HEAT AND MASS TRANSPORT IN NITROGEN ICE WITH APPLICATION TO PLUTO AND TRITON. http://trs-new.jpl.nasa.gov/dspace/bitstream/2014/22228/1/97-0696.pdf

Abstract

Classical,and’’super”solid-state,greenhouses,have,been,suggested,asmech- anismsfor the solar energy,supply,to Tritonian geyser-like plumes,(Brown et al. 1990). In this work we evaluate solar and internal (owing to radioactive decay of U, 7’hand 401{in Pluto’s inferred core) heat,sources,and,their corresponding,mechanisms,for predicted,erup- tive activity on Pluto. For the internal energy supply, a model of conductive-convective heat,and,mass,transport,on Triton (Duxburyand,Brown,1997) is applied to Pluto’s solid N2 layer. Previous,models,of the solid-state convection,for different,celestial bodies,con- sidered,only simpllayer. Convection,in a seasonal,N2 layer on Triton and Pluto is plausible even without the solid-state greenhouse effect, because N2 on these bodies,is so close to its melting,temperature,of= 63.148 K(at zero pressure) that an upper,stagnant,layer does not form. Since d~r~ ~~ < 0.3pmfor the fresh transparent,N2

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HEAT AND MASS TRANSPORT IN NITROGEN ICE
WITH APPLICATION TO PLUTO AND
TRITON
N. S.
Duxbury
Jet Propulsion Laboratory, California Institute of Technology
R,
H. Brown
University of Arizona, Tucson .
J.
D. Goguen
Jet Propulsion Laboratory, California Institute of Technology
Corresponding author:
Natalia
S.
Duxbury
183-501
Jet Propulsion Laboratory
California Institute of Technology
4800 Oak Grove Dr.
Pasadena 91109
tel 8183545516
Fax 8183540966
e-mail
nsd@scnl.jpl.nasa.
gov
suggested key words:
Pluto, Triton, nitrogen, geysers, subsurface, internal

Abstract.
Classical and’’super” solid-state greenhouses have been suggested asmech-
anisms
for the solar energy supply to Tritonian geyser-like plumes (Brown et al. 1990). In
this work we evaluate solar and internal (owing to radioactive decay of U,
7’h
and
401{
in
Pluto’s inferred core) heat sources and their corresponding mechanisms for predicted erup-
tive activity on Pluto. For the internal energy supply, a model of conductive-convective
heat and mass transport on Triton
(Duxbury
and Brown 1997) is applied to Pluto’s solid
N
2
layer. Previous models of the solid-state convection for different celestial bodies con-
sidered only simple parametrized convect ion in H
2
O ice. We have solved numerically the
FJavier-Stokes
system in the
Boussinesque
approximation. Whether solid-state convection
occurs in Pluto’s and Triton’s N
2
ice (and if it does, then its intensity) depends upon the
solid nitrogen grain size and the thickness of the solid N
2
layer. We have computed the
average N
2
grain size sufficient for the onset of convection as a function of the N
2
thickness
in the case of perennial nitrogen deposits and corresponding
Nabarro-Hering
creep (volume
diffusion). We have also performed computations for the case when convection is coupled
with the “super” greenhouse in a seasonal N
2
layer. Convection in a seasonal N
2
layer on
Triton and Pluto is plausible even without the solid-state greenhouse effect, because N
2
on these bodies is so close to its melting temperature of= 63.148
K
(at zero pressure) that
an upper stagnant layer does not form. Since
d~r~
~~
<
0.3pm
for the fresh transparent N
2
deposits on Pluto
(Duxbury
et al. 1997),
Coble
creep (boundary diffusion) dominates in
these deposits until grains grow to the critical size. We have generalized our conclusion for
plausible convection in a seasonal layer for other bodies, on whose surfaces Van der
Waals
solids are near their melting temperatures (e.g.
,CH4
on Titan). We have also proposed a
method for estimating the thickness of Pluto’s perennial
Pluto flyby mission (Pluto Express).
INTRODUCTION
solid N
2
layer during the future
Pluto is the smallest planet in the Solar System with an average radius between 1137
and 1206 km
(Yoder
1995). A Pluto flyby mission is currently planned to be launched
in March 2001, It will take 10 to 12 years to get to the planet at the outskirts of the
Solar System.
Having a highly elliptic orbit with
an
eccentricity of about
0.25
(Yoder
2

