MBCAA Observatory

Transformations for differential photometryM. Bonnardeau

bv = (1/Tbv)*(BV)+cst  (1) 
Bb = Tb*(BV)+cst  (2) 
BV = Tbv*(bv)+k1  (3) 
Bb = Tb*Tbv*(bv)+k2  (4) 
B = b+Tb*Tbv*[(bv)(BcVc)] 
V = B(BcVc)Tbv*[(bv)(BcVc)] 
A variant:
Instead of Tb one may use Tv:
0ne gets then the transformation equations:
V = v+Tv*Tbv*[(bv)(BcVc)] 
B = V+(BcVc)+Tbv*[(bv)(BcVc)] 
A field of several stars whose magnitudes are precisely known is observed. Usually this is a
Landolt's standard field. The intensities of these stars
are measured and fitted with Pogson's law:
The measured magnitudes of the stars are then derived.
The transformation coefficients Tbv and Tb can be derived by fitting
the measurements with linear functions:
Actual observations of the field L95 that give Tbv=1.008 and Tb=0.008.
To determine Tbv, one wants to minimize:
where Cbv is a constant. The denominator is a weight factor: the larger
the error on a measurement, the less it contributes to the determination
of Tbv. (The errors on B,V are negligeable). I use m=1.
This gives:
To determine Tb, one minimizes:
with Cb a constant. The result is:
The wider the BV range of the field, the better is the determination of the slopes and of the transformation coefficients.
The transformation coefficients for my setup:
B, V transformations
V, Rc transformations
V, Ic transformations.
B. Gary's web page on CCD transformations equations for use with single image photometry.
Another method is to derive the uncertainties on the coefficients and to use them as weights to compute the averages and standard deviations.









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