TECHNICAL NOTE

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TECHNICAL NOTE D-1042

DENSITY

IN

A PLANETARY

EXOSPHERE

4.

Jackson

Herring and

Herbert

L.

Goddard Space Greenbelt,

NATIONAL WASHINGTON

AERONAUTICS

Kyle

Flight Center Maryland

AND

SPACE

ADMINISTRATION July

1961

rj

T

DENSITY

A PLANETARY

IN

EXOSPHERE

by Jackson

Herring and

Herbert Goddard

Space

L.

Kyle

Flighl

Center

SUMMARY

A discussion a planetary permits

exosphere the

the

correctness

the

density

method

ballistic

of the

of their

that

of the

components

has

directly the

theory Their

of the

basic

theorem

but for

exosphere,

the

also

be included.

Since

formula an

valid a complete

ionized

for

alternate

is given.

seems

exosphere,

of

formula

exosphere.

on Liouville's

formula

density

density

of the

questioned,

of the

must

of the

been

Opik-Singer

planetary

depth

derivation

distribution

component

scription orbit

Opik-Singer is presented.

calculation

based

concluded

of the

and

It is for

the de-

bound-

¢q

CONTENTS

Summary

..............................

INTRODUCTION

..........................

EQUATIONS FOR THE DENSITY OF THE EXOSPHERE ......................... CONCLUDING

REMARKS

ACKNOWLEDGMENT References

................... ......................

.............................

Ill

DENSITY

IN

A PLANETARY

EXOSPHERE

by

Jackson

Herring and

Herbert Goddard

L. Kyle

Space

Flight

Center

INTRODUCTION

Recently listic

Opik

component

assumes least

that as far

particles

Singer

of the above

with

neutral

the

base

base

particles

implies

eventual

build-up

this

prevents

development

been

developed This

First,

since

tribution

(however,

will

the

theorem.

bution

can be obtained

the

The

number

to infinity. phere.

density

formula

quantity

per

density

escape

at some

distance

law beyond exosphere

the

are

Other

exosphere. original

on the

_pik-Singer

of their

derivation

of the

basic

been

questioned

formula

simple

for

analytic

expression

the

permits

a calculation

of the

in a column

extending

from

to calculations

of the

escape

unit

area

integration

have

(Reference

4).

for

(Reference density,

of the the

exosphere.

density

dis-

5), it is worthwhile based

for

the

required

depth

The Opik-

exosphere

theory

formula

replaces

is relevant

numerical

the

and

in the

comments

1) has

pre-

of the

several

distriof such

level;

base

of the

The

which

escape

omitted theories

theory -- at

absence

planet,

the

3) and by Chamberlain

of their

Their

Maxwellian

the

above

bal-

is concerned.

The

from

the

neglected

profile

velocities.

distribution

gives

exosphere.

to be in a truncated

than

2).

which

be entirely

(Reference

a relatively

which

of particles This

Fish

derivation

Second,

assumed

Reference

12 of Reference

an alternate

ville's

theory.

present

correctness

(Equation

to present

and

may

neutral

Maxwellian

of the

see

in a planetary

of the

barometric

components

by Johnson

paper

located

of a full of the

and ionized

are

a theory

collisions

at greater

a sink,

developed

distribution

exosphere,

exosphere

an extension

bound-orbit Singer

density of the

particles

vents

1) have

calculation

of the

no incoming

incoming the

(Reference

as an approximate

at the

bution,

and

of the the

directly density in the

exosphere, base

of the

of a planetary

on LioudistriOpik-Singer that

is,

exosphere atmos-

EQUATIONS The

FOR

formula

of Liouville's

for

THE

the density

theorem,

is constant

along

DENSITY

which

particle

p(r)

OF THE EXOSPHERE

can

states

that

be derived the

directly

density

from

of particles

the

one-particle

in phase

form

space,

f(r,_)

trajectories:

I b=a O t_

f(?,_)

where

_ is the

particle

had

lisions,

_. _

velocity

at the and

of a particle

base _o,_

of the are

= f(R,_0),

at position

exosphere,

related

v

=

_/Vv02

M is the mass

angles

e and

of the planet, G

-

_o

is the

on a sphere

conservation

v sin 8

where

? and

located

by the

(1)

2_

at

of energy

velocity _. and

that

In the

the

absence

of angular

same of col-

momentum:

(1 - £)

T

(2a)

= voY sin 8o

(2b)

is the gravitational constant, and

8o are the angles the trajectory makes

Y is

P,/r. The

with the radius vector passing through

the center of the planet. These angles are defined with respect to the orbital plane. We shall assume exosphere

in the following discussion that the density and temperature

at the base of the

are constants and therefore independent of the angular co-ordinates of _. The

spacial density p(r) is then

p(r)

Equation

1 then

allows

with at J(v,

3 and

Equations _

and

F.

4, the

2a and In order

8/v o, 8 o) , which

f

f(F,_)

d_.

(3)

us to write:

p(r)

In Equations

:

range

2b; that

only

to evaluate

the

p(r)

the

=

f

f(R,_0)

of integration is,

transforms

=

over

integral

integration

v 2 sin

extends

d_

those

8 f(R'_o)

(4)

d_.

orbits in Equation over

J _

d_

over

all

velocity

intersecting

the

space spherical

4 we introduce to one

over

dv0 d00 "

the

compatible surfaces

Jacoblan,

d_ o :

(5)

The

Jacobian

may

be evaluated

by using

Equations

v, _?

2a and

Y

cos

2b:

_0

(6)

Again using tion 6:

Equations

2a

and

2b,

we eliminate

v and

0 from

_0

_0

Equation

5 and

use

Equa-

3 v0

cos

sin

dvo

dOo

(7) Vvo _ (i- y2 sin2 ;9o)---_2_,tG(1 - Y) Opik and Singer's Equation 12 may be obtained from Equation 7 if we replace a truncated Maxwellian

f(_,_0) by

distributionwhich omits incoming particles with velocities greater

than escape velocity. The

integration

in Equation

7 can be performed

to give

pCr)

in terms

of known

func-

tions:

p(r)

=

I

Po(R)

(1

e "*