1995), Pluto can reach the farthest distance from the Sun of
%
50 AU, with an average
clistance
being about 39.48 AU (Yoder 1995). Presently its distance from the Sun is slightly
less
(X
29.6 AU) than of Neptune and its largest satellite Triton. What is known about
the chemical composition of Pluto’s surface comes mostly from the ground-based spectral
observations led by Cruikshank (1976, 1980), with the recent important cent
ribut
ion by
Owen
et
al. (1993). They revealed that solid nitrogen is the dominant surface volatile
(about 98 %). The derived abundance of solid methane is about 1.5 %, whereas on Triton
it is only 0.05 %
(Cruikshank
et
al.
1993).
The methane absorption band centers in Pluto’s spectrum indicate that much of the
CH
4
on Pluto is present as isolated patches of free CH
4
, whereas on Triton it is more
likely to form a solid solution with N
2
. Some of the methane on Pluto is also diluted in
N2.
The other minor surface volatile constituent on Pluto is solid carbon monoxide (about
0.5 %), which is more volatile than
methane
but
1=s
volatile
than
nitrogen
at
pluto’s
temperatures.
Carbon monoxide is probably present in a form of a solid solution with
nitrogen. The second major difference bet ween the surfaces of Pluto and
‘Jliton
is that,
unlike on Triton, carbon dioxide ice was not detected on Pluto in the near-infrared spectral
region.
Usually the absorption feature at 2.148 microns (on which the nitrogen detections on
Pluto and Triton were based) is such a weak feature in the reflectance spectrum that a
path in solid nitrogen has to be more than a meter in order for nitrogen to be detected at
all. Thick nitrogen ice has many interesting implications because of its thermal insulating
effect. Solid N
2
(as well as other solids:
C’H4,
CO, C0
2
, whose molecules are bonded by
weak Van der
Waals
forces) in the vicinity of 40 K has much lower thermal conductivity
than the water ice. This means that temperatures are elevated at the base of a nitrogen
layer,
By virtue of the latent heat of sublimation effects, the nitrogen surfaces on Pluto
and
Triton
are nearly isothermal. Using ground-based spectroscopy, Tryka
et
al.
( 1994)
estimated the surface temperature of Pluto’s N
2
ice to be
40t~
K, which is close to the
current Triton’s N
2
surface temperature of
38~~
(Tryka et al,, 1993; Grundy et al., 1993).
The higher mean N
2
surface temperature of Pluto compared to Triton can be explained by
3

the
fact that Pluto
hasalower
overall
albedo,
thus presently absorbing
nlore
solar energy.
The currentconstraint
sonthesurface
pressure on Pluto are3t060pbar. The lower
value was derived from stellar occultation measurements (Hubbard
et
al. 1990,
Yelle
and
Lunine1989), but was later criticized by
Stansberryet
al. (1994), who considered that the
occultation measurements did not probe alltheway down to the surface. The highest value
in the pressure interval is obtained under the assumption that the surface and atmospheric
nitrogen are in vapor pressure equilibrium.
TRITONIAN
AND TERRESTRIAL GEYSERS
The above mentioned compositional and temperature analogies with
Triton
made it
worthwhile calculating the value of an internal heat flow on Pluto in order to determine
the applicability of our Triton’s model
(Duxbury
and Brown 1997) for the subsurface
convection in Pluto’s solid nitrogen. The model was initially developed as an alternative
to the solid-state greenhouse mechanism of energy supply to the Tritonian geyser-like
plumes. The idea of internal energy as a source for Tritonian geysers was proposed long
ago (e.g.,
Kargel
and Strom 1990), but the corresponding numerical models, justifying it,
were not presented.
The term “geyser” is commonly used for the liquid H
2
0 (and entrained C02,
H2S,
etc., gases) eruptions on Earth.
It may be more correct to call the
Tritonian
eruptions
fumaroles,
since they were nitrogen GAS with dust rising to about 8 km. The mechanism
for
volcanos
on Earth is thought to be the same as for geysers with the only difference
being in erupting material.
Terrestrial geysers are mostly of the boiling type with H
2
0
and
C02
as the primary erupting material, e.g. geysers in
Karnchatka
and Old Faithful
in Yellowstone Park. Condensing geyser volcanism (which is the main type on
Triton)
of
H
2
O is rare on Earth.
Remarkably, solid N
2
and silicate rocks behave similarly (and opposite to that of
H
2
0 ice) with respect to pressure induced melting: their melting temperatures drop if the
pressure is reduced. Hence the formation of a deep crack (vent) on Earth may bring rock’s
melting temperature down to the rock’s ambient temperature. Therefore, a mechanism
of terrestrial volcanic eruptions without magma chambers is possible
(Valerii
Drozdov,
4

I{amchatka
Volcanology Institute, personal communications).
For
Nz
icethe63.148Kis
the melting temperature at zero pressure and thus is the minimum melting temperature.
At the base ofal-km nitrogen layer on Pluto and Triton, thepressure is only about 7.9
bar, which does not influence the melting temperature noticeably.
ESTIMATES FOR PLUTO’S INTERNAL HEAT FLOW
Following
McKinnon
(1989) and Stern (1989), we deem Pluto to be completely dif-
ferentiated, We adopt thesame model of
Pluto’s
internal structure as was suggested for
Triton by Smith
et
al. (1989). The model implies a rocky core, overlaid by a water-ice
mantle and a surface veneer of volatile (mostly nitrogen) ices. The bulk density estimate
for Pluto is very close to Triton’s average bulk density of 2.054
g/cm3 (Smith
et
al. 1989).
But even if the densities of these two bodies were equal, Pluto’s smaller size may preclude
.
it from having a surface value of the internal heat flow as high as that on Triton, since sur-
face heat flow is proportional to a body’s radius, Hence, separate calculations are needed
for Pluto.
Using the conclusions of Stansberry et al. (1994), we assume that a strong thermal
inversion layer prevented the underlying atmosphere from being sensed by the stellar oc-
cult ation. Thus, we adopt Pluto’s radius of 1150 km, as was derived by
Tholen
and Buie
(1990)
from mutual events, rather than 1195 km calculated from the stellar occultation
data (Minis
et
al. 1993). We took the value of Pluto’s mass from Young et
al.
(1994)
astrometrically
determined range of 119.08
*
10
20
– 128.83* 10
20
kg. The main uncertainty
stems from the determination of Charon’s orbital semimajor axis. For the average mass
,.
value from this interval,
~mecn
=
1.95
g/cm3. This is consistent with
~m~a~
= 2.00
g/cnz3,
derived by
Foust
et
al. (1995).
The next formula relates the mass of the rocky core of a body to the body’s mean
bulk density:
where
~roCk
= 3
g/f3m3
is
M
core
1
–
(Pwater
ice
–
Pmean)
-
=
Mbodv
1
–
(Pwater
ice
–
Pcore)
‘
the average rock bulk density and
~ti,a~er
ice
=
1
g/~~~3
is
the
water ice bulk density, Thus a rocky core
mass. This estimate is consistent with
comprises as much as
x
73$%0
of the total Pluto’s
the historical considerations of McKinnon and
5

