Equations of condition

Assured of a reliable method of solving the normal equations, we now consider the equations of condition. The observed magnitude m of a star at M air masses with true instrumental magnitude is modelled as ([10], [28])

where is a nightly zero-point. The effective extinction coefficient depends on the effective spectral gradient [29] of the star, for which we use some color index, affected by half the atmospheric reddening ([26], [10]):

Here is the extinction coefficient for the effective wavelength of the passband; W is proportional to the second central moment of the instrumental passband; is the extra-atmospheric color of the star (obtained by differencing the values in the appropriate pair of passbands); and R is the atmospheric reddening per unit airmass, obtained by differencing the corresponding pair of values.

On general numerical-analysis grounds, we initially assume the best pair of passbands to use for and R to be the pair that flank the passband in which m was observed. Thus, for the B band of UBV, we use the (U-V) color. This is not conventional practice; however, it provides a general rule that works reasonably well for all photometric systems. Such a rule is required in a general-purpose reduction program. For a well-sampled system, this produces a strikingly accurate representation of the extinction correction [30]. For undersampled systems, we can use a somewhat more accurate gradient estimator, using a linear combination of the two adjacent color indices, as described above (see subsection , ``Choosing a gradient estimator''). For bands at the extreme wavelengths of our system, we adopt just the neighboring color index (e.g., (U-B) for the U magnitude and (B-V) for the V magnitude of UBV).

Regarding the use of (U-V), we may note Bessell's remark [2] (in connection with the problems created by the wide and variable B band) that ``in retrospect, it would have been much better had (U-V) rather than (U-B) been used by Johnson in his establishment of the UBV system.'' In any case, there are special problems in UBV, both due to the inadequate definition of the original system, and the neglect of the transformation from inside to outside the atmosphere entirely in the (U-B) index, which made the ``B'' of (B-V) and the ``B'' of (U-B) have different effective wavelengths; in principle, it is incorrect to add the two indices to a V magnitude and come up with a ``U magnitude'' as a result. Consequently, one must be very careful in doing UBV photometry, no matter how it is reduced; the only safe course is to observe at least the 20 standard stars recommended by Johnson (more would be preferable), and to look very carefully for systematic trends in the transformation residuals.

Although, for astrophysical reasons, there may be partial correlations of the true stellar gradient at a given band with color indices that are remote in wavelength from the band in question, such correlations will depend on metallicity, reddening, and other peculiarities of individual stars. If correlations obtained from one group of stars are applied to another group, the results may easily be worse than if the additional correlation had been ignored in the first place. Thus, it is exceedingly dangerous to employ distant color indices unless the calibration and program stars are very similar in all these respects. We cannot rely on such good matching in general, so these partial correlations are not used in the present package. However, one can expect that very good results will be obtained if the program and extinction/standard stars cover the same region of parameter space. That is, they should have the same ranges of spectral types, reddening, chemical composition, etc.

Because filter photometers observe different bands at different times, we have to do the reduction in terms of magnitudes (which are measured), rather than colors (which are not --- see [10], pp. 152 -- 154). This also allows isolated observations of extinction stars in a single filter, or a partial subset of filters, to be used. Furthermore, the best estimate of the extra-atmospheric color is used to reduce every observation, so that errors in individual measurements have the least effect on the reduction of each particular observation.

Of course, it may be necessary to allow the extinction to be a function of time; we always solve for individual values on different nights, as many careful investigations have demonstrated the inadequacy of ``mean extinction''. The instrumental zero-point terms Z may likewise be functions of time, temperature, or relative humidity. Finally, the zero-points may be kept the same in each passband for all nights in a run if the instrument is sufficiently stable.