I have struggled to understand and simplify color transformations of variable star observations taken with multiple filters. This is a summary of my understanding and a description of some tools I have developed to make it much more user friendly.
As caveats let me state up front that I will only be dealing with CCD measurements, differential photometry and Johnson-Cousin filter specifications (although the math should be adaptable to other systems). Some level of basic understanding about how CCD measurements are made (especially calibration of those images - bias, dark and flat field corrections) is assumed along with use of AAVSO standard definitions.
Differential photometry means two or more stars in the same image (assumed relatively small to provide essentially equal sky conditions across the field of view) are compared as to relative brightness and the difference in magnitude between them is calculated. This, and most other calculations discussed here, will usually be handled by software but the underlying principles will be discussed. If one of the stars is a calibration star (ie, its brightness value is known in various filter bands) the instrumental brightness of the target star can be calculated (by the software) by adding the difference between the target star and the calibration star to the catalog value of the calibration star. Thus,
m0 = m0 - mc + Mc, where 
When we refer to measurements in a specific filter (eg, B, V, R, I , etc) we will use, eg, V0 (or v0), Vc (or vc), etc.
Because it is not possible that all measurement systems (telescope optics, chip response, filters, etc) will give the same results for the same stars it is necessary to adjust (or transform) the instrumental values provided by software to the standard system so that values reported by different observers may be compared more accurately. To see why this is necessary, and to provide a clue as to how to do it, it is instructional to compare measurements of well-known stars to the standard values of those stars. The recommended method is to make well-exposed images in two or more filters of standard fields, the field of M67 being one (there are others). The essence of such fields is that not only are they well measured by experts but they have a wide range of colors (eg, B-V or V-I). Below is a graph of instrumental colors of the M67 cluster measured with my current system which shows some interesting facts about my systsem.
What we notice is that there is a fairly linear relationship between the observed colors of the stars and the "true" or accepted standard colors of the same stars. "Fairly linear" means the measured colors are not the same as the standard colors so something needs to be done to adjust or transform the measured colors. Such data is easily represented by a least squares linear fit which is easily accomplished in a modern spread sheet. For this data set, I determined the the data can be represented by the equation
(v-i) = 0.026 = 0.980*(V-I)
Had my system copied exactly the standard system the 0.026 would have been 0.000 and the 0.980 would have been 1.000 (to three decimal places). However, this neat relationship implies that we can "go backwards" to find what (V-I) would produce the observed (v-i) (ie, invert the equation). Substituting more general terms and solving for (V-I) we get
(V-I) = C + Tvi*(v-i), where
To get rid of the C, we notice that the equation applies to calibration stars as well as unknown stars, thus
(V0 - I0) = C + Tvi*(v0 - i0), and
(Vc - Ic) = C + Tvi*(vc - ic)
subtracting the bottom equation from the top equation give us
(V0 - I0) - (Vc - Ic) = Tvi*[(v0 - i0) - (vc - ic)]
rearranging we get
(V0 - I0) = (Vc - Ic) + Tvi*[(v0 - vc) - (i0 - ic)]
But we note from eq.(1) above that
(v0 - vc) = v0 - Vc and
(i0 - ic) = i0 - Ic
So, substituting these and rearranging we finally get
(V0 - I0) = (Vc - Ic) + Tvi*[(v0 - i0) - (Vc - Ic)] 
Thus, we have a way to get the standard color for a target star using the instrumental constant Tvi and the measured values of V and I.
But we still need to figure out how to get individual values for V0 and I0. To do this we need to look how our system handles (V-v) vs (V-I) which, for my system is shown below
It's kind of sloppy (to my eye) but, nevertheless, we can fit a least squares straight line to it to get
(V - v) = K + Tv*(V - I), where
Again, we can eliminate the unwanted K by noting that the equation applies to both the target star and the calibration star. Thus,
(V0 - v0) - (Vc - vc) = Tv*[(V0 - I0) - (Vc - Ic)] and
V0 = v0 - vc + Vc + Tv*[(V0 - I0) - (Vc - Ic)]
but, again using eq.(1) we get
V0 = v0 + Tv*[(V0 - I0) - (Vc - Ic)] 
Having (V0 - I0) and V0 makes it easy to get I0
I0 = V0 - (V0 -I0) 
It is worth noting that my V filter seems to give an average difference from the standard filter of some -0.154 magnitudes Does this mean my glass is a little clearer (or more opaque) than the standard? Seems reasonable. On the other hand the slope of 0.036 is not too far from the ideal of 0.000 so my glass is probably aligned with the pass band of the standard filter pretty well.