Brackett
(1994).
Usillga
collisional
formation model with water icejetting and subsequent
ice
loss for Pluto, they justified Pluto’s mass ratio:
75?Z0
of a rocky core versus 25910 of ice.
We have calculated the range of Pluto’s total
radiogenic
heat
flow
to be 4.93*
1010
–
8.32*101°
W.
Themaximum corresponds tothespecific (per unit mass)
radiogenic
heat
production of 9.2 *
10-12
W/kg measured for the lunar samples, and the minimum cor-
responds to the specific heat production of 5.45 * 10
–1
2 W/kg, measured for
chondritic
meteorites (Schubert et
d.
1986). We have calculated the final range of Pluto’s internal
heat flow at the surface as 2.97* 10
-3
– 5.00 * 10
-3
W/m2.
Since the total temperature
difference across a layer is the same for pure conduction as that for conduction with
con-
8T
The thermal conductivity
J
of solid
vection, we equate the calculated heat flow and
~
~.
N
2
in the temperature range possible in Pluto’s N
2
layer is about
0.2
W/rnK.
Hence, the
temperature difference across the N
2
layer is 14,85-25 K, which is only slightly lower than
the range obtained for Triton. A high temperature gradient in nitrogen ice on these bodies
is due to good thermal insulating property of solid nitrogen. The low absolute value of the
upper boundary temperature is influencing the convective model only indirectly through
the strongly temperature dependent viscosity of solid nitrogen. In this dependence the
proximity of the nitrogen temperature to the N
2
melting temperature is important.
SOLID NITROGEN EQUATORIAL BAND OR POLAR CAPS ON PLUTO?
According to Ward’s (1974) calculations, the critical obliquity, above which the min-
imum of the mean annual insolation per unit area is at the equator rather than at the
poles, is about 54 deg. Currently there are a few objects in the Solar System satisfying
this condition: Uranus with the obliquity of 97.86 deg (and its satellites) and Pluto with
the obliquity of about
122
deg.
Probably Mars’ obliquity at certain times could have
overcome the critical value, since it was shown that on time scales greater than 107 Earth
years Martian obliquity behaves chaotically reaching as high as 60 deg (e.g.,
Jakosky
et
al.
1995). Spencer et al. (1996) calculated that the current annual mean insolation at the
Pluto equator is about 230
erg/cm2s, which is slightly less than about 260
erg/cm2s
at
the poles.
As it was shown by Brown and Kirk (1994), PERMANENT solid N
2
deposits are
6

stable
in
the regions of long-term minima in the ABSORBED solar flux. Therefore, a
perennial solid nitrogen equatorial “belt” may be expected for Pluto. However, strong
surface
albedo
inhomogeneities were revealed by mutual event observations
(Buie
et
al.
1992; Young and
Binzel
1993) and by more recent images obtained by Stern
et
a~.
1997 in
June 1994 using the Faint Object Camera on the
Hubble
Space Telescope (HST). Owing
to these surface
albedo
inhomogeneities, a perennial nitrogen deposit can be STABLE
ANYWHERE on Pluto. There is a discrepancy between the HST images and the mutual
event observations
(Buie
et
ai.
1992). According to the HST images, the brightest spot is
on the equator, whereas the mutual events show it on the south cap. A definitive answer to
this disagreement will likely have to be preceded by spacecraft exploration. Two spacecraft
are currently planned to be launched to Pluto in March 2001.
In order to determine Pluto’s regions with the minimum ABSORBED insolation,
albedos
at different locations have to be known with much higher precision than they are
known today. Especially, because currently the incoming solar flux at Pluto’s POLES is
ONLY SLIGHTLY GREATER than that at the equator (Spencer et
a2.
1996), and a darker
equator (higher value of (l-A)) may cause the ABSORBED insolation to be higher there.
Nevertheless, the application of our model depends on the thickness of these deposits but
not on their locations.
“old”
ice on Pluto may look darker than on Triton because Pluto has more solid
CH4.
This reconciles the existence of a bright permanent polar capon Triton with the possibility
of the existence of dark PERENNIAL contaminated nitrogen ice on Pluto. Photon and
charge particle bombardment of
CH4
inclusions in N
2
ice produce dark organics, making
“old” contaminated nitrogen ice look dark. In this case, ONLY the conductive-convective
mechanism of energy supply to the plausible eruptions is feasible.
If the rate of nitrogen condensation is low enough, as appears to be the present
case for Triton and Pluto, N
2
condenses as a transparent layer (laboratory experiments
and calculations by
Duxbury
et
al. 1997). The Pluto
albedo
maps derived by Buie
et
al.
(1992), Young and
Binzel
(1993) from mutual events, show that the planet has an extensive
bright southern polar cap and a small or non-existent northern polar cap. (Here we use
the angular momentum vector for the definition of Pluto’s North Pole). Therefore, it is
7

