Large Mass Inflow Rates in Saturn’s Rings due to Ballistic Transport and Mass Loading

Saturn’s rings are continuously subjected to bombardment by extrinsic micrometeoroids. This affects the rings in two principal ways. First, incoming impactors can lead to structural evolution of the rings because the copious ejecta produced by impacts cause mass and angular momentum exchanges between different ring regions [50, 51, 19, 23, 21, 22, 29, 30, hereafter I83, I84, D84, D89, D92, D96, E15 and E18]. The meteoroids can also deposit mass and exert net torques on a global scale (I84; D96; E15; E18). Second, the predominantly icy rings become more and more polluted because the incoming material, which has a large non-icy component, darkens the rings over time [15, hereafter CE98]. In particular, lower mass surface density $\Sigma$ and/or optical depth $\tau$ regions such as the C ring and Cassini division darken faster than the denser, more massive A and B rings. The susceptibility of the rings to the effects of micrometeoroid bombardment comes about in no small measure because of the rings’ massive surface-area-to-mass ratio, about a factor of $\sim 10^{4}-10^{5}$ times more than for a moon of equal mass (E15).

Key parameters, such as the rate of deposition of extrinsic material, the ring mass, and the current level of ring pollution, as well as an understanding of the physics of how material is redistributed in the rings over time, provide a powerful age-dating tool for the rings. Prior to the Cassini Grand Finale, some of these parameters remained the subject of debate and speculation. That the rings might in fact be younger than previously thought was first suggested by Voyager measurements showing that the albedo in the A and B rings were inconsistent with the fraction of pollutant they would have obtained over the age of the Solar System given the accepted micrometeoroid influx at the time [18, CE98]. The idea was considered unpopular because there was no apparent statistically plausible way to form the rings so long after the Late Heavy Bombardment if the rings were only a few hundred million years old [see, e.g., 17, 8]. Thus, the view that the rings cannot be young persisted until Cassini could provide definitive evidence.

Analyzing Cassini radiometer measurements, Zhang et al. [92, 93] found that the non-icy material volume fractions in the C ring and Cassini division were $\sim 1-2\%$, whereas the fractions in the relatively more massive A and B rings were $\sim 0.1-0.5\%$ . These fractions are consistent with HST [16] and VIMS results [9] if the pollutant is intramixed in fine grains and shadowing is allowed for. While visible and IR wavelengths only sample the surface layers of the particles, the microwave measurements sample the bulk of the ring particles, so that the radial variation of pollutants across the rings indeed indicates that the rings overall are not very polluted and, moreover, exhibit a distinct contrast in the level of pollution between low- and high-$\tau$ regions. Although low-$\tau$ regions darken faster than high-$\tau$ regions, this contrast diminishes over long times because the darker regions become the dominant source for darkening of the brighter higher-$\tau$ regions due to ballistic transport [E15; 28]. While there is evidence that the middle C ring is polluted with non-icy material to higher levels than the rest of the C ring, Zhang et al. [93] showed that this can be explained by rubble from a captured Centaur that has become mixed into this region over the last few to ten million years or so. It does not affect the estimate of an overall pollution age.

In another caution about pollution ages, the Conclusions Section of Buratti et al. [7] asserts that their evidence for icy E ring dust accumulating on the surfaces of outer ring moons may invalidate young ring ages. Actually, what these authors find is that Pan in the Keeler gap has the same red color as the surrounding rings and that the reddening diminishes outward systematically as the distance of ring moons from Saturn increases. This suggests that the icy E ring dust, or whatever else may be the source of the icy material, is being deposited only in the outermost ring region and does not penetrate as far as Pan. Pollution age analyses are not sensitive to the outer few thousand kilometres of the rings but rely on comparisons between optically thick and thin regions much closer to Saturn. The orbital dynamics of E ring dust [38, 39, 40] make it unlikely that it would penetrate deeply into the main rings. So, while a refined treatment of pollution should probably consider sources of icy material, it is unlikely to affect ages based on meteoroid pollution for the bulk of the rings (see, however, Section 6.4.2).

Another key piece of data for estimating pollution ages is the input rate from meteoroids. Several authors estimated the mass flux of meteoroids at Saturn prior to Cassini [e.g., 65, 37, 14, hereafter CD90]. After a careful review of this work, CE98 showed in their Figure 17 that their improved estimate (the solid histogram in the figure) agreed with Pioneer and Ulysses data for 10 $\mu$m sized particles and smaller. They concluded that the CD90 value for a one-sided interplanetary flux of 4.5 $\cdot$ 10^-17 g cm^-2 s^-1 for small meteoroids at Saturn, the value used in ballistic transport and pollution calculations up to that time, was good to within a factor of three up or down. Here, “one-sided” means the flux measured passing through an area only from one side to the other. The Cassini Cosmic Dust Analyzer (CDA) experiment has now measured the meteoroid flux at Saturn and infers an interplanetary flux at 10 AU a factor of two less than the CD90 value [2, 1, 55, 54, see Section 5.1].

Furthermore, CDA finds, through orbit reconstruction, that the population of impactors is consistent with Edgeworth-Kuiper Belt (EKB) projectiles, not the Oort Cloud source that had been previously assumed (CD90; CE98). Particles from this dynamical population have much lower velocities as they enter Saturn’s Hill sphere (which we call “at infinity”) than cometary particles, and thus are about ten times more focused gravitationally by the planet. In other words, for a given micrometeoroid flux at infinity, the impact flux on the rings is an order of magnitude higher than previously assumed, thus lowering age estimates by about the same factor. This is illustrated by Figure 9.5 of E18, where the computation of CD90 was redone using a meteoroid velocity at infinity at Saturn of 3 km s^-1 instead of the 14.4 km s^-1 used in CD90. The increase in the ejecta intensity is an order of magnitude¹¹1The actual mean velocity at infinity as derived by CDA is $4.3$ km s^-1 lowering the estimate of Estrada et al. [30, who estimated a factor of $\sim 25$ higher] by a factor of $\sim 2-3$, but still overall an order of magnitude in ejecta intensity.. A similar factor can be obtained by applying the various gravitational focusing formulae referenced in Appendix B of CE98. All these computations involve approximations. A more careful treatment of gravitational focusing is needed but would be a substantial project. However, the order of magnitude difference in focusing between the two meteoroid populations is almost certainly valid and has been accepted for some time [e.g., 65].

Finally, the total ring mass plays a critical role in determining the exposure age of the rings. The post-Voyager estimate for the rings was roughly a Mimas mass [25], but if the ring mass surface density, say in the opaque, dense B ring core were an order of magnitude higher – an idea that seemed plausible from discovery of the rings’ clump-and-gap structure in which density waves could hide large amounts of mass [86, 79] – then the rings would be much more resistant to pollution and could be ancient [24]. However, from gravity field measurements in the final orbits of the Cassini Grand Finale, the mass of the rings has been determined to be $\sim 0.4$ Mimas masses [49], even lower than the post-Voyager estimate and similar to estimates from wavelet-based analyses of several density waves in the B ring [45].

Taken together, the direct bulk pollution measurements, the characterization of the meteoroid impactors, and the determination of the ring mass unambiguously suggest a youthful age for the rings. As discussed in detail by E18, with the new observational constraints, the ages deduced for the rings from meteoroid bombardment based on various global, structural, and pollution considerations all seem to agree on an age on the order of at most a few $\cdot\;10^{8}$ yrs. Rings with ages $\lesssim$ few $\cdot\;10^{8}$ yrs are consistent with the ring age calculated by Kempf et al. [54].

