times 10^{-0.4(24-22)} = 0.8 t$. Background is given by $B=5\times 10^{-0.4(21-22)} = 12.56 t A$, where $A$ is the area of the sky over which you integrate your measurement. For FWHM of 2 arcseconds, I'd say $A=4 \pi \sim 13$.
   

   $${S\over N} = {0.8 t \over \sqrt{0.8 t + (12.56)(13)t}} \sim 0.028 \sqrt{t}$$
   
   
   Thus, to get S/N=10, you would need $\sim$ 26,000 seconds! If the image quality were 0.8 arcsec, the background counts go down, and the exposure time is shortened to about 4000 sec.