Unusual origin, evolution and fate of icy ejecta from Hyperion

Planetary and Space Science 49 (2001) 1265–1279

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Unusual origin, evolution and fate of icy ejecta from Hyperion Alexander V. Krivova; ∗;1 , Marek Banaszkiewiczb a Nonlinear

Dynamics Group, Institute of Physics, University of Potsdam, P.O. Box 601553, D-14415 Potsdam, Germany b Space Research Centre, Bartycka 18A, 00-716 Warsaw, Poland Received 5 January 2001; received in revised form 5 June 2001; accepted 18 June 2001

Abstract We readdress the idea that Hyperion may act as an e6ective source of dust in the outer Saturnian system. Hypervelocity impacts of dust particles coming from the outer irregular moons (Phoebe and several others recently discovered) and, to less degree, bombardment by interplanetary micrometeoroids should eject surface material of Hyperion to planetocentric space. Unlike Hyperion, whose motion is stabilized by a strong 4 : 3 mean motion resonance with the neighbouring Titan, so that encounters with this satellite are prohibited, a signi:cant fraction of the Hyperion debris should be fast enough to be out of resonance. For slower ejecta, resonant locking may be destroyed later by the plasma drag and solar radiation pressure forces. The orbits liberated from the resonance become unstable and experience multiple close approaches to Titan. Using numerical integrations, we performed a statistical study of the grain trajectories to construct a spatial distribution of dust in the Hyperion–Titan system and to :nd out the eventual fate of the debris. Particles locked in resonance form an arc-like structure along the Hyperion orbit centred on Hyperion’s position; this “Hyperion swarm” is populated by grains of tens of micrometres in size and might be dense enough to be detected by the Cassini spacecraft during its Ayby of Hyperion. The whole dust cloud in the Hyperion–Titan system is tilted o6 the equatorial plane of Saturn and has a structure that depends on the particle radii. No particular dust concentration in the vicinity of Titan was found. Most of the grains larger than ∼ 5
m in size :nally collide with Titan, whereas smaller particles are either lost in the inner part of the Saturnian system or hit Saturn. Our estimates of the dust inAux to Titan show that the incoming rate of Hyperion particles may exceed the direct inAux of interplanetary dust particles. The inAux of icy c 2001 (H2 O) particles from Hyperion might help to explain the observed abundance of CO and CO2 molecules in Titan’s atmosphere.  Elsevier Science Ltd. All rights reserved.

1. Introduction We consider the dust environment in the outer part of the Saturnian system, outside the orbit of Titan. This region includes Hyperion, Iapetus and—much farther from Saturn, at nearly 200Rs (Saturnian radii)—also Phoebe, the moon that, until very recently, was the only Saturnian satellite known to have a retrograde orbit. In October–December 2000, a discovery of 12 other small saturnian moons was reported (Gladman et al., 2001., see also http:==www.obs-pm.fr=saturn and references to IAU Circulars therein). These new moons, from several to several tens of kilometers in size, have semimajor axes between about 200 and 400Rs and appreciable ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (A.V. Krivov), [email protected] (M. Banaszkiewicz). 1 On leave from: Astronomical Institute, St. Petersburg University, Stary Peterhof, 198504 St. Petersburg, Russia.

eccentricities. Some of them are prograde, while others, like Phoebe, have retrograde orbits. Soter (1974) :rst suggested that many of the dust grains ejected from Phoebe would be swept out by the inner moons, primarily Iapetus, possibly explaining the observed albedo asymmetry of this moon (Wilson and Sagan, 1996; Hamilton, 1997). About 20% of the Phoebe material may reach Hyperion, the next satellite of Saturn (see Burns et al., 1996). Since the orbits of Phoebe particles are retrograde while that of Hyperion is prograde, the impact velocities could be as large as ∼ 10 km s−1 . Hence, the Phoebe dust grains are quite energetic and may produce a considerable amount of ejecta from Hyperion’s surface. The same should apply to dust from the other small retrograde moons. Another population of impactors that hit Hyperion and contribute to the ejecta production, is interplanetary dust particles (IDPs). At the heliocentric distance of Saturn, most of them probably come from the Edgeworth–Kuiper belt objects, being produced by their mutual collisions (Jewitt and Luu, 1995) and possibly also by impacts of

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Fig. 1. Possible ways of the dust transport in the outer Saturnian system. I—dust ejected by IDPs from Hyperion and arriving at Titan; II—dust ejected by IDPs from Phoebe, impacting Hyperion and ejecting secondary particles which also arrive at Titan; III—dust ejected by IDPs from Phoebe and reaching Titan directly.

interstellar grains onto their surfaces (Yamamoto and Mukai, 1998). Hyperion, a 250 km-sized icy satellite, is not massive enough to reaccrete all of the ejecta from its surface. A sizeable fraction of the icy debris, ejected by projectiles from the Saturnian irregulars and interplanetary impactors, start travelling in circumplanetary space. Thus Hyperion is likely to act as a signi:cant dust source in the Saturnian system. In our previous research (Banaszkiewicz and Krivov, 1997a, b), we have considered the complex dynamics of Hyperion icy ejecta, driven by gravitational, radiative and plasma drag forces, to show that many of the ejecta :nally arrive at Titan, contaminating its dense nitrogen atmosphere with water ice. Preliminary analysis of the dust distribution in the Hyperion–Titan system has also been made and estimates of the dust inAux to Titan have been given. In this paper, we describe an elaborate new numerical model and use it to check our preliminary results and to address a number of new topics. First, we investigate in-depth the role of resonance with Titan for the Hyperion ejecta dynamics. Second, we construct and analyse 3D distributions of the Hyperion dust with respect to Saturn, Titan and Hyperion. Third, we calculate number densities of dust and discuss possible measurements with the Cassini dust detector. Fourth, we estimate how much dust from di6erent sources (Fig. 1) arrives at Titan, what is the distribution of entry points and speeds of the dust grains in Titan’s atmosphere and how the inAux of exogenic dust could a6ect the chemistry of the atmosphere of Titan. 2. Simulation technique 2.1. Force model The following forces are taken into account: Titan’s gravity, solar radiation pressure and the plasma drag force. In our previous studies (Banaszkiewicz and Krivov, 1997a, b), the