The end of the Cassini mission provided a unique opportunity to determine additional constraints for the rings. Observations conducted in its final 22 orbits, when the spacecraft flew through the 2000 km-wide region that separates the rings from Saturn’s upper atmosphere, measured the amount of ring material falling into the planet, thereby providing another means by which to constrain not only how old the rings are but also estimate how long their remaining lifetime may be. These final orbits also allowed for the determination of the composition of the ring material and to test the so-called “ring rain” hypothesis. In this scenario, some fraction of the ejected material resulting from micrometeoroid impacts (in the form of ions or nanometer-to-sub-micron-sized grains), which can be charged either by micrometeoroid impact-produced plasma or photoionization, travel along magnetic field lines and fall into Saturn’s (preferentially southern) atmosphere [11, 69, 53, 52]. The transport of this ejected material, presumed to be mostly water, into the Saturnian atmosphere is thought to explain the latitudinal pattern of ionospheric H${}_{3}^{+}$ infrared emission [e.g., 71, 64].

Cassini’s Magnetospheric Imaging Instrument (MIMI) which measured neutral atoms, ions and electrons was deployed, in part, to test the ring rain hypothesis. MIMI detected a very small flux of $\sim 1-2$ nm radius grains (assuming the density of ice) precipitating into Saturn’s atmosphere through the process of atmospheric drag due to kinetic collisions with exospheric H and H₂. Modeling of the data indicate that the grains, likely originating from the D68 ringlet at the inner edge of the D ring and mostly confined within $\pm 2^{o}$ of the ring plane, precipitate at a rate of $\sim 5.5$ kg s^-1 [63]. These authors also find that the charged component of these grains is relatively low in the midplane, but reaches a maximum in the wings of the distribution at higher latitudes ($\pm 5^{o}$) consistent with transport along magnetic field lines associated with ring rain. This flux is small, but may not be surprising given that MIMI only samples the low end of the particle size distribution, whereas larger particles dominate the mass density [48].

The CDA, which is sensitive to larger particles²²2 The CDA instrument has multiple impact targets that allow for sensitivity to a range of particle size [see 85]., was also active in the final orbits and found that the primary flux into the planet arises from grains $\gtrsim 10-40$ nm in radius entering at moderate latitudes. Larger grains (hundreds of nm) are also seen confined to a band within a few hundred km of the midplane and with a lower abundance [48]. Their simulations indicate that the observed population of nanograins are fast ejecta released from the surfaces of the rings due to micrometeoroid impacts. These nanoparticles travel along magnetic field lines and are deposited in the atmosphere over a latitudinal range of $\pm 40^{o}$. Hsu et al. [48] estimate an ejection rate for these tens of nanometer-sized particles to be 1800 to 6800 kg s^-1 of which 18% arrive at Saturn as ring rain, or roughly 320 to 1200 kg s^-1. Of this fraction, 70% is deposited near the midplane, while 30% (100 to 380 kg s^-1) is deposited in the mid-latitude ring rain regions.

The measured mass inflow rate is over an order of magnitude more than the 3 to 20 kg s^-1 of water product initially thought to be needed to account for the H${}_{3}^{+}$ emission [64]. However, a new analysis of the ground-based Keck II data [71], in which the temperature, density, and cooling rate parameters for H${}_{3}^{+}$ were directly derived, now estimates that the required mass flux is 432 to 2800 kg s^-1 of charged water products [70]. This is interesting in that the composition of the nanograins measured by CDA were found to be that of water and silicates, with the measured silicate mass fractions ranging from 8 to 30% and the silicate fraction being maximum approaching the ring rain latitudes. Since the rings are by mass $>95$% water ice, the depletion of water in the CDA data suggests a mechanism at work to erode water ice from the grains, but the ultimate fate of that water is not clear. These recent results suggest that the water lost from these grains may indeed be ending up in the atmosphere³³3The measured mass fraction in the midplane is close to C ring composition in terms of volume fraction ($\sim 3$%) as derived by Zhang et al. [93]..

The Cassini Ion Neutral Mass Spectrometer (INMS) experiment, which measured the composition of Saturn’s equatorial upper atmosphere and its interactions with material originating from the rings, found a surprisingly large mass influx of between 4800 and 45000 kg s^-1 within a latitude band 8^o near the equator [90]. In addition to observing the infall of water, substantial amounts of volatiles and impact fragments of organic nanoparticles were also measured. These authors concluded that this large mass flux likely requires the regular transfer of material from the C ring to the D ring. Unlike CDA and MIMI, which were sensitive to a specific range of particle radii, INMS sampled remnants of infalling material regardless of original particle size and so may offer a more appropriate measure of the total mass influx from the rings into the planet. The minimum and maximum values of the equatorial mass flux, which were derived from three low altitude revs, are significantly larger than the influxes estimated by MIMI and CDA and larger than the amount of influx needed to explain the ring rain effect.

It is the source of this large overall mass influx that we endeavor to explain in this paper as a natural by-product of outward angular momentum transport by BT and of meteoroid mass loading. Specifically, our aim is, for the first time, to quantify the inward drift of ring material from the main rings into regions close to the planet due to micrometeoroid bombardment and to determine how this compares with the overall mass influx of material measured in the final orbits of the Cassini Grand Finale. If they are comparable, it would provide another confirmation of the importance of meteoroid bombardment for ring dynamics and a consistency check for age dating the rings.

Saturn’s rings are subject to impacts by small meteoroids, with the bulk of the incoming mass in the form of micron-to-tens-of-micron-sized particles of interplanetary origin [CD90; 55, 1, 54]. The dust strikes the ring particles at hypervelocity with speeds of 10’s km s^-1. The primary result of impact is release of a high yield of small ejecta particles with speeds relative to their ring particle of origin that range up to 100’s m s^-1 or more. This is true whether the impacts are cratering events or cause disruption of the ring particle. The fragments of a disrupted ring particle can formally be treated in BT as ejecta. The ejecta that escape their radius of origin arc above the rings on ballistic elliptical trajectories which carry them to ring regions separated in radius by a few to a few thousands of km. Depending on the relative optical depths of the regions where the ballistic trajectories intersect the ring plane, some or all of the ejecta are absorbed by the rings at a radius different from their radius of origin.

A characteristic time scale associated with BT, called the “gross erosion time” (D84), is defined as the time it would take a ring region to erode away completely if all locally produced ejecta were lost and no ejecta were gained from elsewhere. It is given by

BT usually manifests itself on a time scale longer than $t_{\rm{G}}$ because not all ejecta from $r$ are lost and ejecta are also absorbed from other ring regions. It is the net effect of ejecta exchanges between ring regions that produces transport of mass and angular momentum. Net mass exchange leads to what we call a “direct” change in the surface mass density of the rings. In addition, because the ejecta generally have specific angular momentum different from their parent ring particles and also different from the ring particles that absorb them, ejecta exchanges cause the specific orbital angular momentum of the local ensemble of ring particles to change. The result is a slow drift to a different ring radius. The divergence of this local mass flux also changes the ring surface density.

The following equation from E18 describes the result of these two effects on the evolution of the ring surface density $\Sigma$,

The radial drift speed $v_{\rm{r}}$ in Eq. (2)

The radial drift due to mass loading and associated torques is given by

There are a few features of the expected angular and speed distributions of ejecta that are worth noting before we consider approximations to the gain and loss integrals.

Predominance of Prograde Ejecta. Because of the aberration of the meteoroid influx by the ring particle orbital motion (CD90), impacts tend to occur on the leading hemispheres of ring particles. So the ejecta velocities from non-disruptive impacts tend to have a prograde sense with respect to the ring particle orbital velocity. The vertical thickness of the ensemble of ring particles is only tens of meters [10] with random speeds of less than a cm s^-1, much smaller than the ejecta speeds. So, as discussed in D84, D89, and D92, prograde ejecta follow elliptical orbits that arc above the ring plane and re-intersect it at a larger radial distance $r_{\rm{int}}$. Prograde ejecta carry away more specific angular momentum than required for a circular orbit at $r$ but arrive at $r_{\rm{int}}$ with less specific angular momentum than needed there for a circular orbit. Retrograde ejecta have the opposite signs of angular momentum differences. Because the cratering ejecta are predominantly prograde, they represent a net outward transport of angular momentum. However, both prograde ejecta lost from $r$ to $r_{\rm{int}}$ and those gained at $r$ from an interior radius decrease the specific angular momentum at $r$ and $r_{\rm{int}}$, and so cause an inward drift at both locations.

