Part 4: Differential Photometry
The end-game is straightforward, but requires some care.
We want to use the two comparison stars, C1 and C2, to remove any variations in atmospheric transparency among the different night, and even perhaps between images on a given night during the 20+ minutes of observing. This is the power of differential photometry: it lets us remove effects due to variable airmass and even salvage data from nights where thin clouds intervened and made absolute photometry impossible. Since the variable and comparison stars are observed simultaneously on each image, they are affected the same, and so the effects of clouds is factored out.
Step 1: Convert Observed Signal into Instrumental Magnitudes
Convert each of the instrumental counts in ADU and their uncertainties for the variable and two comparison stars into instrumental B magnitudes:
Var = 26 - 2.5 * log[Counts(Var)] C1 = 26 - 2.5 * log[Counts(C1)] C2 = 26 - 2.5 * log[Counts(C2)]and also compute the uncertainties for each of these quantities.
Step 2: Derive the Differential Magnitudes
From the instrumental magnitudes, compute the following differential magnitudes, and their uncertainties, for each observation:
Var-0.5*(C1+C2) C1-C2The first is the differential magnitude of the variable star relative to the two comparison stars. We use this rather odd-looking combination to attempt to reduce the effects of random errors on one of the measurements. There are fancier (and arguably better) ways of doing this, but they are more computationally involved.
The second number, C1-C2, tests to see if the comparison stars truly do not vary. At the very least it lets you set quantitative limits on their variability. This is used as a sanity check here because the variability in the target is so large. In cases where the variability is weaker, the statistics of the check-star differences let assess the significance of any variability you detect, or let you place limits on variability if none is apparent.
Step 3: Prepare a data table of your differential photometry for submission to the class database.
Now compile your results into a 5-column ASCII-format data table with these entries for each measurement:
HJDmid-2450000 Var-0.5*(C1+C2) Err C1-C2 ErrThe data should be given to the precision shown in the following example:
1813.12345 -5.123 0.056 0.535 0.065Please use spaces (no TABS or commas) to separate the data values for a given HJD time, and plase do not put a header on the file (the submission form will complain if you do). The "reduced" HJD leaves just the 9 significant figures that are of interest to us.
Step 4: Submit your differential photometry
Deadline: Tuesday, 2002 Dec 10, 5pm
I have created a central database on the Astro350 web server for all of the data. A special Data Entry Form is used to do the submission. The form gives instructions on how to use it. The form then passes your data on to a "data checker" program, and if everything is OK, it then processes it as follows:
To ensure that everyone has access to these data in time to complete their final reports, all of your final data must be submitted via the entry form by 5pm, Tuesday, 2002 December 10.
Step 5: View the combined light curve
Now comes the fun part. Each of you have only 20-minute chunks of a light curve for an object which has a roughly 88 minute variability period (actually, P=0.061038612 days). Each individual data set will therefore only cover about a quarter of 1 cycle, and since the data are taken on different nights, subsequent data sets will show a (hopefully) different part of the light curve. By phasing the light curves to the variability period and combining them together, we can reconstruct the entire pattern of variability over one complete cycle. The first few people who submit their data won't see much, but as more people submit their data, the overall variability pattern will begin emerge.
Your data can be plotted along with everyone else's data using the Light Curve Plotter. The light-curve plotter is your entry point into the class photometry data base. It plots the raw light curve (brightness versus time), and the phased light curve folded on the variation period of CY Aquarii.
Folding or phasing a light curve works as follows. Suppose you had an object which varies periodically with 1 cycle lasting "Period" days. At any given time, HJD, you are at some fraction of a cycle, called the "Phase", which is given by:
Phase = frac [ (HJD - HJD0) / Period ]where "frac" is the fractional part of the ratio shown, and HJD0 is an arbitrary starting time, called the "Epoch" of the variability (yet another use of the word "epoch" in astronomy - beware!). For example, if an observation at time HJD occurred 3/4 of the way through the 127th cycle since HJD0, then
(HJD - HJD0) / Period = 127.75and the Phase would be 0.75, the fractional part after the whole number 127 is subtracted. Here is a bit of C code that shows how this is done (a program like this is embedded in the data processor invoked by the data entry form above).
Go back to Part 3: Measure the Images
Copyright © Richard W. Pogge, All Rights Reserved.