Optimal low-thrust transfers between libration point orbits

Over the past three decades, ballistic and impulsive trajectories between libration point orbits (LPOs) in the Sun–Earth–Moon system have been investigated to a large extent. It is known that coupling invariant manifolds of LPOs of two different circular restricted three-body problems (i.e., the Sun–Earth and the Earth–Moon systems) can lead to significant mass savings in specific transfers, such as from a low Earth orbit to the Moon’s vicinity. Previous investigations on this issue mainly considered the use of impulsive maneuvers along the trajectory. Here we investigate the dynamical effects of replacing impulsive ΔV’s with low-thrust trajectory arcs to connect LPOs using invariant manifold dynamics. Our investigation shows that the use of low-thrust propulsion in a particular phase of the transfer and the adoption of a more realistic Sun–Earth–Moon four-body model can provide better and more propellant-efficient solution. For this purpose, methods have been developed to compute the invariant tori and their manifolds in this dynamical model.     

Cassini RPWS observations of dust in Saturn's E Ring

The Cassini radio and plasma wave science (RPWS) instrument is sensitive to few-micron dust grains impacting on the spacecraft at relative speeds of order 10 km/s. Through the first year or so of operations in orbit at Saturn, the RPWS has made a number of both inclined and equatorial crossings of the E ring, particularly near the orbit of Enceladus. Assuming water ice grains, the typical size particle detected by the RPWS has a radius of a few microns. Peak impact rates of about 50 s⁻¹ are found near the orbit of Enceladus corresponding to densities of order 5×10⁻⁴ m⁻³. The variation of dust fluxes as a function of height above or below the equator is well described by a Gaussian distribution with a scale height of about 2800 km although there is usually some non-Gaussian variation near the peak fluxes suggesting some structure in the core of the ring. Offsets of the peak number densities are typically of the order of a few hundred km from the geometric equator. A near-equatorial radial profile through the orbit of Enceladus shows a sharply peaked distribution at the orbit of the moon. A size distribution averaged over several passes through the orbit of Enceladus is determined which varies as m^−2.80. The peak in dust number density at the orbit of Enceladus is consistent with previous optical measurements and strongly supports the suggestion that Enceladus is a primary source for E ring particles.     

Two stages of dust delivery from satellites to planetary rings

Faint rings of micrometre-sized dust particles embrace many planets in the Solar system. As a rule, they are replenished by ejecta from embedded atmosphereless moons. On a number of occasions, the ejecta are generated by hypervelocity meteoroid impacts into the moons. Small ejecta fragments are then swiftly shifted into rings by an array of non-gravitational forces, e.g. radiation pressure or plasma drag. A significant fraction of ejecta mass, however, is contained in relatively big, multi-micrometre fragments which are subject to gravity only. Having escaped from the satellite, they stay close to its orbit and form a belt around planet. This belt is itself a source of ring dust through collisional disruption of its particles. Here the contributions of belts to the respective rings are estimated for selected satellites of Jupiter and Saturn. The belts under review could supply substantially more dust to rings than the direct ejecta from satellites and should be taken into account when estimating ring dust budgets. The belts are very difficult to observe, however, and some of them remain a theoretical proposition. We find an appealing evidence for the belts due to Amalthea and Thebe around Jupiter, and for the belt due to Enceladus around Saturn.     

Three tenuous rings/arcs for three tiny moons

Using Cassini images, we examine the faint material along the orbits of Methone, Anthe and Pallene, three small moons that reside between the orbits of Mimas and Enceladus. A continuous ring of material covers the orbit of Pallene; it is visible at extremely high phase angles and appears to be localized vertically to within ±25 km of Pallene's inclined orbit. By contrast, the material associated with Anthe and Methone appears to lie in longitudinally confined arcs. The Methone arc extends over ∼10° in longitude around the satellite's position, while the Anthe arc reaches ∼20° in length. The extents of these arcs are consistent with their confinement by nearby corotation eccentricity resonances with Mimas. Anthe has even been observed to shift in longitude relative to its arc in the expected manner given the predicted librations of the moon.     