As shown in Fig. 12 and Eqs. (41) to (46) of CD90, the prograde ejecta outweigh retrograde ejecta by a considerable factor when the impactors are predominantly Oort Cloud cometary dust. The ratio can be seen to be about 2.5 to 1 for the approximations used in Fig. 3 of Latter et al. [58]. The Cassini Dust Analyzer (CDA) results [55, 1, 54] show that the bulk of impactors on Saturn’s rings are not cometary in origin but have the dynamics of dust originating in the Edgeworth-Kuiper Belt, i.e., low-to-moderate eccentricity and low, prograde inclinations in the heliocentric frame. Unlike the cometary population that is isotropic in the heliocentric frame, the EKB population tends to be isotropic in the frame of the planet and is characterized by a much lower speed $v_{\infty}$ at the Hill radius of Saturn compared to Oort Cloud projectiles. As a result, the EKB micrometeoroids are much more gravitationally focused at the rings (Section 2.1). E18 redid the CD90 analysis for EKB impactors in an approximate manner and found that the resultant angular distribution of the ejecta is even more prograde than the cometary one, with a ratio of prograde to retrograde ejecta of more like 7 to 1 (see Figs. 9.4 and 9.5 of E18), thus accentuating the radial inward drifts.

Power-Law Speed Distribution. We characterize ejecta speeds $v_{\rm{ej}}$ by the parameter $x=v_{\rm{ej}}/v_{\rm{c}}$, where $v_{\rm{c}}$ is the local circular orbit speed. As discussed in E15 and E18, the speed distribution from non-disruptive hypervelocity impacts is likely to be a steep power law $x^{-n}$ with $n$ between 2 and 3, and the speeds in this distribution are likely to span the range from a few to up to several 100 m s^-1. For example, the cumulative ejecta speed distributions from laboratory experiments [47] suggest values of $n\sim$ 2.1 to 2.7 for a wide range of target strengths (higher $n$’s) and porosity (lower $n$’s), and the hypervelocity impact experiments of Koschny and Grün [56, 57] using ice and ice/silicate targets produce maximum velocities of order several 100 m s^-1.

As shown by the $n=$ 2 and $n=$ 3 curves in Figure 9.10 of E18, such $n$-values combined with a speed distribution ranging up to a few$\cdot$100 m s^-1 are probably required to produce the observed features called “ramps”, which are linear in normal optical depth $\tau$ and exist just inside the steep A and B ring inner edges (see Figs. 9.1, 9.2, and 9.10 of E18). Production of ramps for $n$ = 3 was also discussed extensively in D92 and E15.

One possible complication is that Koschny and Grün [56, 57] found the maximum velocity of ejection depended on the angle of ejection. CD90 included an ejecta cone model for impacts, including how the cone varied with angle of incidence of impact. The CD90 results ended up being averaged over all possible impacts, and much of this detail washes out. CD90 did not include variation of the ejecta speed with ejection angle. This could be done but would wash out by the time impacts at all incidence angles were considered. Thought should be given to effects like this when the CD90 computations are redone in the future, but what probably matters more is the average overall ejecta velocity distribution. The BT mass inflow results of Section 4.2 are more sensitive to the lower ejecta speed cutoff of the power law, not the upper cutoff. The upper cutoff matters more for explaining the extent of the observed ramps.

Although disruptive impacts are also likely to occur, we have not yet tried to characterize them or their BT implications in detail. Such impacts will probably include a cratering component of high-speed ejecta from the initial impact plus a high yield of slower moving, possibly more retrograde ejecta due to the breakup of the ring particle. In this paper, we focus on the higher-speed prograde yield from cratering events, which in fact will be most effective at producing radial mass influx in the rings. We offer the ramps as observational evidence that a prograde power-law component does dominate BT at higher speeds.

The gain and loss terms in Eqs. (2) and (4) are given by Eqs. (13), (14), (16), and (17) of E15. We describe them in this subsection only qualitatively. They involve triple integrals, two integrals over the solid angle of the angular ejecta emission function and one over the ejecta speed distribution. The emission angles and speeds determine, for the $\Gamma$’s, where ejecta that are absorbed at $r$ have come from and, for the $\Lambda$’s, where ejecta emitted at $r$ re-intersect the ring plane. In addition, these integrals include a probability function that uses the optical depths at the radii of emission $r_{\rm{ej}}$ and re-intersection $r_{\rm{int}}$ to determine what fraction of ejecta get reabsorbed at $r_{\rm{ej}}$ rather than at $r_{\rm{int}}$. In general, this function depends on the direction in which the ejecta are launched, because the probabilities are determined by slant path optical depths. At low to moderate $\tau$’s, ejecta may execute more than one orbit before being absorbed at one or the other of these radii.

The loss and gain integrals also contain an ejecta emission rate function which depends on $r$ and $\tau$. As given by Eq. (9.6) of E18, CD90 found that a good approximation to the $r$ and $\tau$ dependence of the local ejecta mass emission rate per unit area⁴⁴4This function was determined for Oort Cloud projectiles. It will probably differ somewhat for the EKB population (see Sec. 5.1). is

To an approximation good enough for our purposes,

We first address BT by considering the special case of a disk with constant $\Sigma(r)$ and $\tau(r)$. As demonstrated in D89, E15, and E18, this results in a steady-state behavior to a high degree of approximation away from boundaries, with deviations only to second order in $x$. Even the largest ejection speeds in the power-law distribution are expected to be only 100’s m s^-1, leading to $x\sim{\rm{few}}\cdot 10^{-3}$ or $x^{2}\sim 10^{-6}$ to $10^{-5}$. In other words, it would take about $10^{5}$ to $10^{6}$ $t_{\rm{G}}$ to see large deviations from the steady state, a time comparable to or greater than a Solar System lifetime. Just as for a steady-state gaseous accretion disk, the uniform ring case is only an idealization and is significantly affected by the imposition of realistic boundary conditions.

We explore this analytically by making some approximations similar to those made by Lissauer [61] and Durisen [20, herafter D95]. First suppose that there is only one ejecta speed, characterized by $x$, and that all the ejecta are prograde. Suppose further that the re-intersection radii of the prograde ejecta are distributed uniformly from their radius of origin $r_{\rm{ej}}$ to $r_{\rm{ej}}+\delta r_{\rm{max}}$, where $\delta r_{\rm{max}}$ is the maximum “throw distance” from $r_{\rm{ej}}$ that an ejectum can reach. To first order in $x$, D89 showed, using Keplerian orbits, that $\delta r_{\rm{max}}\approx 4xr_{\rm ej}$. This drastically simplifies the angle and speed integrations in the $\Gamma$’s and $\Lambda$’s.

The gain and loss integrals also contain relative probabilities of absorption at $r_{\rm{ej}}$ and $r_{\rm{int}}$. As discussed in D95 [see also 58], a reasonable approximation to this probability function for a uniform ring is

There are two limiting cases of Eq. (9). For $\tau\rightarrow 0$ the probabilities are 1/2. An ejectum emitted at $r_{\rm{ej}}$ is just as likely to be reabsorbed at $r_{\rm{ej}}$ or $r_{\rm{int}}$. For $\tau\rightarrow\infty$, on the other hand, the probability of absorption at $r_{\rm{int}}$ is unity and the probability of reabsorption at $r_{\rm{ej}}$ is zero. We will consider these two limiting cases, and then the results for a general $\tau$.

