Geoscience Reference
In-Depth Information
The derivation of Eq. 3.21 is simple. Indeed,
@f
@x i D
@f
@E
@E
@x i C
@f
@L 2
@L 2
@x i
@L 2
@ v i
@f
@ v i D
@f
@E
@E
@ v i C
@f
@L 2
Substituting the evident equalities
@E
@x i D
@U
@x i ;
@E
@ v i D m v i ;
@L 2
@x i D
@L 2
@ v i D 0;
and
m v DĖ p L 2 .r/ L 2
into Eq. 3.21 gives Eq. 3.23 .
The ion flux toward the particle is expressed in terms of f as follows:
Z d 3 v Z . v dS /f. r ; v /:
J D
(3.25)
The integrals on the right-hand side of this equation are taken over all v and the
surface of a sphere of radius r . The sign ā ā in the definition of the flux makes J
positive. In spherical coordinates, Eq. 3.24 is rewritten as
J D 8 2 r 2 Z
Z
1
v 3 d v
f.r; v ;/d:
(3.26)
0
1
Now let us change the variables. The rule for replacing the variables .r; v ;/ !
.r;E;L/readily follows from definition ( 3.22 )ofthevariables E and L :
m 2 r X
s
dEdL 2
p L 2 .r/ L 2 :
2 v 2 d v qd !
(3.27)
The restrictions on the intervals of integration over E and L 2 are defined by two
conditions, L 2
L 2 .r/ and L 2 .r/ 0. The latter is equivalent to E U.r/.In
what follows we do not specify the limits of integration except for the cases, where
they play a decisive role.

Search WWH ::

Custom Search