## Publicações

Artigos publicados em periódicos.

### 2018

**Rings under close encounters with the giant planets: Chariklo versus Chiron**

Araujo, R. A. N.; Winter, O. C & Sfair, R. (2018)

*Monthly Notices of the Royal Astronomical Society*

In 2014, the discovery of two well-defined rings around the Centaur (10199) Chariklo was announced. This was the first time that such structures were found around a small body. In 2015, it was proposed that the Centaur (2060) Chiron may also have a ring. In a previous study, we analysed how close encounters with giant planets would affect the rings of Chariklo. The most likely result is the survival of the rings. In this work, we broaden our analysis to (2060) Chiron. In addition to Chariklo, Chiron is currently the only known Centaur with a presumed ring. By applying the same method as Araujo, Sfair and Winter, we performed numerical integrations of a system composed of 729 clones of Chiron, the Sun and the giant planets. The number of close encounters that disrupted the ring of Chiron during one half-life of the study period was computed. This number was then compared to the number of close encounters for Chariklo. We found that the probability of Chiron losing its ring due to close encounters with the giant planets is about six times higher than that for Chariklo. Our analysis showed that, unlike Chariklo, Chiron is more likely to remain in an orbit with a relatively low inclination and high eccentricity. Thus, we found that the bodies in Chiron-like orbits are less likely to retain rings than those in Chariklo-like orbits. Overall, for observational purposes, we conclude that the bigger bodies in orbits with high inclinations and low eccentricities should be prioritized.

doi: http://dx.doi.org/10.1093/mnras/sty1770

arxiv: http://arxiv.org/abs/1807.02096

**The journey of Typhon-Echidna as a binary system through the planetary region**

Araujo, R. A. N.; Galiazzo, M. A.; Winter, O. C & Sfair, R. (2018)

*Monthly Notices of the Royal Astronomical Society*

In 2014, the discovery of two well-defined rings around the Centaur (10199) Chariklo was announced. This was the first time that such structures were found around a small body. In 2015, it was proposed that the Centaur (2060) Chiron may also have a ring. In a previous study, we analysed how close encounters with giant planets would affect the rings of Chariklo. The most likely result is the survival of the rings. In this work, we broaden our analysis to (2060) Chiron. In addition to Chariklo, Chiron is currently the only known Centaur with a presumed ring. By applying the same method as Araujo, Sfair and Winter, we performed numerical integrations of a system composed of 729 clones of Chiron, the Sun and the giant planets. The number of close encounters that disrupted the ring of Chiron during one half-life of the study period was computed. This number was then compared to the number of close encounters for Chariklo. We found that the probability of Chiron losing its ring due to close encounters with the giant planets is about six times higher than that for Chariklo. Our analysis showed that, unlike Chariklo, Chiron is more likely to remain in an orbit with a relatively low inclination and high eccentricity. Thus, we found that the bodies in Chiron-like orbits are less likely to retain rings than those in Chariklo-like orbits. Overall, for observational purposes, we conclude that the bigger bodies in orbits with high inclinations and low eccentricities should be prioritized.

doi: https://doi.org/10.1093/mnras/sty583

arxiv: https://arxiv.org/abs/1803.00634

**Analytical study of the powered Swing-By maneuver for elliptical systems and analysis of its efficiency**

Ferreira, A. F. S.; Prado, A. B. A.; Winter, O. C. & Santos, P. S. (2018)

*Astrophysics and Space Science *

Analytical equations describing the velocity and energy variation of a spacecraft in a Powered Swing-By maneuver in an elliptic system are presented. The spacecraft motion is limited to the orbital plane of the primaries. In addition to gravity, the spacecraft suffers the effect of an impulsive maneuver applied when it passes by the periapsis of its orbit around the secondary body of the system. This impulsive maneuver is defined by its magnitude δV and the angle that defines the direction of the impulse with respect to the velocity of the spacecraft (α). The maneuver occurs in a system of main bodies that are in elliptical orbits, where the velocity of the secondary body varies according to its position in the orbit following the rules of an elliptical orbit. The equations are dependent on this velocity. The study is done using the “patched-conics approximation”, which is a method of simplifying the calculations of the trajectory of a spacecraft traveling around more than one celestial body. Solutions for the velocity and energy variations as a function of the parameters that define the maneuver are presented. An analysis of the efficiency of the powered Swing-By maneuver is also made, comparing it with the pure gravity Swing-by maneuver with the addition of an impulse applied outside the sphere of influence of the secondary body. After a general study, the techniques developed here are applied to the systems Sun-Mercury and Sun-Mars, which are real and important systems with large eccentricity. This problem is highly nonlinear and the dynamics very complex, but very reach in applications.

**Particles Co-orbital to Janus and to Epimetheus: A Firefly Planetary Ring**

Winter, O. C.; Souza, A. P. S.; Sfair, R.; Giuliatti Winter, S. M.; Mourão, D. C. & Foryta, D. W. (2018)

*The Astrophysical Journal*

The *Cassini* spacecraft found a new and unique ring that shares the trajectory of Janus and Epimetheus, co-orbital satellites of Saturn. Performing image analysis, we found this to be a continuous ring. Its width is between 30% and 50% larger than previously announced. We also verified that the ring behaves like a firefly. It can only be seen from time to time, when *Cassini*, the ring, and the Sun are arranged in a particular geometric configuration, in very high phase angles. Otherwise, it remains “in the dark,” invisible to *Cassini*‘s cameras. Through numerical simulations, we found a very short lifetime for the ring particles, less than a couple of decades. Consequently, the ring needs to be constantly replenished. Using a model of particle production due to micrometeorites impacts on the surfaces of Janus and Epimetheus, we reproduce the ring, explaining its existence and the “firefly” behavior.

doi: http://dx.doi.org/10.3847/1538-4357/aa9c7f

arvix: https://arxiv.org/abs/1801.01909

**Poincaré surfaces of section around a 3D irregular body: the case of asteroid 4179 Toutatis**

Winter, O. C. & Borderes-Motta, G. (2018)

*Monthly Notices of the Royal Astronomical Society*

In general, small bodies of the Solar system, e.g. asteroids and comets, have a very irregular shape. This feature affects significantly the gravitational potential around these irregular bodies, which hinders dynamical studies. The Poincaré surface of section technique is often used to look for stable and chaotic regions in two-dimensional dynamic cases. In this work, we show that this tool can be useful for exploring the surroundings of irregular bodies such as the asteroid 4179 Toutatis. Considering a rotating system with a particle, under the effect of the gravitational field computed three dimensionally, we define a plane in the phase space to build the Poincaré surface of section. Despite the extra dimension, the sections created allow us to find trajectories and classify their stabilities. Thus, we have also been able to map stable and chaotic regions, as well as to find correlations between those regions and the contribution of the third dimension of the system to the trajectory dynamics as well. As examples, we show details of periodic (resonant or not) and quasi-periodic trajectories.

doi: http://dx.doi.org/10.1093/mnras/stx2958

arvix: https://arxiv.org/abs/1711.06506

**Production and fate of the G ring arc particles due to Aegaeon (Saturn LIII)**

Giuliatti Winter, S. M.; Madeira, G.; Sfair, R. & Mourão, D. C. (2018)

*Monthly Notices of the Royal Astronomical Society*

The G ring arc hosts the smallest satellite of Saturn, Aegaeon, observed with a set of images sent by Cassini spacecraft. Along with Aegaeon, the arc particles are trapped in a 7:6 corotation eccentric resonance with the satellite Mimas. Due to this resonance, both Aegaeon and the arc material are confined to within 60° of corotating longitudes. The arc particles are dust grains which can have their orbital motions severely disturbed by the solar radiation force. Our numerical simulations showed that Aegaeon is responsible for depleting the arc dust population by removing them through collisions. The solar radiation force hastens these collisions by removing most of the 10 μm sized grains in less than 40 yr. Some debris released from Aegaeon’s surface by meteoroid impacts can populate the arc. However, it would take 30 000 yr for Aegaeon to supply the observed amount of arc material, and so it is unlikely that Aegaeon alone is the source of dust in the arc.

doi: http://dx.doi.org/10.1093/mnras/sty179

arvix: https://arxiv.org/abs/1801.04652

**Growth and evolution of satellites in a Jovian massive disc**

Vieira Neto, E.; Kley, W. & Moraes, R. A. (2018)

*Monthly Notices of the Royal Astronomical Society*

The formation of satellite systems in circum-planetary discs is considered to be similar to the formation of rocky planets in a proto-planetary disc, especially super-Earths. Thus, it is possible to use systems with large satellites to test formation theories that are also applicable to extrasolar planets. Furthermore, a better understanding of the origin of satellites might yield important information about the environment near the growing planet during the last stages of planet formation. In this work, we investigate the formation and migration of the Jovian satellites through *N*-body simulations. We simulated a massive, static, low-viscosity, circum-planetary disc in agreement with the minimum mass sub-nebula model prescriptions for its total mass. In hydrodynamic simulations, we found no signs of gaps, therefore type II migration is not expected. Hence, we used analytic prescriptions for type I migration, eccentricity and inclination damping, and performed *N*-body simulations with damping forces added. Detailed parameter studies showed that the number of final satellites is strong influenced by the initial distribution of embryos, the disc temperature, and the initial gas density profile. For steeper initial density profiles, it is possible to form systems with multiple satellites in resonance while a flatter profile favours the formation of satellites close to the region of the Galilean satellites. We show that the formation of massive satellites such as Ganymede and Callisto can be achieved for hotter discs with an aspect ratio of *H*/*r* ∼ 0.15 for which the ice line was located around 30*R _{J}*.

doi: http://dx.doi.org/10.1093/mnras/stx3268

arvix: https://arxiv.org/abs/1712.05327

**Asteroid families interacting with secular resonances**

Carruba, V.; Vokrouhlický, D. & Novakovic, B. (2018)

*Planetaty and Space Science*

Asteroid families are formed as the result of collisions. Large fragments are ejected with speeds of the order of the escape velocity from the parent body. After the family formation, the fragments’ orbits evolve in the space of proper elements because of gravitational and non-gravitational perturbations, such as the Yarkovsky effect. Disentangling the contribution to the current orbital position of family members caused by the initial ejection velocity field and the subsequent orbital evolution is usually a difficult task. Among the more than 100 asteroid families currently known, some interact with secular resonances. Linear secular resonances occur when there is a commensurability between the precession frequency of the longitude of the pericenter (g) or of the longitude of node (s) of an asteroid and a planet, or a massive asteroid. The linear secular resonance most effective in increasing an asteroid eccentricity is the ν6, that corresponds to a commensurability between the precession frequency g of an asteroid and Saturn’s g6. Non-linear secular resonances involve commensurabilities of higher order, and can often be expressed as combinations of linear secular resonances. This is the case, for instance, of the zk=k(g−g6)+(s−s6) resonances. Asteroid families that are crossed by secular resonances are of particular interest in dynamical astronomy. First, they often provide a clear evidence of asteroid orbit evolution due to the Yarkovsky effect. Second, conserved quantities of secular dynamics can be used to set valuable constraints on the magnitude of the original ejection velocity field. Finally, by changing the value of inclination of family members nodal secular resonances with massive asteroids or dwarf planets can cause the i distribution to become more and more leptokurtic (i.e., more peaked and with larger tails than that of a Gaussian distribution).

doi: http://dx.doi.org/10.1016/j.pss.2018.03.009

arvix: https://arxiv.org/abs/1804.00505

**On the age of the Nele asteroid family**

Carruba, V.; Vokrouhlický, D.; Nesvorny, D. & Aljbaae, S. (2018)

*Monthly Notices of the Royal Astronomical Society*

The Nele group, formerly known as the Iannini family, is one of the youngest asteroid families in the main belt. Previously, it has been noted that the pericentre longitudes ϖ and nodal longitudes Ω of its largest member asteroids are clustered at the present time, therefore suggesting that the collisional break-up of parent body must have happened recently. Here, we verify this conclusion by detailed orbit-propagation of a synthetic Nele family and show that the current level of clustering of secular angles of the largest Nele family members requires an approximate age limit of 4.5 Myr. Additionally, we make use of an updated and largely extended Nele membership to obtain, for the first time, an age estimate of this family using the backward integration method. Convergence of the secular angles in a purely gravitational model and in a model including the non-gravitational forces caused by the Yarkovsky effect are both compatible with an age younger than 7 Myr. More accurate determination of the Nele family age would require additional data about the spin state of its members.

doi: http://dx.doi.org/10.1093/mnras/sty777

arvix: https://arxiv.org/abs/1803.08429

**The quest for young asteroid families: new families, new results**

Carruba, V.; Oliveira, E. R.; Rodrigues, B. & Requena, I. (2018)

*Monthly Notices of the Royal Astronomical Society*

Asteroid families form as a result of collisions. The fragments resulting from the family-forming event are ejected into orbits near that of the parent body, and then start dynamically migrating because of gravitational and non-gravitational effects, such as the Yarkovsky force. Families that formed less than 20 Myr ago are special, since their secular angles, the longitudes of pericenter and nodes, may still converge with respect to those of the putative parent body when integrated backward in time, at the moment of family formation. This allows for obtaining age estimates and family membership with a precision not allowable for other, more evolved asteroid families. This method of family dating, the Backward Integration Method, or BIM, has been, so far, successfully applied to the case of eight asteroid families. In the last years, however, because of the astounding rate of new asteroid discoveries, several new small and compact asteroid families have been identified. In this work, we apply the BIM to 28 asteroid families not previously studied with this method. We identified four families for which we observe a possible convergence of the angles. For three of them, we obtained age estimates: at a 68.3 per cent confidence level, (3152) Jones should be

1.9−1.9+4.3

“>1.9+4.3−1.91.9−1.9+4.3

, (7353) Kazuya should be

2.2−2.2+1.4

“>2.2+1.4−2.22.2−2.2+1.4

, and (108138) 2001 GB_{11} should be

4.6−1.1+1.6

“>4.6+1.6−1.14.6−1.1+1.6

Myr old. (909) Ulla might be younger than ≃ 6 Myr.

**Building an -Escape Portal- with Tethers Fixed in Asteroids**

Gomes, V. M.; Prado, A. F. B. A. & Chanut, T. G. G. (2018)

*Journal of the Astronautical Sciences*

The main idea of this paper is to propose the construction of an “Escape Portal” to send a spacecraft to the exterior planets, or even to make it escape from the Solar System, using a Tethered Sling Shot Maneuver (TSSM) with an asteroid. The construction of this portal allows an unlimited number of maneuvers with the same tether, which is very interesting when considering a possible use for small satellites. This structure would be formed by a tether that remains fixed in an asteroid. At the other end of the tether, a large net is fixed, such that the only action required from the spacecraft to make the TSSM is to hit the net. The net can have a mechanism to open a passage to release the spacecraft when the desired rotation is obtained. This technique would avoid some of the problems that appear when assuming that the spacecraft needs to carry a tether on board that would be released to hit the asteroid just before the maneuver.

**Retrograde and Direct Powered Aero-Gravity-Assist Trajectories Around Mars**

Gomes, V. M.; Prado, A. F. B. A. & Murcia, J. (2018)

*Revista Mexicana de Astronomia y Astrofisica*

A Gravity-Assist maneuver is used to reduce fuel consumption and/or trip times in interplanetary missions. It is based in a close approach of a spacecraft to a celestial body. Missions like Voyager and Ulysses used this concept. The present paper performs a study of a maneuver that combines three effects: the gravity of the planet, the application of an impulsive maneuver when the spacecraft is passing by the periapsis and the effects of the atmosphere of the planet. Direct and retrograde trajectories are considered, with particular attention to the differences due to the higher relative velocity between the spacecraft and the atmosphere, which increases the effects of the atmosphere. The planet Mars is used for the numerical examples.

### 2017

(24 artigos)

**Mapping stable direct and retrograde orbits around the triple system of asteroids (45) Eugenia**

Araujo, R. A. N.; Winter, O. C.; Moraes, R. V. & Prado, A. F. B. A. (2017)

*Monthly Notices of the Royal Astronomical Society*

It is widely accepted that knowing the composition and the orbital evolution of asteroids might help us to understand the process of formation of the Solar system. It is also known that asteroids can represent a threat to our planet. Such an important role has made space missions to asteroids a very popular topic in current astrodynamics and astronomy studies. Taking into account the increasing interest in space missions to asteroids, especially to multiple systems, we present a study that aims to characterize the stable and unstable regions around the triple system of asteroids (45) Eugenia. The goal is to characterize the unstable and stable regions of this system and to make a comparison with the system 2001 SN263, which is the target of the Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) mission. A new concept was used for mapping orbits, by considering the disturbance received by the spacecraft from all perturbing forces individually. This method has also been applied to (45) Eugenia. We present the stable and unstable regions for particles with relative inclination between 0° and 180°. We found that (45) Eugenia presents larger stable regions for both prograde and retrograde cases. This is mainly because the satellites of this system are small when compared to the primary body, and because they are not close to each other. We also present a comparison between these two triple systems, and we discuss how these results can guide us in the planning of future missions.

doi: http://dx.doi.org/10.1093/mnras/stx2230

arxiv: https://arxiv.org/abs/1711.06324

**The asteroid belt outer region under jumping-Jupiter migration**

Gaspar, H. S.; Winter, O. C. & Vieira Neto, E. (2017)

*Monthly Notices of the Royal Astronomical Society*

The radial configuration of the outer region of the main asteroid belt is quite peculiar, and has much to say about the past evolution of Jupiter. In this work, we investigate the dynamical effects of a jumping-Jupiter-like migration over a more extended primordial asteroid belt. Jupiter’s migrations are simulated using a fast jumping-Jupiter synthesizer. Among the results, we highlight non-negligible fractions of primordial objects trapped in 3:2 and 4:3 mean motion resonances (MMRs) with Jupiter. They survived the whole truculent phase of migration and originated populations that are like Hildas and Thules. Fractions ranging from 3 to 6 per cent of the initial distribution remained trapped in 3:2 MMR, and at least 0.05 per cent in 4:3. These results show that the resonance trapping of primordial objects may have originated these resonant populations. This theory is consistent even for Jupiter’s truculent evolution.

