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Satellite formation establishment may consume considerable fuel if cross-track distance (CTD) is a mission requirement. This work proposes a strategy for significantly reducing the required fuel. I...

Deployment of multi-plane satellite constellations has become a fresh challenge in the space program missions since all the direct solutions including plane changing maneuvers and separate launches are energy and time-consuming. A proposed solution to this challenge is launching several satellites simultaneously and deploying them using gravitational perturbations of the Earth. Utilizing this solutions, only in-plane satellite maneuvers are sufficient to deploy a constellation of satellites in several orbital planes. The aim of this study is to extend and develop the present methods that use the concept of nodal precession to deploy a multi-plane satellite constellation with only one launch in order to cover a wider range of mission specifications, provide mathematical expressions for the total deployment time and the fuel expenditure of these methods, optimize the methods multi-objectively and provide a thorough analysis and comparison between them. Two general methods are proposed in this study; one that places all the satellites in a parking orbit and injects them into their final orbits sequentially, and one that first places them in different drifting orbits and after a certain amount of time, maneuvers them to their final orbits. The findings of the optimization process suggest that the method which involves all the satellites in the process of deployment simultaneously and by this means exploits the full nodal precessing potential of all of them provides better results in comparison with the method with a more passive approach, which considers the deployment a sequential process. Finally, since this mathematical modeling is performed for the first time in the literature, the mathematical expressions for both methods are analyzed to provide beneficial insights for the designers.

A new analytical model (R-toroid), representing a 3D generalization of the precessing Gaussian ring, is constructed for the study of secular perturbations in celestial mechanics. Our approach is based on triple averaging of the motion of a material point and is reduced to a chain of transformations: 1D Gaussian ring–2D R-ring–3D R-toroid. The figure, structure and gravitational potential of the R-toroid are studied. We obtained the expression for the mutual energy of the R-toroid and the outer Gaussian ring to study the motion of bodies in the gravitational field of the model in two forms (in the integral and in the form of a power-law series). Two equation systems of the secular evolution of osculating orbits (Gaussian rings), in the gravitational field of an R-toroid and in the field of a central precessing star, are derived using the mutual energy. The periods of nodal TΩ and apsidal Tω orbital precession were obtained. Examples of three hot Jupiters with a known period of nodal precession are considered. For the exoplanet Kepler-413b, the R-toroid describes the evolution of any orbit with a ≥ 5.48 AU, and for the exoplanet PTFO 8-8695b, the critical value of the semi-major axis turned out to be only amin ≈ 0.2 AU. The frequency profile of the precession of the test orbit in the field of the star and planet PTFO 8-8695b has been calculated. The minimum value of the period of nodal precession was TΩ ≈ (26.1 ± 3.0) × 103 years.

The problem of the secular evolution of a thin ring around a rapidly rotating triaxial celestial body is formulated and solved. The technology for calculating secular perturbations is based on two formulas: the azimuthally averaged force field of the central body and the mutual energy $${{W}_{{{\text{mut}}}}}$$ of this body and a Gaussian ring. With $${{W}_{{{\text{mut}}}}}$$ instead of the usual perturbing function, a system of differential equations for the osculating elements of the ring is obtained. An equation is obtained that allows one to find the coefficients of the zonal harmonics of the azimuthally averaged potential of an inhomogeneous ellipsoid using a unified scheme. The method is applied to dwarf planet Haumea with refined masses of the rocky core and the ice shell and the coefficients $${{C}_{{20}}}$$ and $${{C}_{{40}}}$$ of the po-tential’s zonal harmonics. According to new data, the ring around Haumea has a slight obliquity and must precess. It was established that the period of the retrograde nodal precession of the Haumea’s ring (without regard to self-gravity) is $${{T}_{\Omega }} = 12.9 \pm 0.7$$ days and the period of the forward of the apside line precession is $${{T}_{\omega }} \approx 8.{\text{08}}\;{\text{days}}$$ . It is proven that the 3:1 orbital resonance for the particles of the Haumea’s ring is fulfilled only approximately and the averaging time of additional perturbations at a nonsharp resonance turned out to be an order of magnitude smaller than $${{T}_{\Omega }}$$ . This confirms the adequacy of the method.

$\pi$ Men hosts a transiting super Earth ($P\approx6.27$ d, $m\approx4.82$ $M_{\oplus}$, $R\approx2.04$ $R_{\oplus}$) discovered by TESS and a cold Jupiter ($P\approx2093$ d, $m \sin I\approx10.02$ $M_{\rm{Jup}}$, $e\approx0.64$) discovered from radial velocity. We use Gaia DR2 and Hipparcos astrometry to derive the star's velocity caused by the orbiting planets and constrain the cold Jupiter's sky-projected inclination ($I_b=41-65^{\circ}$). From this we derive the mutual inclination ($\Delta I$) between the two planets, and find that $49^{\circ}< \Delta I < 131^{\circ}$ (1$\sigma$), and $28^{\circ} < \Delta I < 152^{\circ}$ (2$\sigma$). We examine the dynamics of the system using $N$-body simulations, and find that potentially large oscillations in the super Earth's eccentricity and inclination are suppressed by general relativistic precession. However, nodal precession of the inner orbit around the invariable plane causes the super Earth to only transit between 7-22 per cent of the time, and to usually be observed as misaligned with the stellar spin axis. We repeat our analysis for HAT-P-11, finding a large $\Delta I$ between its close-in Neptune and cold Jupiter and similar dynamics. $\pi$ Men and HAT-P-11 are prime examples of systems where dynamically hot outer planets excite their inner planets, with the effects of increasing planet eccentricities, planet-star misalignments, and potentially reducing the transit multiplicity. Formation of such systems likely involves scattering between multiple giant planets or misaligned protoplanetary discs. Future imaging of the faint debris disc in $\pi$ Men and precise constraints on the stellar spin orientation would provide strong tests for these formation scenarios., Comment: Corrected Fig 7 and Sec 6.1.1 (HAT-P-11 also weak planet-star coupling). Published as MNRAS 497, 2096-2118 (2020)

An analytical method is developed for finding the coefficients of zonal spherical harmonics of the azimuthally averaged potential of a rotating stratified inhomogeneous triaxial ellipsoid. Two models are considered: i) an ellipsoid of discrete layers of finite thickness, including the two-component Core – Shell model; ii) an inhomogeneous ellipsoid consisting of infinitely thin equidensity layers with arbitrary elliptic profiles from the center to the periphery. For both types of models, equations were obtained that allow calculating the coefficients of zonal spherical harmonics of any degree according to a single scheme. Along with the general stratification, special cases of ellipsoids consisting of homeoids and focaloids were also considered. It is proved that for confocal stratification the coefficients of zonal spherical harmonics of homogeneous and congruent inhomogeneous ellipsoids coincide. The method is applied to the construction of a near-equilibrium model of the dwarf planet Haumea. It has been established that the nodal precession period of the Haumea’s ring is $T_{\mathit{prec}} = 12.9 \pm 0.7~\text{days}$ .

