\chapter{Chapter 1 Introduction}
\chapter{Chapter 2 Collision Physics}
\chapter{Chapter 3 Interaction with Surfaces}
\chapter{Chapter 4 Plasma Erosion and Modification of Natural Surfaces and
Atmospheres}

\section{Introduction}
\label{sec:1}
\cite{monograph}.

Energetic ions and electrons, associated with planetary magnetospheres, the
magnetosphere of the Sun, shock-heated interstellar gases, or cosmic rays,
can impinge on the surfaces of interplanetary dust grains and planetary ring
particles, the surfaces of planetary satellites, and the atmospheres or
surfaces of the planets themselves. This bombardment can modify these
surfaces and atmospheres via the collisional processes described in the
previous chapters.

Although the plasmas have no net charge, the local electron fluxes are
generally larger than the ion fluxes because of their higher mean
velocities. When these ions and electrons impinge nonisotopically on a
conducting surface or medium (e.g., an atmosphere with an ionosphere),
currents are set up to accommodate the different bombardment rates. A
non-conducting surface will eventually charge to the point where the
electron and ion fluxes become equal. This does not require a large amount
of charge per unit area on a satellite; for example, on the surface of an
object the size of Enceladus an electron surface density of 0.3 elec/cm$^{2}$
will stand off electrons with energies less than 10 eV. On a 2 $\mu $m
E-ring ice grain, however, a charge of 3 x 10$^{4}$ elec/cm$^{2}$ would be
required, ignoring secondary electron ejection. Hence, small particles may
be charged to the point where \textit{their} motion is affected by the
planetary (solar) magnetic field and the flow of the plasma onto the object
may also be affected. Objects composed of materials that conduct can also
have fields associated with intrinsic magnetic moments or induced fields
which affect the charge flow onto the surface. Based on scaling of the
fields with current densities, this is generally important for relatively
large objects. Therefore in the next section plasma flow onto large and
small objects will be considered.

Electrons and ions impinging on surfaces or atmospheres differ considerably
in the momentum they carry, with ions generally penetrating to greater
depths. Hence, on an object with a tenuous atmosphere, ions are more likely
to reach the surface. This is especially so as the incident ions will
eventually be neutralized by charge exchange with atoms of the medium. This
not only increases their ability to penetrate, because the interaction cross
sections are smaller for neutrals, but the ions cease to be contained by the
local electric or magnetic fields. On the other hand, electrons and ions
having the same velocities ionize and excite atoms and molecules with
comparable efficiencies. Therefore, electrons having much smaller masses
produce ionization at much lower temperatures than is required by ions.
These differences will be used in the following, although the role of ions
will be emphasized throughout this chapter.

After the discussion of the plasma flow onto an object, the amount of
sputter erosion and the changes induced in its surface will be considered.
Sputtering will then be compared to sublimation of objects in various
regions of space, and the effects produced on large objects for which escape
is restricted by gravity will be discussed. Finally, the fate of those
molecules which escape into the magnetic confinement region of the plasma
will be examined. These processes are described generally and applied only
occasionally to specific objects, particularly Io.As these applications are
often subject to large uncertainties in both the plasma and the surface
composition (Johnson and,Matson 1988), the material is presented in a way
that should be useful as new information becomes available. Note: in this
chapter many of the subscripts distinguishing incident and target particles
are dropped (e.g., E$_{\mathrm{A}}$ $\rightarrow $ E, M$_{\mathrm{A}%
}\rightarrow $ M, n$_{\mathrm{B}}$ $\rightarrow $ n).

\section{Plasma Bombardment}

The description of the plasma flow onto an object in a space plasma can be
complicated. Here a few idealized cases are discussed and the reader is
referred to the more detailed literature (Vasyliunas 1983; Neubauer et al.
1984; Herbert 1985; Luhman 1986; Wolf-Gladrow et al. 1987; Linker et al.
1988). In the following, differences between gases and solids are ignored
when discussing the primary events caused by an incident ion or electron
using only the atomic or molecular column density.

\subsection{\textit{Effect of Fields}}

In Fig. 4.1 is a schematic diagram showing three cases for plasma flow
around an object. To describe the characteristics of this flow requires
knowledge of the flow speed v$_{\mathrm{i}}$, the plasma temperature T$_{%
\mathrm{i}}$ and density n$_{\mathrm{i}}$ and the confining magnetic field
B. It also requires knowledge of the magnetic field of the object and/or a
description of its conducting region. The relative importance of various
effects can be described in terms of three pressures (Kennel et al. 1979;
Wolff and Mendis 1983): the momentum flux carried by the plasma (the ram
pressure) M$_{\mathrm{i}}$n$_{\mathrm{i}}$v$_{\mathrm{i}}^{2}$; the thermal
pressure in the plasma n$_{\mathrm{i}}$k$_{\mathrm{i}}\mathrm{T}_{\mathrm{i}%
} $; and the ability of the fields to confine or accelerate particles (the
magnetic pressure) B$^{2}$/2$\mu _{\mathrm{0}}$, where $\mu _{\mathrm{0}}$
is the magnetic permeability of free space. These are given in Table 4.1 for
the solar wind and the plasmas of Jupitur and Saturn at certain satellites.
When the ram pressure is much larger than the thermal pressure, the flow is
said to be supersonic. If this flow speed is also larger than that velocity
at which signals are transported in the plasma [the Alfven velocity = B/($%
\mu _{\mathrm{0}}$M$_{\mathrm{i}}$n$_{\mathrm{i}}$)$^{1/2}$] the flow is
called super Alfvenic (Hasegawa and Uberoi 1982; Kennel et al. 1979). The
latter is equivalent to the ram pressure being much greater than the
magnetic pressure. Both of these conditions apply to the solar wind, which,
on intercepting an object, produces a bow shock (Russell and Greenstadt
1983; Lanzerotti 1987; Connerney 1987). Behind this shock the impinging
plasma thermalizes and the particles flow (at higher densities and
temperatures) in a region called the magnetosheath. In this region, the flow
is subsonic around the object or onto the object. At a region called the
magnetopause, the flow onto the object is essentially blocked, the
stagnation region of the flow. At the magnetopause, the ram pressure is
effectively deflected by the intrinsic field, i.e., M$_{\mathrm{i}}$n$_{%
\mathrm{i}}$v$_{\mathrm{i}}^{2}$ $\sim $ B$_{\mathrm{p}}^{2}$/2$\mu _{%
\mathrm{0}}$, with B$_{\mathrm{p}}$ the planet's (object's) field at the
magnetospause. The location of this region determines the ability of ions to
bombard the planet or the planet's atmosphere as indicated in Fig. 4.1a and
b. Distances are given for the solar wind incident on the planets in Table
4.2. However, this boundary is described only on the average, as it is
permeable. Ions having gyroradii much large than the distance between the
planet's surface (atmosphere) and the magnetopause can still bombard the
surface (atmosphere). The energies of such ions are associated with solar
flare ions and cosmic ray particles. In addition, ions can be scattered onto
planetary field lines and into the polar cusp in the planetary field and,
thereby, penetrate onto the object or its atmospheric gas. Finally,
neutralized ions can always penetrate.

For an object with no intrinsic field or a very weak one, the interaction
with the plasma and carrier fields can induce a magnetic barrier if the
object is conducting (Herbert 1985; Wolf-Gladrow et al. 1987). For an object
with a significant component of an atmospheric gas this is always the case,
as photons and the incident plasma electrons or ions can produce a
conducting ionosphere. This barrier (stagnation point) is referred to as an
ionopause. The formation of an induced barrier depends bn the conductivity, 
K$^{\prime }$, of the medium (the ionosphere or the surface material) and the
scale of the object. Roughly, K$^{\prime}$ > ($\mu_0$R$_{\mathrm{s}}$
v$_{\mathrm{i}}$)$^{-1}$ for a magnetopause (ionopause) to from on an object 
of radius R$_{\mathrm{s}}$ (Lewis and Prinn 1984), so that very
small objects are not likely to create such a barrier. For an atmosphere, 
K$^{\prime }$ is proportional to atmospheric density; hence, sufficient
densities are required. A bow shock is also formed downstream if the flow is
supersonic. For a comet-like interaction in which a gas (source rate 
$\mathrm{\dot{\eta}}_{\mathrm{s}}$ and molecular mass M) flows outward
against the plasma `wind', a rough lower limit to the stand-off distance is
given by $\dot{\eta}_{\mathrm{s}}$M$\nu _{\mathrm{i}}$/[4$\pi \langle$
v$_{\mathrm{es}}\rangle $ $\langle \rho $v$\rangle _{\mathrm{w}}$ ] where 
$\nu_{\mathrm{i}}$ is the ionization rate for the molecules. Here $\langle$
v$_{\mathrm{es}}\rangle $ is the escape speed and $\langle \rho $v$\rangle_
{\mathrm{w}}$ is the plasma `wind' mass flux (Schmidt and Wegmann 1982).

For a weakly ionized gas a first-order model can be used to calculate the
average plasma flow downstream (Appendix 4A). In this model the ratio of K$%
^{\prime }$ to the conductivity of the medium in which the object is
imbedded, K$_{\mathrm{m}}^{\prime }$, determines the ability of the plasma
to penetrate the conducting region characterized by a radius, R$_{\mathrm{c}%
} $, from the center of the object. The fraction of the incident flux onto
the conducting sphere is [2K$_{\mathrm{m}}^{\prime }$/(K$^{\prime }$ + 2K$_{%
\mathrm{m}}^{\prime }$)]$^{2}$ (Appendix 4A). Estimates at Io are K$_{%
\mathrm{m}}^{\prime }$/K$^{\prime }$ $\sim $ 1/3 (Hill et al. 1983), giving
a fraction $\sim $ 15\%. As this region is often well below the exobase, the
plasma can interact with the tenuous regions of the atmospheric gas (McGrath
1987; Linker et al. 1988). In fact, the thermalized ions in the
magnetosheath can neutralize in weak collisions with the extended
atmospheric gas and then penetrate into the gaseous envelope below the
ionopause. The ions produced above the ionopause, both by the charge
exchange and ionization by the electrons, are added to the flow of the
plasma around the object a process referred to as scavenging (Michel 1971;
Cloutier et al. 1969). This removes energy from the flowing magnetic field.
Limits on the amount of scavenging are determined by the magnetic and plasma
pressures and by the scale of the object.

Finally, if the object has little atmosphere, is not conducting, and/or is
small, the plasma can bombard the surface as in Fig. 4.1c. In describing
this bombardment, the gyroradius (the orbital radius about a field line) and
the pitch angle (determined by the component of velocity along the field
line) characterize the particle's motion.

\subsection{\textit{Flux Distribution}}

Ignoring the fields associated with an object, the flow of ions onto the
surface (or atmospheric gas) of a stationary object is simply described.
However, for a plasma with a relative motion with respect to the object (or
vice versa), the usual problems pertain to any flow onto the downstream
surface. That large objects sweep out the energetic plasma and produce wakes
was well established by the Pioneer and Voyager fly-bys of the satellites of
Jupiter (Paonessa and Cheng 1986). This would imply that the rate of
filling-in downstream was not always fast. This filling-in process is
controlled by the size of the object compared to the largest of the
gyroradius, the scale of the turbulence or the collisional mean free path of
the ions, and the pitch angle distribution of the plasma (Lanzerotti et al.
1981). (Gyroradii are given in Tables 4.1 and 1.2 for ions near a number of
objects.) As we are often not interested in the instantaneous bombardment
rate, the various diffusive processes, fluctuations in the local fields and
nonzero pitch angle, cause an effective in-flow on the downstream side when
averaged over long time periods, even if the gyroradius is not large.
Therefore we first consider the case in which all surfaces are bombarded.

Assuming flow occurs onto every surface according to the local plasma
temperature and the surface motion, the effect of the plasma on an area of
the surface is 

\begin{equation}
\mathrm{ \langle F\Phi \rangle = \int_{+} n_{+}(\vec{u} + \vec{v})\cdot
(- \hat{n})F(\vec{v} + \vec{u})f(\vec{v})d^3 v.}
\end{equation}
Here f($\mathrm{\vec{v}}$) is the velocity distribution of the
plasma ions of density n$_{+}$, \^{n} is the direction of the surface
normal, and $\mathrm{\vec{u}}$ is the \textit{motion of the plasma} past the
object, the streaming flow. The plus on the integrand implies that particles
must cross the surface opposite to the direction \^{n} (i.e., integrand
positive) and F is the effect of interest (e.g., sputtering, secondary
electron ejection, momentum transfer, etc.). For instance, in describing the
effect of implantation on the reflectance, F = 1 for those ions with
penetration depths less than photon penetration depths of interest in remote
sensing. The presence of an atmosphere, local fields, etc., all effects we
are ignoring at present, can be handled via F (Appendix 4B). In the
following we consider all incident ions by setting F = 1.

Using a Maxwellian distribution to describe the thermal motion, Eq. (4.1)
can be integrated. (Note: often the word convected Maxwellian is used; this
is essentially $\mathrm{\vec{v}}$ times the Maxwellian). This gives a flux

\begin{subequations}
\begin{align}
& \mathrm{ \Phi(\theta) = (n_+ \bar{v}/4)exp( - x_0^2 cos^2\theta) + (n_+ /2)
u cos\theta[ - 1 + erf(x_0 cos\theta)],} \notag\\
\end{align}
where n$_{+}$ is the ion number density, and erf(x) is the error function 
[(2/$\pi ^{1/2}$) $\int_{0}^{x}$exp(- y$^{2}$)dy] with x$_{0}$ = [(2/$\pi^{1/2}$)
u/v]. In Eq. (4.2a) \={v} is the mean speed of a Maxwellian distribution, 
\={v} = (8kT/M$\pi $)$^{1/2}$, and $\theta $ is the angle between the surface 
normal and the flow direction (\^{n} $\cdot $ \^{u} = cos$\theta $). In order 
to understand Eq. (4.2a), we consider the limiting cases. When the mean 
thermal speed is much larger than the flow speed, then
\begin{align}
& \mathrm{ \Phi(\theta) = (n_+ \bar{v}/4) - nu cos\theta/2;\hspace{0.1in} 
\bar{v} \gg  u.} \notag\\
\end{align}
In the limit that u 0, every region of the surface receives the same average 
flux n$_{+}$ v/4. When the flow speed is much greater than the mean thermal 
speed, then 
\begin{align}
& \mathrm{ \Phi(\theta) = n_+ u [\vert cos\theta\vert - cos\theta]/2; 
\hspace{0.1in}u \gg \bar{v}.}
\end{align}
\end{subequations}
That is, flow occurs onto the upstream side only ($\theta  > \pi$/2). The 
latter condition applies to many of the plasma interactions in the solar 
system (see Table 4.1). Finally, if the ions have small gyroradii relative 
to the size of the object and very small pitch angles, and we ignore turbulence 
and ion scattering, then the flux is again very simple. That is, even though 
the plasma speed, v, is large compared to the overall motion u, only the 
up-stream hemisphere is bombarded, as in Eq. (4.2c).

Of interest, also, is the net bombardment of the object, averaged over the
full surface. The flux averaged over the sphere is 

\begin{subequations}
\begin{align}
& \mathrm{\Phi = \int\Phi(\theta)d\Omega/4\pi = (n_+ \bar{v}/8)exp(- x_0^2) + 
(n_+ u/8)(2 + x_0^{-2})erf(x_0).} \notag\\
\end{align}
In the two limits discussed above the usual results are obtained,
\begin{align}
& \mathrm{\Phi = n_+ \bar{v}/4;\hspace{0.1in} \bar{v} \gg u.} \notag\\
\end{align}
\begin{align}
& \mathrm{\Phi = n_+ u/4; \hspace{0.1in}u \gg \bar{v}.}
\end{align}
\end{subequations}
In the last result the flow is onto a surface area $\pi $R$_{\mathrm{s}}^{2}$ 
but averaging is over 4$\pi $R$_{\mathrm{s}}^{2}$, where R$_{\mathrm{s}}$ 
is the object radius.

As the plasmas may be nonisotropic (Lanzerotti et al. 1981; Richardson and
Sittler 1989), the pitch angle distribution can complicate the bombardment
pattern ($\alpha^{\prime}$ $\equiv $ pitch angle; angle the ions' motion
makes with the field line). In addition, the gyromotion is not well
represented by a Maxwellian flux so that for large gyroradius, r$_{\mathrm{g}}
\gtrsim$ R$_{\mathrm{s}}$, but high temperatures (\={v}/u > r$_{\mathrm{g}}$/ 
R$_{\mathrm{s}}$) the effective collection area of the
object can increase considerably. Therefore, particle tracking is required
to estimate $\Phi $($\theta $) for a large object (Pospieszalska and Johnson
1989). For field lines perpendicular to the flow, the case at the icy
Saturnian satellites, using a power of sin$\alpha ^{\prime }$ for the pitch
angle distribution for the \textit{hot} oxygen plasma at Dione, the
effective flux is about \textit{twice} that in Eq. (4.3b) due to the
gyromotion. The particle bombardment pattern is given in Fig. 4.2a for the 
\textit{cold} heavy ion plasma at Dione. Both pitch angle distribution and
temperature (i.e., gyroradius) are important for particles reaching the
leading (downstream) hemisphere. A calculation for the S$^{+}$ in the plasma
at Europa is given in Fig. 4.2b, to be discussed shortly.

