THE ORTHO-PARA HYDROGEN EQUILIBRIUM
IN THE URANUS ATMOSPHERE
by
GAIL ANNE HEFFNER KAHN
S.B., Massachusetts Institute of Technology
(1978)
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September, 1979
----- ------- --
II
Signature of Author
I J L IJ
Department of Earth and Planetary Sciences, September 10, 1979
-
Certified by
0"
-c
-
Thesis Supervisor
Accepted by
)artmental Committee on Theses
'3 fE.P"T
1979
MIT LIB ,ES
-I.-ls-X.XII^-L.III111 I
.-IC-~----.
.^illl~L1II
ABSTRACT
THE ORTHO-PARA HYDROGEN EQUILIBRIUM
IN THE URANUS ATMOSPHERE
by
Gail Anne Heffner Kahn
Submitted to the Department of Earth and Planetary Sciences at
the Massachusetts Institute of Technology on September 10, 1979
in partial fulfillment of the requirements for the degree of Master of Science.
ABSTRACT
Ortho-para conversion may be an important heating mechanism
in the Jovian planets. In order to calculate the o-p equilibrium,
an atmospheric model must be developed. Temperature, pressure,
scale height, eddy diffusion coefficient, and number density of
H2 , the dominant gas in the atmosphere, are all defined as a
function of altitude above the 450 0 K level. Ortho-para conversion
mechanisms are considered, with the conclusion that the ionmolecule recombination H' + o-H, = p-H, + HT is important in the
the
dominant in
upper atmosphere, while H + o-Hx = p-H, + H is
mechanisms
loss
differing
the
of
Consideration
lower atmosphere.
in the atmosphere leads to conclusions that, although cosmic rays
are an important source of ionization, the source occurs at a
level where the ion loss is fast due to interactions with
methane. UV ionization very high in the atmosphere above the
methane level leads to high proton densities and subsequent o-p
conversion of Ha. The lower atmosphere mechanism, on the other
hand, is too slow compared to loss mechanisms for H.
In order for the above conclusions to fit with observations,
there must be essentially no Ha transport across the tropopause.
An argument is made for the feasibility of this condition. The
percentage of para-hydrogen is graphed as a function of altitude
in the Uranus atmosphere.
THESIS SUPERVISOR: John S. Lewis
TITLE: Associate Professor
_I^
_
~~~
/i~_ _~
i~_ll_^____ Iil
TABLE OF CONTENTS
1. Introduction
1.1.
The Uranus atmosphere
4
1.2.
A model of the Uranus atmosphere
6
1.3. Ortho- and para- hydrogen
8
1.4.
9
Ortho-para conversion mechanisms
2. The Ortho-Para Equilibrium
13
2.1. Sources of ions in the Uranus atmosphere
13
2.2. Assessment of ion reactions
14
2.3.
18
Conditions for observation of o-p conversion
2.4. Assessment of lower atmosphere conversion
mechanisms
3.
Conclusions
20
22
3.1. Ortho-para equilibrium in the Uranus
atmosphere
3.2. Future work
22
22
Table 1
24
Table 2
25
Figures
26
Bibliography
34
___1_111____
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-4-
1. Introduction
1.1 The Uranus atmosphere
axis
the
Uranus is unique in the solar system in that
of its rotation is inclined 980 to the normal of its orbital
five
system
a
and
satellites
of
plane.
The planet has
rings,
all of whose axes of rotation are also inclined 980.
or
Jupiter
than
less
much
on
but its density is large in comparison
Saturn,
with solar composition material.
50% by mass heavy elements.
that
are
Uranus
The internal pressures on
Uranus is estimated to
be
Therefore, it is thought likely
containing
core
differentiated
Uranus
possesses
a
mass
elements.
Most models of the planet (Hubbard,
higher
1975; Podolak & Cameron, 1974) suppose
a
core
sur-
emission
from
rocky
rounded by an extensive ice layer.
Brown (1976) detected non-thermal radio
the
direction
of
Uranus with IMP-6.
field intensity of 0.6 - 0.9 gauss.
Uranian
magnetic
spin axis.
field
He estimated a polar
The dipole axis of
any
would probably be aligned with its
A magnetic field oriented 980 would have
unique
properties as outlined by Siscoe (1975).
