SPECTRAL OBSERVATIONS OF THE ENCELADUS PLUME WITH CASSINI-VIMS

Since VIMS does not resolve individual jets, these data can only constrain the total rate of particles launched from the surface ∫f(v_o, s)dv_ods, where v_o is the radial component of the velocity at the moon's surface and s is the size (radius) of a particle. The function f specifies the distribution of particle velocities as a function of particle size. The primary goal of this analysis is to constrain f because this function can, in principle, be compared with predictions from various physical models for the origin and acceleration of the particles (e.g., Schmidt et al. 2008).

Given the low optical depth of the plume, it is reasonable to assume that after the particles leave the surface they follow purely ballistic trajectories. In this case, the density of the plume versus height above the surface is determined entirely by f. Furthermore, since the VIMS data discussed here only detect the plume up to altitudes of 300 km above the moon's surface (or 550 km from the moon's center), which is well inside Enceladus' Hill sphere radius of 1440 km, we can assume that the moon's gravity dominates the particle trajectories. Conservation of energy then requires that the following quantity is a constant along any particle's trajectory:

$$\begin{equation} v(r)^2-\frac{2GM_E}{r}, \end{equation} \tag{ 3 }$$

where r is the particle's distance from the moon's center, v(r) is the radial component of the particle's velocity, G is Newton's constant, and M_E is the mass of Enceladus (GM_E = 7.21 km³ s⁻² from Jacobson et al. 2006 and Rappaport et al. 2007). We can, therefore, relate the radial component of the particle's velocity at a given radius v(r) to the radial component of the particle's velocity when it is launched from the moon's surface v_o:

$$\begin{equation} v(r)^2=v_o^2-\left[\frac{2GM_E}{r_o}-\frac{2GM_E}{r}\right]=v_o^2-2\Delta \Phi (r), \end{equation} \tag{ 4 }$$

where r_o = 248 km is the radius of Enceladus (Thomas et al. 2007) and ΔΦ(r) is the difference in gravitational potential between r_o and r.

Provided the plume is nearly in a steady state, then there is a straightforward relationship between f and the density of plume particles as a function of height. First, consider particles launched from the surface with an initial speed greater than the escape speed $v_{\rm esc}=\sqrt{2GM_E/r_o} \simeq 240\,{\rm m}\,{\rm s}^{-1}$. When the plume is in a steady state, the flux of particles through a given surface at a distance r from the moon's center is independent of r. Thus, for v_o > v_esc, we can define

$$\begin{equation} d_{\rm escape}(r,s)ds=\int _{v_{\rm esc}}^\infty \frac{f(v_o,s)dv_o}{v(r)}ds =\int _{v_{\rm esc}}^\infty \frac{f(v_o,s)dv_o}{\sqrt{v_o^2-2\Delta \Phi (r)}}ds, \end{equation} \tag{ 5 }$$

where d_escape(r, s) is the differential number density of escaping particles of size s at a radius r. Note this is the differential number density integrated over the surface of a spherical shell, so d(r, s)ds has units of length⁻¹ and d(r,s) has units of length⁻².

Particles with v_o < v_esc, by contrast, will reach a maximum altitude where v(r)² = v²_o − 2ΔΦ(r) = 0 and then fall back to the surface. Thus, at a given r only particles launched with a velocity v_o greater than the critical velocity $v_{\rm crit}(r) =\sqrt{2\Delta \Phi (r)}$ will contribute to this population, and the integrated density distribution d at a given r of trapped particles will be

$$\begin{equation} d_{\rm trapped}(r,s)ds= 2\int _{v_{\rm crit}(r)}^{v_{\rm esc}}\frac{f(v_o,s)dv_o}{\sqrt{v_o^2-2\Delta \Phi (r)}}ds; \end{equation} \tag{ 6 }$$

the factor of 2 in this expression arises because the particles pass through the given radius twice as they rise and fall through the plume.