\subsection{\textit{Ionization Effects}}

We now consider a stationary object imbedded in a plasma, and we continue to
ignore the magnetic field effects. In this case the bombardment is isotropic
and an ionosphere (or ionized region) will be produced. Further, the object
will attain a net charge. As the recombination rate in the ionized region
depends on the square of the charge density, the equilibrium number density
of ionized species, n$_{+}$, is determined from\smallskip

\begin{equation}
\mathrm{ \partial n_+ / \partial t = F_e(z)\Phi/W_e - \alpha_r n_{-} n_{+} 
- \nabla \cdot \vec{j} = 0,}
\end{equation}
where $\Phi $ is the flux of plasma particles with electronic energy loss 
rate F$_{\mathrm{e}}$ at the depth z, $\alpha _{\mathrm{r}}$ is the 
electron-ion recombination coefficient and j is the local current density 
(charge flux). The last term in Eq. (4.4) can be very complicated. For example, 
in a gas, diffusion of ions can occur as well as electron transport, as 
discussed in the many texts on ionospheric processes (Chamberlain and Hunten 
1987). In the following, we assume charge equilibration has occurred, which 
implies j = 0 for the case of isotropic bombardment and no diffusion. Assuming, 
also, a rough charge neutrality (n$_{\_}$ $\approx $ n$_{+}$ ), the fractional
ionization of the irradiated material versus penetration depth is 

\begin{equation}
\mathrm{ n_+ / n = [F_e(z)\Phi / n^2 \alpha_r W_e]^{1/2},}
\end{equation}
where n is the neutral number density. Such results have been applied to 
describe ionospheres on satellites produced by energetic electrons (Kumar 1985; 
McElroy and Yung 1975). As F$_{\mathrm{e}}$ is proportional to nS$_{\mathrm{e}}$, 
where S$_{\mathrm{e}}$ is the electronic stopping cross section, the fraction of 
the material in an ionized state depends inversely on n$^{1/2}$. Therefore, 
although the number of ionizations per incident particle is the same for a 
given flux, the average fractional ionization at any time in a solid is much 
less than in a low density gas of the same material.

For isotropic bombardment with electrons and ions at the same temperature,
an object will generally attain a net charge. The amount of charge depends
on its size, on the plasma temperature, and on the flux of UV photons. The
net voltage attained is calculated by considering the relative bombardment
rates of ions and electrons, and the secondary electron ejection rate due to
the incident electrons, ions, and photons. Draine and Salpeter (1979a) have
reviewed these processes for astrophysical environments and Hill and Mendis
(1979) and Morfill et al. (1980a, b) have considered them for planetary
magnetospheres in which charging affects the motion of the grains (Griin et
al. 1984).

When UV photons are the dominant charging process, the grains will charge to
a \textit{positive} potential ($\sim $ 5---10 V) sufficient to suppress
further secondary electron loss. The voltage dependence of the flux of 
\textit{ejected} electrons (Draine 1978), for a number of metals, is

\begin{equation} 
\mathrm{ \Phi_e^{\prime}} = 
\begin{cases}
    [1, \quad \mathrm{ U_g \le 0}] \\  
    \mathrm{ \Phi_0^{\prime} [(l - U_g / 5.6eV)^3, \quad 0 < U_g < 5.6eV]} \\ 
    [0, \quad \mathrm{ U_g < 5.6eV} ]
\end{cases}
\end{equation}
where U$_{\mathrm{g}}$ is the potential at the grain surface in electron volts. 
For a very small grain this should be multiplied by $\sim $ a/(a + 0.01 $\mu$m), 
where a is the grain radius. The value for $\Phi_0^{\prime}$ depends on the
photon flux and the material. Draine (1978) gives $\Phi_0^{\prime}$ $\sim$ 
2.4 x 10$^{6}$ elec/(cm$^{2}\cdot $s) for the galactic UV back ground on 
metallic solids. In interplanetary space, Wyatt (1969) uses a slightly different 
but equivalent functional form with $\Phi_0^{\prime}$ $\sim$ 2.5 x 10$^{10}$ 
(1 AU/R$_{\mathrm{os}}$)$^{2}$ elec/(cm$^{2}\cdot $s) for metals where 
R$_{\mathrm{os}}$ is the distance from the sun in AU, and Grun et al. (1984) 
assume about one-tenth of this of dielectrics.

In a plasma, the flux of electrons and ions on-to the surface of a grain
with u zero is affected by U$_{\mathrm{g}}$ (Appendix 4B) 

$$
\mathrm{ \Phi_e = (v_e n_e / 4)(\xi_e - \delta_e^e g(- U_g/kT_e)} 
$$

\begin{equation}
\mathrm{ \Phi_j = (v_i n_i/4)(\xi_i + \delta_e^i - \delta_i^i)g(- U_g/kT_i),}
\end{equation}
where \\
g(y) = exp (y),\hspace{0.1in} y $<$ 0 \\
\quad\quad\quad = 1 + y,\hspace{0.1in} y $>$ 0. \\
In these expressions the $\xi $'s are the sticking probabilities ( $\sim $
1); the $\delta ^{\prime }$s are the loss of either secondary electrons or
ions due to the incident particles; and n, T, and v are the respective
number densities, temperatures, and mean velocities of the plasma particles.
For low temperature plasmas the $\delta ^{\prime }$s are zero; therefore,
when the photo-electron ejection is small \textit{and} T$_{\mathrm{e}}$ $%
\approx $ T$_{\mathrm{i}}$, charge equilibration implies that U$_{\mathrm{g}%
} $ $\approx $ - 2.5 kT for protons and U$_{\mathrm{g}}$ $\approx $ - 3.6 kT
for an oxygen ion plasma. This charging \textit{reduces} the net electron
flux by factors of 0.08 and 0.03 respectively while \textit{increasing} the
ion flux by factors of 3.5 and 4.6 respectively, thereby achieving
equilibrium.

At higher plasma temperatures (kT > 20eV) the grains are
generally positive because of secondary electron ejection primarily by the
incident electrons. Any UV photon flux would add to this. In general, the
charging achieved is not large. However, if a highly charged grain is
produced, the above considerations will be limited by electron or ion field
emission. Ion emission can produce rapid destruction of very small grains,
as the electrostatic forces exceed the atomic bonding and/or tensile
strengths of the material. The maximum field strengths against ion field
emission, based on Muller and Tsong (1969), are of the order of $\sim$ 1 to
3 x 10$^4$ V/$\mathrm{\mu}$m for fluffy aggregates, $\sim$ 10$^2$ to
10$^3$ V/$\mu$m for ices and silicates, and $\sim$ 10$^4$ V/$\mu$m
for metals. From the earlier discussion of charging, such effects tend to be
important primarily for small, submicron particles.

After this cursory treatment of the bombardment of large and small objects,
we presume in the following that the flux onto the surface is known. As can
be seen from the above, this is a significant assumption.

\section{Sputter Erosion}

Consider the bombardment of a surface in a plasma with an ion temperature T$%
_{+}$ and density n$_{+}$ but flow velocity u = 0, and ignore the
gyroradius. The flux of ions impinging on the surface from direction $\theta 
$ to the normal is [(n$_{+}$ \={v}/2) cos$\theta $ d cos$\theta $ ] giving a
net flux, $\Phi $ = (n$_{+}$ \={v}/4) with \={v} = (8kT$_{+}$/$\pi 
$M$_{\mathrm{A}}$)$^{1/2}$ as obtained from Eq. (4.3b). Using a sputter yield Y 
($\theta $) = Y/(cos $\theta $)$^{1.6}$ for $\theta < \theta_
{\mathrm{c}}$, where Y is the yield for normal incidence and Y($\theta $) 
$\approx $ 0 otherwise (Chap. 3), the averaged yield $\langle $Y$\rangle $
from Eq. (4.1) is defined 

\begin{subequations}
\begin{align}
& \mathrm{ \langle Y\Phi \rangle \equiv \langle Y\rangle \Phi.} \notag\\
\end{align}
For isotropic bombardment this gives 

\begin{align}
& \mathrm{ \langle Y\rangle = 5[1 - (cos\theta_c)^{0.4}]Y.}
\end{align}
\end{subequations}
The \textit{same} average yield is also obtained for an unidirectional flux 
of particles incident on a hemi-spherical surface [see. Eq. (4.3)]. Although 
values of $\theta_{\mathrm{c}}$ differ between experiments due to sample 
preparation, typical values of $\theta _{\mathrm{c}}$ for keV ions are of the 
order of 80$^{\circ}$ (cos $\theta_{\mathrm{c}}$ $\sim$ 0.17) for relatively
smooth surfaces. Hence, quite large enhancements in the averaged yield over
the yield at normal incidence might occur.

Astrophysical surfaces, however, are \textit{not} very smooth on a
microscopic scale. Although sputtering for isotropic bombardment can smooth
surfaces that are macroscopically rough (Carey and McDonnell 1976), it can
produce very irregular surfaces on a microscopic scale (Johnson et al.
1985). On a rough surface, molecules that are sputtered from one feature can
strike and stick to a neighbor feature (Clark et al. 1983), and, in
addition, surface features can shadow each other from the incident particles
(Carey and McDonnell 1976). Both of these effects tend to suppress the
angular enhancement in the yield. On a planetary regolith composed of grains
(Hapke 1986) this occurs along with sputter aging (McDonnell 1977).
Therefore, Sieveka and Johnson (1982) ignored the enhancements discussed
above and use $\langle $Y$\rangle $ $\approx $ cY with c $\approx $1 as a
rough estimate of the erosion yield. For a regolith of large, spherical
grains or microcraters 1 $\gtrsim$ c $\gtrsim$ 0.3 (Johnson 1989b) with
the larger value for the larger yields. Using the plasma fluxes in the
introduction and assuming $\langle $Y$\rangle $ $\approx $ Y, we give in
Table 4.3 condensed H$_{2}$O erosion rates on objects in various
astrophysical environments. The major uncertainty in these estimates is the
plasma composition.

The erosion rates in Table 4.3 are significant for very small objects. For
instance, E-ring particles in the Saturnian magnetosphere are effectively
eroded in the order of 10$^{3}$ to 10$^{4}$ years (Haff et al. 1983; Hill
1984; Johnson et al. 1984a). This puts a rather definite constraint on their
lifetimes and implies that there exists a source of material to create these
particles or that they resulted from an event $\sim $ 10$^{4}$ years ago.
For the main rings of Saturn micrometeorite erosion followed by plasma
sweeping dominates (Northrup and Connerney 1987; Ip 1986b). For surfaces
composed of grains containing ices \textit{and} a non-volatile material,
plasma erosion can produce a sputter-resistant residue (Chap. 3). Finally,
the erosion rates in Table 4.3 for the satellites are not large enough to
affect major geologic features as originally suggested (Lanzerotti et al.
1978, 1983). However, the material supply rate to the magnetosphere is
significant, and, in the absence of other geologic processes with relatively
short time scales, the rates are large enough to modify the spectral
reflectance properties of these surfaces. These topics will be addressed
shortly.

For very small grains for which the ion penetration depth is comparable to
or larger than the size of the grain, an ion can eject atoms through the
side and back surfaces, as shown in Fig. 4.3a. Those ions penetrating the
grain sputter both at the entrance surface (back sputterin: enhancing the
erosion rate. In addit greater than the net cohesive energy This occurs
for\smallskip

$$
\mathrm{ D_g < (6S / \pi U)^{1/2}; \ \ \ \ \ nSD_g \ll E }
$$

\begin{equation}
\mathrm{ D_g < (6E / \pi nU)^{1/3}; \ \ \ \  \bar{R}_p(E) < D_g,}
\end{equation}
where D$_{\mathrm{g}}$ is the grain diameter, U is the cohesive energy per 
molecule (roughly the sublimation energy), S is the total stopping power, 
and we neglect the radiative cooling of the grain, a much slower process. 
The total stopping power, including the electronic energy deposited, should 
be used even for metallic grains. Using Eq. (4.9), a 1 MeV proton fully 
erodes water ice grains (S $\sim $ 10$^{-14}$ eV cm$^2$/ molecule U $\sim$ 
0.5 eV/molecule) of radii < 2 $\times$ 10$^{-13}$ $\mu $m. An
energetic O$^{+}$ in the vicinity of the Saturanian E-ring can completely
erode grains that are < 10$^{-2}$ $\mu$m, affecting the very
small particles in the ring. Note: fluffy hrains may be made up of smaller
grains in which case D$_{\mathrm{g}}$ applies to the submit.

\section{Changes in Surface Albedos}

With the exception of the returned lunar rock, soil, and core samples, and
collected meteorites and cosmic dust, data on planetary surfaces are
obtained from ground-based telescopes and spacecraft by remote sensing.
Therefore, before considering the changes induced in the surfaces by ions,
we first consider the nature of the reflectance spectra.

When incident on an object, photons are scattered and absorbed by the
material in the surface, with a fraction eventually backscattered
(reflected). This reflectance, averaged over the surface, is often given as
a normalized albedo, defined as the fraction of the photons scattered
backward along the line of sight for a given illumination angle relative to
the line of sight (Hapke 1981). As the surfaces of objects are
agglomerations of grains or have high densities of defects causing light to
scatter efficiently, the penetration depth of the photons is affected. If
the average grain diameter, D$_{\mathrm{g}}$, is very small compared to the
photon wavelength, then the scattering is roughly isotropic, as in the long
wavelength limit for ion scattering Eq. (2A.11). When the grains are large
compared to the wavelength (see Chap. 2), geometric optics can be applied.
Photons are refracted on entering a grain, internal reflections occur, and
the photon again is refracted on exiting. This is described by a ``single
particle'' (grain) albedo, w, and expressions are given by Hapke (1981)
(Clark and Roush 1984; Burratti 1985). The effect of scattering is to limit
the amount of the surface ``sampled'' by the light. Therefore, small grain
surfaces are brighter because of smaller penetration depths and path lengths
in the sample, hence, less absorption. Although the photon transport
equations, like those for the ion (Appendix 3A), are complex simple
approximations are useful.

For normal incidence, the normal albedo, A$_{\mathrm{N}}$, and the
laboratory reflectance at zero degrees to the normal are equivalent. 
A$_{\mathrm{N}}$ is often given in terms of w, but for a given laboratory
absorption length, $\alpha _{\mathrm{a}}^{-1}$,\smallskip

$$
\mathrm{ A_N \approx exp(- \alpha_a L),}
$$
where L is defined to be the mean path length in the sample. This merely 
changes the problem from knowing w to determining L. However, on bright 
surfaces, [weak absorption and/or small grain size (1 $\gg$ $\alpha_{\mathrm{a}}$
D$_{\mathrm{g}}$)] Hapke (1981) writes w $\approx $ (1 + $\alpha_{\mathrm{a}}$
D$_{\mathrm{g}}$)$^{-1}$ and L $\approx$ 4(2D$_{\mathrm{g}}$ /$\alpha_{\mathrm{a}}$)
$^{1/2}$, giving 

\begin{equation}
\mathrm{ A_N \approx exp [-4(2\alpha_a D_g)^{1/2}].} 
\end{equation}
This explicitly indicates that the reflectance depends both on the physical
state, at the $\mu $m level, of the surface via D$_{\mathrm{g}}$ and the
absorption properties via $\alpha _{\mathrm{a}}$. For instance, weakly
absorbing species in very large grains drastically limit the amount of
reflected light. Therefore, Nash and Fanale (1977) considered irradiation
effects on reflectance of possible materials on Io, an extension of earlier
ideas about lunar soils (Hapke and Cassidy 1978; Nash 1967). On Europa, ion
implantation (Eviatar et al. 1983; O'Shaughnessy et al. 1988a) or
preferential sputtering [see Eq. (3.22)] may also modify surface
reflectance. Changes in the UV (0.34 $\mu $m) absorption coefficient
relative to that at 0.59 $\mu $m (OR), extracted from the Voyager filter
data (Johnson et al. 1988), vs. angle from the apex of the trailing
hemisphere are seen in Fig. 4.2b to closely correlate with the S$^{+}$
plasma bombardment profile (Nelson et al. 1986; Pospieszalska and Johnson
1989), suggesting such alterations occur. These take place in competition
with other processes (Nelson et al. 1987) and any burial processes (Johnson
and Matson 1988; Crawford and Stevenson 1988; Eviatar et al. 1985; Squyres
et al. 1983).

