DIRECT MONTE CARLO SIMULATIONS OF THERMALLY DRIVEN ATMOSPHERIC
ESCAPE: APPLICATIONS TO TITAN AND PLUTO
Orenthal J. Tucker, R.E. Johnson
Engineering Physics, University of Virginia, Charlottesville, Va., 22904
Recent models of the atmospheres of both Pluto and Titan estimate large thermal escape rates of the principal
atmospheric species in comparison to their respective Jeans theoretical rates. However, these continuum models
were applied region of the atmosphere that transitions from being collisional to collisionless. In particular, the ‘slow
hydrodynamic escape’ model requires assumptions about the thermal conduction and temperature in the exosphere
which favors large escape rates. The difficulties with the slow hydrodynamic approach are related to the model
assumptions concerning the atmospheric structure at infinity. Here a kinetic model is used to account for nonequilibrium collisions in the exosphere, and it is found that thermal escape of N2, CH4 and H2 from Titan and N2
from Pluto should occur similar to Jeans rate. In addition a hybrid fluid/DSMC approach has been developed to
obtain consistent results for the escape rate and macroscopic properties of Pluto’s atmosphere between a continuum
and kinetic approach. A summary on the DSMC model is provided with current results for escape from Titan and
Pluto.
1. Introduction
The evolution of a planetary body is
directly coupled to the atmosphere. Whether
or not a planet can support a significant
atmosphere over time depends on its gravity
and the atmospheric temperature. For example
if the thermal energy of the atmosphere is
greater than the planetary binding energy the
atmosphere can escape to space and, over
time, will deplete a planet’s inventory of
volatile elements. Therefore, many studies are
aimed at understanding the mechanisms that
control atmosphere loss in order to infer about
the original inventory and evolution of
planetary bodies. Here we consider loss to
space by thermal escape of the atmospheres of
Titan and Pluto.
Both the Cassini and the New
Horizons mission in the outer solar system are
allowing scientists to critique previous ideas
concerning atmospheric evolution and escape.
While orbiting the Saturn system the Cassini
satellite has been collecting data from Titan’s
atmosphere for over 7 years, and the New
Horizons spacecraft will collect data on
Pluto’s atmosphere in 2015 as it travels to the
Kupier Belt. Titan, Saturn’s largest moon, has
an N2 rich atmosphere with a few percent
CH4, >1/10 of a percent of H2 and a sparse
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population of organic molecules. Pluto also
has a N2 rich atmosphere with a few percent
CH4 but there is also a sparse population of
CO and its surface pressure is nearly (20-200)
millionth of the surface pressure at Titan
(Olkin et. al. 2003).
Terrestrial atmospheres are dominantly
heated by adsorbing solar radiation and
transporting that heat by convection and
conduction. Thermal escape from an
atmosphere occurs when conduction
distributes the kinetic energy, associated with
temperature, to a molecule with a favorable
velocity such that it gains energy greater than
the planetary binding energy. Typically
thermal escape is considered between two
extremes one being an evaporative type of
loss, so-called Jeans escape, and the other
being a bulk type of loss referred to as
hydrodynamic escape. Several decades worth
of continuum models for the upper
atmosphere of Pluto have concluded that the
atmosphere is escaping hydrodynamically due
to its low gravity (e.g.: Hunten and Watson
1982; McNutt 1989; Krasnolpolsky 1999;
Strobel 2008a). However recently, Titan’s
atmosphere has also been suggested to be
undergoing hydrodynamic escape (Strobel
2008b) even though it has 5 times the
1
gravitational binding energy of Pluto at the
surface.
The hydrodynamic escape from Titan
and Pluto is suggested to be slow. It is
assumed that conduction will efficiently
convert thermal energy of molecules into bulk
energy for escape. This results in escape rates
that would be orders of magnitude larger than
the corresponding Jeans rates. However, this
research has shown that applying continuum
models to the upper atmosphere where
collisions are non equilibrium events leads to
incorrect escape estimates and macroscopic
properties for the atmosphere (e.g.
temperature and density) (Tucker and Johnson
2009; Tucker et al. 2011; Volkov et al.