The total density in the plume is the sum of these two terms:

$$\begin{eqnarray} d(r,s)ds&=& \int _{v_{\rm esc}}^\infty \frac{f(v_o,s)dv_o}{\sqrt{v_o^2-2\Delta \Phi (r)}}ds\nonumber\\ && +\, 2\int _{v_{\rm crit}(r)}^{v_{\rm esc}}\frac{f(v_o,s)dv_o}{\sqrt{v_o^2-2\Delta \Phi (r)}}ds. \end{eqnarray} \tag{ 7 }$$

Given the integrated density of particles at a given height, the brightness of the plume as a function of wavelength can be computed using the appropriate scattering theory. Say the particles are illuminated by flux πF of radiation with wavelength λ. The power scattered by an individual particle per unit solid angle is described by the function dP/dΩ (note in this context P indicates power, not phase function), which depends on F, λ, the scattering angle (180° − phase angle) θ, the radius of the particle s, the optical properties of the particles, etc. A variety of sophisticated models have been developed over the years to calculate the light scattering properties of small grains (e.g., van de Hulst 1957). However, as discussed in detail in Section 4.1, simpler diffraction-based analytical calculations can also provide useful estimates of dP/dΩ in certain cases.

Regardless of how it is calculated, once dP/dΩ is determined for particles of range of different sizes s, the reflectivity of a collection of particles can be evaluated by integrating dP/dΩ over the appropriate size distribution. For example, the reflectivity of a region filled with a population of particles described by the differential size distribution n(s)ds at a given wavelength will be

$$\begin{equation} \frac{I}{F}=\frac{1}{F}\int \frac{dP}{d\Omega }\left[\int n(s)dl\right]ds, \end{equation} \tag{ 8 }$$

where ∫n(s)dl is the particle size distribution integrated along the line of sight. Similarly, the equivalent width of the plume (assuming it is sufficiently collimated) is given by

$$\begin{equation} E(r,\lambda)\simeq \frac{1}{F}\int \frac{dP}{d\Omega }d(r,s)ds, \end{equation} \tag{ 9 }$$

where we identify d as the integral of n(s) along the line of sight and horizontally across the plume (given by Equation 7) and we again assume that the observed light is dominated by radiation scattered from particles near the minimum altitude along the line of sight.

A completely generic set of model size distributions can be generated by specifying the particle density distribution at a discrete set of particle sizes s_i, and then interpolating between these values to all values of s. In principle, an infinite number of s_i are needed to specify a completely arbitrary size distribution. In practice, however, the available spectral data can only constrain the particle size distribution over a limited range of s and cannot resolve infinitely sharp features in the size distribution. Thus, there is an optimal set of s_i that contains the smallest number of parameters needed to reproduce any observed spectrum.

In this case, we are fortunate because the plume data were obtained at such high phase angles that the observed light is predominantly due to diffraction around individual particles. Elementary Fraunhofer diffraction theory can, therefore, provide insights into the connection between the measured spectra and the particle size distribution and help us find the optimal choice of s_i.

Consider a circular conducting disk of radius s illuminated by flux πF of radiation with wavelength λ. The power per unit solid angle scattered by this disk is described by the following function (slightly modified from Equation 9.168 in Jackson 1975):

$$\begin{equation} \frac{dP}{d\Omega }=\pi F s^2 \frac{J_1^2(ks\sin \theta)}{\sin ^2\theta }, \end{equation} \tag{ 10 }$$

where k = 2π/λ, θ is the scattering angle (180° − the phase angle), and J₁(z) is a Bessel function of z.