\section{Alteration of Surfaces}

Ion bombardment alters the reflectance of a surface by producing roughness.
Fresh, low-temperature deposits of water molecules from the gas phase are
generally observed to be translucent and nonreflecting unless they are
deposited rapidly in a turbulent stream. The relatively clear deposits
become visually opaque, light-scattering surfaces after ion bombardment
(Johnson et al. 1985; Strazzulla et al. 1988) and evidence of dendritic
crystal growth (Brown et al. 1978) is seen. This indicates that the surface
has become rough on a microscopic scale ($\sim $ 0.1 $\mu $m scale).

Clark et al. (1983) have discussed the competition between grain growth due
to sublimation and preferential destruction of small grains due to ion
bombardment (Fig. 4.3b). They pointed out that the observed depths of the
1.04 $\mu $m water ice band of the reflected light from Europa, Ganymede,
and Callisto are consistent with increasing sublimation rates and decreasing
ion bombardment rates (Clark et al. 1984). They also noted definite
leading/trailing differences in the band depth consistent with enhanced
bombardment on the trailing side on both the icy Galilean and Saturnian
satellites. In addition there are very clear latitudinal changes in the
visual reflectance spectrum on Ganymede. The polar surfaces are observed to
scatter light more efficiently than the surface near the equator. This is
attributed to a thin (few mm) layer of fluffy material at the polar caps. As
the sublimation rate is sufficient to reprocess all freshly deposited
material near the equator but not in the polar regions, these polar regions
may be characteristic of an ion bombarded (sputtered), hence efficiently
light-scattering surface (Johnson 1985a), although other scenorios have been
suggested.

Very long-term bombardment of films of water ice observed in a transmission
electron microscope shows evidence of the production of micron-sized
features (Johnson et al. 1985). These features are produced by the
coalescence of defects forming voids which eventually are exposed and eroded
at the surface. The presence of such features will also change the
scattering properties of the surface, producing a phase angle dependence
which is very different from a snowy surface. Veverka and Gradie (1983) have
pointed out that the surface of Enceladus, for example, does not reflect
like a Lambert surface. The observed phase angle dependence could be
explained by the existence of ion-produced, micron-sized holes, as well as
by the presence of impurities in the icy surface. Recently, such changes
have been shown to occur dramatically when methane is irradiated by
penetrating ions (Strazzulla et al. 1988), as discussed in Chapter 3. On the
other hand, Nash (1987) showed that vacuum weathering (loss of volatiles by
sublimation from a mixed medium) can produce a very porous surface.

Sputtering preferentially removes the more volatile species from a surface,
as discussed in the previous chapter. Therefore, an object having a mixture
of ices and less volatile species subjected to ion bombardment can, in
principle, form a surface dominated by the less volatile species (Haff et
al. 1979). If this happens, a bright, icy surface containing a ``dark''
impurity can become visibly darkened, as indicated in Fig. 4.4a,b. However,
if a small percentage of an impurity is mixed at a \textit{molecular} level
into a volatile material having a very high sputtering yield, the impurity
molecules may be carried off at the \textit{same} rate as the volatile
substance (Y$_{1}$ = Y$_{2}$ below) and the surface concentration will not
change. On a large object the formation of a darkened surface,
sputter-resistant also depends on whether the sputtered volatile is actually
lost to space or gravitationally return to the surface. If Y$_{1}$ $\neq$ 
Y$_2$ and the ``bright'' volatile material does return to the surface, then
the ``dark'' impurity may be buried by the redeposited volatile material, as
indicated in Fig. 4.4c (Sieveka and Johnson 1982; Spencer 1987a, b). Such
processes may be occurring on Pluto (Stern et al. 1988).

With the above qualifications, we showed in Chapter 3 that the rate of loss
of the two species comprising a surface is proportional to the concentration
of each (c$_{1}$, c$_{2}$) and to their sputter yields (Y$_{1}$ and Y$_{2}$). 
For \textit{independent} sputtering of each species [Eq. (3.22)], the
effective yield is Y$_{1}^{\prime }$ = c$_{\mathrm{i}}$Y$_{\mathrm{i}}$,
giving a net yield Y = c$_{1}$Y$_{1}$ + c$_{2}$Y$_{2}$ with 1 = c$_{1}$ + 
c$_{2}$. Therefore, the rate of change of the concentration of a single
species can be written 

\begin{subequations}
\begin{align}
& \mathrm{ d(c_1 N_s) / dt = -\Phi (c_1 Y_1) + \Phi (c_{1}^{0}Y). } \notag\\
\end{align}
The first term on the right in Eq. (4.11a) is the loss of species one from 
the sputter layer, ignoring redeposition, the second term is the addition of 
fresh material from under the sputter layer of depth N$_{\mathrm{s}}$, as
indicated in Fig. 4.4b, and c$_{1}^{0}$, c${}_{{2}}^{{0}}$ are the initial
concentrations. The equilibrium concentrations become 

\begin{align}
& \mathrm{ (c_1/c_2) = (c_1^0 Y_2/c_2^0 Y_1); \ \ \ c_1 + c_2 = c_1^0 + c_2^0 = 1.}
\end{align}
\end{subequations}
This equilibrium is produced in a time of order of 

\begin{subequations}
\begin{align}
& \mathrm{ t_eq \approx (N_s /\Phi)/(c_2^0 Y_1 / c_1^0 Y_2).} \notag\\
\end{align}
As material is removed from the surface and the ions penetrate into fresh 
material, the relative loss of the species must eventually become equal to 
the \textit{initial} concentration ratio in the fresh material [Eq. (3.22d)]. 
The corresponding equilibrium (sputter-aged) yield becomes 

\begin{align}
& \mathrm{ Y = Y_1 Y_2 / (c_2^0 Y_1 / c_1^0 Y_2). }
\end{align}
\end{subequations}
Therefore, for a volatile material containing a dark impurity which has a 
negligible sputtering rate (e.g., Y$_2$ $\rightarrow$ 0), the surface will 
be dominated by the dark material [c$_1$ $\rightarrow 0,$Y$\rightarrow$ 
(Y$_{2}$/c$_{2}^{0}$)] in a time determined by the sputtering rate of the 
volatile species (t$_{\mathrm{eq}}$ $\propto $ Y$_{1}$). Whether or not 
this ``darkening'' is important for remote sensing of the objects depends 
on the thickness of the modified layer, N$_{\mathrm{s}}$, compared to photon 
penetration depths.

In the sputtering of a grain containing a mixture of water and ammonia ice
by 1 MeV proton bombardment the yields are Y$_{\mathrm{H}_{2}\mathrm{O}}$ =
0.2 and Y$_{\mathrm{NH}_{3}}$ = 10. If the mixture is initially 50/50, the
ratio of ammonia to water molecules, based on the above, becomes $\sim $
0.02 in the surface region ($\sim $ 50 $\mu $m, Fig. 3.3a) on an object like
Enceladus for which the escape fraction is large (Lanzerotti et al. 1984).
Although sputter ejection occurs from the first monolayer, the ``surface''
can extend to the depth of penetration of the ion, \={N}$_{\mathrm{p}}$,
becomes the bombardment produces mixing and enhances diffusion of species
along the ion tracks, as discussed in Chapter 3.

\section{Formation of New Molecules in the Surface}

Ion bombardment can in principle make very dramatic changes in the surface
albedos and reflectance spectra of astrophysical objects by chemical
alterations. The incident ions and electrons break bonds forming a surface
with new molecular species, both more volatile and less volatile species. On
such a surface the volatile molecular species (e.g., H$_{2}$, O$_{2}$, N$%
_{2} $, etc.) are eroded preferentially leaving a permanently altered
surface. Therefore, as discussed in Chapter 3, continuous loss of H$_{2}$
from CH$_{4} $ will result in a dark surface enhanced in carbon. On the
other hand, the surface region of water remains predominantly H$_{2}$O, with
an equilibrium mix of radicals and some trapped O$_{2}$ and H$_{2}$.
Similarly, NH$_{3}$ produces both H$_{2}$ and N$_{2}$ as well as hydrazine H$%
_{2}$N-NH$_{2}$. The production of the more volatile species strengthens the
arguments given above for loss of NH$_{3}$ from ice grains or small bodies
like Enceladus. A flux of 30 keV protons or oxygen ions bombarding the
surface of Enceladus will produce a depletion of ammonia to depths $\sim $10$%
^{-2}$ $\mu $m, which are too shallow to affect the reflectance spectra as
the grain sizes are of the order of $\mu $'s. However, from Fig. 3.3a, it is
seen that 100 keV protons and 500 keV O$^{+}$ ions penetrate to more
relevant depths, $\sim $ 5 $\mu $m, although at much lower fluxes. For
example, a 5 $\mu $m layer is depleted in $\sim $3 x 10$^{3}$ years, a time
of the order of the lifetime of the E-ring grains. Therefore if the E-ring
was produced by a meteor impact on Enceladus $\sim $ 10$^{4}$ years ago, the
``fresh'' surface of Enceladus would be depleted in NH$_{3}$ as observed. In
mixed ices other new species form. For example, H$_{2}$O and CO (or CO$_{2}$%
) lead to formaldehyde (Brown et al. 1982b; Pirronello et al. 1982), and
induced polymers of such materials may have been seen in the comet effluent
(Mitchell et al. 1987; Heubner 1987). Ejection of polymers of such materials
is also feasible (Moore and Tanabe 1989).

More dramatic changes in the albedo may occur because of irradiation. Lane
et al. (1981) observed an absorption band on the trailing hemisphere of
Europa at $\sim $ 0.26 $\mu $m which they attributed to SO. O'Shaughnessy et
al. (1988a) observed an increased absorption due to implanted sulfur at 0.4 $%
\mu $m, which they could not correlate with the leading/trailing differences
in absorption, leaving this question open for the time. A sputter-induced
absorption feature in pure ice which only occurred at low temperatures was
found also (Fig. 4.5).

The polar caps of Io are noticeably darker than the equatorial regions
(McEwen et al. 1988). In the laboratory, freshly deposited (from the gas
phase) sulfur, S$_{8}$, becomes darkened under ion (Chrisey et al. 1987) and
UV photon (Steudel 1982; Steudel et al. 1986) irradiation due at least in
part to the production of S$_{2}$ fragments in the surface. In addition, on
warming, the surface again becomes bright because of annealing. It is,
therefore, possible that Io's poles are darkened by ion bombardment, whereas
the equatorial regions are, on the average, brighter due to both annealing
and the redeposition of sulfur and SO$_{2}$ in a band extending to about 30${%
{}^{\circ }}$ latitude north and south. Recently, condensed H$_{2}$S has
been observed on Io (Nash and Howell 1989; Salama et al. 1990). This may be
intrinsic or produced by proton implantation (Nash and Fanale 1977) and may
contribute to the darkening at the poles (Matson et al. 1988).

Organic materials suffer even more dramatic effects because of ion
bombardment changing from bright, clear surfaces to dark, sputter-resistant
surfaces under long-term ($\sim $ 10$^{17}$ protons/cm$^{2}$) irradiation
(Chap. 3). Such doses are large but are easily achievable when describing
geologically stable surfaces, e.g., a grain in the interstellar (Wdowiak et
al. 1988, 1989) or interplanetary medium and the outer layer of a comet
nucleus in the Oort cloud. Therefore, this process has been proposed for
conversion of loosely bound aggregates of grains (Greenberg 1982) into a
cohesive aggregate of refractory material as in Fig. 4.6 (Johnson and
Lanzerotti 1986) which is similar to the Brownlee particles (Bradley et al.
1984). This process can also lead to the initiation of a dark crust on a
comet (Donn 1976; Strazzulla 1985; Johnson et al. 1987) assuming the nucleus
is composed of grains formed in the interstellar medium and precipitated in
the outer solar system. Such grains are thought to have a significant
organic component ($\sim $ 40\% by mass) and silicates or carbon cores ($%
\sim $10\% by mass) Kissel and Krueger 1987), so that long-term cosmic ray
or solar energetic particle irradiation segregates the condensed volatiles
(e.g., CO, H$_{2}$O) and newly produced volatile species (H$_{2}$ and O$_{2}$%
) from the carbonized residue produced. In Fig. 4.7a is shown the estimated
cosmic ray dose, D$_{\mathrm{M}}$, vs. depth into an object such as a comet
(Strazzulla and Johnson 1990) or Pluto (Johnson 1989d). On warming the
irradiated mix, the residual volatiles are lost and the residual radicals
recombine, leaving behind a black porous residue. Hence a comet entering the
inner solar system may have an outer crust ($\sim $100g/cm$^{2}$) of
carbonized and silicate materials in which adhesion has also been enhanced
by the irradiation. As this process occurs on an irregular object, there
will be active regions in which this crust did not fully develop or was
unstable and lost on first entry into the inner solar system. The dark crust
with active regions is consistent with the picture of a comet nucleus seen
in the Halley missions, and differences in the amount of ortho to para-water
for new comets have been attributed to an irradiation effect (Mumma et al.
1989). In describing the production of the modified layer the role of other
surface alterations needs to be considered (Stern and Shull 1988; Stern
1988, 1989).

The outer solar system is replete with dark, probably organic materials.
This is suggestive of radiation processing of the materials comprising these
objects. This irradiation could have occurred in situ, after formation, or
it could have occurred during an active, T-Tauri phase in the early solar
system (Strazzulla et al. 1985), or may have occurred in the interstellar
cloud which was the precursor of the solar system (d'Hendecourt et al.
1986). Possibly the black ring particles at Uranus are materials which were
processed in the Uranian magnetosphere by energetic ions and electrons
(Cheng and Lanzerotti 1978; Calcagno et al. 1985a). As the times for
darkening of any organic containing surface materials on the Uranian
satellite are short compared to geologic processes, Fig. 4.7b, in situ
darkening can occur (Lanzerotti et al. 1987b). If the carbon component is
predominantly in CO (T.V. Johnson et al. 1987; Veverka et al. 1987), then
the irradiation `darkening' effects are slower, and carbon sub-oxide
(Chrisey et al. 1989), formaldehyde (Pirronello 1985) or carbonates (Moore
and Khanna 1990) may be products. Although models have also been proposed
for radiation darkening of Iapetus (Strazzulla 1986), the possibility of
`dark' \textit{meteoritic} materials impacting icy surfaces from Callisto to
Umbriel is high (Bell et al. 1985). Indeed, in addition to the preferential
trailing hemisphere absorbing component at Europa (see Fig. 4.2b) there is a
more uniformly distributed unidentified component (McEwen 1986; Nelson et
al. 1986; Brown et al. 1986) for which the absorption coefficient in the
visible has been extracted (Johnson et al. 1988) and is unrelated to
irradiation. There is, however, an apparent trend found in the asteroid belt
of objects devoid of organics, objects containing highly processed
carbonized materials, and finally redder organic materials (Bell et al.
1985; Andronico et al. 1987). This is a trend that may be correlated with
the exposure of fresh organic material by other surface weathering processes
followed by solar wind ion processing.

\section{Atmospheric Alterations}

Photo-absorption in a planetary atmosphere results in the formation of new
species, an ionosphere, and a thermosphere, and also leads to escape. These
same effects are initiated by charge particle bombardment of an atmosphere.
In the latter case the effects are often given specific names, e.g., the
aurora, as the magnetic fields may control the distribution of the
bombarding species. In the outer solar system Voyager data have shown that
particle radiation plays an important role in affecting the nature of the
atmospheric envelope of a number of objects, a role which parallels that of
photon absorption. The atmospheric heating and escape produced by impacting
charged particles are treated latter. Here, only a very brief discussion of
the chemical alterations is given as they closely parallel photo-effects
discussed above and extensively discussed elsewhere (see Bibliography).

To determine the rate of alteration of an atmospheric gas the quantity F$_{%
\mathrm{e}}$(or Se) divided by W$_{\mathrm{e}}$ (Chap. 3), roughly replaces
the photo-absorption cross section, as indicated in Eq. (4.4). A principal
problem, except for the very energetic cosmic-ray particles, is the
calculation of the particle paths which are needed for determining F$_{%
\mathrm{e}}$(z). These paths are affected by the magnetic and conducting
properties of the object, in addition to the energy degradation processes
considered in Chapter 3. Beyond this, the energy deposited in ionization or
excitation leads to alterations like those occurring in surfaces. R\"{o}%
ssler and co-workers (R\"{o}ssler 1986) have examined the differences
between solids and liquids, and from the gas-phase G-values in Table 3.3 it
is seen that the G-values depend on the phase. In an atmosphere, in
addition, the reactions occur in competition with transport and mixing
processes.