Satellite relative motion around the Earth has been thoroughly studied during the last two decades. However, considerably less attention has been given to the study of satellite relative motion around Mars. As the cost of space technologies decreases and more space missions are within reach, formation flying missions around Mars have the potential to benefit future exploration missions launched to the Red Planet. A key parameter in such missions will be the frequency at which the spacecraft need to perform formation-keeping maneuvers to compensate for unwanted drifts due to differential perturbations. The Martian $$J_3$$ and $$J_4$$ gravitational harmonics are significant enough to warrant a dedicated investigation of bounded satellite relative motion configurations. In this study, we derive conditions for bounded satellite relative motion in non-critical inclinations around Mars, while considering its gravitational harmonics up to $$J_4$$ . We first introduce a family of stable frozen orbits facilitating the implementation of formation flying and then apply differential nodal precession negation and differential periapsis rotation negation methods while considering the gravitational harmonics up to $$J_4$$ . Using this procedure, we demonstrate how the secular growth of the relative distance can be arrested during long time intervals.

Even sub-millimeter-size debris could cause a fatal damage on a spacecraft. Such tiny debris cannot be followed up or tracked from the ground. Therefore, Kyushu University has initiated IDEA the project for In-situ Debris Environmental Awareness, which conducts in-situ measurements of sub-millimeter-size debris. One of the objectives is to estimate the location of on-orbit satellite fragmentations from in-situ measurements. The previous studies revealed that it is important to find out the right nodal precession rate to estimate the orbital parameters of a broken-up object properly. Therefore, this study derives a constraint equation that applies to the nodal precession rate of the broken-up object. This study also establishes an effective procedure to estimate properly the orbital parameters of a broken-up object with the constraint equation.

The discovery of numerous circumprimary planets in the last few years has brought to the fore the question of planet formation in binary systems. The significant dynamical influence, during the protoplanetary disk phase, of a binary companion on a giant planet has previously been highlighted for wide binary stars. In particular, highly inclined binary companion can induce perturbations on the disk and the planets, through the Lidov-Kozai resonance, which could inhibit the formation process. In this work, we aim to study how the disk gravitational potential acting on the planet and the nodal precession \textbf{induced by the wide binary companion with separation of 1000 AU} on the disk act to suppress the Lidov-Kozai perturbations on a migrating giant planet. We derive new approximate formulas for the evolution of the disk's inclination and longitude of the ascending node, in the case of a rigidly precessing disk with a decreasing mass and perturbed by a wide binary companion, which are suitable for $N$-body simulations. We carry out 3200 simulations with several eccentricity and inclination values for the binary companion. The gravitational and damping forces exerted by the disk on the planet tend to keep the latter in the midplane of the former, and suppress the effect of the binary companion by preventing the planet from getting locked in the Lidov-Kozai resonance during the disk phase. We also confirm that because of nodal precession induced by the binary, a primordial spin-orbit misalignment could be generated for circumprimary planets with an inclined binary companion., Comment: 11 pages, 10 figures, to be published in MNRAS

Carri, B., et al., [Context] Dynamical models of Solar System evolution have suggested that the so-called P- and D-type volatile-rich asteroids formed in the outer Solar System beyond Neptune's orbit and may be genetically related to the Jupiter Trojans, comets, and small Kuiper belt objects (KBOs). Indeed, the spectral properties of P- and D-type asteroids resemble that of anhydrous cometary dust. Aims. We aim to gain insights into the above classes of bodies by characterizing the internal structure of a large P- and D-type asteroid. [Methods] We report high-angular-resolution imaging observations of the P-type asteroid (87) Sylvia with the Very Large Telescope Spectro-Polarimetric High-contrast Exoplanet REsearch (SPHERE) instrument. These images were used to reconstruct the 3D shape of Sylvia. Our images together with those obtained in the past with large ground-based telescopes were used to study the dynamics of its two satellites. We also modeled Sylvia's thermal evolution. [Results] The shape of Sylvia appears flattened and elongated (a/b 1.45; a/c 1.84). We derive a volume-equivalent diameter of 271 ± 5 km and a low density of 1378 ± 45 kg m-3. The two satellites orbit Sylvia on circular, equatorial orbits. The oblateness of Sylvia should imply a detectable nodal precession which contrasts with the fully-Keplerian dynamics of its two satellites. This reveals an inhomogeneous internal structure, suggesting that Sylvia is differentiated. [Conclusions] Sylvia's low density and differentiated interior can be explained by partial melting and mass redistribution through water percolation. The outer shell should be composed of material similar to interplanetary dust particles (IDPs) and the core should be similar to aqueously altered IDPs or carbonaceous chondrite meteorites such as the Tagish Lake meteorite. Numerical simulations of the thermal evolution of Sylvia show that for a body of such a size, partial melting was unavoidable due to the decay of long-lived radionuclides. In addition, we show that bodies as small as 130-150 km in diameter should have followed a similar thermal evolution, while smaller objects, such as comets and the KBO Arrokoth, must have remained pristine, which is in agreement with in situ observations of these bodies. NASA Lucy mission target (617) Patroclus (diameter ≈140 km) may, however, be differentiated., This work has been supported by the Czech Science Foundation through grants 20-08218S (J. Hanuš, J. Ďurech) and by the Charles University Research program No. UNCE/SCI/023. The work of TSR was carried out through grant APOSTD/2019/046 by Generalitat Valenciana (Spain). This work was supported by the MINECO (Spanish Ministry of Economy) through grant RTI2018-095076-B-C21 (MINECO/FEDER, UE). This material is partially based upon work supported by the National Science Foundation under Grant No. 1743015.

To fulfill its operational function, the Sun-synchronous satellite needs to have a specific nodal precession value of 360°/year. This nodal precession value is mainly affected by semi-major axis, eccentricity, and inclination. However, in fact, semi-major axis and inclination of an orbit often drift over a long period of operation. Thus, the nodal precession of the orbit will be no longer 360°/year. This paper discusses the simulation of inclination maneuver to correct the nodal precession of Sun-synchronous satellite by using a non-impulsive thruster approach. Data from LAPAN Sun-synchronous satellite was used for simulation. The results of the simulation show that there are no significant changes to other orbital parameters for inclination maneuver for every 1, 2, 3, 4, and 6 months. Also, the results of the simulation are compared to the analytical results (impulsive thruster approach), which turns out that the analytical approach still gives a good accuracy of inclination changes for that maneuver interval with accuracy up to 0.001°. The results can be used as a recommendation for the thruster operation of the next LAPAN Sun-synchronous satellite.

