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<H3><A NAME="SECTION001855400000000000000">
Standard stars</A>
</H3>
<P>
Standard stars can be included in the extinction solution, if one is careful.
The problem is that there are usually appreciable ``conformity errors'' due to
mismatch between the instrumental and the standard passbands [<A
HREF="node307.html#MS92">16</A>].
These are due to the integrated influence of features in the spectra of
individual stars, which differ in age, metallicity, reddening, etc. at a given
color index value.
Thus, it is wrong to assume that the transformation between the instrumental
system and the standard system is exact, even if there were no measurement
error in either system [<A
HREF="node307.html#BN89">1</A>].
<P>
Usually, one finds that the instrumental system can be reproduced
internally with much more
precision than one can transform between it and the standard system.
This means we should regard the standard values as imprecise estimates of the
extra-atmospheric instrumental values,
after transforming from standard to instrumental system.
In effect, we regard the standard values as noisy pseudo-observations of the
instrumental, extra-atmospheric magnitudes.
<P>
Therefore, a reasonable equation of condition to use for the standard values
is:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{displaymath}
m_{std} = m_0 + a C_0
\end{displaymath} -->
<I>m</I><SUB><I>std</I></SUB> = <I>m</I><SUB>0</SUB> + <I>a C</I><SUB>0</SUB>
</DIV>
<BR CLEAR="ALL">
<P></P>
where we effectively regress the noisy (because of conformity errors) standard
values on the internally precise instrumental system.
No zero-point term appears, as we already took it out in conjunction with
the extinction.
As before, <I>m</I><SUB>0</SUB> is the extra-atmospheric instrumental magnitude,
and <I>C</I><SUB>0</SUB> is the estimator of stellar gradients across the passband
being transformed; the same gradient estimator should be used in both
extinction and transformation equations.
The transformation coefficient <I>a</I> is then estimated in the general solution.
In general, we expect a higher-order polynomial in <I>C</I><SUB>0</SUB> to be
needed [<A
HREF="node307.html#Y92">29</A>]; it can be determined if enough standard stars are available.
This equation then gives an explicit transformation from the instrumental to
the standard system.
<P>
One problem is to know what weights to assign such pseudo-observations.
The conformity errors in the transformation are usually much larger than the
internal errors of measurement in the standard system, so that quoted
uncertainties in well-observed standard stars are usually
irrelevant to our problem.
For convenience, unit weight in the extinction solution is intended to
correspond to errors of 0.01 magnitude.
This is a typical order of magnitude of conformity errors as well, so we can
start with unit weight for these values, and adjust the weights after a trial
solution, followed by examination of the residuals.
This process assigns self-consistent weights to the standard values.
<P>
Alternatively, we can omit the standard stars from the extinction solution,
determine the extinction entirely from the observations, and then determine the
transformation in a separate step.
In this case, a separate set of zero-points must be determined in the
transformation solution, as the nightly zero-points in the extinction solution
are no longer coupled to the transformation.
However, this does not increase the number of unknowns, as we must then
fix one night's zero points to prevent indeterminacy.
(In principle, one should do this using Lagrange multipliers; in practice, it
hardly matters, because the nightly zero points are always determined much more
precisely than other parameters in the solution.)
This process is preferable if there are enough extinction observations, as it
does not propagate systematic errors from the transformation into the
extinction (see pp. 178 - 179 of [<A
HREF="node307.html#Y74">10</A>], and [<A
HREF="node307.html#P82">19</A>]).
<P>
However, observers sometimes do not get enough extinction data to permit solving
for extinction directly.
Those who follow Hardie's [<A
HREF="node307.html#H62">10</A>] bad advice sometimes observe each
standard star just once; then, if there are no real extinction stars,
the standard values have to be used to obtain any extinction solution at all.
But usually,
in combined solutions, the extinction coefficients will be slightly
more precise, but less accurate, than in separate solutions.
The reduction program will tell you if one method or the other seems
preferable; in any case, it lets you decide which method to use.
<P>
There has been much splitting of hairs in the astronomical literature over the
direction in which the standard-star regression should be performed.
Unfortunately, the arguments have all assumed that the regression model is
functional rather than structural; that is, they assume there is an exact
relation connecting the two systems, in the absence of measurement noise.
In practice, conformity errors are usually larger than measurement errors, so
the functional-regression model is incorrect.
In any case, the problem we have here is a calibration problem: given
measurements of some stars in the standard system A and the instrumental system
B, we want to predict the values that <I>would</I> have been observed in A
from the values we <I>have</I> observed in B, for the program stars.
In this case, the regression of A on B provides the desired relationship
[<A
HREF="node307.html#FB92">5</A>].
<P>
While the conformity errors make it reasonable to do the regression
in the sense described above, it is also clear that they involve significant
effects that are not accounted for in the usual treatments of transformation.
In particular, substantial <I>non-linear</I> terms are to be expected
in transformations, and the neglect of higher-order cross-product terms in
general makes transformations appear to be <I>multivalued</I> [<A
HREF="node307.html#Y92">29</A>].
The correct treatment of these problems requires a well-sampled system.
Trying to correct for missing information after the fact is a lost cause;
effort should instead go into minimizing the conformity errors in the first
place.
<P>
In general, one cannot emphasize too strongly the need to measure the
instrumental spectral response functions, and to choose filters -- by
actual transmission, not by thickness! -- that reproduce the standard
system as accurately as possible.
The difference between instrumental and standard response functions represents
a serious disparity that cannot be made up by any reduction algorithm, no
matter how cleverly contrived.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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