Fig. 2. Two-dimensional isolines of the calculated plasma drag force strength in a meridional plane perpendicular to the equatorial plane of Saturn. Also shown are the plasma drag force strengths near locations of Titan (bold dot) and Hyperion (double arrow with the ends at pericentre and apocentre of its orbit). All strengths are given in percent of the radiation pressure force strength exerted on a like-sized dust grain (the ratio of the moduli of the two forces is independent on the grain’s radius).

latter was computed using the formulae of Banaszkiewicz et al. (1994) for the direct drag and those by GrOun et al. (1984) for the indirect one, with the plasma data taken from Lazarus and McNutt (1983) and Sittler et al. (1983). We shall use a two-dimensional axisymmetric model of plasma extending from L = 4 to Hyperion’s orbit (L = 25). Following Jurac et al. (1995), we consider four charged species: thermal and hot electrons, hydrogen and oxygen ions. For each of them density and temperature in the equatorial plane is introduced: in the inner magnetosphere (4 ¡ L ¡ 10) according to the Richardson and Sittler (1990) model, in the outer region from Richardson’s (1986) interpretation of Voyager data, and near Titan from Hartle et al. (1982) analysis. The vertical distribution of plasma along the magnetic :eld lines is then found from the momentum equation, with gravity, centrifugal force, pressure and ambipolar electric :eld taken into account, similar to the approach of Bagenal and Sullivan (1981) applied to the Jupiter magnetosphere. Electrons and ions contribute to the dust grain charging, either directly or by secondary electron emission. In addition, reAected electrons and photoelectrons are included into the current balance as in Jurac et al. (1995). The equilibrium potential is obtained and the corresponding grain charge is calculated on a two-dimensional grid in (; z) coordinates,  and z being the distances from the planet and from the equatorial plane, respectively. Finally, the plasma drag force is found from the Northrop and Birmingham (1990) formula, that results in a much smaller e6ect (by an order of magnitude) than previously calculated by us (Banaszkiewicz and Krivov, 1997b) employing an approximate GrOun et al. (1984) expression. Therefore, plasma drag amounts to only 0.1%– 0.3% of the radiation pressure for most of the locations near Hyperion and Titan orbits. Calculated strengths of the plasma drag force in the (; z) plane are shown in Fig. 2. In fact, the

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Fig. 3. Tests for accuracy of numerical integrations: conservation of the Jacobi constant in the restricted circular three-body problem “grain–Titan–Saturn”. The mass of Titan in these tests is seven times its actual mass. Left and right: integrations in Saturn-centred and Titan-centred coordinate systems. Top and bottom: integrations with lower and higher accuracy. The latter is speci:ed by LL, the number of signi:cant digits, and NOR, the order of the integration method (input parameters of the integrating routine).

plasma drag would be even weaker if we adopted a more recent plasma environment model by Richardson (1995), which predicts a smaller corotation fraction and therefore lower relative velocities between the plasma and the dust grains. 2.2. Numerical integrations In our problem, a major technical diQculty of numerical simulations is posed by a signi:cant loss of accuracy during close encounters of grains with Titan (Farinella et al., 1997; Banaszkiewicz and Krivov, 1997b). Just using a good integrator (we employ the Everhart (1974) routine with adaptive stepsize control) does not ensure the validity of the results. To solve this problem, we employ a classic technique of the three-body problem and transform the equations of motion

to a coordinate system centred on Titan and rotating around Saturn with the angular velocity of the satellite’s orbital motion. To see the e6ect, we have performed a series of tests. We have set the perturbing forces other than Titan’s gravity to zero and arti:cially increased the mass of Titan by a factor of 7, thereby enhancing the frequency of encounters. Then we integrated a set of 100 trajectories in two coordinate systems (the traditional Saturn- and Titan-centred one just described) and for two adjustments of accuracy control (5 and 9 signi:cant digits). To estimate the accuracy, we calculated the Jacobi constant of the Saturn–Titan–grain system, which, as far as the orbit of Titan is treated as circular, must be exactly conserved. The results are quite impressive (Fig. 3). Integration in the titanocentric coordinate system increases the accuracy by one or even two orders of magnitude (cf. left and right panels). On the other hand, diminishing step size is not so e6ective (cf. top and bottom

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graphs), but slows down the computations appreciably. For the main bulk of integrations discussed below, we therefore adopt the Titan-centred coordinate system and a moderate integration accuracy (:ve signi:cant digits).

3.2. Are the grains locked in the resonance initially?

2.3. Initial and output data With the technique described above, we integrated the trajectories of grains of three sizes—100, 10 and 2
m— over a maximum time interval of 100 Saturnian years (s.y.) instead of 10 s:y: (Banaszkiewicz and Krivov, 1997b). The starting velocities (at the boundary of Hyperion’s sphere of inAuence) were calculated randomly in accordance with a plausible ejecta velocity distribution at the surface (see Section 3.2). With a given printout step (0:1 s:y:), we stored the following dynamical data: positions and velocities of grains relative to Saturn, Titan and Hyperion, osculating orbital elements of grains and Jacobi constant. For each trajectory, we also calculated statistics of encounters with Titan and, in the case of a collision with Titan, the parameters of that collision: the :nal velocity vector relative to Titan and the location of impact point on its surface.

