--_i
(,!i_.
NASA
iN D-1042
¢Xl
z.
,
_
.¢,
Z
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 "*<l-Y)
v_
_
1
_
--
y2
-_-erf
vax_
o(
e l÷Y
1
-
)
1
(8)
"_ err
where
a = R/H,
H = scale
po(R)
erf
height,
= density
x
= 2--_
V_"
_x
at the
law for
;0(R),
the
the
density
and
REMARKS
considering
density
level,
e-Y2dy.
CONCLUDING
When
critical
the
above
variation
just
below
distribution,
is obtained.
the
base
we first
Secondly,
of the
exosphere.
notice
we observe
The
that
as
that
reason
a -, _,
;(R)
the
barometric
is not equal
for this
is that
to
we have
omittedall
tinuity
incoming
particles
from
arbitrarily
results
more
realistic
zone
above
The
theory
which
density
profile,
to gain
replace
some
This
insight
conventional
p0 H , with
d/P0H
H being
is plotted
exosphere
the
for
of 1500°K
H approaches
large
infinity,
values
atmosphere
of
above
(Reference
scale
at the
d
this
matter,
:
of
For
a.
base
and
of the
approaches
level.
This
studies
of the
escape
extent
escape
level,
level
and
a
by a diffuse
a = IUH, but not
of the
the
atmosphere,
exosphere,
quantity
and
in
in or-
d:
we may
In the
of a, and
note
that
limit
there
been
used
of atmospheres,
for
oxygen,
as
p aecomes
words,
has
to be
In Figure
atomic
exosphere.
result
d is taken
exosphere.
a = 72 for
Poll; in other
escape
escape
discon-
(O)
of the
independent
the
of :he
This
dr.
orientation,
hydrogen
above
calculated
YR p(r)
base
velocity.
by a parameter
the
of a planetary
at the
escape
likely.
we have
height
becomes
a, d
the
6) in their
atomic
less
determines
of escape
as a function
a = 4.5
temperature
For
theory
boundary
8, is characterized
parameter
into
than
collisions
sharp
become
d
In the
all
the
gradually
Equation
a way.
greater
neglecting
would
collisions
too transparent
der
with velocities
equal
1, the
the
we take
_ approaches
proportional
is one
but their
ratio
earth's
provided
a
zero,
to
scale
by Various
to
height
1/r
2.
of
investigators
justification
for
its "
16
14
12
10
:0o8
O6
04
0.2
0
0
I
I
I
1
I
I
I0
BO
50
70
90
I10
_=-Figure
1 - Variation
of the
exosphere
density
is given
of a planetary
atmosphere,
ratio
by
d
d is
I
..
130
150
R
H
d/Poll
with
the
parameter
in Equation
8.
In the
set
equal
to
p0 H.
a.
con,'entional
The
integrated
theory
of
the
neutral
escape
use
applies
seems
the
to stellar
to provide
exosphere;
must
a valid
for
(Reference
above
1800
ponent
of the
a complete
7) has
of the
is unimportant
major
its use
that
earth's
(Reference
H +.
orbit
The
case
the
earth's
Opik
but
may
component
example,
dominant
recently
findings
components
published
a sup-
that
com-
indicate
be important
of
components
For
the
have
Their
theory
bound-orbit
exosphere.
and Singer
exosphere,
and
exosphere
component.
Opik-Singer
of the ballistic
ionized
of a planetary
in the
0 + and
bound
in the
that
description
be
in the
planets
for
atmospheres.
be stressed
suggested
may
discussion
to planetary
it should
kilometers
plemental
than
justification
however,
be included
Singer
rather
in the
this
exospheres
2).
ACKNOWLEDGMENT
I
The
cal
authors
integration
wish
to thank
of Equation
R. Roth
9 on the
of the
Goddard
IBM 7090
Space
Flight
Center
for
the
numeri-
computer.
REFERENCES
1.
Opik,
E.J.,
Physics
2.
Opik,
E.J.,
4.
and Singer,
F.S.,
March
5.
Physics
of Fluids
7.
Singer,
S.F.,
September
"Distribution
R.A.,
"The
in a Planetary
Exposphere,"
1959
of Density
February
"Interplanetary
131:47-56,
J.C.,
S.K.,
_^_
J.W.,
Brandt,
Mitra,
S.F.,
Fish,
of Density
November--December
4(2):221-233,
and
J.
6.
"Distribution
in a Planetary
Exosphere.
II,"
1961
Telluric
Hydrogen
Corona,"
Astrophys.
J. 131:502-
1960
Chamberlain,
Astrophys.
S.F.,
2:653-655,
of Fluids
Johnson,
515,
Singer,
of Fluids
Physics
3.
and
and
"The
Chamberlain,
3:485-486,
Upper
"Structure
1960
- L_._ey n_Id, w.
January
D-1042
Gas.
II Expansion
of a Model
Solar
Corona,"
1960
J.W.,
May--June
"Density
of Neutral
in a Planetary
Exosphere,"
1960
Atmosphere,"
Calcutta:
of the
Exosphere,"
Earth's
Gas
The
Asiatic
J. Geophys.
Society,
Res.
1952
65:2577-2580,