Hot Oxygen in the Exosphere of Venus
Bernie D. Shizgal
Department of Chemistry, Unversity of British Columbia
2036 Main Mail, Vancouver, B.C. V6T 1Z1 Canada
Abstract. It has now been firmly established from both theoretical calculations and observations, that the
exospheres of the terrestrial planets have extended coronae of translational energetic oxygen atoms. The
current model generally accepted as the source of these hot atoms is the dissociative recombination of Oj,
that is, Oj + e~ —> O* + O* where the product O* atoms are translationally energetic. The determination
of the extent of this population of superthermal atoms is an important endeavor. The present paper considers
a simple model based on the Boltzmann equation for the energy and altitude dependence of the oxygen atom
distribution function. The density and temperature distributions are determined with this distribution.
INTRODUCTION
There are numerous processes in the terrestrial atmosphere and in the atmospheres of other planets, notably
Mars and Venus, which involve the production of translational energetic atoms with energies considerably
above thermal energies [1-3]. These "hot" atoms can play an important role in enhanced reaction rates [4-6]
nonthermal emissions [7], and in particular the enhanced nonthermal escape of atmospheric species [1,2,8].
Photodissociation of O2 and Oa produces a population of energetic O(1D) [4,5] which can react with other
atmospheric species producing an abundance of other free radicals. Energetic O(1S) atoms are produced by
dissociative recombination of Oj. Metastable O(1S) are the source of the 5577 A2 green line emission in the
F-region of the thermosphere [7,9]. Similarly, ion-molecule reactions can produce energetic nitrogen atoms and
these can play an important role in atmospheric nitrogen chemistry [6,10,11]. Exothermic chemical reactions
in the Martian atmosphere can produce nonthermal distributions of oxygen and nitrogen atoms and lead to
enhanced escape of these species [12,13].
On the terrestrial planets, there are several processes that lead to the production of substantial populations
of energetic atoms that can have energies above the escape speed of the planet. These nonthermal escape
mechanisms play an important role in the evolution of the atmospheres of these planets [1,2]. The charge
exchange process of energetic H+ or D+ in collision with neutral H can produce a significant population of H
and D atoms with energies well above thermal. The dissociative recombination of Oj can produce energetic
oxygen atoms on all three terrestrial planets, and substantial coronae of hot oxygen is expected on Earth, Mars
and Venus [14-19]. These energetic oxygen atoms can transfer their energy to H and D and create additional
energetic populations of H and D [20,21].
The existence of extended corona of energetic H and O in the atmospheres of the terrestrial planets is now
well established both from theoretical models and observations. There is a continued interest in a better
understanding of the physics of the processes that produce and maintain these steady state nonequilibrium
distributions. In the rarefied atmosphere of the high altitude portions of these planetary atmospheres, collisional
relaxation of nonthermal distributions is slow. The extent of the departure from equilibrium distributions
depends on the strengths of the processes that perturb the distributions from equilibrium and the collisional
relaxation processes that restore the distributions to Maxwellians. If there is a significant population of energetic
atoms with speeds in excess of the escape speed of the planet, these extended coronae can have an important
effect on the rate of loss of atmospheric species, both directly and indirectly. The indirect loss usually involves
the interaction of the neutral atoms with the plasma of the solar wind. A detailed review of these nonthermal
processes has been presented recently [2].
In this paper, we consider the dissocative recombination of Oj producing energetic oxygen atoms, O*, that
CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. Gallis
© 2001 American Institute of Physics 0-7354-0025-3/01/$18.00
119
600
500
400
300
200
100
FIGURE 1. The day and night density profiles of atomic oxygen and of Oj on Venus
is, O2" + e~ —> O* + O* where there are several different channels possible with the product oxygen atoms in
different electronic states [13,22]. This is an important nonthermal process that has been considered in the
modelling of the hot oxgen coronae in the atmospheres of the terrestrial planets [14,15,23,24]. The theoretical
models employed to date to study the production of energetic atoms in planetary exospheres are exclusively
Monte-Carlo simulations [14,15,25,27]. The main objective of the present paper is to construct a simple model
based on the Boltzmann equation. This is an important endeavor to complement the previous Monte-Carlo
simulations.