3. Gross features of the Hyperion ejecta dynamics 3.1. The role of resonance with Titan According to the previous work (Banaszkiewicz and Krivov, 1997b), the dynamical evolution of the particles can be qualitatively outlined as follows. Hyperion is known to be locked in a strong 4 : 3 mean motion resonance with Titan. The resonance is characterized by the critical argument = 4 − 3 T − $;

and eQciencies of the resonance breaking mechanisms described in our previous papers. Let us discuss these mechanisms on the basis of the new numerical results.

(1)

where and T are the longitudes of Hyperion and Titan and $ is the longitude of pericentre of Hyperion’s orbit. ◦ The critical argument librates around 180 with amplitude ◦ ≈ 36 . Resonant locking ensures that conjunctions of both satellites always occur near Hyperion’s apocentre and therefore prevents close approaches between the two moons. The same is thought to be true for the dust particles originating from Hyperion; it would strictly be true, were the particles ejected at low speeds and were the other forces absent. Thus, one may expect the Hyperion ejecta to form a dust swarm near Hyperion. However, we pointed out several mechanisms that have the ability to destroy the resonant locking. These mechanisms are of major interest because breaking of the resonance would have immediate and major consequences for the dynamics. Once a particle is out of the resonance, it gets an opportunity to experience essentially stochastic encounters with Titan. This leads, :nally, to collisions with Titan or Saturn or escape from the system. The new modelling introduces essential corrections to the comparative importance

The :rst question to answer is: are all the particles starting from Hyperion locked in the resonance immediately after ejection? The answer is negative. If a particle is ejected with an appreciable speed, of the order of several tens m s−1 relative to the parent moon, the above-mentioned resonant locking does not occur (Farinella et al., 1983, 1990). Another possibility is that the particles may get into orbits which are more distant from the centre of the “resonant stability island” than the orbit of Hyperion. For such orbits, the li◦ bration amplitude is much higher that Hyperion’s 36 . For brevity, we will refer to this case as shallow resonance. The ◦ ◦ ◦ amplitude of 100 (or the libration interval [80 ; 280 ]) was reported by previous workers (see Farinella et al., 1983) as a critical one: with this amplitude, the grain’s orbital pericentre at conjunction with Titan is too close to the satellite to allow stability. In our problem, the numerical value of the critical amplitude may be di6erent, as our force model includes non-gravitational forces, whereas the cited paper has dealt with the purely gravitational three-body problem. The ability of initial velocities to break or weaken the resonance raises the question of how many of the ejecta are fast enough to either be unlocked from the resonance at all or at least to :nd themselves in a shallow resonance. Consider a speed distribution of the impact ejecta at the Hyperion surface. The mass fraction of ejecta with speeds ¿ u is F(¿ u) = (u=u0 )− ;

(2)

where u0 ∼ 10 m s−1 and  ∼ 2 (Banaszkiewicz and Krivov, 1997b). Distribution (2) suggests velocities just above 10 m s−1 to be typical for most of the ejecta. This is de:nitely not suQcient to unlock the grains from the resonance. Does it mean that the initial velocity e6ect is rather unimportant, as was assumed by Banaszkiewicz and Krivov (1997b)? Not necessarily so! What we need to analyze is the velocities at the boundary of the Hyperion action sphere, and not just the surface distribution. The mean escape velocity for Hyperion is uesc ≈ 140 m s−1 . By virtue of (2), the mass fraction of ejecta that overcome the satellite gravity is (u0 =uesc ) ≈ 2 × 10−3 . From now on let us consider only these, escaping particles. Transforming the distribution at surface (2) to the velocity distribution of the escaping particles at the boundary of the action sphere (see Eq. (6) of Krivov, 1994) yields
F(¿ u) =

2 uesc 2 + u2 uesc

=2 :

(3)

For  = 2, the median velocity of the escaping grains (i.e., the grains that have u ¿ uesc at the surface) is equal to

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Fig. 4. Breaking resonance by the plasma drag force. Shown is a trajectory of a 10
m test particle launched from the edge of the action sphere of Hyperion with zero initial velocity. The particle is exposed to the plasma drag force of various strengths: 30 (left), 10 (middle) and 3 (right) times its actual (modeled) value. Top: critical argument, bottom: semimajor axis.

uesc ≈ 140 m s−1 —more than enough for the grains to avoid resonance locking! 3.3. Can the resonance be broken later? Now let us consider particles that are ejected at relatively low speeds and start their planetocentric motion, being locked in the resonance. A mechanism which has a potential to destroy the stable resonant motion is a plasma drag force. In the earlier studies, we estimated its strength to range from some 4% to 6% of the radiation pressure force. The new model described above reduces typical values to a few tenths of a percent. This brings up a question: is this much smaller dissipative force still able to release the particles from the resonance and, if so, how e6ective is it? To :nd an answer (to be con:rmed by the main bulk of integrations), we have performed a series of numerical experiments. A 10
m-sized particle was launched from the edge of the action sphere of Hyperion with zero initial velocity. First, we followed this trajectory with the normal strength of the PD force predicted by our model. In the subsequent runs, we did the same, but with the plasma drag force arti:cially magni:ed by the factors of 3, 10 and 30. Note that the latter value is nearly as large as that used in our previous modelling (Banaszkiewicz and Krivov, 1997b). The results

of these test runs are shown in Fig. 4. The plasma drag magni:ed by a factor of 30 breaks down the resonance in 1 s:y:, which is seen as a sharp transition of the critical argument from librational to rotational regime. This is accompanied by an expected slow monotonic increase in the semimajor axis during the :rst 1 s:y: of the motion. The time interval of the resonance breaking is nearly the same as in our previous model. Our plasma drag, which is 10 times the actual value, breaks down the resonance as well, but more gently. After 2 s:y:, the libration band starts to grow and after 5 s:y: ◦ ◦ broadens to [80 ; 280 ] which, as discussed above, leads to instability. The inevitable consequence of this is that again the resonance is broken: after 7 s:y:, the critical argument gets into the rotational mode. Interestingly, the semimajor axis no longer shows a secular trend. Instead, the continuous action of the plasma drag compels the semimajor axis to oscillate with an increasing amplitude around a nearly constant mean value. The next run was one with the PD force strength exceeding the actual value by the factor of 3. The picture is qualitatively similar, but the time scales are considerably longer. The libration band starts to grow after 6 s:y: and after 25 s:y: the resonance is broken down. The :nal run—with the actual values of PD force—resulted in a strong resonance locking over the interval of 100 s:y:! These tests suggest the existence of a certain threshold in the plasma drag strength. Below a certain minimum value,