While the above equation applies to conducting obstacles, a similar expression can be used to approximate the diffracted signal from dielectric particles like those in the plume, accounting for the (possibly wavelength-dependent) index of refraction of the material. Mie theory calculations show that to first order the optical properties of the dielectric do not strongly affect the shape of this part of the phase function (i.e., the θ-dependent terms in dP/dΩ, see Fymat & Mease 1981). Thus, to first order, we only need to multiply the above expression by a scaling factor that is independent of scattering angle but does depend on wavelength. It turns out that the most elementary way to do this is to exploit the optical theorem, which relates the imaginary part of the forward-scattered signal to the total cross section of the obstruction (see Jackson 1975, p. 453; also van de Hulst 1957, pp. 30–36): this means that the desired scaling factor can be derived directly from the so-called extinction factor Q_ext, which is defined as the ratio of the total cross section of the particle to its geometrical cross section and can be computed from Mie theory. For perfect conductors Q_ext = 2, but in general Q_ext is a function of ks and the index of refraction m_r + im_i of the object (van de Hulst 1957, p. 179):

$$\begin{eqnarray} Q_{\rm ext}&=&2-4e^{-\rho \tan \beta }\frac{\cos \beta }{\rho } \left[\vphantom{\frac{\cos \beta }{\rho }}\sin (\rho -\beta)\right.\nonumber\\ &&\left. +\,\frac{\cos \beta }{\rho }\cos (\rho -2\beta)\right] +4\frac{\cos ^2\beta }{\rho ^2}\cos 2\beta, \end{eqnarray} \tag{ 11 }$$

where tan β = m_i/(m_r − 1) and ρ = 2ks(m_r − 1). This function declines to zero as ks approaches zero and asymptotes to 2 when ks > >1.

In the limit where the Mie scattering amplitude function S is purely real, the optical theorem indicates that the scattering law for a dielectric can be approximated as the above expression for a conductor scaled by Q²_ext/4 (Fymat & Mease 1981):

$$\begin{equation} \frac{dP}{d\Omega }=\frac{\pi Fs^2}{4\sin ^2\theta }J_1^2(ks\sin \theta) Q_{\rm ext}^2(ks,m). \end{equation} \tag{ 12 }$$

Thus, the equivalent width of the plume is given by (from Equation 9)

$$\begin{equation} E(r,\lambda)=\int \frac{\pi s^2d(r,s)}{4\sin ^2\theta }J_1^2(ks\sin \theta) Q_{\rm ext}^2(ks,m)ds. \end{equation} \tag{ 13 }$$

Figure 3 compares the spectra derived using this method with the results of full Mie calculations for spheres and irregular particles computed using the techniques outlined in Pollack & Cuzzi (1980) and Showalter et al. (1992) for the case of pure ice grains with d(r, s) ∝ s⁻³ and s⁻⁴ viewed at a scattering angle θ = 20°. The spectra are scaled to match at short wavelengths for clarity. If the simple model yielded spectra with the same shape as the Mie calculations, the fractional difference would be constant. This is approximately true outside of the 3 μm band, which indicates that the simple model can adequately approximate the shape of the spectrum in these regions. However, in the vicinity of the strong ice band, the fractional differences change rapidly. This is because the large imaginary component of the refractive index causes the assumption that S is purely real to be violated. Note that for the irregular particle models (which are better fits to the observed spectra on account of their weaker 1.5 and 2 μm ice bands), the simple model agrees with the more exact calculations to within 50% everywhere outside of 2.5–3.5 μm, which should be sufficient for the purposes of this paper.

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Given this analytical approximation of particles' scattering properties does roughly reproduce numerical calculations, we can now use Equation (13) to determine the best range and spacing of s_i as a basis for a generic suite of particle size distributions. The Bessel function J²₁(kssin θ) has its tallest peak where kssin θ ≃ 2, which at phase angles of 160° (θ ≃ 20°), corresponds to s/λ ≃ 1. Furthermore, outside of the strong 3 μm absorption feature, water ice has a real index of refraction of approximately 1.3, which means that Q_ext(ks) peaks where ks ≃ 6, which also corresponds to s/λ ≃ 1. Thus, for this particular viewing geometry, particles of a size s are most efficient at scattering radiation into the instrument if the radiation's wavelength is comparable to s.