The expressions in Eqs. (10) to (15) below are cited by reference to equations from appendices in D92 and E18. The reader who wishes to verify these results can use the linearized gain and loss integrals given, in slightly different notation, by Eqs. (22) to (30) of D95⁵⁵5A factor of $h_{\rm{c}}$ is erroneously missing from the RHS’s of the angular momentum loss and gains integrals in Eqs. (24) and (25) of D95. This omission is a typo and does not affect any other results in D95.. Set $A$ and $B=0$ in those equations and adopt $g(\cos\theta)=1$ from $\cos\theta=$ 0 to 1 and $g(\cos\theta)=0$ otherwise. The $P_{0}$ in D95 is $P(\tau)$ given above, with 1/2 for very low $\tau$ and 1 for very high $\tau$. The zero-order terms cancel when the gain and loss integrals are subtracted, and the first-order terms lead to the RHS’s of Eqs. (10) to (15) below.

Insight into the physics can be gained from understanding the $x\,{\rm cos}\,{\theta}$ terms in the equations of D95. The factor $1-8x\,{\rm cos}\,\theta$ in $\Gamma_{\rm m}$ comes from the flat disk geometry of the rings, which causes the ring area elements for emitted ejecta to be smaller than the ring area elements that they are transported to. Hence, for prograde ejecta, $\Lambda_{\rm m}<\Gamma_{\rm m}$ to order $x$ because ejecta emission annuli have less area than the absorption annuli. In other words, less mass is gained from inner ring regions than is lost to outer regions for purely prograde ejecta, leading to a decrease in $\Sigma$ due to direct mass exchange. The difference in the angular momentum exchange integrals is that there is an extra factor $1-x{\rm cos}\,\theta$ in $\Gamma_{\rm h}$ due to the $x$ and ${\rm cos}\,\theta$ dependence of the ejecta angular momentum. So the radial mass influx is non-zero to order $x$. For a uniform ring, the divergence of the radial flux exactly cancels the mass deposition to order $x$.

Low Optical Depth. For a uniform ring, far from any boundaries, with these approximations, Appendix B of D92 and Appendix A of E18 show that we get, with $\tau\rightarrow 0$, to first order in $x$,

High Optical Depth. In the limit $\tau\rightarrow\infty$, these same expressions become, to first order in $x$,

General Optical Depth. The general case has the $\tau$-dependent $P$ function.

If we multiply (15) on both sides by $\Sigma$, we get mass inflow in units of mass per unit length of circumference per unit time. So, multiplying by $2\pi r$ gives the total mass inflow rate $\dot{M}_{\rm{bt}}$,

Strictly speaking, Eq. (16) is valid only for a single value of $x$, but, when there is an $x$-distribution, each $x$ will contribute to the influx according to Eq. (16) but with appropriate weighting. Let’s assume that $x$ obeys an $x^{-3}$ power law from a lower limit $x_{b}$ up to an upper value $x_{t}$ much like what is assumed in D92 and E15. Then, for $x_{t}>>x_{b}$, the average $x$ weighted by the power law is $2x_{b}$, and we get the case for general $\tau$ and an $n=3$ power law as

While these mass inflow rates $\rightarrow 0$ as $\tau\rightarrow 0$, it is important to realize that the BT radial inward drift speed $v_{\rm{r}}^{\rm{ball}}$ does not $\rightarrow 0$ as $\tau\rightarrow 0$. The terms involving the $\Gamma$’s and $\Lambda$’s in the second bracket on the RHS of Eq. (4) are proportional to $\tau$ at low $\tau$ and hence, at constant opacity, to $\Sigma$, while the first factor on the RHS contains a $1/\Sigma$. So, as $\tau\rightarrow 0$, the $\tau$ and $\Sigma$-dependences cancel out in the expression for $v_{\rm{r}}^{\rm{ball}}$. This same effect applies to $v^{\rm load}_{\rm r}$.

The uniform ring resembles a steady-state viscous accretion disk [see, e.g., 42]. In both cases, angular momentum is transported outward while mass flows inward without changing the local surface density. In the BT uniform ring, as illustrated in Figure 1, ejecta flying mostly outward above the ring plane carry mass and angular momentum outward and cause changes in the local surface density and specific angular momentum. Because angular momentum is transported inexorably outward by the prograde ejecta, the specific angular momentum of the ensemble of ring particles in the ring plane decreases so that it drifts inward en masse. The combined effect is that $\Sigma$ remains constant locally in a uniform ring system to a high degree of approximation. The entire ensemble of ring particles consequently moves inward like a conveyor belt because angular momentum is being transported outward. The analogy is not perfect of course. In a viscous accretion disk, it is viscous stresses that carry angular momentum outward, and the steady-state surface density profile depends on the viscosity law.

Realistically, there will be inner and outer boundaries where the BT uniform ring approximation will not apply. Still, the inner boundary region, whatever it is – perhaps the atmosphere of the planet or a region where other mechanisms operate effectively to remove mass – must accommodate the mass inflow. Similarly, the angular momentum carried outward by the ejecta must accumulate somewhere or be removed by other processes. Similar boundary condition issues arise for a viscous accretion disk.

The direct deposition of mass and angular momentum by meteoroids sets a lower bound on the radially inward drift speed and mass inflow rate that is independent of uncertain parameters like $x_{b}$ and $<Y>$. Fig. 1 of D96, which is based on CD90, demonstrates the non-zero inward radial drift speed as $\tau\rightarrow 0$ due to mass loading and meteoroid torque. It also shows that the meteoroid torque effect on radial drift at low $\tau$ is only about 1/4 that of mass loading alone. So, for a first estimate of inflow rate and speed due to meteoroid deposition, we consider only the direct deposition of mass and not the meteoroid torque. In other words, we ignore the $\dot{J}_{\rm{im}}$ term in Eq. (5).

D96 made analytic fits, given by their Eqs. (7) and (8), to the numerical results of CD90 for cometary dust to get $v^{\rm load}_{\rm r}$ as a function of $\tau$ and $r$. Here we adopt the D96 work but use simpler analytic forms that still reflect what we know about the expected dependences on variables $\Sigma$, $\tau$ and $r$ and on the parameter $F_{\rm g}$. We replace the $\tau^{-1}$ with $\Sigma^{-1}$ by calling out the assumption in D96 that the ring opacity has a constant value of about 0.01 cm² g^-1. Also, the $\tau$-dependence of $v^{\rm{load}}_{\rm{r}}$ should be the same as the $\tau$-dependence of $\dot{\sigma}_{\rm{im}}$ given by Eq. (2) in E15 (see also Eq. [1] of D96), also based on fits to CD90 computations. We have simplified the D96 $\tau$-dependence by setting the $k$ and $k_{ml}$ parameters that appear in the D96 equations equal to one. The linear $r$-dependence in Eq. (8) of D96 is actually just a crude approximation to the power-law $r$-dependence of the focussing. Here we more accurately reflect that $r$-dependence of focussing explicitly.

The result of all these modifications is a more generally applicable and versatile expression, which represents the CD90 computations as accurately as do Eqs. (7) and (8) of D96. The revised equation is

The RHS’s of Eqs. (17), (18), and (19) depend on location in the rings $r$, the optical depth $\tau$ at $r$, the typical impact yield $<Y>$, the $x_{b}$ characterizing the low end of the power-law ejecta speed distribution, the meteoroid influx at large distance from Saturn $\dot{\sigma}_{\infty}$, and the strength of the gravitational focusing $F_{\rm{g}}$, which in turn depends on the meteoroid source population. It is worth noting that the RHS’s depend only on $\tau$, not on $\Sigma$. Parametrically, what matters for mass inflow is the efficiency of meteoroid absorption, through $\tau$, not the particle size distribution, particle porosity, ring texture, or ring opacity. This inflow rate is also independent of viscosity.