Whereas the ionosphere on Io (Kumar 1985) and, more recently, Triton may be
produced by energetic electrons [see Eq. (4.5)], of considerable recent
interest is the chemistry induced in the atmospheres of Triton (Thompson et
al. 1989), Titan (Sagan et al. 1984), and Pluto (Stern et al. 1988). In
these atmospheres methane, in the presence of N$_{2}$ and other species, is
a precursor of a number of carbonaceous species. These species, produced by
the radiation, may exist as large molecules or aerosols which in turn may
produce a haze that is eventually deposited on the surface. This
irradiation-induced precipitation occurs in competition with regolith mixing
processes (Stern et al. 1988) and can produce a change in the albedo.
Because the irradiation of organics tends chemically toward heavier species,
to first order the precipitation rate is roughly equal to the energy
deposition rate times the gas phase G-value (Khare et al. 1984). Although
the products can form a carbonaceous precipitate, care must be taken as to
the molecular environment in which the irradiation occurs in trying to
determine the chemical character of the end products. The availability of
ion-molecule reaction rates (Atreya 1986) allows the possibility of
calculating the molecular composition under a given level of particle
bombardment by solving the rate equations rather than simply using G-values
(Kumar 1985).

\section{Sputtering vs. Sublimation}

In Fig. 1.9 the characteristic temperatures in the solar system were given.
The solid line, the black-body radiation equilibrium temperature on a
rapidly rotating object with an albedo of zero, is calculated by setting the
solar energy absorbed by the disk of the object, (1 - A)$\pi $R$_{\mathrm{o}%
}^{2}$ $\Phi _{0}$/R$_{\mathrm{os}}^{2}$, equal to the black-body emission
from the full surface, $\varepsilon $4$\pi $R$_{\mathrm{o}}^{2}$ $\sigma _{%
\mathrm{b}}$T$^{4}$. Here $\sigma _{\mathrm{b}}$ is the Stefan-Boltzmann
constant, A, $\varepsilon $, and R$_{\mathrm{o}}$ are the visible albedo,
the IR emissivity and the radius of the object and $\Phi _{0}$/R$_{\mathrm{os%
}}^{2}$ is the solar flux at a distance R$_{\mathrm{os}}$ from the object to
the sun in AU. ($\Phi _{0}$ = 1.38 kW/m$^{2}$.) The resulting temperature
is 

\begin{equation}
\mathrm{ T = (279K)[(1 -A) / \varepsilon]^{1/4}/R_{os}^{1/2}.}
\end{equation}

For the results in Figure 1.9, A was set equal to zero and $\varepsilon $ to
one. The effective temperatures of the planets using measured albedos in Eq.
(4.13) are shown as the effective average temperatures of objects composed
of H$_{2}$O, NH$_{3}$, or CH$_{4}$ ice. At distances close to the sun, the
melting temperatures are controlled by the sublimation rates, which increase
rapidly with temperature, and the objects lose mass rapidly.

Not only are the objects in solar systems heated by photons, they may be
bombarded by solar ions and micrometeoroids. The sublimation flux from an
icy surface can be written as P(T)/(2$\pi $ MkT)$^{1/2}$, where P(T) is the
equilibrium vapor pressure of the material at temperature T and M is the
mass of sublimated molecule. Writing P(T) = P$_{\mathrm{o}}$(T) exp ( -
U/kT), where P${}_{\mathrm{o}}$(T) is a slowly varying function of T, then
at those distances at which radiation loss dominates, the sublimation rate
decreases as R$_{\mathrm{os}}^{1/4}$ exp ( - R$_{\mathrm{os}}^{1/2}$ U/0.024
eV). On the other hand, sputter erosion is directly proportional to the
solar ion flux, which decreases as 1/R$_{\mathrm{os}}^{2}$. (Recently, it
has been found that the decay in the flux of the energetic ions may be much
slower). Therefore, the sputter erosion rate for icy objects must eventually
exceed thermal erosion (Lanzerotti et al. 1981; Mukai and Schwehm 1981).
Using sputtering yields characteristic of 1 keV/amu protons and alpha
particles, the surface loss rates due to sputtering of objects composed of
either H$_{2}$O or CO$_{2}$ ice are calculated vs. distance from the Sun. It
is seen in Fig. 4.8 that, at those distances for which charged-particle
erosion dominates, the loss rates are relatively small under present solar
wind conditions, however, chemical alterations may be relevant (Pirronello
and Lanzafame 1989). For an icy body in a reasonably circular orbit, such as
Chiron, the amount of matter lost per orbit (Appendix 4C) is only $\sim $10$%
^{-5}$g/cm$^{2}$ for an H$_{2}$O surface or $\sim $ 2 x 10$^{-4}$ g/cm$^{2}$
for a CO$_{2}$ surface at $\sim $ 50 AU. However, in the early history of
the solar system, if the Sun went through a T-Tauri phase in which residual
inner solar system volatiles were driven off, sputtering could have played a
very important role (Strazzulla et al. 1983). Micrometeoroid bombardment is
probably more effective as a darkening agent than as an ejection process on
a large object (Hall and Eviatar (1986), but may be important for eroding
Saturn's main rings (Northrop and Connerney 1987) and has recently been
proposed as a source of Na at Mercury (Morgan et al. 1988).

For objects imbedded in a magnetosphere the sputtering rate may also be
important (Table 4.3). Figure 4.9 compares the total surface erosion rates
due to sublimation and sputtering for water ice on Europa, Ganymede, and
Callisto and SO$_{2}$ on Io. In obtaining the sputter rates both the plasma
and LECP results for Voyager 1 were used. In Fig. 4.9a erosion rates are
calculated under the assumption that the LECP data is predominantly O$^{+}$
or H$^{+}$, with the former more likely at Europa and the latter at
Callisto, whereas H$^{+}$, He$^{+}$, O$^{+}$, and S$^{+}$ are considered in
Fig. 4.9b. It is seen that there are important regions of the satellite
surface for which sputtering dominates sublimation. These rates are
determined ignoring redeposition, which we will discuss shortly.

Finally, in molecular clouds, the background cosmic ray flux can be
important for returning molecular species to the gas phase (Draine 1985;
Leger et al. 1985), for dissociating molecules (Sternberg et al. 1987), or
producing molecular species from H2 (Pirronello and Averna 1988) to large
organic molecules (Sundqvist 1989; de Vries et al. 1984b). In a cloud in
which gas-phase species dominate condensed species, the cosmic-ray energy
loss in the gas will result in predominantly ionizations (e.g., Fig. 4.7a).
Subsequent recombination [e.g., Eq. (3.7)] produces excited species, which
results in a flux of photons that can produce photo-stimulated desorption
and alteration of condensed species.

\section{Sputtering in the Presence of Gravity}

In calculating the loss of material to space due to ion bombardment of large
objects, such as satellites, comets, or planets, three effects must be
included: the possible presence of an intrinsic magnetic field, the
existence of an atmosphere, and the gravitational attraction of the
satellite. Earlier we noted that an intrinsic magnetic field can alter the
flow of ions and electrons about the object changing the amount and
direction of the bombardment. The presence of an atmosphere can also shield
the surface or restrict loss to space of the molecules sputtered from the
surface. Even in the absence of an atmosphere or an intrinsic field,
molecules sputtered from the surface of a satellite having a large mass are
less likely to escape to space. Those sputtered molecules which do not
escape contribute to an atmosphere and become redistributed across the
surface of the satellite.

The schematic diagrams in Fig. 4.10 show the fate of molecules ejected from
a surface of an object exposed to ion or micrometeorite bombardment or
simply sublimed from a surface having a high temperature (Cheng et at. 1986;
Cheng and Johnson 1988). These scenarios were originally described by Matson
et al. (1974) for understanding the observation of free Na at Io (Brown and
Chaffee 1974). In Fig. 4.10a, if the molecule is ejected with a very low
energy relative to the escape energy on an object with little or no
atmosphere, then it returns to the surface on a ballistic trajectory. While
in orbit it contributes to a weak atmospheric corona on the object. More
energetic molecules are ejected to space. In Fig. 4.10b, a molecule is
ejected from the surface of an object having an atmospheric column density
sufficient to inhibit escape. This species becomes equilibrated in the
atmosphere and, if it is nonreactive, it eventually returns to the surface
by atmospheric transport processes. The composition of the gaseous envelope,
therefore, is affected by this process. In Fig. 4.10c, a molecule ejected
with an energy greater than the escape energy enters the plasma region,
where it is eventually ionized by a photon or plasma electron. It becomes
confined by the plasma magnetic field and becomes part of the plasma,
thereby affecting the local plasma composition. In Fig. 4.10d, a molecule
ejected from the surface is ionized and picked up by the local field. If it
is on the up-stream side of the plasma flow it can immediately be driven
back into the surface enhancing the sputtering rate. The ''surface'', as has
been stated previously, can be the physical surface or the exobase of an
atmosphere. Therefore, most of the subsequent discussion applies both to a
planetary surface and a planetary corona, stimulated by photons as well as
ions or electrons.

\subsection{\textit{Gravitational Escape}}

When the surface ejection flux of molecules is relatively low, the ejected
molecules can be treated as traveling in ballistic trajectories
uninterrupted by collisions with other molecules. In Fig. 4.11 we use the
energy distributions for water molecules ejected in the electronic and
collisional sputtering regime (Fig. 3.24) to calculate that fraction of
molecules which are sputtered from a surface of a satellite and which have
energies greater than the minimum escape energy, U$^{\mathrm{s}}$. For a
particle of mass, M, U$^{\mathrm{s}}$ is 

\begin{equation}
\mathrm{ U^s = G(MM_s)/R_s = M_gR_s,}
\end{equation}
where R$_{\mathrm{s}}$ is the satellite radius, M$_{\mathrm{s}}$ is the 
satellite mass, G is the gravitational constant, and g is the acceleration 
of gravity at the satellite's surface. On satellites which are close to a 
present body (e.g., Io with Jupiter), this quantity is reduced by the 
gravitational effect of the parent. One imagines that around the satellite 
there is a region in which the field of the parent body dominates, and the 
ejected molecules can no longer be thought of as attached to the satellite. 
The boundary of this region is referred to as the Hill (or Lagrange) ``sphere''
and is determined by that distance from satellite center, R$_{\mathrm{L}}$,
for which the gravitational force of the parent planet minus the centrifugal
force is equal to the satellite attractive force (Chamberlain and Hunten
1987); R$_{\mathrm{L}}$ $\approx $ R$_{\mathrm{sp}}$ (M$_{\mathrm{s}}$/3 
M$_{\mathrm{p}}$)$^{1/3}$ (Table 1.6). Such a region is not, of course, a
sphere, as different distances apply to the sides facing and opposite to the
parent planet. The ``mean'' values of RL given in Table 1.6 apply at the
satellite orbital radii. The corresponding escape energies we will call 
U$^{\mathrm{L}}$.

Using values for U$^{\mathrm{L}}$ appropriate to the icy Galilean
satellites, Fig. 4.11 shows that the escape fractions determined from
collision cascade energy distributions are only significant for material
binding energies, U, comparable to or greater than U$^{\mathrm{L}}$. In
fact, the escape fraction increases monotonically to unity for 
U$^{\mathrm{L}}$ $\ll$ U. This is somewhat misleading, however, for as 
U becomes larger, the sputter yield, Y, decreases. Multiplying the collisional 
cascade ejection yield in Eq. (3.15) for planar surface binding by the escape
fraction [U (2U$^{\mathrm{L}}$ + U)/(U$^{\mathrm{L}}$ + U)$^{2}$] gives a
net yield for ejection to space for normal incidence

\begin{equation}
\mathrm{ Y_{es} = (3/2\pi^2)(\alpha S_n/\bar{\sigma}_d)(2U^L + U)/(U^L + U)^2.}
\end{equation}
This yield obtains its largest value at U = 0, i.e., only one barrier to 
overcome. When U$^{\mathrm{L}}$ $\gg$ U the surface binding energy, U, is 
replaced by (U$^{\mathrm{L}}$/ 2) in the expression for the sputter yield 
in Eq. (3.15). This is equivalent to treating escape from the ``spherical'' 
gravitational barrier U$^{\mathrm{L}}$ [see Eq. (3.14b)] and it gives an 
upper limit to the yield in the absence of detailed knowledge of the surface 
binding. In fact, as the electronic sputtering yield is small for a surface 
with U $\sim$ 2 eV (Chap. 3), the collisional yield in Eq. (3.15) with 
U $\rightarrow $ U$^{\mathrm{L}}$/2 can be used in order to estimate the total 
yield for sputtering from large satellites (e.g., Ganymede) or a planet 
(e.g., Mercury) with appropriate corrections for a regolith of grains, 
discussed earlier.

In Fig. 4.11b we also give escape fractions using a Maxwellian distribution
to characterize the flux of sublimed molecules or molecules ejected on
micro-meteorite impact of a surface (Haff and Eviatar 1986; Morgan et al.
1988). Escape fractions equivalent to those for either collision cascade or
electronic sputtering require very large temperatures, indicative of the
fact that sputtering is an energetic ejection process. The relative
importance of various ejection processes depends, of course, not only on the
escape fractions but on the stimulating fluxes. Therefore, photon heating
(sublimation) is often dominant, and micrometeorites can be important.

In determining the loss of volatiles from a large body, gravity acts as a
mass filter (Haff et al. 1981a; Hunten et al. 1989). The discussion in the
section on surface alteration remains valid on such objects if the yields, Y$%
_{\mathrm{i}}^{\prime }$, are multiplied by the escape fractions for each
species. Therefore a \textit{more} massive molecular species may be
preferentially retained even if it is more volatile! This would be the case
for sputtering of the O$_{2}$ produced in an H$_{2}$O ice on Europa.
Although O$_{2}$ once formed is readily ejected from the surface, it has a
much higher gravitational energy than H$_{2}$O and would accumulate on the
surface of Europa (Johnson et al. 1982; Cheng and Johnson 1988). On the
other hand, the very light volatiles produced, such as H$_{2}$, are lost
with near unit efficiency. Therefore, the surface layer of Europa may be
deplected in H relative to O, and should be slightly oxidizing. This
depletion may be mitigated by the bombardment and implantation of H$^{+}$
from the plasma torus and the presence of NH$_{3}$ in the ice (Brown et al.
1985) can effect this also.

Because D$_{2}$, HDO and D$_{2}$O are heavier than H$_{2}$ and H$_{2}$O,
there is a tendency on a surface with gravity to accumulate a higher D to H
ratio in the \textit{sputter layer} than the normal isotopic abundance.
Ignoring the surface binding in Eq. (4.15), the ratio of the H$_{2}$ to HD
escape fraction is 3/2 and the HDO to H$_{2}$O escape ratio is 19/18. As the
dominant H(D) escape at Europa is via formation and loss of H$_{2}$(HD),
then the D/H ratio in the \textit{sputter layer} is 3/2 the original
abundance, using Eq. (4.11b). Possible observation of this effect has been
reported by R.H. Brown.

\subsection{\textit{Redeposition and Ballistic Transport}}

In describing preferential erosion, we assumed that any sputtered species
could escape to space from a small object. That is, the Y$_{\mathrm{i}%
}^{\prime }$ in Eq. (3.22) are the true surface yields. However, under
gravity, molecules may return to the surface affecting whether the surface
darkens or not (see Fig. 4.4). Therefore, small objects may rapidly darken
by bombardment (e.g., Uranian rings), but large objects might do so only
very slowly compared to other geologic processes or micrometeorite impacts.
If the escape fraction is nearly zero, volatile material is plumbed from the
ion penetration depth (Chap. 3), sputtered into the atmosphere and
redeposited (McGrath et al. 1986). If this bombardment is uniform, then the
layers can brighten, not darken, (assuming the condensed volatile forms a
``bright'' material). That is a layer of volatiles forms on the surface of
the involatile material as indicated in Fig. 4.4c. If the irradiation is
nonuniform, then the bright material may migrate preferentially to ``cold
trapping'' regions (Sieveka and Johnson 1982). In the case of solar photon
radiation volatiles preferentially migrate to polar regions brightening the
poles (e.g., Mars) or to other already bright (cooler) close-by regions
[e.g., Callisto (Spencer 1987a); Io (McEwen et al. 1988)]. On a very porous
surface they can also migrate into colder subsurface regions [e.g. Io
(Matson and Nash 1983)].

On Mercury (Potter and Morgan 1985) and the Moon (Potter and Morgan 1988a),
the more volatile Na can be made mobile in the rocky material by solar wind
and solar flare ions, cosmic rays, and energetic photons, and diffuse to the
surface so that adsorbed Na forms on the surface (Hapke 1986; McGrath et al.
1986; Cheng et al. 1987; Hunten et al. 1988) as in Fig. 4.12. This Na has a
low binding energy (Hunten et al. 1988) and is desorbed thermally as well as
by the UV photons and ions. Because Mercury has a weak magnetic field, the
ion flux at the solid surface is reduced, depending on the solar activity
(except at the polar regions). Therefore, on the subsolar hemisphere when
the surface temperatures are high a thermally stimulated atmosphere is
formed of Na atoms in low ballistic trajectories. The ion flux, including
locally ionized and accelerated Na, produces a much weaker neutral extended
corona (McGrath et al. 1986), to be discussed shortly, from which loss of Na
to space occurs efficiently. Such a corona has recently been observed on the
Moon (Potter and Morgan 1988b). Loss also occurs at Mercury due to ionized
Na reaching open field lines (Cheng et al. 1987; Ip 1986a, 1987). These
processes may deplete Mercury's Na content down to the depth of penetration
of the most-energetic radiation. This depletion must compete with a supply
process which is presumably micrometeorite gardening of the surface exposing
fresh materials or deposition of Na by meteorites (Morgan et al. 1988).
Diffusion from depth has also been discussed (Sprague et al. 1989; Killen
1989). Oxygen may be ejected as O$_{2}$ or O, with the latter reacting at
the surface (Cheng et al. 1987).

