SYNCHROTRON AND SYNCHROTRON SELF-ABSORPTION FOR A POWER-LAW PARTICLE DISTRIBUTION: ASYMPTOTIC FORMS FOR FINITE ENERGY RANGE

We calculate and plot the synchrotron power, P(nu), the absorption coefficient, alpha(nu), and the source function, S(nu), for a power-law distribution of charged particles with Lorentz parameter values gamma(1) <= gamma <= gamma(2). For this purpose, we define parametric functions Z(p) (x, eta), H(p)(x, eta), and Y(p) (x, eta) with eta = gamma(2)/gamma(1), such that P(nu) proportional to Z(p) (gamma(-2)(1)nu/nu(0), eta), alpha(nu) proportional to H(p)(gamma(-2)(1)nu/nu(0), eta), and S(nu) proportional to Y(p)(gamma(-2)(1)nu/nu(0), eta). Corresponding asymptotic forms are also calculated and plotted for three frequency ranges, i.e., x << 1, 1 << x << eta(2), and x >> eta(2), especially in the case of finite parameter eta. Asymptotic forms of the middle range are possible for functions Z(p) and Y(p) for p > 1/3, and for function H(p) for all positive values of index p. A characteristic value, eta(c)(p, epsilon) (with epsilon << 1), is then defined for each of the above functions so that for eta greater than or similar to eta(c)(p, epsilon) the middle range asymptotic forms could be considered. Further calculation details are also presented and discussed.