The effective temperature of Uranus has
to
be
been
58± 30 K (Fazio, Traub, and Wright, 1976).
observed
Since the
calculated temperature of a gray body at Uranus' distance is
58 ± 2cK,
this
leads
some
to
internal heat source, while the
conclude that Uranus has no
other
Jovian
planets
do.
II-~II1.1.11
~XY____
-5However, the effects of the planet's unique orbital inclina-
source.
this
Fazio, Traub, and
problem
and
have
measuring
Wright
(1976)
concluded
that
source should have a maximum value of "10
Uranus is
planet which is
the first
effective
a value for the internal heat
determining
and
temperature
when
account
tion must be taken into
have
considered
the internal heat
3
.
ergs cm'~sec'
far enough from the
Earth so that dynamical activity manifested by clouds cannot
be observed with
there
is
no
direct
Consequently
observations.
ground-based
information concerning motions in the
atmosphere, although motions must be present because of differential solar heating.
There is visual evidence for broad
The
exist.
bands in the atmosphere, and discrete features
planet appears greenish-blue with faint dusky belts.
Tables 1 and 2 summarize the
orbital
atmospheric
and
parameters for Uranus.
Two molecules have been identified in the Uranus atmosphere; H, and CH* (Newburn and Gulkis, 1973).
The 1:~ ratio
has been estimated to be between 1 and 300 times
value (Trafton, 1976).
solar
the
The high mean density indicates that
other, heavier molecules may be
found
in
the
atmosphere.
Prinn and Lewis (1973) and others have suggested other other
gases which are likely to be
NH3 ,
HO,
HS,
present.
CH,,, and CH,,.
These
include
Prinn and Lewis (1973)
concluded that cloud layers of CHI, NH,, NH4SH, and HaO
possible.
HIS,
He,
have
are
H,0, and NHjSH would primarily be confined
~_~
~--~)YP--LX) ~Y-*~L- -~
____
___
-6to the deeper layers ( P Z 10 bars).
gests
that
there
exists
sug-
(1976)
Trafton
a thin CHf haze at P ~ 0,2 - 0.7
bars and a thick NH3 cloud at P - 4 - 10 bars
1.2 A
gmel
Uranus atmosohere
_t
f.
Any derivation of the ortho-para
hydrogen
equilibrium
in the Uranus atmosphere must have as a basis an atmospheric
The model atmosphere used here is
model.
the
models
of
(P
=
2.3
In
(1973).
x 10'
T3
addition,
shown in
an
adiabat
model
atmosphere
was
is
Figure 1.
Graphs of pressure and temperature, P(z) and T(z),
derived
of
) above T = 400cK and P = 158
The P-T diagram for the Uranus
bars.
compilation
Trafton (1976), Wallace (1975), and Weiden-
schilling & Lewis
assumed
a
are
from models of Trafton (1967) and Weidenschilling &
Lewis (1973),
and are shown in Figures 2 and 3.
Scale height H(z) was calculated using the equation
H(z) = RT(z) ,
ug
R = gas constant
u = molecular weight of atmosphere
g = acceleration of gravity
Values were supplied by Table 2 and Figure 3.
The H(z) pro-
file is shown in Figure 4.
The eddy diffusion coefficient K(z) is shown in
5.
K(z) was calculated using the equation
Figure
UL_
-7K(z) = H(z)
[Au)jRT(z)]
[3uP(z)
I,
few bars).
which applies in the case of free convection (P Z
of P(z), T(z), and H(z) were supplied from Figures 2
Values
- 4.
The internal heat source
4
was
using
calculated
the
conservative estimate that the long-lived radionuclides U, K
and Th are the only internal heat sources.
value for 0 of ~ 10a ergs cm'-
This provides
a
sec"'.
High in the atmosphere (PA 0.1 bars),
K(z)
[uP(z)
While the proportionality constant is unknown, it
that
K
"
10
is
known
Thus for higher alti-
in the stratosphere.
tudes, the values for K(z) derived from the
above
equation
were adjusted accordingly.
The number density of Ha was derived from the equation
_,
kT(z)
k = Boltzmann's constant = 1.38 x 10"1b ergCK-I.
nH (z) =
nl4x(z) is shown in Figure
observations
6.
Trafton
(1976)
notes
that
of the hydrogen spectrum indicate that Ha dom-
inates all other gases in the atmosphere.