On the other hand, the factor $<Y>$ is likely to depend on ring particle properties, but exactly how is not well understood. Let $Y$ without brackets denote the nominal yield for cratering impacts with normal incidence at a speed of 14.4 km s^-1. Previous models of the rings’ structural evolution showed that $Y\gtrsim 10^{6}$ was required to maintain the sharpness of the inner B ring edge and to form the linear ramp that connects it to the C ring (D92; E15). These calculations, however, employed the old cometary flux. The most recent models using the more focused EKB flux adopted in this paper only require $Y\gtrsim 10^{5}$ [27, 28, E18]. Moreover, analyses of ring pollution ages (E15; E18) seem roughly consistent with $Y\gtrsim 10^{4}$ to $10^{5}$. Thus we adopt $<Y>\;=10^{5}$ as our fiducial value for this work, and this is equivalent to choosing $Y$ to be about the same value.

In BT simulations, we have typically adopted $x_{b}=10^{-4}$. This parameter is not well constrained, but is roughly consistent with matching structures near the inner edge of Saturn’s B ring (E15). In terms of the nature of impacts on the rings, both $x_{b}$ and especially $Y$ are uncertain by up to an order of magnitude, possibly more, and could vary with location depending on ring particle properties like shape and porosity.

The CDA observations [55, 1, 54] indicate that dust-sized meteoroid impacts are dominated by an EKB source and that the value of the micrometeoroid flux at infinity is $\dot{\sigma}_{\infty}=\dot{\sigma}_{\infty}{\rm{(CDA)}}=2.2\cdot 10^{-17}$ g cm^-2 s^-1. This value is lower by a factor of two compared to earlier estimates for the influx of small meteoroids at Saturn of $\dot{\sigma}_{\infty}=\dot{\sigma}_{\infty}({\rm{CD}})=4.5\cdot 10^{-17}$ g cm^-2 s^-1 (Grün et al., 1985; CD90; CE98; Sec. 2.1), where again “CD” refers to the value adopted in CD90, and is generally consistent with measurements of the interplanetary meteoroid flux by the New Horizons Student Dust Detector [73, 74, 75]. For simplicity, in this paper, we assume the influx at Saturn is constant over relevant time scales.

The CDA measurements also give information about the velocity distribution of the meteoroids as they enter the Saturn system. We have not yet performed a detailed re-computation of the focusing and aberration for EKB meteoroids along the lines of CD90. However, E15 and E18 estimated that the focusing factor $F_{\rm{g}}$ is $\sim 30$ for these objects rather than the three or four assumed by CD90 and CE98 for cometary dust, because the EKB particles enter Saturn’s Hill sphere with smaller speeds relative to Saturn (see CD90; CD98; and the discussion in Section 2.1). The exact functional form of $\mathcal{R}(\tau)$, based here on CD90, is likely to change due to a different degree of aberration for EKB particles, but we suspect this probably only affects results by factors of order unity.

The values of $<Y>$ used here are rather large but, as explained in E18, are adopted based on pollution models and fits to ring structure, especially the outer C ring and inner B ring regions. As noted above, using cometary meteoroids, D92 and D96 argued for $<Y>\sim 10^{6}$ to match the edge/ramp structure. In their laboratory experiments for impacts onto ice and ice/silicate targets, Koschny and Grün [56] found it was difficult to justify $Y$ being as large as $10^{6}$ and suggested that other highly uncertain parameters in ballistic transport modelling might have to be adjusted to bring the required yield down.

The CDA results already move in the right direction. The increase in $F_{\rm g}$ due to the EKB nature of the impactors has now reduced the estimated $Y$ for structural fits by about an order of magnitude (E15;E18). With remaining uncertainties in viscosity, in the upper and lower limits of the ejecta velocity power law, and in the exact value of the power law index $n$, plus the absence of a proper and complete treatment of meteoroid focusing and aberration, the value of the structurally required $Y$ could drop further and be brought more in line with laboratory results.

Another concern raised by Koschny and Grün [56] is illustrated by their Eq. (7) and Table III. Using parameters appropriate for icy projectiles and targets, their experimentally determined yields for a normal impact at 14 km s^-1 can be approximately fit by $1.2\cdot 10^{5}(r_{\rm i}/1{\rm cm})^{0.69}$, where $r_{\rm i}$ is the radius of the impactor. The particles detected by CDA are generally less than 100 $\mu$m⁶⁶6One grain considerably larger than 100 $\mu$m was detected which would increase the magnitude of $\dot{\sigma}_{\infty}$, but was not included in the estimate of the flux for reasons discussed in Kempf et al. [54]. These authors also discuss the probabilities of non-detection of larger grains due to insufficient detector exposure time., implying yields from this fit of less than about 5$\cdot$10³. Here we adopt values of $<Y>$ compatible with the inner edge results of BT evolutions. It should be noted that it is actually the combination $\dot{\sigma}_{\infty}<Y>/\nu$ that matters for matching inner edge structures (E15; E18). Any deficiency in the actual $<Y>$ could be compensated by an increase in $\dot{\sigma}_{\infty}/\nu$. We should also keep in mind that, despite best efforts, the laboratory impact results may not apply well to ring particles with regoliths and with complex substructure that are in free-fall at cryogenic temperatures.

Another concern is how well our inflow models may apply to parts of the B ring where observations suggest high porosity of the ring particles [e.g., 78, 92], and similarly for observations in the A and C rings [76, 66, 93]. High porosity may imply lower yields so that possibly the inflow rate in the central part of the B ring is actually lower. Pollution models and BT structural arguments do not constrain $Y$ in the middle B ring. However, this would not affect our explanation of how mass flows from the inner B ring and through the C ring toward the planet, which is the main focus of this paper. On the other hand, the yield in regions of porous particles may well be just as large as elsewhere if the typical compacted components of the porous particle aggregates are significantly larger than the size of the impacting meteoroids. Then impacts by meteoroids could produce yields corresponding to the individual components of the aggregate, not the porous aggregate as a whole.

So, with all these caveats, we here consider a range of $<Y>$ from $10^{4}$ to $10^{5}$.

Altogether, Eqs. (18) and (19) then become

More generally, Figure 2 shows Eq. (17) computed for a variety of $\tau$’s as a function of $r$ for the same nominal parameter choices used in Eqs. (22) and (23). Note that the inflow rate asymptotes to a finite value as $\tau\rightarrow\infty$. The result is that we expect a rather constant high value of $\dot{M}_{\rm bt}$ for a region where optical depths are generally greater than unity, like the B ring. Figure 3 shows how well the low-$\tau$ and high-$\tau$ limiting expressions match the inflow results for general $\tau$.

As a supplemental result, which will aid in later discussion, the gross erosion time in the inner B ring for the same parametrization is

Eq. (20) gives the mass loading influx rate for cometary dust. For EKB impacting meteoroids, we make the approximation that the gravitational focusing should be about a factor of ten higher (E18). Adopting $F_{\rm g}\approx 30$ instead of 3 for EKB impactors, as well as a correction factor to account for the CDA flux $\dot{\sigma}_{\infty}{\rm{(CDA)}}/\dot{\sigma}_{\infty}{\rm{(CD)}}$, Eqs. (20) and (21) become

Figure 4 shows the result of evaluating Eq. (26) for a variety of $\tau$’s as a function of $r$. Comparing this with Fig. 2 reveals that $\dot{M}_{\rm{load}}$ is roughly an order of magnitude smaller than $\dot{M}_{\rm{bt}}$, and is, by itself, slightly smaller than the lower estimate of the mass inflow into Saturn observed by INMS. If we reduce our choice of $<Y>$ from $10^{5}$ to $10^{4}$, the inflow rates for BT and mass loading are comparable in magnitude, and in tandem are consistent with the lower bound of the mass inflow. Overall, the total mass inflow rate $\dot{M}_{\rm{total}}=\dot{M}_{\rm{bt}}+\dot{M}_{\rm{load}}$ is quite large for $\tau>0.1$ and $<Y>\;=10^{5}$ and is consistent with the INMS upper estimate for the mass inflow.