Because the sputtered particle orbits are large, nonisotropic bombardment
will result in the transport of material across the surface of the
satellite. The rate of transport can be roughly estimated from the escape
fraction, the mean flight time of the particles, and the mean angular
excursion across the surface. For a sputtered or sublimated molecule in a
ballistic orbit the angular excursion distance, $\delta $, is given as

\begin{subequations}
\begin{align}
%\begin{equation}
& \mathrm{ cos\delta = 1 - 8(E/U^s)^2 sin^2\theta cos^2\theta {1 - 4[1 - (E/U^s)] 
(E/U^s)sin^2\theta}^{-1}.} \notag\\
%\end{equation}
\end{align}
Values for an average angular excursion, cos$^{-1}$ $\langle$ cos $\delta$ 
$\rangle $, are given in Table 4.4 for sputtering. Typical angular
excursions for water molecules on the Galilean are of the order of 60$^{\circ}$. 
That is, the bound particles travel over a large fraction of
the hemisphere. Fig. 4.13 gives the redistribution of material due to
sputter ejection for a collision cascade distribution and thermal ejection
according to a Maxwellian distribution. The results are shown for
bombardment (illumination) of the trailing hemisphere for a number of ratios
of U/U$^{\mathrm{s}}$ and kT/U$^{\mathrm{s}}$ respectively. The nonuniform
bombardment can produce the same result as nonuniform heating-latitudinal or
longitudinal variations in brightness. Although polar caps are clearly
visible on Ganymede, it has not been definitely identified with sputtering
(Sieveka and Johnson 1982) or thermal transport (Purves and Pilcher 1980)
and may be, as discussed earlier, a competition between surface roughening
and annealing by solar heating (Johnson 1985a).

\subsection{\textit{Ballistic Corona}}

In the absence of a significant atmosphere those molecules sputtered from
the surface which \textit{do not} escape follow large ballistic trajectories
across the surface of the satellite, as shown in Fig. 4.10a. By keeping
track of the time a ballistic particle spends in any region of space about
the satellite the column densities of the collisionless neutral corona
produced by sputtering can be calculated. For a particle, M, leaving the
surface of a satellite with a speed v, energy E and direction $\theta $ to
the surface normal, the time spent before returning to the surface
is 

\begin{align}
& \mathrm{ t = 2 \int\limits_{Rs}^{R_max} dr/(dr/dt)} \notag\\
\end{align}
with [e.g., Eq. (2.16b)] 
\begin{align}
& \mathrm{ (dr/dt) = v \left[ 1 - \left(\frac{R_s}{r}\right)^2 sin^2\theta - 
(U^s/E)\left( 1-\frac{R_{\dot{s}}}{r}\right)\right]^{1/2}. }
\end{align}
\end{subequations}
If the angular and energy distribution of the sputtered particles is 
f(E,$\theta $)dEdcos$\theta $ then the average time is

\begin{equation}
\mathrm{ \langle t \rangle_b = (1 - f_{es})^{-1} \int \int f(E, \theta) t 
dE dcos\theta.}
\end{equation}
Here the integral is over the fraction in bound trajectories, (1 - 
f$_{\mathrm{es}}$), and t is the time for return to the surface (see Table
4.4). Writing f(E,$\theta $)dEdcos$\theta $ = f(E)dE(2 cos $\theta $)(d cos 
$\theta $) the angular averaged, $\langle $t$\rangle _{\mathrm{b}}$(E), can
be carried out analytically (Watson 1982). This is given in Fig. 4.14; when
E $\ll $ U$^{\mathrm{s}}$ (``flat'' object) $\langle $t$\rangle _{\mathrm{b}} 
$(E) $\rightarrow $ (4/3) (2R$_{\mathrm{s}}$/g)$^{1/2}$, where 
(2R$_{\mathrm{s}}$/g)$^{1/2}$ is an intrinsic time scale for the object.

The coronal column density is determined by the time the molecules spend in
orbit. For isotropic bombardment and using Eq. (4.17), the column is

\begin{subequations}
\begin{align}
& \mathrm{ N = \langle t\rangle_b \langle Y\rangle \Phi (l - f_{es}).} \notag\\
\end{align}
The number density of sputtered molecules at a radial distance r above the 
surface is given by the average time molecules spend in a volume about r 
weighted inversely by the expanding conical area above the surface, 

\begin{align}
& \mathrm{ n_b(r) = \langle 2(R_s/r)^2/(dr/dt)\rangle_b \langle Y\rangle \Phi 
(1 - f_{es}),} \notag\\
\end{align}
\begin{align}
& \mathrm{ n_e(r) = \langle (R_s/r)^2/(dr/dt) \rangle_e \langle Y\rangle \Phi f_{es}.}
\end{align}
\end{subequations}
Here n$_{\mathrm{b}}$(r) and n$_{\mathrm{e}}$(r) are the two contributions 
to n(r) due to particles in bound orbits and escaping particles. The integral 
for the particles in bound orbits is carried out only over those ejected 
particles reaching height r. The thickness of the atmosphere (radial, 
line-of-sight column density) is then obtained by integrating the contributions 
in Eq. (4.18) [i.e., N = $\int $n(r) dr]. In Appendix 4D, Eqs. (4.18b) and 
(4.18c) are integrated for an energy distribution appropriate for atmospheric 
sputtering.

To obtain the number density close to the surface the satellite can be
treated as flat (see Fig. 4.14), in which case much simpler analytic
expressions can be used (Johnson et al. 1982). For a collisional cascade
distribution for sputtering of particles from a ``flat'' satellite, t = 2 v
cos$\theta $ /g in Eq. (4.17). With f$_{\mathrm{es}}$ = 0 and assuming a
planar ``barrier'' for sputtering then 
%%%(4.19)
\begin{subequations}
\begin{align}
& \mathrm{ n(z) = (N/2H_c) (1 + z/H_c)^{-3/2}, \ \ \ N = \langle Y\rangle \Phi 
(4v_c/3g),} \notag\\
\end{align}
with H$_{\mathrm{c}}$ = U/Mg, and v$_{\mathrm{c}}$ = (3$\pi $/4)(2U/M)$^{1/2}$. 
For a spherical surface ``barrier'', 
$$
\mathrm{ n(z) = (3N/2H_c)\left[ \frac{(1+2z/H_c)}{(1+z/H_c)^{1/2}} - 
2(z/H_c)^{1/2}\right],}
$$
\begin{align}
& \mathrm{ N = \langle(2Y)\rangle \Phi(4 v_c^s/3g)}
\end{align}
\end{subequations}
with v$_{\mathrm{c}}^{\mathrm{s}}$ = ($\pi $/2) (2U/M)$^{1/2}$ and (2Y) is the
spherical binding yield [Eq. (3.14b)]. It is seen that H$_{\mathrm{c}}$ is a
rough scale height and the column densities have been put in the form
generally used for a Maxwellian flux distribution of ejected particles. That
is, writing H$_{\mathrm{M}}$ = kT/Mg, 

\begin{equation}
\mathrm{ n(z) = (N/H_M)exp( - z/H_M), \ \ \ N = \Phi_s (4v_M/3g),}
\end{equation}
where $\Phi$, is the surface flux and v$_{\mathrm{M}}$ = (3$\pi $/4)(2 kT/$\pi 
$M)$^{1/2}$. In these expressions we assumed f(E, $\theta $) $\propto$ 
cos$\theta$, which give the factor (4/3) in each case. The v$_{\mathrm{c}}$, 
v$_{\mathrm{c}}^{\mathrm{s}}$, and v$_{\mathrm{M}}$ are the mean speeds of 
molecules leaving the surface for each distribution. For cascade sputtering 
it is seen that the effective cohesive energy, U, roughly replaces kT. Based 
on values for U, sputtering is a ``high temperature'' process. At altitudes 
greater than H the densities of planar and spherical sputtered collision 
cascade coronae, Eq. (4.19), become identical, independent of the material 
binding, U, as $\langle $Y$\rangle $ $\propto $ U$^{-1}$ [Eq. (3.15)], and
equivalent to a corona above an exobase, to be discussed shortly (Appendix
4D). The collision cascade corona also decays much more slowly (z$^{-3/2}$)
than the Maxwellian corona, Eq. (4.20). These differences are enhanced if
the surface curvature is large. The expressions above are useful for regions
where H and z are much less than R$_{\mathrm{s}}$. They can be corrected for
escape using (1 - f$_{\mathrm{es}}$) from Fig. 4.11 and used at larger
altitudes letting z $\rightarrow $ (r - R$_{\mathrm{s}}$) and multiplying by
(R$_{\mathrm{s}}$/r)$^{2}$ (Appendix 4D). This gives a corona which decays
in a manner similar to the dependence measured for Na at Io by Schneider et
al. (1987), r$^{-3.5}$. For the escaping species at radii larger than R$_{%
\mathrm{L}}$ the ``atmosphere'' decays, of course, as r$^{-2}$ (Appendix
4D), which is roughly what was found for Na by Trafton (1975a).

In most instances the bombardment is not isotropic. If the gradients in the
flux across the surface are small over the transport distances described in
the previous section, then the results above are valid using the local flux.
If this is not the case, then even if the orbit is ballistic, the number
density must be calculated numerically (Sieveka and Johnson 1985, 1986). The
space above the satellite is divided into volumes about the axis of symmetry
of the ion flux. For a given choice of initial conditions the time a
molecule spends in each region, $\Delta $t, is determined. The incremental
contribution to the number density of bound molecules [Eq. (4.18b)] due to
the transit of a test particle though a region $\Delta $r in time $\Delta $t
is calculated as $\Delta $n = ($\Delta $t/$\Delta $r) (R$_{\mathrm{s}}$/r)$%
^{2}$ $\Delta $ [$\langle $Y$\rangle $ $\Phi $(1 - f$_{\mathrm{e}}$)]. The
last factor is the fraction of the total flux represented by the particular
choice of initial conditions for the ejected test molecule. Using a Monte
Carlo selection of initial conditions weighted according to the appropriate
energy distribution, the molecular number density profile for SO$_{2}$ for
co-rotating ion bombardment is shown in Fig. 4.15, using a collision cascade
spectrum with the experimentally measured value of the effective binding
energy, U $\sim $ 0.05 eV and $\langle $Y$\rangle $ $\sim $ Y. These results
for the night-side of Io should be scaled to the appropriate flux
(Wolf-Gladrow et al. 1987; Linker et al. 1988) and the appropriate SO$_{2}$
coverage (Howell et al. 1984; McEwen et al. 1988). It is seen that the
sputter-produced atmospheric densities indeed fall off slowly with distance
from the satellite, forming a giant corona over all regions of the surface.
There will also be an S$_{2}$ component to the night-side corona due to
sulfur sputtering ($\sim $ 20\%) (Chrisey et al. 1987) and a small component
of sodium and sodium contain species due to Na$_{2}$S$_{\mathrm{x}}$
sputtering (Chrisey et al. 1988). A sputtered Na corona has also been
observed on the moon (Potter and Morgan 1989).

In all the above the sticking coefficient of the surface was assumed to be
unity, which is a reasonable assumption for H$_{2}$O or SO$_{2}$ on the cold
surfaces expected at the distant satellites. On the other hand, those O$_{2}$
produced and ejected by sputtering from a surface with temperature > 90 K may 
not condense with unit efficiency. Further, their escape rate is negligible on 
the Galilean satellites. Therefore, in the important temperature regime for 
which the sputter yield for H$_{2}$O and SO$_2$ produces significant amounts 
of O$_2$ (> 100 K, Chap. 3), an atmosphere can build up which is larger than 
the ballistic atmosphere. Such a prospect was initially proposed for Ganymede 
(Yung and McElroy 1977), where sublimation was the surce of H$_{2}$O and
photochemistry produced O$_{2}$. In the present case, the O$_{2}$ is
directly produced in the sputtering process, and atmospheric chemistry does
not play a role. The effect of photo-absorption in such an atmosphere would
be to produce an O corona, but these species stick efficiently to the
surface.

Determining the nature of the O$_{2}$ atmosphere is fairly complex. The cold
polar regions can act as sinks for O$_{2}$, resulting in a net transport of O%
$_{2}$ poleward, the O$_{2}$ may react on the surface, or ``cold trapping''
on a porous surface may occur (Matson and Nash 1983). Those O$_{2}$
molecules striking the surface which do not react will accommodate to the
surface temperature yielding a thermal energy distribution. Finally, ions
and electrons bombarding the colder regions can desorb any condensed O$_{2}$
or dissociate it, and implanted protons can cause H$_{2}$O to reform. Column
densities of O$_{2}$ ($\sim $ 10$^{16}$O$_{2}$/cm$^{2}$ condensed or
atmospheric) may be achievable on Europa (Johnson et al. 1982; Cheng and
Johnson 1988).

In the above discussion the results for sputtering are proportional to the
incident particle flux. The very presence of an atmosphere, however, can
limit this flux. For micrometeroid ejecta and sublimation due to solar
heating, attenuation by the presence of an atmosphere is not as severe. That
is, small energetic particles and ``visible'' photons require considerable
gas densities in order to restrict the flux reaching the surface. On the
other hand, incident slow ions ($\sim $ 10$^{2}$ eV) will be impeded by
column densities, N, of the order of 3 x 10$^{15}$ atoms/cm2 and fast
protons ($\sim $ 1 MeV) by N $\approx $ 10$^{20}$ atoms/cm$^{2}$ (see Fig.
3.3a). This slowing of the ions means the atmospheric column density
achievable by sputtering has an upper limit (Lanzerotti et al. 1978; Watson
1982) determined by the mean projected range of the particles. That is,
independent of the sticking coefficients, accommodation coefficient,
etc., 

\begin{equation}
\mathrm{ N_{max} \approx \bar{N}_p, }
\end{equation}
where \={N}$_{\mathrm{p}}$ is the mean projected range expressed as a column 
density. (For lower energy particles with large straggling N$_{\mathrm{max}}$ 
$\approx $ 2 \={N}$_{\mathrm{p}}$ is appropriate.) Using the characteristic 
temperature for the Voyager LECP flux at Ganymede (kT $\approx$ 30 keV) 
sputter column densities of the order of 10$^{18}$ O$_{2}$ molecules/cm$^{2}$ 
are allowable (Johnson et al. 1982). This is three orders of magnitude larger 
than an exospheric column density, and in apparent disagreement with the 
Voyager ultraviolet limit, $\sim $ 2 $\times $ 10$^{15}$ O$^{2}$/cm$^{2}$ for
Ganymede (Broadfoot et al. 1981). In the following, the plasma bombardment
of a thick (N > \={N}$_{\mathrm{p}}$) atmosphere is described.

\section{Atmospheric Escape}

The upper regions of an atmospheric vapor subjected to solar UV radiation
produces, eventually, dissociated species (Fig. 4.16a) which in addition to
forming a corona can (1) escape directly, (2) cause ejection of other
species, and (3) cause densities are N > $\sigma_{\mathrm{a}}^{-1}$, where 
$\sigma _{\mathrm{a}}$ is the average UV photoabsorption
cross section, 10$^{-17}$ --- 10$^{-19}$ cm$^{2}$. The UV absorption, which
is very sensitive to the molecular content of the gas, is well discussed in
the literature. It is responsible for production of ionospheres as well as
for heating of the upper atmospheres (thermospheres) of the planets.
Therefore, it is also responsible for the loss of atoms and molecules to
space and, thereby, affects the composition of the atmospheres (Kumar et al.
1983; Chamberlain and Hunten 1987; Lewis and Prinn 1984). Here the heating
and molecular ejection processes are treated separately even though they are
closely coupled, particularly as the former determines the exobase altitude
and thermal (Jeans) escape rate. In this section the direct ejection
processes associated with ion bombardment are first considered after briefly
describing direct ejection induced by UV absorption. Subsequently the
expansion of the upper atmosphere due to heating is considered.