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the force in question is no longer able to free the debris from the resonance, despite the tangential directionality and non-conservative nature of this perturbation. The absolute value of the threshold should depend on the sizes and initial velocities of the grains. As a rule, the modeled strengths of the plasma drag seem to be insuQcient for 10
m-sized and larger grains to leave the zone of resonant stability. Nevertheless, they still have a chance to avoid resonance trapping, if their velocities are high enough, as discussed above. 3.4. What happens to smaller particles? For particles with radii of several micrometres, the above scenarios need substantial modi:cations, as the dynamics of these grains are controlled by radiation pressure perturbations. At this size regime the plasma drag gets stronger as well. The combined inAuence of the two forces allows most of the ejecta less than several micrometres in size to leave the resonant zone in a small fraction of one s.y., regardless of their initial velocities. 4. Eventual fate of the Hyperion ejecta From a general qualitative overview of the dynamics, we turn to the results of the main set of numerical integrations. We will proceed from the largest grains to the smallest ones, giving the statistics of various scenarios and analyzing them from the dynamical viewpoint. 4.1. 100
m-sized grains Over the time interval considered, as many as 54% of 100
m-sized particles ended their histories at Titan. The remaining 46% are the particles which are still in Aight when 100 s:y: elapsed; following Banaszkiewicz and Krivov (1997b), we call these trajectories clear. We now discuss both the collisional and clear trajectories in more details. The collisional trajectories are those where the particles are either free of the resonance or locked in a shallow resonance. In accordance with our expectations, both cases trace back to relatively large initial velocities of ejection. It is important to analyze the collisional times of particles. Fig. 5 (line with pluses) gives the probabilities of collisions with Titan versus time interval. The collisional probability increases monotonically with time, but is not likely to approach 100% asymptotically (the insuQcient number of trajectories, unfortunately, does not allow us to draw :rm conclusions). We can just suppose that a certain portion of trajectories will remain clear even on much longer intervals than 100 s:y: Now let us look at the clear trajectories. To clarify the eventual fate of such grains, we have analyzed the evolution of the orbital elements. In some cases, the trajectories are still locked in the resonance with Titan after 100 s:y:; such

Fig. 5. Collisional times for Hyperion grains. Shown are percentages of collisions against Aight time of particles. Pluses: 100
m, collisions with Titan; crosses: 10
m, collisions with Titan; :lled circles: 2
m, collisions with Saturn.

particles are likely to stay for a long time in the extended swarm around Hyperion. A particular trajectory of this kind is shown in the left panels of Fig. 6. Some other clear trajectories are trapped in shallow resonances. This is exempli:ed by the middle column of Fig. 6, which presents a trajec◦ ◦ tory with a libration band extending from 110 to 250 (i.e., ◦ with the libration amplitude of 70 , twice as large as Hyperion’s). The remaining cases are the grains in non-resonant, but “safe” trajectories, whose sizes and orientations make approaches to Titan hardly possible. This is most typical for grains initially ejected at very high speeds (the tail of the velocity distribution). The trajectory of a particle occasionally ejected with velocity of 785 m s−1 is shown in the right column of Fig. 6. The orbit is large and eccentric; it lies wholly outside Hyperion’s and Titan’s orbits and does not approach Titan. We can speculate that orbits of this type, being nearly Keplerian, would steadily evolve under a large array of weak gravitational and non-gravitational forces, including ones not taken into account in our analyses (e.g., Poynting–Robertson drag). Whatever happens eventually to these particles, the involved time scales are large and the spatial densities low, so that their destinations are out of the range of interest; such grains may be regarded as a minor part of the sporadic background which exists everywhere in interplanetary space. 4.2. 10
m-sized grains Over the time interval of 100 s:y:, 60% of 10
m-sized grains collided with Titan. This fact, along with the 54% result for 100
m-sized ones, con:rms our earlier conclusion (Banaszkiewicz and Krivov, 1997a, b): the most probable fate of grains larger than ∼ 5
m is collision with Titan.

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Fig. 6. Three particular “clear” trajectories of 100
m-sized particles (columns). From top to bottom: critical argument, semimajor axis, eccentricity and inclination. See text for discussion.

An analysis of both collisional and clear trajectories of 10
m-sized yields results similar to those for 100
m particles. Again, the collisional trajectories mostly correspond to higher ejection velocities and are either unlocked or locked in shallow resonances. The collisional times of particles are plotted in Fig. 5 (crosses) giving nearly the same result as for 100
m grains. Three types of clear trajectories—locked in a “standard” or in a shallow resonance and unlocked “safe”

trajectories—are also the same. One more type of behavior, which we did not meet for larger grains, is unlocked trajectories with numerous encounters with Titan. In one of the cases, the grain made as many as 304 Aybys of Titan at distances less than the radius of its sphere of inAuence, but had not collided with Titan during 100 s:y:! Such trajectories will most likely be lost to Titan after a longer time interval.