DENSITY PROFILES AND HOT OXYGEN PRODUCTION
The recombination of O J proceeds along several different channels, that is,
3
3
0( P)-t- 0( P) + 6.99eV
1
3
0( P)-)-O( D) + 5.02eV
1
X
0( D ) -f O ( D ) 4 3.05eV
O(1S) + O(3P) + 2.80eV
X
1
0( D)-f O ( S) + 0. 83eV
f = 0.25
f = 0.39
f = 0 .27
f = 0.00
f = 0.09
The branching ratios, /, listed above have been recently determined [28]. The exothermicity of the first two
reactions is just above the escape energy of atomic oxygen from Venus, so that these reactions can be responsible
for the loss of atomic oxygen provided the thermalization is sufficiently inefficient. There is considerable
evidence for the existence of a hot population of atomic oxygen on Venus as well as on Mars and Earth.
The hot population depends on the strength of the production rate and the rate of collisions of the hot
atoms with the ambient atmosphere, which is largely thermal (cold) atomic oxygen and some carbon dioxide.
With an increase in altitude, collisions become infrequent and the energy distribution function of hot atoms
departs from the local Maxwellian. The concentration of O J (assumed equal to the electron concentration)
determines the rate of production of energetic atomic oxygen. In the present paper, we asssume that the
thermalization is by thermal atomic oxygen and is controlled by the O density. The cold oxygen densities for
day and night conditions are taken from Brinton et al. [29] and are given by the approximate analytic fits,
log ATc(day)(z) = 13.118 - 0.02670z and log JVc(night) (z) = 18.9235 - 0.06858z, with Nc(z) in units of cm"3 and
z in km. These atomic oxygen densities and of Oj, taken from Nagy et al [30], are shown in Figure 1.
The source distribution of hot oxygen atoms produced from dissociative recombination is required. Gurwell
and Yung [20] employed a Monte Carlo technique to calculate the product velocity distributions of the oxygen
atoms following dissociative recombination but did not include any subsequent thermalization. Fox and Hac
120
0.3
0.2
0.1
0.0
10
4
6
v (km/s)
FIGURE 2. The source distribution of hot oxygen atoms resulting from dissociative recombination of Oj. Consult
the text for further details.
[13] carried out a Monte Carlo simulation for the O atom distributions resulting from dissociative recombination
of O J including the dependence on the OJ vibrational state. In this work as well, there was no account taken
of subsequent thermalization of the product O atoms. Hodges [25] used a Monte-Carlo simulation to study the
nature of the product oxygen atom speed distribution functions in the exosphere of Venus with and without
collisions with the background thermal oxygen. In an earlier paper [26], Hodges found a significant reduction
in the population of energetic oxygen atoms at high energies and concluded that energetic oxygen may not
play an important role in the activation of H and D [21].
In this paper, the product velocity distribution function is based on the previous paper by Shizgal [21] and a
similar development long ago by Whipple [31]. For a hard sphere reactive cross section the analytic distribution,
jS)(erf (x + x0) +
x x
- °)2 + (x - x0)e-(x+x°>2
erf (x - XQ)) + (x
(1)
was reported previously [21,31]. In Eq. (1), v and m are the atomic O speed and mass, respectively,
x = y m v B , XQ =
^ T = 3000ET (the ion and electron temperature), ER is the reaction
exothermicity listed on the previous page, and erf(x) is the error function. The factor A is determined by
normalizing the distribution to unity. We take the same branching ratios and EH values that correspond to
Fig. 3 in the paper by Gurwell and Yung [20]. In Fig. 1, the dashed lines and symbols are the results from
Fig. 3 of Gurwell and Yung [20] obtained with a Monte-Carlo simulation. The solid curves are the results with
Eq. (1). The curves labelled A-D are for four different channels with finite branching ratio, /, as previously
discussed. The curve labelled sum is the sum of A-D weighted with the branching ratio. As can be seen
from the figure, the agreement of the analytic result with the Monte-Carlo simulations is remarkable, although
there is a slight shift in position of the peaks. The distributions in Fig. 1 are the instantaneous distributions
following a dissociative recombination collision. What is required is the altitude variation of the steady distribution including the effects of elastic collisions. The density and temperature profiles of hot atoms can then
be determined.