The dawn of "New Space" in recent years is changing the landscape of the space industry. In particular, the shift to smaller satellites, requiring shorter development time s and using off-the-shelf-components and standardized buses, has led to a continuing reduction in spacecraft cost. However, launch costs remain extremely high and frequently dominate the total mission cost. Additionally, many small satellites are designed to operate as part of a larger constellation, but traditional launch methods require a difference dedicated launch for each orbit plane to be populated. This need for multiple costly launches can stifle, and even prohibit, some missions requiring numerous orbit planes as the launch cost increases beyond what can be justified for the mission. As of 2014, most smallsats, including CubeSats, have been launched on opportunistic ‘rideshare’ or ‘piggy-back’ launches, in which the spacecraft shares its launch with other craft, often as a secondary payload. This has the advantage of providing a cheaper launch but restricts the operator’s choice of orbit, which will affect the system performance.

We investigate the formation mechanism for the observed nearly polar aligned (perpendicular to the binary orbital plane) debris ring around the eccentric orbit binary 99 Herculis. An initially inclined nonpolar debris ring or disc will not remain flat and will not evolve to a polar configuration, due to the effects of differential nodal precession that alter its flat structure. However, a gas disc with embedded well coupled solids around the eccentric binary may evolve to a polar configuration as a result of pressure forces that maintain the disc flatness and as a result of viscous dissipation that allows the disc to increase its tilt. Once the gas disc disperses, the debris disc is in a polar aligned state in which there is little precession. We use three-dimensional hydrodynamical simulations, linear theory, and particle dynamics to study the evolution of a misaligned circumbinary gas disc and explore the effects of the initial disc tilt, mass, and size. We find that for a wide range of parameter space, the polar alignment timescale is shorter than the lifetime of the gas disc. Using the observed level of alignment of 3 deg. from polar, we place an upper limit on the mass of the gas disc of about 0.014 M_sun at the time of gas dispersal. We conclude that the polar debris disc around 99 Her can be explained as the result of an initially moderately inclined gas disc with embedded solids. Such a disc may provide an environment for the formation of polar planets., Comment: 13 pages, 12 figures, accepted for publication in MNRAS

When a star passes close to a supermassive black hole (BH), the BH's tidal forces rip it apart into a thin stream, leading to a tidal disruption event (TDE). In this work, we study the post-disruption phase of TDEs in general relativistic hydrodynamics (GRHD) using our GPU-accelerated code H-AMR. We carry out the first grid-based simulation of a deep-penetration TDE ($\beta$=7) with realistic system parameters: a black-hole-to-star mass ratio of $10^6$, a parabolic stellar trajectory, and a nonzero BH spin. We also carry out a simulation of a tilted TDE whose stellar orbit is inclined relative to the BH midplane. We show that for our aligned TDE, an accretion disk forms due to the dissipation of orbital energy with $\sim$20 percent of the infalling material reaching the BH. The dissipation is initially dominated by violent self-intersections and later by stream-disk interactions near the pericenter. The self-intersections completely disrupt the incoming stream, resulting in five distinct self-intersection events separated by approximately 12 hours and a flaring in the accretion rate. We also find that the disk is eccentric with mean eccentricity e$\approx$0.88. For our tilted TDE, we find only partial self-intersections due to nodal precession near pericenter. Although these partial intersections eject gas out of the orbital plane, an accretion disk still forms with a similar accreted fraction of the material to the aligned case. These results have important implications for disk formation in realistic tidal disruptions. For instance, the periodicity in accretion rate induced by the complete stream disruption may explain the flaring events from Swift J1644+57., Comment: Accepted to MNRAS on November 23rd, 2021, 23 pages, 25 figures, uses mnras.cls. Comments welcome. Movies available at https://www.youtube.com/playlist?list=PL7YbfRC6zxzAqIPXWbJgXpr4Wq_nwHiDu

Black hole and neutron star X-ray binary systems routinely show quasi-periodic oscillations (QPOs) in their X-ray flux. Despite being strong, easily measurable signals, their physical origin has long remained elusive. However, recent observational and theoretical work has greatly improved our understanding. Here, we briefly review the basic phenomenology of the different varieties of QPO in both black hole and neutron star systems before focusing mainly on low frequency QPOs in black hole systems, for which much of the recent progress has been made. We describe the detailed statistical properties of these QPOs and review the physical models proposed in the literature, with particular attention to those based on Lense-Thirring precession. This is a relativistic effect whereby a spinning massive object twists up the surrounding spacetime, inducing nodal precession in inclined orbits. We review the theory describing how an accretion flow reacts to the Lense-Thirring effect, including analytic theory and recent numerical simulations. We then describe recent observational tests that provide very strong evidence that at least a certain type of low frequency QPOs are a geometric effect, and good evidence that they are the result of precession. We discuss the possibility of the spin axis of the compact object being misaligned with the binary rotation axis for a large fraction of X-ray binaries, as is required for QPOs to be driven specifically by Lense-Thirring precession, as well as some outstanding gaps in our understanding and future opportunities provided by X-ray polarimeters and/or high throughput X-ray detectors., Comment: Accepted for publication in New Astronomy Reviews; 67 pages, 22 figures

The Space Propulsion Laboratory (SPL) of the Massachusetts Institute of Technology (MIT) is developing the Ion Electrospray Propulsion System (iEPS), designed to address a current need in CubeSat technology: miniaturized electric thrusters. These could be used for different applications, ranging from attitude control to interplanetary flights. In this work, performed together with the Space Systems Laboratory of the Costa Rica Institute of Technology (SETEC Lab), we explore a case study in which the iEPS is used for constellation management in Low Earth Orbit (LEO) when integrated in a 3U CubeSat. We analyze how a 180° separation in the Right Ascension of the Ascending Node (RAAN) between two CubeSats (SatA and SatB) starting in the same orbit can be achieved by modifying one of the spacecraft's orbital altitude, resulting in a difference in their rate of nodal precession (defined as the drift rate) due to the J 2 effect, and therefore a difference in their relative RAAN. The method consists of SatB increasing its semi-major axis, drifting in a higher orbit with a lower drift rate, and returning to the original semi-major axis once the desired difference in RAAN in achieved relative to the other spacecraft. SatA will stay in its original orbit, using its thruster to compensate for orbital energy loss due to atmospheric drag, therefore demonstrating another application of iEPS for constellation management. Three different simulations were studied, defined as the minimum time trajectory, minimum propellant trajectory and a hybrid trajectory, consisting of reaching a higher altitude orbit, but actively changing the RAAN using the propulsion system instead of drifting. It was observed that the difference in this orbital element could be achieved using 85 g of propellant in as little as 164 days for the minimum time trajectory. The same difference could also be achieved using only 44 g of propellant in 245 days for the minimum propellant trajectory. Furthermore, the results of the hybrid trajectory showed that the goal could be achieved in 161 days, but using 158 g of propellant mass, demonstrating the benefit of using a drift orbit. The results proved the feasibility of implementing iEPS for constellation management using 3U CubeSats in LEO.

Finding satellite relative orbits that are resilient to differential gravitational perturbations has received much attention in the literature. In particular, detecting “invariant” relative orbits ...