**Periodic orbits for space-based reflectors in the circular restricted three-body problem**

Salazar, F. J. T.; Winter, O. C. & McInnes, C. R. (2017)

*Celestial Mechanics & Dynamical Astronomy*

The use of space-based orbital reflectors to increase the total insolation of the Earth has been considered with potential applications in night-side illumination, electric power generation and climate engineering. Previous studies have demonstrated that families of displaced Earth-centered and artificial halo orbits may be generated using continuous propulsion, e.g. solar sails. In this work, a three-body analysis is performed by using the circular restricted three body problem, such that, the space mirror attitude reflects sunlight in the direction of Earth’s center, increasing the total insolation. Using the Lindstedt–Poincaré and differential corrector methods, a family of halo orbits at artificial Sun–Earth

“>L2L2 points are found. It is shown that the third order approximation does not yield real solutions after the reflector acceleration exceeds 0.245 mm

“>s−2s−2, i.e. the analytical expressions for the in- and out-of-plane amplitudes yield imaginary values. Thus, a larger solar reflector acceleration is required to obtain periodic orbits closer to the Earth. Derived using a two-body approach and applying the differential corrector method, a family of displaced periodic orbits close to the Earth are therefore found, with a solar reflector acceleration of 2.686 mm

“>s−2s−2.

**Effects of the eccentricity of the primaries in powered Swing-By maneuvers**

Ferreira, A. F. S.; Winter, O. C.; Prado, A. F. B. A. & Santos, D. P. S. (2017)

*Advances in Space Research*

The present paper studies the powered Swing-By maneuver when performed in an elliptical system of primaries. It means that there is a spacecraft travelling in a system governed by the gravity fields of two bodies that are in elliptical orbitsaround their center of mass. The paper particularly analyzes the effects of the parameters relative to the Swing-By (Vinf-,rp,ψ

“>Vinf-,rp,ψ), the orbit of the secondary body around the primary one (e,ν

“>e,ν) and the elements that specify the impulse applied (δV,α

“>δV,α) to the spacecraft. The impulse is applied when the spacecraft passes by the periapsis of its orbit around the body, where it performs the Swing-By, with different magnitudes and directions. The inclusion of the orbital eccentricity of the primaries in this problem makes it closer to reality, considering that there are many known systems with eccentricities different from zero. In particular, there are several moons in the Solar System which orbits are not circular, as well as some smaller bodies, like the dwarf planet Haumea and its moons, which have eccentricities of 0.25 or even larger. The behavior of the energy variation of the spacecraft is shown in details, as well as the cases where captures and collisions occur. The results show the conditions that optimize this maneuver, according to some given parameters, and how much can be obtained in terms of gains or losses of energy using the best conditions found by the algorithm developed here.

**A numerical mapping of energy gains in a powered Swing-By maneuver**

Ferreira, A. F. S.; Winter, O. C. & Prado, A. F. B. A. (2017)

*Nonlinear Dynamics*

The present paper studies the effects of a powered Swing-By maneuver, considering the particular and important situations where there are energy gains for the spacecraft. The objective is to map the energy variations obtained from this maneuver as a function of the three parameters that identify the pure gravity Swing-By with a fixed mass ratio (angle of approach, periapsis distance and velocity at periapsis) and the three parameters that define the impulsive maneuver (direction, magnitude and the point where the impulse is applied). The mathematical model used here is the version of the restricted three-body problem that includes the Lemaître regularization, to increase the accuracy of the numerical integrations. It is developed and implemented by an algorithm that obtains the energy variation of the spacecraft with respect to the largest primary of the system in a maneuver where the impulse is applied inside the sphere of influence of the secondary body, during the passage of the spacecraft. The point of application of the impulse is a free parameter, as well as the direction of the impulse. The results make a complete map of the possibilities, including the maximum gains of energy, but also showing alternatives that can be used considering particularities of the mission.

** Planar powered Swing-By maneuvers to brake a spacecraft**

Ferreira, A. F. S.; Winter, O. C. & Prado, A. F. B. A. (2017)

*Computacional & Applied Mathematics*

The Swing-By maneuver is a technique used in many space mission to modify the trajectory of a spacecraft. The most usual goal is to increase the energy of the spacecraft, but it is also possible to reduce this energy. An important application is to break a spacecraft coming to the Earth using a Swing-By with the moon, which is the example used in the present paper. Other possibilities also exist, such as reducing the velocity of a spacecraft going to the planets Mercury or Venus. The goal is to help a possible capture by the planet, or at least to provide a passage with smaller velocities to allow better observations during the passage. Therefore, the goal of the present paper is to study the energy loss that a spacecraft may have during a powered Swing-By maneuver, which is a maneuver that combines a close approach by a celestial body with the application of an impulsive maneuver. The behavior of the energy variation is analyzed as a function of the parameters related to the pure gravity maneuver: periapsis radius, angle of approach and approach velocity; and the parameters related to the impulsive maneuver: the location of application of the impulse and its direction and magnitude. The maneuver is performed in a system composed by two bodies, such as the Earth–moon system, around the secondary body, and the energy is measured with respect to the primary body of the system. This problem is solved by developing a mathematical algorithm that guides larger efforts in terms of computer simulations. The results show maps of conditions made from the numerical simulations for different points of application and direction of the impulse, where the maneuver is advantageous and how much more energy can be removed from the spacecraft.

**Collecting solar power by formation flying systems around a geostationary point**

Salazar, F. J. T.; Winter, O. C. & McInnes, C. R. (2017)

*Computacional & Applied Mathematics*

Terrestrial solar power is severely limited by the diurnal day–night cycle. To overcome these limitations, a Solar Power Satellite (SPS) system, consisting of a space mirror and a microwave energy generator-transmitter in formation, is presented. The microwave transmitting satellite (MTS) is placed on a planar orbit about a geostationary point (GEO point) in the Earth’s equatorial plane, and the space mirror uses the solar pressure to achieve orbits about GEO point, separated from the planar orbit, and reflecting the sunlight to the MTS, which will transmit energy to an Earth-receiving antenna. Previous studies have shown the existence of a family of displaced periodic orbits above or below the Earth’s equatorial plane. In these studies, the sun-line direction is assumed to be in the Earth’s equatorial plane (equinoxes), and at

“>23.5∘23.5∘ below or above the Earth’s equatorial plane (solstices), i.e. depending on the season, the sun-line moves in the Earth’s equatorial plane and above or below the Earth’s equatorial plane. In this work, the position of the Sun is approximated by a rectangular equatorial coordinates, assuming a mean inclination of Earth’s equator with respect to the ecliptic equal to

“>23.5∘23.5∘. It is shown that a linear approximation of the motion about the GEO point yields bounded orbits for the SPS system in the Earth–satellite two-body problem, taking into account the effects of solar radiation pressure. The space mirror orientation satisfies the law of reflection to redirect the sunlight to the MTS. Additionally, a MTS on a common geostationary orbit (GEO) has been also considered to reduce the relative distance in the formation flying Solar Power Satellite (FF-SPS).

**Applications of celestial mechanics in natural objects and spacecrafts**

Gomes, V. M.; De Mello, C. F.; Macau, E. E. N.; Prado, A. F. B. A. & Winter, O. C. (2017)

*Computacional & Applied Mathematics*

Studies related to Celestial Mechanics started long ago, and it is one of the oldest fields in Astronomy. It started to try to explain the motions of the stars in the sky, in particular the irregular motion of some of those of then, which were really the planets of the Solar System. In the 20th century, with the arrival of the “Space Age”, many applications related to the motion of artificial spacecrafts appeared. This new field was called “Astrodynamics”, to designate the use of Celestial Mechanics in man-made objects. Several aspects, like orbit determination, maneuvers to change the orbit of the spacecraft, etc., are covered by this topic. The present Focus Issue in Celestial Mechanics publishes a list of papers in topics related to applications in Celestial Mechanics to both situations: natural and artificial satellites.

doi: http://dx.doi.org/10.1007/s40314-017-0499-9

**Solar Power Satellite system in formation on a common geostationary orbit**

Salazar, F. J. T. & Winter, O. C. (2017)

*Journal of Physics: Conference Series*

The diurnal day-night cycle severely limits the Terrestrial solar power. To overcome this limitation, a Solar Power Satellite (SPS) system, consisting of a sunlight reflector and a microwave energy generator-transmitter in formation, is presented in this work. The microwave transmitting satellite (MTS) is placed on a common geostationary orbit (GEO) in the Earth’s equatorial plane, and the sunlight reflector uses the solar radiation pressure to achieve quasi-periodic orbits about the MTS, so that the sunlight is always redirected to the MTS, which converts the solar energy in electromagnetic power and transmits it by microwaves to an Earth-receiving antenna. Assuming the sun line direction constant at dierent seasons (i.e. autumn/spring equinoxes and winter and summer solstices), previous studies have shown the existence of a family of displaced ecliptic orbits above or below the equatorial plane of the Earth around a GEO. In this study, the position of the Sun is assumed on the ecliptic plane with a mean obliquity (inclination of Earth’s equator with respect to the ecliptic) of 23.5◦. A linear solution as an initial condition for the full equations of motions about a GEO, which yields bounded orbit for the sunlight reflector about the MTS in the Earth-satellite two-body problem with solar radiation pressure. To redirect the sunlight to the MTS, the law of reflection is satisfied by the space mirror attitude.

**Studying the energy variation in the powered Swing-By in the Sun-Mercury system**

Ferreira, A. F. S.; Prado, A. F. B. A. & Winter, O. C. (2017)

*Journal of Physics: Conference Series*

A maneuver where a spacecraft passes close to Mercury and uses the gravity of this body combined with an impulse applied at the periapsis, with different magnitudes and directions, is presented. The main objective of this maneuver is the fuel economy in space missions. Using this maneuver, it is possible to insert the spacecraft into an orbit captured around the Sun or Mercury. Trajectories escaping the Solar System are also obtained and mapped. Maps of the spacecraft energy variation relative to the Sun and the types of orbits resulting from the maneuver are presented, based in numerical integrations. The results show that applying the impulse out of the direction of motion can optimize the maneuver due to the effect of the combination of the impulse and the gravity.

doi: http://dx.doi.org/10.1088/1742-6596/911/1/012007

**Long-term Evolution and Stability of Saturnian Small Satellites: Aegaeon, Methone, Anthe, and Pallene**

Gutierrez, M. M. & Giuliatti Winter, S. M. (2017)

*Monthly Notices of the Royal Astronomical Society*

Aegaeon, Methone, Anthe and Pallene are four Saturnian small moons, discovered by the Cassini spacecraft. Although their orbital characterization has been carried on by a number of authors, their long-term evolution has not been studied in detail so far. In this work, we numerically explore the long-term evolution, up to 10^{5} yr, of the small moons in a system formed by an oblate Saturn and the five largest moons close to the region: Janus, Epimetheus, Mimas, Enceladus and Tethys. By using frequency analysis, we determined the stability of the small moons and characterize, through diffusion maps, the dynamical behaviour of a wide region of geometric phase space, *a* versus *e*, surrounding them. Those maps could shed light on the possible initial number of small bodies close to Mimas, and help to better understand the dynamical origin of the small satellites. We found that the four small moons are long-term stable and no mark of chaos is found for them. Aegaeon, Methone and Anthe could remain unaltered for at least ∼0.5 Myr, given the current configuration of the system. They remain well trapped in the corotation eccentricity resonances with Mimas in which they currently librate. However, perturbations from nearby resonances, such as Lindblad eccentricity resonances with Mimas, seem responsible for largest variations observed for Methone and Anthe. Pallene remains in a non-resonant orbit and it is the more stable, at least for 64 Myr. Nonetheless, it is affected by a quasi-resonance with Mimas, which induces long-term orbital oscillations of its eccentricity and inclination.

doi: http://dx.doi.org/10.1093/mnras/stx1537

arxiv: https://arxiv.org/abs/1706.05393

**The Rafita asteroid family**

Aljbaae, S.; Carruba, V.; Masiero, J.; Domingos, R. C. & Huaman, M. (2017)

*Monthly Notices of the Royal Astronomical Society*

The Rafita asteroid family is an S-type group located in the middle main belt, on the right-hand side of the 3J:-1A mean-motion resonance. The proximity of this resonance to the family left-hand side in the semimajor axis caused many former family members to be lost. As a consequence, the family shape in the (*a*, 1/*D*) domain is quite asymmetrical, with a preponderance of objects on the right-hand side of the distribution. The Rafita family is also characterized by a leptokurtic distribution in inclination, which allows the use of methods of family age estimation recently introduced for other leptokurtic families such as Astrid, Hansa, Gallia and Barcelona. In this work, we propose a new method based on the behaviour of an asymmetry coefficient function of the distribution in the (*a*, 1/*D*) plane to date incomplete asteroid families such as Rafita. By monitoring the time behaviour of this coefficient for asteroids simulating the initial conditions at the time of the family formation, we were able to estimate that the Rafita family should have an age of 490 ± 200 Myr, in good agreement with results from independent methods such as Monte Carlo simulations of Yarkovsky and YORP dynamical induced evolution and the time behaviour of the kurtosis of the sin (*i*) distribution. Asteroids from the Rafita family can reach orbits similar to 8 per cent of the currently known near-Earth objects. During the final 10 Myr of the simulation, ≃1 per cent of the simulated objects are present in NEO space, and thus would be comparable to objects in the present-day NEO population.

doi: http://dx.doi.org/10.1093/mnras/stx184

arxiv: https://arxiv.org/abs/1705.08354

**Scattering V-type asteroids during the giant planets instability: A step for Jupiter, a leap for basalt**

Brasil, P. I.; Roig, F.; Nersvorny, D. & Carruba,V. (2017)

*Monthly Notices of the Royal Astronomical Society*

V-type asteroids are a taxonomic class whose surface is associated with a basaltic composition. The only known source of V-type asteroids in the Main Asteroid Belt is (4) Vesta, which is located in the inner part of the Main Belt. However, many V-type asteroids cannot be dynamically linked to Vesta, in particular, those asteroids located in the middle and outer parts of the Main Belt. Previous works have failed to find mechanisms to transport V-type asteroids from the inner to the middle and outer belts. In this work, we propose a dynamical mechanism that could have acted on primordial asteroid families. We consider a model of the giant planet migration known as the jumping Jupiter model with five planets. Our study is focused on the period of 10 Myr that encompasses the instability phase of the giant planets. We show that, for different hypothetical Vesta-like paleo-families in the inner belt, the perturbations caused by the ice giant that is scattered into the asteroid belt before being ejected from the Solar system are able to scatter V-type asteroids to the middle and outer belts. Based on the orbital distribution of V-type candidates identified from the Sloan Digital Sky Survey and the VISTA Survey colours, we show that this mechanism is efficient enough provided that the hypothetical paleo-family originated from a 100 to 500 km crater excavated on the surface of (4) Vesta. This mechanism is able to explain the currently observed V-type asteroids in the middle and outer belts, with the exception of (1459) Magnya.

doi: http://dx.doi.org/10.1093/mnras/stx529

arxiv: https://arxiv.org/abs/1703.00474

**The asteroid population in g-type non-linear secular resonances**

Huaman, M.; Carruba, V.; Domingos, R. C. & Aljbaae, S. (2017)

*Monthly Notices of the Royal Astronomical Society*

Non-linear secular resonances of *g*-type, i.e. involving the frequency of precession g of the asteroid pericentre, can affect the proper eccentricities of asteroids in resonant or near-resonant configurations. We first identified objects that could potentially be affected by non-linear secular resonances of this type. We then numerically integrated these objects and checked for their resonant argument. We identified a population of 1517 asteroids in *g* − 2*g*_{6} + *g*_{5} librating states, and of 128 objects in *g* − 3*g*_{6} + 2*g*_{5} resonant configurations. While secular resonances are rather extended structures and many objects from different and unrelated parts of the main belt could be encountered within, we found that *g* − 2*g*_{6} + *g*_{5} librators are predominantly of the S taxonomical type (56 per cent of the total), but with a significant fraction of other spectral types. No spectral type dominates in the population of *g* − 3*g*_{6} + 2*g*_{5} resonators. Several asteroid families are affected by the *g* − 2*g*_{6} + *g*_{5} secular resonance. The Astraea group is cut into two by this resonance, while the Tirela and Brasilia groups are on the resonance centre and on the left side, respectively. The *g* − 2*g*_{6} + *g*_{5}secular can significantly affect the shape of families inside the resonance, such as Astraea. It can also increase the flux of asteroids to nearby powerful mean-motion resonances, such as the 5J:-2A and the 2J:-1A. As expected, the long-term effect of *g*-type resonances on inclinations is essentially negligible.