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4.3. 2
m-sized and smaller grains Smaller grains (rg = 2
m) behave di6erently than our earlier results predicted. Instead of expected numerous escapes from the Saturnian system discussed in Banaszkiewicz and Krivov (1997a, b), we see dominant collisions with Saturn (81% of grains); the probabilities of collisions with Saturn as a function of time interval are depicted in Fig. 5 (circles). Occasional collisions with Titan are not improbable as well (10%). The remaining trajectories (9%) are clear. The di6erence from the statistics reported earlier is due to essential changes in our plasma drag model. The tiniest particles (rg . 1
m) evolve similarly. With rare exceptions, they collide with Saturn in a fraction of Saturnian year after ejection. Note that our model may not apply to submicrometre-sized grains. They are subject to the Lorentz force which is able to modify their dynamics drastically. However, it seems diQcult to quantify the electromagnetic force e6ect in view of large uncertainties in the magnetic :eld strength and geometry, plasma parameters and especially electrostatic charges acquired by the dust grains in the Hyperion–Titan region. 5. Spatial distribution of dust Dynamics of the Hyperion ejecta determine the geometry and dimensions of the steady-state dust cloud in the Hyperion–Titan system. In an attempt to :nd spatial inhomogeneities of the dust distributions—either the regions of increased dust concentrations or zones nearly devoid of the dust material—we have constructed snapshots of the dust complex in several reference frames. 5.1. Distribution of dust with respect to Hyperion Fig. 7 shows a typical distribution of large (10 – 100
m-sized) dust in a Hyperion-centred frame. As shown in Section 4, almost half of the grains in this size range stay locked in resonance with Titan, like Hyperion itself. Such particles :ll an arc-shaped region of an enhanced number density around Hyperion, which is clearly seen in the :gure. 5.2. Distribution of dust with respect to Saturn Fig. 8 depicts snapshots of the partial dust clouds formed by di6erent-sized grains in the Saturn-centred reference frame. Most of 100
m grains reside in a narrow and Aat ring between the pericentric and apocentric distances of Hyperion. The 10
m particles are located in a torus which is more extended both radially and vertically. The 2
m grains form a di6use spheroidal dust cloud extending from the innermost part of the Saturnian system to far beyond Hyperion’s orbit. An interesting feature seen in the plots is a tilt of the dust clouds with respect to the equatorial plane of Saturn. The plane of symmetry of a cloud always in-

Fig. 7. Spatial distribution of 10 –100
m-sized dust in the XY -projection of the Hyperion-centred coordinate system (x-axis along Saturn–Hyperion line, y-axis parallel to Hyperion’s orbital velocity vector). In this frame, Saturn oscillates because of Hyperion’s orbital eccentricity; solid circles depict two extreme positions of the planet. Coordinates are in Titan’s semimajor axis units.

tersects the equatorial plane along the direction toward the Saturnian vernal equinox point. The tilt angle increases with ◦ decreasing grain size and reaches nearly 30 for 2
m-sized grains. These features are exactly the same as those predicted for the presumed dust torus around the orbit of Deimos, the outer moon of Mars (Hamilton, 1996; Krivov et al., 1996; Krivov and Hamilton, 1997). In the cited papers, the planetocentric motion of a particle, perturbed by solar radiation pressure, was investigated. This suggests that the radiation pressure is a signi:cant, if not the dominant, dynamical factor that determines the geometry of dust clouds in the Hyperion– Titan system as well. This conclusion is con:rmed by other analyses of the orbital evolution that we made—short-term variations in eccentricity and long-term high-amplitude variations in inclination. However, the strong gravity of Titan, which acts as a continuous disturber and occasionally modi:es the orbits during close encounters, causes a tangible scatter of orbits in the space of orbital elements. This restricts the applicability of the analytic tools and does not allow us to proceed with the analytic approach to quantify the long-term evolution of the Hyperion ejecta. 5.3. Distribution of dust with respect to Titan Fig. 9 is similar to Fig. 8, except that it shows the snapshots in the Titan-centred frame. A clustering of grains seen ◦ at equal angles of 120 is just a direct consequence of the 4 : 3

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Fig. 8. Spatial distribution of dust in the Saturn-centred equatorial system (x-axis toward vernal equinox, z-axis toward Saturn’s north pole). From top to bottom: 100, 10 and 2
m. Left and right: XY - and YZ-projections. Coordinates are in Titan’s semimajor axis units. Solid circle in the left panels is the orbit of Titan, dashed circles correspond to the apocentric and pericentric distances of Hyperion and therefore limit the zone of Hyperion’s motion.

commensurability between the mean motions of Hyperion and Titan. Hyperion, surrounded by the dust grains moving in similar orbits, spends more time near the apocentre. Let us look at the consecutive positions of the Hyperion apocentre in the Titan-centred frame of Fig. 9. Suppose that at a certain time instant Hyperion is near the apocentre of its orbit. As ◦ the critical argument is nearly 180 , Titan is on the opposite side of Saturn. It means that the dust swarm around Hyperion’s position, when viewed in the Titan-centred frame, is ◦ located at 180 relative to the Saturn–Titan line. When Hyperion completes one circuit about Saturn to arrive at the

apocentre next time, Titan makes 4=3 revolutions; therefore, in our coordinate system the position of the apocentre (and ◦ the dust swarm) moves by 120 clockwise, which explains the second clustering point in the plots. Similarly, the next ◦ orbital revolution of Hyperion gives one more 120 clockwise shift of the clustering point. After three orbital circuits of Hyperion, we return to the initial geometry: the Hyperion apocentre and the dust cluster location are again to the left of the plots. We were looking for an enhanced dust density in the vicinities of Titan, which could result from the gravitational

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Fig. 9. Spatial distribution of dust in the Titan-centred rotating equatorial system (x-axis along Saturn–Titan line, z-axis toward Saturn’s north pole). Panels are the same as in Fig. 8. Coordinates are in Titan’s radii (including the atmosphere). Small circle and cross in the left panels are the position of Saturn and Titan, respectively, and dashed circles have the same meaning as in Fig. 8.

