Transformation Coefficients for Photometry

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(F_M/F₀) and

log(F₀/F_M) = 0.4*M.

Raising both sides to a power of 10 gives

F₀/F_M = 10^0.4M

and

F₀ = 10^0.4M*F_M

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 within 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 v_i and b_i. We know, of course, what the reference values are from the spreadsheet V_i and B_i. (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 B_i-b_i vs. B_i-V_i, V_i-v_i vs B_i-V_i and b_i-v_i vs B_i-V_i. 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 (b_c -v_c) = (B_c - V_c) = j₁ + j₂*(B_c-V_c)

(note: b_c = B_c and v_c = V_c by definition of the comparison star)

subtracting we get

(b-v) - (B_c-V_c) = j₂*[(B-V) - (B_c-V_c)] (thus j₁ disappears)

(4) [(B-V) - (B_c-V_c)] = (1/j₂)*[(b-v)-(B_c - V_c)]

Now, consider eq. (1) and note that

(B-b) - (B_c-B_c) = h₂*[(B-V) - (B_c-V_c)]

using eq. (4)

(B-b) - (B_c-B_c) = (h₂/j₂)*[(b-v) - (B_c-V_c)]

A simple adjustment gives

(5) B = b + (h₂/j₂)*[(b-v) - (B_c-V_c)]

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) -(B_c-V_c)]

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).