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Log f

Figure 11.7. Schematic of a power-law spectrum, I a va where a = constant. On a log-log plot, as shown here, the spectrum is a straight line with slope a.

leads to a radiated photon spectrum that is also a power law. The spectrum of the radiation can thus be represented, of course, by a power-law function, where I is the specific intensity, K is a constant, and a is a constant exponent. In nature, power-law spectra usually decrease with frequency, in which case a is negative. Because of this, a positive spectral index f = -a is often defined so that I = K v-f. It is important to specify which convention you use; we use the convention (20).

A power law is most conveniently plotted in a log-log format (Fig. 7). In this format, a power law is a straight line with a slope equal to the exponent, a. This is apparent if one takes the logarithm of (20),

The exponent a is thus the logarithmic slope. From it, one can deduce the logarithmic slope of the energy spectrum of the radiating electrons.

The slope is obtained from a spectrum by using logarithmic intervals. For instance, in Fig. 7,

The slope can be measured as a ratio of distances if the ordinate and abscissa both have the same scale factors, that is, a decade is the same distance on both axes. It is often helpful to construct log-log plots with equal scales because the slope which equals a can readily be estimated by eye.

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