Many protostellar gapped and binary discs show misalignments between their inner and outer discs; in some cases, $\sim70$ degree misalignments have been observed. Here we show that these misalignments can be generated through a "secular precession resonance" between the nodal precession of the inner disc and the precession of the gap-opening (stellar or massive planetary) companion. An evolving protostellar system may naturally cross this resonance during its lifetime due to disc dissipation and/or companion migration. If resonance crossing occurs on the right timescale, of order a few Myrs, characteristic for young protostellar systems, the inner and outer discs can become highly misaligned ($\gtrsim 60$ degrees). When the primary star has a mass of order a solar mass, generating a significant misalignment typically requires the companion to have a mass of $\sim 0.01-0.1$ M$_\odot$ and an orbital separation of tens of AU. The recently observed companion in the cavity of the gapped, highly misaligned system HD 142527 satisfies these requirements, indicating that a previous resonance crossing event misaligned the inner and outer discs. Our scenario for HD 142527's misaligned discs predicts that the companion's orbital plane is aligned with the outer disc's; this prediction should be testable with future observations as the companion's orbit is mapped out. Misalignments observed in several other gapped disc systems could be generated by the same secular resonance mechanism., 12 pages, 8 figures, submitted to MNRAS

Accreting stellar mass black holes (BHs) routinely exhibit Type-C quasi-periodic oscillations (QPOs). These are often interpreted as Lense-Thirring precession of the inner accretion flow, a relativistic effect whereby the spin of the BH distorts the surrounding space-time, inducing nodal precession. The best evidence for the precession model is the recent discovery, using a long joint XMM-Newton and NuSTAR observation of H 1743-322, that the centroid energy of the iron fluorescence line changes systematically with QPO phase. This was interpreted as the inner flow illuminating different azimuths of the accretion disc as it precesses, giving rise to a blue/red shifted iron line when the approaching/receding disc material is illuminated. Here, we develop a physical model for this interpretation, including a self-consistent reflection continuum, and fit this to the same H 1743-322 data. We use an analytic function to parameterise the asymmetric illumination pattern on the disc surface that would result from inner flow precession, and find that the data are well described if two bright patches rotate about the disc surface. This model is preferred to alternatives considering an oscillating disc ionisation parameter, disc inner radius and radial emissivity profile. We find that the reflection fraction varies with QPO phase (3.5 sigma), adding to the now formidable body of evidence that Type-C QPOs are a geometric effect. This is the first example of tomographic QPO modelling, initiating a powerful new technique that utilizes QPOs in order to map the dynamics of accreting material close to the BH., Comment: Accepted for publication in MNRAS

We analyse the evolution of a mildly inclined circumbinary disc that orbits an eccentric orbit binary by means of smoother particle hydrodynamic (SPH) simulations and linear theory. We show that the alignment process of an initially misaligned circumbinary disc around an eccentric orbit binary is significantly different than around a circular orbit binary and involves tilt oscillations. The more eccentric the binary, the larger the tilt oscillations and the longer it takes to damp these oscillations. A circumbinary disc that is only mildly inclined may increase its inclination by a factor of a few before it moves towards alignment. The results of the SPH simulations agree well with those of linear theory. We investigate the properties of the circumbinary disc/ring around KH 15D. We determine disc properties based on the observational constraints imposed by the changing binary brightness. We find that the inclination is currently at a local minimum and will increase substantially before setting to coplanarity. In addition, the nodal precession is currently near its most rapid rate. The recent observations that show a reappearance of Star B impose constraints on the thickness of the layer of obscuring material. Our results suggest that disc solids have undergone substantial inward drift and settling towards to disc midplane. For disc masses $\sim 0.001 M_\odot$, our model indicates that the level of disc turbulence is low $\alpha \ll 0.001$. Another possibility is that the disc/ring contains little gas., Comment: 16 pages, 16 figures; accepted for publication in MNRAS

A Neptune-sized exomoon candidate was recently announced by Teachey & Kipping, orbiting a 287 day gas giant in the Kepler-1625 system. However, the system is poorly characterized and needs more observations to be confirmed, with the next potential transit in 2019 May. In this Letter, we aid observational follow up by analyzing the transit signature of exomoons. We derive a simple analytic equation for the transit probability and use it to demonstrate how exomoons may frequently avoid transit if their orbit is larger than the stellar radius and sufficiently misaligned. The nominal orbit for the moon in Kepler-1625 has both of these characteristics, and we calculate that it may only transit roughly 40% of the time. This means that approximately six non-transits would be required to rule out the moon's existence at 95% confidence. When an exomoon's impact parameter is displaced off the star, the planet's impact parameter is displaced the other way, so larger planet transit durations are typically positively correlated with missed exomoon transits. On the other hand, strong correlations do not exist between missed exomoon transits and transit timing variations of the planet. We also show that nodal precession does not change an exomoon's transit probability and that it can break a prograde-retrograde degeneracy., Accepted at ApJ Letters

Recent studies suggest that ultra-short period planets (USPs), Earth-sized planets with sub-day periods, constitute a statistically distinct sub-sample of {\it Kepler} planets: USPs have smaller radii ($1-1.4R_\oplus$) and larger mutual inclinations with neighboring planets than nominal {\it Kepler} planets, and their period distribution is steeper than longer-period planets. We study a "low-eccentricity" migration scenario for the formation of USPs, in which a low-mass planet with initial period of a few days maintains a small but finite eccentricity due to secular forcings from exterior companion planets, and experiences orbital decay due to tidal dissipation. USP formation in this scenario requires that the initial multi-planet system have modest eccentricities ($\gtrsim 0.1$) or angular momentum deficit. During the orbital decay of the inner-most planet, the system can encounter several apsidal and nodal precession resonances that significantly enhance eccentricity excitation and increase the mutual inclination between the inner planets. We develop an approximate method based on eccentricity and inclination eigenmodes to efficiently evolve a large number of multi-planet systems over Gyr timescales in the presence of rapid (as short as $\sim 100$~years) secular planet-planet interactions and other short-range forces. Through a population synthesis calculation, we demonstrate that the "low-$e$ migration" mechanism can naturally produce USPs from the large population of {\it Kepler} multis under a variety of conditions, with little fine tuning of parameters. This mechanism favors smaller inner planets with more massive and eccentric companion planets, and the resulting USPs have properties that are consistent with observations., Comment: submitted to Monthly Notices

We measured the Rossiter-McLaughlin effect of WASP-107b during a single transit with Keck/HIRES. We found the sky-projected inclination of WASP-107b's orbit, relative to its host star's rotation axis, to be $|\lambda| = {118}^{+38}_{-19}$ degrees. This confirms the misaligned/polar orbit that was previously suggested from spot-crossing events and adds WASP-107b to the growing population of hot Neptunes in polar orbits around cool stars. WASP-107b is also the fourth such planet to have a known distant planetary companion. We examined several dynamical pathways by which this companion could have induced such an obliquity in WASP-107b. We find that nodal precession and disk dispersal-driven tilting can both explain the current orbital geometry while Kozai-Lidov cycles are suppressed by general relativity. While each hypothesis requires a mutual inclination between the two planets, nodal precession requires a much larger angle which for WASP-107 is on the threshold of detectability with future Gaia astrometric data. As nodal precession has no stellar type dependence, but disk dispersal-driven tilting does, distinguishing between these two models is best done on the population level. Finding and characterizing more extrasolar systems like WASP-107 will additionally help distinguish whether the distribution of hot-Neptune obliquities is a dichotomy of aligned and polar orbits or if we are uniformly sampling obliquities during nodal precession cycles., Comment: 13 pages, 6 figures, to be published in The Astronomical Journal