To get transformations for other filters, the same process will yield the needed results.
An occultation of the star 32 Lyncis by asteroid (372) Palma was predicted by the folks at IOTA (International Occultation Timing Association) for the early morning (CST) of January 26, 2007. The predicted path from which the event could be observed passed (relatively) near to my home in Wentzville, Missouri. Wayne Clark and I decided to try to observe the event.
Accompanied by or wives, Yvonne and Nancy respectively, we drove 110 miles north on US 61 to LaGrange, Missouri (about 30 miles north of Hannibal). Our original goal was Wakonda State Park which is just off the highway at the turn off to LaGrange, but it was closed. So we journeyed a little farther and pulled off at a rural intersection to see if we wanted to try it out - we didn’t as it was still ice and snow covered from the previous snow (they got a lot more up north than we did in St. Charles County). When we tried to back up, we couldn’t. Very slick!. After a good half hour of rocking, women and children pushing, pulling some vines off a nearby tree to put under the wheels (sure hope it wasn’t poison ivy) we finally got free. Did any of you know that astronomy has cardio-pulmonary benefits?
Heading on in to LaGrange we discovered that Terribles Casino had cleared their huge parking lot although the level of wattage on their parking lot lights probably could have melted the snow and ice without any atmospheric warming. (As we pushed and struggled and slipped trying to free our vehicles from winters icy grasp, we noticed a lot of traffic coming from the direction of LaGrange - at 1:00 am to 1:30 am - but those locals didn’t seem interested to stop and give city slickers (Freudian slip?) a helping hand/push. The sight of Terribles Casino cleared up the mystery: they were broke gamblers who were probably afraid we’d beg them for money that they didn’t have (anymore)).
We checked out the parking lot and decided that we could see our target star and, maybe, get a nice tan at the same time even if it was 22 deg. F. But like trolls who crave the darkness, we decided to explore just a little more and found a small public park where the street-side parking had been scraped (sorta) and there were only two streetlights, so we set up there. Both of our GPS receivers put the location at 40 02.679 N, 91 30.124 W and 611 feet elevation.
The night patrolman soon spotted us but drove on by, so we figured he was benign. But he circled the block and came back after deciding he couldn’t believe his first observation. He rolled down his window (staying inside the nice warm patrol vehicle) and allowed as how he recognized those funny objects as telescopes (at least mine, but he wasn’t too sure about Wayne’s) but, even so, were we crazy to be in LaGrange, Missouri at 2:30 am in frigid weather? We said yes (seconded by Yvonne and Nancy - who also stayed inside the nice, warm(er) vehicles), and he went on his way.
We finished setting up the 8-inch SCTs (Meade LX-200 and Celestron whatever) and found our target star quickly (we had actually found it from my backyard in Wentzville with 7×50s before we set out). Of course, as there were two similar stars in the field of view and the field was upside-down, reversed and otherwise convoluted, we debated over whether up in the eyepiece was east or west.
With some 30 or so minutes to spare, I studied the candidate star and finally detected the asteroid just east (up) of the target. I guess I was surprised at how faint 10.5 mag appeared in the 8-inch, but then we were no ways dark adapted with the two street lights nearby.
Wayne had brought along a couple of portable WWV receivers and we set them up on a folding table (unfolded, of course) and placed our sound recorders nearby. Then we sat down to keep steady, if teary, eyes on the target as the time ticked away. At shortly after 9:45 “Coordinated Universal Time.” the star disappeared. Wow. We hollered out “one” to record the event on the recorders although Wayne beat me by a fraction of a second. When the star reappeared, Wayne hollered out “two” and I missed it. While waiting for these events to happen, one purses the lips to be absolutely ready, but I think mine froze, or at least stiffened in the cold.