Now that the Uranus atmosphere has been
modeled,
the next step is to consider the ortho-para hydro-
gen question.
between
quantitatively
Section
1.3
deals
with
the
distinctions
ortho- and para- hydrogen, and section 1.4 looks at
the different ortho-para conversion mechanisms.
~
II
_~_
_L~_1^
.C~1I~UI~--C.~CII
-8-
hydrogen
1.3 0rtho- and a-
The two protons in an Ha molecule
parallel or anti-parallel spins.
can
either
possess
An H. molecule with paralrota-
lel nuclear spin is called ortho-hydrogen and has odd
tional
quantum
numbers.
Para-hydrogen
nuclear spin and even quantum numbers.
has anti-parallel
Spontaneous
conver-
sion of the two types of molecules practically does not take
either by radiative or
place,
of
lifetime
collisional
Thus
is
(p-H 2 )
ortho-
and
of
molecules
The
a radiative transition from the j=1 rotational
state of ortho-hydrogen (o-Ha)
hydrogen
processes.
two
1.3
para-
to the
x 10
j=O
state
of
sec (Raich and Good,
hydrogen
can
be
para1964).
as
considered
different gases, which differ in certain
optical and thermal properties.
The thermodynamic equilibrium between ortho- and
hydrogen
is
governed
temperatures near OcK,
energy
states.
para-hydrogen.
brium
Thus,
nuclear spin.
by Boltzmann's distribution law.
all molecules pass into their
to
the
This equilibrium is
room temperatures,
lowest
statistical
the
equili-
weights due to
practically
and thus "normal" hydrogen is
attained
of
at
composed of
3/4 para-hydrogen and 1/4 ortho-hydrogen (Farkas, 1935).
graph
At
at low temperatures, all Hx will be
At high temperatures (kT >> B),
correspond
will
para-
A
para-hydrogen percentage vs. temperature is shown
in Figure 7.
It
would be useful to know the ortho-para ratio in
the
_
-9-
para-
is
ortho- and para- hydrogen are coupled neither
nor
since
mechanism,
heating
a
Even though
higher energy state.
a
in
The transition from
planets.
is
hydrogen
to ortho-
ortho-hydrogen
Jovian
the
of
atmospheres
radiative
by
collisional mechanisms, interconversion between the two
Conversion can be accomplished by
species can occur.
atom-
atom or ion-atom interchanges, and these can be catalyzed by
grains or paramagnetic substances. Section 1.4 looks at each
of
these
interconversion
and
mechanisms
estimates their
importance in the atmosphere of Uranus.
conversion mechanisms
1.4 Ortho-ara
The atom-atom interchange conversion
H + o-H
p-H, + H
.
cm 3sec'
has a rate constant k of 6.3 x 10ture
of
1000'K
Thus, at
to
temperatures
corresponding
the effective temperature of Uranus (580 K) and the upper
troposphere of the atmosphere (100
rate
tempera-
cmsec - ' at 283CK (Hirsch-
and 3.5 x 10 - 7
felder et al., 1936).
at a
will
less
be
c
- 300'K),
reaction
than or on the order of 10-'7 cmsec " .
Thus, this reaction is very slow in
Furthermore,
the
the
upper
atmosphere.
the temperatures at which the reaction becomes
fast enough to be an important factor in o-p conversion
temperatures
so
high
that the o-p equilibrium is
are
3/4 para
and 1/4 ortho hydrogen.
Ortho-para conversion can
also
be
catalyzed
on
the
-
10 -
surface of grains:
o-H, + g ---
g + p-H,
The "grains" present in the Uranus atmosphere would be solid
CHq
particles, which are present at pressures of 0.2 to 0.7
bars.
H,
This corresponds to temperatures of 90
densities of ~ 101 0 cm "3 .
number
-
1600K
At such temperatures,
the collision of HA molecules with grains is likely to
to
the
loss
of
temperatures.
10- 17
is
n,
probably
It
much
less
The
at
such
cannot exceed the collision rate of
or 103 see - ' (Dalgarno et. al.,
sec'
lead
kinetic energy of the gas particles.
efficiency of the catalysis is unlikely to be high
low
and
than
that.
Since
knowledge of grain surface conditions, no
1973),
there
definite
and
is
no
conclu-
sions can be reached about the efficiency of the process.