This paper proposes a method for deploying a nanosatellite constellation to several orbital planes from a single launch vehicle. The method is based on commercially available deorbit devices that are used to lower the initial orbit, and that are discarded after the correct altitude has been reached. Nodal precession of the right ascension of the ascending node at different altitudes results in spreading the orbital planes of the satellites. Maneuvering all satellites to a similar final altitude freezes the relative separation of the orbital planes. Calculations and simulations of the method are presented, and the results indicate that with a launch of 6 satellites to an initial 800 km sun-synchronous orbit, orbital plane separation of approximately 30° between each satellite can be achieved within 5 years, with each satellite in its own final 600 km orbital plane. Such a constellation could provide continuous global coverage, while requiring only one launch vehicle. Due to the timescales required by the method, it is best suited for nanosatellite missions designed for long lifetimes. Possible applications of such constellations are also discussed.

Aims: We aim to determine whether Jupiter's obliquity is bound to remain exceptionally small in the Solar System, or if it could grow in the future and reach values comparable to those of the other giant planets. Methods: The spin axis of Jupiter is subject to the gravitational torques from its regular satellites and from the Sun. These torques evolve over time due to the long-term variations of its orbit and to the migration of its satellites. With numerical simulations, we explore the future evolution of Jupiter's spin axis for different values of its moment of inertia and for different migration rates of its satellites. Analytical formulas show the location and properties of all relevant resonances. Results: Because of the migration of the Galilean satellites, Jupiter's obliquity is currently increasing, as it adiabatically follows the drift of a secular spin-orbit resonance with the nodal precession mode of Uranus. Using the current estimates of the migration rate of the satellites, the obliquity of Jupiter can reach values ranging from 6{\deg} to 37{\deg} after 5 Gyrs from now, according to the precise value of its polar moment of inertia. A faster migration for the satellites would produce a larger increase in obliquity, as long as the drift remains adiabatic. Conclusions: Despite its peculiarly small current value, the obliquity of Jupiter is no different from other obliquities in the Solar System: It is equally sensitive to secular spin-orbit resonances and it will probably reach comparable values in the future., Comment: Published in Astronomy & Astrophysics

While mantle convection is a fundamental ingredient of geodynamics, the driving mechanism of plate tectonics remains elusive. Are plates driven only from the thermal cooling of the mantle or are there further astronomical forces acting on them? GPS measurements are now accurate enough that, on long baselines, both secular plate motions and periodic tidal displacements are visible. The now >20 year-long space geodesy record of plate motions allows a more accurate analysis of the contribution of the horizontal component of the body tide in shifting the lithosphere. We review the data and show that lithospheric plates retain a non-zero horizontal component of the solid Earth tidal waves and their speed correlates with tidal harmonics. High-frequency semidiurnal Earth's tides are likely contributing to plate motions, but their residuals are still within the error of the present accuracy of GNSS data. The low-frequency body tides rather show horizontal residuals equal to the relative motion among plates, proving the astronomical input on plate dynamics. Plates move faster with nutation cyclicities of 8.8 and 18.6 years that correlate to lunar apsides migration and nodal precession. The high-frequency body tides are mostly buffered by the high viscosity of the lithosphere and the underlying mantle, whereas low-frequency horizontal tidal oscillations are compatible with the relaxation time of the low-velocity zone and can westerly drag the lithosphere over the asthenospheric mantle. Variable angular velocities among plates are controlled by the viscosity anisotropies in the decoupling layer within the low-velocity zone. Tidal oscillations also correlate with the seismic release.

Planet-planet perturbations can cause planets' orbital elements to change on secular timescales. Previous work has evaluated the nodal precession rate for planets in the limit of low $\alpha$ (semi-major axis ratio, 0$, Comment: 18 pages, 11 figures, submitted to AJ

We study the orbital evolution of black hole (BH) binaries in quadruple systems, where the tertiary binary excites large eccentricity in the BH binary through Lidov-Kozai (LK) oscillations, causing the binary BHs to merge via gravitational radiation. For typical BH binaries with masses $m_{1,2}\simeq 20M_\odot-30M_\odot$ and initial semimajor axis $a_0\sim100$ AU (such that the binaries have no chance of merging by themselves within $\sim10^{10}$ yrs), we show that binary-binary interactions can significantly increase the LK window for mergers (the range of companion inclinations that allows the BH binary to merge within 10~Gyrs). This increase arises from a secular resonance between the LK oscillation of the BH binary and the nodal precession of the outer (binary-binary) orbit driven by the tertiary binary. Therefore, in the presence of tertiary binary, the BH merger fraction is increased to $10-30\%$, an order of magnitude larger than the merger fraction found in similar triple systems. Our analysis (with appropriate scalings) can be easily adapted to other configurations of systems, such as relatively compact BH binaries and moderately hierarchical triples, which may generate even higher merger fractions. Since the occurrence rate of stellar quadruples in the galactic fields is not much smaller than that of stellar triples, our result suggests that dynamically induced BH mergers in quadruple systems may be an important channel of producing BH mergers observed by LIGO/VIRGO., 12 pages, 8 figures, accepted for publication in MNRAS

HAT-P-11 is a mid-K dwarf that hosts one of the first Neptune-sized planets found outside the solar system. The orbit of HAT-P-11b is misaligned with the star's spin --- one of the few known cases of a misaligned planet orbiting a star less massive than the Sun. We find an additional planet in the system based on a decade of precision radial velocity (RV) measurements from Keck/HIRES. HAT-P-11c is similar to Jupiter in its mass ($M_P \sin{i} = 1.6\pm0.1$ $M_J$) and orbital period ($P = 9.3^{+1.0}_{-0.5}$ year), but has a much more eccentric orbit ($e=0.60\pm0.03$). In our joint modeling of RV and stellar activity, we found an activity-induced RV signal of $\sim$7 m s$^{-1}$, consistent with other active K dwarfs, but significantly smaller than the 31 m s$^{-1}$ reflex motion due to HAT-P-11c. We investigated the dynamical coupling between HAT-P-11b and c as a possible explanation for HAT-P-11b's misaligned orbit, finding that planet-planet Kozai interactions cannot tilt planet b's orbit due to general relativistic precession; however, nodal precession operating on million year timescales is a viable mechanism to explain HAT-P-11b's high obliquity. This leaves open the question of why HAT-P-11c may have such a tilted orbit. At a distance of 38 pc, the HAT-P-11 system offers rich opportunities for further exoplanet characterization through astrometry and direct imaging., Comment: 16 pages, 11 figures, 4 tables. Accepted to AJ