**Dynamics in the vicinity of (101955) Bennu: Solar radiation pressure effects in equatorial orbits**

Chanut, T.; Prado, A. F. B. A.; Aljbaae, S. & Carruba, V. (2017)

*Monthly Notices of the Royal Astronomical Society*

Here, we study the dynamical effects of the solar radiation pressure (SRP) on a spacecraft that will survey the near-Earth rotating asteroid (101955) Bennu when the projected shadow is accounted for. The spacecraft’s motion near (101955) Bennu is modelled in the rotating frame fixed at the centre of the asteroid, neglecting the Sun gravity effects. We calculate the SRP at the perihelion, semimajor axis and aphelion distances of the asteroid from the Sun. The goals of this work are to analyse the stability for both homogeneous and inhomogeneous mass distribution and study the effects of the SRP in equatorial orbits close to the asteroid (101955) Bennu. As results, we find that the mascon model divided into 10 equal layers seems to be the most suitable for this problem. We can highlight that the centre point E8, which was linearly stable in the case of the homogeneous mass distribution, becomes unstable in this new model changing its topological structure. For a Sun initial longitude ψ_{0} = −180°, starting with the spacecraft longitude λ = 0, the orbits suffer fewer impacts and some (between 0.4 and 0.5 km), remaining unwavering even if the maximum solar radiation is considered. When we change the initial longitude of the Sun to ψ_{0} = −135°, the orbits with initial longitude λ = 90° appear to be more stable. Finally, when the passage of the spacecraft in the shadow is accounted for, the effects of SRP are softened, and we find more stable orbits.

doi: http://dx.doi.org/10.1093/mnras/stx1204

arxiv: https://arxiv.org/abs/1705.09564

**Detection of the Yarkovsky effect for C-type asteroids in the Veritas family**

Carruba, V.; Vokrouhlický, D.; & Nesvorny, D. (2017)

*Monthly Notices of the Royal Astronomical Society*

The age of a young asteroid family can be determined by tracking the orbits of family members backward in time and showing that they converge at some time in the past. Here we consider the Veritas family. We find that the membership of the Veritas family increased enormously since the last detailed analysis of the family. Using backward integration, we confirm the convergence of nodal longitudes Ω, and, for the first time, also obtain a simultaneous convergence of pericentre longitudes ϖ. The Veritas family is found to be 8.23-_{0.31}^{+0.37 }Myr old. To obtain a tight convergence of Ω and ϖ, as expected from low ejection speeds of fragments, the Yarkovsky effect needs to be included in the modelling of the past orbital histories of Veritas family members. Using this method, we compute the Yarkovsky semimajor axis drift rates, d*a*/d*t*, for 274 member asteroids. The distribution of d*a*/d*t* values is consistent with a population of C-type objects with low densities and low thermal conductivities. The accuracy of individual d*a*/d*t*measurements is limited by the effect of close encounters of member asteroids to (1) Ceres and other massive asteroids, which cannot be evaluated with confidence.

doi: http://dx.doi.org/10.1093/mnras/stx1186

arxiv: https://arxiv.org/abs/1705.04333

**An automatic approach to exclude interlopers from asteroid families**

Radovic, V.; Novakovic, B.; Carruba, V. & Marceta, D. (2017)

*Monthly Notices of the Royal Astronomical Society*

Asteroid families are a valuable source of information to many asteroid-related researches, assuming a reliable list of their members could be obtained. However, as the number of known asteroids increases fast it becomes more and more difficult to obtain a robust list of members of an asteroid family. Here, we are proposing a new approach to deal with the problem, based on the well-known hierarchical clustering method. An additional step in the whole procedure is introduced in order to reduce a so-called chaining effect. The main idea is to prevent chaining through an already identified interloper. We show that in this way a number of potential interlopers among family members is significantly reduced. Moreover, we developed an automatic online-based portal to apply this procedure, i.e. to generate a list of family members as well as a list of potential interlopers. The Asteroid Families Portal is freely available to all interested researchers.

doi: http://dx.doi.org/10.1093/mnras/stx1273

arxiv: https://arxiv.org/abs/1705.09226

**The Maria asteroid family**

Aljbaae, S.; Carruba, V.; Masiero, J.; Domingos, R. C. & Huaman, M. (2017)

*Monthly Notices of the Royal Astronomical Society*

The Maria asteroid family is a group of S-type asteroids. Its location adjacent to the left side of the 3J:–1A mean-motion resonances could be the reason for the absence of the left side of the ‘V’ shape in the (*a*, 1/*D*) domain. This family can be considered as a likely source of ordinary chondrite-like material. In this work, we make use of the time dependence of the asymmetric coefficient *A*_{S} describing the degree of asymmetry of the *C* distribution of a fictitious Maria family generated with the value of the ejection velocity parameter *V*_{EJ} = 35 m s^{−1} to obtain an age estimate of

“>1750−231+5371750+537−231

Myr, in good agreement with the family age found in the literature. Analysing the contribution to the near-Earth object (NEO) population, we found that about 7.6 per cent of presently known near-Earth asteroids (NEAs) have orbits similar to asteroids from the Maria family. Only ∼1.7 per cent of our simulated family can stay in NEO space for more than 10 Myr, while only five asteroids become NEOs in the last 500 Myr of the simulation.

**Searching for some natural orbits to observe the double asteroid 2002CE26**

Mescolotti, B. Y. P. M.; Prado, A. F. B. A.; Chiaradia , A. P. M. & Gomes, V. M. (2017)

*Astrophysics and Space Science*

Knowledge of the Solar System is increasing with data coming from space missions to small bodies. A mission to those bodies offers some problems, because they have several characteristics that are not well known, like their shapes, sizes and masses. The present research has the goal of searching for trajectories around the double asteroid 2002CE_{26}, a system of Near-Earth Asteroids (NEAs) of the Apollo type. For every trajectory of the spacecraft, the evolution of the distances between the spacecraft and the two bodies that compose the system is crucial, due to its impact in the quality of the observations made from the spacecraft. Furthermore, this study has a first objective of searching for trajectories that make the spacecraft remain as long as possible near the two bodies that compose the asteroid system, without the use of orbital maneuvers. The model used here assumes elliptical orbits for the asteroids. The effect of the solar radiation pressure is also included, since it is a major perturbing force acting in spacecrafts traveling around small bodies. The natural orbits found here are useful for the mission. They can be used individually or combined in several pieces by orbital maneuvers. Another point considered here is the importance of the errors in the estimation of the physical parameters of the bodies. This task is very important, because there are great uncertainties in these values because the measurements are based on observations made from the Earth. It is shown that a variation of those parameters can make very large modifications in the times that the spacecraft remains close to the bodies of the system (called here “observational times”). Those modifications are large enough to make the best trajectories obtained under nominal conditions to be useless under some errors in the physical parameters. So, a search is made to find trajectories that have reasonable observation times for all the assumed error scenarios for the two bodies, because those orbits can be used as initial parking orbits for the spacecraft. We called these orbits “quasi-stable orbits”, in the sense that they do not collide with any of the primaries nor travel to large distances from them. From these orbits, it is possible to make better observations of the bodies in any scenario, and a more accurate estimation of their sizes and masses is performed, so giving information to allow for other choices for the orbit of the spacecraft.

**Orbital transfers in an asteroid system considering the solar radiation pressure**

Oliveira, G. M. C.; Prado, A. F. B. A.; Sanchez, D. M. & Gomes, V. M. (2017)

*Astrophysics and Space Science*

The present paper studies the effects of the radiation pressure in the trajectories of a spacecraft in transfers between the collinear Lagrange points of a double asteroid system. The system considered is this paper is formed by the double asteroid 1996FG_{3} and the maneuvers are always assumed to be bi-impulsive. In a system formed by asteroids, the solar radiation pressure has a significant influence in the transfers paths. This occurs because the gravitational forces in these systems are smaller if compared with systems formed by larger bodies. Solutions with lower and higher fuel consumption can be found by adding the solar radiation pressure. The radiation pressure was not used as a control but its effects over the transfers were measured. For a small system of primaries such as an asteroid system, it is very important to take into account this force to make sure that the spacecraft will reach the desired point.

**Searching for orbits around the triple system 45 Eugenia**

Mescolotti, B. Y. P. M.; Prado, A. F. B. A.; Chiaradia, A. P. M. & Gomes, V. M. (2017)

*Journal of Physics: Conference Series*

Asteroids are small bodies that raises high interest, because they have unknown characteristics. The present research aims to study orbits for a spacecraft around the triple asteroid 45 Eugenia. The quality of the observations made by the spacecraft depends on the distance the spacecraft remains from the bodies of the system. It is used a semi-analytical model that is simple but able to represent the main characteristics of that system. This model is called “Precessing Inclined Bi-Elliptical Problem” (PIBEP). A reference system centered on the main body (Eugenia) and with the reference plane assumed to be in the orbital plane of the second more massive body, here called Petit-Prince, is used. The secondary bodies are assumed to be in elliptical orbits. In addition, it is assumed that the orbits of the smaller bodies are precessing due to the presence of the flattening of the main body (J_{2}). This work analyzes orbits for the spacecraft with passages near Petit-Prince and Princesses, which are the two smaller bodies of the triple system.

**Searching for orbits around the triple asteroid 2001SN 263**

Cavalca, M. P. O.; Prado, A. F. B. A.; Formiga, J. K. S. & Gomes, V. M. (2017)

*Journal of Physics: Conference Series*

The asteroid 2001SN_{263} is one of the possible targets of a proposed mission that would be the first Brazilian exploration in deep space, the Aster Mission. This asteroid is composed by three bodies: Alpha, Beta, and Gamma, in decreasing order of mass. For this study, it is proposed to split this triple system in two double systems: Alpha-Beta-spacecraft and Alpha-Gamma-spacecraft, all of them considered to be points of mass, such that it is possible to use the circular planar restricted three-body problem as the mathematical model. The goal is to find orbits that can be used by a spacecraft to observe the bodies Beta and Gamma. Each orbit can be identified by the initial conditions of the spacecraft with respect to Beta or Gamma: position and velocity. These orbits are classified by the minimum average distance spacecraft-celestial body. The results showed stable orbits around Beta and Gamma, with an average distance below 1.5 km, under the influence of the gravity of Alpha and the solar radiation pressure.

**Orbital effects in a cloud of space debris making a close approach with the earth**

Formiga, J. K. S.; De Moraes, R. V. & Gomes, V. M. (2017)

*Computacional and Applied Mathematics*

The present paper has the goal of studying the changes of the orbital parameters of each individual element of a cloud of particles that makes a close approach with the Earth. Clouds of particles are formed when natural or man-made bodies explode for some reason. After an explosion like that, the center of mass of the cloud follows the same orbit of the body that generated the explosion, but the individual particles have different trajectories. The cloud is specified by a distribution of semi-major axis and eccentricity of their particles. This cloud is assumed to pass close to the Earth, making a close approach that modifies the trajectory of every particle that belongs to the cloud. The present paper makes simulations based in the “Patched-Conics” model to obtain the new trajectories of each particle. Then, it is possible to map the new distribution of the Keplerian elements of the particles that constituted the cloud, using the previous distribution as initial conditions. These information are important when planning satellite missions having a spacecraft passing close to a cloud of this type, because it is possible to obtain values for the density and amplitude of the cloud, so finding the risks of collision and the possible maneuvers that need to be made in the spacecraft to avoid the collisions.

**Mapping trajectories for a spacecraft to hit an asteroid to avoid a collision with the Earth**

Oliveira, G. M. C.; Prado, A. F. B. A.; Sanchez, D. M.; Nascimento, J. M. & Gomes, V. M. (2017)

*Advances in the Astronautical Sciences*

### 2016

(21 artigos)

**Formation of terrestrial planets in disks with different surface density profiles**

Winter, O. C. & Haghihipour, N. (2016)

*Celestial Mechanics & Dynamical Astronomy*

We present the results of an extensive study of the final stage of terrestrial planet formation in disks with different surface density profiles and for different orbital configurations of Jupiter and Saturn. We carried out simulations in the context of the classical model with disk surface densities proportional to r−0.5,r−1

“>r−0.5,r−1r−0.5,r−1 and r−1.5

“>r−1.5r−1.5, and also using partially depleted, non-uniform disks as in the recent model of Mars formation by Izidoro et al. (Astrophys J 782:31, 2014). The purpose of our study is to determine how the final assembly of planets and their physical properties are affected by the total mass of the disk and its radial profile. Because as a result of the interactions of giant planets with the protoplanetary disk, secular resonances will also play important roles in the orbital assembly and properties of the final terrestrial planets, we will study the effect of these resonances as well. In that respect, we divide this study into two parts. When using a partially depleted disk (Part 1), we are particularly interested in examining the effect of secular resonances on the formation of Mars and orbital stability of terrestrial planets. When using the disk in the classical model (Part 2), our goal is to determine trends that may exist between the disk surface density profile and the final properties of terrestrial planets. In the context of the depleted disk model, results of our study show that in general, the ν5

“>ν5ν5 resonance does not have a significant effect on the dynamics of planetesimals and planetary embryos, and the final orbits of terrestrial planets. However, ν6

“>ν6ν6 and ν16

“>ν16ν16 resonances play important roles in clearing their affecting areas. While these resonances do not alter the orbits of Mars and other terrestrial planets, they strongly deplete the region of the asteroid belt ensuring that no additional mass will be scattered into the accretion zone of Mars so that it can maintain its mass and orbital stability. In the context of the classical model, the effects of these resonances are stronger in disks with less steep surface density profiles. Our results indicate that when considering the classical model (Part 2), the final planetary systems do not seem to show a trend between the disk surface density profile and the mean number of the final planets, their masses, time of formation, and distances to the central star. Some small correlations were observed where, for instance, in disks with steeper surface density profiles, the final planets were drier, or their water contents decreased when Saturn was added to the simulations. However, in general, the final orbital and physical properties of terrestrial planets seem to vary from one system to another and depend on the mass of the disk, the spatial distribution of protoplanetary bodies (i.e., disk surface density profile), and the initial orbital configuration of giant planets. We present results of our simulations and discuss their implications for the formation of Mars and other terrestrial planets, as well as the physical properties of these objects such as their masses and water contents.

doi: http://dx.doi.org/10.1007/s10569-015-9663-y

arxiv: https://arxiv.org/abs/1512.02852

**Dynamics of rotationally fissioned asteroids: non-planar case**

Winter, O. C.; Boldrin, L. A. G. & Scheeres D. J. (2016)

*Monthly Notices of the Royal Astronomical Society*

The rotational fission of asteroids has been studied previously with simplified models restricted to planar motion. However, the observed physical configuration of contact binaries leads one to conclude that most of them are not in a planar configuration and hence would not be restricted to planar motion once they undergo rotational fission. This motivated a study of the evolution of initially non-planar binaries created by fission. Using a two-ellipsoid model, we performed simulations taking only gravitational interactions between components into account. We simulate 91 different initial inclinations of the equator of the secondary body for 19 different mass ratios. After disruption, the binary system dynamics are chaotic, as predicted from theory. Starting the system in a non-planar configuration leads to a larger energy and enhanced coupling between the rotation state of the smaller fissioned body and the evolving orbital system, and enables re-impact to occur. This leads to differences with previous planar studies, with collisions and secondary spin fission occurring for all mass ratios with inclinations θ_{0} ≥ 40^{o}, and mimics a Lidov–Kozai mechanism. Out of 1729 studied cases, we found that ∼14 per cent result in secondary fission, ∼25 per cent result in collisions and ∼6 per cent have lifetimes longer than 200 yr. In Jacobson & Scheeres stable binaries only formed in cases with mass ratios *q* < 0.20. Our results indicate that it should be possible to obtain a stable binary with the same mechanisms for cases with mass ratios larger than this limit, but that the system should start in a non-planar configuration.
doi: http://dx.doi.org/10.1093/mnras/stw1607

**Intervening in Earth’s climate system through space-based solar reflectors**

Winter, O. C.; Salazar, F. J. T. & McInnes C. R. (2016)

*Monthly Notices of the Royal Astronomical Society*

Several space-based climate engineering methods, including shading the Earth with a particle ring for active cooling, or the use of orbital reflectors to increase the total insolation of Mars for climate warming have been considered to modify planetary climates in a controller manner. In this study, solar reflectors on polar orbits are proposed to intervene in the Earth’s climate system, involving near circular polar orbits normal to the ecliptic plane of the Earth. Similarly, a family of displaced polar orbits (non-Keplerian orbits) are also characterized to mitigate future natural climate variability, producing a modest global temperature increase, again to compensate for possible future cooling. These include deposition of aerosols in the stratosphere from large volcanic events. The two-body problem is considered, taking into account the effects of solar radiation pressure and the Earth’s J2

“>J2 oblateness perturbation.

**The Rings of Chariklo Under Close Encounters with the Giant Planets**

Winter, O. C.; Araujo, R. A. N. & Sfair, R. (2016)

*Astrophysical Journal*

The Centaur population is composed of minor bodies wandering between the giant planets that frequently perform close gravitational encounters with these planets, leading to a chaotic orbital evolution. Recently, the discovery of two well-defined narrow rings was announced around the Centaur 10199 Chariklo. The rings are assumed to be in the equatorial plane of Chariklo and to have circular orbits. The existence of a well-defined system of rings around a body in such a perturbed orbital region poses an interesting new problem. Are the rings of Chariklo stable when perturbed by close gravitational encounters with the giant planets? Our approach to address this question consisted of forward and backward numerical simulations of 729 clones of Chariklo, with similar initial orbits, for a period of 100 Myr. We found, on average, that each clone experiences during its lifetime more than 150 close encounters with the giant planets within one Hill radius of the planet in question. We identified some extreme close encounters that were able to significantly disrupt or disturb the rings of Chariklo. About 3% of the clones lose their rings and about 4% of the clones have their rings significantly disturbed. Therefore, our results show that in most cases (more than 90%), the close encounters with the giant planets do not affect the stability of the rings in Chariklo-like systems. Thus, if there is an efficient mechanism that creates the rings, then these structures may be common among these kinds of Centaurs.

doi: http://dx.doi.org/10.3847/0004-637X/824/2/80

arxiv: https://arxiv.org/abs/1604.07323

**The asteroid belt as a relic from a chaotic early solar system**

Winter, O. C.; Izidoro, A.; Raymond, S. N.; Pierens, A.; Morbidelli, A. & Nesvorný, D. (2016)

*Astrophysical Journal*

The orbital structure of the asteroid belt holds a record of the solar system’s dynamical history. The current belt only contains ~10^{−3} Earth masses yet the asteroids’ orbits are dynamically excited, with a large spread in eccentricity and inclination. In the context of models of terrestrial planet formation, the belt may have been excited by Jupiter’s orbital migration. The terrestrial planets can also be reproduced without invoking a migrating Jupiter; however, as it requires a severe mass deficit beyond Earth’s orbit, this model systematically under-excites the asteroid belt. Here we show that the orbits of the asteroids may have been excited to their current state if Jupiter’s and Saturn’s early orbits were chaotic. Stochastic variations in the gas giants’ orbits cause resonances to continually jump across the main belt and excite the asteroids’ orbits on a timescale of tens of millions of years. While hydrodynamical simulations show that the gas giants were likely in mean motion resonance at the end of the gaseous disk phase, small perturbations could have driven them into a chaotic but stable state. The gas giants’ current orbits were achieved later, during an instability in the outer solar system. Although it is well known that the present-day solar system exhibits chaotic behavior, our results suggest that the early solar system may also have been chaotic.

doi: http://dx.doi.org/10.3847/1538-4357/833/1/40

arxiv: https://arxiv.org/abs/1609.04970

**Exploring the Moon gravity to escape from the Earth-Moon system**

Winter, O. C.; Santana, S. H. S.; de Melo, C. F. & Macau, E. E. N. (2016)

*Computational & Applied Mathematics*

Escape trajectories from the Earth–Moon system can be obtained through a single gravity assist maneuver of a spacecraft with the Moon. This paper presents a semi-analytical study of variations in the semi-major axis and the eccentricity of a spacecraft for a geocentric frame—before and after a close encounter of the spacecraft with the Moon. Studies on the swing-by dynamics between a spacecraft and the Moon were performed in order to establish a set of initial conditions in terms of eccentricities and semi-major axes of geocentric initial orbits. This way, it was possible identify which orbits, launched from Earth parking orbits, accomplish swing-bys with the Moon and generate escape trajectories.

