Daniel J Scheeres - Profile on Academia.edu (original) (raw)

Papers by Daniel J Scheeres

Stability of Relative Equilibria for Coupled Rotational and Translational Motion

Aas Division of Dynamical Astronomy Meeting 40, May 1, 2009

The stability of relative equilibria for two non-spherical mass distributions is studied. We acco... more The stability of relative equilibria for two non-spherical mass distributions is studied. We account for the full rotational/translational interaction between the two bodies by using a mutual gravitational potential expanded to 2nd order in both bodies. This allows us to restrict the system to planar motion, thus simplifying our analysis. We find that, for a given value of angular momentum, there are several possible relative equilibria for the system -- however only one is energetically stable and corresponds to the minimum energy configuration of the system at a given value of angular momentum. Applying these analytical results to the fission of a contact binary asteroid, we find that all equal density bodies that are spun to fission will enter directly into an unstable relative equilibrium, with the conjugate stable relative equilibrium lying at a greater distance. Thus, any contact binary asteroid spun to fission will immediately enter a strongly evolving phase of dynamical evolution. These results extend to the stability of nearly spherical bodies in contact, spun to fission, leading to the surprising result that a circular Keplerian orbit may actually be unstable when rotational coupling is allowed.

Special issue: Selected papers from the 38th annual meeting of the Division of Dynamical Astronomy of the American Astronomical Society, Ann Arbor, MI, USA, May 6–10, 2007

Celestial Mechanics and Dynamical Astronomy, 2008

Journal of Guidance Control and Dynamics, Mar 1, 2007

In this paper, we derive sufficient conditions for local and global boundedness of spacecraft mot... more In this paper, we derive sufficient conditions for local and global boundedness of spacecraft motion inside a prescribed region subject to a dead-band hovering thrust control law in time-invariant Lagrangian dynamical systems. Using the conservative properties of the system, we define a zero-velocity restriction on the spacecraft motion, then show that a dead-band controller exists that bounds nearby trajectories arbitrarily close to the desired hovering position. The minimum number of independent directions that the dead band must restrict is well defined as a function of hovering position and divides the position space into distinct dynamical regions. We present numerical plots of these regions in the two-body, restricted three-body, and Hill problems and find hovering near a central body usually requires dead-band control in only one or two independent directions to be bounded. The effects of uncertainty in the initial hovering state on the zero-velocity surface are evaluated and the largest allowable perturbations in the Jacobi constant that maintain boundedness are formulated. Nomenclature A = set of allowable spacecraft positions (localized formulation) B = set of allowable spacecraft positions (global formulation) C L = value of Jacobi constant in the Lagrangian formulation C 0 = nominal value of Jacobi constant C = value of perturbed Jacobi constant c = unit vector in dead-band thrust direction D = set of positions on the dead-band surface d = function that defines the dead-band surface f db = dead-band function G = function that defines the zero-velocity surface J = Jacobi constant J bf = Jacobi constant for the two-body problem J Hill = Jacobi constant in the Hill three-body problem J R3BP

This paper studies the dynamics of temporarily captured trajectories in the Hill 3-body problem, ... more This paper studies the dynamics of temporarily captured trajectories in the Hill 3-body problem, with application to orbits about Jupiter's satellite Europa. Using a model that includes the tidal force of Jupiter, the phase space of capture trajectories is numerically determined and trajectories are identified that don't impact with the planetary satellite or escape over long periods of time. These 'safe zones' in phase space are mapped out over different energy levels, identifying regions in phase space where an uncontrolled spacecraft would remain in orbit without impact or escape. The 'distances' of these safe trajectories from various frozen orbit trajectories are determined and the cost to transfer into these orbits is evaluated. Resulting from this analysis is the identification of robust capture trajectories and criteria on placing them into long-term stable orbits.

A long-term stable equilibrium for synchronous binaries including tides and the byorp effect

We present theoretical evidence for the existence of a long-term stable equilibrium solution for ... more We present theoretical evidence for the existence of a long-term stable equilibrium solution for synchronous binary asteroids accounting for mutual body tides, the binary YORP (BYORP) effect and dynamics. Synchronous binary asteroid systems consist of a rapidly spinning primary and a tidally-locked secondary, analogous to the Earth-Moon system. Tidal evolution of these systems leads to growth in the semi-major axis. Evolution from the BYORP effect can lead to both contraction and growth of the semi-major axis. There are two scenarios for joint evolution of a synchronous binary when both effects are considered: expansive and opposing evolution. During joint expansive evolution, both effects grow the semi-major axis. The system will either grow to the Hill sphere and disrupt if tidally dominated, or the mutual orbit will be de-stabilized due to runaway eccentricity growth if BYORP dominated. During joint opposing evolution, tidal and BYORP evolution act to evolve the system to a stable equilibrium. The location of this equilibrium to first order depends on just the tidal parameters, specific tidal dissipation number Q and the tidal Love number k, as well as the BYORP shape coefficient. If the observed population of small (0.1 - 10 km diameter), synchronous binaries are assumed to be in this static configuration, then our analysis shows that a monolithic geophysical model is not satisfactory, whereas the ``rubble pile'' model proposed by Goldreich & Sari (2009) is sufficient to prevent runaway eccentricity growth. The existence of this equilibrium and a secondary shape model built from observations enables direct study of asteroid geophysics through tidal theory. The existence of this equilibrium would be confirmed by a lack of migration in observational tests for the BYORP effect. Goldreich, P. & R. Sari, ApJ, 691:54-60 (2009)