conventional to
REDISTRIBUTION
speak about N
2
polar caps on Pluto.
OF THE INTERNAL HEAT FLOW OWING TO CONVECTION IN
Nz
ICE
Accordingly, our modeling for Triton maybe applicable to convection in Pluto’s upper
frozen nitrogen layer, provided this layer is thick enough and the solid nitrogen grain size
is sufficiently small. Therefore we may expect to observe the nitrogen geyser-like plumes
on Pluto as well. Methane geysers are unlikely to operate on Pluto because the CH
4
equilibrium vapor pressure of 15
p
bar requires a temperature of about 53 K, and the
CH4
equilibrium vapor pressure of 3
p
bar (the lower estimate for Pluto’s lower atmospheric
pressure) requires a temperature of about 50 K (Brown and Ziegler 1979). This is much
higher than the surface temperature of Pluto’s volatile ice of about 40 K (Tryka
et
al.
1994). Our modeling
(Duxbury
and Brown 1997) was performed for the case
when
the
basal temperature (63 K) was lower than the nitrogen melting temperature (63.148 K at
zero pressure and increases with pressure, see Scott 1976).
As in any problem with a strongly temperature dependent viscosity, a question arises
about the temperature at which to evaluate the parameters (especially viscosity)
consti-
t
ut
ing the
Rayleigh
number. This is equivalent to the problem of the formation of an icy
lithosphere on top of convecting N
2
layers, whose viscosity contrast is large. The viscosity
contrast for Pluto’s perennial
Nz
ice is caused by the temperature difference of about 23
K between the lower and the upper boundary temperatures.
All
techniques to take into
account this conducting overlayer in a SINGLE REPRESENTATIVE temperature (at
which
thermophysical
characteristics of a convecting layer are evaluated) were developed
either for terrestrial planets or for the H
2
O-ice satellites (mostly
Galilean
and Saturnian
satellites). The absence of convection in the H
2
O-ice upper layer is substantiated by ob-
servations
(McKinnon
et
al. 1996) and the fact that water ice on Pluto and Triton is too
far from its melting temperature.
Here we argue that, though these considerations may
be applicable to the water-ice mantle convection on Pluto and Triton (McKinnon
e-t
al.
1996), they are not applicable to convection of nitrogen ice on these bodies.
First, the peculiarity of N
2
ice is that it is much closer to its melting point on Pluto
and Triton than H
2
O ice on most bodies in the Solar System. Actually, N
2
on Pluto
8

and Triton is in the same vicinity of its melting point as water ice in terrestrial glaciers.
The base of the lithosphere is defined by the MINIMUM temperature T
1
sufficient for
creep to occur on a geological time scale. Creep in solids is caused by self-diffusion, which
is a thermally activated process.
It is a common phenomena in solids, at least in the
vicinity of their melting points (see experiments by Esteve and Sullivan (1981) with pure
and cent
aminated
lV
2
). The approach by Ellsworth and Shubert (1983) for the mid-sized
satellites of Saturn uses the formula
T1
=
0.6 *
Tmelting.
This formula is based on the
creep data for metals that show the occurrence of solid-state creep at temperatures
>
than
0.6 *
Tme/ting.
The application of this approach gives the temperature of= 37.89 K at the
base of the assumed Pluto’s
N2-ice
lithosphere. This is even less than the lower limit of
the spectrally measured temperature of
40~~
K at Pluto’s surface. Hence, we must take
T
1
=
Tup.
Though laboratory experiments gave 0.6*
Tme(t
as the MINIMUM temperature
SUFFICIENT for creep to occur during laboratory times, on GEOLOGICAL time scales
creep may occur at lower temperatures. This additionally substantiates our concept that
the very UPPER N
2
layer is convecting on Pluto and
Triton.
Therefore, we evaluate
the
thermophysical
coefficients in Pluto’s
IV2
layer at the average of the upper and lower
boundary temperatures, as we did for Triton
(Duxbury
and Brown 1997).
Secondly, from a theoretical standpoint, the bonds between water-ice molecules in
a crystalline lattice are strong hydrogen bonds, (with the two kinds of H-bonds being
randomly distributed,
Jichen
and Ross 1993) whereas in N
2
ice intermolecular bonds are
weaker (electrical) Van der
Waals
bonds.
Thirdly, there are no sufficiently resolved observations of Pluto’s surface to make a
conclusion about the involvement of the N
2
surface layer in convection. For Triton’s surface
there are only Voyager
2
observations in August 1989 (its surface is not resolved from the
Earth), thus it is impossible to compare the state of the surface at two different epochs.
Moreover, there is some morphological evidence from the Voyager 2 images for
diapirism
on
Triton (cantaloupe terrain, Schenk and Jackson 1993). Presumably, this
crustal
overturn
is driven by compositional layering of Triton’s crust. The authors have noted that thermal
convection may also be important as the driving mechanism.
Using the dimensionless formulation corresponding to the formulation from
Duxbury
9