\subsection{\textit{Escape Induced by Photon Flux}}

Loss of atoms and molecules can occur via Jeans escape and, directly, via
chemical processes. Whereas Jeans escape generally requires very high
temperatures (see Fig. 4.11), the latter effect can occur if molecular ions
exist high in the ionosphere (Stewart 1972). For example, on Mars the
ionosphere is predominantly O$_{2}^{+}$. Therefore, electron recombination, O%
$_{2}^{+}$ + e $\rightarrow $ O + O, results in each O atom achieving
energies of the order of 0.5 - 2 eV contributing to the O corona (Ip 1988).
At the upper limit of this range such atoms can escape from the upper
atmosphere of Mars where the escape energy required is lower than that at
the surface, (see Table 1.4). This process was also considered for a
possible atmosphere on Ganymede (Yung and McElroy 1977), but for larger
planets such loss processes are feasible only for the lighter constituents
(e.g., H$_{2}^{+}$ + e $\rightarrow $ H + H) residing at very high
altitudes. This repulsive recombination process is similar to the repulsive
decay in the solid which energized atoms and molecules in the electronic
sputtering process. However, there can be a distinctly different effect in
the two media. For example, following N$_{2}^{+}$ + e $\rightarrow$ N + N
in an N$_2$ gas \textit{or} solid, the N atoms either escape or energize
neighboring N$_2$ molecules. In the solid the N$_{2}$ molecules have a
much lower surface binding energy than the N atoms so that N$_{2}$ escapes
preferentially. In an atmosphere, the gravitational ``binding'' of N is half
that of N$_{2}$ and, therefore, N is generally the dominant ejecta. The
presence of NH$_{3}$ and/or CH$_{4}$ (e.g., CH$_{4}$ + e $\rightarrow $ CH$%
_{3}$ + H, etc.) leads to loss of H and precipitation of heavier species
such as organics (Hunten et al. 1984; Sagan et al. 1984; Delitsky and
Thompson 1987; Thompson et al. 1989).

The upper regions of an O$_{2}$ atmosphere subjected to solar radiation will
be dominated by O. As any O$^{+}$ created by photo-ionization does not
recombine with electrons effectively, O$_{2}^{+}$ forms (i.e., O$^{+}$ + O$%
_{2}$ $\rightarrow $ O + O$_{2}^{+}$) after which recombination occurs
efficiently. Any O atoms energized by recombination (or directly by
photo-dissociation of O$_{2}$) are likely to see a predominantly O
atmosphere producing a cascade of O atoms which contributes to the escape
yield in addition to direct O ejection (see Fig. 4.16a). Therefore the
description of the escape yield per recombination event is identical to our
discussion of electronic sputtering. That is, the spectrum of O atoms
energized by an O atom of initial energy E$_{\mathrm{o}}$ using Eq. (3.5)
is 

\begin{equation}
\mathrm{ N(E_O, E) \approx \delta (E_O - E) + (6/\pi^2)(E_O/E^2) }
\end{equation}
where the first term is a delta function which has a value only when the 
integrand is zero. If the cascade is isotropic and initiated at random depths, 
then, using Eq. (4.22), the direct escape yield is enhanced by a factor of 

\begin{equation}
\mathrm{ \approx {1 + (6/\pi^2) [(E_O / U_O^L) - 1]}, \ \ \ \ \ E_O > U_O^L. }
\end{equation}
Therefore, for large E$_{\mathrm{o}}$/U$_{\mathrm{o}}^{\mathrm{L}}$ multiple 
collisions dominate. Those atoms not escaping but ejected away from the exobase 
produce a nonthermal component for the neutral corona, and all the atoms not 
escaping eventually return to the exobase, contributing to the heating of the 
upper atmosphere and Jeans escape. (For O$_{2}^{+}$ + e recombination in an 
O$_2$ exosphere, the cascade contribution above would be reduced by the increased 
gravitational attraction for the struck O$_2$ molecules.)

The net loss of O via O$_{2}^{+}$ + e is proportional to the local
electron-ion recombination rate ($\partial $n$_{\mathrm{i}}$/$\partial $t)$_{%
\mathrm{r}}$ in the exospheric region [see Eq. (4.4)] and the escape yield, Y%
$_{\mathrm{x}}$. In the same manner as in Chapter 3, the rate of loss of O
is written as 

\begin{equation}
\mathrm{ (\partial \eta_o/\partial t) = \int \int R^2dR d\Omega (\partial n_i/
\partial t)_r Y_x(R).}
\end{equation}
Here Y$_{\mathrm{x}}$(R) is averaged over the possible initial orientations
of the ejected O and is determined for spherical binding. It depends on the
column density above R and the relative size of the local gravitational
energy compared with the initial energy of the O atom (E$_{\mathrm{o}}$ $%
\sim $ $\Delta $E/2, where $\Delta $E is the net repulsive energy). The
recombination rate can depend on transport processes (ambi-polar diffusion)
as well as the local production rate. For simplicity we consider an example
in which each ionization produced in the exospheric region eventually
results in a recombination in the vicinity of the exobase where the electron
density is highest, and that $\Delta $E/2 is greater than the gravitational
energy at the exobase. Ignoring the multiple collision contribution, the
loss rate of O atoms is 

\begin{equation}
\mathrm{ (\partial \eta_o/\partial t) \approx J_i (N_x A_{eff}). }
\end{equation}
Here N$_{\mathrm{x}}$ is the exospheric column density for O (N$_{\mathrm{x}}$ 
$\sim $ $\sigma _{\mathrm{d}}^{-1}$ $\approx $ 3 $\times $ 10$^{15}$/cm$^{2}$ 
for O + O collisions (Yee and Dalgarno 1987)), J$_{\mathrm{i}}$ is the 
ionization source rate, and A$_{\mathrm{eff}}$ is the effective area illuminated. 
If the scale height in the exosphere is small compared to the exospheric radius, 
as is generally the case, then A$_{\mathrm{eff}}$ $\approx $ 2$\pi 
$R$_{\mathrm{x}}^{2}$ [1 + 2H/R$_{\mathrm{x}}$ + 2(H/R$_{\mathrm{x}}$)$^{2}$]. 
Including multiple collisions in the region just below the O exosphere increases 
the result above by the factor in Eq. (4.23).

\subsection{\textit{Escape Induced by Plasma Bombardment}}

Before calculating the corresponding plasma-induced processes, the schematic
diagram in Fig. 4.16b shows the events initiated by ion bombardment of a
gravitationally bound gas based on the particle penetration processes
discussed in Chapter 3 (Cheng and Johnson 1988). These include processes
like those induced by UV absorption but, in addition, momentum transfer
events occur. One of the most remarkable features of the interaction of the
Jovian magnetospheric plasma with Io is that almost all of the processes
shown in Fig. 4.16b have been observed as features in the neutral sodium
cloud, which we will discuss shortly. Ions passing through the edge of the
atmosphere eject molecules in single collisions (prompt sputtering) (1).
Ions can sputter the atmosphere (i.e., multiple collision ejection) on
entrance (backsputtering) (3), and exit (transmission sputtering) (2). They
can produce new ions via collisional ionization (2) or charge exchange (2,
6, 7). In the latter case, if the atmosphere is thin, the incident ion may
leave as an energetic neutral. Ions can modify the atmosphere by
implantation (3) or sputtering of the surface (4, 5). In addition, very low
energy atoms may be scattered back out, often as neutrals (reflection in
Chap. 3), or if the path length is long enough they may neutralize \textit{%
and} scatter (6). The presence of the magnetic field which transports the
plasma, as discussed earlier, affects these processes in that it controls
the path length in the medium and may sweep up newly created ions.

The ionization contribution to the loss processes can be estimated for the
incident plasma by replacing the UV value of J$_{\mathrm{i}}$ in Eq. (4.25)
by J$_{\mathrm{i}}$ $\approx $ $\Phi _{\mathrm{ions}}$(S$_{\mathrm{e}}$/W$_{%
\mathrm{e}}$), where the W$_{\mathrm{e}}$ value is defined following Eq.
(3.4). For solar wind proton bombardment of an O$_{2}$ atmosphere J$_{%
\mathrm{i}}$ $\sim $ (6 x 10$^{-8}$/s)/R$_{\mathrm{os}}^{2}$(AU), where R$_{%
\mathrm{os}}^{2}$(AU) is the distance from the sun in AU. Further, as keV
protons will deposit all of their energy in a column density $\sim $ 10$%
^{18} $ mol/cm$^{2}$ the stopping of solar wind protons in an atmosphere
provides a ``heat'' source for molecules in this region of $\sim $ 2 $\times 
$ 10$^{-7}$ eV/[molecule R$_{\mathrm{os}}^{2}$(AU)] as compared to typical
energy absorption rate for UV of $\sim $ 2 $\times $ 10$^{-3}$ eV/[molecule R%
$_{\mathrm{os}}$(AU)]. It is clear, therefore, that for ejection due to
electronic-recombination or Jeans escape, the solar contribution is totally
dominated by photons. Including the plasma electron bombardment rates does
not change this significantly.

For satellite atmospheres in the distant solar system exposed to
mangetospheric ion bombardment, the relative sizes of plasma and UV
excitation rates change. For example, for bombardment of a possible O$_{2}$
atmosphere on Ganymede by the ions measured by the Voyager LECP instrument
the source term in Eq. (4.25) is J$_{\mathrm{i}}$ $\sim $ (10$^{-7}$/s) and J%
$_{\mathrm{i}}$ $\sim $ (3 x 10$^{-8}$/s) for the ions and UV respectively,
in which case the plasma bombardment dominates. Including the effect of
plasma electron bombardment (Kumar 1985; Kumar and Hunten 1982) further
favors plasma-induced formation of an ionosphere, although these electrons
are easily deflected by fields and cooled by collisions with molecules
(McGrath and Johnson 1987; Summers et al. 1989).

The loss of ions from a rarefied atmosphere occurs via magnetic field
sweeping ``scavenging'' with unit efficiency (ignoring the blocking area of
the satellite) and by electron recombination. A source of fresh ions, J, is
produced by the hot plasma ions at a rate J $\sim $ J$_{\mathrm{i}}$ N 4$\pi 
$R$_{\mathrm{s}}^{2}$. Here N is that column density of gas in the
atmosphere from which the ions can escape, averaged over the satellite
surface. N is determined by the column of gas available, by N$_{\mathrm{x}}$%
, or by the charge exchange cross section between a freshly produced ion and
a neutral (McGrath and Johnson 1987). If we assume this atmosphere is also
produced by ion sputtering of the icy surface so that N $\sim $ 10$^{14}$/cm$%
^{2}$ at Ganymede, then ion bombardment alone produces a significant source
of new water product ions J $\sim $ 10$^{25}$ ions/s at Ganymede.

Although the heating rate of solar wind ions is small compared to the UV
heating rate, the quality of the energy deposited is very different. Whereas
the energy impulses deposited by repulsive dissociation initiated by UV
absorption are of the order of an eV, the collision cascades initiated by
the nuclear elastic energy losses were seen to have a continuous spectrum of
energies up to the maximum energy transferable to a target atom by the
primary, $\gamma $E$_{\mathrm{A}}$. Therefore, even when electronic ejection
or Jeans escape are improbable processes, collisional ejection can occur in
an atmosphere which is not shielded.

The linear-cascade sputtering expression in Eq. (3.15) is independent of the
target number density, requiring only that the target (here an atmosphere)
must contain an amount of material greater than the scale of the cascades;
that is, N > 10$^{16}$ atoms/cm$^{2}$. (For lower column
densities a rough estimate can be obtained by scaling down the atmospheric
yield and including a contribution from the physical surface.) For
atmospheric sputtering, U in Eq. (3.15) is replaced by the effective
gravitational binding energy of the satellite, U$^{\mathrm{L}}$, evaluated
at the sputter exobase and the spherical binding applies, Eq. (3.14b).
Finally, for ``back-sputtering'' of an atmosphere the enhancement in the
yield due to angular incidence is clearly \textit{not} affected by the
surface regolith problem discussed earlier. Using a conservative estimate of
the incident angle dependence, Y $\propto$ cos$\theta^{-1}$, in Eq.
(4.8) then $\langle $Y$_{\mathrm{x}}\rangle $ $\approx $ 2Y$_{\mathrm{x}}$ 
$\approx$ (6/$\pi ^{2}$)($\alpha$S$_{\mathrm{n}}$/$\bar{\sigma}_{\mathrm{d}} 
$U$^{\mathrm{L}}$).

When the upper atmosphere is atomic in nature (e.g., dissociated O$_{2}$),
the sputtering expression above will give a reasonably accurate yield. The
cascade in an undissociated O$_{2}$ atmosphere is initiated by a binary
collision in which the incident ion energizes an O atom. Since most of the
incident ion-O-atom collisions are glancing collisions, the O atom is highly
energized only occasion-ally and, therefore, is not effective in
dissociating additional O$_{2}$. In the initial collision of the incident
ion and one O of an O$_{2}$ molecule, the energy transfer, T, is shared
between the CM motion of the two O atoms (T/2) and the motion of their
center of mass (T/2). (For atom B of mass M$_{\mathrm{B}}$ which is part of
a molecule of total mass M these fractions are [1 - (M$_{\mathrm{B}}$/M)]T
and (M$_{\mathrm{B}}$/M)T respectively (e.g., BEA Chap. 2; Sieveka and
Johnson 1984). If the CM energy is less than the dissociation energy, D, of
the struck atom, then the molecule moves as a whole molecule with kinetic
energy T/2 and the corresponding momentum. If the CM energy is much greater
than D the molecule is dissociated and the struck atom moves with energy $%
\sim $ T and its initial momentum, leaving the other atom behind.

The multiple collision (sputter) contribution to the loss rate (McGrath and
Johnson 1987) for an O atmosphere is 

\begin{subequations}
\begin{align}
& \mathrm{ (\partial \eta_o/\partial t) \approx \Phi_i 2\pi R_x^2 \langle 
Y_x\rangle \approx \Phi_i(2\pi R_x^2)(6/\pi^2)(\alpha S_n/\bar{\sigma}_d 
U_o^L)} \notag\\
\end{align}
using Eq. (4.25), where S$_{\mathrm{n}}$ and $\alpha $ are determined by 
the ion-O-atom collisions, U$_{\mathrm{o}}^{\mathrm{L}}$ is the escape energy 
for an O atom, and $\bar{\sigma}_{\mathrm{d}}$ is the average O + O diffusion 
cross section. This expression also gives the total O atoms lost as O$_{2}$
molecules from an undissociated atmosphere by replacing $\bar{\sigma}_{%
\mathrm{d}}$ for an O + O collision ($\approx $3.6 $\times $ 10$^{-16}$ cm$%
^{2}$) by ad for an O$_{2}$ + O$_{2}$ ($\approx $ 10$^{-15}$ cm$^{2}$)
collision and multiplying the expression by 2 (i.e., two atoms per molecule
ejected). The latter cross section is nearly four times the former (i.e.,
each exiting 0 atom in the exiting molecule interacts with either O atom in
another molecule), S$_{\mathrm{n}}$ and U$^{\mathrm{L}}$ are both doubled
and therefore, the yield of O atoms from an O$_{2}$ atmosphere is roughly
one half that from an O atmosphere.

If the initiating collisions are produced in the exosphere, they can also
lead to direct, single-collision ejection, process (1) in Fig. 4.16b.
Because the struck atoms predominantly have velocity vectors close to 90${%
{}^{\circ }}$ to the incident ion direction, we can see from the geometry in
Fig. 4.16b that many of those struck in the exosphere can escape or leave by
backscattering near the exobase. The source term in Eq. (4.24) is replaced
by the number density per unit time of atoms (molecules) which receive
kinetic energies greater than the escape energy (Sieveka and Johnson 1984).
Therefore, the \textit{single collision} source of O atoms ejected from an O
exobase with gravitational energy U$_{\mathrm{o}}^{\mathrm{L}}$ is given as
in Eq. (4.25) 

\begin{align}
& \mathrm{ (\partial\eta_o/\partial t) \approx \Phi_i\sigma (T > U_O^L)
(N_x A_{eff}).} \notag\\
\end{align}
Here $\Phi_{\mathrm{i}}$ is the averaged incident flux from Eq. (4.3a), 
$\sigma $(T > U$_{\mathrm{o}}^{\mathrm{L}}$) is the collision 
cross section for energy transfer greater than U$_{\mathrm{o}}^{\mathrm{L}}$ 
(Chap. 2), with A$_{\mathrm{eff}}$ and N$_{\mathrm{x}}$ as in Eq. (4.25). 
For an O$_{2}$ atmosphere, if U$_{\mathrm{o}}^{\mathrm{L}}$ > D, then molecular 
ejection is unlikely and Eq. (4.26a) applies. If D > U$_\mathrm{O}^{\mathrm{L}}$ 
the quantity $\sigma $(T > U$_{\mathrm{o}}^{\mathrm{L}}$) is
replaced by [$\sigma $(T $\geq$ 2D) + 2$\sigma $(2D $\geq$ T $\geq$ 
2U$_{\mathrm{o}}^{\mathrm{L}}$)], where the second term is the
molecular contribution.

