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] |
___________________________________________________________________
WebMaster: Bruce
L. Gary.
Nothing on this
web page is copyrighted. This site opened: 2009.03.10. Last Update: 2009.03.10