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\s+3IRAF PHOTOMETRY PACKAGE\s-3
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Differential Reduction Equations
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Paul C. Schmidtke
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August 3, 1983
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Last Revision: August 24, 1983
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This paper discusses
the standard reduction equations for differential photometry
as used in the IRAF Photometry Package.
Specific examples for the UBV photometric system are presented.
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IRAF PHOTOMETRY PACKAGE
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Differential Reduction Equations
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As presented in
\fIIRAF Photometry Package: Reduction Equations and Methods
of Calculation\fP (hereafter Paper I),
the standard equations for photometric data reduction are:
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.EQ
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define = '~"="~'
define + '~"+"~'
define - '~"-"~'
define x '^\(bu^'
define Qi 'Q sub i'
define Qn 'Q sub n'
define Qs 'Q sub s'
define Ci 'C sub i'
define Cs 'C sub s'
define k1 'kappa sub 1'
define k2 'kappa sub 2'
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.EQ I (1)
Qn = Qi - k1 x X - k2 x X x Ci
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.EQ I (2)
Qs = eta x Qn + xi x Cs + zeta
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where
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@Qi@, @Qn@, and @Qs@ are the instrumental, natural,
and standard system magnitudes of a particular photometric quantity.
This quantity can be the result of a measurement through a single filter
(i.e. a magnitude),
the ratio of measurements through two filters
(i.e. a color index),
or a combination of measurements through three or more filters
(i.e. a complex index).
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@Ci@ and @Cs@ are the instrumental and standard system magnitudes
of another photometric quantity, usually a color index.
@Ci@ may refer to the same quantity as @Qi@,
but @Cs@ cannot refer to the quantity as @Qs@.
Examples are given the in next section.
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@X@ is the air mass of a observation.
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@k1@ and @k2@ are the first- and second-order extinction coefficients,
respectively.
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@eta@, @xi@, and @zeta@ are the transformation coefficients
(@zeta@ is the nightly zero point).
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Equations (1) and (2) are used for the reduction of all-sky photometry.
In differential photometry, however,
two stars are observed: a variable (i.e. program) star and
a comparison (i.e. a psuedo-standard) star.
They ought to be relatively close to each other in the sky,
have approximately the same photometric colors,
and be observed in rapid succession.
If these conditions are met,
it can be assumed that the atmospheric and instrumental conditions
effecting both stars are nearly identical
and that the accuracy of the results is increased by
measuring the variable star relative to the comparison.
That is, differential magnitudes between the two stars can be
measured more precisely than separate all-sky magnitudes for each star.
The differential reduction equations, found by subtracting the
all-sky relationship for one star from the other, are:
.EQ I (3)
DELTA Qn = DELTA Qi - k1 x DELTA X - k2 x X bar x DELTA Ci
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.EQ I (4)
DELTA Qs = eta x DELTA Qn + xi x DELTA Cs
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where the order of subtraction is not important,
but it must be consistent.
In Paper I, an equation similar to equation (3) is used
in the determination of @k2@.
In that case, the stars (i.e. blue-red pairs) are sufficiently close
in the sky that the term @k1 x DELTA X@ can be ignored.
In the general case, however, the term must be included.
Values for all coefficients are found by means of the methods
described in Paper I,
using data explicitly obtained for such calculations.
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It is common to insert an additional star into the observing
sequence: a check (i.e. another psuedo-standard) star.
This star is also reduced differentially, relative to the designated
comparison star.
If the differential magnitudes are not constant
(from night to night or from observing run to observing run),
then one of these supposed-constant stars is in fact variable.
That is, the check star is used to check the constancy of the comparison star.
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Examples of Differential Reduction Equations
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In this section,
examples of differential reduction equations are given for
the UBV photometric system.
The notation is adopted from Hardie
(1962, \fIBasic Astronomical Techniques\fP, p. 178).
The differential extinction equations are:
.EQ I (5.a)
DELTA v sub 0 = DELTA v - k prime sub v x DELTA X
- k prime prime sub v x X bar x DELTA (b-v)
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.EQ I (5.b)
DELTA (b-v) sub 0 = DELTA (b-v) - k prime sub bv x DELTA X
- k prime prime sub by x X bar x DELTA (b-v)
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.EQ I (5.c)
DELTA (u-b) sub 0 = DELTA (u-b) - k prime sub ub x DELTA X
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The transformation equations are:
.EQ I (6.a)
DELTA V = DELTA v sub 0 + epsilon x DELTA (B-V)
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.EQ I (6.b)
DELTA (B-V) = mu x DELTA (b-v) sub 0
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.EQ I (6.c)
DELTA (U-B) = psi x DELTA (u-b) sub 0
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These sets of extinction and transformation equations are
readily expanded to include the additional colors in
UBVR or UBVRI differential photometry.

