GEOPHYSICAL RESEARCH LETTERS, VOL. 36, L23103, doi:10.1029/2009GL041030, 2009
Plasma mass loading from the extended neutral gas torus of Enceladus
as inferred from the observed plasma corotation lag
D. H. Pontius Jr.1 and T. W. Hill2
Received 17 September 2009; revised 25 October 2009; accepted 29 October 2009; published 2 December 2009.
[1] At Saturn, significant plasma mass loading and radial
transport occur together throughout the extended neutral
torus, both producing corotation lags. We present a theory
that depends upon the ratio of two classes of process: those
injecting new mass (simple ionization) and those effectively
removing momentum without changing plasma mass
density (elastic ion-neutral collisions, like-species charge
exchange). Both induce ionospheric torques, but only the
former increases total outward mass flux. From an observed
angular velocity, our model calculates the distribution of
mass loading with radial distance. We present solutions
based on Cassini Plasma Spectrometer measurements and
theoretical chemistry models for Saturn’s magnetosphere.
We find a pronounced interaction peak near Enceladus’
orbit and a broad ionization region between 5 and 8 Saturn
radii (Rs). The total mass outflux at 8Rs is 100 kg/s (S/0.1
S), where S is Saturn’s ionospheric Pedersen conductance.
The reported decrease in lag beyond 8Rs requires that S 6¼
const or reducing the total mass outflux. Citation: Pontius,
D. H., Jr., and T. W. Hill (2009), Plasma mass loading from the
extended neutral gas torus of Enceladus as inferred from the
observed plasma corotation lag, Geophys. Res. Lett., 36, L23103,
doi:10.1029/2009GL041030.
1. Introduction
[2] Plasma injection and outflow have important dynamical consequences in a magnetosphere where ionospheremagnetosphere coupling is strong enough to enforce a
degree of corotation between magnetospheric plasma and
the planet itself. The strength of this coupling depends on
the difference between the angular velocities of the magnetosphere and atmosphere. In the absence of competing
dynamical processes and given sufficient time, the magnetosphere and ionosphere would eventually come into corotation with each other. Both radial plasma transport and
interaction with a collocated neutral gas population lead to
changes in the plasma’s rotation rate, which trigger coupling
with the ionosphere that opposes the departure from corotation. In a steady state, the radial profile of angular velocity
depends on a balance between the rate at which the
ionosphere can supply angular momentum and the rate at
which mass is injected, moves radially, and interacts with
neutral gas.
1
Department of Physics, Birmingham-Southern College, Birmingham,
Alabama, USA.
2
Department of Physics and Astronomy, Rice University, Houston,
Texas, USA.
Copyright 2009 by the American Geophysical Union.
0094-8276/09/2009GL041030
[ 3 ] Hill [1979] examined how rotational dynamics
depend on outward transport, neglecting any local sources
of new mass or ion-neutral collisions. His approach
balanced the rate of angular momentum addition in the
magnetosphere with the planet’s capability to exert the
required torque, which depends upon the magnetic field
and Pedersen conductance S in the ionosphere (see Hill
[1979, equation 12] for details). Under steady conditions,
the plasma’s angular velocity declines monotonically with
increasing distance from the planet. For a spin-aligned
dipole magnetic field, the solution depends simply on radius
scaled by a single parameter determined by the rate of mass
transport divided by the ionospheric Pedersen conductance.
This approach does not depend on the nature of the
transport process, except to assume that the simple azimuthal
average of radial velocity maintains a constant ratio to the
mass-outflux-weighted average [Vasyliūnas, 1983; Pontius,
1997]. The electrical currents coupling the ionosphere to the
magnetospheric plasma were not treated explicitly by Hill’s
paper, though some later works have reproduced his result
using that approach [e.g., Vasyliūnas, 1983; Cowley et al.,
2002]. The methods are entirely equivalent.
[4] Pontius [1997] extended Hill’s [1979] formalism to
non-dipolar magnetic fields that are still symmetric about
the spin axis. He demonstrated that the radial profile of
angular velocity is not significantly changed when magnetic
field lines are stretched out radially in the manner inferred
from spacecraft measurements at Jupiter. Later work has
shown that this stretching does have important consequences
for the ionospheric location and magnitude of the Birkeland
(magnetic-field aligned) coupling currents and resulting
auroral emissions [Cowley et al., 2002]. However, these do
not appear to require field-aligned potential drops large
enough to alter the flow solution substantially [Nichols and
Cowley, 2005].
