Previous method, with the transformation coefficients derived from the average (not weighted) and the standard deviation (not weighted) of several measurements.

Transformations for differential photometry (with weights)

M. Bonnardeau
27 Sept 2007
Updated 18 Apr 2012

Abstract

An introduction to transformation coefficients and equations for differential photometry. The transformation coefficients are derived from the weighted averages and the weighted standard deviations of several measurements.

The transformation equations

The transformation is a correction to differential photometry measurements to take into account the instrument (telescope, filter, camera) and atmospheric transmission. This transformation can be made when two filters are used, for example the B,V filters.

For this, the transformation coefficients Tbv and Tb are necessary:



B,V are the exact magnitudes, b,v are the measured magnitudes. A first order approximation is to assume that the responses of the setup (instrument + atmosphere) are linear. Then Tbv and Tb are constant slopes. For a perfect setup, Tbv=1 and Tb=0.

The astronomer uses a comparison star with known magnitudes Bc,Vc and measures the magnitudes b,v for a star. One wants the transformed magnitudes B,V. One has:

b-v = (1/Tbv)*(B-V)+cst	(1)
B-b = Tb*(B-V)+cst	(2)

This gives:

B-V = Tbv*(b-v)+k1	(3)
B-b = Tb*Tbv*(b-v)+k2	(4)

with k1 and k2 constants. For the comparison star, (4) gives:

0 = Tb*Tbv*(Bc-Vc)+k2

Then the transformation equation:

B = b+Tb*Tbv*[(b-v)-(Bc-Vc)]

For the comparison star, (3) gives:

Bc-Vc = Tbv*(Bc-Vc)+k1

Then:

V = B-(Bc-Vc)-Tbv*[(b-v)-(Bc-Vc)]

A variant:

Instead of Tb one may use Tv:



0ne gets then the transformation equations:

V = v+Tv*Tbv*[(b-v)-(Bc-Vc)]

B = V+(Bc-Vc)+Tbv*[(b-v)-(Bc-Vc)]

Tb and Tv are connected:

1/Tbv = 1 + Tv -Tb

Derivation of the transformation coefficients

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+/-0.029 and Tb=0.008+/-0.006.

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:



with the error:

To determine Tb, one minimizes:

with Cb a constant. The result is:


with the error:

The wider the B-V range of the field, the better is the determination of the slopes and of the transformation coefficients.

Let be N measurements of a coefficient T and of its error, from different nights and different Landolt's fields. I adopt the weighted average of the measurents as the value of T:


and as the error the weighted standard deviation:

Actual derivation of the coefficients

The transformation coefficients for my setup:
B, V transformations
V, Rc transformations
V, Ic transformations
Rc, Ic transformations.

Link

B. Gary's web page on CCD transformations equations for use with single image photometry.