While this result indicates that there could be a relatively direct mapping from observed wavelength to particle size, such a simple result will only occur if we place certain constraints on the overall slope of the size distribution. These constraints are necessary because in addition to the primary peak at kssin θ ≃ 2, the Bessel function has secondary peaks at other values of kssin θ, which correspond to locations where s/λ = 2.8, 4.3, .... Thus, radiation with a wavelength λ could be efficiently scattered by particles of size λ, 2.8λ, 4.3λ, etc. Only if the size distribution is sufficiently steep, and the larger particles sufficiently rare, can we neglect the contribution of larger particles to the scattered light at a given wavelength. Fortunately, this condition does not place a very restrictive constraint on the particle size distribution, since even with a d(s) as shallow as s⁻² the peak at s/λ = 2.8 is 6 times lower than the peak at s/λ = 1. We, therefore, expect that this condition will be satisfied everywhere in the plume.

If wavelength translates directly into particle size, then the VIMS IR spectra will primarily be sensitive to the particle size distribution in the 1–5 μm range. Furthermore, our ability to resolve structures in the size distribution will be limited by the finite width of the peak in the function J²₁(kssin θ)Q²_ext(ks), which has a full width at half-maximum of approximately 0.7 in s/λ. Based on these considerations, we determined that interpolating a size distribution based on the specified densities at s_i = 0, 1, 2, 3, 4, and 5 μm would be sufficient for this analysis (the 0 and 5 μm values are used to provide well-defined endpoints for the interpolation). Note that while the evenly spaced s_i values used here are a reasonable choice, a slightly more optimal spacing would have the ratio of adjacent values s_i be a value specified by the width of the peak.

With the s_i selected, the next step is to determine the array of density values to assign to the s_i that will yield a reasonably complete suite of models. Here it is important to note that in the limit where the index of refraction is constant with wavelength, Equation (13) indicates that if d ∝ s⁻³ then the integrated reflectivity is independent of wavelength. This means the variations in the brightness level across the VIMS IR band measure deviations from this simple power-law form. The size distributions used in this analysis therefore all have the form d = C(s, s_i, c_i)s⁻³, where C(s, s_i, c_i) is a function that varies between 0 and 1. C(s, s_i, c_i) is calculated by assigning a set of values c_i to the selected particle sizes s_i = [0, 1, 2, 3, 4, 5] μm; interpolating between these discrete points to all other particle sizes s; and finally smoothing the size distribution with a moving boxcar average with a smoothing length equal to 1 μm. For this analysis, we computed size distributions for all possible combinations of c_i where each c_i could have any of the values 0, 0.25, 0.5, 0.75, or 1. This yielded a total of 15,625 (5⁶) model size distributions.

In principle, we could calculate the spectrum for each of the 15,625 different size distributions using numerical Mie scattering or physical optics codes. However, the number of models involved makes this task computationally intensive. In this paper, we will instead approximate the spectra using Equation (13), which has the advantage of being faster to compute. While such an approximate method may affect the estimates of the particle size distribution somewhat, we expect that any errors introduced by this simplification will be comparable to uncertainties associated with the particles' shape and composition. For example, Figure 3 indicates that the shapes of the spectra derived with the analytic calculation differ from those of the proper numerical methods by of order 50%, which is comparable to the differences among models with different assumed particle shapes. To further verify the validity of this approach, we also compute spectra for the various best-fit size distributions obtained below using both the above approximate analytic theory and numerical scattering codes for Mie spheres and irregular particle models (see below).

For the purposes of this analysis, we assume the index of refraction values for water ice to evaluate Q_ext (provided by D. P. Cruikshank). This is reasonable, given that E-ring particles are known to be largely water ice (Postberg et al. 2008) and no spectral features beyond the strongest water ice band are apparent in the spectra. If any contaminants are present, they would therefore probably only affect the overall albedo of the particles, which will change the overall normalization of the density distributions, but not the trends with height. Similarly, the use of Equation (13) to approximate the scattering properties of the particles may also affect the density estimates, but should not have a large effect on the shape of the density profiles versus height.