Although most of the circumbinary planets detected by the Kepler spacecraft are on orbits that are closely aligned with the binary orbital plane, the systems Kepler-413 and Kepler-453 exhibit small misalignments of $\sim 2.5^\circ$. One possibility is that these planets formed in a circumbinary disc whose midplane was inclined relative to the binary orbital plane. Such a configuration is expected to lead to a warped and twisted disc, and our aim is to examine the inclination evolution of planets embedded in these discs. We employed 3D hydrodynamical simulations that examine the disc response to the presence of a modestly inclined binary with parameters that match the Kepler-413 system, as a function of disc parameters and binary inclinations. The discs all develop slowly varying warps, and generally display very small amounts of twist. Very slow solid body precession occurs because a large outer disc radius is adopted. Simulations of planets embedded in these discs resulted in the planet aligning with the binary orbit plane for disc masses close to the minimum mass solar nebular, such that nodal precession of the planet was controlled by the binary. For higher disc masses, the planet maintains near coplanarity with the local disc midplane. Our results suggest that circumbinary planets born in tilted circumbinary discs should align with the binary orbit plane as the disc ages and loses mass, even if the circumbinary disc remains misaligned from the binary orbit. This result has important implications for understanding the origins of the known circumbinary planets., accepted in MNRAS

We apply the relativistic precession (RP) model with its variants and the resonance epicyclic model with its variants, based on the frequencies of the geodesic epicyclic motion in the field of a Kerr black hole, to put limits on the mass of the black hole in the ultraluminous X-ray source M82 X-1 demonstrating twin high-frequency quasi-periodic oscillations (HF QPOs) with the frequency ratio near 3:2. The mass limits implied by the geodesic HF QPO models are compared to those obtained due to the model of string loop oscillations around a stable equilibrium position. Assuming the whole range of the black hole dimensionless spin, 0 < a < 1, the restrictions on the black hole mass related to the twin HF QPOs are widely extended and strongly model dependent; nevertheless, they give the lower limit M_{M82 X-1} > 130 M_{sun} confirming existence of an intermediate black hole in the M82 X-1 source. The upper limit given by one of the variants of the geodesic twin HF QPO models goes up to M_{M82 X-1}

Accretion discs are common in binary systems, and they are often found to be misaligned with respect to the binary orbit. The gravitational torque from a companion induces nodal precession in misaligned disc orbits. We calculate whether this precession is strong enough to overcome the internal disc torques communicating angular momentum. For typical parameters precession wins: the disc breaks into distinct planes that precess effectively independently. We run hydrodynamical simulations to check these results, and confirm that disc breaking is widespread and generally enhances accretion on to the central object. This applies in many cases of astrophysical accretion, e.g. supermassive black hole binaries and X--ray binaries., 8 pages, 6 figures, accepted for publication in MNRAS

A dynamical model of interacting nuclear stellar rings in the central parsec of our Galaxy is constructed. We discuss the physical sources of nodal precession and of the associated time scales. For approximate study of the mutual orbital precession, we replace broad nuclear rings by weighted average narrow circular rings. The model with narrow circular rings is shown to adequately describe the nodal precession. The period of relativistic apsidal precession in the center of the Galaxy, Tap•≈5⋅108 yr$T_{{\rm{ap}}}^ \bullet \approx 5 \cdot 10^8 \, {\rm{yr}}$, is almost an order of magnitude longer that the period of nodal precession, Tnod ≈ 7 · 107 yr, due to gravitational perturbations of nuclear disks (or rings). An important property of the nodal precession of nuclear rings is established: the lines of nodes of the two rings rotate uniformly with the same angular velocity, but in different directions. This explains the important observational fact that the lines of nodes of nuclear disks are not collinear, but are directed at large angles to each other.

We present analytical formulae for the orbital and epicyclic frequencies in orbits around Maclaurin spheroids in Newtonian gravity. The Laplace equation for the gravitational potential implies that the orbital frequency squared is the arithmetic mean of the squares of the epicyclic frequencies, $\omega _r^2 + \omega _z^2 = 2\Omega _{\rm o}^2$. The radial epicyclic frequency has a maximum at radius $r=\sqrt{2}ae$ for spheroid ellipticities $e>1/\sqrt{2}$, while for e = 0.834 583 18 it vanishes at the stellar equator (at r = a). For still larger ellipticities the innermost stable circular orbit (ISCO) is separated from the surface of the spheroid by a gap and has radius r ms = 1.198 203 ae. The vertical epicyclic frequency is always larger than the orbital one, and always by a factor of $\sqrt{2}$ in the marginally stable orbit. The presence of periastron motion, nodal precession (whose sense is the same as in retrograde orbits in the Kerr metric) and of the ISCO makes the properties of orbital motion around Maclaurin spheroids analogous to those in the Kerr metric. We find that the condition for the existence of circular orbits in test-particle motion is $\omega _r^2 + \omega _z^2 >0$, equally for the Maclaurin spheroid and for the Kerr metric.

NSV 1907, formerly listed as an irregular variable in variability catalogues, was classified as an Algol-type eclipsing binary in the Catalina Surveys Periodic Variable Star Catalogue. We have identified NSV 1907 as an ultraviolet (UV) bright source using measurements from the GALEX space telescope and detected obvious out-of-eclipse variability in archival photometric data from the Catalina Sky Survey, which instigated a closer examination of the object. A spectrum and extensive multicolour photometric observations were acquired, from which we deduce that NSV 1907 is a deeply eclipsing, nova-like cataclysmic variable. Apart from the orbital variations (deep eclipses with a period of P ~ 6.63 hours), changes in mean brightness and irregular short-term variability (flickering) were observed. The presence of a secondary minimum at phase phi ~ 0.5 was established, which indicates a significant contribution of the companion star to the optical flux of the system. We find possible evidence for sinusoidal variations with a period of P ~ 4.2 d, which we interpret as the nodal precession period of the accretion disc. No outbursts or VY Scl-like drops in brightness were detected either by the CSS or during our photometric monitoring. Because of its spectral characteristics and the observed variability pattern, we propose NSV 1907 as a new moderately bright long-period SW Sextantis star. Further photometric and spectroscopic observations are encouraged., 7 pages, 12 figures

Frequencies of the three quasi-periodic oscillation (QPO) modes observed simultaneously in the accreting black hole GRO J1655-40 are compared with the predictions of models. Models in which all three QPO signals are produced at the same radius are considered: these include different versions of relativistic precession, epicyclic resonance, tidal disruption, and warped disc models. Models that were originally proposed to interpret only the twin high-frequency QPOs are generalized here to interpret also the low-frequency QPO in terms of relativistic nodal precession. Inferred values of the black hole mass and spin from each QPO model are compared with the mass measured from optical observations and the spin inferred from X-ray spectroscopy techniques. We find that along with the relativistic precession model predicting $M=(5.3\pm0.1)~M_{\odot}, a=0.286\pm0.004$, the so-called total precession model ($M=(5.5\pm0.1)~M_{\odot}, a=0.276\pm0.003$), and the resonance epicyclic model with beat frequency ($M=(5.1\pm0.1)~M_{\odot}, a=0.274\pm0.003$) also satisfy the optical mass test. We compare our results with those inferred from X-ray spectral measurements.