2011a,b). A Monte Carlo approach, as
opposed to a fluid (continuum) approach, is
necessary to model thermal escape from a
rarefied atmosphere. In the following sections
the application of a Direct Simulation Monte
Carlo model applied to thermal escape is
explained, with summaries presented on the
current results for the atmosphere of Titan and
Pluto.
2. Exobase, Jeans Escape & Hydrodynamic
Escape
Escape occurs most efficiently in the
exosphere, a tenuous region in the upper
atmosphere, where molecules can travel
planetary scale distances and not collide with
other molecules (Johnson et. al. 2008). The
exobase is referred to as the lower boundary to
this region, and it is defined as the altitude
where the mean free path for collisions, l =
(21/2nσ)-1, is comparable to the atmospheric
scale height H = kT/mg (n – density, σ –
collision cross-section, k – Boltzmann
constant, T – temperature, m – molecule mass,
g – gravity). In rarefied gas dynamics the ratio
l/H is referred to as the Knudsen number Kn,
and it provides the criterion for when a gas
flows similarly to a fluid and can be modeled
using solutions to the hydrodynamic equations
(Kn << 1), or when it becomes increasingly
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collisionless and must be modeled
stochastically (Kn > 0.1). At the exobase Kn ~
1, and it is the altitude in the atmosphere
where the column density is N = σ-1.
Thermally driven atmospheric escape is often
characterized by the Jeans parameter; it is the
ratio of the gravitational potential energy to
the thermal energy at the exobase λexo =
GMpm/(rexokTexo) where Mp is the mass of the
planet. Using inferred temperatures obtained
from Cassini density measurements for Titan
and occultation data for Pluto the Jeans
parameter are λexo ~ 50, and ~10 respectively.
For large Jeans parameter values, the
gravitational energy of the planet dominates
the thermal energy of the atmosphere.
Therefore, only a small fraction of particles
which have speeds near the tail of distribution
can escape. In this case the escape rate
obtained from kinetic theory is:
φ =π rexo2 nexo<vexo>exp[-λexo.](1+λexo)
(Jeans)
where <vexo> is the mean thermal speed (Jeans
1916). In the above we assume that molecules
have speeds according to the Maxwell
Boltzmann speed distribution.
If the gravitational energy of the planet
is comparable to the thermal energy of the
atmosphere at the exobase, λexo = 2, then the
atmosphere can escape the planet in bulk, like
a fluid, and requires consideration with a
hydrodynamic approach.
Slow hydrodynamic escape is
suggested as an intermediate case between
hydrodynamic and Jeans escape (Chamberlain
1961; Parker 1963a, b). It is defined as an
organized flow produced by thermal
conduction which produces an expansion in
the upper atmosphere in which the bulk flow
speed can exceed the escape speed (Johnson
et. al. 2008). In this formulation conduction is
assumed to occur efficiently beyond the
exobase and all of the heat is used to lift the
2
atmosphere out of the planetary gravitational
well.
The thermal escape problem at first
glance appears intuitive, nevertheless it
remains a controversial topic (Jeans 1916;
Chamberlain 1960, 1961; Watson 1981;
Parker 1963a, b; Johnson 2010; Grusinov
2011; Volkov et al. 2011a,b). The problem is
formulated by considering a spherically
symmetric atmosphere in which the dominate
heating of the atmosphere by solar radiation
occurs below some lower boundary ro. At ro
the temperature To and density no are
maintained at constant values. The lower
boundary is chosen at an altitude considered to
be in approximate radiative equilibrium, and
escape as the result of non-thermal external
processes is neglected. The goal is to describe
how heat transferred from ro by thermal
conduction and convection drive escape. To
this end, two methods have been developed to
obtain the escape rate and macroscopic
properties (i.e. n(r) and T(r)) of the
atmosphere. In the first approach solutions are
obtained using the hydrodynamic equations by
placing restrictions on density and temperature
at infinity (Chamberlain 1961; Parker 1963a,
b). The second method consists of explicitly
modeling the atmospheric flow with a
representative sample of modeling particles
which in effect solves the Boltzmann
equation. These methods are discussed further
below.