Paramagnetic gases provide another
conversion
small
can
magnetic
Conversion
is
be
catalyzed.
moment,
in
Ortho-hydrogen
while
para-hydrogen
which
o-p
has a very
has
none.
caused by the non-homogeneous magnetic field
of the paramagnetic molecules
Thus,
way
paramagnetic
involved
the
collision.
gases will catalyze the para- to ortho-
hydrogen conversion (Farkas, 1935).
p-Ha +
in
MpC.~Aq-> M ?%.
has a rate constant of 5 x 10"
The reaction
+ o-H,
cm3 sec - '
(Farkas 1935).
Paramagnetic gases which may be present in
the
Uranus
atmosphere include include free radicals such as PH,, H,
HS,
-
CH., and NHI.
calculated
instance,
-
The abundance of each species present can
by
of
rate
equals
For
destruction.
of H can be roughly determined by
abundance
The
be
atmospheric steady state, i.e.
an
assuming
production
of
rate
11
using the major loss mechanism
H + H + HA
->
Ha
+ H
which.has a rate of reaction k = 9.8 x 10 3 3
m'sec -
at T
=
By scaling Jovian data from Stro-
700CK (Kondratiev, 1972).
bel (1969) appropriately to account for Uranus' greater distance from the Sun, the production rate of H , as determined
by
production rate = flux / scale height,
can be found to be approximately 103 cm-3 see ' at the
level.
700cK
Thus, the equation
d[H]/dt = klH]'[H,] = production rate
can be solved for [H]. The number
density
of
H
from
the
above calculations is nw = 3x10" emi
The
involved
number
are
densities
acquired
of
the
other
free
in the same fashion, using as their
major loss mechanisms interactions with H.
number
cm10"
3
.
radicals
At
700CK,
the
densities of the free radicals range from 106 to 104
At 1000K, their number densities are
to 10" cm' 3 .
higher,
from
Any recombinations of these free radicals
have comparable or lower number densities.
At these
levels
in the atmosphere, the dynamical lifetime, as calculated by
- 12 -
K
is approximately 2.5 x 10
number
density
sec.
calculated
Even
using
the
largest
for the paramagnetic gases, the
chemical lifetime, as determined using
kni,
is approximately 2 x 10"I sec, much longer than the dynamical
lifetime.
Thus the paramagnetic substances will not be able
to produce ortho-para conversion.
The single remaining method for o-p conversion
is
the
ion-molecule interchange,
H4 + o-Hj___ p-H
+ Ht
.
The rate constant for this reaction is given by Dalgarno et.
al.
(1973)
as k = 101
cem3 sec - 1 at a temperature of
which they calculated from quantal statistics.
Dalgarno,
170.5
0
K.
reaction
50OK,
According to
the energy defect corresponds to a temperature of
Thus,
at temperatures much less
forms p-Hz and destroys o-Ha.
tion rates alone,
fastest
-
in
it
than
170 0 K,
By looking at reac-
seems that the ionic mechanism
the upper atmosphere.
the amount of ions present is
the
is
the
However, an assessment of
required before a
of the chemical lifetime can be made.
calculation
_1_Y11_^_____
~--I~~L~I~IIIL
- 13 -
2. The Ortho-Pa a Eauilibrium
2.1 Soues
of ions j
Atmospheric
sources.
Jthe- Urau Q atmosphere
derive
can
ions
possible
three
At
high altitudes, photolysis produces ions.
At
great depths, ions can be a result of
And
from
be a source of ions, assuming
can
lightning
finally,
equilibrium.
thermal
that such a phenomenon occurs in the Uranus atmosphere.
Molecules in the Uranus atmosphere can
two
be
by
ionized
of radiation: solar extreme ultra-violet (EUV)
sources
radiation and cosmic rays.
Since the intensity of solar EUV
radiation decreases with the square of the distance from the
is
Sun, at Uranus' distance cosmic ray ionization
more important than EUV ionization (Capone, et.
actually
al.,
1977).
According to McElroy (1973), the most important source of H+
comes
In addition,
from the dissociative ionization of H,,.
protons recombine very slowly in
the
atmosphere,
so
that
most atmospheric models assume that protons are removed only
about
105
cm
3.
and
occurs
hydrogen density of ~ 10'
high
in the atmosphere, at a
cm-3 (Capone, et.
al. , 1977).
Lightning is unlikely to be a major factor in ion
duction.