focussing of dust by Titan. No such e6ect has been found. As suggested by the plots, the focussing is not strong enough to produce a marked enhancement in the dust density near Titan. 5.4. Absolute number densities of dust The Cassini spacecraft, equipped with a specialized dust detector (Cosmic Dust Analyzer, CDA, Srama et al., 2001), is now en route to Saturn. After Saturn orbit insertion in the mid-2004, Cassini will traverse the region between the Hy-

perion and Titan orbits many times. In addition, a targeted Ayby of Hyperion at the minimum distance from Hyperion’s center of about 1000 km is scheduled for September 26, 2005 (R. Srama, priv. comm.). However, to reveal the dust structures discussed here, a special e6ort is required. Unlike the dust detectors onboard Galileo and Ulysses that detect dust impacts from a large solid angle because of the fast rotation of both spacecraft, the Cassini CDA must be pointed to a direction from which dust is expected to come. Therefore, quantitative estimates of dust Auxes are necessary to see whether these are high enough to justify speci:c

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CDA pointing. To facilitate the CDA experiment planning, we estimate now the expected absolute spatial densities of dust and detectability of the dust swarms shown in Figs. 7–9. Consider 10 –100
m-sized dust free from resonance with Titan and con:ned to a toroidal domain around the Hyperion orbit (the region between the two dashed circles in Figs. 8 and 9). The volume occupied by these particles is estimated as V ∼ 8a3H eH i, where aH and eH are the semimajor axis and eccentricity of Hyperion, and i is the typ◦ ical inclination of the dust grains. Assuming i ∼ 10 , we 3 18 :nd V ∼ 10 km . For interplanetary meteoroids as projectiles, the injection rate of dust with rg & 10
m into the Hyperion–Titan space was estimated from the model of KrOuger et al. (2000) to be N˙ ∼ 8 × 107 s−1 . Assume that half of these grains are free from the resonance and are kept in the torus for typical time intervals of T ∼ 100 s:y: ∼ 1011 s. Therefore, the mean number density inside the swarm is n ∼ 1=2 × N˙ T=V ∼ 3 km−3 . This is only ∼ 102 – 103 times larger than the number density of like-sized IDPs expected near Saturn (Banaszkiewicz and Krivov, 1997b). The corresponding geometrical normal optical depth of the torus is  ∼ 5 × 10−10 . One consequence is that the grain– grain collisions are unimportant. The collisional lifetime of grains is estimated as (e.g., Artymowicz and Clampin, 1997) Tcoll ∼ P=(12), where P is the orbital period of particles. For 10
m grains, Tcoll ∼ 3×105 s:y. So a low optical depth also shows clearly that there is no chance for success for any search based on remote sensing measurements. Nevertheless, there might be a hope to detect the Hyperion debris by the Cassini CDA. For the assumed velocities of 5 km s−1 and a sensitive area of the CDA’s impact ionization target of 0:08 m2 (Srama et al., 2001), the number density found above corresponds to ∼ 40 hits of ≈ 10
m-sized grains per year. The probability of detection could be higher, if the spacecraft made a passage through the presumed dust swarm near Hyperion seen as an arc in Fig. 7. Consider again the interplanetary impactors and assume that half of the escaping ejecta from Hyperion with rg & 10
m, i.e., N˙ ∼ 4 × 107 s−1 , are kept in this resonant cloud around the moon, the volume of which we estimate as V ∼ 6 × 1016 km3 . The major loss mechanism for these grains is probably reaccretion by Hyperion. Our numerical integrations (Section 2) give the mean random speed of particles with respect to Hyperion of v ∼ 1 km s−1 . The timescale of reimpact with Hyperion is then estimated as T ∼ VSH−1 v−1 ∼ 103 s:y: (SH is the cross section of the satellite). Using these estimates to scale the dust density distribution of Fig. 7, we estimated the absolute number densities at various spatial locations within the Hyperion swarm. Along the orbit of Hyperion (ahead or behind it), n ∼ 3 × 105 × r −1 km−3 , where r is the distance from the moon in 1000 km. In the directions perpendicular to the moon’s orbit, the density decreases faster. Assume that the Cassini spacecraft intersects the Hyperion orbit at a minimum distance of 103 km, which is close to the value

1275

currently planned for the targeted Ayby. With an e6ective detector area of 0:08 m2 and a relative spacecraft velocity of 5 km s−1 , the expected impact rate is ∼ 6 min−1 , so the detection is likely. It should be noted that we discuss now the detectability of quite large dust grains, & 10
m in size, which may favour the charging and chemical composition measurements by the Cassini CDA. Another population of dust that could be detected with the Cassini CDA during the Hyperion Ayby is the satellite ejecta that have just left the Hyperion surface. Most of these debris should move in ballistic trajectories and fall back to the surface shortly after ejection, while others spread in circumplanetary space, giving rise to the Hyperion torus and Hyperion swarm just described. First successful measurements of such “fresh” ejecta have been recently made with the dust detector onboard the Galileo spacecraft near the Jovian moons Ganymede (KrOuger et al., 1999, 2000), Europa, and Callisto (KrOuger et al., 2001). These detections demonstrated the eQciency of the impact ejecta production via micrometeoroid bombardment. Furthermore, the Galileo data turned out to be consistent with the model of an impact-generated dust cloud around a moon (KrOuger et al., 2000). Applying the same model to Hyperion, we estimated the number densities of the Hyperion ejecta with radii rg ¿ 0:3
m (close to the CDA threshold). The result is n∼6×105 ×r −2:5 km−3 , where r is the distance from Hyperion in thousand km. This translates to an impact rate at the closest approach of ∼10 min−1 , showing a possibility of detection. Two remarks should be made about all absolute estimates given above—for a non-resonant torus around Hyperion’s orbit, the resonant swarm around Hyperion and the ejecta cloud around the satellite. First, the estimates are very uncertain. Most of the uncertainty comes from the vaguely known dust production rate from the Hyperion surface, N˙ , which, in turn, is caused by poor knowledge of the mass Aux of interplanetary micrometeoroids at Saturn, ejecta yield and ejecta speed distribution (see Section 6 for more discussion). We note, however, that the model used here, including the way the poorly known parameters were chosen, is similar to the one tested earlier at Jupiter against real data of the Galileo spacecraft (KrOuger et al., 2000). Our second remark is that these estimates were obtained for interplanetary impactors and therefore are rather conservative. If dust grains from Phoebe and=or recently discovered other irregular satellites are also eQcient in ejecting material from Hyperion (see Section 6), the number densities and Auxes in all dust structires under study may be about one order of magnitude higher than the estimates above suggest. 6. Hyperion dust at Titan 6.1. Dust inAux to Titan We wish to assess quantitatively the mechanisms of dust delivery to Titan. Reverting back to a general scheme of