2. Slow hydrodynamic model
When the hydrodynamic equations are
applied to a 1D radial atmosphere, the
continuity equation leads to a constant
atmospheric flow vs. radial distance from a
lower boundary: 4π r2 n(r)u(r) = , with n(r)
the number density and u(r) the flow speed vs.
r the radial distance from the center of the
planet. The radial momentum (pressure)
equation for molecules of mass m in which the
viscous term is typically dropped, can be
written
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dp/dr = n [dΦ/dr] – n d(mu2/2)/dr
(1a)
where p is the pressure (nkT) and Φ the
gravitational potential energy (GMpm/r). The
corresponding energy equation can be written:
d{ (mu2/2 + CpT - Φ)- 4π r2κ(dT/dr)}/dr = 4π r2Q(r)
(1b)
where Cp is the heat capacity per molecule and
κ = κ(T) the thermal conductivity (e.g.,
Johnson et al. 2008).
The standard procedure in the slow
hydrodynamic model is to use  ≠ 0, but set u
= 0 in Eq. (1a) and (1b) below an upper
boundary that is above the exobase. In the
approach described in Strobel (2008a, b) Eq.
1a and 1b are solved iteratively using assumed
values of  and dT/dr|ro with the conditions n
and T 0 as r  . Throughout the
remainder of the paper we will refer to that
slow hydrodynamic escape model as the SHE
model.
The main difficulties with the applying
the SHE model are the constraints placed on
density and temperature at infinity without
prior knowledge of what they should be.
Furthermore the She model is unable properly
account for the non-equilibrium effects in the
exosphere. As shown in Eq. 1b temperature is,
therefore, assumed to be defined even in the
exosphere and the expression for thermal
conductivity is essentially, independent of
density. This procedure gives solutions of n(r)
and T(r) for a range of escape rates. Strobel
(2008a) obtains a ‘best’ solution by using
density data as a constraint, and other models
only require a solution that conserves energy
with asymptotically requiring n and T 0 as r
  (Parker 1963b). However, Tucker and
Johnson (2009) have used Monte Carlo
techniques to test the SHE model result and
obtained very different escape fluxes while
3
producing solutions for density profiles
consistent with data from Titan’s atmosphere.
3. Direct Simulation Monte Carlo
Applying a continuum model it beyond
the exobase where Kn > ~1 must be done with
care. To describe the transition region in the
atmosphere from below the exobase to beyond
Kn >~ 0.1, solutions to the Boltzmann
equation or Monte Carlo simulations are
required. We use the Direct Simulation Monte
Carlo model, DSMC, (Bird, 1994). The
atmosphere is described using a set of
representative molecules and its evolution is
calculated by following the motion of these
particles subject to gravity and collisions.
Thermal conduction is explicitly included and
depends on the cross sectional area for
collisions as well as the local density. A 1D
simulation is carried out, consistent with the
1D continuum models being tested. In the
main flow direction, radially outward from
Titan and Pluto, the space is divided into cells
with heights less than the local mean free path
for collisions. To accurately describe an
atmosphere using such simulations three
conditions must be satisfied (Shematovich
2004): there should be a sufficient number of
representative particles to describe the nature
of the flow; the molecular motion between
collisions should be independent of the nature
of the collisions; and on average molecules
should experience less than one collision
during a time step. The upper boundary is
placed many scale heights above the exobase
where it is safe to assume further collisions
will not significantly affect the escape rates.
At the upper boundary upward moving
molecules with speeds greater than the escape
speed are assumed to escape while the others
were specularly reflected back to the
simulation region, allowing a shorter time to
achieve steady state. The reflected, nonescaping molecules represent ballistic
particles that will eventually return to the
simulation region. If chosen high enough the
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upper boundary should not affect the results.
Typically, on the order of a few hundred
thousand particles were used to represent the
atmosphere in order to have at least a few
hundred particles in the very top cells. The
weights for the representative molecules can
be adjusted in order to meet that condition.
