7.7 Radiation detection
In astronomy we detect the radiation from large numbers of electrons, rather than being able to distinguish the contributions of individual electrons. The electrons will have a range of velocities and of orientations with respect to the magnetic field, so the synchrotron spectrum we observe will be the sum of lots of individual spectra with varying values of max. The resulting observed synchroton spectrum is illustrated in Figure 16.
The observed synchrotron emission from a cloud of electrons in Figure 16 is the sum of the emission from many individual electrons, each with a different energy and each with velocity at a different orientation to the magnetic field. Hence each individual electron produces a contribution peaking at its own value of max. At each frequency, , the emission comes predominantly from the electrons which have max ≈ . Consequently, if the distribution of electron energies follows a power law then the observed synchrotron spectrum will also be a power law.
How much radiation is emitted at frequency depends predominantly on how many electrons have max ≈ . That is to say the shape of the summed spectrum in Figure 16 depends on the distribution of electron energies. Mathematically, this distribution can be described by saying N (E)dE is the number density (i.e. number per unit volume) of electrons with energies between E and E + dE. For many sources of non-thermal emission, a power-law distribution of electron energies seems to hold, i.e.
where N 0 is a constant of normalization, and s is a constant known as the particle exponent. When astrophysical sources of synchrotron emission are studied, it is found that typically s lies between 2 and 3.
Question: If the number of electrons per unit volume is ne, write down a mathematical expression relating ne and N(E).
The total number of electrons is equal to the sum over all possible energies, E, of the number of electrons with energy
Notice that the relationship between the electron energy, E, and the frequency, , at which it produces most synchrotron radiation is described by Equation 15, and if we want to use the distribution of electron energies to deduce the distribution of radiation with frequency (i.e. the spectrum), we must remember to include the relationship between an energy interval dE and a frequency interval d which is implicit in Equation 15. That is to say, differentiating Equation 15, treating B as constant, we obtain