## Photometric Systems

The reason that the magnitude scale is defined in terms of standard stars is that it is extremely difficult to measure the absolute flux of light with any precision, but it is fairly easy to compare the flux of one source with that of another. Why should this be so?

The answer lies in another question: What do we mean by "flux?" In theory, of course, we can easily define it as the arrival of some number of photons per second in some well-defined range of wavelengths—that is, in fact, the goal in measuring flux. However, what comes out of the detector, whether that detector is a photomultiplier tube or a CCD image, is an instrumental response to the starlight—a meter reading, a chart deflection, a count in electrons per second, a total pixel value—one number. We have no idea how many photons at each wavelength contributed to the response of the detector; we see only its integrated response to all wavelengths. When we write an equation to describe the factors that affect detector response, ¿/star, the problem becomes clear:

where X denotes wavelength, Fstar(X) is the flux from the star reaching the top of Earth's atmosphere as a function of wavelength, Ax(?i) is the transmission of Earth's atmosphere for an air-mass of X as a function of wavelength, T(X) is the transmission of the telescope optics as a function of wavelength, f(X) is the transmission of the filter used in the observation as a function of wavelength, Q(k) is the quantum efficiency of the detector as a function of wavelength, and the notation dX indicates that the integration is performed over wavelength.

This equation says that the response of your CCD depends on a large number of factors, all of which vary significantly with wavelength; and the total response that you obtain is the sum of the detector responses at each wavelength. The dilemma of photometry turns out to be figuring out how much each wavelength of light contributes to the flux.

The solution to this dilemma is not to measure all of the wavelengths at once, but to use filters to divide the spectrum into short segments, and measure each color individually. If the segments can be kept narrow enough, the wavelength dependencies of the factors disappear, and the flux equation becomes much simpler:

The dependence on wavelength has disappeared because each spectrum segment under consideration, AX, is made small enough that each of the parameters is constant across such a narrow wavelength span. As a matter of practical concern, however, if we make the range of wavelengths too narrow, the total signal is smaller and more difficult to measure.

By measuring two close-together stars in rapid succession, the atmospheric properties are the same, the telescope optics remain unchanged, the color filter doesn't change, and the detector quantum efficiency is constant—so the like terms in the numerator and denominator cancel:

¿ref FVQiAxTfQAX Fref

All this equation says is that all of the factors cancel out for a star observed with the same telescope, filter, and detector through the same air mass. Thus, we can plug the two instrumental responses into Equation 10.2 and find the ratio of the flux of a star relative to that of a reference star, and then compute the difference in magnitude. To accomplish this, we have placed significant constraints on how we make the observation. We must observe stars through the same atmospheric path; and even more importantly, we must measure them through color filters that

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