An analytic form of the accelerations due to the luni-solar perturbations resolved along the rotating Euler-Hill frame is devised by using the expansion method. The addition of higher order terms to the main gravity gradient term linear in the spacecraft radial distance, carried out to the third order, provides a very high level of accuracy in accounting for the gravity perturbations experienced by a vehicle in orbit due to the sun and the moon. The nodal precession as well as the perigee advance of the lunar orbit is taken into account analytically by using the analytic lunar theory of de Pontecoulant. The analytic description of the apparent solar orbit and the motion of the moon remove the need to call an epherneris generator at each integration step during the numerical integration of the spacecraft trajectory, leading to the self-contained software for rapid and efficient optimal trajectory generation through iterations. Equinoctial elements are used to describe the spacecraft state and the luni-solar accelerations are given in terms of the apparent solar and lunar longitudes as well as Eulerian angles of the spacecraft orbit with respect to the inertial ecliptic system. The analysis is useful in optimal low-thrust orbit transfers complementing previous analyses carried out by this author, in which thrust and Earth zonal perturbations such as J2, J3 and J4 in terms of the nonsingular equinoctial elements are included.

We explore the effects of an undetected outer giant planet on the dynamics, observability, and stability of Systems with Tightly-packed Inner Planets (STIPs). We use direct numerical simulations along with secular theory and synthetic secular frequency spectra to analyze how analogues of Kepler-11 and Kepler-90 behave in the presence of a nearly co-planar, Jupiter-like outer perturber with semi-major axes between 1 and 5.2 au. Most locations of the outer perturber do not affect the evolution of the inner planetary systems, apart from altering precession frequencies. However, there are locations at which an outer planet causes system instability due to, in part, secular eccentricity resonances. In Kepler-90, there is a range of orbital distances for which the outer perturber drives planets b and c, through secular interactions, onto orbits with inclinations that are $\sim16^\circ$ away from the rest of the planets. Kepler-90 is stable in this configuration. Such secular resonances can thus affect the observed multiplicity of transiting systems. We also compare the synthetic apsidal and nodal precession frequencies with the secular theory and find some misalignment between principal frequencies, indicative of strong interactions between the planets (consistent with the system showing TTVs). First-order libration angles are calculated to identify MMRs in the systems, for which two near-MMRs are shown in Kepler-90, with a 5:4 between b and c, as well as a 3:2 between g and h., 17 pages, accepted for publication in AJ

We analyze photometry of the dwarf nova MN Dra carried out using various instruments at four observatories on 18 nights between May 20 and June 28, 2009. The observations cover a variety of activity states of the system: a superoutburst, three normal outbursts, and quiescence. Analysis of the system’s light curve during the superoutburst decline reveals positive superhumps that recur, on average, with a period of 0.105 days and are due to the direct apsidal precession of the accretion disk. These are observed until the end of the superoutburst, but their period decreases at a rate of −24.5 × 10−5 of the period per period. Both the positive-superhump period and its derivative are in good agreement with estimates made during previous superoutbursts. At the brightness minimum and in normal outbursts, MN Dra displays brightness variations with a period of 0.096 days, whose amplitude is much larger during the brightness minimum (0.8m–1.5m) than during normal outbursts (0.1m–0.2m). We suggest that these brightness variations could be negative superhumps due to nodal precession of the oblique accretion disk.

Expressions for the fundamental natural frequencies associated with J 2 -perturbed relative motion of satellites in near-circular orbits are derived. Special values of the orbit inclination, dependent on initial conditions, are obtained, for which the in-plane and out-of-plane fundamental frequencies remain equal to each other over an extended period of time, resulting in nonprecessing relative orbits. This result validates and generalizes a similar finding, based on numerical investigations by other researchers. The analysis is extended to the developments of accurate prediction and control models for fuel-optimal formation maintenance and intersatellite fuel balancing. Numerical simulation results are presented to demonstrate the accuracy of the developed models and the effects of frequency matching on control requirements.

The minimum-fuel low-thrust transfer between circular orbits is formulated using the Edelbaum's averaged dynamics with the addition of the nodal precession due to the first zonal term. The extremal analysis shows that an optimal transfer is composed of three sequences in the regular case. The optimal control problem is solved by a shooting method with a costate guess derived from an approximate solution.

We study the circularization of tidally disrupted stars on bound orbits around spinning supermassive black holes by performing three-dimensional smoothed particle hydrodynamic simulations with Post-Newtonian corrections. Our simulations reveal that debris circularization depends sensitively on the efficiency of radiative cooling. There are two stages in debris circularization if radiative cooling is inefficient: first, the stellar debris streams self-intersect due to relativistic apsidal precession; shocks at the intersection points thermalize orbital energy and the debris forms a geometrically thick, ring-like structure around the black hole. The ring rapidly spreads via viscous diffusion, leading to the formation of a geometrically thick accretion disk. In contrast, if radiative cooling is efficient, the stellar debris circularizes due to self-intersection shocks and forms a geometrically thin ring-like structure. In this case, the dissipated energy can be emitted during debris circularization as a precursor to the subsequent tidal disruption flare. The possible radiated energy is up to ~2*10^{52} erg for a 1 Msun star orbiting a 10^6 Msun black hole. We also find that a retrograde (prograde) black hole spin causes the shock-induced circularization timescale to be shorter (longer) than that of a non-spinning black hole in both cooling cases. The circularization timescale is remarkably long in the radiatively efficient cooling case, and is also sensitive to black hole spin. Specifically, Lense-Thirring torques cause dynamically important nodal precession, which significantly delays debris circularization. On the other hand, nodal precession is too slow to produce observable signatures in the radiatively inefficient case. We also discuss the relationship between our simulations and the parabolic TDEs that are characteristic of most stellar tidal disruptions., 23 pages, 18 figures, 1 appendix, accepted for publication in MNRAS (with significant improvement)

This work shows the dynamical instability that can happen to close-in satellites when planet oblateness is not accounted for in non-coplanar multiplanet systems. Simulations include two secularly interacting Jupiter-mass planets mutually inclined by 10 degrees, with the host planet either oblate or spherical. With a spherical host planet, moons within a critical planetocentric distance experience high inclinations and in some cases high eccentricities, while more distant moons orbit stably with low inclinations and eccentricities, as expected. These counter-intuitive dynamical phenomena disappear with an oblate host planet, in which case the moons' Laplace plane transitions from the host planet's equatorial plane to the host planet's precessing orbital plane as their semi-major axes increase, and all moons are dynamically stable with very mild changes in orbits. Direct perturbation from the perturbing planet has been investigated and ruled out as an explanation for the behavior of the innermost satellites, therefore leaving the central star's perturbation as the cause. Instability occurs while the nodal precession of the satellite and the central star (as seen from the host planet's frame) approaches the 1:1 secular resonance. In non-coplanar systems, around a non-oblate planet, the nodal precession of the moon becomes slow and comparable to that of the planet, giving rise to resonant configurations. The above effect needs to be taken into account in setting up numerical simulations., Comment: 7 pages, 6 figures. Accepted for publication in MNRAS