BOLTZMANN EQUATION FOR HOT OXYGEN
The distribution function for hot oxygen arising from the dissociative recombination of O J is given by the
Boltzmann equation,
- + v • Vr/ + F • Vvf = J[f] + KrN0+S(v)
121
(2)
tofl
O
-2
8
12
16
20
X = V/VQ
FIGURE 3. The atomic oxygen distribution (F = /, day conditions) with a large high energy tail. The distributions
from top to bottom correspond to altitudes at 192km, 208km, 225km, 260km and 293km.
where F is the force per unit mass due to the gravitational field, J[f] represents the action of the collision
operator on the distribution function, and S(v) (see Fig. 2) is the source distribution of hot oxygen from
dissociative recombination of O^. Kr = 10~7cm~3s~1 [22] is the dissociative recombination rate coefficient,
and from quasi-neutrality Ne- = NQ+. The rigorous solution of the Boltzmann equation, Eq. (2), is a
formidable 7-dimensional problem in time, position r, and velocity v. We greatly simplify this Boltzmann
equation to construct a simple model that retains the essential physics of the problem. We assume a steady
distribution so that the time derivative in Eq. (2) can be deleted. We seek the isotropic energy distribution
function and the manner in which it varies with altitude z. In this way, we obtain the simpler equation,
U
+
(3)
where g is the acceleration of gravity. If there is no source of hot atoms and the distribution function is a
Maxwellian at altitude z, that is f ( z , v ) « JV(z)exp(—mv 2 /2fc J BT), then the collison term vanishes in Eq. (3)
and it is easy to show that the result gives the variation of N(z) with altitude, that is,
dz
(4)
kT
The density varies according to the barometric distribution,
N(z) = N(zmin)exp[—
H
(5)
where H = ksT/mg is the scale height, and the distribution function remains Maxwellian with increasing
altitude.
Equation (3) is solved for the distribution function, f ( z , v ) , by discretizing in speed with a finite difference
method. The cross section in the collision operator is taken to be a hard sphere cross section, for which the
collision operator, J[/], can be expressed as an integral operator with an analytic kernel which is well known
[32]. The discretized form of Eq. (3) is,
N
(6)
dz
where / = v2f. The actual speed variable used is the reduced speed x = V/VQ where VQ =
altitude is also scaled with the scale height so that z = z/H is actually used in the calculations.
122
. The
25000
20000
15000
10000
5000
800
400
1200
z (km)
FIGURE 4. Temperature profiles for hot atomic oxygen in the exosphere of Venus for day (a) and night (b) conditions.
The solid lines are the present solution of the Boltzmann equation, Eq. (2) whereas the dashed lines are the BGK solution
of the local equation, Eq. (7). The symbols are the temperatures determined from the distributions reported by Hodges,
J. Geophys. Res. 98, 10833-10838 (1993).
THE BGK MODEL
Shizgal [21] introduced a very simple model to determine the temperature profile of hot oxygen in the
exosphere of Venus. This model is reporoduced here for completeness and to permit a comparison with the
present approach. Instead of Eq. (1) with diffusion and gravity, a local Boltzmann equation is considered of
the form,
- S(v)
(7)
Ul
and a solution is considerd with the approximation of the collision operator introduced by Bhatnager et al.
[33] and referred to as the BGK approximation. The collision operator is written as
J\f\ = Nh(z)J[F] = -Nh(z)
T Z
(8)
where / = Nh(z)F and the second equality is the BGK approximation of the collision operator. The quantity
r(z) is a relaxation time and is chosen as the reciprocal of the elastic collision frequency, that is,
TTfJ,
r(z =
8kBTeff(z)
(9)
where Te/f = (Tc -f T/ l )/2 with Tc and Th the temperatures of cold and hot oxygen, resepctively. The BGK
approximation effectively replaces the spectrum of relaxation times of the collision operator with one representative value. We use a Chapman-Enskog type solution of the Boltzmann equation [34,35] and assume that the time
dependence of the distribution function is implicit through the density; that is, d f / d t = (df/dNh)(dNh/dt),
where N^(z) is the density of "hot" oxygen, dN^/dt = KrNf, and Ni(z) is the Oj density. Here, we have
divided the oxygen atom population into two components, one cold and the other hot. This assumes that the
oxygen distribution function has two very distinct features that permit the identification of a hot component
distinct from a cold component. This will be illustrated later.
If these assumptions are introduced into Eq. (7), a steady state distribution function is obtained in the form
(10)
123
1OOO
500
Night
Day
8OO
4OO
6OO
300
400
-6
_4
200
-2
O
logN
2
logN
FIGURE 5. Density profiles for hot atomic oxygen in the exosphere of Venus. The solid line is obtained with the
present solution of the Boltzmann equation, Eq. (2), whereas the dashed line is the barometric density for the background
thermal oxygen.
where N*(z) = r(z)KrN2(z) + Nh(z). The temperature of the energetic oxygen atoms is defined by T(z) =
3^— J F(z, v)mv2dv and is given by
T(z) =
r(z)KrN?(z),
NJz)
(11)
RESULTS AND DISCUSSION
The Boltzmann equation for a hard sphere cross section (?rd2) for O-O* collisions of 14 A2 is solved with
a finite difference method as previously discussed. Typically, 50-100 velocity points on a uniform grid were
used. Equation (6) is integrated in altitude with an explicit Euler scheme. This is not the most efficient
approach and the altitude step size was of the order of 10~5 in z units. The integration in altitude is begun
at an altitude below the peak in the Oj density shown in Fig. 1. The distribution at this altitude is chosen
to be a Maxwellian normalized to the oxygen density also shown in Fig. 1. Initially the distribution remains
Maxwellian with increasing altitude with a density that varies barometrically owing to the gravity term in Eqs.
