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ideal circumstances, e.g., full quantum efficiency, the errors are Poisson-distributed and are determined by photon statistics. This means that the expected uncertainty in an observation of n photons is *Jn; therefore, b = 0.5. If the errors are constant on the magnitude scale, such as those due to scintillation noise, the major source of noise for bright stars in an otherwise perfect sky, one should use b = 1. This arises because of the relation between magnitude differences and light fluctuations: Am = 1.086.41/1 so that if Am = const we have Al a l. Finally, if the major source of the noise is "read noise" from a CCD chip, or detector noise from a photoconductive device, the errors are independent of light level and b = 0. See Young et al. (1991) for a detailed assessment of the sources of photometric error.

In many cases, the appropriate choice of flux-dependent weight is not clear because several sources of noise can occur together. Variable sky transparency can be a major source of noise and may contribute in more than one way, especially if the observations are made near a bright city sky, where increased cloudiness means also an increase in the sky brightness and the shot noise associated with it. In such a case, not even clever instruments such as the Rapid Alternate Detection System (see Chap. 2) or other two-star photometers may be able to improve much on the error in the data for the following reason: As the star dims, it also undergoes irregular fluctuations as cloud transparency varies, and, on top of this, the fluctuations in sky brightness as well as the shot noise in the sky flux all contribute to the noise level. Clearly the separation of the comparison and program stars on the sky and the spatial and temporal scales of cloud variation play critical roles in determining how well these compensatory photometry systems can overcome the intrinsic difficulties. This having been said, we hasten to note that we would not be without one! Many, many nights of useful data collected on otherwise useless nights attest the value of RAD systems because we can sometimes make up deficits in precision by increases in integration time or through the superposition of data. The main difficulty arises when the data are phase limited to an extent that longer integration or superposition is impractical.

The use of a nonzero value for b assumes that the data have not been binned or otherwise averaged. If they have been binned, and the standard deviations of the means are used to establish the weights of each averaged or binned point, the resulting weight should have already contained some light-dependent factor; in such a case, the appropriate value for b is 0. Otherwise, the weighting factors will no longer be independent, and systematic trends in the residuals may result.

The curve-dependent weights wc are important for the simultaneous analysis of either a set of several light curves or possibly a set of light and radial velocity curves (see Sect. 4.1.1.6). Each light or velocity curve has its own variance which can be used to weight the data in that curve. Usually, but not always, light curves in shorter wavelengths, such as the Johnson U, show larger scatter than those in longer wavelengths, such as V or I. The scatter may be intrinsic, such as active region emission, or arise from variable atmospheric extinction, to which ultraviolet photometry is especially susceptible, or may also come from auroral emission. The latter explanations are less likely if the data are obtained from the variable and comparison stars simultaneously, as in a CCD or multichannel photometer, but spatial variations in auroral emission are possible. In any case, we need to provide a weight or a measure of accuracy for the light curve as a whole. In the Wilson-Devinney program, a variance ac is required. Usually, ac is the standard deviation of a single observation (rather than of the mean), and thus the associated weight follows (see Appendix A.3 for reasoning) as c 1

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