With the above estimates, the net O atom yield from the dissociated
atmosphere can be written as 

\begin{align}
& \mathrm{ (\partial \eta_o/\partial t) \approx \Phi_i(2\pi R_x^2 [\sigma 
(T > U_O^L) + (6/\pi^2)(\alpha S_n/U_O^L)] /\bar{\sigma}_d, }
\end{align}
\end{subequations}
where N$_{\mathrm{x}}$ $\sim$ $\bar{\sigma}_{\mathrm{d}}^{-1}$ and 
A$_{\mathrm{eff}}\sim $ 2$\pi $R$_{\mathrm{x}}^{2}$ in Eq. (4.26a). The term 
in brackets divided by $\bar{\sigma}_{\mathrm{d}}$is a ratio of effective cross 
sections: the stimulation cross sections divided by the cross section for 
collisions between atoms in the gas. For energetic incident ions ($\varepsilon$ 
> 1 in Appendix 2B) $\sigma $(T > U$_{\mathrm{o}}^{\mathrm{L}}$) 
$\propto $ 1/U$_{\mathrm{o}}^{\mathrm{L}}$ if U$_{\mathrm{o}}^{\mathrm{L}}$ 
is large compared to the screening energy. The loss rate of atoms depends on R. 
via the target size and the binding (U$_{\mathrm{o}}^{\mathrm{L}}$ $\propto$ 
1/R$_{\mathrm{x}}$). Therefore, the loss rate is very nearly proportional to 
R$_{\mathrm{x}}^{3}$ (McGrath and Johnson 1987). The rate also depends on the
mass of the ejected atom predominantly through this binding (U$_{\mathrm{o}}^
{\mathrm{L}}$ $\propto $ M$_{\mathrm{o}}$). For U$_{\mathrm{o}}^{\mathrm{L}}$ 
small compared to the screening energy, $\sigma $ is very nearly independent of 
U$_{\mathrm{o}}^{\mathrm{L}}$. The other quantities are atomic quantities, which 
depend on the \textquotedblleft sizes\textquotedblright\ of the incident and 
target atoms.

The result above is a conservative estimate of collisional atmospheric
ejection. If the scale height of the atmosphere at the sputter exobase is
much larger than the satellite radius, the single collision yields are
enhanced via A$_{\mathrm{eff}}$ and a large gyro-radius can also increase A$%
_{\mathrm{eff}}$ (Pospieszalska and Johnson 1989) as discussed earlier.

\section{Plasma Heating and Production of an Extended Corona}

The energetic processes discussed above can act to change the character of
the upper atmosphere. Whereas solids are good conductors of heat,
atmospheres are not. Therefore, the majority of the energy of photons and
fast ions which is deposited below the exobase provides a heat source which
raises the temperature and expands the upper atmosphere. If this energy is
deposited very close to the exobase, where the collision frequency is small,
the change in the scale height of the exosphere can be dramatic. This occurs
in the thermosphere of the earth, in which the atoms have high effective
temperatures due to UV photo-absorption (Chamberlain and Hunten 1987). Such
an expansion gives a larger target, A$_{\mathrm{eff}}$, and lower U$^{%
\mathrm{L}}$ for collisional ejection. We use Io as an example in the
following, assuming that SO$_{2}$ is the dominant volatile, although Chrisey
et al. (1987) show S$_{2}$ contributes and recently H$_{2}$S has been
identified (Nash and Howell 1989).

Ignoring, initially, the increase in the radial value of the exobase by
heating from below, the neutral corona produced by plasma bombardment is
similar to the corona sputtered from the surface [see Eq. (4.16) and
following]. If the flux of stimulating radiation is small the sputter
component of the corona can be determined by tracking particles which are
ejected at the exobase along ballistic paths and determining the time spent
by these particles in any region of the atmosphere, as was done for
sputtering from the surface. This plasma or photon energized corona (McGrath
and Johnson 1987) is superimposed on a Maxwellian component of the corona,
as shown in Fig. 4.17a, having a temperature determined by atmospheric
transport processes. If the number of particles per unit volume energized by
the incident radiation is large then the temperature of the exobase and the
coronal densities are coupled. The extended nature of such a corona will
also act to reduce U$^{\mathrm{L}}$. An average U$^{\mathrm{L}}$, which
differs by terms of the order of H/U$^{\mathrm{L}}$, is 

\begin{equation}
\mathrm{ \bar{U}^L = \int\limits_{R_x}^{R_L} n(r)U^L(r)r^2 dr/\eta_x,}
\end{equation} 
where $\eta_{\mathrm{x}}$ is the number of particles in the exosphere, 
$\eta_{\mathrm{x}} $ = $\int $n(r)r$^{2}$ dr.

The location and temperature of the exobase in Fig. 4.17a is determined by
the energy deposited below the exobase where the molecular collision rate is
large compared to the simulation rate. in this ``thermalized'' region the
one-dimensional heat conduction equation for a flat atmosphere is 

\begin{subequations}
\begin{align}
& \mathrm{ o = d[K(T) dT / dz] / dz + \Phi F(z) - R(z). } \notag\\
\end{align}
Here K is the thermal conductivity, F is the energy deposited per unit depth 
by the incident radiation (see Chap. 3) and R(z) is the radiative loss rate. 
This integrates to 
\begin{align}
& \mathrm{ - K(T)[dT/dz] = - \int\limits_{z}^{\infty}\Phi_i F(z^{\prime})
dz^{\prime} + \int\limits_{z}^{\infty}\mathrm{R}(z^{\prime})dz^{\prime} }
\end{align}
\end{subequations}
giving a downward heat flux. Because the radiative loss depends on the
density and type of molecules, the second term is often treated as if all
the loss comes from some radiative layer at z$_{\mathrm{o}}$ or a transport
process removes the heat at z$_{\mathrm{o}}$. Therefore, this equation is
often solved by specifying a boundary condition at z$_{\mathrm{o}}$ and
dropping the last term at larger z (Chamberlian and Hunten 1987). The energy
deposition integral in Eq. (4.28b) depends on the column density of
atmosphere, N(z), above z. The pressure equation with g approximately
constant is 

\begin{subequations}
\begin{align}
& \mathrm{ (dN/dz)/N \approx - Mg/kT(z),} \notag\\
\end{align}
giving 
\begin{align}
& \mathrm{ N(z) \approx N_o exp\left[ -\int\limits_{z_o}^{z} (Mg/kT)dz^{\prime}
\right].}
\end{align}
\end{subequations}
The column density at z$_{\mathrm{o}}$, N$_{\mathrm{o}}$, is determined either 
directly by the local density (N$_{\mathrm{o}}$ = n$_{\mathrm{o}}$kT$_{\mathrm{o}}$
/Mg) or the surface vapor pressure (N$_{\mathrm{o}}$ = MgP$_{\mathrm{o}}$) or 
indirectly, by the supply rate (flux) of the particles at z$_{\mathrm{o}}$. 
Given F in Eq. (4.28a), the temperature and column density profile can be 
extracted from Eqs. (4.28b) and (4.29b) by an iterative procedure. This has 
been treated extensively for photo-dissociation and photo-ionization heating 
of the thermospheres of the planets. In the following we give approximate 
solutions for the plasma heating of an atmosphere.

If the ions penetrate to the surface and we assume the collisional heat
transport below the exobase is fast, an average temperature \={T} is
obtained. The heat flux into the atmosphere is that deposited by the
incident ions and that due to sublimated and/or sputtered particles entering
the atmosphere from the surface. The deposited energy flux can be written
approximately as $\Phi _{\mathrm{i}}$\={S}N$_{\mathrm{o}}$ $\approx $ $\Phi
_{\mathrm{i}}$\={S}n$_{\mathrm{o}}$\={H}, where \={S} is the average,
stopping cross section in the atmosphere of column density N$_{\mathrm{o}}$(N%
$_{\mathrm{o}}$ < \={N}$_{\mathrm{p}}$) having a surface
density no and average scale height \={H} = k\={T}/Mg. Ignoring the heat due
to sublimated particles, the surface sputter source (e.g., collisional
sputtering, planar binding) is approximately $\Phi _{\mathrm{i}}$ $\langle $Y%
$\rangle $ [2U ln (E$_{\mathrm{M}}$/U)], where the term in brackets is the
average sputter energy for the distribution in Eq. (3.20a) with E$_{\mathrm{M%
}}$ the maximum sputtered particle energy. Heat is lost by collisions of the
atmospheric particles with the surface, c$_{\mathrm{T}}$(n$_{\mathrm{o}}$\={v%
}/4)(2k\={T}) with c$_{\mathrm{T}}$ the thermal accommodation coefficient,
and escape at the top of the atmosphere. If the escape and surface sputter
contributions are small we can ignore the difference between them, in which
case 

\begin{subequations}
\begin{align}
& \mathrm{ \bar{v} = (8k \bar{T}/M\pi)^{1/2} \approx 2\Phi_i\bar{S}/Mgc_T.} \notag\\
\end{align}
\={T} is seen to be roughly independent of the column density, N$_{\mathrm{o}}$, 
which can be estimated by balancing the particle fluxes into and out of
the atmosphere. On a satellite for which the sputtering dominates, the
molecular flux balance is 

$$
\mathrm{ (cn_o\bar{v}/4) + \Phi_i\langle Y_x\rangle = \Phi_i\langle Y\rangle, }
$$
where c is the sticking coefficient and $\langle$Y$_{\mathrm{x}}\rangle $ and 
$\langle $Y$ \rangle$ are the effective exosphere and surface sputter yields. 
Using Eq. (4.30a) 
\begin{align}
& \mathrm{ N_o \approx 2k\bar{T}(\langle Y\rangle - \langle Y_x\rangle)c_T/
c\bar{S},}
\end{align}
\end{subequations}
which applies for N$_{\mathrm{x}}$ < N$_{\mathrm{o}}$ < 
\={N}$_{\mathrm{p}}$. (If N$_{\mathrm{o}}$ < N$_{\mathrm{x}}$ 
then the ballistic calculations in the previous section apply.) For the flux 
at Io this gives a night side coronal column density close to that discussed 
earlier (a few times 10$^{14}$ SO$_{2}$/cm$^{2}$) for c$_{\mathrm{T}}$ 
$\approx $ c $\approx $ 1. It is seen from Eq. (4.30a) that k\={T} hence 
N$_{\mathrm{o}}$, depends quadratically on the flux of ions. Therefore, on 
any object for which the yields are large, the column density can approach 
the value \={N}$_{\mathrm{p}}$, resulting in a self-limiting atmosphere.

As fresh ions produced in the atmosphere by charge exchange or electron
impact ionization can be accelerated, the effective sputtering rate at the
surface can be enhanced as described earlier (see Fig. 4.10d). Therefore,
the limiting value of the column density may be approached even for modest
ion fluxes. Defining P$_{\mathrm{i}}$ to be the ionization probability for a
sputtered neutral molecule during its mean residence time in the exosphere,
the effective sputter yield, including the freshly ionized species,
is 

\begin{equation}
\mathrm{ \langle Y\rangle \rightarrow \langle Y\rangle (1 - P_i)\langle 
Y\rangle)^{-1} }
\end{equation}
for P$_{\mathrm{i}}$ $\langle$ Y$\rangle$ < 1. This replaces $\langle 
$Y$\rangle $ in Eq. (4.30b). If the lifetime against ionization, 
$\tau_{\mathrm{i}}$, is primarily due to the incident plasma then 
$\tau _{\mathrm{i}}^{-1}$ $\propto $ $\Phi _{\mathrm{i}}$ and 
P$_{\mathrm{i}}$ $\approx $ 1 - exp (- $\langle $t$\rangle $/$\tau_
{\mathrm{i}}$), where $\langle $t$\rangle $ is the mean ballistic flight 
time [see Eq. (4.17) and Fig. 4.14]. Sieveka and Johnson (1985, 1986) 
calculated a sputter atmospheric column density $\sim$ 3 $\times$ 10$^{14}$ 
SO$_{2}$/cm$^{2}$ on the night side of Io for 50\% coverage of SO$_{2}$ with 
$\langle $Y$\rangle $ $\approx $ 50 (Fig. 4.15). They also estimated that 
(P$_{\mathrm{i}}$ $\langle $Y$\rangle $) approaches one if a large fraction 
of the plasma flux is incident on the exobase. Therefore, more than an 
exopheric component is attainable on the night side of Io.

For an atmosphere thick enough to stop the plasma particles, the net heat
flux into the upper layer of column density \={N}$_{\mathrm{p}}$ is $\Phi _{%
\mathrm{i}}$E$_{\mathrm{A}}$. The average temperature in this region is
determined by conduction of heat to the radiating layer. For many
atmospheric molecules K(T) $\approx $ K$_{\mathrm{0}}$T$^{3/4}$ with K$_{0}$ 
$\approx $ 30 ergs/cm/s/K$^{7/4}$ (Chamberlain and Hunten 1987) whereas for
SO$_{2}$ Kumar (1982) uses K(T) $\approx $ K$_{0}$T$^{1.33}$ with K$_{0}$ $%
\approx $ 0.49 ergs/cm/s/K$^{2.33}$. From Eq. (4.28b) the average
temperature in the absorbing layer of Io is 

\begin{subequations}
\begin{align}
& \mathrm{ T_p^{2.33}\approx (2.33)(\Phi_i E_A/K_0)(z_p - z_o) + T_o^{2.33}  
\equiv \xi (z_p - z_o) + T_o^{2.33}, } \notag\\
\end{align}
where T$_{\mathrm{o}}$ is the temperature at the radiating layer, z$_{\mathrm{o}}$, 
z$_{\mathrm{p}}$ is the altitude at which the column density is $\sim$ \={N}$_
{\mathrm{p}}$ ($\sim $ 2\={N}$_{\mathrm{p}}$ for slow ions with large range 
straggling, Chap. 3) and $\xi$ is a lapse rate. [The radial solution (Johnson 
1989b) is approximated by the replacement (z - z$_{\mathrm{o}}$) $\rightarrow$ 
R$_{\mathrm{s}}$(1 - R$_{\mathrm{s}}$/r)]. The atmospheric column density, N, 
is obtained from Eq. (4.29b). Using Eq. (4.32a) and an absorbing layer (2 \={N}
$_{\mathrm{p}}$ $\approx $ 6 $\times $ 10$^{15}$ SO$_{2}$/cm$^{2}$) gives

\begin{align}
& \mathrm{ ln(N/2\bar{N}_p) = 1.75 \left(\frac{Mg}{k\xi}\right)(T_p^{1.33} - 
T^{1.33}).}
\end{align}
\end{subequations}
Therefore, for T$_{\mathrm{o}}$ $\sim $ 130 K and an energy flux of \={E}$_
{\mathrm{A}}\bar{\Phi}_{\mathrm{i}}$ $\approx $ 8 $\times $ 10$^{11}$ eV/
(cm$^{2}$ s) ($\sim $ 20\% of maximum) (Wolf-Gladrow et al. 1987; Linker et 
at. 1988) of half sulfur and half oxygen results in $\xi $ $\approx $ 6.1 
K$^{2.33}$/cm giving the temperatures and densities in Fig. 4.17b, c (Johnson 
1989c). For a column of SO$_{2}$ over the subsolar region $\sim$ 6 $\times $ 
10$^{16}$ SO$_{2}$/cm$^{2}$ (Fanale et al. 1982), the average temperature in 
the absorbing layer is $\sim $ 3800 K which is achieved at about 300 km above 
the surface assumed to be z$_{\mathrm{o}}$. This is a temperature much larger 
than that produced by photo-absorption, even ignoring heating by the electrons 
in the plasma. It is also a significant fraction of escape energy for SO$_{2}$. 
However, kT$_{\mathrm{p}}$ is reduced by the \textit{horizontal transport} of 
molecules because of nonisotropic bombardment and the high temperature. The 
Fanale et at. (1982) atmosphere spread over the illuminated hemisphere corresponds 
to an average N$_{\mathrm{o}}$ $\approx $ 3 $\times $ 10$^{16}$/cm$^{2}$ with 
T$_{\mathrm{p}}$ $\rightarrow $ 2900 K at about 200 km above the surface. The 
resulting large scale heights may limit the possible diffusive separation of
dissociated species (Hunten 1985) so that 0 does not dominate the corona.
Further these large temperatures are consistent with the line-of-sight Na
column densities across Io's atmosphere measured by Schneider et al. (1987)
(see Fig. 4.18a).

