Scintillations in astrophysics. I - an analytic solution of the second-order moment equation

We present an analytic solution to the second-order moment equation which describes the correlated behavior of two wave amplitudes at two different spatial positions and at two different wavenumbers. The solution is valid for waves from a common source passing through a medium containing electron density fluctuations with a Gaussian spectrum. A comparison with results obtained by numerical solution of the same equation is given. It is shown that the analytic solution leads to a pulse shape and a power spectrum, both of which differ significantly from those given previously by numerical calculations. Consideration of the global energy conservation theorem for the correlated behavior of four wave amplitudes shows that the square of the second moment cannot, in general, equal the fourth moment, as has occasionally been assumed, unless energy conservation is violated. It is also shown that the scintillation index would necessarily be precisely unity if such an assumption were in force. These results bear directly on the use of pulsar signals as probes to unravel the turbulent structure of the interstellar medium.