On
Earth,
approximately
energy is converted to energy in
currents
and
acoustic waves).
conversion factors, 10-
is
density
The maximum proton
by radiative recombination.
10-
the incident
of
lightning
pro-
(as
electrical
On Uranus, assuming similar
of the solar flux is
converted
to
- 14
electrical
10 -
and
acoustical
-
energy.
of the incident solar flux is
Therefore,
violet radiation.
However a fraction of
in
there is
the
is
of
ultra-
103 times more energy
available from UV radiation to be used in
bonds.
form
breaking
chemical
In addition, a large part of the energy in lightning
involved in
ciation.
heating the atmosphere without causing disso-
Thus,
lightning
is
a
minor factor in
the bulk
chemistry of a planet (Prinn and Owen, 1976).
Thermochemical equilibrium, the last
in
ion
production,
process
is difficult to quantify in the Uranus
atmosphere since so little is known about the
phere.
work.
Any
estimates
range
from 10"
lower
atmos-
of ion density would be pure guess-
In the lower atmosphere of Earth, ion
ties
involved
to 10£ cm-3
number
densi-
(Wallace and Hobbs,
1977).
So, for lack of any other data, the abundance of ions in the
Uranus
cm
- 3
atmosphere
will be taken to be approximately 10
.
2.2 Assessment of ijn reactions
To calculate the chemical lifetime of the reaction
H
+ o-H,
.=
p-H
the abundance of H+ is needed.
the
ion
+ H+ ,
H+ abundance will depend
on
production rate and on the various loss mechanisms
for H+ and H3 +:
- 15 -
(1) H
-> CH' + H.
-> CH4* + H
+ CH
ki = 3.8 x 100
cm'sec "' (Huntress, 1977)
(2) H* + e-> H. + H
ka = 3.8 x 10'7 cm'sec'
(Atreya & Donahue, 1976)
-> H + hi
(3) H- + ek 3 = 6.6 x 10" cm'sec'
(Atreya & Donahue, 1976)
H +, H5+, etc. are not considered to be losses of H+,
these species can also produce o-p conversion.
rate.
also depends on the ion production
section
2.1,
ions
can
be
since
H+ abundance
As
discussed
in
produced by two major sources:
solar EUV radiation and cosmic rays.
According to the model
of Capone et. al. (1977) for the Uranus model atmosphere, UV
production of H+ reaches a peak at n,
rays
have
flux
a
= 10"
0.8 cm -~see '
of
cm -3 .
Cosmic
in interstellar space
Typical cosmic rays have energies of
(Ginzburg, 1969).
100
Therefore they will penetrate to greater depths in the
MeV.
atmosphere than UV radiation.
Capone
et.
the
al.,
= 101'
occurs at nmH
In the ionospheric
model
of
peak ion production from cosmic rays
cmS .
In the Uranus ionosphere, diffusive unmixing creates an
upper
region in which there exists no methane.
This region
can be divided into two layers: an upper layer (layer I)
in
which H+ is the dominant H ion, and a lower layer (layer II)
in which H3 + dominates. In the Capone model, layer I
at
nj .
4x10"
< nt,
cm "3 .
Layer II
l
< 3x10
'
cm" .
is
occurs
very thin, existing at
III
4x10"
cm-3
where
the methane abundances become significant, and occurs
at n 4 -:- 3x10
*
cm -3 .
Layer
is
the
The turbopause occurs at n,,
layer
=
10 '3
- 16 -
cm'
3
The three layers are characterized by their rates of
.
catalysis, their dominant loss mechanisms for
H+
and
H3 +,
and their rates of ion loss.
The dominant loss mechanism in
interaction
CHq,
with
a
lifetime for neutralization of H+ for
this region (nI = 101 4 cm
"3
= 4x10
(Note that CHs is calculated using
10
over
times
point
in
) is
tLke,(H+) = 1 / k,n,i
enriched
typical
H+
The
as given above.
(1)
reaction
the
is
III
layer
solar
a
_
sec.
value
of
CHz
/
Hz
proportions: n~q4 = 7x10-3
the
nH,.) H+ is the dominant H,+ ion, since
conversion
to
H3 + via the reaction
H + + 2H, ->
has a rate constant
Donohue,
1976)
into H3 + is 3x10
k=3.2
H+ + H,
10- 'l
x
cm 3
sec-'
(Atreya
&
so that the lifetime of H+ until conversion
"e
sec,
longer
than
time.
neutralization
The collision lifetime at this point is calculated from
JId.