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dust production in the outer Saturnian system (see Introduction), we have three mechanisms that may sustain continuous dust inAux to Titan (marked I to III in Fig. 1). The :rst one is hypervelocity impacts of interplanetary grains onto Hyperion’s surface. The second is hits of Hyperion by retrograde projectiles from the outer satellites of Saturn; for the sake of simple estimates and because the knowledge of the orbital and physical parameters of newly discovered outer moons is as yet poor, we con:ne our consideration to Phoebe. Finally, the third possible source of dust for Titan is the Phoebe grains that avoid trapping by both Iapetus and Hyperion and come to Titan directly. Accordingly, let us consider three dust populations at Titan: (i) the Aux of Hyperion dust ejected by interplanetary impactors FHT , (ii) the Aux of Hyperion dust ejected by the Phoebe projectiles FPHT , (iii) the Aux of Phoebe particles FPT . Each of these should be compared to the direct interplanetary Aux FT at Titan. Hereafter, indices T, H and P refer to Titan, Hyperion and Phoebe, respectively. We can write FHT H H H HT = "HT geom "yield "esc "size "coll ; FT

(4)

FPHT P P P PH H H H HT = "PT geom "yield "esc "size "coll "yield "esc "size "coll ; FT

(5)

FPT P P P PT = "PT geom "yield "esc "size "coll : FT

(6)

There are :ve types of factors in the right-hand sides. The :rst factor, "AB geom , is the ratio of surface areas of bodies A and B: −3 "HT geom = SH =ST ≈ 2 × 10

and

−3 "PT geom = SP =ST ≈ 1 × 10 :

(7)

The factor "A yield is the characteristic yield, de:ned as the ratio of the ejected mass to the impacting mass. Plausible estimates for hypervelocity impacts into icy (Hyperion) and regolith-like (Phoebe) targets are (Koschny and GrOun, 1996) 3 "H yield ≈ 3 × 10

and

"Pyield ≈ 1 × 102 :

(8)

Our estimates for the third factor, "A esc , the mass fraction of ejecta that overcomes the gravity of body A, are (Banaszkiewicz and Krivov, 1997b) "H esc

∼ 0:08

and

"Pesc

∼ 0:7:

(9)

The fourth factor, "A size , represents the mass fraction of the dust ejecta from body A that falls in the size regime rg & 5
m (the smaller grains are unlikely to reach Titan). For a power-law ejecta size distribution with the slope of ≈ 3:5 (KrOuger et al., 2000), most of the mass is in the large fragments, so that P "H size ≈ "size ∼ 1:

(10)

Finally, the :fth factor, "AB coll , de:nes the portion of the particles which, if escaped from body A, eventually collide with body B. Estimates by Burns et al. (1996) suggest that "PH coll ≈ 0:2

and

"PT coll ≈ 0:1

(11)

and our numerical integrations (for 5
m . rg . 100
m) show that "HT coll ∼ 0:6:

(12)

Combining the above formulae, we :nally obtain FHT ≈ 0:3; FT

(13)

FPHT ≈ 2; FT

(14)

FPT ≈ 7 × 10−3 : FT

(15)

The uncertainties of these estimates come from the uncertainties in the "-factors, most notably "yield and "esc . These two are not independent, however: their product is roughly proportional to f = Ke =Ki , the fraction of the impact energy Ki that goes to the ejecta kinetic energy Ke (see Eq. (6) of KrOuger et al., 2000). The uncertainty in f is a factor of several, and therefore the uncertainty in (13) and (15) is probably about one order of magnitude. Estimate (14) is more uncertain, because the underlying Eq. (5) contains H P P PH both "H yield "esc and "yield "esc , as well as another factor, "coll , whose estimate is based solely on the modelling results. We see that the direct Aux of Phoebe particles at Titan FPT makes little contribution to the dust budget and can be neglected. In contrast, the inAux of Hyperion dust to Titan (FHT ; FPHT ) is most likely comparable with the direct inAux of IDPs to Titan. It is also evident that the bombardment of Hyperion’s surface by Phoebe grains (FPHT ) is probably more e6ective than that by IDPs (FHT ), therefore most of the Hyperion ejecta are likely to be due to the impacts of Phoebe projectiles. Were the other retrograde moons eQcient as dust sources (which largely depends on the "-factors for these satellites that are yet unknown), this would further strengthen the e6ect. 6.2. Hyperion dust and Titan’s atmosphere One of the intriguing problems in the photochemistry of the Titan’s atmosphere is the source and the abundances of oxygen-bearing compounds. Two of them, CO and CO2 , have been detected by Voyager and, more recently, the water lines have been observed by ISO (Coustenis et al., 1998). The CO2 molecules are formed in the reaction CO + OH → CO2 + H and CO, in turn, is either released from the surface or produced from OH and CH3 , OH and CH2 (Yung et al., 1984; Lara et al., 1996), therefore a source of OH is