Upon reviewing my recording I found good data (haven’t heard from Wayne, I suppose he is still sleeping as I write this - but one recording will get us through if need be). The time signal and “one” and “two” can be heard plainly. Using a stop watch and repeating the measurements several times to get an average, I put the start of the event at 09:45:05.03 (UTC) and the end at 09:45:18.27 (UTC) so the event lasted just over 13 seconds at our location. We will submit the timings to IOTA along with our latitude and longitude as determined and cross checked by our two GPS units. Hopefully other observers across the country got good data and these data will be combined to yield an estimate of the cross section of the asteroid and improved orbit parameters.
Usually, a plot is prepared showing all the chords of the profile and I will post it when it becomes available.
This post may be too technical for the casual reader, but is intended to document my personal derivation of coefficients to transform instrumental photometry measurements to the standard UBVRI system used by professional astronomers.
First, a brief background. I measure the brightnesses of various stars (variables) using a telescope and CCD camera and various standard filters. In particular I have Johnson B and V filters and a Cousins I filter. These filters isolate certain wavelength bands (”colors”) and the information of just how bright a given star is in the different colors is very useful to astronomers. Lots of astronomers do this kind of work but the trouble is unavoidable differences in equipment (due to manufacturing variances) makes it impossible to combine results without correcting each individual set up to some standard which is the UBVRI system. There is a process to determine the necessary correction factors but I have failed to understand any of the explanations I have found in the literature so I decided to derive the coefficients myself. Here goes.
The astronomical magnitude system is based on the fact that it is much easier to measure brightness ratios of stars than it is to measure absolute values. Thus, Pogson suggested that a brightness ratio of 100 should be assigned to a difference of 5 magnitudes. This leads to the relation
m1 - m0 = -2.5*log(F1/F0)
where F1 is the “flux” of star 1 and F0 is the flux of star 2. In the case of CCD detectors, the flux is the number of electrons captured by the device that can be attributed to the star light. With most software I am aware of the flux is measured and the ratio taken to calculate the magnitude difference between the two stars. The flux F0 that would be produced by a zero magnitude star can be determined by selecting a star of known magnitude M, measuring its flux and noting:
M - zero = -2.5*log(FM/F0) and
log(F0/FM) = 0.4*M.
Raising both sides to a power of 10 gives
F0/FM = 100.4M
F0 = 100.4M*FM
Using this value of F0 for a known comparison star in our image, we (actually our software) can determine the magnitude of any other star in our image. Unfortunately, this value will not match the values obtained by others even though our one comparison star will. This is due to unavoidable differences between equipment set ups. Thus, we must call our magnitude measurements instrumental magnitudes and find a way to transform them to the standard system. We do that by using a procedure to determine a set of transformation coefficients that will permit us to transform our instrumental values into the standard system.
The “real world” has been established by professionals by the establishment of standard stars to which all others are referenced. It involves a technique called all sky photometry which takes into account such parameters as differing air masses above the respective stars, etc. However, I (and most other amateurs, I believe) don’t do this. Instead, we take advantage of professional work that has established reference stars that fit withing the field of view of our cameras for objects we want to measure. We use the technique of differential photometry. The idea is, if our target star and the reference star(s) are on the same image, any differences in sky conditions are negligible and can be ignored.
An example will help. M67 is an open cluster for which some 65 stars have been determined to be not variable and for which standard brightnesses in U, B, V, R and I have been determined. (An Excel spreadsheet file with these stars is available here ). In this example we will only work with B and V but the other colors will follow exactly the same procedure.
Consider two images of M67 taken with B and V filters. We can use appropriate software to determine the instrumental values for the reference stars which we will call vi and bi. We know, of course, what the reference values are from the spreadsheet Vi and Bi. (Matching the image stars to the stars in the spreadsheet is not a trivial exercise. I will defer my techniques to the end of this writeup to avoid breaking the train of thought.) Putting all of these data in a spreadsheet greatly aids the calculations.
We can plot Bi-bi vs. Bi-Vi, Vi-vi vs Bi-Vi and bi-vi vs Bi-Vi. Here are the example plots from my own images of M67.