= mean free path / speed
= 10 - cm / 10'cm sec-' = 10-6 sec.
The number of collisions per H+ lifetime is equal to
-
(H+)
/t ,ot. = 4 collisions/H+.
Assuming that each collision
this
an
c-p
conversion,
means that every H+ will convert 4 Hg molecules before
becoming neutralized.
will
produces
The total number of H+
ions
be due to cosmic ray ionization at this level.
present
Assum-
1_
- 17 -
ing 100 MeV cosmic ray energies and an ionization
of
14
eV
cm- sec ble
for Ha (Huntress, 1977),
H+.
then there exists 7x106
Thus, the total number of conversions
4 x 7x10' = 3x10
is
potential
cm" sec'' conversions.
possi-
The upper
limit of the o-p conversion lifetime is the dynamical
chemical lifetimes must be less than dynamical
since
time,
life-
lifetimes for chemical reactions to occur. At this level
in
the atmosphere the dynamical lifetime is 8x10' sec.
x total number of conversions
o<Cro
(8x10' sec)(3x10' cm
2x10"
This means that total conversion of Hx will
sec
)
conversions/cm .
occur
down
to
number densities of H. calculated from
n,
no, = 5x10
cm-3,
= 2x10"
a number
cm
1
/
density
scale height .
much
lower
than
that
present in layer III. The conclusion drawn from this is that
little
present,
ortho-para
conversion
occurs
where
methane
is
because of the fast rate of loss of H+ ions due to
methane interactions.
In
layer II
H3 + is
model of Capone et. al.
10
cm"3
reaction
at n"L = 10 'I
(2)
above,
the
dominant
(1977),
cm- 3 .
the
electron
abundance is
The major
be
10a
cm.
the
loss
mechanism
is
electron- H3+ recombination. The
3x10
sec,
equal to the H3 + abundance.
collisional lifetime is 10- 1 sec,
to
From
the peak abundance of H3 + is
lifetime of H3+ due to neutralization is
the
species.
taking the mean free
where
The
path
Therefore, there are 3x105 collisions/H 3 +
II_1141~~--L1~1-4Y1 -~-~..
- 18 -
cm
lifetime, and the total number of collisions is 3x10
nH
Since
This is due to the electron-ion
does not occur in layer II.
of
which removes many H3+ ions before
mechanism
they can effect o-p
ratio
.
cm-3 in this region, total o-p conversion
= 10'
recombination
3
Note
conversion.
conversions/total
however,
is higher than in
density
HR
the
that
layer III.
mechanism
imum of 10
al.,
1977).
is given by reaction (3).
cm
3
to
the
by
cm-"
sec' .
is
electrons
et.
(Capone
erg cm" sec'
, the
The lifetime of H+ due
2x10
sec,
taking ng,
The mean free path at this level is 103 cm, and
collisional
lifetime is 10 -
is
t
sec.
Therefore,
collisions/H+ lifetime.
conversions
cImr3
Taking a UV flux of 3.8x10-3
neutralization
equals n+.
H+ density is at a max-
at an H2 density of 10"
flux of H+ becomes 2x10
loss
major
In layer I, H+ is the dominant ion and its
cm7" sec - '.
4x10'~
This produces 2x10
total
the
number
of
The conversion lifetime
must be less than or equal to the dynamical lifetime ta ,
=
H /D (D=molecular diffusion coefficient).
x total number of conversions = 8x10
T
H.
Thus, total conversion of
/scale
height = 2x10
3
cm-
3
.
will
occur
clown
to
8x101
Note that this is greater than
region.
the number density of H, present in the
total conversion of Ha occurs in layer I.
Therefore
conv./cm'
-
19 conversion
m observation 9f --
2.3 Conditions
Many observations have been made
(ex.
Trafton,
1976)
that indicate the presence of an ortho-para conversion above
the tropopause.
2.2,
the
According to calculations made
o-p
H.
section
conversion occurs at levels much higher than
the tropopause.
verted
in
In order to fit with observations, the con-
must be rapidly mixed down below the turbopause,
but must not be mixed into the lower
tropopause.
atmosphere
below
the
In order for this to occur, the transport rate
across the turbopause must be relatively
large,
transport
must be relatively
small.
rate
across
the
tropopause
while
the
The transport rate can be calculated from
w
jx = transport rate of H,
= vertical speed
w = K / H.