**Formation of the G-ring arc**

Vieira Neto, E.; Araujo, N. C. S. & Foryta, D. W. (2016)

*Monthly Notices of the Royal Astronomical Society*

Since 2004, the images obtained by the *Cassini* spacecraft’s on-board cameras have revealed the existence of several small satellites in the Saturn system. Some of these small satellites are embedded in arcs of particles. While these satellites and their arcs are known to be in corotation resonances with Mimas, their origin remains unknown. This work investigates one possible process for capturing bodies into a corotation resonance, which involves increasing the eccentricity of a perturbing body. Therefore, through numerical simulations and analytical studies, we show a scenario in which the excitation of Mimas’s eccentricity could capture particles in a corotation resonance. This is a possible explanation for the origin of the arcs.

doi: http://dx.doi.org/10.1093/mnras/stw1055

arxiv: https://arxiv.org/abs/1606.05190

**Constraints on the original ejection velocity fields of asteroid families**

Carruba, V. & Nesvorny, D. (2016)

*Monthly Notices of the Royal Astronomical Society*

Asteroid families form as a result of large-scale collisions among main belt asteroids. The orbital distribution of fragments after a family-forming impact could inform us about their ejection velocities. Unfortunately, however, orbits dynamically evolve by a number of effects, including the Yarkovsky drift, chaotic diffusion, and gravitational encounters with massive asteroids, such that it is difficult to infer the ejection velocities eons after each family’s formation. Here, we analyse the *inclination* distribution of asteroid families, because proper inclination can remain constant over long time intervals, and could help us to understand the distribution of the component of the ejection velocity that is perpendicular to the orbital plane (*v*_{W}). From modelling the initial break up, we find that the distribution of *v*_{W}of the fragments, which manage to escape the parent body’s gravity, should be more peaked than a Gaussian distribution (i.e. be leptokurtic) even if the initial distribution was Gaussian. We surveyed known asteroid families for signs of a peaked distribution of *v*_{W} using a statistical measure of the distribution peakedness or flatness known as kurtosis. We identified eight families whose *v*_{W} distribution is significantly leptokurtic. These cases (e.g. the Koronis family) are located in dynamically quiet regions of the main belt, where, presumably, the initial distribution of *v*_{W} was not modified by subsequent orbital evolution. We suggest that, in these cases, the inclination distribution can be used to obtain interesting information about the original ejection velocity field.

doi: http://dx.doi.org/10.1093/mnras/stw043

arxiv: https://arxiv.org/abs/1602.04486

**Characterizing the original ejection velocity field of the Koronis family**

Carruba, V.; Nesvorny, D. & Aljbaae, S. (2016)

*Icarus*

An asteroid family forms as a result of a collision between an impactor and a parent body. The fragments with ejection speeds higher than the escape velocity from the parent body can escape its gravitational pull. The cloud of escaping debris can be identified by the proximity of orbits in proper element, or frequency, domains. Obtaining estimates of the original ejection speed can provide valuable constraints on the physical processes occurring during collision, and used to calibrate impact simulations. Unfortunately, proper elements of asteroids families are modified by gravitational and non-gravitational effects, such as resonant dynamics, encounters with massive bodies, and the Yarkovsky effect, such that information on the original ejection speeds is often lost, especially for older, more evolved families.

It has been recently suggested that the distribution in proper inclination of the Koronis family may have not been significantly perturbed by local dynamics, and that information on the component of the ejection velocity that is perpendicular to the orbital plane (*v _{W}*), may still be available, at least in part. In this work we estimate the magnitude of the original ejection velocity speeds of Koronis members using the observed distribution in proper eccentricity and inclination, and accounting for the spread caused by dynamical effects. Our results show that (i) the spread in the original ejection speeds is, to within a 15% error, inversely proportional to the fragment size, and (ii) the minimum ejection velocity is of the order of 50 m/s, with larger values possible depending on the orbital configuration at the break-up.

doi: http://dx.doi.org/10.1016/j.icarus.2016.01.006

arxiv: https://arxiv.org/abs/1602.04491

**Footprints of a possible Ceres asteroid paleo-family**

Carruba, V.; Nesvorny, D.; Marchi, S. & Aljbaae, S. (2016)

*Monthly Notices of the Royal Astronomical Society*

Ceres is the largest and most massive body in the asteroid main belt. Observational data from the Dawn spacecraft reveal the presence of at least two impact craters about 280 km in diameter on the Ceres surface, that could have expelled a significant number of fragments. Yet, standard techniques for identifying dynamical asteroid families have not detected any Ceres family. In this work, we argue that linear secular resonances with Ceres deplete the population of objects near Ceres. Also, because of the high escape velocity from Ceres, family members are expected to be very dispersed, with a considerable fraction of km-sized fragments that should be able to reach the pristine region of the main belt, the area between the 5J:-2A and 7J:-3A mean-motion resonances, where the observed number of asteroids is low. Rather than looking for possible Ceres family members near Ceres, here we propose to search in the pristine region. We identified 156 asteroids whose taxonomy, colours, albedo could be compatible with being fragments from Ceres. Remarkably, most of these objects have inclinations near that of Ceres itself.

doi: http://dx.doi.org/10.1093/mnras/stw380

arxiv: https://arxiv.org/abs/1602.04736

**On the oldest asteroid families in the main belt**

Carruba, V.; Nesvorny, D.; Domingos, R. C.; Huaman, M. & Aljbaae, S. (2016)

*Monthly Notices of the Royal Astronomical Society*

Asteroid families are groups of minor bodies produced by high-velocity collisions. After the initial dispersions of the parent bodies fragments, their orbits evolve because of several gravitational and non-gravitational effects, such as diffusion in mean-motion resonances, Yarkovsky and Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effects, close encounters of collisions, etc. The subsequent dynamical evolution of asteroid family members may cause some of the original fragments to travel beyond the conventional limits of the asteroid family. Eventually, the whole family will dynamically disperse and no longer be recognizable. A natural question that may arise concerns the time-scales for dispersion of large families. In particular, what is the oldest still recognizable family in the main belt? Are there any families that may date from the late stages of the late heavy bombardment and that could provide clues on our understanding of the primitive Solar system? In this work, we investigate the dynamical stability of seven of the allegedly oldest families in the asteroid main belt. Our results show that none of the seven studied families has a nominally mean estimated age older than 2.7 Gyr, assuming standard values for the parameters describing the strength of the Yarkovsky force. Most ‘paleo-families’ that formed between 2.7 and 3.8 Gyr would be characterized by a very shallow size–frequency distribution, and could be recognizable only if located in a dynamically less active region (such as that of the Koronis family). V-type asteroids in the central main belt could be compatible with a formation from a paleo-Eunomia family.

doi: http://dx.doi.org/10.1093/mnras/stw533

arxiv: https://arxiv.org/abs/1603.00818

**Detection of the YORP effect for small asteroids in the Karin cluster**

Carruba, V.; Nesvorny, D. & Vokrouhlický, D. (2016)

*The Astronomical Journal*

The Karin cluster is a young asteroid family thought to have formed only Myr ago. The young age can be demonstrated by numerically integrating the orbits of Karin cluster members backward in time and showing the convergence of the perihelion and nodal longitudes (as well as other orbital elements). Previous work has pointed out that the convergence is not ideal if the backward integration only accounts for the gravitational perturbations from the solar system planets. It improves when the thermal radiation force known as the Yarkovsky effect is accounted for. This argument can be used to estimate the spin obliquities of the Karin cluster members. Here we take advantage of the fast growing membership of the Karin cluster and show that the obliquity distribution of diameter km Karin asteroids is bimodal, as expected if the YORP effect acted to move obliquities toward extreme values (0° or 180°). The measured magnitude of the effect is consistent with the standard YORP model. The surface thermal conductivity is inferred to be 0.07–0.2 W m^{−1} K^{−1} (thermal inertia J m^{−2} K^{−1} s). We find that the strength of the YORP effect is roughly of the nominal strength obtained for a collection of random Gaussian spheroids. These results are consistent with a surface composed of rough, rocky regolith. The obliquity values predicted here for 480 members of the Karin cluster can be validated by the light-curve inversion method.

doi: http://dx.doi.org/10.3847/0004-6256/151/6/164

arxiv: https://arxiv.org/abs/1603.09612

**On the Astrid asteroid family**

Carruba, V. (2016)

*Monthly Notices of the Royal Astronomical Society*

Among asteroid families, the Astrid family is peculiar because of its unusual inclination distribution. Objects at *a* ≃ 2.764 au are quite dispersed in this orbital element, giving the family a ‘crab-like’ appearance. Recent works showed that this feature is caused by the interaction of the family with the *s* − *s*_{C} nodal secular resonance with Ceres, that spreads the inclination of asteroids near its separatrix. As a consequence, the currently observed distribution of the *v*_{W} component of terminal ejection velocities obtained from inverting Gauss equation is quite leptokurtic, since this parameter mostly depends on the asteroids inclination. The peculiar orbital configuration of the Astrid family can be used to set constraints on key parameters describing the strength of the Yarkovsky force, such as the bulk and surface density and the thermal conductivity of surface material. By simulating various fictitious families with different values of these parameters, and by demanding that the current value of the kurtosis of the distribution in *v*_{W} be reached over the estimated lifetime of the family, we obtained that the thermal conductivity of Astrid family members should be ≃0.001 W m^{−1} K^{−1}, and that the surface and bulk density should be higher than 1000 kg m^{−3}. Monte Carlo methods simulating Yarkovsky and stochastic Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) evolution of the Astrid family show its age to be *T* = 140 ± 30 Myr old, in good agreement with estimates from other groups. Its terminal ejection velocity parameter is in the range

VEJ=5−5+17

“>VEJ=5+17−5VEJ=5−5+17

m s^{−1}. Values of *V _{EJ}* larger than 25 m s

^{−1}are excluded from constraints from the current inclination distribution.

doi: http://dx.doi.org/10.1093/mnras/stw1445

arxiv: https://arxiv.org/abs/1606.05189

**On the highly inclined leptokurtic asteroid families**

Carruba, V.; Domingos, R. C.; Huaman, M. & Aljbaae, S. (2016)

*Monthly Notices of the Royal Astronomical Society*

*v*_{W} leptokurtic asteroid families are families for which the distribution of the normal component of the terminal ejection velocity field *v*_{W} is characterized by a positive value of the γ_{2} Pearson kurtosis, i.e. they have a distribution with a more concentrated peak and larger tails than the Gaussian one. Currently, eight families are known to have γ_{2}(*v*_{W}) > 0.25. Among these, three are highly inclined asteroid families, the Hansa, Barcelona, and Gallia families. As observed for the case of the Astrid family, the leptokurtic inclination distribution seems to be caused by the interaction of these families with node secular resonances. In particular, the Hansa and Gallia family are crossed by the *s* − *s*_{V} resonance with Vesta, that significantly alters the inclination of some of their members. In this work we use the time evolution of γ_{2}(*v*_{W}) for simulated families under the gravitational influence of all planets and the three most massive bodies in the main belt to assess the dynamical importance (or lack of) node secular resonances with Ceres, Vesta, and Pallas for the considered families, and to obtain independent constraints on the family ages. While secular resonances with massive bodies in the main belt do not significantly affect the dynamical evolution of the Barcelona family, they significantly increase the γ_{2}(*v*_{W}) values of the simulated Hansa and Gallia families. Current values of the γ_{2}(*v*_{W}) for the Gallia family are reached over the estimated family age only if secular resonances with Vesta are accounted for.

doi: http://dx.doi.org/10.1093/mnras/stw2047

arxiv: https://arxiv.org/abs/1606.05189

**The dynamical environment of asteroid 21 Lutetia according to different internal models**

Carruba, V.; Chanut, T.; Souchay, j.; Prado, A.F. B. A.; Amarante, A. & Aljbaae, S. (2016)

*Monthly Notices of the Royal Astronomical Society*

One of the most accurate models currently used to represent the gravity field of irregular bodies is the polyhedral approach. In this model, the mass of the body is assumed to be homogeneous, which may not be true for a real object. The main goal of the this paper is to study the dynamical effects induced by three different internal structures (uniform, three- and four-layered) of asteroid (21) Lutetia, an object that recent results from space probe suggest being at least partially differentiated. The Mascon gravity approach used in the this work consists of dividing each tetrahedron into eight parts to calculate the gravitational field around the asteroid. The zero-velocity curves show that the greatest displacement of the equilibrium points occurs in the position of the *E*4 point for the four-layered structure and the smallest one occurs in the position of the *E*3 point for the three-layered structure. Moreover, stability against impact shows that the planar limit gets slightly closer to the body with the four-layered structure. We then investigated the stability of orbital motion in the equatorial plane of (21) Lutetia and propose numerical stability criteria to map the region of stable motions. Layered structures could stabilize orbits that were unstable in the homogeneous model.

doi: http://dx.doi.org/10.1093/mnras/stw2619

arxiv: https://arxiv.org/abs/1610.02338

**On the Hoffmeister family**

Carruba, V.; Novakovic, B. & Aljbaae, S. (2016)

*Monthly Notices of the Royal Astronomical Society*

The Hoffmeister family is a C-type group located in the central main belt. Dynamically, it is important because of its interaction with the ν_{1C} nodal secular resonance with Ceres, which significantly increases the dispersion in inclination of family members at a lower semimajor axis. As an effect, the distribution of inclination values of the Hoffmeister family at a semimajor axis lower than its centre is significantly leptokurtic, and this can be used to set constraints on the terminal ejection velocity field of the family at the time it was produced. By performing an analysis of the time behaviour of the kurtosis of the *v _{W}* component of the ejection velocity field [γ

_{2}(

*v*)], as obtained from Gauss’ equations, for different fictitious Hoffmeister families with different values of the ejection velocity field, we were able to exclude that the Hoffmeister family should be older than 335 Myr. Constraints from the currently observed inclination distribution of the Hoffmeister family suggest that its terminal ejection velocity parameter

_{W}*V*

_{EJ}should be lower than 25 m s

^{−1}. Results of a Yarko-YORP Monte Carlo method to family dating, combined with other constraints from inclinations and γ

_{2}(

*v*), indicate that the Hoffmeister family should be

_{W}“>220+60−40220−40+60

Myr old, with an ejection parameter *V*_{EJ} = 20 ± 5 m s^{−1}.

doi: http://dx.doi.org/10.1093/mnras/stw3022

arxiv: https://arxiv.org/abs/1611.06176

**Developing the "Precessing Inclined Bi-Elliptical Four-Body Problem with Radiation Pressure" to search for orbits in the triple asteroid 2001SN263**

Gomes, V. M.; Masago, B. Y. P. L.; Prado, A. F. B. A. & Chiaradia, A. P. M. (2016)

*Advances in Space Research*

Space missions to visit small bodies of the Solar System are important steps to improve our knowledge of the Solar System. Usually those bodies do not have well known characteristics, as their gravity field, which make the mission planning a difficult task. The present paper has the goal of studying orbits around the triple asteroid 2001SN_{263}, a Near-Earth Asteroid (NEA). A mission to this system allows the exploration of three bodies in the same trip. The distances reached by the spacecraft from those three bodies have fundamental importance in the quality of their observations. Therefore, the present research has two main goals: (i) to develop a semi-analytical mathematical model, which is simple, but able to represent the main characteristics of that system; (ii) to use this model to find orbits for a spacecraft with the goal of remaining the maximum possible time near the three bodies of the system, without the need of space maneuvers. This model is called “Precessing Inclined Bi-Elliptical Problem with Radiation Pressure” (PIBEPRP). The use of this model allow us to find important natural orbits for the exploration of one, two or even the three bodies of the system. These trajectories can be used individually or combined in two or more parts using orbital maneuvers.

**Studying the lifetime of orbits around Moons in elliptic motion**

Gomes, V. M. & de Cássia Domingos, R. (2016)

*Computational & Applied Mathematics*

The main goal of the present paper is to study the lifetime of orbits around moons that are in elliptic motion around their parent planet. The lifetime of the orbits is defined as the time the orbit stays in orbit around the moon without colliding with its surface. The mathematical model used to solve this problem is the second order expansion of the potential of the disturbing planet, assumed to be in an elliptical orbit. The results are presented in maps showing the lifetime of the orbit as a function of its initial inclination and eccentricity. The only perturbation acting on the orbit of the spacecraft is assumed to be the gravity of the planet, so the problem is solved by studying the orbital evolution of the spacecraft perturbed by a third body in an elliptical orbit. The region of inclination above the critical value of the third-body perturbation (around 63

“>∘∘) is studied, since below that value the orbits survive for a long time. The influence of the eccentricity of the primaries is also investigated, assuming a hypothetical system that has the same mass parameter and sizes of the Earth–Moon system, but the eccentricity can be in the range 0.0–0.2.

**Close approach of a cloud of particles around an oblate planet**

Gomes, V. M.; Oliveira, G. M. C.; Prado, A. F. B. A. & Sanchez, D. M. (2016)

*Computational & Applied Mathematics*

The goal of the present paper is to study close approaches of a cloud of particles with an oblate planet, which means that there is a

“>J2J2 term in the gravitational potential of the planet. This cloud of particles is assumed to be created during the passage of a spacecraft by the periapsis of its orbit, by an explosion or any other disruptive event. The system is formed by two large bodies (Sun and planet), assumed to be in circular orbits around the center of mass of the system, and the cloud of particles. The particles that belong to the cloud make a close approach to the flat planet and then they are dispersed by the gravitational force of the planet. The motion is governed by the equations of motion given by the planar restricted circular three-body problem plus the effects of the oblateness of the planet. Jupiter is used for numerical simulations. The results show the differences between the behavior of the cloud after the passage, considering or not the effects of the oblateness of the planet. The results show that the oblateness of the planet is equivalent to an increase in the mass of the planet.

**Atmospheric close approaches with the Earth considering drag and lift forces**

Gomes, V. M.; Piñeros, J. O. M.; Prado, A. F. B. A. & Golebiewska, J. (2016)

*Computational & Applied Mathematics*

A maneuver called “Aero-Gravity Assisted” is known in the literature to increase the energy gains given by a close approach between a spacecraft and a planet using the atmosphere of the planet. In a sequence of studies related to this problem, the present paper studies close approaches between a spacecraft and the Earth, in situations where the passage is close enough to the surface of the Earth such that the spacecraft crosses its atmosphere. The dynamical model considers the atmosphere of the Earth, in terms of drag and lift, the gravitational fields of the Earth and the Sun, assumed to be points of mass, and the spacecraft. The Earth and the Sun are assumed to be in circular coplanar orbits around their common center of mass. The equations of motion are the ones given by the circular planar restricted three-body problem with the addition of the forces generated by the atmospheric drag and lift. The primary objective is to map the variations of energy of the orbits of the spacecraft due to this close approach. The results show how the atmosphere affects the trajectory of the spacecraft, generating situations where the variation of energy changes sign with respect to the gravity part of the maneuver or where they have a zero net result, based in the equilibrium between atmospheric and gravity forces. This result opens the possibility of changing only the eccentricity of the orbit, keeping fixed its semi-major axis.