Multiple-satellite-aided capture trajectories at Jupiter using the Laplace resonance

Satellite-aided capture is a mission design concept used to reduce the delta-v required to capture into a planetary orbit. The technique employs close flybys of a massive moon to reduce the energy of the planet-centered orbit. A sequence of close flybys of two or more of the Galilean moons of Jupiter may further decrease the delta-v cost of Jupiter orbit insertion. A Ganymede-Io sequence can save 207 m/s of delta-v over a single Io flyby. A phase angle analysis based on the Laplace resonance is used to find triple-satellite-aided capture sequences involving Io, Europa, and Ganymede. Additionally, the near-resonance of Callisto and Ganymede is used to find triple-satellite-aided capture sequences involving Callisto, Ganymede, and another moon. A combination of these techniques is used to find quadruple-satellite-aided capture sequences that involve gravity-assists of all four Galilean moons. These sequences can save a significant amount of delta-v and have the potential to benefit both NASA’s Jupiter Europa orbiter mission and ESA’s Jupiter Ganymede orbiter mission.     

The use of invariant manifolds for transfers between unstable periodic orbits of different energies

Techniques from dynamical systems theory have been applied to the construction of transfers between unstable periodic orbits that have different energies. Invariant manifolds, trajectories that asymptotically depart or approach unstable periodic orbits, are used to connect the initial and final orbits. The transfer asymptotically departs the initial orbit on a trajectory contained within the initial orbit’s unstable manifold and later asymptotically approaches the final orbit on a trajectory contained within the stable manifold of the final orbit. The manifold trajectories are connected by the execution of impulsive maneuvers. Two-body parameters dictate the selection of the individual manifold trajectories used to construct efficient transfers. A bounding sphere centered on the secondary, with a radius less than the sphere of influence of the secondary, is used to study the manifold trajectories. A two-body parameter, κ, is computed within the bounding sphere, where the gravitational effects of the secondary dominate. The parameter κ is defined as the sum of two quantities: the difference in the normalized angular momentum vectors and eccentricity vectors between a point on the unstable manifold and a point on the stable manifold. It is numerically demonstrated that as the κ parameter decreases, the total cost to complete the transfer decreases. Preliminary results indicate that this method of constructing transfers produces a significant cost savings over methods that do not employ the use of invariant manifolds.     

Accretion of Saturn’s mid-sized moons during the viscous spreading of young massive rings: Solving the paradox of silicate-poor rings versus silicate-rich moons

The origin of Saturn’s inner mid-sized moons (Mimas, Enceladus, Tethys, Dione and Rhea) and Saturn’s rings is debated. Charnoz et al. [Charnoz, S., Salmon J., Crida A., 2010. Nature 465, 752–754] introduced the idea that the smallest inner moons could form from the spreading of the rings’ edge while Salmon et al. [Salmon, J., Charnoz, S., Crida, A., Brahic, A., 2010. Icarus 209, 771–785] showed that the rings could have been initially massive, and so was the ring’s progenitor itself. One may wonder if the mid-sized moons may have formed also from the debris of a massive ring progenitor, as also suggested by Canup [Canup, R., 2010. Nature 468, 943–946]. However, the process driving mid-sized moon accretion from the icy debris disks has not been investigated in details. In particular, Canup’s (2010) model does not seem able to explain the varying silicate contents of the mid-sized moons (from 6% to 57% in mass). Here, we explore the formation of large objects from a massive ice-rich ring (a few times Rhea’s mass) and describe the fundamental properties and implications of this new process. Using a hybrid computer model, we show that accretion within massive icy rings can form all mid-sized moons from Mimas to Rhea. However in order to explain their current locations, intense dissipation within Saturn (with Q_p < 2000) is required. Our results are consistent with a satellite origin tied to the rings formation at least 2.5 Gy ago, both compatible with either a formation concurrent to Saturn or during the Late Heavy Bombardment. Tidal heating related to high-eccentricity post-accretional episodes may induce early geological activity. If some massive irregular chunks of silicates were initially present within the rings, they would be present today inside the satellites’ cores which would have accreted icy shells while being tidally expelled from the rings (via a heterogeneous accretion process). These moons may be either mostly icy, or, if they contain a significant amount of rock, already differentiated from the ice without the need for radiogenic heating. The resulting inner mid-sized moons may be significantly younger than the Solar System and a ∼1 Gyr formation delay is possible between Mimas and Rhea. The rings resulting from this process would evolve to a state compatible with current mass estimates of Saturn’s rings, and nearly devoid of silicates, apart from isolated silicate chunks coated with ice, interpreted as today Saturn’s rings’ propellers and ring-moons (like Pan or Daphnis).     

