DERIVATION OF DIFFENTIAL PHOTOMTERY SIMPLIFIED TRANSFORMATION EQUATIONS
Bruce L. Gary, Hereford, Arizona, USA

INTRODUCTION

This web page describes a derivation of the "simplified transformation equations" that are presented on the web page where their use is described (link). The first part of the next section repeats some of the definitions that are also presented in the other web page.

BASIC PHOTOMETRY EQUATION

Mag = Z - 2.5 × LOG10 (Flux / g) - K' × AirMass + S × StarColor                                                                                                                                (1)
where Z is a zero-shift constant, specific to each telescope system (which should remain the same for many months),
Flux is the star's flux (sum of counts associated with the star). It's called "Intensity" in MaxIm DL,
g is exposure time ("g" is an engineering term meaning "gate time"),
K' is zenith extinction (units of magnitude),
S is "star color sensitivity." S is specific to each telescope system (and should remain the same for amny months).

This is a general equation that is true for all filter bands (even unfiltered), though there are different values for the constants for each filter. For example,

V = Zv - 2.5 × LOG (Flux / g) - Kv' × AirMass + Sv × StarColor                                                                                                                                  (2)
R = Zr - 2.5 × LOG (Flux / g) - Kr' × AirMass + Sr × StarColor                                                                                                                                   (3)

Similar equations exist for bands B and I (the rest of this web page uses only V and R to illustrate concepts).

Note that some observers like to add the term SM * AirMass * StarColor, where SM is a constant specific to each telescope system and filter band. The value of SM is usually so small that I will ignore it in the following.

Note that I haven't defined StarColor. It can be defined using any two filter bands, with B-V, V-R, V-I in common use. For this web page I'll use:

StarColor, C = V - R - 0.4

Why, you may ask, do I subtract 0.4 form V-R in defining C. By doing this I've defined C in such a way that for typical stars C ~ zero, and this color term can be ignored if I don't know a satr's color and I don't mind having errors of ~ 0.03 mmagnitude due to that assumption.

To illustrate eqns 2 & 3 using my 14-inch telescope:

V = 19.81 - 2.5 × LOG (F/g) - 0.16 × m - 0.07 × C                                                                                                                                                    (4)
R = 19.95 - 2.5
× LOG (F/g) - 0.11 × m - 0.10 × C                                                                                                                                                    (5)
where m = AirMass, Kv' ~ 0.16 [mag/airmass] and Kr' ~ 0.11 [mag/airmass] typically, and F = Flux for large photometry aperture

Similar equations exist for B, I and C (clear). I find these equations handy for quickly calculating magnitudes when I don't want to bother consulting a star catalog for setting MaxIm DL's magnitude scaling. For example, suppose I have a V-band image with F = 10,000 counts, g = 60 seconds, m = 1.5. Let's set C = 0, and solve for V = 19.81 - 2.5 * LOG (10,000/60) - 0.11 * 1.5, or V = 14.09. I've checked stability of the constants Z, K' and S for all bands and indeed they are pretty constant for many months, provided I don't change optical configuration (or let the corrector plate collect dust). I think this equation alone can produce results with an uncertainty of ~0.06 magnitude.

SUMMARY OF SYMBOL DEFINITIONS

V and v represent true and instrument V-band magnitudes. R and r represent true and instrument R-band magnitudes.
The second letter for a magnitude refers to either the target star or calibration star. Thus, Vt is the true magnitude of the target star.
C is star color, which I define to be V - R - 0.4. Ct is star color for the target star, and Cc is star color for the calibration star.
S is star color sensitivity, which is a constant unique to a specific telescope system (which should remain the same for many months). S depends on filter band, so Sv and Sr correspond to V-band and R-band star color sensitivities.
F stands for Flux, called "Intensity" by MaxIm DL. It should be measured using a large photometry aperture. Fv is an V-band flux, Fr is
an R-band flux. Fvt is V-band flux for the target star, etc.
g stands for exposure time [seconds].
K' is zenith extinction, or extinction rate per air mass. Each filter band has a different extinction rate, so Kv' is for V-band and Kr' is for R-band.
m is shorthand for air mass.
Instrumental magnitudes (with zero-shift values that will be seen to be unimportant) can be written: v = - 2.5
× LOG (Fv/g) + Zv and r = - 2.5 × LOG (Fr/g) + Zr, where Zv and Zr are instrumental magnitude zero-shift values that will eventually cancel and become irrelevant in the following derivation.

DERIVATION

In the following I'll use eqns (4) and (5) for derivations. The use of specific constants will facilitate knowing which parameters have known values and which have unknown values. Also, I'll use red to indicate which parameters have "not yet" been evaluated.