In the plasma heated region below the exobase a nonequilibrated component of
fast recoils is superimposed on the equilibrated (Maxwellian) component as
shown in Fig. 4.17a. The corona also consists of a ``thermal'' component of
neutrals for which the temperature depends nonlinearly on $\Phi _{\mathrm{i}%
} $, and a sputter component. The effective flux of thermal particles at the
exobase [ $\approx $ (n$_{\mathrm{x}}$\={v}/4) $\approx $ N$_{\mathrm{x}}$Mg%
\={v}/4 kT] decreases with increasing plasma heating while the exobase
height increases. On the other hand the ``sputter-produced'' component of
energetic recoils increases as $\Phi _{\mathrm{i}}$ increases. These recoils
have an energy distribution 1/E$^{2}$, for E > kT$_{\mathrm{p}}$%
. When a large plasma flux is incident the sputter-produced component is
dominant in the exosphere, in which case the number density in the corona
can be estimated by truncating the sputtered particle energy distribution
when the total column density is $\sim $ N$_{\mathrm{x}}$. The truncation is
consistent with the values of T$_{\mathrm{p}}$ above. Calculating such a
corona for Io (McGrath and Johnson 1987) the line-of-sight column densities
integrated along the plasma flow direction are displayed vs. ion impact
parameter for SO$_{2}$ molecules in \textit{bound orbits} only (Fig. 4.18b)
for the full plasma flux. The corresponding line-of-sight column densities
for the atomic Na component are normalized to the data of Schneider et al.
(1987) in Fig.4.18a. Finally, the corresponding sputter-loss rates are given
in Fig. 4.18c assuming a variety of exobase altitudes for an SO$_{2}$
atmosphere and the full flux. This increases as $\sim $ R$_{\mathrm{x}}^{3}$
as discussed earlier. Also given is the ejection rate from the corona
including exospheric processes: single collision ejection [Eq. (4.26b)] and
sweeping of species ionized by charge transfer and electron impact outside
the disk $\pi $R$_{\mathrm{x}}^{2}$. Using an average column N$_{\mathrm{o}}$
$\approx $ 3 $\times $ 10$^{15}$ SO$_{2}$/cm$^{2}$ and 20\% of the flux
incident on R$_{\mathrm{x}}$ (e.g. Eq. 4.A5) then from Fig. 4.17b R$_{%
\mathrm{x}}$ $\approx $ 1.37 R$_{\mathrm{Io}}$ and from Fig. 4.18c a supply
rate of 10$^{3{{}0}}$ amu/s is obtained roughly consistent with expectations.

\section{Production of a Neutral Torus}

Atoms and molecules ejected from the surface or atmosphere of a satellite
which have sufficient energy to escape require much larger energies in order
to escape the gravitational field of the parent planet. Those not energetic
enough to escape the planet's field will orbit the planet until they are
ionized (Smyth and McElroy 1978) or they impact and stick to an object.
These ejected neutrals have the orbital velocity of their satellite source
superimposed on their escape velocity. When the satellite orbit velocity is
large compared to typical exit velocities, the orbits of the neutrals will
differ only slightly from that of the satellite. The orbiting neutrals form
an atmospheric component gravitationally attached to the parent planet which
roughly co-orbits with the satellite (Smyth 1979; Smyth and Combi 1988).
This neutral cloud is also an extension of the satellite's atmosphere, as
the choice of a boundary [i.e., the Hill (Lagrange) sphere] is arbitrary.
Therefore, the satellite's effective neutral envelope can become enormous
compared to its size. If these neutrals have lifetimes very long compared to
the orbital period, then the neutral cloud becomes a torus about the parent
planet.

The most striking example of such an extended atmosphere is the Na cloud
about Io detected first by R.A. Brown (1974). He observed sodium D line
emission which was correlated with Io's position. Trafton (1975a) showed
that this feature had a large spatial extent by observing Na using a slit
several Io radii away from Io (Bergstralh et al. 1975), and he also observed
a potassium atmosphere (Trafton 1975b). More recently, Ballester et al.
(1987) observed neutral oxygen and sulfur. Brown divided the Na feature into
three regions of varying brightnesses (Brown and Schneider 1981): a bright
A-cloud corresponding to a region close to Io, material probably
gravitationally bound to Io and recently described by Schneider et al.
(1987); the less intense B-cloud consisting of low velocity escaping atoms;
and the very weak C-cloud at considerable distance from Io.

The observed sodium atoms are thought to be produced by the incident plasma
bombardment of the atmosphere and/or the surface of Io (Matson et al. 1974;
Chrisey et al. 1988). Even though Na is likely to be a small fraction of the
material escaping from Io ($\sim $ a few percent) its resonance line is
easily excited by the solar radiation. Because sodium is also rapidly
ionized by plasma electrons and solar photons the co-orbiting Na cloud is
not toroidal but has a morphology strongly dependent on the nature of the
source and the sink. That is, the more energetic atoms remain neutral at
larger distances from Io. Therefore, the classifications which Brown derived
roughly correspond to the various ejection processes described in Fig. 4.16b.

The ejection processes discussed in the previous section have distinct
energy distributions (see Fig. 4.19). Jeans escape processes have low
average energy (1) in Fig. 4.19, various sputter processes (2,4,5) have
harder spectra, and single collisions (3) a very hard spectrum (i.e., larger
number of fast particles).

The B cloud is thought to correspond to atoms leaving Io with average
velocities $\sim $ 3 km/s at an altitude $\sim $ 700 km above the surface
according to the model of Smyth (1979, 1983). Using the sputtered particle
energy distribution for a spherical barrier the mean ejected particle speed
is ($\pi $/2) (2U$^{\mathrm{L}}$/M)$^{1/2}$ [e.g., Eq. (4.19b)] which for Na
ejected from near the surface of Io is about $\sim $ 2-3 km/s consistent
with the above. Even though the sputtering occurs predominantly on the
trailing hemisphere, allowing for the effect of Io on the particle's orbit
gives a source term that consists of particles moving outward from Io in all
directions (Smyth and McElroy 1978). The morphology of the cloud, as
indicated by the spatial distribution in the emission, also depends strongly
on the local plasma electron ionization rate. Therefore, the cloud appears
to extend preferentially inward from Io, where the plasma electron
temperatures are lower (Johnson and Strobel 1982).

The faster component (C cloud) probably consists of particles which are
ejected in single collisions and direct charge exchange interactions
(Sieveka and Johnson 1984; Ip 1982). These processes have quite distinct
directional spectra, as indicated in Fig. 4.20. The charge exchange
component is predominantly ``forward'' directed (i.e., along plasma flow),
and the collisional component is predominantly directed at right angles. The
latter feature was consistent with the fast Na observed by Pilcher et al.
(1984) and Schneider et al. (1988). These speeds are generally estimated
using the Na ionization and fluorescence rates and then tracking the Na back
to Io. Trauger (1985) has observed the Doppler shift in the ejected Na
indicating speeds up to the corotation speed of the magnetosphere. Such
speeds can occur due to the charge exchange interaction, as indicated in
Fig. 4.20 and described in Chapter 2.

The charge exchange process at the low velocities appropriate here is state
selective, as discussed in Chapter 2. That is, at a relative speed of v 
$\sim$ 60 km/s, the uncertainty in the energy levels during the collision is 
$\Delta$ E $\sim$ $\hbar$ v/a $\sim $ 0.5 eV where a is the radius of a
Na atom ($\sim $ 1.5 a$_{\mathrm{0}}$). Therefore, differences in the total
electronic energy before and after charge exchange (i.e., Na$^{+}$ + B $%
\rightarrow $ Na + B$^{+}$) must be of the order of or less than this value
for the cross sections to be significant. In Fig. 2.16a are given cross
sections for Na$^{+}$ on a number of atomic species indicating Na$^{+}$ only
charge exchanges efficiently with Na or K, molecular species containing
these, or highly excited oxygen or sulfur. These species account only for a
small fraction (few percent) of the total atmosphere. As free sodium and
sodium-containing molecules comprise the largest fraction, the very fast
sodium atoms produced represent a profile of the Na in the atmosphere.

In contrast to Na at Io, a calculation of the neutral torus of ``heavy''
atoms and molecules produced by the sputtering of the icy satellities of
Saturn (Cheng et al. 1982) is shown in Fig. 4.21a (Johnson et al. 1989a).
The species represented are H$_{2}$O, OH, O (depending on the dissociation
level) associated with ejected H$_{2}$O in orbit about Saturn in a region in
which the plasma densities and photon fluxes are much lower than at lo.
Therefore, each molecule, on the average, makes a number of orbits about
Saturn before it is ionized by photons, electron impact, or charge exchange
(Fig. 4.21b) giving a flattened torus enclosing the five icy satellite
orbits. In determining these densities, the net sputter rates in Table 4.3
and the escape fractions in Table 4.4 are used along with the measured
ejected particle energy and angular distributions. In this calculation an
isotropic distribution (Pospieszalska and Johnson 1989) from each satellite
was assumed as the keV O$^{+}$ dominates the sputtering (Lanzerotti et al:
1983, 1984). The molecules were tracked in ballistic trajectories until they
were ionized (Smyth and Shemansky 1983). The ionization rates in Fig. 4.21b
were determined by 

\begin{equation}
\mathrm{ J_i = k_1 n_{+} + k_2 n_{-} + J_{UV}}
\end{equation}, 
where k$_{1}$ is the charge-transfer rate coefficient ($\sim $ $\sigma_
{\mathrm{ct}}$ v$_{+}$ ), k$_{2}$ is the electron ionization rate, which is 
strongly dependent on electron temperature (Fig. 2.13b), and J$_{\mathrm{UV}}$ 
is the UV ionization rate. Dissociation was ignored as it primarily energized 
H atoms into large orbits (i.e., H$_{2}$O + $\hbar \nu $ $\rightarrow $ OH +
H with H receiving 17/18 of the excess energy). As the neutral cloud
co-exists with the plasma it is \textquotedblleft heated\textquotedblright\
not only by the electrons (e.g., dissociation) but by collisions with the
ions. Charge exchange produces very fast neutrals and, in addition, elastic
collisions of the ions with the neutrals can transfer momentum and can heat
this atmosphere at a rate n$_{\mathrm{i}}$v$_{\mathrm{i}}S$, which for the
ionization time near Dione (Fig. 4.21b) gives a net energy transfer to an 0
atom of the order of the orbital energy.

The net line-of-sight column density of these species is not large enough to
be detected from earth; however, it is a source of the heavy ion plasma
observed in this region by Voyager (Eviatar et at. 1983; Richardson and
Sittler 1989). (Micrometeorite erosion of the main rings also contributes
inside the orbit of Enceladus.) Of interest in such calculations are the
scale height of the atmosphere perpendicular to the orbit plane, H, and mean
radial extent, $\Delta $R, of the distributed particles. These are directly
dependent on the ejection speeds and can be estimated (Johnson et al. 1984a)
as $\Delta $R/R$_{\mathrm{ps}}$ $\sim $ 4 $\langle $ v$_{\mathrm{es}}\rangle 
$ / v$_{\mathrm{s}}$ and H/R$_{\mathrm{ps}}$ $\sim $ $\langle $ v$_{\mathrm{%
es}}$ $\rangle $ / 2v$_{\mathrm{s}}$, where R$_{\mathrm{ps}}$ is the
distance between the planet and the satellite, v$_{\mathrm{s}}$ is the
orbital speed of the satellite, and < v$_{\mathrm{es}}$%
> is the mean ejection velocity from Table 4.4. Based on this,
all of the satellites contribute to the neutral densities in the vicinity of
Enceladus, Table 4.5. Therefore neutrals may be swept up by a satellite,
providing a transfer of material over large distances, an extension of the
transport across the surfaces of individual satellites described earlier.
For example, transfer of sulfur from Io may account for certain spectral
properties for Amalthea.

Assuming the neutrals are distributed uniformly over a limited volume V [ $%
\approx $ 2$\pi $R$_{\mathrm{ps}}$ (2H) (2$\Delta $R)], an equation for the
mean density of heavy neutrals, n$_{\mathrm{o}}$, can be written (Huang and
Siscoe 1987; Eviatar 1984) 

\begin{subequations}
\begin{align}
& \mathrm{ dn_o / dt = n_{+} \langle v_{+} Y\rangle_{es} \pi R_s^2/V - J_i n_o.} 
\notag\\
\end{align}
The first term is the sputter supply rate discussed earlier divided by V. In 
the denser regions of the torus generally J$_{\mathrm{i}}$ $\approx$ 
k$_{1}$n$_{+}$ + k$_{2}$n$_{-}$. Writing n$_{-}$ $\approx $ n$_{+}$ the 
equilibrium value of the average neutral number density is independent of 
the ion density! That is 
\begin{align}
& \mathrm{ n_o = \langle v_{+} Y\rangle_{es} \pi R_s^2 /[V(k_1 + k_2)], }
\end{align}
\end{subequations}
so that the neutral density in this self-sustaining cloud depends primarily 
on the temperatures of the ions and electrons (via v$_{+}$, k$_{1}$, and 
k$_{2}$).

In the Io torus, k$_{1}$ + k$_{2}$ $\approx $ 3 $\times $ 10$^{-8}$ cm$^{3}$%
/s for S and O, $\langle $v$_{+}$ Y$\rangle _{\mathrm{es}}$ $\sim $ 2 $%
\times $ 10$^{8}$ atoms cm/s, $\Delta $R $\approx $ 0.9 R$_{\mathrm{J}}$,
and H $\approx $ 0.1 R$_{\mathrm{J}}$ determined by the electron-ion
collisions and the tilt of the magnetic axis to the satellite orbit. These
give n$_{\mathrm{o}}$ $\approx $ 70 for S and O. Smyth and Shemansky (1983)
have described the distribution of the neutral oxygen at Io. They find this
torus has a form intermediate to the Na torus and the heavy neutral torus at
Saturn. That is, the oxygen atoms live long enough to distribute
significantly, but still show a higher density close to the source, Io.

\section{Accumulation of a Magnetospheric Plasma}

The neutrals in the extended satellite atmospheres are eventually ionized,
as described above (Brown and Ip 1981; Eviatar and Podolak 1983), in a
manner similar to a cometary coma (Ip et al. 1987). Therefore, the volcanoes
on Io, caused by tidal heating, are the ultimate source of the Io sulfur and
oxygen plasma (Kupo et al. 1976; Pilcher and Morgan 1979; Cheng 1980). The
fresh ions produced from the orbiting neutrals have a relative velocity, $%
\mathrm{\vec{v}}_{\mathrm{r}}$ = $\mathrm{\vec{v}}_{\mathrm{es}}$ + $\mathrm{%
\vec{u}}$ with respect to the planetary magnetic field, where v$_{\mathrm{es}%
}$ is the ejection speed and u is the speed of the satellite through the
plasma. Hence, these ions see a q $\mathrm{\vec{v}}_{\mathrm{r}}$ $\times $ $%
\mathrm{\vec{B}}$ force in a coordinate system fixed to the field lines
(Belcher 1983; Acuna et al. 1983). The net result is that the ions gyrate
around the magnetic field lines having a gyroradius r$_{\mathrm{g}}$ = M(v$_{%
\mathrm{r}}$)$_{\bot }$/qB (see Tables 4.1 and 1.2), where (v$_{\mathrm{r}}$)%
$_{\bot }$ is the component perpendicular to B. With respect to the field
lines, these ions have a mean energy M(v$_{\mathrm{r}}$)$^{2}$/2, which is
superimposed on the corotation velocity. In addition they move along the
field lines with a speed (v$_{\mathrm{r}}$)$_{//}$, where we assume the
planetary rotation is roughly perpendicular to the field lines in the
subsequent discussion. The motion is often defined by a pitch angle [$\alpha
^{\prime }$ = tan$^{-1}$ (v$_{\mathrm{r}}$)$_{//}$/(v$_{\mathrm{r}}$)$_{\bot
}$] and the energy Mv$_{\mathrm{r}}^{2}$ /2 is given as a thermal component
of the plasma superimposed on the corotation of the ions. The fresh ions,
therefore, have an initial pitch angle distribution which reflects the
neutral speed distribution at the point of ionization.

The gyro-energy and pitch angle distribution are subsequently modified by
cooling to the electrons and by scattering among the ions (Smith and Strobel
1985; Kennel et al. 1979). The acceleration of the fresh ions results in a
drag on the magnetic field rotation (Hill 1980) and the ``thermal'' energy
heats the electron component of the plasma (Sullivan and Siscoe 1982). The
plasma so produced increases in density until various loss processes
dominate. Ions are lost by being scattered into smaller pitch angles (travel
along field lines) and by various processes (centrifugal field line loading,
plasma wave scattering, etc.) which can be lumped together as transport
processes (Siscoe and Summers 1981; Cheng 1988; Barbosa 1989). Ions are also
replaced in the plasma by charge exchange with neutrals in the extended
satellite atmospheres. Huang and Siscoe (1987) have discussed cases in which
these various loss processes dominate. The equation for the mean density in
volume V of a single ion type corresponding to the ejected neutrals is 

\begin{equation}
\mathrm{ dn_+ / dt = k_2 n_{-} n_o + J_{UV} n_o - \alpha_r n_{+} n_{-} - 
transport.}
\end{equation}
Here $\alpha_{\mathrm{r}}$, is the