and Brown (1997), we have computed that the closest approach to the surface of the
Tup
+
1
A’
isotherm is about 16
m
for
l?a/RaC.itiC.l
= 5, where
RaCritiC=l
= 1100.65
(Chandrasekhar
1981). This number is the minimum Rayleigh number required for the
onset of convection for the upper stress-free and lower rigid boundary conditions. The
minimum is taken with respect to all horizontal wavenumbers. The parameters used in
our computations are summarized in Table 1.
For nitrogen ice the
Prandtl
number is
essentially infinite (see Table 1 and
cf.
the Pr number for liquid nitrogen). This significantly
simplifies the hydrodynamic part of the problem in consideration by reducing the
Navier-
Stokes system to the Stokes system.
Even a small decrease in the average grain size gives a significant rise to the ratio
Ra/RaC,iti~~l.
Therefore, we investigated Pluto-and-Triton-like conditions, which will give
the needed
Ra/RaCritiCa/.
The results for thick perennial
Nz
layers are shown in Fig. 1.
Plots in Fig. 1 present the thickness of a perennial nitrogen layer versus nitrogen grain
diameter. These layers are thick enough to start thermal convection (the curves marked
by filled circles) and to achieve
Ra
=
5*
~acritimr
in thick
Nz
layers (the curves marked by
open triangles). A pure conductive thermal gradient of 0.0225 K/m is assumed (no
solid-
state greenhouse effect ).
For the grain
diaineters
corresponding to 800 - 1000-m-thick
layers on these curves and for Triton’s -
Pluto’s temperatures, Nabarro-Herring volume
diffusion dominates
(Duxbury
and Brown 1997). For the higher pair of curves the volume
self-diffusion coefficient of nitrogen ice is assumed to be equal to the average estimate
from the measurements of Esteve and Sullivan (1981), For the lower pair the self-diffusion
coefficient is taken at the upper limit from the same experimental measurements.
The nitrogen grain size is an important parameter.
If we measure the rate of nitrogen
grain growth in a laboratory under the Pluto-like conditions, in order to make a conclusion
about the grain size at the end of Pluto’s winter season, we need to know the original size.
The dynamic viscosity is proportional to the 2nd power (for the Nabarro-Herring volume
diffusion creep) or to the 3rd power (for the
Coble
boundary diffusion creep) of the grain
diameter.
CONVECTION IN A SEASONAL NITROGEN LAYER?
10

Earlier we examined the conditions sufficient for the onset of convection in a thick
perennial
lV2
ice. Itisinteresting tonotethat convection mayoccur evenin aSEASONAL
N
2
ice. Though its thickness is much smaller than that of perennial deposits, the sufficient
condition
Ra
>
RaCritiCai
may still be satisfied because of the small original grain diameter
and not enough time lapsed for grains to anneal into larger grains. As we have shown
in
Duxbury
et
al.
(1997), the initial grain size on Pluto (and moreover on
Triton)
is
<
0.3pm,
This estimate, can be improved by using the
emissivity
e
= 0.8 for the
/?
phase
of nitrogen
(Stansberry
et
ai.
1996) instead of the nominal value
c
= 1 originally used by
Eluszkiewicz
(1991). Additionally, some evidence for small particles on Triton is provided
by the scattering properties of discrete clouds, with particle sizes 0.2 -0.4
pm
in radius
(Hillier and Veverka 1994).
At SMALL grain sizes under consideration and Triton’s temperatures the
Coble
(boundary) diffusion creep dominates (a letter to N.
S.
Duxbury
dated March
31,”
1995).
Thus, we have used in our calculations for Pluto’s SEASONAL N
2
layers the dynamic vis-
cosity proportional to the 3rd power of the grain size instead of the 2nd power as for volume
diffusion used for PERENNIAL layers (Nabarro-Herring creep). We have calculated the
nitrogen dynamic viscosity using the grain-boundary diffusion coefficient (Goodman
et
al.
1981):
where the
preexponential
factor
DOb
=
DoU
= 1.6 x 10
-7
m
2
/s (Ashby and
Verrall
1978,
Eluszkiewicz
1991),
Eb
= (2/3) x
E.
= 5.7 x 10
3
J/mol is the activation energy for the
boundary diffusion
(Eluszkiewicz
1991, pp.
19219-
19q~o),
the universal gas constant
R=8.3145
J/(mol
K) and T is the temperature.
We used this coefficient in the dynamic viscosity formula for
polycrystalline
solids
with
Coble
diffusion creep (Raj and Ashby 1971, p.
112!1):
d~kBTauerage
~b(~at~erage)
=
,
Paos,
141
X 6 X
Db(Tauerage)
x
~
where
0
=
4.7
x
10-29/m3
is
the
NZ
molecular volume,
6
=
2f2113
zv
7.2 x
101°
m is the
11

width of
a
nitrogen grain boundary
(Eluszkiewicz
1991),
kB
= 1.381 x 10
-23
J/K is the
Boltzmann
constant,
dg
is the grain diameter (assuming spherical grains), and
T.V,To~,
is
an average of the upper and lower boundaries temperature,
In our program we calculated
Z’.V,..~.
as a function of the N
2
layer’s thickness, of a
pure conductive gradient and of the upper boundary temperature.
In our computations we have used the temperature dependence of N
2
thermophysical
characteristics from Scot t (1976). We have used a seventh-order polynomial for the ap-
proximation of density, second-order polynomials for the heat
resistivity
(the inverse of the
heat conductivity), and third and second-order polynomials to approximate the
N@
and
the
NP
heat capacity, respectively. Note that for a fixed
VZ’CO.~UCtiv,
the average of the
upper and lower boundary temperature (at which coefficients are taken in the
Ra
number)
is a function of depth.
.,
The maximum thickness of a seasonal N
2
layer on Triton is about 10 m. On Pluto
it can be even more, if we extrapolate the current condensation rate on these bodies over
their winter seasons. Results of our computations for seasonal N
2
layers are shown in
Fig. 2. They demonstrate N
2
grain diameters sufficiently small for thermal convection
to start versus the thickness of a seasonal
Nz
layer (the curve marked by filled circles).
The second curve, marked by open triangles, shows nitrogen grain diameters sufficiently
small for the Rayleigh number to be 5 times that of the critical for the free-rigid boundary
case
(RaCr~~iC~~
= 1100.65). Solely
radiogenic
heating is assumed. The average internal
heat flow of 4.6 * 10
-3
W/m2
is used for Pluto. This corresponds to the average thermal
gradient of 0.023 K/m. If N
2
ice on Pluto and Triton would not be so close to the melting
temperature, a stagnant layer would form on top, possibly preventing seasonal transient
convection.
COUPLING THE SOLID-STATE GREENHOUSE EFFECT AND CONVECTION IN N
2
ICE
The thickness of the N
2
layer sufficient for the onset of convection in a seasonal layer is
much smaller if we calculate the temperature difference between the upper and the lower
boundaries considering the solid-state greenhouse effect. Since it
initially forms a transparent layer (experiments by
Duxbury
et
al.
12
is likely that nitrogen
1997) on top of a dark