To determine which of the model particle size distributions best reproduces the observed spectra at various locations in the plume, each observed spectrum is compared to the complete array of model spectra. First, all the model spectra are renormalized so that they produce the same mean flux between 0.88 and 1.53 μm as the observed data, and the particle size distributions scaled appropriately. The goodness of fit of the observed shape of the spectra to the normalized models is then evaluated using χ² statistics. These χ² statistics are computed excluding the data from spectral channels associated with the filter gaps and known hot pixels, as well as the channels beyond 4.75 μm because of their extremely low S/N. We also exclude the data between 2.5 and 3.5 μm because our simplistic models do not reproduce the observed spectra well here (see Section 4.1). In total, 162 of the original 256 spectral channels are used to compute the χ² statistics.

The reduced χ² statistics for even the best-fit models are poor (typically around 5). This is probably not due to a deficiency in the models, but instead because the primary data reduction underestimates the error bars on the spectra. As can be seen in Figure 2, the scatter among nearby data points in a single spectrum is often larger than the error bars, so no smooth curve can fit these data well. The reasons for this are still not fully understood, but it may be due to slight offsets between spectral channels induced by the incomplete background subtraction. In spite of this issue, the χ² statistics still provide a useful measure of how well the various models fit the data relative to one another, and the best-fit models should still be the ones with the smallest χ².

In general, the above calculations do not produce a single uniquely favored model, but instead yield multiple models with comparably low values for χ². This is partially because there are some redundancies in the array of models (i.e., c_i = [0.5, 0.5, 0.5, 0, 0, 0] and [1, 1, 1, 0, 0, 0] will produce the same normalized spectrum) and the sampling of model size distributions is relatively coarse (so that the true best-fit model probably lies between some of the calculated models). However, we also find that models with similar densities between 1 and 4 μm but radically different densities below 1 μm and above 4 μm have roughly the same χ² statistics. This should not be surprising, given the above arguments that the data would not be very sensitive to these regions of the size distribution. Thus, to prevent these outlying regions of particle size distribution from having an undue affect on the overall fit, we impose a constraint that the c_i values for 0 and 5 μm must be less than or equal to those for 1 and 4 μm, respectively. After applying this constraint, a single estimate for the size distribution responsible for generating the observed spectrum is computed as the weighted mean of the individual models with the lowest χ² statistics. To obtain the weights for the models, the χ² statistics are first rescaled so that the best-fit model has a reduced χ² = 1 (this amounts to rescaling the error bars by a fixed factor). The probability to exceed the rescaled χ² then provides the appropriate weight for each model.

Figure 2 (as well as the figures in the Appendix) includes curves showing the best-fit model spectra overplotted on the data. Outside the 3 μm band, these models do a good job in reproducing the spectral slopes observed in the data, which implies that our models are sufficiently generic to reproduce all the spectral variations observed in the plume. Also plotted in these figures are the predicted spectra of the best-fit size distributions computed using numerical scattering codes for Mie spheres and irregular particle models 3 "Cubes" and 5 "Plates" from Pollack & Cuzzi (1980). The numerical spectra for the spheres, cubes, and plates are divided by factors of 2.2, 1.6, and 2.3, respectively, to match the analytical model spectra. The different calculations thus give somewhat different values for the absolute density of the plume, which limits our ability to precisely constrain the total flux of particles from the surface. However, after normalizing each set of the numerically computed spectra by a single factor, they all show very similar spectral slopes outside the strong 3 μm band and nearly identical trends in brightness with altitude as the spectra derived from the approximate analytical method. We can, therefore, infer that this method provides reliable information about the shape of the particle size distribution and the relative particle density at different altitudes in the plume.