The DSMC model does not make any
assumptions about density and temperature at
infinity and the results depend on how
realistically molecular collisions can be
represented.
4. Results
Before presenting the results from the
DSMC simulations a brief overview is given
of the current SHE model. For a ‘baseline
case’ in which all the heat input into the
atmosphere is deposited below the lower
boundary [Eq. 1b Q(r) = 0 above ro in (Strobel
2008a, b)], solutions with a significant escape
flux (φH =1x1027 s-1 and 5x1026 s-1) were
obtained for both the atmospheres of Titan and
Pluto respectively. Even though Titan has a
Jeans parameter 5 times larger than that for
Pluto, the SHE model concluded that Titan’s
atmosphere should escape at a higher rate.
Furthermore the exobase temperatures and
densities derived from the SHE model suggest
the Jeans rates would be several orders of
magnitude smaller (φJ ~109 s-1 and ~1019 s-1)
for Titan and Pluto respectively.
DSMC is ideally suited to test these
results from SHE model since no assumptions
are made about the temperature or density at
infinity, and thermal conduction is inherently
a function of the local density. Therefore a
series of simulations were done in Tucker and
Johnson (2009) to test the SHE model results
for Q(r) = 0 for Titan. In this result the
temperature and density at the lower boundary
(3875 km) used in the DSMC simulation was
normalized to the solution from the SHE
model in (Strobel 2008b). A comparison n(r)
and T(r) of the DSMC and SHE model results
are shown in Fig. 1. The density profiles
4
-3
Density (cm )
4
10
5
6
10
10
7
10
8
9
10
10
10
10
140
145
10
11
4600
4500
Radial Distance (km)
between the methods are similar, but the
temperature for the DSMC result is nearly
isothermal throughout the simulation region.
The simulations were performed for a suitable
amount of time such that if 1 test particle
escaped during the simulation, the escape rate
would have been 6 orders of magnitude less
than the SHE model result. Since no test
particles escaped during our DSMC
simulation, our upper bound to the escape rate
is ~1021 s-1. The gravitational binding energy
at the DSMC exobase (4000km) is ~ 0.6eV,
and the mean energy of the N2 molecules at
141K is ~0.01eV. Only a very small fraction
of particles at the tail of the distribution will
have sufficient energy to escape, therefore, the
rate should be similar to the Jeans rate. In our
simulations, achieving escape for such a large
Jeans parameter is limited by the capabilities
of the random number generator.
However, by artificial increasing the
temperature to 600K at the lower boundary for
Titan we were able to force escape to occur
(Tucker and Johnson 2009). We found that
even at λexo ~ 11 escape still occurred similar
to the Jeans theoretical values (1.5*Jeans) as
opposed to the hydrodynamic values obtained
in Strobel (2008a) which are orders of
magnitude larger than the Jeans rates.
Therefore, we considered escape from Pluto
using the DSMC approach where λexo ~ 10.
4400
4300
4200
4100
4000
3900
100
105
110
115
120
125
130
135
Temperature (K)
Figure (1): N2 n (solid curves (triangles) - top axis) and
T (dotted curves (inverted triangles) –bottom axis) vs.
radial distance in Titan’s atmosphere: DSMC (open
triangles) (Tucker and Johnson (2009)) and SHE model
(filled triangles) using lower boundary conditions at
3875 km for Q = 0 from in Strobel (2008b) (rexo = 4000
km represented by horizontal line).
5. Combined fluid/DSMC Approach
The DSMC simulations of thermal
escape from Pluto’s atmosphere, where λexo ~
10, obtained a rate ~1.6*Jeans similar to the
artificial Titan result discussed in section (4)
(Tucker et al. 2011). Many studies of gas
flows have shown for the limit Kn  0 the
continuum and kinetic approach should give
the same result (Bird 1994). Therefore we
developed a fluid/DSMC approach to obtain
consistent solutions for n(r) and T(r) with the
hydrodynamic equations (fluid) and kinetic
approach (DSMC) for Pluto’s atmosphere.