(3) and (6). The density initially, that is for small altitude changes, maintains the form given by Eq. (5) shown
in Fig. 1. With increasing altitude, the source function increases dramatically and the distribution departs
from Maxwellian, particularly in the high energy tail.
The energy distribution function of atomic oxygen versus altitude is shown in Fig. 3 for day conditions.
These distributions are qualitatively similar to the distribution functions shown in Fig. (7) reported by Hodges
[26], although the C>2 density profile employed by Hodges is not realistic.
For the BGK approach, we have with the chosen cross section that r(z)Kr = 628 s~l/Nc(z) with Nc in
cm~3. In this model, we also we require the three density profiles, Nc(z), Nh(z), and Ni(z). Notice that this
formalism is local, and the density Nh(z) is a parameter. The temperature Th is determined from the average
of mv2/2 with S(v) and is 21,450 K. The temperature Tc is taken from the scale height corresponding to Nc
124
and is 284 K for day and 132 K for night. The hot oxygen densities are taken from Nagy et al. [23] and are given
by the approximate analytic fits, log A^day) (>) = 5.762 - 0.001273;? and log A^night) ( z ) = 3.283 - 0.001599z
with Nh(z) in units of cm~3 and z in km.
The temperature profiles determined with Eq. (11) are shown in Fig. 4 (dashed curves) in comparison with
the result obtained from the solution of the Boltzmann equation, Eqs. (3) and (6) (solid curves). Since the
BGK local approach is very approximate it is surprising that it is still qualtitatively similar to the temperature
profiles obtained from the solution of the Boltzmann equation versus altitude. The curve with the symbols is
the temperature profile determined from the distributions in Fig. 6 in the paper by Hodges [26]. The agreement
with the present result is remarkable. Another surprising result is that these profiles are not very sensitive
to the choice of hard sphere elastic cross section. This is also verified with the BGK local approach. The
BGK method used here is a local approach and cannot be used to determine the density profile. With the
BGK approach, two distinct populations of atomic oxygen are assumed to exist with separate density profiles
that parametrize the solution.
The oxygen density profiles determined from the solution of the Boltzmann
equations, Eqs. (3) and (6), are shown in Fig. 5 for day and night condition. In this calculation, there
is no distinction made between hot and cold components. The integration at low altitudes is begun with a
Maxwellian normalized to the (cold) oxygen density at that altitude. As mentioned previously and shown in
Fig. 5, the density distribution remains barometric, Eq. (5), with a scale height determined by the (cold)
oxygen temperature. At some altitude, where the dissociation recombination reaction becomes significant, the
density distribution becomes rapidly barometric but at a much higher temperature. This is consistent with
the temperature profiles shown in Fig. 4. Unfortunately, the actual densities at these high altitudes are much
lower than those reported by other workers [14,23,24]. The reason for this discrepancy is not well understood.
Nevertheless, the simple models introduce here clearly show that there is a hot oxygen corona on Venus. The
ratio of hot to cold oxygen density is 2.67x 108 at 500km for day conditions is 2.67x 108. This ratio is 1.47x
107 at 1000km for night conditions.
SUMMARY
The present paper has considered a solution of the Boltzmann equation for the altitude and speed dependence
of the atomic oxygen distribution function in the exosphere of Venus including the production of hot atoms
via dissociative recombination. The distribution function is assumed to be isotropic and the background
atmosphere is thermal oxygen with a specified barometric density distribution at a fixed temperature. The
Boltzmann equation is integrated in altitude with a finite difference discretization of the speed variable. The
solution with the Boltzmann equation has provided temperature profiles that show a rapid change from a cold
temperature to a hot temperature. These temperature profiles are in good agreement with the results with
an approximate BGK approach and in remarkable agreement with Monte Carlo simulations by Hodges [26].
However, the densities of hot oxygen were lower than the densities reported by other workers [14,23,24]. Work
is in progress to contruct more realistic models based on the Boltzmann equation.
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