Rotational Dynamics of a Comet Nucleus Subject to Outgassing Jets

A theory for the evolution of comet nucleus rotation states is presented, verified, and used to d... more A theory for the evolution of comet nucleus rotation states is presented, verified, and used to develop realistic simulations. If the comets are axisymmetric with multiple jets, ~50% have an spin up end state and over 50% tend to a relaxed state.

Applications of the Generalized Model for Solar Sails

Proceedings of the International Astronomical Union, 2007

The dynamics of binary asteroids in the Near Earth Object (NEO) space are discussed and constrain... more The dynamics of binary asteroids in the Near Earth Object (NEO) space are discussed and constraints on their evolution presented. The dynamics of binary NEO involve coupling between their rotational states and the orbital dynamics of the system to an extent not found elsewhere in the current solar system. Due to this close coupling classical approximations that can be used for the dynamics of planetary satellites no longer apply and force more general considerations. In addition, due to the small total mass of these binary systems they are susceptible to external perturbations from the sun, planetary flybys, and even from solar irradiation via the YORP effect. Also, the evolutionary dynamics of these systems depend on the poorly understood mechanisms by which granular material aggregates dissipate energy. Despite these complexities, several basic rules governing the evolution of binary asteroid systems can be established and their implications considered in light of the observed state of binary asteroid systems. These also lead to competing theories of binary asteroid formation and subsequent evolution that still require formal validation. These theories primarily involve the balance between planetary flybys and solar irradiation effects on these small systems, although the solar tide can also become significant for some systems. We explore the implications of each of these perturbation sources and their possible effect on NEO binary systems, along with possible signatures that they could leave on these systems as they form and evolve.

Satellite dynamics about a planet with a narrow ring

The dynamics of a satellite attracted by a planet with a ring are investigated. A single model fo... more The dynamics of a satellite attracted by a planet with a ring are investigated. A single model for the gravitational potential of a ring is used which extracts the fundamental gravitational effects on a satellite. Analytical and qualitative results are presented in cases where the satellites orbit is in the plane of the ring and where it is out of the plane of the ring. As final results we present equations expressing the perturbation of a body from two body motion when the ring is narrow. These effects include the advance of the argument of the periapsis and the precession of the plane of the orbit (when the satellite orbit is not contained in the ring plane). Additionally, inequalities on the angular momentum of the satellite (for the planar case) are found which guarantee that the orbits are bounded and do not intersect the ring.

Granular Mechanics in the Asteroid Regime

We study the granular mechanics properties of asteroid regolith and of asteroids modeled as gravi... more We study the granular mechanics properties of asteroid regolith and of asteroids modeled as gravitational aggregates using soft-sphere molecular simulation codes. For definiteness we assume parameters similar to the asteroid Itokawa, for which we have detailed observational data. Essential questions that can be studied using the techniques of granular mechanics are why large blocks dominate 80% of the surface of Itokawa and why the remaining 20% is uniformly covered with smaller particles, indicating global segregation mechanisms at work on this body. The prime energy source proposed for the segregation of granular materials on asteroids has been seismic shaking due to hypervelocity impacts with asteroids much smaller than the target body. We analyze the detailed mechanics of segregation physics in the asteroid environment due to such interactions. First we analyze the so-called Brazil Nut Effect (BNE), which preferentially causes larger particles to rise to the highest potential energy in a granular material. We note that the regions of highest potential on Itokawa are dominated by larger blocks, while the potential lows are dominated by smaller blocks. We verify and characterise the BNE effect in an asteroid environment under a variety of boundary and shaking conditions. We also extend our analyses to a global-scale simulation of aggregates, modeling the response of self-gravitating granules of a mixture of sizes to impacts. Analysis of such global-scale systems show additional mechanics that may account for the exposure of large blocks on the surface. Specifically we find that hypervelocity impacts are more effective in removing and transporting smaller regolith, exposing sub-surface larger blocks that might otherwise be covered in finer grained material. We discuss the scaling of granular mechanics effects from local regolith to global aggregate scale.

Cohesion in ``Rubble-Pile'' Asteroids

ABSTRACT Fine regolith in rubble pile asteroids may act as a sort of "van der Waals conc... more ABSTRACT Fine regolith in rubble pile asteroids may act as a sort of "van der Waals concrete" that forms bridges that bind larger boulders and strengthens small asteroids, allowing them to rotate more rapidly. We test these ideas using DEM simulations.

