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.