**Transfers between the Lagrangian points and the primaries considering radiation pressure**

Gomes, V. M.; Oliveira, G. M. C.; Prado, A. F. B. A. & Sanchez, D. M. (2016)

*Advances in the Astronautical Sciences*

### 2015

(25 artigos)

**Natural formations at the Earth-Moon triangular point in perturbed restricted problems**

Winter, O. C.; Salazar, F. J. T.; Macau, E. E.; Masdemont, J. J. & Gómez, G. (2015)

*Advances in Space Research*

Previous studies for small formation flying dynamics about triangular libration points have determined the existence of regions of zero and Minimum Relative Radial Acceleration with respect to the nominal trajectory, that prevent from the expansion or contraction of the constellation. However, these studies only considered the gravitational force of the Earth and the Moon using the Circular Restricted Three Body Problem (CRTBP) scenario. Although the CRTBP model is a good approximation for the dynamics of spacecraft in the Earth–Moon system, the nominal trajectories around equilateral libration points are strongly affected when the primary orbit eccentricity and solar gravitational force are considered. In this manner, the goal of this work is the analysis of the best regions to place a formation that is flying close a bounded solution around L4

“>L4, taking into account the Moon’s eccentricity and Sun’s gravity. This model is not only more realistic for practical engineering applications but permits to determine more accurately the fuel consumption to maintain the geometry of the formation.

**Chaotic Dynamics in a Low-Energy Transfer Strategy to the Equilateral Equilibrium Points in the Earth-Moon System**

Winter, O. C.; Salazar, F. J. T. & Macau, E. E. N.; (201

*Internacional Journal of Bifurcation and Chaos in Applied Sciences and Engineering*

In the frame of the equilateral equilibrium points exploration, numerous future space missions will require maximization of payload mass, simultaneously achieving reasonable transfer times. To fulfill this request, low-energy non-Keplerian orbits could be used to reach L4 and L5 in the Earth–Moon system instead of high energetic transfers. Previous studies have shown that chaos in physical systems like the restricted three-body Earth–Moon-particle problem can be used to direct a chaotic trajectory to a target that has been previously considered. In this work, we propose to transfer a spacecraft from a circular Earth Orbit in the chaotic region to the equilateral equilibrium points L4 and L5 in the Earth–Moon system, exploiting the chaotic region that connects the Earth with the Moon and changing the trajectory of the spacecraft (relative to the Earth) by using a gravity assist maneuver with the Moon. Choosing a sequence of small perturbations, the time of flight is reduced and the spacecraft is guided to a proper trajectory so that it uses the Moon’s gravitational force to finally arrive at a desired target. In this study, the desired target will be an orbit about the Lagrangian equilibrium points L4 or L5. This strategy is not only more efficient with respect to thrust requirement, but also its time transfer is comparable to other known transfer techniques based on time optimization.

**Stable retrograde orbits around the triple system 2001 SN263**

Winter, O. C.; Araujo, R. A. N. & Prado, A. F. B. A. (2015)

*Monthly Notices of the Royal Astronomical Society*

The near-Earth Asteroid 2001 SN263 is a triple system of asteroids and it is the target of the *ASTER* mission – First Brazilian Deep Space Mission. The announcement of this mission has motivated a study aimed to characterize regions of stability of the system. Araujo et al., characterized the stable regions around the components of the triple system for the planar and prograde cases. Through numerical integrations they found that the stable regions are in two tiny internal zones, one of them placed very close to Alpha and another close to Beta, and in the external region. For a space mission aimed to place the probe in the internal region of the system those results do not seem to be very interesting. Therefore, knowing that the retrograde orbits are expected to be more stable, here we present a complementary study. We now considered particles orbiting the components of the system, in the internal and external regions, with relative inclinations between 90° < *I* ≤ 180°, i.e. particles with retrograde orbits. Our goal is to characterize the stable regions of the system for retrograde orbits, and then detach a preferred region to place the space probe. For a space mission, the most interesting regions would be those that are unstable for the prograde cases, but stable for the retrograde cases. Such configuration provide a stable region to place the mission probe with a relative retrograde orbit, and, at the same time, guarantees a region free of debris since they are expected to have prograde orbits. We found that in fact the internal and external stable regions significantly increase when compared to the prograde case. For particles with *e* = 0 and *I* = 180°, we found that nearly the whole region around Alpha and Beta remain stable. We then identified three internal regions and one external region that are very interesting to place the space probe. We present the stable regions found for the retrograde case and a discussion on those preferred regions. We also discuss the effects of resonances of the particles with Beta and Gamma, and the role of the Kozai mechanism in this scenario. These results help us understand and characterize the stability of the triple system 2001 SN263 when retrograde orbits are considered, and provide important parameters to the design of the ASTER mission.

doi: http://dx.doi.org/10.1093/mnras/stv592

arxiv: https://arxiv.org/abs/1503.07546

**Formation of the Janus-Epimetheus system through collisions**

Winter, O. C.; Treffenstädt, L. L. & Mourão, D. C. (2015)

*Astronomy & Astrophysics*

*Context. *Co-orbital systems are bodies that share the same mean orbit. They can be divided into different families according to the relative mass of the co-orbital partners and the particularities of their movement. Janus and Epimetheus are unique in that they are the only known co-orbital pair of comparable masses and thus the only known system in mutual horseshoe orbit.

*Aims. *We aim to establish whether the Janus-Epimetheus system might have formed by disruption of an object in the current orbit of Epimetheus.

*Methods. *We assumed that four large main fragments were formed and neglected smaller fragments. We used numerical integration of the full *N*-body problem to study the evolution of different fragment arrangements. Collisions were assumed to result in perfectly inelastic merging of bodies. We statistically analysed the outcome of these simulations to infer whether co-orbital systems might have formed from the chosen initial conditions.

*Results. *Depending on the range of initial conditions, up to 9% of the simulations evolve into co-orbital systems. Initial velocities around the escape velocity of Janus yield the highest formation probability. Analysis of the evolution shows that all co-orbital systems are produced via secondary collisions. The velocity of these collisions needs to be low enough that the fragments can merge and not be destroyed. Generally, collisions are found to be faster than an approximate cut-off velocity threshold. However, given a sufficiently low initial velocity, up to 15% of collisions is expected to result in merging. Hence, the results of this study show that the considered formation scenario is viable.

doi: http://dx.doi.org/10.1051/0004-6361/201425543

arxiv: https://arxiv.org/abs/1509.06933

**On the Erigone family and the z 2 secular resonance**

Carruba, V.; Aljbaae, S. & Winter, O. C. (2015)

*Monthly Notices of the Royal Astronomical Society*

The Erigone family is a C-type group in the inner main belt. Its age has been estimated by several researchers to be less then 300 Myr, so it is a relatively young cluster. Yarko-YORP Monte Carlo methods to study the chronology of the Erigone family confirm results obtained by other groups. The Erigone family, however, is also characterized by its interaction with the *z*_{2} secular resonance. While less than 15 per cent of its members are currently in librating states of this resonance, the number of objects, members of the dynamical group, in resonant states is high enough to allow us to use the study of dynamics inside the *z*_{2}resonance to set constraints on the family age. Like the ν_{6} and *z*_{1} secular resonances, the *z*_{2} resonance is characterized by one stable equilibrium point at σ = 180° in the *z*_{2} resonance plane

“>(σ,dσdt)(σ,dσdt)

, where σ is the resonant angle of the *z*_{2} resonance. Diffusion in this plane occurs on time-scales of ≃ 12 Myr, which sets a lower limit on the Erigone family age. Finally, the minimum time needed to reach a steady-state population of *z*_{2} librators is about 90 Myr, which allows us to impose another, independent constraint on the group age.

**Terrestrial planet formation constrained by Mars and the structure of the asteroid belt**

Izidoro, A.; Raymond, S. N.; Morbidelli, A. & Winter, O. C. (2015)

*Monthly Notices of the Royal Astronomical Society*

Reproducing the large Earth/Mars mass ratio requires a strong mass depletion in solids within the protoplanetary disc between 1 and 3 au. The Grand Tack model invokes a specific migration history of the giant planets to remove most of the mass initially beyond 1 au and to dynamically excite the asteroid belt. However, one could also invoke a steep density gradient created by inward drift and pile-up of small particles induced by gas drag, as has been proposed to explain the formation of close-in super-Earths. Here we show that the asteroid belt’s orbital excitation provides a crucial constraint against this scenario for the Solar system. We performed a series of simulations of terrestrial planet formation and asteroid belt evolution starting from discs of planetesimals and planetary embryos with various radial density gradients and including Jupiter and Saturn on nearly circular and coplanar orbits. Discs with shallow density gradients reproduce the dynamical excitation of the asteroid belt by gravitational self-stirring but form Mars analogues significantly more massive than the real planet. In contrast, a disc with a surface density gradient proportional to *r*^{−5.5}reproduces the Earth/Mars mass ratio but leaves the asteroid belt in a dynamical state that is far colder than the real belt. We conclude that no disc profile can simultaneously explain the structure of the terrestrial planets and asteroid belt. The asteroid belt must have been depleted and dynamically excited by a different mechanism such as, for instance, in the Grand Tack scenario.

doi: http://dx.doi.org/10.1093/mnras/stv1835

arxiv: https://arxiv.org/abs/1508.01365

**A numerical study of powered Swing-Bys around the Moon**

Ferreira, A. F. S.; Prado, A. F. B. A. & Winter, O. C. (2015)

*Advances in Space Research*

The present research studies Swing-By maneuvers combined with the application of an impulse at the spacecraft periapsis passage. The studies were made for different values of

“>rp (periapsis distance),

“>δV (magnitude of the impulse),

“>Ψ (angle of approach) and

“>α (angle that defines the direction of the impulse). The results show the best direction to apply the impulse for each specific geometry of the passage, maximizing the gains or energy losses. The results show, as an example, that an angle

“>α near 20° gives the best solution to maximize the energy gains for the situation where the periapsis distance is 1.1 Moon’s radius and

“>Ψ=90°. This value goes to near

“>-20° when

“>Ψ=270°. In the case of maximizing the energy losses, two families with impulses against the direction of the spacecraft motion are found to be the best solutions. Conditions where the impulse generates a capture around the Moon or a collision are also mapped. For values larger than 1.1 Moon’s radius for the periapsis distance, the angle that maximizes the energy variation increases. Empirical analytical equations are obtained that express the energy variation as a function of the angle of approach, which replaces well-known equations obtained from the two-body approximation

** Astronautics & Astronomy - a Profitable Partnership**

Winter, O. C. (2015)

*Journal of Aerospace Technology and Management*

**The sailboat island and the New Horizons trajectory**

Giuliatti Winter, S. M.; Winter, O. C.; Vieira Neto, E. & Sfair, R. (2015)

*Icarus*

**Planet formation in a triple stellar system: implications of the third star’s orbital inclination**

Winter, O. C.; Domingos, R. & Izidoro, A. (2015)

*International Journal of Astrobiology*

**Zero drift regions and control strategies to keep satellite in formation around triangular libration point in the restricted Sun-Earth-Moon scenario**

Winter, O. C.; Salazar, F. J. T.; Macau, E. E.; Masdemont, J. J. & Gómez, G. (2015)

*Advances in Space Research*

In this work, we are interested in avoiding large variations in the mutual distances among multiple satellites and also in controlling their geometric configuration around an Earth–Moon triangular point. Previous studies about triangular libration points have determined the existence of zero drift regions with respect to the nominal trajectory, in which the expansion or contraction of the formation never take place. Our goal is to carry out two different control strategies for a formation near a given nominal trajectory around L4

“>L4: a bang-off-bang control and a minimum weighted total ΔV

“>ΔV consumption. A linearization relative to the reference trajectory around the triangular libration point is carried out, and different geometrical possibilities in the zero drift regions are studied. To investigate the influence of the gravitational force of the Sun, the BiCircular Four Body Problem is considered here. According to the results obtained, some meaningful insights to allow a proper design of the geometric configuration of the formation are drawn.

**Celestial mechanics: from the errant stars to guidance of spacecrafts**

Winter, O. C.; de Melo, C. F.; Macau, E. E. & Prado, A. B. A. (2015)

*Matemática Aplicada e Computacional *

Celestial mechanics is one of the most ancient science. It is dedicated to the study of the motion of planets, moons, asteroids, comets and other celestial bodies. It probably started when humans discovered that some special stars differentiate from the others in the sense that they move on the celestial sphere. Currently, it is responsible for successfully guiding spaceships to distant objects in our solar system aiming to explore them. As an introduction to this Focus Issue in Celestial Mechanics, we make here a historical overview of developments in this area and present the articles that comprise this special issue.

**Exoplanets in binary star systems: on the switch from prograde to retrograde orbits**

Winter, O. C.; Carvalho, J. P. S.; Mourão, D. C.; de Moraes, R. V. & Prado, A. F. B. A. (2015)

*Celestial Mechanics & Dynamical Astronomy*

The eccentric Kozai–Lidov mechanism, based on the secular theory, has been proposed as a mechanism that plays an important role in producing orbits that switch from prograde to retrograde. In the present work we study the secular dynamics of a triple system composed of a Sun-like central star and a Jupiter-like planet, which are under the gravitational influence of another perturbing star (brown dwarf). The perturbation potential is developed in closed form up to the fifth order in a small parameter (α=a1/a2

“>α=a1/a2α=a1/a2), where a1

“>a1a1 is the semimajor axis of the extrasolar planet and a2

“>a2a2 is the semimajor axis of the perturbing star. To eliminate the short-period terms of the perturbation potential, the double-average method is applied. In this work we do not eliminate the nodes, a standard method in the literature, before deriving the equations of motion. The main goal is to study the effects of the higher-order terms of the expansion of the perturbing force due to the third body in the orbital evolution of the planet. In particular, we investigate the inclination and the shape (eccentricity) of these orbits. We show the importance of the higher-order terms in changing the inversion times of the flip, i.e., the times where the inclination of the inner planet flips from prograde to retrograde trajectories. We also show the dependence of the first flip with respect to the semimajor axis and eccentricity of the orbit of the planet. The general conclusion is that the analytical model increases its accuracy with the inclusion of higher-order terms. We also performed full numerical integrations using the Bulirsch–Stoer method available in the Mercury package for comparison with the analytical model. The results obtained with the equations developed in this work are in accordance with direct numerical simulations.

**Analysis of the orbital evolution of exoplanets**

Winter, O. C.; Carvalho, J. P. S.; Mourão, D. C.; de Moraes, R. V. & Prado, A. F. B. A. (2015)

*Computational & Applied Mathematics*

An exoplanet, or extrasolar planet, is a planet that does not orbit the Sun, but is around a different star, stellar remnant, or brown dwarf. Up to now, about 1900 exoplanets were discovered. To better understand the dynamics of these exoplanets, a study with respect to possible collisions of the planet with the central star is shown here. We present an expanded model in a small parameter that takes into account up to the fifth order to analyze the effect of this potential in the orbital elements of the extrasolar planet. Numerical simulations were also performed using the *N*-body simulations, using the software Mercury, to compare the results with the ones obtained by the analytical model. The numerical simulations are presented in two stages: one considering the celestial bodies as point masses and the other one taking into account their dimensions. This analysis showed that the planet collided with the central star in the moment of the first inversion for orbits with high inclinations in various situations. The results of the simulations of the equations developed in this study are consistent with the *N*-body numerical simulations. We analyze also the flip of the inclination taking into account the coupling of the perturbations of the third body, effect due to the precession of periastron and the tide effect. In general, we find that such perturbations combined delay the time of first inversion, but do not keep the planet in a prograde or retrograde orbit.

**Pareto Frontier for the time-energy cost vector to an Earth-Moon transfer orbit using the patched-conic approximation**

Winter, O. C.; Salazar, F. J. T. & Macau, E. E. N. (2015)

*Matemática Aplicada e Computacional*

In this work, we present a study about the determination of the optimal time–energy cost vector, i.e., flight time and total ΔV

“>ΔVΔV (velocity change) spent in an orbital transfer of a spacecraft from an Earth circular parking orbit to a circular orbit around the Moon. The method used to determine the flight time and total ΔV

“>ΔVΔV is based on the well-known approach of patched conic in which the three-body problem that involves Earth, Moon and spacecraft is decomposed into two ‘two bodies’ problems, i.e., Earth–spacecraft and Moon–spacecraft. Thus, the trajectory followed by the spacecraft is a composition of two parts: The first one, when the spacecraft is within the Earth’s sphere of influence; The second one, when the spacecraft enters into the Moon’s sphere of influence. Therefore, the flight time and total ΔV

“>ΔVΔV to inject the spacecraft into the lunar trajectory and place it around the Moon can be determined using the expressions for the two-body problem. In this study, we use the concept of Pareto Frontier to find a set of parameters in the geometry of patched-conic solution that minimizes simultaneously the flight time and total ΔV

“>ΔVΔV of the mission. These results present different possibilities for performing an Earth–Moon transfer where two conflicting objectives are optimized.