[5] The above models are applicable at Jupiter outside the
Io plasma torus, where approximately a ton of new plasma
is injected each second. Pontius and Hill [1982] showed
that the ongoing need to accelerate that new material
requires a persistent lag from corotation of about 4%, as
observed by Brown [1994]. About three quarters of that
mass loading takes place through charge exchange, which
ionizes a slow neutral while neutralizing a fast ion. The
former must be accelerated, while the later is ejected
ballistically as a fast neutral, and the net plasma density
remains unchanged for particles of the same species. The
remaining mass injection takes place though processes such
as electron impact ionization, which increases the plasma
density and contributes to outward transport.
[6] Those mathematical models are applicable separately
in regions where either mass transport or mass loading is
significant, but not both. This presents a complication at
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Saturn, where both processes are expected to contribute
significantly in the same region of space [e.g., Eviatar and
Richardson, 1986; Saur et al., 2004]. Here, we develop an
analytic formalism relating the angular velocity profile to
coincident mass loading and transport. We apply this
method to Cassini observations of plasma velocity at Saturn
combined with theoretical estimates of parameters describing chemical interactions among particles.
conductivity. It also includes both hemispheres in parallel.
[9] From a specified profile of the distribution of mass
loading, one can solve (3) for the resulting angular velocity.
Alternatively, we can adopt the opposite approach and
calculate the distribution of mass loading from an observed
angular velocity profile. In either case, m(L) appears as an
input determined from considerations of particle populations and their chemistry.
2. Mathematical Derivation
3. Deriving the Source From Observed Angular
Velocities
[7] We will assume that plasma is confined close to the
equator so that the distance to the spin axis varies negligibly
along a magnetic field line. The magnetic field is represented by a spin-aligned dipole of equatorial magnitude Bp,
and radial currents in the magnetosphere provide an increase of angular momentum. In a steady state, the torque T
exerted per unit L equals the convective time derivative of
angular momentum [Hill, 1979]
dT
d _
dwL2
Mt þ M_ x þ M_ t R2p
¼ ðw wk ÞL2 R2p
dL
dL
dL
ð1Þ
where L is the equatorial field line crossing distance in units
of planetary radius Rp. The first term on the right represents
the rate of angular momentum addition required to bring
newly injected plasma up to the local plasma angular speed
w from the local angular speed wk of a circular Keplerian
orbit due to the two distinct types of processes described
above. Interactions that change the plasma density, such as
electron impact ionization, are quantified by a local source
rate dM_ t/dL, and the total outward flux of mass is M_ t(L). In
contrast, dM_ x/dL represents processes such as elastic ionneutral collisions and like-species charge exchange where
angular momentum must be supplied to the plasma without
a net change in plasma density [e.g., Pontius and Hill,
1982; Eviatar and Richardson, 1986; Saur et al., 2004].
These are not mass loading processes per se; rather, dM_ x/dL
represents the equivalent mass loading rate associated with
a given momentum loading rate. The final term is the
angular momentum required to maintain the existing
plasma at the ambient angular velocity as it moves outward.
[8] To facilitate analysis, define m(L) as the fraction of the
mass loading rate that contributes to transport relative to the
total rate including charge exchange and elastic collisions,
i.e.,
d _
d _
Mt ¼ m
Mt þ M_ x
dL
dL
ð2Þ
With this substitution, equate (1) to the torque per unit L by
Hill [1979, equation 12]:
dT
ð1 1=LÞ1=2
dw
¼ 4p S R4p B2p
L3
dL
1
d
dwL2
¼ ðw wk ÞL2 R2P M_ t þ M_ t R2p
dL
m
dL
ð3Þ
where dw = wp w is the lag relative to the planetary rate
wp. The appropriate value of S here includes slippage
within the neutral atmosphere [Huang and Hill, 1989;
Pontius, 1995], so it is less than the true Pedersen
[10] If the angular flow velocities w(L) are known from
observation, then the distribution of mass loading can be
determined. Rearrange (3) to get:
1=2
d _
m
1 dwL2 _
dw
2 2 ð1 1=LÞ
M
Mt þ
¼
4p
SR
B
m
t
p
p
L5
dL
w wk L2 dL
w wk
ð4Þ
The solution is
M_ t ð LÞ ¼ 4p SR2p B2p eI ðLÞ
ZL
meI ðxÞ
ð1 1=xÞ1=2 dw
dx
x5
w wk
Li
I ð LÞ
þe
_i
M
I ð LÞ ¼
ð5Þ
ZL
m
d 2
wx dx
ðw wk Þx2 dx
ð6Þ
Li
where M_ i is the net flux of outward moving mass at the
lower limit of integration. The local source rates dLM_ t(L)
and dLM_ x(L) are then found by differentiation. If no mass is
transported, then m = 0 and the source can be determined
from the lag directly using (1) and (3), giving
1=2
d M_ x 4p SR2p B2p ð1 1=LÞ
dw
¼
dL
L5
w wk
ð7Þ
which is equivalent to Pontius and Hill’s [1982] solution.
