Differential Photometry and Color Transformations
Post date: Apr 23, 2013 7:21:25 PM
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 
0 refers to the target star,
c refers to the calibration star, and
lower case applies to instrumental values and upper case applies to standard values
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
C is a constant that we need to eleminate
Tvi = reciprocal of the slope of (v-i) vs (V-I) (1/0.980 = 1.02 in my case)
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
K = a constant we need to get rid of (again), in my case from the data above K = -0.154,
Tv = slope of (V-v) vs (V-I), in my case Tv = 0.036
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.