Obtaining Transformation Coefficients

Photographers sometimes include a color chart in their photographs. The chart helps them get proper color balance in the final prints. Photometrists must worry about color as well, and they adjust for imbalance via "transformation." Transformation is effected via a coefficient called "epsilon" (ε), and PEP observers must establish epsilons for the different bands in which they take data.

An epsilon is determined from raw photometry of a pair of stars having a high color contrast - a reddish one and a bluish one - for which the standard magnitudes are well-established. The "instrumental" magnitudes from your telescope/filter/photometer combination will differ from the standard magnitudes to some extent. The epsilon coefficient is used to adjust your instrumental data to the standard system. The closer your instrumental system is to the standard, the smaller your coefficient will be.

Bright red/blue pairs of calibration stars are hard to come by. We have a list of twelve, of which some are under review. To establish epsilon, we treat the blue star as if it were a variable, and the red star as if it were a comparison. An observation run consists of seven samples of the blue star bracketed by eight of the red.

Stars in a pair are close together, but they are not "doubles." We want to observe them quickly and as high in the sky as possible to minimize extinction effects.

Calculations

If we are calibrating V band, the math looks like this:

epsilonV = (ΔV - Δv)/Δ(B-V)

Where ΔV is the standard magnitude different between blue and red, and Δ(B - V) is the standard color contrast. Δv is the instrumental magnitude difference you measure.

The table below contains standard magnitude and color differences for BV photometry. We are slowly adding values for VI photometry. For calibrating I band, the math is:

epsilonI = (ΔI - Δi)/Δ(V-I)

B band is more complicated on account of "second order" extinction:

epsilonB = (ΔB - Δb - k"*Δ(B-V)*X)/Δ(B-V)

Where k" is the second-order extinction coefficient and X is airmass. For guidance, look to the PEP manual.

The Δv (or Δb or Δi) is an average differential magnitude of the seven individual red/blue bracketings:

-2.5*log(2*blue_counts/(red_before_counts+red_after_counts))

Where counts are the net counts for each star. A standard deviation for the average should be computed (a deviation over 0.01 is not very good).

An epsilon should be established every year, or after any cleaning of the optical surfaces of your telescope (aside from blowing off dust). Calibrations from at least two nights should be averaged together (weighted by standard deviation) for a final value.

How it's used - V band example

When reducing normal photometry, you first obtain an instrumental differential magnitude, Δv. Transformation is effected by adding the product of epsilon and the color contrast between the variable and comparison:

standard ΔV = Δv + epsilon*Δ(B-V)

If your measurement of Δv for the red/blue pair was very close to the standard ΔV, then epsilon will be very small. If the color contrast between your variable and comparison is reasonable, then the transformation adjustment will be modest. This is why we try to choose comparisons of color similar to the variable.

Notes:

1. Various sources claim that some of the calibration stars are variable (see below). We're gradually sorting that out.

2. We currently do not attempt transformation for U band, and transformation for R band presently involves all-sky photometry, which is beyond the scope of this document.

3. There also exist transformations for color indexes such as B-V and U-B. They are known as mu (μ) and phi (φ), but we do not presently work with them.

4. If you are using a German equatorial mount, do not perform a meridian flip in the middle of a calibration sequence. Take all of your data before or after transit.

5. These magnitudes are preliminary and subject to change.