**The evolution of a Pluto-like system during the migration of the ice giants**

Giuliatti Winter, S. M.; Pires, P. & Gomes, R. S. (2015)

*Icarus*

The planetary migration of the Solar System giant planets in the framework of the Nice model (Tsiganis, K., Gomes, R., Morbidelli, A., Levison, H.F. [2005]. Nature 435,459–461; Morbidelli, A., Levison, H.F., Tsiganis, K., Gomes, R. [2005]. Nature 435, 462–465; Gomes, R., Levison, H.F., Tsiganis, K., Morbidelli, A. [2005]. Nature 435, 466–469) creates a dynamical mechanism which can be used to explain the distribution of objects currently observed in the Kuiper belt (e.g., Levison, H.F., Morbidelli, A., Vanlaerhoven, C., Gomes, R., Tsiganis, K. [2008]. Icarus 196, 258–273). Through this mechanism the planetesimals within the disk, heliocentric distance ranging from beyond Neptune’s orbit to approximately 34 AU, are delivered to the belt after a temporary eccentric phase of Uranus and Neptune’s orbits. We reproduced the mechanism proposed by Levison et al. to implant bodies into the Kuiper belt. The capture of Pluto into the external 3:2 mean motion resonance with Neptune is associated with this gravitational scattering model. We verified the existence of several close encounters between the ice giants and the planetesimals during their outward radial migration, then we believe that the analysis of the dynamical history of the plutonian satellites during this kind of migration is important, and would provide some constrains about their place of formation – within the primordial planetesimal disk or *in situ*. We performed *N*-body simulations and recorded the trajectories of the planetesimals during close approaches with Uranus and Neptune. Close encounters with Neptune are the most common, reaching approximately 1200 in total. A Pluto similarly sized body assumed the hyperbolic trajectories of the former primordial planetesimal with respect to those giant planets. We assumed the current mutual orbital configuration and sizes for Pluto’s satellites, then we found that the rate of destruction of systems similar to that of Pluto with closest approaches to Uranus or Neptune <0.10 AU is 40%, i.e. these close approaches can lead to ejections of satellites or to changes in the satellites eccentricities at least 1 order of magnitude larger than the currently observed. However, we also found that the number of closest approaches which the minimum separation to Uranus or Neptune <0.10 AU is negligible, reaching 6%. In the other 60% of close encounter histories with closest approaches >0.10 AU, none of the systems have been destroyed. The latter sample concentrates 94% of closest approaches with the ice giants. Recall that throughout the early history of the Solar System giant impacts were common (McKinnon, W.B. [1989]. Astrophys. J. 344, L41–L44; Stern, A. [1991]. Icarus 90; Canup, R.M. [2005]. Science 307, 546–550). Also, impacts capable of forming a binary like Pluto-Charon can occur possibly prior to 0.5–1 Gyr (Kenyon, S.J., Bromley, B.C. [2014]. Astron. J. 147, 8), and small satellites such as Nix and Hydra can grow in debris from the giant impact (e.g., Canup, R.M. [2011]. Astron. J. 141, 35). Thus, we conclude that if Pluto and its satellites were emplaced into the KB from lower heliocentric orbits, then the Pluto system could survive the encounters that may have happened for emplacement of the Plutinos through the mechanism proposed by Levison et al.

**Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites 2014**

Giuliatti Winter, S. M.; Prado, A. F. B. A.; Masdemont, J. J.; Zanardi, M. C.; Yokoyama, T. & Gomes, V. M. (2015)

**Study of the influence of computational parameters on the formation of a giant gaseous planet**

Neto, E. V. & Moraes, R. A. (2015)

*Journal of Physics*

Today, computational simulatios are the best method to describe the formation of our Solar System, however this kind of simulations contain numerical errors caused by boundary conditions and choises of the grid, for instance. In this paper we will present results from several hidrodynamical simulations of the formation of a planet like Jupiter able to migrate using different computational parameters. We will compare the density profile, the semimajor axis and the eccentricity of the planet for those different parameters and measure the effect of them. We will not extend our results for planets with initial eccentric orbits in this paper.

**Evasive Maneuvers in Route Collision With Space Debris Cloud**

Neto, E. V.; Jesus, A. D. C. & Sousa, R. R. (2015)

*Journal of Physics*

Collisions between operational vehicles and space debris can completely derail the continuity of space missions, especially if there is chain collisions between debris, which generate even smaller fragments. In this paper, we investigate the dynamics on between an operational vehicle and space debris that form a cloud, considering the possibility of collisions between debris during an evasive maneuver the vehicle. For a radius of 3 km celestial sphere, we find possibilities of collision between debris up to 10 m, while the vehicle performs an evasive maneuver in time 3,000 s range. These results depend on the time collision, the angular positions of the collisional objects and the amount of debris that form the cloud.

**Mascon gravitation model using a shaped polyhedral source**

Carruba, V.; Chanut, T. & Aljbaae, S. (2015)

*Monthly Notices of the Royal Astronomical Society *

In the last two decades, new computational tools have been developed in order to aid space missions to orbit around irregular small bodies. One of the techniques consists in rebuilding their shape in tetrahedral polyhedron. This method is well suited to determine the shape and estimate certain physical features of asteroids. However, a large computational effort is necessary depending on the quantity of triangular faces chosen. Another method is based on a representation of the central body in terms of mascons (discrete spherical masses). The main advantage of the method is its simplicity which makes the calculation faster. Nevertheless, the errors are non-negligible when the attraction expressions are calculated near the surface of the body. In this work, we carry out a study to develop a new code that determines the centre of mass of each tetrahedron of a shaped polyhedral source and evaluates the gravitational potential function and its first- and second-order derivatives. We performed a series of tests and compared the results with the classical polyhedron method. We found good agreement between our determination of the attraction expressions close to the surface, and the same determination by the classical polyhedron method. However, this agreement does not occur inside the body. Our model appears to be more accurate in representing the potential very close to the body’s surface when we divide the tetrahedron in three parts. Finally, we have found that in terms of CPU time requirements, the execution of our code is much faster compared with the polyhedron method.

**Dynamical evolution of the Cybele asteroids**

Carruba, V.; Nesvorny, D.; Huaman, M. & Aljbaae, S. (2015)

*Monthly Notices of the Royal Astronomical Society *

The Cybele region, located between the 2J:-1A and 5J:-3A mean-motion resonances, is adjacent and exterior to the asteroid main belt. An increasing density of three-body resonances makes the region between the Cybele and Hilda populations dynamically unstable, so that the Cybele zone could be considered the last outpost of an extended main belt. The presence of binary asteroids with large primaries and small secondaries suggested that asteroid families should be found in this region, but only relatively recently the first dynamical groups were identified in this area. Among these, the Sylvia group has been proposed to be one of the oldest families in the extended main belt. In this work we identify families in the Cybele region in the context of the local dynamics and non-gravitational forces such as the Yarkovsky and stochastic Yarkovsky–O’Keefe–Radzievskii–Paddack (YORP) effects. We confirm the detection of the new Helga group at ≃3.65 au, which could extend the outer boundary of the Cybele region up to the 5J:-3A mean-motion resonance. We obtain age estimates for the four families, Sylvia, Huberta, Ulla, and Helga, currently detectable in the Cybele region, using Monte Carlo methods that include the effects of stochastic YORP and variability of the solar luminosity. The Sylvia family should be *T* = 1220 ± 40 Myr old, with a possible older secondary solution. Any collisional Cybele group formed prior to the Late Heavy Bombardment would have been most likely completely dispersed in the jumping Jupiter scenario of planetary migration.

doi: http://dx.doi.org/10.1093/mnras/stv845

arxiv: https://arxiv.org/abs/1505.03745

**The Euphrosyne family’s contribution to the low albedo near-Earth asteroids**

Carruba, V.; Masiero, J.; Mainzer, A.; Bauer, J. M. & Nugent, C. (2015)

*Astrophysical Journal*

The Euphrosyne asteroid family is uniquely situated at high inclination in the outer Main Belt, bisected by the secular resonance. This large, low albedo family may thus be an important contributor to specific subpopulations of the near-Earth objects. We present simulations of the orbital evolution of Euphrosyne family members from the time of breakup to the present day, focusing on those members that move into near-Earth orbits. We find that family members typically evolve into a specific region of orbital element-space, with semimajor axes near AU, high inclinations, very large eccentricities, and Tisserand parameters similar to Jupiter family comets. Filtering all known Near-Earth objects (NEOs) with our derived orbital element limits, we find that the population of candidate objects is significantly lower in albedo than the overall NEO population, although many of our candidates are also darker than the Euphrosyne family, and may have properties more similar to comet nuclei. Followup characterization of these candidates will enable us to compare them to known family properties, and confirm which ones originated with the breakup of (31) Euphrosyne.

doi: http://dx.doi.org/10.1088/0004-637X/809/2/179

arxiv: https://arxiv.org/abs/1507.07887

**Dynamical dispersal of primordial asteroid families**

Carruba, V.; Brasil, P. I.; Roig, F.; Nesvorny, D.; Huaman, M. & Aljbaae, S. (2015)

*Icarus*

Many asteroid families are identified and well characterized all over the main asteroid belt. Interestingly, however, none of them are older than 4 Gyr. Many mechanisms have been proposed to disperse such old primordial asteroid families that presumably have existed, but only very few have really worked. Here we present a plausible mechanism for dispersing primordial asteroid families that is based on the 5-planet instability model known as jumping Jupiter. Using two different evolutions for the jumping-Jupiter model, we have numerically integrated orbits of eight putative primordial families. Our results show that the most important effect on the asteroid families’ eccentricity and inclination dispersal is that of the secular resonances, in some cases associated with the mean motion resonances. As for the semimajor axes spreading we find that the principal effect is that of close encounters with the fifth giant planet whose orbit briefly overlaps with (part of) the main belt. Therefore, the existence of a fifth giant planet with the mass comparable with that of Uranus’ or Neptune’s could contribute in important ways to dispersal of the primordial asteroid families. To have that effect, the interloper planet should go into and considerably interact with the asteroids during the instability phase.

**Orbital evolution of planet around a binary star**

Mourão, D. C.; de Moraes, R. V.; Carvalho, J. P. S. & Prado, A. F. B. A. (2015)

*Advances in the Astronautical Sciences*

We study the secular dynamics of hierarchical (if there is a clearly defined binary and a third body which stays separate from the binary) triple systems composed by a Sun-like central star and a Jupiter-like planet, which are under the gravitational influence of a further perturbing star (brown dwarf). The main goal is to study the orbital evolution of the planet. In special, we investigate the orientation (inclination) and the shape (eccentricity) of its orbit. One key feature explored is the time needed for the first flip in its orientation (prograde to retrograde). The gravitational potential is developed in closed form up to the third order. We have compared the secular evolution of systems with and without the third order term of the disturbing potential. The R2 (quadrupole) and R3 (octupole) terms of the disturbing potential are developed without using the elimination of nodes. Numerical simulations were also performed to compare with the analytical model using the N-body simulations with the Mercury code. The results show that the analytical model are in agreement with the numeric simulations.

**Searching for capture and escape trajectories around Jupiter using its Galilean satellites**

Gomes, V. M. & Prado, A. F. B. A. (2015)

*Computational & Applied Mathematics*

This paper has the goal of searching for natural trajectories that can be used for a particle or a spacecraft coming from a region of the space far from Jupiter system to be captured into this system by making close approaches with the Galilean satellites of the planet. The opposite situation is also possible and escape trajectories can also be found. This type of maneuver is called “Swing-By” and it is usual in astrodynamics. It was used in many space missions to reduce the fuel consumption by gaining or loosing energy from the gravity of a celestial body. Several famous examples are the Voyager, Cassini, Galileo and other missions. The idea of the present research is to study this type of maneuver using the Galilean satellites of Jupiter, to search for trajectories that change the two-body energy (particle or spacecraft)–(Jupiter) from positive to negative (a capture trajectory) or from negative to positive (an escape trajectory). Those trajectories can be used for a spacecraft going or leaving the planet Jupiter or to explain how particles can be captured or escape from Jupiter system by close approaches with the Galilean satellites. Initial conditions are varied to cover the whole possible alternatives and then small regions of captures and escapes are identified. After that, a study is made to see the accuracy of the Tisserand’s method when applied to those close approach trajectories.

### 2014

(16 artigos)

**Terrestrial planet formation in a protoplanetary disk with a local mass depletion: a successful scenario for the formation of Mars**

Winter, O. C.; Izidoro, A.; Haghighipour, N. & Tsuchida, M. (2014)

*The Astrophysical Journal*

Models of terrestrial planet formation for our solar system have been successful in producing planets with masses and orbits similar to those of Venus and Earth. However, these models have generally failed to produce Mars-sized objects around 1.5 AU. The body that is usually formed around Mars’ semimajor axis is, in general, much more massive than Mars. Only when Jupiter and Saturn are assumed to have initially very eccentric orbits (*e* ~ 0.1), which seems fairly unlikely for the solar system, or alternately, if the protoplanetary disk is truncated at 1.0 AU, simulations have been able to produce Mars-like bodies in the correct location. In this paper, we examine an alternative scenario for the formation of Mars in which a local depletion in the density of the protosolar nebula results in a non-uniform formation of planetary embryos and ultimately the formation of Mars-sized planets around 1.5 AU. We have carried out extensive numerical simulations of the formation of terrestrial planets in such a disk for different scales of the local density depletion, and for different orbital configurations of the giant planets. Our simulations point to the possibility of the formation of Mars-sized bodies around 1.5 AU, specifically when the scale of the disk local mass-depletion is moderately high (50%-75%) and Jupiter and Saturn are initially in their current orbits. In these systems, Mars-analogs are formed from the protoplanetary materials that originate in the regions of disk interior or exterior to the local mass-depletion. Results also indicate that Earth-sized planets can form around 1 AU with a substantial amount of water accreted via primitive water-rich planetesimals and planetary embryos. We present the results of our study and discuss their implications for the formation of terrestrial planets in our solar system.

doi: http://dx.doi.org/10.1088/0004-637x/782/1/31

arxiv: https://arxiv.org/abs/1312.3959

**3D stability orbits close to 433 Eros using an effective polyhedral model method**

Winter, O. C.; Chanut, T. G. G. & Tsuchida, M. (2014)

*Monthly Notices of the Royal Astronomical Society*

One of the techniques used in the past decade to determine the shape with a good accuracy and estimate certain physical features (volume, mass, moments of inertia) of asteroids is the polyhedral model method. We rebuild the shape of the asteroid 433 Eros using data from 1998 December observations of the probe *Near-Earth-Asteroid-Rendezvous-Shoemaker*. In our computations, we use a code that avoids singularities from the line integrals of a homogeneous arbitrary shaped polyhedral source. This code evaluates the gravitational potential function and its first- and second-order derivatives. Taking the rotation of asteroid 433 Eros into consideration, the aim of this work is to analyse the dynamics of numerical simulations of 3D initially equatorial orbits near the body. We find that the minimum radius for direct, equatorial circular orbits that cannot impact with the Eros surface is 36 km and the minimum radius for stable orbits is 31 km despite significant perturbations of its orbit. Moreover, as the orbits suffer less perturbations due to the irregular gravitational potential of Eros in the elliptic case, the most significant result of the analysis is that stable orbits exist at a periapsis radius of 29 km for initial eccentricities *e*_{i} ≤ 0.2.

**Alternative transfer to the Earth-Moon Lagrangian points L4 and L5 using lunar gravity assist**

Winter, O. C.; Salazar, F. J. T. & Macau, E. E. N. (2014)

*Advances in Space Research*

Lagrangian points L4 and L5 lie at 60° ahead of and behind the Moon in its orbit with respect to the Earth. Each one of them is a third point of an equilateral triangle with the base of the line defined by those two bodies. These Lagrangian points are stable for the Earth–Moon mass ratio. As so, these Lagrangian points represent remarkable positions to host astronomical observatories or space stations. However, this same distance characteristic may be a challenge for periodic servicing mission. This paper studies elliptic trajectories from an Earth circular parking orbit to reach the Moon’s sphere of influence and apply a swing-by maneuver in order to re-direct the path of a spacecraft to a vicinity of the Lagrangian points L4 and L5. Once the geocentric transfer orbit and the initial impulsive thrust have been determined, the goal is to establish the angle at which the geocentric trajectory crosses the lunar sphere of influence in such a way that when the spacecraft leaves the Moon’s gravitational field, its trajectory and velocity with respect to the Earth change in order to the spacecraft arrives at L4 and L5. In this work, the planar Circular Restricted Three Body Problem approximation is used and in order to avoid solving a two boundary problem, the patched-conic approximation is considered.

**Near-Earth asteroid binaries in close encounters with the Earth**

Winter, O. C. & Araujo, R. A. N. (2014)

*Astronomy & Astrophysics*

The asteroids that may cross, or approach, the orbit of the terrestrial planets compose the NEAs population. In the present work we studied the effects of close encounters between hypothetical NEAs’ binaries and the Earth to determine the limiting stability of their satellites as a function of the encounter conditions. We were able to estimate the frequency of such encounters, thus, the expected lifetime of the NEAs binaries. Through numerical simulations, all the encounters between asteroids and the Earth that were closer than 100 Earth radii were recorded. The next step consisted of simulating a representative sample of those encounters considering the Earth, and the asteroid with a cloud of non-interacting satellites around it. The largest radial distance for which all the satellites survive (no collision or disruption) was defined as the critical radius *R*_{C}. We present a statistical analysis of the recorded encounters, and the critical radius given as a function of the impact parameter and of the relative velocity, defining the stable and unstable encounter conditions for the NEAs satellites. In all these simulations and analyses three distinct satellite masses were considered: massless satellites, satellites with ten percent of the asteroid mass, and satellites with the same mass of the asteroid. We found that the close encounters that are able to remove the most internal satellites are quite frequent. For the NEAs of the sample initially belonging to the Atens group, we found that ≈93% of them suffer an encounter with such characteristics within 10 Myr and that 50% of these encounters happen within approximately 330 thousand years. Our results confirm that, in fact, the close encounter of binaries with the Earth is a powerful mechanism for disrupting these systems and that their lifetimes are strongly influenced by the planetary close encounters.

**The triple near-Earth asteroid (153591) 2001 SN263: an ultra-blue, primitive target for the Aster space mission**

Winter, O. C.; Perna, D.; Alvarez-Candal, A.; Fornasier, S.; Giuliatti Winter, S. M. & Vieira Neto, E. (2014)

*Astronomy & Astrophysics*

*Context. *The Brazilian Aster project plans a space mission to rendezvous and characterize (153591) 2001 SN263, one of the only two known triple near-Earth asteroids (NEAs). Improving the knowledge of its physical properties is necessary to optimize the mission planning and science return.

*Aims. *We study the surface composition and physical nature of 2001 SN263 by analyzing and comparing its reflectance spectra with laboratory spectra of minerals and meteorites.

*Methods. *We performed spectroscopic observations of 2001 SN263 using the UV-to-NIR X-Shooter spectrograph at the ESO Very Large Telescope (VLT). Complementary photometric observations of the target were acquired with the FORS2 instrument.