Figure 4 illustrates the size distributions derived from the integrated spectra for the three different combined cubes. We caution against taking these size distributions too literally, given the approximations used in computing the model spectra, which are only good to within 50%. However, we do wish to point out that all three cubes show similar features and all show consistent and repeatable trends in the size distributions with altitude. This provides some evidence that the above fitting procedures can yield useful information about the underlying particle size distribution. Furthermore, we may note that all the distributions indicate all observed regions of the plume are strongly depleted in 4 μm particles relative to a s⁻³ power-law size distribution.

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Figure 5 shows the density profiles for 1, 2, and 3 μm particles. (There is insufficient signal in the 4 μm data to produce useful profiles.) Recall the altitudes used here correspond to the minimum altitude for each row that was integrated to produce a single spectrum (see Section 2). In this plot, there are systematic differences between the profiles derived from the different cubes, specifically the later cubes give systematically higher densities than the earlier cubes. The observed differences are of order 25%–50% in the worse case, which seem too large to be due to uncertainties in the pixel scale. More likely, uncertainties in the altitude scale, which would allow the curves to slide back and forth by ∼15 km may be responsible. We present the curves here without any attempt to adjust the scaling or altitudes to bring the data sets into alignment in order to illustrate the range of uncertainty in the normalization of these curves.

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In spite of these uncertainties in the amplitude of the density profiles, the three data sets show very consistent trends with height. Between 50 and 300 km above the surface of Enceladus, the 1 μm particle density declines by only a factor of 2, while the 2 μm particle density declines by about a factor of 4, and the 3 μm particle density declines by over a factor of 6. Again, the consistent trends among the different observations indicate that these data can yield useful information about the distribution and dynamics of the plume. Specifically, these data indicate that bigger particles are launched at lower velocities on average than smaller particles.

As discussed in Section 3.1, the density of particles at any given altitude is an integral over particles traveling with a range of speeds. However, since particles must be launched faster than a certain speed at the surface to reach a certain altitude, we can use an onion-peeling deconvolution algorithm somewhat analogous to those used to derive radial profiles of diffuse edge-on rings (see e.g., Showalter et al. 1987; de Pater et al. 1999, 2004) to derive the velocity distribution from the density profile. Starting from the top of the plume, we use the density of the plume at a given altitude to determine how many particles were launched fast enough to reach that altitude. The contribution of these particles to the rest of the profile can then be computed and removed before using the particle densities at lower altitudes to determine the number of particles that were launched at lower speeds. Because this method amounts to taking the derivative of the density profile, small errors in the density can be greatly magnified with this sort of iterative procedure. In the current situation, however, this problem is ameliorated because particles launched at a particular speed produce a density profile with a strong peak where they turn around and fall back to the surface. This means that particles detected at high altitudes do not contribute as much to the density at lower altitudes, which improves the stability to this inversion procedure.

The specific algorithm used for this analysis begins with a mean density measurement for plume particles of a given size between radii r₁ and r₂ from the moon's center (r₂ > r₁). Let us assume that the contribution from all particles with $v_o>v_2=v_{\rm crit}(r_2)=\sqrt{2GM_E/r_o-2GM_E/r_2}$ has already been subtracted from the density measurement. Furthermore, particles must be launched with $v_o>v_1=v_{\rm crit}(r_1)=\sqrt{2GM_E/r_o-2GM_E/r_1}$ to reach the observed radial range. Assuming that for v₂ > v_o > v₁ the velocity distribution function f(v_o) is constant, the density of particles as a function of radius in the bin is given by (from Equation 7)

$$\begin{equation} d(r)=2f(v_o)\int _{v_{\rm crit}(r)}^{v_2}\frac{dv_o}{\sqrt{v_o^2-2\Delta \Phi (r)}}, \end{equation} \tag{ 14 }$$

which can be evaluated to give

$$\begin{equation} d(r)=2f(v_o)\ln \left[\frac{\sqrt{1-r_o/r_2}+\sqrt{r_o/r-r_o/r_2}}{\sqrt{1-r_o/r}}\right]. \end{equation} \tag{ 15 }$$