[11] Wilson et al. [2008, 2009] have presented the radial
profile of angular velocity w(L) based on their analysis of
Cassini Plasma Spectrometer (CAPS) data. Figure 1 shows
their angular velocity data as a fraction of the local
corotation speed, along with an eighth-order polynomial
fit to those data that we used in our calculations. The radial
profiles of mass loading and momentum loading rates
follow immediately from those data given a model for m(L).
[12] However, a first-principles derivation of m(L) is well
beyond the scope of the present paper. Instead, we use
reaction rates derived from a model of Saturn’s neutral gas
cloud developed by Johnson et al. [2006]. Those authors
computed the trajectories of neutrals ejected from a narrow
source at the orbit of Enceladus. Using measured or
computed cross sections and observed particle fluxes to
describe the ion-neutral chemistry, their calculations predicted a narrow torus encircling Saturn around the orbit of
Enceladus (L = 3.95) composed mainly of water-group
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Figure 1. Angular velocities from CAPS measurements
[Wilson et al., 2008, 2009], along with a fitted curve used
for the integrations of equations (5) and (6).
neutral species. Subsequent charge exchange and other
reactive collisions with sub-corotating plasma produced a
broader OH torus, in accord with Hubble Space Telescope
(HST) observations [Jurac and Richardson, 2005]. Combining this neutral cloud model with empirically-based
estimates of the neutral lifetimes against charge exchange
and electron-impact ionization, R. E. Johnson (personal
communication, 2008) produced the charge exchange and
ionization rates shown in Figure 2a. Combining these rates
with equation (2), we obtain the discrete values for m(L)
shown in Figure 2b. This approach essentially represents all
interactions as being between neutrals and ions of the same
species, e.g., H2O and H2O+, so that particle masses cancel
in (2). This approximation omits possibly relevant interactions between unlike species that would be incorporated into
a more complete model, such as charge exchange reactions
between O+ and H2O that would change the plasma mass
density. However, the fundamental result should be robust,
that charge exchange strongly dominates over electronimpact ionization in the region L ] 5, so that only a small
fraction of particle interactions there contribute to outward
mass transport.
[13] The solid curve in Figure 2b is a sigmoidal fit to
Johnson’s data that we used in integrating (5) and (6). Our
results are shown in Figure 3, assuming zero outflow at an
inner computational boundary at Li = 3. While the initial
outflow at Li = 3 may not be zero, the resulting profiles for
mass loading differ only slightly unless M_ i is comparable to
the total mass calculated to be added within the entire
integration domain. The peak of the inferred charge exchange rate near L = 3.6 is a direct consequence of the
observed peak of the fractional corotation lag near L = 4.2
(Figure 1). The difference in L value between these two
peaks results almost entirely (in our model) from the L5
factor on the right-hand side of (4). Because electron impact
ionization is strongly suppressed in this region (Figure 3),
very little new plasma has accumulated there to move
outward, and most of the corotation lag is caused by
charge-exchange momentum loading rather than radial
transport. Indeed, solutions calculated inside of L = 5 for
zero transport using (7) are essentially identical to those
shown here.
Figure 2. (a) Charge exchange and ionization rates from
models of Saturn’s neutral-plasma interactions based on the
model of Johnson et al. [2006], the latter multiplied by a
factor of ten for clarity. (b) The ratio of net (source minus
loss) to gross (total source) mass addition rate based on that
model, along with a sigmoidal fit.
4. Discussion
[14] The peak of the inferred momentum-loading profile
(L 3.6, Figure 3) is well inside of the peak of the observed
fractional corotation-lag peak (L 4.2, Figure 1) from
which it derived, and this peak displacement is fully
explained by the L5 factor in (8). This factor, however,
does not explain the displacement between the inferred
momentum-loading (charge-exchange) peak near L = 3.6
(Figure 3) and the peak location L = 3.95 expected from
Figure 3. Solutions for the mass injection rate from
exchange processes and net ionization, the latter multiplied
by a factor of ten.
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Johnson et al.’s [2006] model (Figure 2a). This difference
is well outside the combined error bars of the two models
and of the underlying data of Wilson et al. [2008, 2009]. It is
possible that an asymmetry of the polar plume of Enceladus
produces a neutral-density peak inside the orbital radius of
Enceladus (L = 3.95), even though Enceladus is clearly the
ultimate source for neutrals. This is an interesting area for
future research.
[15] The inferred broad maximum of electron impact
ionization between L 4 and 7 (Figure 3) corresponds
roughly with that derived by Johnson et al. [2006] (Figure
2a). Those particles originate near Enceladus as ions that lag
corotation sufficiently that, following neutralization by
charge exchange, their speed is less than the gravitational
escape speed. Increasing their residence time as neutrals in
regions with sufficiently hot electrons allows them to be reionized and subsequently transported outward as plasma.