*Results. *We find B-type, featureless convex spectra (Themis- or Polana-like). 2001 SN263 presents the bluest visible spectrum ever observed for small bodies in the solar system, even bluer than NEAs Phaethon and Bennu. The spectra suggest that the surface composition is organic- and magnetite-rich, similar to that of heated CI carbonaceous chondrites. Phyllosilicates may be abundant as well. We find hints of a coarse-grained surface and composition variety within the triple system.

*Conclusions. *Both the large grain size and surface variability might be connected to the formation of the triple system. The Aster mission will have the intriguing possibility of checking current models of asteroid binary formation.

**A peculiar stable region around Pluto**

Winter, O. C.; Giuliatti Winter, S. M.; Sfair, R. & Vieira Neto, E. (2014)

*Monthly Notices of the Royal Astronomical Society*

Giuliatti Winter et al. found several stable regions for a sample of test particles located between the orbits of Pluto and Charon. One peculiar stable region in the space of the initial orbital elements is located at *a* = (0.5*d*, 0.7*d*) and *e* = (0.2, 0.9), where *a* and *e* are the initial semimajor axis and eccentricity of the particles, respectively, and *d* is the Pluto–Charon distance. This peculiar region (hereafter called the sailboat region) is associated with a family of periodic orbits derived from the planar, circular, restricted three-body problem (Pluto–Charon–particle). In this work, we study the origin of this stable region by analysing the evolution of such family of periodic orbits. We show that they are not in resonances with Charon. The period of the periodic orbit varies along the family, decreasing with the increase of the Jacobi constant. We also explore the extent of the sailboat region by adopting different initial values of the orbital inclination (*I*) and argument of the pericentre (ω) of the particles. The sailboat region is present for *I* = [0°, 90°] and for two intervals of ω, ω = [−10°, 10°] and (160°, 200°). A crude estimative of the size of the hypothetical bodies located at the sailboat region can be derived by computing the tidal damping in their eccentricities. If we neglect the orbital evolution of Pluto and Charon, the time-scale for circularization of their orbits is longer than the age of the Solar system for bodies smaller than 500 m in radius.

**Zero, minimum and maximum relative radial acceleration for planar formation flight dynamics near triangular libration points in the Earth-Moon system**

Winter, O. C.; Salazar, F. J. T.; Masdemont, J. J.; Gómez, G. & Macau, E. E. (2014)

*Advances in Space Research*

Assume a constellation of satellites is flying near a given nominal trajectory around

“>L4 or

“>L5 in the Earth–Moon system in such a way that there is some freedom in the selection of the geometry of the constellation. We are interested in avoiding large variations of the mutual distances between spacecraft. In this case, the existence of regions of zero and minimum relative radial acceleration with respect to the nominal trajectory will prevent from the expansion or contraction of the constellation. In the other case, the existence of regions of maximum relative radial acceleration with respect to the nominal trajectory will produce a larger expansion and contraction of the constellation. The goal of this paper is to study these regions in the scenario of the Circular Restricted Three Body Problem by means of a linearization of the equations of motion relative to the periodic orbits around

“>L4 or

“>L5. This study corresponds to a preliminar planar formation flight dynamics about triangular libration points in the Earth–Moon system. Additionally, the cost estimate to maintain the constellation in the regions of zero and minimum relative radial acceleration or keeping a rigid configuration is computed with the use of the residual acceleration concept. At the end, the results are compared with the dynamical behavior of the deviation of the constellation from a periodic orbit.

**A ring system detected around the Centaur (10199) Chariklo**

Sfair, R.; Braga-Ribas, F.; Sicardy, B.; Ortiz, J. L.; Snodgrass, C.; Roques, F.; Vieira-Martins, R. Camargo, J. I.; Assafin, M.; Duffard, R.; Jehin, E.; Pollock, J.; Leiva, R.; Emilio, M.; Machad, D. I.; Colazo, C.; Lellouch, E.; Skottfelt, J.; Gillon, M.; Ligier, N.; Maquet, L.; Benedetti-Rossi, G.; Gomes, A.; Ramos Kervella, P. & Monteiro, H. (2014)

*Nature*

Hitherto, rings have been found exclusively around the four giant planets in the Solar System^{1}. Rings are natural laboratories in which to study dynamical processes analogous to those that take place during the formation of planetary systems and galaxies. Their presence also tells us about the origin and evolution of the body they encircle. Here we report observations of a multichord stellar occultation that revealed the presence of a ring system around (10199) Chariklo, which is a Centaur—that is, one of a class of small objects orbiting primarily between Jupiter and Neptune—with an equivalent radius of 124 9 kilometres (ref. 2). There are two dense rings, with respective widths of about 7 and 3 kilometres, optical depths of 0.4 and 0.06, and orbital radii of 391 and 405 kilometres. The present orientation of the ring is consistent with an edge-on geometry in 2008, which provides a simple explanation for the dimming^{3} of the Chariklo system between 1997 and 2008, and for the gradual disappearance of ice and other absorption features in its spectrum over the same period^{4,5}. This implies that the rings are partly composed of water ice. They may be the remnants of a debris disk, possibly confined by embedded, kilometre-sized satellites.

**A statistical model of aggregate fragmentation**

Vieira Neto, E.; Spahn, F.; Guimarães, A. H. F; Gorban, A. N. & Brilliantov, N. V. (2014)

*New Journal of Physics*

A statistical model of fragmentation of aggregates is proposed, based on the stochastic propagation of cracks through the body. The propagation rules are formulated on a lattice and mimic two important features of the process—a crack moves against the stress gradient while dissipating energy during its growth. We perform numerical simulations of the model for two-dimensional lattice and reveal that the mass distribution for small- and intermediate-size fragments obeys a power law, *F*(*m*)∝*m*^{−3/2}, in agreement with experimental observations. We develop an analytical theory which explains the detected power law and demonstrate that the overall fragment mass distribution in our model agrees qualitatively with that one observed in experiments.

doi: http://dx.doi.org/10.1088/1367-2630/16/1/013031

arxiv: https://arxiv.org/pdf/1106.2721.pdf

**A statistical model of aggregate fragmentation**

Vieira Neto, E.; Spahn, F.; Guimarães, A. H. F; Gorban, A. N. & Brilliantov, N. V. (2014)

*New Journal of Physics*

A statistical model of fragmentation of aggregates is proposed, based on the stochastic propagation of cracks through the body. The propagation rules are formulated on a lattice and mimic two important features of the process—a crack moves against the stress gradient while dissipating energy during its growth. We perform numerical simulations of the model for two-dimensional lattice and reveal that the mass distribution for small- and intermediate-size fragments obeys a power law, *F*(*m*)∝*m*^{−3/2}, in agreement with experimental observations. We develop an analytical theory which explains the detected power law and demonstrate that the overall fragment mass distribution in our model agrees qualitatively with that one observed in experiments.

doi: http://dx.doi.org/10.1088/1367-2630/16/1/013031

arxiv: https://arxiv.org/pdf/1106.2721.pdf

**Dynamical evolution of V-type asteroids in the central main belt**

Carruba, V.; Huaman, M.; Domingos, R. C.; Santos, C. R. & Souami, D. (2014)

*Monthly Notices of the Royal Astronomical Society*

V-type asteroids are associated with basaltic composition, and are supposed to be fragments of crust of differentiated objects. Most V-type asteroids in the main belt are found in the inner main belt, and are either current members of the Vesta dynamical family (Vestoids), or past members that drifted away. However, several V-type photometric candidates have been recently identified in the central and outer main belt.

The origin of this large population of V-type objects is not well understood. Since it seems unlikely that Vestoids crossing the 3J:-1A mean-motion resonance with Jupiter could account for the whole population of V-type asteroids in the central and outer main belt, origin from local sources, such as the parent bodies of the Eunomia, and of the Merxia and Agnia asteroid families, has been proposed as an alternative mechanism.

In this work, we investigated the dynamical evolution of the V-type photometric candidates in the central main belt, under the effect of gravitational and non-gravitational forces. Our results show that dynamical evolution from the parent bodies of the Eunomia and Merxia/Agnia families on time-scales of 2 Byr or more could be responsible for the current orbital location of most of the low-inclined V-type asteroids.

doi: http://dx.doi.org/10.1093/mnras/stu192

arxiv: https://arxiv.org/abs/1401.6332

**Peculiar Euphrosyne**

Carruba, V.; Souami, D. & Aljbaae, S. (2014)

*The Astrophysical Journal*

The asteroid (31) Euphrosyne is the largest body of its namesake family, and it contains more than 99% of the family mass. Among large asteroid families, the Euphrosyne group is peculiar because of its quite steep size-frequency distribution (SFD), significantly depleted in large- and medium-sized asteroids (8 < *D* < 12 km). The current steep SFD of the Euphrosyne family has been suggested to be the result of a grazing impact in which only the farthest, smallest members failed to accrete. The Euphrosyne family is, however, also very peculiar because of its dynamics: near its center it is crossed by the ν_{6} = *g* – *g* _{6} linear secular resonance, and it hosts the largest population (140 bodies) of asteroids in ν_{6} antialigned librating states (or Tina-like asteroids) in the main belt. In this work we investigated the orbital evolution of newly obtained members of the dynamical family, with an emphasis on its interaction with the ν_{6} resonance. Because of its unique resonant configuration, large- and medium-sized asteroids tend to migrate away from the family orbital region faster than small-sized objects, which were ejected farther away from the family center. As a consequence, the SFD of the Euphrosyne family becomes steeper in time with a growing depletion in the number of the largest family members. We estimate that the current SFD could be attained from a typical, initial SFD on timescales of 500 Myr, consistent with estimates of the family age obtained with other independent methods.

**Dynamical evolution of V-type photometric candidates in the outer main belt**

Carruba, V.; Huaman, M. & Domingos, R. C. (2014)

*Monthly Notices of the Royal Astronomical Society*

V-type asteroids, characterized by two absorption bands at 1.0 and 2.0 μm, are usually thought to be portions of the crust of differentiated or partially differentiated bodies. Most V-type asteroids are found in the inner main belt and are thought to be current or past members of the Vesta dynamical family. Recently, several V-type photometric candidates have been identified in the central and outer main belt. While the dynamical evolution of V-type photometric candidates in the central main belt has been recently investigated, less attention has been given to the orbital evolution of basaltic material in the outer main belt as a whole. Here, we identify known and new V-type photometric candidates in this region, and study their orbital evolution under the effect of gravitational and non-gravitational forces. A scenario in which a minimum of three local sources, possibly associated with the parent bodies of (349) Dembowska, (221) Eos, and (1459) Magnya, could in principle explain the current orbital distribution of V-type photometric candidates in the region.

doi: http://dx.doi.org/10.1093/mnras/stu1655

arxiv: https://arxiv.org/abs/1408.7080

**A Permanent Magnet Hall Thruster for Pulsed Orbit Control of Lunar Polar Satellites**

Mourão, D. C.; Moraes, B. S.; Ferreira, J. L.; Ferreira, I. S. & Winter, O. C. (2014)

*Journal of Physics*

Future Moon missions devoted to Lunar surface remote sensing, for example, will require very fine and accurate orbit control. It is well known that Lunar satellites in polar orbits will suffer a high increase on the eccentricity due to the gravitational perturbation of the Earth. Without proper orbit correction the satellite lifetime will decrease and end up in a collision with the Moon surface. It is pointed out by many authors that this effect is a natural consequence of the Lidov-Kozai resonance. We studied different arcs of active lunar satellite propulsion, centered on the orbit apoapsis or periapsis, in order to be able to introduce a correction of the eccentricity at each cycle. The proposed method is based on an approach intended to keep the orbital eccentricity of the satellite at low values.

**Effects of the Eccentricity of a Perturbing Third Body on the Orbital Correction Maneuvers of a Spacecraft**

Gomes, V. M..; Domingos, R. C. & Prado, A. F. B. A. (2014)

*Mathematical Problems in Engineering*

The fuel consumption required by the orbital maneuvers when correcting perturbations on the orbit of a spacecraft due to a perturbing body was estimated. The main goals are the measurement of the influence of the eccentricity of the perturbing body on the fuel consumption required by the station keeping maneuvers and the validation of the averaged methods when applied to the problem of predicting orbital maneuvers. To study the evolution of the orbits, the restricted elliptic three-body problem and the single- and double-averaged models are used. Maneuvers are made by using impulsive and low thrust maneuvers. The results indicated that the averaged models are good to make predictions for the orbital maneuvers when the spacecraft is in a high inclined orbit. The eccentricity of the perturbing body plays an important role in increasing the effects of the perturbation and the fuel consumption required for the station keeping maneuvers. It is shown that the use of more frequent maneuvers decreases the annual cost of the station keeping to correct the orbit of a spacecraft. An example of an eccentric planetary system of importance to apply the present study is the dwarf planet Haumea and its moons, one of them in an eccentric orbit.

**Satellite Orbit Determination Using Short Arcs of GPS Data**

Chiaradia, A. P. M..; Kuga, H. K. & Prado, B. Y. P. L. (2014)

*Applied Mechanics and Materials*

This work is concerned with short arcs orbit determination using GPS signals. A special case of truncated arcs assuming that GPS data is only available when the satellite carrying the GPS receiver passes over a ground tracking station is presented. The behaviour of an Extended Kalman filter (EKF) in real time satellite orbit determination using short arcs of data is analysed. The algorithm is a simplified and compact model with low computational cost, and uses the EKF to estimate the state vector, composed of position and velocity components, and GPS receiver clock parameters. The algorithm may use different step-sizes between the GPS signal measurements. Its force model in the motion equations considered the perturbations as being due to the geopotential up to the 10^{th} order and degree of the spherical harmonics. The algorithm has been formerly qualified using raw single frequency pseudorange GPS measurements of the Topex/Poseidon (T/P) satellite, and used as reference in this work. However, the GPS data are truncated as if they had been collected by a single ground tracking station. In other words, the data are obtained only when the satellite T/P is within the viewing area of the station. The research results are presented showing the degradation of performance when compared to a full arc orbit determination.

doi: http://dx.doi.org/10.4028/www.scientific.net/amm.706.206

### 2013

(18 artigos)

**A compound model for the origin of Earth’s water**

Izidoro, A.;Torres, K. S.; Winter, O. C. & Haghighipour, N. (2013)

*The Astrophysical Journal*

One of the most important subjects of debate in the formation of the solar system is the origin of Earth’s water. Comets have long been considered as the most likely source of the delivery of water to Earth. However, elemental and isotopic arguments suggest a very small contribution from these objects. Other sources have also been proposed, among which local adsorption of water vapor onto dust grains in the primordial nebula and delivery through planetesimals and planetary embryos have become more prominent. However, no sole source of water provides a satisfactory explanation for Earth’s water as a whole. In view of that, using numerical simulations, we have developed a compound model incorporating both the principal endogenous and exogenous theories, and investigating their implications for terrestrial planet formation and water delivery. Comets are also considered in the final analysis, as it is likely that at least some of Earth’s water has cometary origin. We analyze our results comparing two different water distribution models, and complement our study using the D/H ratio, finding possible relative contributions from each source and focusing on planets formed in the habitable zone. We find that the compound model plays an important role by showing greater advantage in the amount and time of water delivery in Earth-like planets.

doi: http://dx.doi.org/10.1088/0004-637x/767/1/54

arxiv: https://arxiv.org/abs/1302.1233

**Nebular gas drag and planetary accretion with eccentric high-mass planets**

Chanut, T. G. G.; Winter, O. C. & Tsuchida, M. (2013)

*Astronomy & Astrophysics*

*Aims. *We investigate the dynamics of pebbles immersed in a gas disk interacting with a planet on an eccentric orbit. The model has a prescribed gap in the disk around the location of the planetary orbit, as is expected for a giant planet with a mass in the range of 0.1–1 Jupiter masses. The pebbles with sizes in the range of 1 cm to 3 m are placed in a ring outside of the giant planet orbit at distances between 10 and 30 planetary Hill radii. The process of the accumulation of pebbles closer to the gap edge, its possible implication for the planetary accretion, and the importance of the mass and the eccentricity of the planet in this process are the motivations behind the present contribution.

*Methods. *We used the Bulirsch-Stoer numerical algorithm, which is computationally consistent for close approaches, to integrate the Newtonian equations of the planar (2D), elliptical restricted three-body problem. The angular velocity of the gas disk was determined by the appropriate balance between the gravity, centrifugal, and pressure forces, such that it is sub-Keplerian in regions with a negative radial pressure gradient and super-Keplerian where the radial pressure gradient is positive.

*Results. *The results show that there are no trappings in the 1:1 resonance around the *L*_{4} and *L*_{5} Lagrangian points for very low planetary eccentricities (*e*_{2} < 0.07). The trappings in exterior resonances, in the majority of cases, are because the angular velocity of the disk is super-Keplerian in the gap disk outside of the planetary orbit and because the inward drift is stopped. Furthermore, the semi-major axis location of such trappings depends on the gas pressure profile of the gap (depth) and is *a* = 1.2 for a planet of 1 *M*_{J}. A planet on an eccentric orbit interacts with the pebble layer formed by these resonances. Collisions occur and become important for planetary eccentricity near the present value of Jupiter (*e*_{2} = 0.05). The maximum rate of the collisions onto a planet of 0.1 *M*_{J} occurs when the pebble size is 37.5 cm ≤ *s* < 75 cm; for a planet with the mass of Jupiter, it is15 cm ≤ *s* < 30 cm. The accretion stops when the pebble size is less than 2 cm and the gas drag dominates the motion.
doi: http://dx.doi.org/10.1051/0004-6361/201118629

**Stable regions around Pluto**

Giuliatti Winter, S. M.; Winter, O. C.; Vieira Neto, E. & Sfair, R. (2013)

*Monthly Notices of the Royal Astronomical Society*

In a previous work, Giuliatti Winter et al. found several stable regions for test particles in orbit around Pluto associated with families of periodic orbits obtained in the circular, restricted three-body problem. They have shown that a possible eccentricity of the Pluto–Charon binary slightly reduces but does not destroy any of these stable regions. In this work, we extended their results by analysing the cases with the orbital inclination (*I*) equal to zero and considering the argument of pericentre (ω) equal to 90°, 180° and 270°. We explore the influence of the orbital inclination of the particles in these stable regions. In this case, the initial inclination varies from 10° to 170° in steps of 10°. We also present a sample of results for the longitude of the ascending node Ω = 90°, considering the cases *I* = 20°, 50°, 130° and 180°. Our results show that stable regions are present in all of the inclined cases, except when the initial inclination of the particles is equal to 110°. A sample of 3D trajectories of quasi-periodic orbits were found related to the periodic orbits obtained in the planar case by Giuliatti Winter et al.