Weak stability boundary trajectories for the deployment of lunar spacecraft constellations

Suitable lunar constellation coverage can be obtained by separating the satellites in inclinations and node angles. It is shown in the paper that a relevant saving of velocity variation ΔV can be achieved using weak stability boundary trajectories. The weakly stable dynamics of such transfers allows the separation of the satellites from the nominal orbit to the required orbit planes with a small amount of ΔV. This paper also shows that only one different set of orbital parameters at Moon can be reached with the same ΔV manoeuvre starting from a nominal trajectory and ending at a fixed periselenium altitude. In fact, such a feature is proved to be common to other simpler dynamical systems, such as the two- and three-body problems.     

The disruption of planetary satellites and the creation of planetary rings

The Voyager spacecraft discovered that small moons orbit within all four observed ring systems coincident with the discovery of narrow and dusty rings around Jupiter, Saturn, Uranus and Neptune. These moons can provide the source for new rings if they are catastrophically disrupted by a comet or large meteoroid impact. This hypothesis for ring origins provides a natural mechanism for the ongoing creation of planetary rings. While it relieves somewhat the problem of explaining the continued existence of rings with apparently short evolutionary lifetimes, it raises the problem of explaining the continued existence of small moons, and the coexistence of moons and rings at comparable locations within the Roche zones of the giant planets. This problem has been studied in some detail recently, and the present work is a review of our current understanding of the processes in satellite disruption that pertain to the creation of planetary rings and the collisional cascade of circumplanetary bodies. Significant progress has been made. Narrow rings are produced by disruption of small moons in numerical simulations, and a self-consistent model of the collisional cascade can explain present-day moon populations. Absolute timescales and initial moon populations remain uncertain due to our poor knowledge of the impactor population and uncertainties in the strength of planetary satellites. More pressing are the qualitative issues that remain to be resolved including the nature of reaccretion of the debris and the origin of Saturn's rings.     

Astrometric studies of the results of a new reduction of old photographic observations of the Saturnian System based on the comparison with the modern theories of satellite motion

The paper shows the possibility of increasing the accuracy of the results of photographic observations of Saturn and its moons made in the 1970s and reduced using the old reference star catalogues and semiautomatic measurements. New celestial coordinates of the moons (from the third to the eighth), “satellite minus satellite” relative moon coordinates, and Saturn coordinates by positions of satellites are obtained without measuring its images. The results are stored in the Pulkovo Observatory database on the Solar System bodies and are available online at www.puldb.ru. The efficiency of the reduction method based on digitizing of astronegatives using 21 Mpx Canon digital camera and IZMCCD software is shown. The comparison of new results of old observations with the latest theories of moon motion has revealed a significant increase in satellite positioning accuracy. The investigation of the differences (O–C) of celestial coordinates from satellite positions in their apparent Saturn-centric orbits has revealed a noticeable motion of the differences (O–C) in right ascension depending on their distances from Saturn for all moons.     

Processes forming and sustaining Saturn’s proton radiation belts

Saturn’s proton radiation belts extend over the orbits of several moons that split this region of intense radiation into several distinct belts. Understanding their distribution requires to understand how their particles are created and evolve. High-energy protons are thought to be dominantly produced by cosmic ray albedo neutron decay (CRAND). The source of the lower energies and the role of other effects such as charge exchange with the gas originating from Enceladus is still an open question. There is also no certainty so far if the belts exist independently from each other and the rest of the magnetosphere or if and how particles are exchanged between these regions. We approach these problems by using measurements acquired by the MIMI/LEMMS instrument onboard the Cassini spacecraft. Protons in the range from 500 keV to 40 MeV are considered. Their intensities are averaged over 7 years of the mission and converted to phase space densities at constant first and second adiabatic invariant. We reproduce the resulting radial profiles with a numerical model that includes radial diffusion, losses from moons and interactions with gas, and a phenomenological source. Our results show that the dominating effects away from the moon sweeping corridors are diffusion and the source, while interactions with gas are secondary. Based on a GEANT4 simulation of the interaction of cosmic rays with Saturn’s rings, we conclude that secondary particles produced within the rings can only account for the high-energy part of the source. A comparison with the equivalent processes within Earth’s atmosphere shows that Saturn’s atmosphere can contribute to the production of the lower energies and might be even dominating at the higher energies. Other possibilities to supply the belts and exchange particles between them, as diffusion and injections from outside the belts, or stripping of ENAs, can be excluded.     