substrate, the “super” solid-state greenhouse effect can occur (Brown
et
al, 1990). Note
that the solid-state greenhouse effect does not require continuous sunlight and thus can
occur in Pluto’s southern hemisphere, where the seasonal
iV2
layer is currently growing.
Pluto’s subsolar point has crossed the equator in 1986 moving north at a rate of about 2
deg/earth year, hence the regions from the south pole down to about 70 S are in continuous
darkness. (We again use the angular momentum vector to define Pluto’s North Pole). The
condition for the daily averaged insolation, sufficient for the greenhouse effect to work, can
be found in the Appendix of the article by Matson and Brown (1989).
In our computations described below we have coupled the solid-state greenhouse effect
and convection. Results in Fig. 3 illustrate the grain sizes sufficiently small for convection
to start in a seasonal layer with the solid-state greenhouse effect (the curve marked by
filled circles). The curve marked by open triangles shows
lV2
grain diameters sufficiently
.
small to obtain
Ra
= 5 *
RaCrit
;Cal.
The “super”
solid-state greenhouse effect is assumed
with a dark underlayer having an
albedo
of
570,
thus giving the conductive gradient of 4
K/m (Brown
et
al.
1990). This is about 174 times more than the average thermal gradient
of 0.023 K/m calculated using only radiogenic heating (Fig. 2).
The plots in Fig. 4 represent an intermediate case of nitrogen grain diameters suffi-
ciently small for thermal convection to start in a seasonal layer with the solid-state green-
house effect (the curve marked by filled circles). The “super” solid-state greenhouse effect
is assumed with an intermediate value of 1 K/m for the conductive gradient (the dark
un-
derplayer
having an
albedo
> 5%). The curve with open triangles shows
Nz
grain diameters
sufficient to obtain
Ra
= 5 *
RaCri~iCai.
METHODS TO ESTIMATE N
2
GRAIN DIAMETER AND N
2
ICE THICKNESS
Solid-state convection occurs on Pluto if the solid nitrogen average grain size is small
enough and the N
2
ice layer is sufficiently thick.
The grain diameter in PERENNIAL
nitrogen ice can be assessed using the semiquantitative Zener theory (Kirk, personal
com-
municant
ions; A letter by R. Kirk to N. S.
Duxbury
dated 31 March 1995;
Duxbury
and
Brown 1997). Dust grains are inhibitors for the growth of the ice grains, though, as ex-
periments with metals showed, the cessation of the grain growth may occur even in pure
13

substance (Smith 1948). The dust flux on Triton was estimated by
Pollack
et
al.
(1990)
from the optical atmospheric properties measured during the Voyager
2
flyby in August
1989. Unfortunately, this value is not known for Pluto, since the planet has not been
visited by a spacecraft. The rate of cometary impacts was estimated for the Pluto-
Charon
binary by Weissmann and Stern (1994). Due to a higher than on Triton flux of comets
from the Kuiper belt and from the inner Oort cloud, the dust, brought by these impactors,
is probably more abundant on Pluto. Hence, the estimate for the maximum grain size of
x
1 cm, which we have obtained for Triton
(Duxbury
and Brown 1997) can be applied
and even reduced for Pluto. This makes the solid-state subsurface convection in perennial
N
2
deposits more probable on Pluto than on Triton, assuming the same thickness of the
perennial N
2
ice.
The other parameter important for calculating the
Rayleigh
number is the total thick-
ness of the nitrogen layer. In this connection we propose here a new method of determin-
ing this thickness during the future Pluto flyby mission. We have dubbed it the “crater”
method. By
analizing
the state of relaxation of the ejects we will be able to determine
whether the ejected material is near its melting point. At Pluto’s surface temperature of
about 40 K, N
2
ice will flow readily. If a crater’s depth is great enough to extract H
2
O “
ice, this ice (being very far from its melting temperature) forms rampart craters. It will be
possible to determine a crater’s depth using stereo images. (The resolution of the proposed
Pluto flyby camera is planned to be better than Voyager’s best resolution of
1
km/pixel).
Then by comparing the relaxation of the ejects for different craters, and thus for different
depths, we can estimate the depth of the perennial N
2
layer at different locations. The
second method of determining the depth of a crater is
photoclinometry,
but its application
on Pluto will be hampered by large surface albedo contrasts. Ground-based infrared spec-
troscopic observations also allow to estimate the ratio of
151V2
to the regular isotope
‘*N2
(Brown
et
al. 1997). If this ratio is high, then a significant portion of the regular nitrogen
isotope has escaped and it is likely that the perennial nitrogen ice on Pluto is relatively
thin.
DISCUSSION: NITROGEN GEYSERS ON PLUTO?
14