In� uence of Unstable Manifolds on Orbit Uncertainty

Indeterminacy in the Stable States of 4-Grain Rubble Piles

Reduction of Arbitrary Orbit Perturbation Problems to a Standard Form

Aas Division of Dynamical Astronomy Meeting 38, Jul 1, 2007

A novel coefficient-based method is developed to evaluate the trajectory dynamics of a perturbed ... more A novel coefficient-based method is developed to evaluate the trajectory dynamics of a perturbed particle. The perturbation acceleration vector components are represented as Fourier series in eccentric anomaly. Assuming low force magnitude such that the orbit does not change significantly over any individual revolution, Gauss's variational equations are averaged over one orbit with respect to mean anomaly. The averaged Gauss equations are shifted to eccentric anomaly as an independent parameter, resulting in expressions for the average rate of change of the six orbital elements in terms of the perturbation vector components. Substituting the Fourier series for these perturbation components, orthogonality conditions eliminate all but the 0th, 1st, and 2nd coefficients of each Fourier series. The resulting set of secular equations is a function of only 14 of the original Fourier coefficients, regardless of the order of the original Fourier series. The set of equations resulting from this analysis is sufficient to determine the secular evolution of perturbed motion with significantly reduced computational requirements as compared to integration of the full Newtonian problem. It also provides a standard set of equations to be solved for analytical investigations of perturbed motion. One implication of this result is that arbitrary continuous time perturbations acting on a particle's dynamics over one orbit period can be effectively represented with a simple Fourier series with at most 14 coefficents. Application of this result to spacecraft trajectory control problems allows for continuous time controls to be rigorously described by only 14 constant coefficients. Examples and applications of these reductions will be presented. This material is based upon work supported under a National Science Foundation Graduate Research Fellowship.

Concept of Operations for Deploying a Lander on the Secondary Body of Binary Asteroid 1996 FG3

ABSTRACT

Automatica, May 1, 2006

Given a nonlinear system and a performance index to be minimized, we present a general approach t... more Given a nonlinear system and a performance index to be minimized, we present a general approach to expressing the finite time optimal feedback control law applicable to different types of boundary conditions. Starting from the necessary conditions for optimality represented by a Hamiltonian system, we solve the Hamilton-Jacobi equation for a generating function for a specific canonical transformation. This enables us to obtain the optimal feedback control for fundamentally different sets of boundary conditions only using a series of algebraic manipulations and partial differentiations. Furthermore, the proposed approach reveals an insight that the optimal cost functions for a given dynamical system can be decomposed into a single generating function that is only a function of the dynamics plus a term representing the boundary conditions. This result is formalized as a theorem. The whole procedure provides an advantage over methods rooted in dynamic programming, which require one to solve the Hamilton-Jacobi-Bellman equation repetitively for each type of boundary condition. The cost of this favorable versatility is doubling the dimension of the partial differential equation to be solved. ᭧

This article describes and discusses the navigation fundamentals needed for designing a precision... more This article describes and discusses the navigation fundamentals needed for designing a precision-landing mission to a small solar-system body such as a comet or asteroid. The discussion focuses on the unique aspects of a small-body landing, the roles that ground-based operations and data types have, the roles that body-relative data types have, the constraints under which an autonomous system must operate, and the interaction and cooperation that must exist between all these aspects. Some specific examples of precision-landing capabilities are included for the Near Earth Asteroid Rendezvous (NEAR) spacecraft at the end of its prime mission and for the Champollion spacecraft.

Implications of Outgassing Jets for the Comet Dynamical Environment

Lunar and Planetary Science Conference, Mar 1, 2007

We define and explore a simple model for an outgassing jet of a comet while considering its impli... more We define and explore a simple model for an outgassing jet of a comet while considering its implications, define a methodology for the in situ estimation of jet structures, and investigate particle and rotational dynamics of comet 81/P Wild 2.

The Near-Earth Asteroid Rendezvous (NEAR)-Shoemaker mission was a milestone in that it represente... more The Near-Earth Asteroid Rendezvous (NEAR)-Shoemaker mission was a milestone in that it represented the practical navigation of a spacecraft in the most severely perturbed orbital environment (relative to the standard two-body problem) ever experienced by a spacecraft. Furthermore, all future asteroid orbital missions will encounter environments similar, in some way, to the environment encountered by NEAR at Eros. Thus, it is of interest to discuss the orbital dynamics that the NEAR-Shoemaker spacecraft was subject to at Eros. This article first provides a brief review of the measured force and model parameters of Eros. Next, an evaluation of the resulting orbital dynamics environment in the vicinity of Eros is made using these estimated values. The Eros dynamical environment is discussed in the region relatively far from the asteroid, where solar effects are the dominant perturbation, and in the region close to the asteroid, where the gravity field and rotation of Eros are most important. In particular, we give limits for stable motion about Eros and provide analytical descriptions of the dynamical environment found there. Many of the methods discussed and introduced in this article can be used for future asteroid missions as well.