3. Solving the Îto equations
where is a vector of functions, G is a matrix of functions, is a vector of independent Wiener processes and D is the correlation matrix of the Wiener processes.
The moments of the elements of can be found from the moment ordinary differential equation (ODE), see Soong (1973), Sect. 22.214.171.124(c):
3.1. Solving for moments in µ and y
The system of equations we wish to solve is (3), (4) and (5). Fortunately, only one of the three equations contains an explicit noise term, and further simplification arises because the energy equation is very simple.
Firstly, we derive the statistical properties of . This can be done by using the µ Eq. (5) to construct the moment ODE using (7) with
Together, (8), (9) and (10) can be used to find the moments of . Doing so does not present any further difficulties beyond the tedious task of evaluating multiple integrals over x, where, in general, the integrands consist of terms of the form , l being an integer.
Result (13) is the familiar mean scattering result, re-derived from this new rigorous standpoint. Result (14) is completely new, and gives the variance in column depth of those electrons whose energy has been degraded by a factor : the variance of y is given by .
3.2. Constructing the distribution function
Now we show how the distribution function itself can be expressed analytically. We note that such an exact, analytic expression is mainly of academic interest, since the complete expression can be very complicated, and may prove rather difficult to use in practise. More often, the moments themselves will prove to be of more immediate use - this is demonstrated in Sect. 5, where the second order moment is used to look at how large pitch angle electrons spread away from their point of injection.
The characteristic function of a random variable X that has distribution function is defined to be, see Soong (1973),
It is clear therefore that and g form a Fourier transform pair. Now, the MacLaurin series for is
Differentiating the integral in (15) gives
ie each term of the MacLaurin expansion of is simply related to the moments of X.
Applying this to the random variable we find that its distribution function is given by
where is the distribution f integrated over pitch angle. The pitch angle distribution at x can be calculated in a similar fashion and is known to be given by a Legendre polynomial solution, see Kel'ner and Skrynnikov (1985), Lu and Petrosian (1988) and MC.
© European Southern Observatory (ESO) 1998
Online publication: March 3, 1998