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.