Habitable zones of host stars during the post-MS phase

A star will become brighter and brighter with stellar evolution, and the distance of its habitable zone will become larger and larger. Some planets outside the habitable zone of a host star during the main sequence phase may enter the habitable zone of the host star during other evolutionary phases. A terrestrial planet within the habitable zone of its host star is generally thought to be suitable for the existence of life. Furthermore, a rocky moon around a giant planet may be also suitable for life to survive, provided that the planet–moon system is within the habitable zone of its host star. Using Eggleton’s code and the boundary flux of the habitable zone, we calculate the habitable zone of our Solar system after the main sequence phase. It is found that Mars’ orbit and Jupiter’s orbit will enter the habitable zone of the Solar system during the subgiant branch phase and the red giant branch phase, respectively. And the orbit of Saturn will enter the habitable zone of Solar during the He-burning phase for about 137 million years. Life is unlikely at any time on Saturn, as it is a giant gaseous planet. However, Titan, the rocky moon of Saturn, may be suitable for biological evolution and become another Earth during that time. For low-mass stars, there are similar habitable zones during the He-burning phase as our Solar, because there are similar core masses and luminosities for these stars during that phase.     

A dynamical solution of the triple asteroid system (45) Eugenia

We present the first dynamical solution of the triple asteroid system (45) Eugenia and its two moons Petit–Prince (diameter ∼ 7 km) and S/2004 (45) 1 (diameter ∼ 5 km). The two moons orbit at 1165 and 610 km from the primary, describing an almost-circular orbit (e ∼ 6 × 10⁻³ and e ∼ 7 × 10⁻² respectively). The system is quite different from the other known triple systems in the main belt since the inclinations of the moon orbits are sizeable (9° and 18° with respect to the equator of the primary respectively). No resonances, neither secular nor due to Lidov–Kozai mechanism, were detected in our dynamical solution, suggesting that these inclinations are not due to excitation modes between the primary and the moons. A 10-year evolution study shows that the orbits are slightly affected by perturbations from the Sun, and to a lesser extent by mutual interactions between the moons. The estimated J₂ of the primary is three times lower than the theoretical one, calculated assuming the shape of the primary and an homogeneous interior, possibly suggesting the importance of other gravitational harmonics.     

Determination of the optimal conditions for inclination maneuvers using a Swing-by

The search for methods to reduce the fuel consumption in orbital transfers is something relevant and always current in astrodynamics. Therefore, the maneuvers assisted by the gravity, also called Swing-by maneuvers, can be an advantageous option to save fuel. The proposal of the present research is to explore the influence of some parameters in a Swing-by of an artificial satellite orbiting a planet with one of the moons of this mother planet, with the goal of changing the inclination of the artificial satellite around the main body of the system. The fuel consumption of this maneuver is compared with the required consumption to perform the same change of inclination using the classical approach of impulsive maneuvers.     

Main methods of trajectory synthesis for scenarios of space missions with gravity assist maneuvers in the system of Jupiter and with landing on one of its satellites

The development of a methodology for designing trajectories of spacecraft intended for the contact and remote studies of Jupiter and its natural satellites is considered. This methodology should take into account a number of specific features. Firstly, in order to maintain the propellant consumption at an acceptable level, the flight profile, ensuring the injection of the spacecraft into orbit around the Jovian moon, should include a large number of gravity assist maneuvers both in the interplanetary phase of the Earth-to-Jupiter flight and during the flight in the system of the giant planet. Secondly, the presence of Jupiter’s powerful radiation belts also imposes fairly strict limitations on the trajectory parameters.