**Analysis of 25 mutual eclipses and occultations between the Galilean satellites observed from Brazil in 2009**

Dias-Oliveira, A.; Vieira-Martins, R.; Assafin, M.; Camargo, J. I. B.; Braga-Ribas, F.; Da Silva Neto, D. N.; Gaspar, H. S.; Dos Santos, P. M. P.; Domingos, R. C.; Boldrin, L. A. G.; Izidoro, A.; Carvalho, J.P.S.; Sfair, R.; Sampaio, J. C. & Winter, O. C. (2013)

*Monthly Notices of the Royal Astronomical Society*

The light curves of mutual eclipses and occultations between the natural satellites of a planet allow us to obtain high-precision position and relative motion from differential photometry, enough to detect weak orbital disturbing forces, such as tidal forces. The observations are made during the equinoxes of the planet.

We studied 25 light curves observed in Brazil during the 2009 campaign of the Galilean satellites’ mutual phenomena. A narrow-band filter centred at 890 nm was used, strongly attenuating the Jupiter’s scattered light. We fitted the occultation and eclipse light curves using semi-analytical models that take into account the gradual decrease of light over the shadow, the solar limb darkening and the solar phase angle. The Oren–Nayar reflexive model was used to map the inhomogeneous light scattering on the surface of the satellites. For the first time it is used in a work about mutual events. Here, we include the study that made us decide for this model.

We measured the impact parameter, relative velocity and central instant with average precisions of 7.46 km (2.2 mas), 0.08 km s^{−1} (0.02 mas s^{−1}) and 0.42 s (6.13 km), respectively. The fit precision of the normalized light-curve fluxes ranged between 0.4 and 4.4 per cent.

**Irregular satellites of Jupiter: three-dimensional study of binary-asteroid captures**

Gaspar, H. S.; Winter, O. C. & Vieira Neto, E. (2013)

*Monthly Notices of the Royal Astronomical Society*

Among the hidden pieces of the giant puzzle, which is our Solar system, the origins of irregular satellites of the giant planets stand to be explained, while the origins of regular satellites are well explained by the *in situ* formation model through matter accretion. Once they are not locally formed, the most acceptable theory predicts that they had been formed elsewhere and became captured later, most likely during the last stage of planet formation. However, under the restricted three-body problem theory, captures are temporary and there is still no assisted capture mechanism which is well established. In a previous work, we showed that the capture mechanism of a binary asteroid under the co-planar four-body scenario yielded permanent captured objects with an orbital shape which is very similar to those of the actual prograde irregular Jovian satellites. By extending our previous study to a 3D case, here we demonstrate that the capture mechanism of a binary asteroid can produce permanent captures of objects by itself which have very similar orbits to irregular Jovian satellites. Some of the captured objects without aid of gas drag or other mechanisms present a triplet: semi-major axis, eccentricity and inclination, which is comparable to the already known irregular Jovian objects.

**Powered Swing-By Maneuvers around the Moon**

Silva, A. F.; Prado, A. F. B. A. & Winter, O. C. (2013)

*Journal of Physics. Conference Series*

A Swing-By maneuver occurs when a satellite approaches a celestial body to gain or lose energy from its gravitational field. The present work studies Swing-By maneuvers that are combined with the use of an impulsive thrust in different directions during the passage of the spacecraft by the periapsis of its trajectory around the Moon. The main objective of this type of maneuver is the fuel economy for orbital transfers. From the results, it is visible that the best direction to apply the impulse is not the direction to the motion of the spacecraft, as might be expected. In fact, by using a different direction, it is possible to maximize the effects of the Swing-By by decreasing the periapsis distance and/or increasing the turning angle of the maneuver, that are the key parameters to specify the variation of energy due to the Swing-By. The changes in the periapsis distance and turning angle cause modifications in the geometry of the original Swing-By, generating a maneuver with new parameters. This new Swing-By compensates for the loss of energy transfer that results of applying the impulse in a in a non-tangential direction.

**Analysis of the secular problem for triple star systems**

Carvalho, J. P. S.; De Moraes, R. Vilhena; Prado, A. F. B. A. & Winter, O. C. (2013)

*Journal of Physics. Conference Series*

The long-term dynamics of the three-body problem is studied. The goal is to study the motion of a planet (*m*_{1}) around a star (*m*_{0}) that is perturbed by a third-body (*m*_{2}) (a planet or a brown dwarf star). The gravitational potential is developed in closed form up to the fourth order. Taking into account the triple system, it is shown here the evolution of some orbital parameters of the planet (*m*_{1}). A comparison considering models with different orders for the disturbing potential is presented. We show that the behavior of the orbit of the inner planet can flip from prograde to retrograde trajectories. This is due to the third-order term, which strongly affects the eccentricity and inclination. We show that the effect of the fourth order term is to change the times when the phenomenon occurs.

**Exploring sensitive dependence and transitivity to optimize travel time in chaotic systems**

Santana, S. H. S.; Macau, E. E. N.; Winter, O. C. & França, L. F. A. (2013)

*Journal of Physics. Conference Series*

Transitivity and sensitive dependence on initial conditions are the main characteristics of chaotic behavior. The latter one can be exploited so that small controlled perturbations in system parameters may imply a faster transfer in time from a desired start point to a neighborhood of a desired final state. In this study three targeting approaches are evaluated: The first one uses a geometric approach to find the proper perturbation which allows a faster transfer between two desired points; The second, an evolutionary algorithm called GEO (Generalized External Optimization), is adapted to search for optimized orbits; The third one, uses successive perturbations along the path in order to direct the orbits to the final desired point in a short time interval. These three methods are evaluated regarding performance and implementation complexity.

**Small particles in Pluto’s environment: effects of the solar radiation pressure**

Santos, P. M. S.; Giuliatti Winter, S. M.; Sfair, R. & Mourão, D. C. (2013)

*Monthly Notices of the Royal Astronomical Society *

Impacts of micrometeoroids on the surfaces of the plutonian small satellites Nix and Hydra can generate dust particles. Even in this region so far from the Sun these tiny ejected particles are under the effects of the solar radiation pressure.

In this work, we investigate the orbital evolution of the escaping ejecta from both the small satellites under the effects of the radiation pressure combined with the gravitational effects of Pluto, Charon, Nix and Hydra. The mass production rate of micron-sized dust particles generated by micrometeoroids hitting the satellites is obtained, and numerical simulations are performed to derive the lifetime of the ejecta. These pieces of information allow us to estimate the optical depth of a putative ring, which extends from the orbits of Nix to Hydra.

The ejected particles, between the orbits of Nix and Hydra, form a wide ring of about 16 000 km. Collisions with the massive bodies and escape from the system are mainly determined by the effects of the solar radiation pressure. This is an important loss mechanism, removing 30 per cent of the initial set of 1 μm-sized particles in 1 yr. The surviving particles form a ring too faint to be detectable with the derived maximum optical depth of 4 × 10^{−11}.

doi: http://dx.doi.org/10.1093/mnras/stt076

arxiv: https://arxiv.org/abs/1108.0712

**Mathematical Methods Applied to the Celestial Mechanics of Artificial Satellites 2013**

Prado, A. F. B. A.; Masdemont, J. J.; Zanardi, M. C.; Giuliatti Winter, S. M. & Yokoyama, Tadashi (2013)

*Mathematical Problems in Engineering *

This article has no abstract.

**A study of low velocities regions near the Roche lobe during the gas giant planets formation**

Moraes, R. A. & Vieira Neto, E. (2013)

*Journal of Physics. Conference Series*

In this paper were studied regions close to the Roche lobe of a planet like Jupiter, in order to find regions with low velocities. We simulated a two dimensional and non-self-gravitating disk, where tidal and viscous torques are considered, using the hydrodynamic numerical integrator FARGO 2D. As stated earlier we are interested in find low velocities regions for in future works study the possibility of satellites formation in these regions.

**Chaotic diffusion caused by close encounters with several massive asteroids. II. The regions of (10) Hygiea, (2) Pallas, and (31) Euphrosyne**

Carruba, V.; Huaman, M.; Domingos, R. C. & Roig, F. (2013)

*Astronomy & Astrophysics*

*Context.* Close encounters with (1) Ceres and (4) Vesta, the two most massive bodies in the main belt, are known to be a mechanism of dynamical mobility able to significantly alter proper elements of minor bodies, and they are the main source of dynamical mobility for medium-sized and large asteroids (*D* > 20 km, approximately). Recently, it has been shown that drift rates caused by close encounters with massive asteroids may change significantly on timescales of 30 Myr when different models (i.e., different numbers of massive asteroids) are considered.

*Aims.* So far, not much attention has been given to the case of diffusion caused by the other most massive bodies in the main belt: (2) Pallas, (10) Hygiea, and (31) Euphrosyne, the third, fourth, and one of the most massive highly inclined asteroids in the main belt, respectively. Since (2) Pallas is a highly inclined object, relative velocities at encounter with other asteroids tend to be high and changes in proper elements are therefore relatively small. It was thus believed that the scattering effect caused by highly inclined objects in general should be small. Can diffusion by close encounters with these asteroids be a significant mechanism of long-term dynamical mobility?

*Methods.* By performing simulations with symplectic integrators, we studied the problem of scattering caused by close encounters with (2) Pallas, (10) Hygiea, and (31) Euphrosyne when only the massive asteroids (and the eight planets) are considered, and the other massive main belt asteroids and non-gravitational forces are also accounted for.

*Results.* By finding relatively small values of drift rates for (2) Pallas, we confirm that orbital scattering by this highly inclined object is indeed a minor effect. Unexpectedly, however, we obtained values of drift rates for changes in proper semi-major axis *a*caused by (10) Hygiea and (31) Euphrosyne larger than what was previously found for scattering by (4) Vesta. These high rates may have repercussions on the orbital evolution and age estimate of their respective families.

**An analysis of the Hygiea asteroid family orbital region**

Carruba, V. (2013)

*Monthly Notices of the Royal Astronomical Society *

(10) Hygiea is the fourth largest asteroid of the main belt, by volume and mass, and it is the largest member of its family, that is made mostly by low-albedo, C-type asteroids, typical of the outer main belt. Like many other large families, it is associated with a ‘halo’ of objects, that extends far beyond the boundary of the core family, as detected by traditional hierarchical clustering methods (HCM) in proper element domains. Numerical simulations of the orbital evolution of family members may help in estimating the family and halo family age, and the original ejection velocity field. But, in order to minimize the errors associated with including too many interlopers, it is important to have good estimates of family membership that include available data on local asteroid taxonomy, geometrical albedo and local dynamics.

For this purpose, we obtained synthetic proper elements and frequencies of asteroids in the Hygiea orbital region, with their errors. We revised the current knowledge on asteroid taxonomy, including Sloan Digital Sky Survey-Moving Object Catalog 4th release (SDSS-MOC 4) data, and geometric albedo data from *Wide-field Infrared Survey Explorer* (*WISE*) and *Near-Earth Object WISE* (*NEOWISE*). We identified asteroid family members using HCM in the domain of proper elements (*a*, *e*, sin (*i*)) and in the domains of proper frequencies most appropriate to study diffusion in the local web of secular resonances, and eliminated possible interlopers based on taxonomic and geometrical albedo considerations. To identify the family halo, we devised a new hierarchical clustering method in an extended domain that includes proper elements, principal components *PC*_{1}, *PC*_{2}obtained based on SDSS photometric data and, for the first time, *WISE* and *NEOWISE* geometric albedo. Data on asteroid size distribution, light curves and rotations were also revised for the Hygiea family.

The Hygiea family is the largest group in its region, with two smaller families in proper element domain and 18 families in various frequencies domains identified in this work for the first time. Frequency groups tend to extend vertically in the (*a*, sin (*i*)) plane and cross not only the Hygiea family but also the near C-type families of Themis and Veritas, causing a mixture of objects all of relatively low albedo in the Hygiea family area. A few high-albedo asteroids, most likely associated with the Eos family, are also present in the region. Finally, the new multidomains hierarchical clustering method allowed us to obtain a good and robust estimate of the membership of the Hygiea family halo, quite separated from other asteroids families halo in the region, and with a very limited (about 3 per cent) presence of likely interlopers.

**A multidomain approach to asteroid families’ identification**

Carruba, V.; Domingos, R. C.; Nesvorný, D.; Roig, F.; Huaman, M. & Souami, D. (2013)

*Monthly Notices of the Royal Astronomical Society *

It has been shown that large families are not limited to what found by hierarchical clustering methods in the domain of proper elements (*a*, *e*, sin (*i*)), which seems to be biased to find compact, relatively young clusters, but that there exists an extended population of objects with similar taxonomy and geometric albedo, which can extend to much larger regions in proper elements and frequencies domains: the family ‘halo’. Numerical simulations can be used to provide estimates of the age of the family halo, which can then be compared with ages of the family obtained with other methods. Determining a good estimate of the possible orbital extension of a family halo is therefore quite important, if one is interested in determining its age and, possibly, the original ejection velocity field. Previous works have identified families’ haloes by an analysis in proper elements domains, or by using Sloan Digital Sky Survey-Moving Object Catalog data, fourth release (SDSS-MOC4) multiband photometry to infer the asteroid taxonomy, or by a combination of the two methods. The limited number of asteroids for which geometric albedo was known until recently discouraged in the past the extensive use of this additional parameter, which is however of great importance in identifying an asteroid taxonomy. The new availability of geometric albedo data from the *Wide-field Infrared Survey Explorer* (*WISE*) mission for about 100 000 asteroids significantly increased the sample of objects for which such information, with some errors, is now known.

In this work, we proposed a new method to identify families’ haloes in a multidomain space composed by proper elements, SDSS-MOC4 (*a**, *i* − *z*) colours, and *WISE* geometric albedo for the whole main belt (and the Hungaria and Cybele orbital regions). Assuming that most families were created by the breakup of an undifferentiated parent body, they are expected to be homogeneous in colours and albedo. The new method is quite effective in determining objects belonging to a family halo, with low percentages of likely interlopers, and results that are quite consistent in term of taxonomy and geometric albedo of the halo members.

doi: http://dx.doi.org/10.1093/mnras/stt884

arxiv: https://arxiv.org/abs/1305.4847?context=astro-ph

**Dynamical evolution and chronology of the Hygiea asteroid family**

Carruba, V.; Domingos, R. C.; Huaman, M.; Santos, C. R. & Souami, D. (2013)

*Monthly Notices of the Royal Astronomical Society *

The asteroid (10) Hygiea is the fourth largest asteroid of the main belt, by volume and mass, and it is the largest member of its own family. Previous works investigated the long-term effects of close encounters with (10) Hygiea of asteroids in the orbital region of the family, and analysed the taxonomical and dynamical properties of members of this family. In this paper we apply the high-quality Sloan Digital Sky Survey-Moving Object Catalog data, fourth release (SDSS-MOC4) taxonomic scheme of DeMeo & Carry to members of the Hygiea family core and halo, we obtain an estimate of the minimum time and number of encounter necessary to obtain a 3σ (or 99.7 per cent) compatible frequency distribution function of changes in proper *a* caused by close encounters with (10) Hygiea, we study the behaviour of asteroids near secular resonance configurations, in the presence and absence of the Yarkovsky force, and obtain a first estimate of the age of the family based on orbital diffusion by the Yarkovsky and Yarkovsky–O’Keefe–Radzievsky–Paddack (YORP) effects with two methods.

The Hygiea family is at least 2 Byr old, with an estimated age of

“>T=3200+380−120T=3200−120+380

Myr and a relatively large initial ejection velocity field, according to the approach of Vokrouhlický et al. Surprisingly, we found that the family age can be shortened by ≃25 per cent if the dynamical mobility caused by close encounters with (10) Hygiea is also accounted for, which opens interesting new research lines for the dynamical evolution of families associated with massive bodies. In our taxonomical analysis of the Hygiea asteroid family, we also identified a new V-type candidate: the asteroid (177904) (2005 SV5). If confirmed, this could be the fourth V-type object ever to be identified in the outer main belt.

doi: http://dx.doi.org/10.1093/mnras/stt2040

arxiv: https://arxiv.org/abs/1310.5982

**Studying Close Approaches for a Cloud of Particles Considering Atmospheric Drag**

Gomes, V. M.; Formiga, J. K. S. & Moraes, R. V. (2013)

*Mathematical Problems in Engineering *

The present paper has the goal of studying close approaches between a planet and a group of particles. The mathematical model includes the presence of the atmosphere of the planet. This cloud is assumed to be created by the passage of the spacecraft in the atmosphere of the planet, which can cause the explosion of the spacecraft. The system is assumed to be formed by the Sun, the planet, and the spacecraft that explodes and becomes a cloud of particles. The Sun and the planet are assumed to be in circular orbits and the motion is planar. The equations of motion are the ones given by the circular planar restricted three-body problem combined with the forces given by the atmospheric drag. In the numerical simulations, the planet Jupiter is the celestial body used for the close approaches. The initial positions and velocities of the spacecraft and the particles are specified at the periapsis, because it is assumed that this is the point where the explosion occurs.

**Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth**

Gomes, V. M.; Prado, A. F. B. A. & Golebiewska, J. (2013)

*The Scientific World Journal*

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

**Onboard and Real-Time Artificial Satellite Orbit Determination Using GPS**

Chiaradia, A. P. M.; Kuga, H. K. & Prado, A. F. B. A. (2013)

*Mathematical Problems in Engineering*

An algorithm for real-time and onboard orbit determination applying the Extended Kalman Filter (EKF) method is developed. Aiming at a very simple and still fairly accurate orbit determination, an analysis is performed to ascertain an adequacy of modeling complexity versus accuracy. The minimum set of to-be-estimated states to reach the level of accuracy of tens of meters is found to have at least the position, velocity, and user clock offset components. The dynamical model is assessed through several tests, covering force model, numerical integration scheme and step size, and simplified variational equations. The measurement model includes only relevant effects to the order of meters. The EKF method is chosen to be the simplest real-time estimation algorithm with adequate tuning of its parameters. In the developed procedure, the obtained position and velocity errors along a day vary from 15 to 20 m and from 0.014 to 0.018 m/s, respectively, with standard deviation from 6 to 10 m and from 0.006 to 0.008 m/s, respectively, with the SA either on or off. The results, as well as analysis of the final adopted models used, are presented in this work.