Note that the $r$-dependence for both effects is the same and is due to the combined effects of thin disk geometry and gravitational focusing. While the BT inflow depends on the uncertain parameters $x_{b}$ and $<Y>$, which characterize the lowest speeds in the power-law ejecta speed distribution and the impact yield, respectively, mass loading depends only on the influx rate of meteoroids and their degree of focusing. With BT added, the mass influxes are larger by an order of magnitude for $<Y>\;=10^{5}$. We remind the reader that this value of $<Y>$ is adopted here on the basis of BT structural arguments (Sec. 5.1).

In Figure 5, we show the inward radial drift velocities due to mass loading alone (solid curves) and due to BT alone (dashed curves) for characteristic locations in the C and B rings at 1.4 R_S (red curves) and 1.8 R_S (black curves), respectively, as a function of optical depth. In generating these curves, we use Eq. (25) for mass loading and Eq. (15) integrated over the standard $x$-distribution for BT. We also assume that $\Sigma=\tau/\kappa$ and use an opacity $\kappa=0.05$ cm² g^-1 for the C ring and $0.013$ cm² g^-1 for the B ring [see, e.g., CE98; 4, 43, 92, 93]. For fixed opacity, the radial drift speeds increase to a nonzero asymptotic value as optical depth and surface density decrease, even though the mass inflow rate goes to zero.

Based on Fig. 5, we expect the low-$\tau$ base of the inner edge of an initially high-$\tau$ ring to spread inwards more quickly than the high-$\tau$ bulk of the ring. Such sharp high-$\tau$ inner edges do occur in global viscous evolutions of rings like those of Salmon et al. [81] and Crida and Charnoz [13]. The implication is that the C ring may have been created from the original B ring inner edge due to BT or even to mass loading alone [see 26].

These radial drift speeds can be used to estimate a time scale $t_{\rm{loss}}$ for ring material located at some radial distance $r$ to fall into Saturn. Figure 6 shows this time scale for mass loading (red curves) and BT (black curves) for several $\tau$. To generate these curves, we have again used Eq. (25) and Eq. (15) integrated over $x$ to obtain inward drift speeds. We then integrate the drift speeds over $r$ assuming constant optical depth and surface density and using the C ring opacity in the previous paragraph. Drift times are much shorter for low $\tau$ and consistent with previous estimates that the C ring could be lost (or “recycled”) in $\sim 10^{7}$ years by mass loading alone. These times are much shorter for BT, but would be comparable to mass loading for $<Y>\;=10^{4}$. We also see that $t_{\rm{loss}}$ becomes longer with increasing optical depth (or $\Sigma$), so that, in an initially massive ring evolving due to strong viscosity, the effects of mass loading and ballistic transport are probably overwhelmed by viscosity until the ring mass decreases significantly.

The dashed curves in Fig. 6 are for a case in which we model the C ring and inner B ring as a step function with $\Sigma=2$ g cm^-2 for $r<1.525$ R_S, and $\Sigma=52$ g cm^-2 for $r>1.525$ R_S (Sec. 5.2), representative of current values. For the B ring, this corresponds to taking $\tau\sim 1$. For the dashed curves loss time estimates, $\tau$ and $\Sigma$ at a given $r$ are kept constant with time, but the co-moving values change as the mass elements cross the inner B ring edge. Note that the residence time for a mass element in the inner B ring is quite short – less than a few million years for BT with high $<Y>$ or a few tens of millions of years for mass loading alone.

It is important to notice in Fig. 5 that the inward drift speeds increase as $\tau\rightarrow 0$. So, if we start with a ring of finite mass and radial extent, we expect ML alone to empty the entire B ring onto the planet in a finite time, on the order of a factor of a few times the largest drift times given in Fig. 6 in red (which are the times on which individual annuli of the rings drift all the way to Saturn). Global ring simulations performed in a companion paper verify this [26]. The case for BT alone is more complicated, because angular momentum is conserved, which means that some material must be left behind in orbit, but the bulk of mass elements will flow into Saturn on the times suggested by the black curves for $<Y>=10^{5}$. Also recognise that these drift times do not apply to narrow, dusty rings or sparse debris belts. The drift times presented here are indicative of how long a broad, dense ring can last. The true final behaviour of such a ring system is a separate and interesting question. Perhaps it leads to a long-lived sparse system of dust belts and narrow ringlets.

It is interesting to compare the mass inflow in the rings expected from viscosity with those derived above. Using the approximation of a steady-state viscous accretion disk with a constant kinematic viscosity $\nu$, the mass inflow rate is

The proper value of viscosity in the rings is a complicated story, and expectations vary theoretically depending on the particle size distribution and whether or not gravitational wakes occur [see, e.g., 83, 82]. Values of the kinematic shear viscosity between 1 and 100 cm² s^-1 generally result, with low optical depth regions tending toward the lower end of the range and higher optical depth regions toward the higher end (see discussions in E15 and E18). We adopt values of 10 cm² s^-1 and 100 cm² s^-1, respectively, which probably overestimate the viscosity [see, e.g., 88, 87]. For estimates of $\Sigma$ at low and high $\tau$, we use the same values adopted in Section 6.2 below.

So, for a low-$\tau$ region like the C ring, we get

The mass inflow rates from the rings into Saturn’s atmosphere as measured by various Cassini instruments in the Grand Finale orbits cover a considerable range (Sec. 2.2). Our general picture is that meteoroid effects drive a large inflow rate towards Saturn with the bulk of the inflow delivered equatorially through the rings. Various erosion processes along the way cause some significant flow into Saturn at high Saturn latitude in the form of ring rain as was measured by CDA, but these measurements are restricted to a narrow range of grain sizes. Such erosion mechanisms were suggested originally by Cook and Franklin [12, see review in D84], before there were many relevant observations.

According to Hsu et al. [48], the tens-of-nanometer grains they detected with CDA are ejected from the main rings at speeds of $\sim$ few $\cdot 10^{2}$ - $10^{3}$ m s^-1 with 50% from the C ring, 40% from the B ring, and the remaining fraction from the Cassini Division, A and D rings. The ejecta speeds for these small charged grains are larger than typical speeds assumed for the $\gtrsim$ micron-sized particles which dominate the impact ejecta size distribution in BT (CD90; D92; CE98; Sec. 3.2), but are consistent with experimental [3, 84] and theoretical [69] studies for smaller grains.

It is informative to compare the total ejecta rate of $1800$ to $6800$ kg s^-1 derived by Hsu et al. [48] for these nanograins to the total ejecta production rate due to micrometeoroid impacts for all particle sizes across the rings. To estimate the total impact ejecta emission rate from the main rings, we integrated the impact flux $\dot{\sigma}_{\rm{im}}$ over each ring assuming mean optical depths of $\tau=0.1$ for the C ring and Cassini division, $\tau=2$ for the B ring and $\tau=0.5$ for the A ring to determine impact probabilities. The result is

As pointed out in Section 2.2, the $100-370$ kg s^-1 of the CDA flux deposited at the ring rain latitudes in the form of charged, solid nanograin particles are apparently not sufficient to explain the $432-2800$ kg s^-1 required to account for the observed H${}_{3}^{+}$ infrared emission pattern [70]. However, the silicate-to-ice mass fraction at the ring rain latitudes being as high as $\sim 30$% [48] implies that a significant fraction of water has been liberated from these nanograins. If one imagines “reconstituting” the CDA grain population with sufficient water so that it is similar to C ring material ($\sim 6$% non-icy by mass), this gives an inflow of $\sim 470$ to $1750$ kg s^-1 of additional water, not too dissimilar from the estimate of O’Donoghue et al. [70].⁷⁷7It should be acknowledged that this agreement may be coincidental because the CDA flux originates predominantly from the C and B rings, whereas the mass flux estimated from the O’Donoghue et al. [70] analysis comes overwhelmingly from the C ring.