This is achieved by solving the hydrodynamic
equations (Eq. 1a and 1b) in the region of
atmosphere where Kn < 0.1and applying the
DSMC simulation to regions where Kn > 0.1.
Upon integration of Eq. 1a and 1b with
u = 0 we obtain the following expressions for
pressure and heat flow.
p = po exp[- ∫ro(GMpm/r2)/kT dr]
(2a)
with p = nkT and
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5
[ (CpT - Φ) - 4r2 κ(dT/dr)] = <E>ro + 4ro2 (r)
(2b)
Here <E>ro is the energy flow across ro, and,
as in Strobel (2008a), (r) = ro-2 [ ∫ro r2 Q(r) dr]
with  o as r  .
The conditions we use to solve Eq. 2a
and 2b are a fixed no and To at ro and the heat
flow <E>ro and escape rate φ were
determined iteratively as discussed below and
shown schematically in Fig. 2. We do not
assume, as in SHE, that T0 as r. In the
most dense region of the atmosphere (Kn <<
0.1), we numerically solve Eq. 2a and 2b to
obtain n(r) and T(r) from ro to rod an altitude
below the exobase. The solutions provide (nod
and Tod) at rod and are then used as the lower
boundary conditions to the DSMC model. The
DSMC model discussed in section (3) is
applied to a region from rod to rtop an altitude
many scale heights above the exobase. At rtop
we calculate both the particle escape rate φ,
and the average energy carried off by escaping
molecules, <E>∞=<E>ro. The new <E>ro
and φ are then used to resolve Eq. 2a and 2b,
which in turn provide new conditions at rod for
the DSMC model. This procedure is iterated
until the pair (, <E>ro) is obtained that
produce a convergent n(r) and T(r) in the
region where the fluid equation solution and
the DSMC results overlap 0.1 < Kn < 1. Again
this approach differs from the SHE because no
assumptions are made about the tendency of
density and temperature at infinity.
Furthermore in the SHE model it is customary
to set <E>ro = 0, we directly calculate this
quantity and  from the amount and energies
of escaping molecules.
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Figure (2): Schematic of the fluid/DSMC iterative
procedure.
In Fig. 3 we compare results for solar
minimum conditions at Pluto obtained with
the fluid/DSMC method to the SHE model
The implementation of solar heating will be
discussed further in Tucker et al. (2011), the
following discussion is a summary of the
result. The escape rate obtained from the
fluid/DSMC method 1.2 x1027 s-1 is
fortuitously close the SHE model result, 1.5
x1027 s-1, but the resulting macroscopic
properties for the atmosphere are very
different as shown in Fig. 3. Therefore, using
the corresponding exobase values the
fluid/DSMC model give a rate 1.9*Jeans
while the SHE model suggests the rate is
>103*Jeans. Furthermore, the fluid/DSMC
model determined the energy flow into the
lower boundary to be <E>ro = 1.5 x1013 ergs
s-1and not 0 as assumed by the SHE model.
For solar minimum conditions this energy
amounts to a small fraction of the actual heat
deposited, but it still affects the temperature
gradient from lower boundary.
6
-3
10
10000
1
10
3
10
5
Density (cm )
7
9
11
10
10
10
10
13
10
15
a-fluid/DSMC
b-SHE
9000
Radial Distance (km)
8000
a-(n)
a-(T)
7000
6000
5000
a-(u)
4000
b-(T)
3000
b-(u)
b-(n)
2000
0
10 20 30 40 50 60 70 80 90 100 110
Temperature & Flow Speed (K, m/s)
Figure (3): Result for solar minimum conditions: n(cm3
) (top axis), T(K) & u(m/s) (bottom axis) vs. radial
distance. Comparison of Fluid/DSMC n(dashed
curves), T(solid curves) & u(dotted curves) to SHE
model results from Strobel (2008a) n(circles),
T(triangles) & u(squares): exobase 6200 km
Fluid/DSMC (solid curve on right axis) and 3530 km
SHE (dashed curve).