Consider the V-band image with one target star "t" and one calibrated star "c". The target and calibration star magnitudes can be written:

Vt  = 19.81 - 2.5 × LOG (Fvt/g) - 0.16 × m - 0.07 × Ct                                                                                                                                                (6)
Vc = 19.81 - 2.5 × LOG (Fvc/g) - 0.16 × m - 0.07 × Cc                                                                                                                                                (7)

Let's replace the -2.5 × LOG (F/g) terms by their instrumental magnitude equivalents v and r:

Vt  = 19.81 + (vt - Zv) - 0.16 × m - 0.07 × Ct                                                                                                                                                               (8)
Vc = 19.81 + (vc - Zv) - 0.16 × m - 0.07 × Cc                                                                                                                                                              (9)

Subtracting (9) from (8), and simplifying, yields:

Vt-Vc = (vt - vc) - 0.07 × [Ct-Cc]                                                                                                                                                                                (10)

Notice that the simplification process removed the zero-shift constants (19.81), the extinction terms (0.16 × m) and the instrumental magnitude zero-shift values (Zv).

Now consider the R-band image. Similar operations lead to the following equivalent of eqn (10):

Rt-Rc = (rt - rc) - 0.10 × [Ct-Cc]                                                                                                                                                                                (11)

Recall that the two constants, -0.07 and -0.10, are the star color sensitivity parameters that are assumed determined before the differential photometry task is performed. These star color sensitivity values are obtained by observing a calibrated star field, such as M67 or the many Landolt star fields. The procedure for interpreting those observations to obtain star color sensitivity for each filter band are described on another web page.

Recall how C is related to V and R:

Ct =  Vt - Rt - 0.4                                                                                                                                                                                                        (12)
Cc = Vc - Rc - 0.4                                                                                                                                                                                                        (13)

Let's substitute these versions of Ct and Cc in eqns (10) and (11). First, here's the new (10) with simplifications:

Vt - Vc = (vt - vc) - 0.07 × [ (Vt - Rt - 0.4) - (Vc - Rc - 0.4) ]                                                                                                                                  (14)

Vt = Vc + (vt - vc) - 0.07 × [ (Vt - Rt) - (Vc - Rc) ]                                                                                                                                                    (15)

And here's the new (11), with simplifications:

Rt - Rc = (rt - rc) - 0.10 × [ (Vt - Rt - 0.4) - (Vc - Rc - 0.4) ]                                                                                                                                    (16)

Rt = Rc + (rt - rc) - 0.10 × [ (Vt - Rt) - (Vc - Rc) ]                                                                                                                                                    (17)

Let's subtract eqn (17) from (15):

(Vt -Rt ) = [Vc + (vt - vc)] - [ Rc + (rt - rc)] + (- 0.07 + 0.10) × [ (Vt - Rt) - (Vc - Rc) ]                                                                                           (18)

The only unknown in the above equation is the target star's color, so let's extract it:

(Vt -Rt ) - (- 0.07 + 0.10) × (Vt - Rt) = Vc + (vt - vc) - Rc - (rt - rc) - (- 0.07 + 0.10) × (Vc - Rc)                                                                           (19)

(Vt -Rt ) × [1 - (- 0.07 + 0.10)] = Vc + (vt - vc) - Rc - (rt - rc) - (- 0.07 + 0.10) × (Vc - Rc)                                                                                    (20)

Vt -Rt { Vc + (vt - vc) - Rc - (rt - rc) - (- 0.07 + 0.10) × (Vc - Rc) } / {1 - (- 0.07 + 0.10)}                                                                                  (21)

Ct { Vc + (vt - vc) - Rc - (rt - rc) - (- 0.07 + 0.10) × (Vc - Rc) } / {1 - (- 0.07 + 0.10)} - 0.4                                                                                (22)

This is the solution for the target star's color, Ct.

Since we have a way to evaluate Ct it is now possible to return to eqns (10) and (11) to evaluate Vt and Rt.

Now let's generalize by replacing constants with their general symbols:

Ct { Vc + (vt - vc) - Rc - (rt - rc) - ( Sv - Sr) × (Vc - Rc) } / {1 - ( Sv - Sr)} - 0.4                                                                                               (23)

Vt = Vc + (vt - vc) + Sv × [Ct - Cc]                                                                                                                                                                          (24)

Rt = Rc + (rt - rc) + Sr × [Ct - Cc]                                                                                                                                                                           (25)

In summary, when a telescope system's star color sensitivity is knwon (from earlier observations of calibrated star fields), and given V-band and R-band images that include a target star and calibrated star, the target star's V-band and R-band magnitudes can be obtained using the following procedure:

 Ct = { Vc + (vt - vc) - Rc - (rt - rc) - ( Sv - Sr) × (Vc - Rc) } / {1 - ( Sv - Sr)} - 0.4     Vt = Vc + (vt - vc) + Sv × [Ct - Cc]       Rt = Rc + (rt - rc) + Sr × [Ct - Cc]

where the following definitions are used:

Vc = known V-mag for calibration star,
Rc = known R-mag for calibration star,
Sv = V-band star color sensitivity (slope of plot: V-mag error versus star color C, for several calibrated stars), explained more elsewhere,
Sr =  R-band star color sensitivity (slope of plot: R-mag error versus star color C, for several calibrated stars), explained more elsewhere,
C = star color, defined to be V - R - 0.4,
Ct = star color of target star,
Cc = star color of calibration star,
Vt = V-mag of target star,
Rt = R-mag of target star,
vi = instrumental magnitude = 2.5 × LOG10 (Fi / g) + zero-shift constant (that doesn't matter),
Fi = flux of star i,
g = exposure time