N2 R

To exclude unknown parameter k(r) let us divide Eq. 2.42.4 by Eq. 2.42.5; in this case we obtain equation for determining unknown x = T - Te :


Eq. 2.42.6 can be solved by the iteration method: as a first approximation, we can use x1 = T - Te ~ 500 sec which is the minimum time propagation of relativistic particles from the Sun to the Earth's orbit. Then, by Eq. 2.42.7 we determine T(x1) and by Eq. 2.42.6 we determine the second approximation x2. To put x2 in Eq. 2.42.7 we compute T(x2), and then by Eq. 2.42.6 we determine the third approximation x3 , and so on. After solving Eq. 2.42.6 and determining the time of ejection, we can compute very easily diffusion coefficient from Eq. 2.42.4 or Eq. 2.42.5:

K(R)=_ r2 (2 - T )4xfo - T + x) =- r2 (( - t )4xfc - T + x) (242 8)

'njiiji ((( - * + ^ } "{^(((3 - T, + x)f2 J' ■

After determining the time of ejection and diffusion coefficient, it is easy to determine the SEP source spectrum:

= 2n1/2 N2 (R^d^ - T1 + x))3/2exp(r2 /(4KR)(T2 - T1 + x))) = 2n1/2N3(R)x (T -T + x))3/2exp(r2 /(4KPd^J -T1 + x))). (2.42.9)

2.42.3. The inverse problem for the case when diffusion coefficient depends from particle rigidity and from the distance to the Sun

Let us suppose, according to Parker (M1963), that the diffusion coefficient k(R, r )=k(R )x(r/r f. (2.42.10)

In this case the solution of diffusion equation will be

No (R )x r13P(2-P)(K1(R ) ) (2-P) (2 -p)(4+P)/(2"P)r(3/(2 -P))

x exp

Pr2-P A

r1 r

where t is the time after SEP ejection into solar wind. So now we have four unknown parameters: time of SEP ejection into solar wind Te, p, k1(r), and No (r ). Let us assume that according to ground and satellite measurements at the distance r = r1 = 1 AU from the Sun we know N1 (R), N2 (r), N3 (r), N4 (R) at UT times T1, T2, T3, T4 . In this case t1 = T1 - Te = x, t2 = T2 - T1 + x, t3 = T3 - T1 + x, t4 = T4 - T1 + x , (2.42.12) For each N, (r,r = r1, T) we obtain from Eq. 2.42.11 and Eq. 2.42.12:

x exp

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