     

The E ring in the vicinity of Enceladus: I. Spatial distribution and properties of the ring particles

Saturn's diffuse E ring is the largest ring of the Solar System and extends from about (Saturn radius RS=60,330 km) to at least encompassing the icy moons Mimas, Enceladus, Tethys, Dione, and Rhea. After Cassini's insertion into her saturnian orbit in July 2004, the spacecraft performed a number of equatorial as well as steep traversals through the E ring inside the orbit of the icy moon Dione. Here, we report about dust impact data we obtained during 2 shallow and 6 steep crossings of the orbit of the dominant ring source—the ice moon Enceladus. Based on impact data of grains exceeding 0.9 μm we conclude that Enceladus feeds a torus populated by grains of at least this size along its orbit. The vertical ring structure at agrees well with a Gaussian with a full-width-half-maximum (FWHM) of ∼4200 km. We show that the FWHM at is due to three-body interactions of dust grains ejected by Enceladus' recently discovered ice volcanoes with the moon during their first orbit. We find that particles with initial speeds between 225 and 235 m s⁻¹ relative to the moon's surface dominate the vertical distribution of dust. Particles with initial velocities exceeding the moon's escape speed of 207 m s⁻¹ but slower than 225 m s⁻¹ re-collide with Enceladus and do not contribute to the ring particle population. We find the peak number density to range between 16×¹⁰⁻² m⁻³ and 21×¹⁰⁻² m⁻³ for grains larger 0.9 μm, and 2.1×¹⁰⁻² m⁻³ and 7.6×¹⁰⁻² m⁻³ for grains larger than 1.6 μm. Our data imply that the densest point is displaced outwards by at least with respect of the Enceladus orbit. This finding provides direct evidence for plume particles dragged outwards by the ambient plasma. The differential size distribution for grains >0.9 μm is described best by a power law with slopes between 4 and 5. We also obtained dust data during ring plane crossings in the vicinity of the orbits of Mimas and Tethys. The vertical distribution of grains >0.8 μm at Mimas orbit is also well described by Gaussian with a FWHM of ∼5400 km and displaced southwards by ∼1200 km with respect to the geometrical equator. The vertical distribution of ring particles in the vicinity of Tethys, however, does not match a Gaussian. We use the FWHM values obtained from the vertical crossings to establish a 2-dimensional model for the ring particle distribution which matches our observations during vertical and equatorial traversals through the E ring.     

The fate of ejecta from Hyperion

Ejecta from Saturn's moon Hyperion are subject to powerful perturbations from nearby Titan, which control their ultimate fate. We have performed numerical integrations to simulate a simplified system consisting of Saturn (including optical flattening as well as dynamical oblateness), its main ring system (treated as a massless flat annulus), the moons Tethys, Dione, Titan, Hyperion, and Iapetus, and the Sun (treated simply as a massive satellite). At several different points in Hyperion's orbit, 1050 massless particles, more or less evenly distributed over latitude and longitude, were ejected radially outward from 1 km above Hyperion's mean radius at speeds 10% faster than escape speed from Hyperion. Most of these particles were removed within the first few thousand years, but ∼3% of them survived the entire 100,000-year duration of the simulations. Ejecta from Hyperion are much more widely scattered than previously thought, and can cross the orbits of all of Saturn's satellites. About 9% of all the particles escaped from the saturnian system, but Titan accreted ∼78% of the total, while Hyperion reaccreted only ∼5%. This low efficiency of reaccretion may help to account for Hyperion's small size and rugged shape. Only ∼1% of all the particles hit other satellites, and another ∼1% impacted Saturn itself, while ∼3% of them struck its main rings. The high proportion of impacts into Saturn's rings is surprising; these collisions show a broad decline in impact speed with time, suggesting that Hyperion ejecta gradually spread inwards. Additional simulations were used to investigate the dependence of ejecta evolution on launch speed, the mass of Hyperion, and the presence of the Sun. In general, the wide distribution of ejecta from Hyperion suggests that it does contribute to “Population II” craters on the inner satellites of Saturn. Ejecta which escape from a satellite into temporary orbit about its planet, but later reimpact into the same moon or another one produce “poltorary” impacts, intermediate in character between primary and secondary impacts. It may be possible to distinguish poltorary craters from primary and secondary craters on the basis of morphology.     