This paper addresses the problem of how to optimally rendezvous with a set of satellites in geosynchronous orbit that have small inclinations. The method developed has application to two problems, servicing of geosynchronous satellites and the removal of dead satellites in geosynchronous orbit. It is shown that when inclination changes are required the optimal solution reduces to the solution of the traveling salesman problem. The method was applied to a set of non-operational geosynchronous satellites selected from the Space Object catalog. The method also directly identifies those satellites that are very costly (in fuel) to visit so that they can be removed from the list to be visited. NOMENCLATURE Ω right ascension I inclination λ longitude n mean motion h angular momentum v velocity T R g g , geosynchronous radius and orbit period τ jk jk N , time and number of orbits to travel from satellite j to satellite k Tm mission lifetime Ts servicing time for each satellite INTRODUCTION The servicing of satellites in orbit is a concept that is receiving more and more attention. If this could be achieved economically then the system lifetime could be extended and system costs reduced. The Hubble Space Telescope (HST) has been serviced to repair the solar panels, mirror and gyros. The cost of the HST made it worthwhile even though it was a very expensive mission in that it involved the use of astronauts and the Space Shuttle. For servicing to be economical satellites will have to be designed to be serviced. In addition, the servicing will have to be performed by robotic spacecraft; the use of man in space will make the costs prohibitive in most cases. The best economy will be achieved when numerous satellites can be serviced in one servicing mission. Since plane changes require a lot of fuel two scenarios in which numerous satellites could be serviced are constellations with many satellites in each plane, such as Iridium, and geosynchronous satellites. A few satellites could potentially be serviced in low Earth orbit utilizing differential nodal precession if all the satellites had approximately the same inclination. Even though there is a policy now that satellites are to be removed from the geosynchronous belt at end of life there are many non-operational satellites in geosynchronous orbit that are a hazard to other satellites. If there was a way to rendezvous with these satellites and attach a small engine to raise their orbit then over a period of time the geosynchronous belt could be made safer. The method developed in this paper applies to these two problems. The method determines the minimum fuel solution for visiting (rendezvousing with) a set of satellites in geosynchronous orbit with small inclinations. Since the fuel to rendezvous with a satellite in the same orbit is very small if sufficient time is allowed for the maneuver the method finds the order in which the satellites should be visited to minimize the fuel required for the plane changes. It is shown that the minimum fuel solution is proportional to the minimum distance path through the set of points that are the projections of the angular momentum AIAA/AAS Astrodynamics Specialist Conference and Exhibit 5-8 August 2002, Monterey, California AIAA 2002-4905 Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. 2 American Institute of Aeronautics and Astronautics vectors on the equatorial plane. Thus, the minimum fuel (∆v) solution is the solution of the Traveling Salesman Problem (TSP). The fuel required for the in-plane maneuver is not minimized for several reasons. It is much smaller than the fuel required for the plane change and it can be obtained by vectoring the out of plane ∆v to obtain the necessary small in-plane component. Strategies for the rendezvous include equal time for each rendezvous or equal ∆v for the in-plane portion of the maneuver. Parameters in the problem are the total time and the servicing time at each satellite. The required ∆v is relatively insensitive to the servicing time until it becomes so large that only a short time is allowed for the transfer from one satellite to another. Then the in-plane ∆v begins to increase rapidly and dominate. ANALYSIS Consider a set, S, of N geosynchronous satellites where Si represents the ith satellite with orbital elements ei i i i T e e e = ( ) 1 2 6 , , , . K Find the order of visiting the satellites such that the total fuel ∆v ( ) to rendezvous (visit) is minimized. In addition, identify those satellites that should not be serviced because the fuel to service them is prohibitive. In this preliminary study we restrict the set S to consist of satellites in circular orbits and equal periods of 24 hours, small, but not necessarily zero, inclination and arbitrary right ascension. The circular orbit and 24 hour restrictions could be relaxed to consider near circular and near 24 hour satellites, such as the drifters. First we show that the finding the order of visiting each of the satellites that results in the minimum ∆v solution for the plane change portion of the ∆v to rendezvous with each of the N satellites in S is the solution of the Traveling Salesman Problem (TSP). Consider two satellites of the set S with inclination and right ascensions I I 1 1 2 2 , , Ω Ω ( ) ( ) and . Let X Y Z , , ( ) be an inertial coordinate system with Z along the polar axis, and X and Y define the equatorial plane. As shown in Figure 1 the coordinates of the projection of the angular momentum vector in the equatorial plane are h I j j j j j sin sin , cos , , Ω Ω − ( ) = 1 2. With equal angular momentum the distance d between the two points is

The dynamics of synchronous rotation and physical librations are revisited in order to establish a conceptually simple and general theoretical framework applicable to a variety of problems. Our motivation comes from disagreements between the results of numerical simulations and those of previous theoretical studies, and also because different theoretical studies disagree on basic features of the dynamics. We approach the problem by decomposing the orientation matrix of the body into perfectly synchronous rotation and deviation from the equilibrium state. The normal modes of the linearized equations are computed in the case of a circular satellite orbit, yielding both the periods and the eigenspaces of three librations. Libration in longitude decouples from the other two, vertical modes. There is a fast vertical mode with a period very close to the average rotational period. It corresponds to tilting the body around a horizontal axis while retaining nearly principal-axis rotation. In the inertial frame, this mode appears as nutation and free precession. The other vertical mode, a slow one, is the free wobble. The effects of the nodal precession of the orbit are investigated from the point of view of Cassini states. We test our theory using numerical simulations of the full equations of the dynamics and discuss the disagreements among our study and previous ones. The numerical simulations also reveal that in the case of eccentric orbits large departures from principal-axis rotation are possible due to a resonance between free precession and wobble. We also revisit the history of the Moon's rotational state and show that it switched from one Cassini state to another when it was at 46.2 Earth radii. This number disagrees with the value 34.2 derived in a previous study.

The orbital inclinations of a bounded, self-gravitating particle population subject to secular perturbations from an inclined planet are studied. The particle disk is shown to have discrete modes and pattern speeds in which the wave behavior is very pronounced compared with the nonwave behavior. The response is in the form of a standing wave pattern produced by wave reflection at the finite outer edge of the disk. In the absence of dissipation, the perpendicular torque between disk and perturber vanishes, with equal incoming and outgoing angular momentum fluxes carried by the wave trains. This resonant cavity theory is applied to Neptune perturbations of the primordial Kuiper belt. If the nodal precession rate of Neptune was near one of the disk's mode frequencies, very high inclinations could be produced for this population. There are a number of mechanisms in the early solar system that could have tuned the system to pass through one or more resonant states.