A.V. Krivov, M. Banaszkiewicz / Planetary and Space Science 49 (2001) 1265–1279

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Fig. 10. Distribution of entry points of the Hyperion ejecta over the Titan atmosphere. Longitude is measured in the orbital plane of Titan from the anti-Saturn direction. Circles and crosses: 100 and 10
m grains. Slightly more material is seen to be deposited on the leading hemisphere of Titan. A moderate concentration of entry points toward the equatorial region is also apparent.

required. The most likely mechanism is photolysis of water that is brought to the atmosphere by meteoroids and released from grains by ablation (Ip, 1990). In a numerical study, English et al. (1996) calculated the vertical pro:le of water production rate due to nominal interplanetary dust Aux. They have found, however, that to produce the observed abundances of CO and CO2 the Aux of pure water-ice meteoroids should exceed the measured interplanetary Aux by a factor of 200. This requirement is somewhat relaxed if one allows the grains to contain 10% of CO (as in Halley’s comet), but still a rather substantial Aux, a factor of 20 larger than the nominal one, is necessary. Such enormous interplanetary dust populations are certainly ruled out. In the following paper, Lara et al. (1996) have used a modi:ed mixing ratio of CO2 (from 10−9 at 50 km to 7 × 10−8 at 400 km) and were able to explain the abundances by a meteoroid Aux only 2.5 times more intensive than the nominal one, however with a 20% content of CO in the grains. The recent ISO observations (Coustenis et al., 1998) suggest that the required Aux can be decreased to the level given by English et al. (1996), therefore the local (Saturnian) inventory of water-containing grains is not necessarily needed as a main source. Still, several problems remain: (i) The interplanetary meteoroids in the vicinity of Saturn, most of which may originate from the Edgeworth–Kuiper belt, are probably not composed of pure volatile materials (see, e.g., Barucci et al., 1999). If so, an additional supply of water ice (e.g., ejecta from icy Hyperion) should be introduced. (ii) The interplanetary Aux shows a pronounced asymmetry when deposited on the leading and trailing hemispheres of Titan: an order of magnitude more material is delivered to the leading side. This e6ect is caused by the orbital motion

of Titan, which leads to di6erent relative velocities of grains approaching Titan. In the case of dust originating from Hyperion that asymmetry, according to the collision statistics we obtained, is signi:cantly smaller: the fractions of material, deposited on the leading and trailing hemispheres, are 67% and 33% for 100
m particles, and 56% and 44% for 10
m ones (Fig. 10). (iii) The encounter velocity of Hyperion ejecta that reach Titan is much smaller than that of interplanetary grains (Fig. 11). Hence, the altitude pro:le of water deposition should show a peak at a slightly lower altitude than the pro:le calculated for interplanetary meteoroids. Also the water molecules should be delivered to lower altitudes when the entry velocity of meteoroids is smaller (see Fig. 1 in English et al., 1996). Moreover, Hyperion ejecta contain, most probably, a signi:cant fraction of refractory material. As the result, the water deposition pro:le will change. 7. Conclusions In this paper, we readdressed the idea that Hyperion may emit quite a large amount of icy ejecta. We suggest that debris are produced by hypervelocity impacts of particles coming from Phoebe and other irregular moons, as well as of interplanetary micrometeoroids. A quantitative assessment of dust production mechanisms and detailed numerical modelling of the Hyperion ejecta dynamics lead us to the following conclusions. 1. The dynamics of ejecta from Hyperion is governed by the gravitational resonance with Titan and by non-gravitational forces (radiation pressure and plasma drag). We expect that a majority of particles escaping from

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remote sensing is ruled out. Much better are the chances for in situ detection by the Cassini detector during the planned Ayby of Hyperion. With a minimum encounter distance of about one thousand kilometres, the detection of dust would be very likely. 4. Although the spatial density of dust in the planetocentric space near the orbits of Hyperion and Titan is low, the contamination of Titan by the Hyperion debris may be signi:cant. It is quite probable that the dust inAux from Hyperion to Titan is comparable with, or even greater than, the direct inAux of interplanetary particles to Titan. The inAux of icy (H2 O) particles from Hyperion could be a possible explanation of the observed abundance of CO and CO2 molecules in Titan’s atmosphere. Acknowledgements We thank Michael Belton and Jean-Marc Petit for thorough reviews and Jim Howard for improving the style of the manuscript. Much of this work was done during the stay of A.V.K. in the Space Research Centre (Warsaw), supported by grants from the JXozef Mianowski Fund and Foundation for Polish Science (Poland) and the State Committee for Higher Education (Russia). A.V.K. also acknowledges support from Deutsches Zentrum fOur Luft- und Raumfahrt e.V. (DLR). Fig. 11. Entry speeds of the Hyperion ejecta penetrating Titan’s atmosphere. (a) Speeds of 100
m-sized grains (circles) and 10
m-sized grains (crosses) plotted against longitude at Titan’s surface. No statistical dependence on the longitude is seen. (b) Distributions of entry speeds of 100
m-sized grains (solid line) and 10
m-sized grains (dashed line). The minimum speed is about 2:2 km s−1 , close to the escape velocity at the upper atmosphere. The distribution of 10
m grains is broader than that of 100
m ones.

Hyperion are not locked in the resonance because of their high initial velocities, or are released from the resonant locking by the plasma drag and radiation pressure forces. The orbits liberated from the resonance become unstable and experience multiple close approaches to Titan. As a result, most of the particles larger than ∼ 5
m in size that escape from Hyperion eventually collide with Titan; smaller debris would predominantly strike Saturn. 2. We have constructed and explained steady-state spatial distributions of dust in the Hyperion–Titan system. The dust torus along the Hyperion orbit is tilted o6 the equatorial plane of Saturn and has a structure that depends on the particle radii. The dust density is much higher near Hyperion: a population of relatively big particles locked in the resonance form an arc-like structure along the Hyperion orbit around Hyperion’s position. However, no particular dust concentration in the vicinities of Titan was found. 3. The spatial density of dust everywhere in the Hyperion– Titan system is expected to be so low that the possibility of

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