It was estimated theoretically (under the assumption that all dark streaks on the
surface of Triton’s south polar cap are extinct plumes) that the lifetime of Tritonian geyser-.
like plumes is 1-5 Earth years (Smith
et
al.
1989). This is consistent with our conclusion
that the plumes can form in a SEASONAL
Nz
layer. The initially condensed grains grow
through the formation of a neck between two grains
(pressureless
sistering). The process
is driven by the surface tension and the differences in grain curvatures. When the average
grain size exceeds the critical value, which is coupled with the thickness of the growing
layer via the Rayleigh number, convection ceases.
The probability of internally driven geysers on Pluto is somewhat lower than that on
Triton due to Pluto’s smaller size, which results in a slightly smaller internal heat flow at
the surface (this heat flow is proportional to the body’s radius). Secondly, smaller gravity
acceleration gives a smaller
Ra
(the
Rayleigh
number is proportional to gravity). Thirdly,
the N
2
escape rate on Pluto was estimated to be higher than that on Triton due to the
heating of Pluto’s upper atmosphere
(Trafton
et al. 1996). This possibly results in a
smaller nitrogen amount currently available. Although the extrapolation of the presently
high escape rate over the age of the Solar System may not be possible, since then there
would be no
volatiles
currently on Pluto. Ground-based observations by Owen et
al.
( 1993)
showed the presence of nitrogen, met bane and carbon monoxide on Pluto’s surface.
Continuous sunlight is not needed for the solid-state greenhouse effect. The formula
for the temperature difference between the subsurface and the surface value includes the
insolation averaged over the diurnal cycle (Mat son and Brown 1989). Physically, it means
that the solar energy acquired during the day may be enough to drive the plumes at night.
Whether the solar heating is enough is determined by the solar constant of a body and by
the season. Nevertheless, in at least two cases, only the internal heat can drive the plumes.
These are the existence of the plumes during a continuous night and the opacity of the
surface layer. The N
2
layer is probably darker on Pluto than on Triton, due to the higher
abundance of CH
4
and its
photochemical
darkening. If plumes are detected on Pluto, they
are more likely to be internally powered, because the solar constant at 39,48 AU (Pluto’s
mean dist ante from the Sun) is less than that at 30 AU
(Triton’s
dist ante from the Sun).
An observational test for the solid-state greenhouse involves the moderation of the
15

surface temperatures and a delay in the maximum diurnal surface temperature (maximum
is reached later than at the local noon). An observational test for the internal origin of the
plumes includes a detection of a geyser in a region of continuous darkness. We recommend
to conduct thermal
IR
observations of Pluto’s south polar region, which will be in the polar
night down to about 40 S in 2011 (time planned for the Pluto flyby), Another observational
clue for convection involves morphological evidence, e.g., cantaloupe terrain and volcanic
depressions on Triton
(Kargel
and Strom 1990). Recent spectroscopic studies of
‘5Nz
/14Nz
put additional observational constraints on N
2
thickness on Pluto and Triton (Brown
et
al.
1997). An additional support “for the locally higher internal heat flow is rendered by the
three volcanic
landforms
in Triton’s equatorial region detected in the Voyager 2 images by
the two groups of investigators (Kirk and Brown 1991;
Helfenstein
et al. 1991, 1992).
FUTURE WORK
Since convection in a seasonal layer is possible, it is interesting to
high will a “warm”
solid N
2
parcel rise from the bottom of a seasonal
winter season. More realistic (and thus more complicated) problem is,
understand, how
layer during one
how high it rises
before the average grain size becomes large enough for convection to cease and/or before
the seasonal layer sublimates to the thickness less than the critical (whatever comes first).
Convective velocity could be estimated via the Ra number using the work by McKenzie
et
al.
(1974, p. 494) and an estimate for the rate of the N
2
grain growth can be taken
from
Eluszkiewicz
(1991). In our case the convective velocity will be diminishing gradually
since the Ra number will be attenuating with the increase in the average grain size.
The possibility of convection in a thin N
2
layer under Pluto-like conditions can be
tested in a laboratory.
Since the expected convective velocities for small
supercritical
Rayleigh numbers are on the order of 1 mm/earth year
(Duxbury
and Brown 1997), it
will be reasonable to put a weight on top of the N
2
layer. This will increase the gravity
acceleration, and hence the Ra number and the convective velocity. With a solid weight on
top, the upper stress-free boundary condition changes to the rigid one, thus changing the
critical
Ra
number from 1100.65 to 1707.762 (for symmetrical rigid boundary conditions).
A liquid layer on top of solid nitrogen will allow to avoid the problem.
16