Our analysis supports this interpretation because the increase in fractional corotation lag from L 6 to 8 (Figure 1)
cannot be accounted for by transport alone with constant
M_ t. Johnson’s results from physical chemistry modeling
(Figure 2a) are similar to those of our electrodynamic model
(Figure 3), both qualitatively as well in overall magnitude.
As the two approaches are independent, this agreement
provides mutual confirmation for both.
[16] Using a constant effective Pedersen conductance S =
0.1 S, these solutions imply a total mass outflux of 100
kg/s near L = 7.5. Beyond that distance, the observed
fractional lag begins to decline, which physically implies
that either the ionosphere has become more effective at
exerting torque or that the magnetosphere is requiring less
angular momentum [Pontius, 1997]. The former could arise
from an increase in the conductance, here treated as a
constant. Such an increase could occur if the effective value
is elevated toward the true Pedersen conductivity [Pontius,
1997] or from enhancement of the latter caused by energetic
particle precipitation [Nichols and Cowley, 2004]. Alternatively, the total mass transported outward could be reduced
by dissociative recombination or by charge exchange between light neutrals and heavy ions.
[17] In its present form, our model does not incorporate
any of these effects. Variations in conductance can be
included with a prescription for S(L), but a net loss of
plasma would require modifying our basic formulation. The
total mass transported outward in our model is simply the
accumulation of the net mass loaded everywhere closer to
the planet. Plasma mass addition requires a prograde torque,
but a net loss of plasma mass does not produce a torque in
the opposite sense. However, the first term on the right side
of (1) being negative would imply just that; this term should
include local plasma sources but not losses. Physically,
ionizing a neutral requires giving it the momentum it needs
to attain plasma speed, but neutralizing an ion does not
return its momentum to the plasma. Its momentum is carried
away by the escaping neutral. Hence, the negative values of
mass loading inferred for L ^ 8 indicate that our solutions are
not physical in that region. Future work on this analysis will
include a more thorough investigation of the particle interactions determining m(L), which may include net mass loss.
Expanding the model to include the possibilities described
above is not presently justified given the lack of specific
information available about these processes at this time.
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[18] To gain insight about the magnitude of these omitted
factors, we assume that mass loading is zero in this region
so that the lag is accounted for by transport alone. Setting
the first term on the right side of (3) equal to zero, we use
the observed angular velocity profile to calculate the implied ratio M_ t /S as a function of distance. That calculation
(not shown here) shows a monotonic reduction of M_ t /S by
a factor of roughly 4 from L = 7.5 to 10. Hence, either
decreasing the mass outflow rate or enhancing the Pedersen
conductance by that amount (or a combination of the two)
would account for the data. It is worth noting that, using the
peak value of M_ t /S, we calculate that Lo 8, where Lo is
the distance [Hill, 1979] at which corotation breaks down
due to radial transport alone [Krupp et al., 2007]. The same
value was derived by Eviatar and Richardson [1986] from
Voyager observations.
5. Conclusion
[19] The analysis presented here predicts the spatial
distribution of plasma injection from the observed azimuthal
velocity. In that, it is similar to our previous analysis
[Pontius and Hill, 1982], but the present approach determines
what part of the corotation lag is caused by local injection and
what is cause by outward transport. Our model depends on
the results of models of particle interactions, and one could
calculate the source rates directly from a sufficiently extensive and detailed model, at least in principle. However, our
approach only requires knowing the relative effectiveness of
ionization and exchange processes averaged within each
magnetic shell. The lag from corotation between L = 3 and
5 is mainly caused by charge exchange and elastic ion-neutral
collisions, which peak in intensity slightly inside of Enceladus’ orbit. The net amount of new mass added there and
transported outward is relatively small, which is why the
angular velocity starts to recover toward corotation between 4
and 5.5. However, continued mass loading throughout the
extended neutral torus eventually increases the total mass
outflux until the lag due to transport becomes substantial,
which explains the decline in angular velocity outside of L =
5.5. The decrease in lag beyond L = 8 is surprising and not
explained by the present model.
[20] Acknowledgments. We are indebted to R. E. Johnson for
providing pre-publication model results incorporated in Figure 2 and to
R. J. Wilson for providing pre-publication data incorporated in Figure 1.
This work was supported by NASA grant NNX08AQ78G to BirminghamSouthern College and by NASA JPL contract NAS703001 to Southwest
Research Institute through a subcontract to Rice University.
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T. W. Hill, Department of Physics and Astronomy, Rice University,
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D. H. Pontius Jr., Department of Physics, Birmingham-Southern College,
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