Most of the ejecta mass from impacts (Eq. [30]) should be in larger grains, too large to be charged significantly, which is consistent with the flux measured from CDA (and MIMI) only representing a small fraction of the total inflow. The total midplane loss rate from these instruments is on the order of $225$ to $875$ kg s^-1, far less than the equatorial loss rate of $4800$ to $45000$ kg s^-1 measured by INMS, which should sample the full range of sizes [90]. Because the origin of this material appears to be the D ring, it implies that the C ring must replenish the D ring material at a similar rate assuming a steady-state scenario. Likewise, the C ring must also be replenished from the B ring at a roughly similar rate.

From our analysis, for our nominal choice of parameters, the mass inflow rate through the B ring and the C ring due to ballistic transport and mass loading is consistent with the observed mass inflow rates near Saturn as measured by INMS. For nominal parameters, the C ring mass inflow rate is predicted to be a factor of several lower than that through the B ring. We find that even decreasing one of the critical parameters, like reducing $<Y>$ to 10⁴, which remains in a plausible range of parameters, the mass inflow rates for both rings stay within range of the lower bound estimate for inflow from the INMS measurements. Our main point here is that the observed mass inflow rates toward Saturn through the rings can, in principle, be sustained by ballistic transport plus mass loading without straining parameters. Some complications are addressed below in Section 6.4.

We have argued for many years that the rings must be substantially younger than the Solar System owing to consequences of meteoroid bombardment, based on both structural and pollution effects (D92; D96; CE98; E15; E18). By determining some key parameters required to assess the importance of meteoroid bombardment, the Cassini Grand Finale has substantially strengthened the arguments for a young age (E18). In addition, the measurement of the large inflow from the rings into Saturn combined with a well-determined ring mass now also permits estimates for the future lifetime of the rings (see also the discussion at the end of Section 5.3).

First consider the age. The structural evolution of the rings due to BT occurs on time scales that vary from a few up to hundreds or thousands of $t_{\rm{G}}$ (Eqs. [1] and [24]) depending on the structure being modelled (E18). The ages depend on the uncertain impact yield parameter $<Y>$ but are always less than the direct deposition time scale $t_{\rm{dd}}=\Sigma/\dot{\sigma}_{\rm{im}}$ for reasonable choices of $<Y>$. The time $t_{\rm{dd}}$ itself does not depend on $<Y>$, and we expect the evolution of the rings to their current state of pollution to be some fraction of $t_{\rm{dd}}$. Thus, age estimates based on pollution are more reliable measures of absolute ring age than estimates based on BT structures alone.

As revisited by Kempf et al. [54], an absolute exposure time for the rings can be obtained by assuming they began as pure water ice and darkened to their current state [15, E15, E18]. The rate per unit area at which the ring annulus at some $r$ with surface density $\Sigma$ is impacted is given by $\dot{\sigma}_{\rm{im}}$, so that the mass density of pollutant accumulated in a time $t_{\rm{pol}}$ is $\Sigma_{\rm{pol}}\simeq\eta\dot{\sigma}_{\rm{im}}t_{\rm{pol}}$, where the factor $\eta\leq 1$ allows for the possibility that not all of the impactor survives the impact as absorbing material (CE98; E15). The mass fraction of pollutant after this time is $\Sigma_{\rm{pol}}/(\Sigma+\Sigma_{\rm{pol}})\simeq\Sigma_{\rm{pol}}/\Sigma$. Measured values derived from Cassini RADAR for the amount of non-icy material in the rings are reported as volume fractions of the solid component $\upsilon$ [92, 93]. So, in terms of the mass fraction, $\upsilon\simeq(\rho/\rho_{\rm{pol}})(\Sigma_{\rm{pol}}/\Sigma)\ll 1$ where the mean solid density of a ring particle is $\rho=\rho_{\rm{pol}}\upsilon+\rho_{\rm{ice}}(1-\upsilon)$. The density of ice $\rho_{\rm{ice}}=0.9$ g cm^-3, and we take the solid density of the incoming pollutant to be 2.8 g cm^-3, which represents a volume mixture of 30% carbonaceous and 70% silicate material [91, 54]. The time to acquire the non-icy material then is

For the B ring where $\tau$ is very large, we found that the average surface density is 52 g cm^-2 based on the derived mass for the rings during the Cassini Grand Finale [49, Sec. 5.2], while the measured volume fraction of pollutant is $\sim 0.1$ to $0.5$% [93], with the lowest values found in the very opaque B ring core. Using the median value for $\upsilon$ and the average $\Sigma$, we find that at $r_{0}=1.8$ R_S

For the C ring, we can do a similar calculation, except the low continuum optical depth, $\tau\sim 0.1$, means that the probability of absorption of a meteoroid, which is $\sim 0.2$, must be taken into account. A typical value for the surface density away from plateaus and the anomalous “rubble belt” in the middle C ring is $\Sigma\sim 2$ g cm^-2 [4, 44, 93]. Taking $\upsilon=2$% we get at $r_{0}=1.4$ R_S a similar time scale

Both these time scales are in agreement with those derived in Kempf et al. [54]. These time scales will be modestly longer if the rings began with a larger mass, as implied by the currently observed mass loss rates. Thus, given the complete suite of available Cassini data, the case appears fairly solid that the rings were formed $\lesssim$ a few 100 Myr ago, in agreement with arguments based on pollution and structural models presented in previous works [18, CE98; E15; E18].

We establish this young ring age using the ring mass [49], micrometeoroid flux at Saturn [54], and the volume fractions of pollutants [92, 93]. Combined with measurements of the mass loss rates into Saturn during the Cassini Grand Finale, we can now also estimate the turnover time of the C and D rings and the remaining lifetime of the rings as a whole.

The mass loss rate of nanograins measured by CDA is a not a reasonable gauge for remaining ring lifetime because of the limited particle size range to which it is sensitive. On the other hand, the mass flux of charged water products as calculated recently by O’Donoghue et al. [70] may be more representative because the ring rain effect is probably caused by a contribution from all particle sizes. Based on their estimates for the mid-latitude mass loss rate, these authors get a ring lifetime ranging from $\sim$ 150 to 1,100 Myr. with a likely value of 300 Myr, just from the ring rain. This is an upper limit because it does not include the equatorial losses detected in the INMS data [90].

Waite et al. [90] do not estimate an overall remaining lifetime for the main rings, but they do estimate the lifetime of the D and C rings from their mass inflow rates and obtain $7\cdot 10^{3}$ to $7\cdot 10^{4}$ yr for the D ring, and $\sim 7\cdot 10^{5}$ to $7\cdot 10^{6}$ yr for the C ring. Their calculation for the D ring assumes that its mass is $\sim 1$% of the $\sim 10^{18}$ kg C ring mass. Their D ring lifetime estimate is based on the assumption that the D ring has a similar particle size distribution to the C ring so that the difference in surface density is simply the optical depth ratio between the D and C ring ($\sim 0.01$). However, meter-size particles dominating the mass in the D ring may not be a reasonable assumption. Particles sizes in the D ring of only tens of microns may be more appropriate [46]. This would significantly lower the mass so that the current mass inflow rates would dissolve the D ring in only about a year. If so, this would tend to support the idea of episodic repopulation of the D ring from the C ring. Waite et al. [90] also posit that the D ring could be repopulated sporadically by large impact events like the one that tilted the D and C ring plane [43].