6. Summary/Future Work
Pluto’s atmosphere has been widely
considered to be undergoing hydrodynamic
escape (Hunten and Watson 1982; McNutt
1989; Krasnolpolsky 1999; Strobel 2008a),
and more recently measurements made by the
Cassini spacecraft of density data in Titan’s
upper atmosphere (Waite et al. 2005) have
been used to postulate hydrodynamic escape
from Titan’s atmosphere (Strobel 2008b). In
particular the a recent application of SHE
model to the atmospheres of Pluto and Titan
Strobel (2008a, b) found the mass loss rate of
from Titan (λexo = 50) and Pluto (λexo = 10) to
be several orders of magnitude larger than the
corresponding Jeans rates.
In this research a kinetic approach is
used to consider the thermal escape problem,
and contrary to the slow hydrodynamic results
it is found that escape occurs on molecules by
molecules basis. Specifically we use the socalled DSMC model which in effect solves the
Boltzmann equation. Such simulations include
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thermal conduction explicitly via the
molecular collisions. Therefore they can serve
as a test of the results from the SHE model
which is unable to properly account for the
effect of collisions on temperature and escape
in the exosphere region. For Titan the DSMC
results have shown that in the absence of any
external energy input into the atmosphere by
non-thermal processes N2 escape occurs at
rates similar to Jeans. For Pluto, we used a
combined fluid/DSMC iterative procedure to
obtain consistent escape rates, density and
temperature profiles between the DSMC
model and the solutions to the equilibrium
fluid equations without the assumption that n,
T0 as r . In principle the DSMC can be
used to simulate the entire simulation region
but as Kn becomes << 0.1 the calculation
becomes computationally expensive.
Therefore, we numerically solve the
equilibrium equations in the region of the
atmosphere where Kn<<0.1, and use the
DSMC model in the tenuous region Kn>0.1
(section 5). In the case of escape from Pluto
atmosphere for solar minimum conditions the
fluid/DSMC method again results in an escape
rate similar to Jeans, and the temperatures and
density profiles are significantly different
from the SHE model result. Below we list the
general conclusions from our ongoing studies.
1) Both Pluto and Titan are not undergoing
hydrodynamic escape (Tucker and
Johnson 2009; Tucker et al. 2011).
2) The SHE model significantly
overestimates the heat flux in the
exosphere region (Kn>1) (Volkov et al.
2011a, b).
3) Escape occurs on a molecule by
molecules basis for atmospheres where
λro >~3, where λro is the Jeans parameter
defined at ro (Volkov et al. 2011a, b).
7
At the VSGC 2010 conference the
DSMC results presented only considered hard
sphere collisions and the internal energy of
molecules was neglected. In additions for the
fluid/DSMC method solar heating above ro
was not considered. We note for Titan even
with such assumptions regarding the
collisional model used in the DSMC
simulations were able to reproduce the Cassini
density for the N2 and without requiring
significant mass loss as in the SHE model.
However, the accuracy of the DSMC results
depends upon how well collisions are
described. Since the 2010 meeting we have
performed simulations including internal
energy of molecules, and using a temperature
dependent cross-section (Tucker et al. 2011).
While these changes do slightly change the
quantitative results slightly (e.g. φ, n(r), T(r),
u(r) etc.), the qualitative conclusion remains
that thermal escape for bound planetary
atmospheres λexo >~ 3 occurs on a molecule
by molecules basis.
In moving forward the main goal is to
evaluate thermal escape from multicomponent atmospheres. To this end we are
using the DSMC model with Cassini density
data to model thermal escape of N2, CH4 and
H2 from Titan’s atmosphere, see Fig. 4. It is
important to note the most recent SHE model
results that consider multiple species have
backed off the assertion that that N2 escapes
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4200
4200
4100
r
exo
= 4000 km
4000
3900
Radial Distance km
5) The energy flow <Eφ>ro is not 0 as
assumed in the SHE model (Tucker et al.
2011). This assumption by the SHE
affects solutions obtained by Eq. 2b.