The turbopause in this atmospheric model
cm 3 .
1013
At
cm.
Therefore,
nH,
= 10'
mE
= 5x10
cm
's
cm'sec '
this level, K = 10
3
H, = 2x10'
, K
cm- 2 sec~' .
= 10" cm'sec'
cm-P sec - '.
occurs
'
nH
=
and H = 5x10'
At the
, and H
at
tropopause,
= 2x10e cm.
Thus,
Since the values of K, H, and
nw,
may be off by an order of magnitude, the transport rates can
be said to be essentially equal.
value
of
In particular, the minimum
the eddy diffusion coefficient is uncertain.
The
minimum value calculated in this model is 10q cmzsec -' ,
but
K
could
conceivably
planet whose
minimum
value
have a lower value still.
atmosphere
of
is
much
more
K = 10" cm sec-' .
Jupiter, a
energetic,
has
Uranus, which is
a
much
- 20 -
further from the Sun, will have a more stable atmosphere. It
is
therefore
not unreasonable to expect that K will have a
minimum value of 10 - 1 cm sec
1
.
The high flux of conversions calculated in section
produces
total
o-p
conversion throughout the upper atmos-
phere, down to the tropopause.
low
value
2.2
If, at that point, K
has
as described above, a "bottleneck" will occur at
the tropopause, such that mixing will be minimal across
boundary.
a
Converted
Hx
will
then
be
the
seen in the upper
atmosphere as described by observations.
2.4
Assessmet of lower atmosphere conversin mechanism
Lower atmosphere conversion of hydrogen,
if
it
takes
place at all, will be caused by the reaction
H + o-Hz -
p-Hz + H.
The dominant loss mechanisms for H are
CH 3 + H + M -> CH, + M
k = 8.5 x 10[M] cm' sec -" (Strobel, 1973)
H + H + M -> Hz + M
kL = 3 x 10 " cm' sece 1 (Kondratiev, 1972)
(1)
(2)
In addition, since CH
is an important species, loss of
CH3
must be included:
CH 3 + CH3 + M -> CzH.: + M
k1 = 6 x 10"9 [M] cmI sec-i
(3)
The loss mechanism H + H -> H.
important
tion.
reaction
due
is not considered to
(Strobel, 1973)
be
an
to the problems of energy dissipa-
- 21 -
A steady state is assumed: rate
rate
of
destruction.
of
production
equals
Thus, two simultaneous equations may
be set up:
d[CH3 ]/dt = J[CH] = k,[CH3 1[H][M] + k 3 [CH31'[M].
d[H]/dt = J[CHq] = k,[CH1j][H[M + k:[H];[M].
The CHq photolysis rate, J[CH], is equal to the flux at the
Uranus
distance
/
scale
height.
J[CH4 1 = 500 cm- sece'.
Upon solving for the abundance of H, it is found that
1.0 - 8x10
4x10'
cm- 3 .
This gives chemical lifetimes of 3x10'( -
see, which are much
dynamical lifetimes.
will not occur.
nH
longer
than
the
corresponding
Thus, o-p conversion from this process
- 22 -
3.
Conclusions
equilibrium in thb Uranus atmoshere
3.1 Ortho-ara
From the calculations done in section 2.2, and
tions
minimal transport of H, across the tropopause due
of
to a low eddy diffusion coefficient,
a
emerges
picture
o-p ratio as a function of altitude.
the
assump-
of
The ortho to para
conversion occurring at an ionospheric temperature of
1350K
p-Ha of 30%, from Figure 7.
This
will
give
a
percentage
ratio will be mixed down
o:p
observed
to
ratio
to
the
tropopause,
differ from that expected at upper
atmospheric temperatures if o-p conversion
In
the
causing
not
did
occur.
the lower atmosphere below the tropopause no o-p conver-
sion is able to take place, as determined
Therefore,
lower
in
section
atmospheric o:p ratios will be those pro-
duced by local temperature equilibrium, and will be
cal
to
the
2.4.
in
ratios
identi-
Figure 7. The percentage of para-
hydrogen vs. altitude in the Uranus atmosphere is
shown
in
Figure 8.