Limits on the Orbits of Possible Eccentric and Inclined Moons of Extrasolar Planets Orbiting Single Stars

Limits are placed on the range of orbits and masses of possible moons orbiting extrasolar planets which orbit single central stars. The Roche limiting radius determines how close the moon can approach the planet before tidal disruption occurs; while the Hill stability of the star–planet–moon system determines stable orbits of the moon around the planet. Here the full three-body Hill stability is derived for a system with the binary composed of the planet and moon moving on an inclined, elliptical orbit relative the central star. The approximation derived here in Eq. (17) assumes the binary mass is very small compared with the mass of the star and has not previously been applied to this problem and gives the criterion against disruption and component exchange in a closed form. This criterion was applied to transiting extrasolar planetary systems discovered since the last estimation of the critical separations (Donnison in Mon Not R Astron Soc 406:1918, 2010a) for a variety of planet/moon ratios including binary planets, with the moon moving on a circular orbit. The effects of eccentricity and inclination of the binary on the stability of the orbit of a moon is discussed and applied to the transiting extrasolar planets, assuming the same planet/moon ratios but with the moon moving with a variety of eccentricities and inclinations. For the non-zero values of the eccentricity of the moon, the critical separation distance decreased as the eccentricity increased in value. Similarly the critical separation decreased as the inclination increased. In both cases the changes though very small were significant.     

Arecibo radar observations of Rhea, Dione, Tethys, and Enceladus

We have measured the bulk radar reflectance properties of the mid-size saturnian satellites Rhea, Dione, Tethys, and Enceladus with the Arecibo Observatory's 13 cm wavelength radar system during the 2004 through 2007 oppositions of the Saturn system. Comparing to the better studied icy Galilean satellites, we find that the total reflectivities of Rhea and Tethys are most similar to Ganymede while Dione is most similar to Callisto. Enceladus' reflectivity falls between those of Ganymede and Europa. The mean circular polarization ratios of the saturnian satellites range from ∼0.8 to 1.2, and are on average lower than those of the icy Galilean satellites at this wavelength although still larger than expected for single reflections off the surface. The ratio for the trailing hemisphere of Enceladus may be the exception with a value ?0.56. The 13 cm wavelength radar albedos and polarization ratios may be systematically lower than similar results from the Cassini orbiter's RADAR instrument at 2.2 cm wavelength [Ostro, S.J., and 19 colleagues, 2006. Icarus 183, 479-490]. Overall, these reflectivities and polarization properties, together with the shapes of the echo spectra, suggest subsurface multiple scattering to be the dominant reflection mechanism although operating less efficiently than on the large icy moons of Jupiter. All these saturnian moons and icy jovian moons are atmosphere-less, low temperature water ice surfaces, and any differences in radar properties may be indicative of differences in composition or the effects of various processes that modify the regolith structure. The degree of variation in radar properties with wavelength on each satellite may constrain the thickness and efficiency of the scattering layer.     

The three-dimensional structure of Saturn’s E ring

Saturn’s diffuse E ring consists of many tiny (micron and sub-micron) grains of water ice distributed between the orbits of Mimas and Titan. Various gravitational and non-gravitational forces perturb these particles’ orbits, causing the ring’s local particle density to vary noticeably with distance from the planet, height above the ring-plane, hour angle and time. Using remote-sensing data obtained by the Cassini spacecraft in 2005 and 2006, we investigate the E-ring’s three-dimensional structure during a time when the Sun illuminated the rings from the south at high elevation angles (>15°). These observations show that the ring’s vertical thickness grows with distance from Enceladus’ orbit and its peak brightness density shifts from south to north of Saturn’s equator plane with increasing distance from the planet. These data also reveal a localized depletion in particle density near Saturn’s equatorial plane around Enceladus’ semi-major axis. Finally, variations are detected in the radial brightness profile and the vertical thickness of the ring as a function of longitude relative to the Sun. Possible physical mechanisms and processes that may be responsible for some of these structures include solar radiation pressure, variations in the ambient plasma, and electromagnetic perturbations associated with Saturn’s shadow.