Titan at rates excessively larger than Jeans
(Strobel 2009). These new results attribute the
dominant portion of the escaping mass flux to
CH4 2.5x1027s-1 (λexo = ~30) and H2 9x1027
(λexo = ~4). However, using the corresponding
exobase values for that result the above rates
would are 1010 and ~3 times the Jeans values
respectively. Still CH4 is considered to escape
at large rates even though it has a mass 8
times that of H2. The SHE model results
appear to contradictorily suggest that
hydrodynamic escape of an atmosphere is
more favorable for larger Jeans parameters.
We attribute this discrepancy to the
assumptions inherent to the SHE model
approach.
The preliminary DSMC result shown
in Fig. 4, again suggest that CH4 will not
escape thermally at rates significantly larger
than Jeans similar to (Tucker and Johnson
2009), and for H2 we obtain a rate that is
1.2*Jeans.
Radial Distance (km)
4) A consistent solution for φ, <Eφ> and
the macroscopic properties of an
atmosphere can be obtained between the
hydrodynamic equations and a kinetic
model in the region Kn << 1 to Kn ~1
using the described fluid/DSMC
approach (Tucker et al. 2011).
4100
4000
3900
3800
3700
3600
118 120 122 124 126 128 130 132 134
Temperature (K)
3800
3700
3600
6
10
7
10
8
9
10
10
10
10
-3
Density (cm )
Figure (4): Comparison of n(r) vs. radial distance:
DSMC result (N2 (solid curve,) CH4 (dash dotted
curve), H2 (dotted curve)) to Cassini data (N2
(triangles,) CH4 (squares), H2 (circles)) for the major
species in Titan’s atmosphere. The inserted graph
shows the corresponding single species temperatures
for the DSMC result.
Also as shown in Fig. 4 the current
DSMC results do not match the Cassini
8
density data for H2. Therefore, we are
performing DSMC simulations with more
realistic cross-sections, based on available
experimental data. Below the exobase H2 is a
trace species in Titan’s atmosphere, but above
the exobase it becomes the dominant species,
therefore it is important to describe the
collisions appropriately. Furthermore, these
simulations are very time consuming so we
are evaluating techniques to reduce the
computational time. After a series of
simulations are performed to evaluate how the
choice of collision model used in the DSMC
simulation affects the n(r), T(r) and φ for H2,
the effect of non-thermal processes occurring
in the upper atmosphere on the density
distribution will be considered.
Acknowledgements
This work is supported by the NSF and the
VSGC. Special thanks to Justin Erwin and
Alexey Volkov for discussions, and to Darrell
Strobel for providing solar heating data.
References
Bird, G.A., 1994. DSMC procedures in a
homogenous gas. In:
Molecular Gas Dynamics and the Direct
Simulation of Gas Flows, pp. 218–256.
Clarendon Press, Oxford, England.
Chamberlain, J. W. 1960. Interplanetary Gas.
II. Expansion of a model of the Solar
Corona. Astrophys. J 131, 47-56.
Chamberlain, J. W. 1961. Interplanetary Gas.
III. Hydrodynamic Model of the Corona.
Astrophys. J 133, 675-687.
Chamberlain, J. W., Hunten D., 1987. Theory
of Planetary Atmosphere. Academic Press,
New York.
Cui, J., Yelle, R.V., Volk, K., 2008.
Distribution and escape of molecular
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hydrogen in Titan’s thermosphere and
exosphere. J. Geophys. Res. 113.
J. Cui, R.V. Yelle, V. Vuitton, J.H. Waite Jr.,
W.T. Kasprzak, D.A. Gell, H.B. Niemann,
I.C.F. Müller-Woodarg, N. Borggren, G.C.
Fletcher, E.L. Patrick, E. Raaen, B.A. Magee,
Icarus, 200, 581-615 (2009).
Grusinov, A., 2011. The rate of thermal
atmospheric escape,’ arXiv:1101.1103v1
[astro-ph.EP] 5 Jan 2011.
Hirschfelder, J. O., Curtiss C. F., Bird, R. B.,
1964. Molecular Theory of Gases and Liquids.
New York: Wiley; 2nd corrected printing.