SUMMARY
In our previous work
(Duxbury
et
al.
1997) we estimated the initial grain size at which
N
2
currently condenses on Pluto and Triton to be < 0.3pm. The estimate can be some-
what improved using the
emissivit
y of 0.8
(Stansberry
et
al. 1996) for the hexagonal N
2
.
Additional evidence for small particles on Triton is provided by the scattering properties of
discrete clouds, with particle sizes 0.2- 0.4
pm
in radius (Hillier and Veverka 1994). The-
oretically N
2
ice at Pluto’s temperatures can begin to convect having an arbitrary small
thickness, if
Nz
crystals are small enough, i.e.: Ra
cx
cZ3/d~,~i~
~ con~t(llacritic.l). In the
last condition the 3rd power of the grain diameter was used instead of the 2nd
(Nabarro-
Herring diffusion), since, as we have shown above, in a freshly condensed nitrogen
Coble
diffusion creep dominates Nabarro-Herring diffusion. This allows for larger grains in order
to obtain the same Rayleigh number. Because the upper boundary temperature is greater
than 0.6 *
T~elting,
the stagnant layer is unlikely to form on top of convecting N
2
ice on
Pluto and Triton, making a transient thermal convection possible in a seasonal layer.
A vent (provided by the crystalline phase transition) does not have to be directly
above an
upwelling
plume because there can be a lateral
Nz
vapor transport. It occurs
in horizontal fissures due to the pressure difference, since
P(vent
) is slightly lower than
that away from the vent due to the lower temperature at the vent. The
T(vent
) is lower
because sublimation at the vent absorbs the latent heat.
In conclusion, we would like to underscore that solid nitrogen is a unique material
in the sense that on Pluto and Triton it probably convects all the way up to the surface
without forming an upper stagnant layer because it is a weakly bonded solid near its melting
temperature. This behavior can be generalized for other
Van-der-Waals
solids (C0
2
, NH
3
,
CH
4
, CO) near their melting points.
For example, our conclusions can
be
extended for
solid methane on the Saturnian satellite Titan. The melting temperature of methane is
about 90.67 K, which is very close to the estimated average surface temperature on Titan.
Creep in solid methane occurs at temperatures
a
54.402
K
= 0.6*
T~.~ti.~
~e~~ane.
This
temperature would be considered the temperature at the base of the upper stagnant layer,
if any.
17

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22

FIGURE CAPTIONS
Fig. 1. Thickness of perennial nitrogen layers versus nitrogen grain diameter. These
layers are thick enough to start thermal convection
(Ra =
RaC.ltiC.l
for the curves marked
by filled circles) and to achieve
Ra
=
5*
.RaCritiCal
in thick N
2
layers (the curves marked by
.,
open triangles). For the
free-rlgld
boundary case
Racritical
= 1100.65 A pure conductive
thermal gradient of 0.0225 K/m is assumed (no solid- state greenhouse effect). For the
grain diameters corresponding to 800- 1000-m-thick layers on these curves and for Pluto’s
temperatures, Nabarro-Herring volume diffusion dominates
(Duxbury
and Brown 1997).
For the higher pair of curves the volume self-diffusion coefficient of nitrogen ice is assumed
to be equal to the average estimate from the measurements of Esteve and Sullivan (1981 ).
For the lower pair the self-diffusion coefficient is taken at the upper limit from the same
experiment al measurements.
Fig. 2. Nitrogen grain diameters, assuming spherical grains, sufficiently small for
thermal convection to start (Ra =
RaCrit~eal
= 1100.65)
as
a function of the thickness of a
seasonal
lV2
layer (the curve marked by filled circles). The second curve, marked by open
triangles, shows nitrogen grain diameters sufficiently small for the
Rayleigh
number to be
5 times that of the critical. Solely
radiogenic
heating is assumed. The average internal
heat flow of 4.6 *
10-3W/m2 is used for Pluto.
This corresponds to an average thermal
gradient of 0.023 K/m.
Fig. 3. N
2
grain diameters
~ufficiently
small for thermal convection to start in a
seasonal layer with the solid-state greenhouse effect (the curve marked by filled circles).
The curve marked by open triangles shows
Nz
grain diameters sufficient to obtain
Ra
=
5 *
RaCritiCai.
The “super” solid-state greenhouse effect is assumed with a dark underlayer
having an
albedo
of 5%, thus giving the conductive gradient of 4 K/m. This is about
174
times more than the average thermal gradient of 0.023 K/m calculated using only
radiogenic
heating (Fig. 2). At those small grain sizes and Pluto’s temperatures the
Coble
diffusion creep is dominating.
Thus, we used in our calculations the dynamic viscosity
proportional to the 3rd power of the grain size instead of the 2nd power as for the volume
diffusion (Nabarro-Herring creep).
23

Fig. 4. Nitrogen grain diameters sufficiently small for thermal convection to start (the
curve marked by filled circles) in a seasonal layer with the solid-state greenhouse effect.
The “super” solid-state greenhouse effect is assumed with an intermediate value of 1 K/m
for the conductive gradient (the dark underlayer having an
albedo
> 5%). The curve
marked by open triangles shows N
2
grain diameters sufficient to obtain
Ra
=
5 *
Racritical.
24

Thickness (m)
o
N
o
0
o
0
m
o
0
03
0
0
0
0
0
—
\
—
.
1
I
I
1
I
I
I
I
I
1
I
I
I
I
I
I
I
I
I

Plausible solid
N2
grain diameter (microns)
o
in
o
w
N
cd
.-!-
-<
>
‘“i’.’
k
0
N
0
m
0
m
I
I
I
I
I
I
I
I
I
v
N
$
~
I
I
I
I I
I
I
I
I
I
I
I
I
I
I
I
1
I

,;
I
M
Plausible solid
N2
grain diameter (microns)
.
b
id
&
in
b
o
P
o
I
I
I
I
l“’’I’’’’
1’’’’ 1’’’’
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-i
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\
--l
\
1
-
*
I
I
I
I
‘1’’’
~1””1
i

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Plausible solid
N2
grain diameter (microns)
I
I
I
b
b
b
in
b)
I
I
1
l“’’I’’’’
1’’’’ 1’’”-
p
u-l
t-
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1
1
7
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-1
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lt,
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