Note that there are approximate linear relationships (pretty good in the case of b-v vs B-V). We can fit straight lines of the form y = a + b*x. This is done easily with most spreadsheets, at least those that have slope() and intercept() functions. In particular, we can get
(1) (B-b) = h1 + h2*(B-V)
(2) (V-v) = i1 + i2*(B-V) and
(3) (b-v) = j1 + j2*(B-V).
We will see that the h1, i1 and j1 values are not important.
We start with eq. (3) and note that
(b-v) = j1 + j2*(B-V) and (bc -vc) = (Bc - Vc) = j1 + j2*(Bc-Vc)
(note: bc = Bc and vc = Vc by definition of the comparison star)
(b-v) - (Bc-Vc) = j2*[(B-V) - (Bc-Vc)] (thus j1 disappears)
Now, consider eq. (1) and note that
(B-b) - (Bc-Bc) = h2*[(B-V) - (Bc-Vc)]
using eq. (4)
(B-b) - (Bc-Bc) = (h2/j2)*[(b-v) - (Bc-Vc)]
A simple adjustment gives
(5) B = b + (h2/j2)*[(b-v) - (Bc-Vc)]
All of the terms on the right are either known or measured from the image.
A likewise manipulation of eq. (2) via eq. (4) yields
(6) V = v + (i2/j2)*[(b-v) -(Bc-Vc)]
Eqs. (5) and (6) will thus yield transformed values for the instrumental values measured by typical software. The values I got for my system from the data plotted above are:
h2 = 0.072
i2 = 0.012
j2 = 0.940
I checked these coefficients by choosing one of the reference stars against which to measure and transform the instrumental values from the images above for the rest of the 65 reference stars for which I knew the actual values. The average deviation was -0.013 and -0.004 for the B and V values respectively (with standard deviations of 0.015 and 0.007).
I said I would give some hints to matching image stars to the values in the spreadsheet table. First, I plate solved the V and V images using Pinpoint contained in MaximDL. This process writes the World Coordinate System (WCS) values into the FITS header of each image. I then used Source Extractor ( http://terapix.iap.fr/rubrique.php?id_rubrique=91/ ) to identify all the stars in the image. The output of Source Extractor is configurable, but I chose to output the RA, DEC, mag and magerr for each detected star. The output file can be imported into a spreadsheet. One little trick is to do the first image, then do the second image referenced to the first. The stars in the two lists will then correspond.Source Extractor is a Linux program. For those of you running Windows, a free virtual machine using Ubuntu is available at http://www.vmware.com/appliances/directory/1068 . A free virtual machine player is available fromhttp://www.vmware.com/products/player/ .
GJ 436 (also known as Tycho 1984 2613 1) is a 10th magnitude star in Leo approximately 33 light years away. It is a red dwarf whose luminosity is about 0.5 percent of the Sun’s. A Neptune-sized planet has been detected and another planet is suspected based upon variations in the observed transits of the first.
Based upon previous measurements a transit was predicted for Friday night, February 29, 2008 CST (March 1, 2008 UT) so I set up to observe the star using the 80-cm telescope of the ASEM observatory. Over a two hour span centered on the predicted transit I took 300 images with the I filter and reduced the data in MaximDL using reference stars provided by the AAVSO. The standard deviation of the reference stars over this span was 0.006 magnitude - not bad for 10 second exposures! I subtracted the observed measurements from the average of all of them so that up is brighter and down is dimmer in the plot below.
The transit is subtle, but real. By visual inspection, I selected the regions shown by the thick horizontal bars and calculated the average over that span. The change in brightness is 0.009 magnitudes (0.007 predicted). While the standard deviation of the individual data points is some 0.01 magnitudes, the standard deviation for these averages is some 10-12 times better than that, so on the order of 0.001 magnitudes. While the depth of the transit is fairly reliable, the onset and duration are more suspect given that the values were estimated visually. Nevertheless, the estimated start of the transit was 20080301 3h28m47s (some 27 minutes later than predicted) and the estimated duration was some 46 minutes. It is that lag in the start of the transit that will yield information about a second planet - if it exists.