From our own estimates of the mass inflow rates due to BT alone in Eqs. (22) and (23), we can calculate the remaining ring lifetime. Taking the current ring mass to be $\sim 0.4$ Mimas masses [49], or $\sim 1.5\cdot 10^{19}$ kg, we find that the range of times for the remaining mass of the rings to fall into the planet is $\sim$ 15 to 40 Myrs for the case when $<Y>\;=10^{5}$, or $\sim$ 150 to 400 Myrs for $<Y>\;=10^{4}$. This assumes that the mass inflow rate remains constant over that time. The lifetime estimate would be shorter at the tens of percent level or less, depending on parameter choices, if we include contributions due to mass loading and meteoroid torques. They can also be shorter if the slope of the ejecta velocity distribution is flatter so that more ejecta mass is placed at higher speeds or if there is an accentuation of the inward drift rate of material due to the Goertz et al. [36] mechanism (see Section 6.4.4). Our lower bound estimate for the lifetime is naturally similar to what would be inferred from the upper bound for the mass inflow measured by INMS because BT produces mass inflow rates of the same order as observed for our nominal parameter choices. The future lifetimes estimated here based on ring mass and inflow rates are similar within a factor of a few to the inward drift times in Fig. 6 of Section 5.3, as they should be.

The lower bound estimate of 15 Myr seems exceedingly short, perhaps suggesting that the yield $<Y>$ or other parameters vary across the rings and/or $<Y>$ is generally lower than what we adopt. Even if the influence of BT alone is not as great as computed here, our results show that mass loading alone, which is not far off from the lower bound measured by INMS, represents a relatively firm upper bound on the ring lifetime, similar to the upper bound estimated above for the low-$<Y>$ BT case, namely $\sim$ 400 Myr. It could also be that, if the upper estimate of the mass inflow rate by INMS is correct, the inflow just happened to be exceedingly high at the time of observation due to some recent event [90].

The mass inflow rates rates from our BT analysis indicate that the ring mass was substantially larger in the past. Given our estimate for the current ring age of $\sim 10^{8}$ years based on pollution, this would imply that the initial ring mass was greater by $\sim$ few tenths to a few Mimas masses. The inclusion of mass loading increases these numbers by tens of percent. So, as we argue in more detail in our companion paper [26], the deduced initial ring mass is consistent with a recent origin scenario in which the rings start significantly more massive ($\sim$ one to several Mimas masses) and are polluted by micrometeoroids as they evolve to their current mass and darkened state.

The mass inflow rates measured by Cassini, as well as those that we calculate here, suggest that relatively massive rings are ephemeral when subjected to micrometeoroid bombardment⁸⁸8BT may not be so influential in less massive and sparse ring systems such as Uranus and Neptune in which the rings and arcs are radially thin.. Previously, for evolution by viscosity alone, it was found that a massive ring evolves until its surface density drops to the point where the viscous spreading time becomes exceedingly long [8]. The ring mass then approaches an asymptotic limit [81]. The current ring mass is close to that asymptotic limit. However, this result only applies in the absence of other mass loss mechanisms. Given the results of observations and our analysis here, it seems that the ring mass being similar to the Salmon et al. [81] theoretically-derived asymptotic mass may be a coincidence.

The main conclusion of this paper is that, in light of new determinations of key parameters by Cassini, ballistic transport and mass loading due to meteoroid bombardment can plausibly explain the magnitude of the mass inflow from the rings into Saturn as observed during the Grand Finale. For our nominal choices of parameters, we show that these mechanisms can drive mass inflow through the B and C rings with rates that fall within the range determined by INMS [90].

The mass loading inflow rates computed in Section 5.3 depend only on the product of the Saturn Hill radius meteoroid influx $\dot{\sigma}_{\infty}$ and the gravitational focusing factor $F_{\rm{g}}$. This product is now determined by CDA measurements of dust impacts [54]. So the mass loading results are pretty secure. At moderate to high values of $\tau$, mass loading alone is comparable to (a factor of two smaller) the lower range of the INMS inflow measurement.

For ballistic transport, we use the approximate steady-state solution developed in Section 4 to estimate mass inflow rates. In addition to the factors $\dot{\sigma}_{\infty}$ and $F_{\rm{g}}$, the ballistic transport mass inflow (Section 5.2) depends on the less well-determined factors of typical impact yield $<Y>$ and average normalized ejecta speed $x_{b}$. The ballistic transport inflow rates are roughly compatible with the lower and upper estimates of the observed mass inflow rates reported by INMS for impact yields $<Y>\;\approx 10^{4}$ to $10^{5}$, respectively. While it is unclear how well the steady-state solution applies to the real rings locally, it probably gives a useful global estimate for inflow. The meteoroid-driven mass inflows are several orders of magnitude greater than mass flows due to viscosity (Section 5.4).

Regardless of the various uncertainties, discussed in some detail in Section 6.4, this paper shows that mass should be driven inward toward Saturn in the B ring at rates high enough to explain the observed mass inflows into Saturn. Even though mass inflow rates in the C ring can also be large enough under reasonable assumptions, the situation in the C ring is not as clear because it contains a lot of unexplained complex structure and the detailed loss mechanisms operating close to Saturn are not yet well enough understood.

The mass inflow rates driven by ballistic transport and mass loading do not in themselves dictate a ring age. Instead, in Section 6.2, we used modern data to estimate a ring age due to pollution by meteoroids of only $\sim$ 170 Myr. Also in Section 6.2, we showed that the ballistic transport mass inflow rates in Eqs. (22) and (23) suggest future ring lifetimes of as little as 15 to as much as 400 Myr using the specific examples of impact yields of $10^{4}$ and $10^{5}$. These lifetimes are reduced by tens of percent when mass loading is included. The range of measured mass inflow rates from INMS, which are consistent with our results here, if steady, are large enough to deplete the C ring in only $\sim 0.7$ to 7 Myr.

As discussed in Section 6.3, the ballistic transport mass inflow rates for nominal parameters operating over our estimated $\sim$ 120 Myr age for the rings suggest that $\sim$ a few tenths to several Mimas masses have already been lost into Saturn. Adding mass loading increases the numbers modestly.

Although it will probably make differences of a factor of a few at most, the CD90 analysis of focusing and aberration effects for meteoroid impactors with a cometary source needs to be repeated for the EKB source determined by Kempf et al. [54]. E18 estimated the changes with simple approximations but did not redo the CD90 analysis in detail. The various $\tau$-dependences for impactor absorption probabilities, ejecta emission rates, and mass loading drift are likely to be affected by the change from a cometary to an EKB source. For instance, the $\tau_{0}$ in Eq. (6) will probably be smaller due to the greater aberration of the EKB impactors leading to increased slant path optical depths experienced by the meteoroids at the rings. It is also likely that meteoroid torques will play a larger role for EKB impactors for similar reasons. So, we plan to redo the CD90 analysis for EKB projectiles and derive proper ejecta distribution functions and $\tau$-dependencies.

It is clear, by comparing the viscous mass inflow rates in Eqs. (28) and (29) with the ballistic transport rates in Eqs. (22) and (23) and the mass loading rate in Eq. (26) that the inclusion of mass loading and ballistic transport in global ring simulations is necessary. In our companion paper [26], we show that adding just mass loading to global viscous evolutions leads to finite lifetimes for massive rings rather than the asymptotic mass found in purely viscous simulations [81].

Adding ballistic transport correctly to global simulations is computationally involved but should shorten the expected lifetimes further and have interesting implications for the current mass distribution of the rings because of the outward angular momentum transport involved. Global pollution evolutions could be run simultaneously, and the possibility of a changing composition due to selective retention of water during the inflow process should be considered. Once global simulations with proper ballistic transport become possible, it will be interesting to include satellite torques, especially in scenarios where a proto-Mimas forms outside the Roche radius and then recedes due to resonant torques. Other mass and angular momentum transport processes, like the Goertz et al. [36] mechanism could also be considered. In addition to global simulations, localized ballistic transport simulations will be needed to study how mass inflow is affected by gaps and other ring structures, especially in the C ring.

Acknowledgements

We thank Josh Colwell, Jeff N. Cuzzi, Joe Burns, and Glen Stewart for useful discussions. This work was supported by a grant from NASA’s Cassini Data Analysis Program (PRE) and a Cassini IDS grant (PRE) to Jeff N. Cuzzi. We are also extremely grateful to two anonymous referees for thorough and thoughtful reviews that led to extensive revisions of the manuscript.