3.2 Future work
The atmospheric model described here is a
dimensional
ters.
calculation
conversion
understanding of
o-p
one-
of the various atmospheric parame-
An improved model, since it
ortho-para
crude,
is
calculations,
conversion
in
the
basis
for
the
would greatly improve
the
atmosphere. - Of
- 23 -
course, the atmospheric model must have for its
vational
evidence
parameters
for
different
of
Uranus.
greatly enhance not only
equilibrium,
orbital
and
basis obseratmospheric
Improvement of these figures would
comprehension
of
the
ortho-para
but also understanding of other aspects of the
planet Uranus.
Uranus' distance from the
important
source
of
energy.
Sun
makes
cosmic
rays.
and
accuracy.
mination
the
effect
rate
of
internal
heat
source.
Uranus'
rotation have yet to be measured with
All these quantities have some part in the deterof
the
o-p conversion.
Hopefully, Voyager II
1986 will shed light on these and other questions about
planet
of
Infrared observations are necessary to deter-
mine the value of Uranus'
radius
an
Observations of the planet's
magnetic field are essential in determining
cosmic
rays
Uranus.
Truly
in
the
accurate calculations of the ortho-
para equilibrium cannot be made until that time.
- 24 -
Uranus'
orbital Darameters:
Parameter:
orbital period:
T= 84.01 yrs
1
inclination of equator to orbit:
i= 97.930
1
rotation period:
t= 15.57t 0.80 hr
2
radius:
Rq = 25,400 km
3
mass:
M= 14.6 M &
3
equatorial surface gravity:
g= 830 cm sec'"
3
solar constant:
F= 3.8 x 103 erg em-a sec-'
4
bond albedo:
A= 0.35
3
effective temperature, measured:
Te,= 58 t 3vK
5
effective temperature, predicted:
Te= 58.3± 2.2"K
5
mean density:
S= 1.31 g cm -'
3
References:
1. Levine, Kraemer, & Kuhn (1977)
2.
Brown & Goody (1977)
3.
Newburn & Gulkis (1973)
4.
Stone (1975)
5. Fazio, Traub, & Wright (1976)
- 25 -
TABLE
Uranus'
atmosheric parameters:
Parameter:
Value:
mean molecular weight:
u= 3
ratio of specific heats:
9= 1.6
gas constant:
R= 3.0 x 101 ergs OK-'g
specific heat:
C,= 8 x 107 ergs 0K- g'-
adiabatic lapse rate:
r= 1.0 CK km''
radiative time constant:
t" = 2 x 10'0 sec
(Values are approximate.)
All values taken from Stone (1975).
-26-
300
temperature
(OK)
Itoo
700
?oo
(. o
log pressure
Figure 1.
(dynes cm-x )
0-0
_11~1
-27-
700
altitude
(km)
,ou)
500
k0 0
/00o
q 0
log P (pressure)
(dynes cm-" )
Figure 2.
6. o
_IIIY____II1 __~~*)lL_1
-~11~
1111111^ .
_ly__ I ~_1_.~II~LIX
~-IIII~--LI~l
~i-L--~
50o
700
altitude
(kin)
oo00
330
.00
I0OQ.
0
1oo
200
30o
temperature (CK)
Figure 3.
.Oc
5so0
-29-
700
altitude
(km)
500
Soo
'-fcc
3O
"
f - i
4.20
690
log H
(
(scale height)
(cm)
Figure 4.
0o
.,oo
-30-
0oo0
altitude
(km)
00oO
500
30o
:0 0
3o
9.O
so
.0
7.0
, c
0
/o.,
log K (eddy diffusion coeff.)
(cm' sec-' )
Figure 5.
S0oo
7o0
altitude
(km)
0Go0
Sod
rb~c;
log ng
(cm -3 )
r
tSIO.Q
(hydrogen number density)
Figure 6.
~
/1
-L---~.IIIPI**LUCLIIII11X-I-_C-~l~,_lli
-32-
temperature
(OK)
150
/00
% para-hydrogen (p-H.)
Figure 7.
~~
-33-
7o0
-
altitude
(km)
-
-
-
-
--
TPPUC
q oo
300
loo
O
(,
% para-hydrogen (p-Ha)
Figure 8.
10t
___ .~-LLYYI~LI11II~
Il^..
I
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