The average density in the bin D derived from the spectral inversions is, therefore,

$$\begin{equation} D=\frac{1}{r_2-r_1}\int ^{r_2}_{r_1}d(r)dr =\frac{2f(v_o)}{r_2-r_1}I(r_0,r_1,r_2), \end{equation} \tag{ 16 }$$

where I(r_o, r₁, r₂) is a factor that can be derived by integrating the logarithm in the above expression for d(r). Note that while a closed-form solution of this integral exists, in practice it is easy enough to evaluate numerically. We, therefore, obtain the following estimate for f(v_o):

$$\begin{equation} f(v_o)=\frac{D \times (r_2-r_1)}{2I(r_o,r_1,r_2)}. \end{equation} \tag{ 17 }$$

Given this value for f(v_o), we can compute the contribution of these particles to the remainder of the density profile,

$$\begin{eqnarray} d^{\prime }(v_o,r)&=&2f(v_o)\int _{v_1}^{v_2}\frac{dv_o}{\sqrt{v_o^2-2\Delta \Phi (r)}}\nonumber\\ &=& 2f(v_o) \ln \left[\frac{v_2+\sqrt{v_2^2-2\Delta \Phi (r)}}{v_1+\sqrt{v_1^2-2\Delta \Phi (r)}}\right], \end{eqnarray} \tag{ 18 }$$

which can be averaged as necessary to compute and subtract the contribution of these particles to the density estimates at lower altitudes. The entire procedure can then be repeated for the next lower density measurement, etc.

The highest altitude bin is a special case for this analysis because all particles faster than a threshold speed pass through it, including escaping particles, so there is no way to determine if the particles detected here are near their maximum altitude or if they are moving well beyond the escape speed. If all particles detected in this bin are assumed to reach their highest altitude in this bin and then fall back to Enceladus, then this bin is treated just like all the others. By contrast, if all particles in this bin were launched much faster than the escape speed, then they travel with roughly constant velocity at all altitudes and contribute roughly the same density to all the measurements. To bracket the range of possibilities in the high-velocity end of the size distribution, we will here compute the velocity distributions for both these limiting cases. Intermediate cases between these limits are treated as if the topmost bin contains some fraction of particles near their maximum altitude, while the remainder are traveling well beyond the escape speed.

Recall that due to its finite resolution and field of view, VIMS was only able to measure the plume signal between altitudes of 30 km and 300 km during these observations. These measurements, therefore, constrain the velocity distributions between 80 m s⁻¹ and 180 m s⁻¹, or between one-third and two-thirds of the escape speed. Particles launched from the surface at less than 80 m s⁻¹ do not reach sufficient altitude for VIMS to resolve them from the bright limb, while particles launched much faster than 180 m s⁻¹ sail beyond the edge of the field of view. This analysis, therefore, provides information about particles that fall back to Enceladus' surface.

Figure 6 shows the velocity profiles derived from the density profiles. A range of solutions are shown corresponding to different possible velocity distributions for the material in the highest altitude bin. This uncertainty in the high end of the velocity distribution does not greatly affect the 3 μm velocity profiles because the density in the highest bins is very low. By contrast, the density of 1 μm particles in the highest altitude bin is not much less than the density elsewhere in the plume, so uncertainties in the upper end of the velocity distribution lead to significant uncertainties for the remainder of the profile. Note also that the discrepancies between the density profiles from the different cubes show up again here in the velocity profiles.

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In spite of these issues and uncertainties, the trends with velocity and particle size are reasonably consistent. The velocity distribution for 1 μm particles is almost a constant, while those for the 2 and 3 μm particles are progressively steeper. This confirms that larger particles are launched from the surface at lower speeds on average than smaller particles. In particular, it appears that 3 μm particles are launched from the surface with a maximum speed of only 160–180 m s⁻¹.