# 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 so that there is little or no difference in atmospheric extinction, either between the two of them.

## 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.

16 July 2020 stars Draper RA Dec approx V ΔV ΔB ΔI Δ(B-V) Δ(V-I)
Northern Hemisphere

Andromeda red HD 10307 01 41 47.1 +42 36 48 4.96 under review under review   under review
blue HD 10205 01 40 34.8 +40 34 37 4.94

Perseus red HD 21552 03 30 34.5 +47 59 43 4.36 under review under review   under review
blue HD 21551 03 30 36.9 +48 06 13 5.82

Orion red HD 30545 04 48 44.6 +03 35 19 6.03 1.290 0.045   -1.245
blue HD 30544 04 48 39.4 +03 38 57 7.30

Leo Minor red HD 90040 10 24 08.6 +33 43 07 5.50 0.378 -0.652 1.381 -1.030 -1.003
blue HD 89904 10 23 06.3 +33 54 29 5.90

Serpens red HD 140573 15 44 16.1 +06 25 32 2.63 2.930 1.800   -1.130
blue HD 140775 15 45 23.5 +05 26 50 5.58

Hercules red HD 156283 17 15 02.8 +36 48 33 3.18 1.465 0.071 2.780 -1.394 -1.315
blue HD 156729 17 17 40.3 +37 17 29 4.65

Ophiuchus red HD 161242 17 44 13.1 +05 15 02 7.80 0.510 -0.720   -1.230
blue HD 161261 17 44 15.7 +05 42 51 8.26

Aquarius red HD 210434 22 10 33.7 -04 16 01 5.98 0.291 -0.694   -0.985
blue HD 210419 22 10 21.1 -03 53 39 6.28 see note 5

Pegasus red HD 220657 23 25 22.8 +23 24 15 4.40 under review under review   under review
blue HD 220061 23 20 38.2 +23 44 25 4.58

Southern Hemisphere

Sculptor red HD 10142 01 38 27.5 -36 31 42 5.94 -0.240 -1.295   -1.055
blue HD 10538 01 42 03.0 -36 49 56 5.70

Columba red HD 43899 06 17 01.2 -37 44 15 5.53 0.340 -0.650   -0.990
blue HD 43940 06 17 09.6 -37 15 13 5.87

Centaurus red HD 129456 14 43 39.4 -35 10 25 4.05 -0.050 -1.570   -1.520
blue HD 129116 14 41 57.6 -37 47 37 4.00

## 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. Aquarius magnitudes are preliminary and subject to change.

claims of variation star SIMBAD HIP VSX
Andromeda blue variable no Yes (1918)

Perseus red variable no Yes (1914)

Serpens red     micro var?
Serpens blue suspect var no no

Hercules red variable Yes Yes 1918
Hercules blue no no Yes 1929

Ophiuchus blue no no Yes (1995)

Aquarius red no no Yes (1984)

Pegasus blue variable Yes 70 min period

Columba blue no no Yes (1981)