Hubbard, W.B., Yelle, R.V., Lunine, J.I.,
1999. Nonisothermal Pluto Atmosphere
Models. Icarus 84, 1-11.
Hunten, D. M.,Watson, A.J., 1982. Stability of
Pluto’s Atmosphere. Icarus 51, 655 – 657.
Jeans, J. H., M. A., F. R. S., 1916. The
Dynamical Theory of Gases: The Outer
Atmosphere. Cambridge University Press,
351- 363.
Johnson, R. E., 1990. Energetic ChargedParticle Interactions with Atmospheres and
Surfaces. Springer-Verlag, Berlin.
Johnson, R.E., 2009 Phil. Trans. R. Soc. A.
367,753-771.
Johnson, R.E., M.R. Combi, J.L. Fox, W-H.
Ip, F. Leblanc, M.A. McGrath, V.I.
Shematovich, D.F. Strobel, J.H. Waite Jr,
"Exospheres and Atmospheric Escape",
Chapter in Comparative Aeronomy, Ed. A.
Nagy, Space Sci Rev 139: 355-397, (2008).
Johnson, R.E., Combi, M. R., Fox, J. L., Ip,
W-H., Leblanc, F., McGrath, M. A., Kasting,
J.F., Pollack, J.B., 1983. Loss of water from
9
Venus I. Hydrodynamic Escape of Hydrogen.
Icarus 53 479-508.
Olkin, C.B., Wasserman, L.H., Franz, O.G.,
The mass ratio of Charon to Pluto from the
Hubble Space Telescope astrometry with fine
guidance sensors, Icarus,164, 254-259, (2003).
Shematovich, V. I, Strobel, D. F., Waite J.H.,
Jr., Exospheres and Atmospheric Escape.
Comparative Aeronomy, edited by. A. Nagy et
al., Space Sci. Rev, doi 10.1007/s11214-0089415-3 (2008)
hydrodynamic to Jeans escape. Astrophys. J.,
729, L24
Volkov, A. N., Tucker, O. J., Erwin, J. T. and
Johnson, R. E., 2011b. Kinetic Simulations of
Thermal escape from a single component
atmosphere. submitted.
Waite Jr., J.H., et. al., 2005. Ion neutral mass
spectrometer results from the first flyby of
Titan. Science 308, 982-986.
Krasnopolsky, V.A., 1999. Hydrodynamic
flow of N2 from Pluto. J. Geophys. Res. 104,
5955–5962.
Watson, A.J., Donahue, T.M., Walker, J.C.G.,
1981. The dynamics of a rapidly escaping
atmosphere: Applications to the evolution of
Earth and Venus. Icarus 48, 150–166.
McNutt, R.L., 1989. Models of Pluto’s upper
atmosphere. Geophys. Res. Lett. 16, 1225–
1228.
Yelle, R.V., Borggren, Cui, J., MullerWodarg, I.C.F., 2008. Methane escape from
Titan’s atmosphere. J. Geophys. Res 113.
Strobel, D.F. N2 escape rates from Pluto’s
atmosphere Icarus 193, 612-619 (2008a).
Strobel, D. F., Titan's hydrodynamically
escaping atmosphere. Icarus 193, 588—594
(2008b).
Strobel, D.F. Titan's hydrodynamically
escaping atmosphere: Escape Rates and
Exobase Structure, Icarus 202, 632—641
(2009).
Tucker, O.J. and Johnson R.E., 2009.
Thermally driven atmospheric escape: Monte
Carlo simulations for Titan’s atmosphere. PSS
57, 1889-1894.
Tucker, O.J., Erwin, J.T., Volkov, A.N.,
Johnson, R.E., Deighan, J.I., 2011. Thermally
driven escape from Pluto’s atmosphere: A
fluid/hybrid model. (to be submitted)
Trafton, L., 1980. Does Pluto Have a
Substantial Atmosphere?. Icarus 44, 53-61.
Volkov, A. N., Johnson, R. E., Tucker, O. J.
and Erwin, J. T. 2011a. Thermally-driven
atmospheric escape: Transition from
Tucker
10