279 research outputs found
A note on independent variables for restricted three-body problems
In studies of the elliptic restricted three-body problem, the true anomaly of the motion of the primaries is often used as the independent variable. The equations of motion then show invariancy in form from the circular case. It is of interest whether other independent variables exist, such that the invariant form of the equations is maintained. It is found that true anomaly is the only such variable.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42558/1/10569_2005_Article_BF01231394.pd
Capture of dark matter by the Solar System. Simple estimates
We consider the capture of galactic dark matter by the Solar System, due to
the gravitational three-body interaction of the Sun, a planet, and a dark
matter particle. Simple estimates are presented for the capture cross-section,
as well as for density and velocity distribution of captured dark matter
particles close to the Earth.Comment: 5 page
Extracting Multidimensional Phase Space Topology from Periodic Orbits
We establish a hierarchical ordering of periodic orbits in a strongly coupled
multidimensional Hamiltonian system. Phase space structures can be
reconstructed quantitatively from the knowledge of periodic orbits alone. We
illustrate our findings for the hydrogen atom in crossed electric and magnetic
fields.Comment: 4 pages, 5 figures, accepted for publication in Phys. Rev. Let
Straight Line Orbits in Hamiltonian Flows
We investigate periodic straight-line orbits (SLO) in Hamiltonian force
fields using both direct and inverse methods. A general theorem is proven for
natural Hamiltonians quadratic in the momenta in arbitrary dimension and
specialized to two and three dimension. Next we specialize to homogeneous
potentials and their superpositions, including the familiar H\'enon-Heiles
problem. It is shown that SLO's can exist for arbitrary finite superpositions
of -forms. The results are applied to a family of generalized H\'enon-Heiles
potentials having discrete rotational symmetry. SLO's are also found for
superpositions of these potentials.Comment: laTeX with 6 figure
Trajectory and stability of Lagrangian point in the Sun-Earth system
This paper describes design of the trajectory and analysis of the stability
of collinear point in the Sun-Earth system. The modified restricted three
body problem with additional gravitational potential from the belt is used as
the model for the Sun-Earth system. The effect of radiation pressure of the Sun
and oblate shape of the Earth are considered. The point is asymptotically
stable upto a specific value of time correspond to each set of values of
parameters and initial conditions. The results obtained from this study would
be applicable to locate a satellite, a telescope or a space station around the
point .Comment: Accepted for publication in Astrophysics & Space Scienc
A Motivating Exploration on Lunar Craters and Low-Energy Dynamics in the Earth -- Moon System
It is known that most of the craters on the surface of the Moon were created
by the collision of minor bodies of the Solar System. Main Belt Asteroids,
which can approach the terrestrial planets as a consequence of different types
of resonance, are actually the main responsible for this phenomenon. Our aim is
to investigate the impact distributions on the lunar surface that low-energy
dynamics can provide. As a first approximation, we exploit the hyberbolic
invariant manifolds associated with the central invariant manifold around the
equilibrium point L_2 of the Earth - Moon system within the framework of the
Circular Restricted Three - Body Problem. Taking transit trajectories at
several energy levels, we look for orbits intersecting the surface of the Moon
and we attempt to define a relationship between longitude and latitude of
arrival and lunar craters density. Then, we add the gravitational effect of the
Sun by considering the Bicircular Restricted Four - Body Problem. As further
exploration, we assume an uniform density of impact on the lunar surface,
looking for the regions in the Earth - Moon neighbourhood these colliding
trajectories have to come from. It turns out that low-energy ejecta originated
from high-energy impacts are also responsible of the phenomenon we are
considering.Comment: The paper is being published in Celestial Mechanics and Dynamical
Astronomy, vol. 107 (2010
Nearby low-mass triple system GJ795
We report the results of our optical speckle-interferometric observations of
the nearby triple system GJ795 performed with the 6-m BTA telescope with
diffraction-limited angular resolution. The three components of the system were
optically resolved for the first time. Position measurements allowed us to
determine the elements of the inner orbit of the triple system. We use the
measured magnitude differences to estimate the absolute magnitudes and spectral
types of the components of the triple: =7.310.08,
=8.660.10, =8.420.10, K5,
K9, K8. The total mass of the system is
equal to =1.69. We show
GJ795 to be a hierarchical triple system which satisfies the empirical
stability criteria.Comment: 6 pages, 2 figures, published in Astrophysical Bulleti
Quantum effects on Lagrangian points and displaced periodic orbits in the Earth-Moon system
Recent work in the literature has shown that the one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric functions. By performing the numerical analysis of the exact formulas for the roots, we confirm and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian points the quantum corrections are of the same order of magnitude in the Earth-Moon system. Third, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The present paper investigates therefore, eventually, a restricted three-body problem involving Earth, Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist
Earth-Moon Lagrangian points as a testbed for general relativity and effective field theories of gravity
We first analyse the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid. This scheme allows us to take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. A vehicle initially placed at L4 or L5 will not remain near the respective points. In particular, in the classical case the vehicle moves on a trajectory about the libration points for at least 700 days before escaping away. We show that this is true also if the modified long-distance Newtonian potential of effective gravity is employed. We also evaluate the impulse required to cancel out the perturbing force due to the Sun in order to force the spacecraft to stay precisely at L4 or L5. It turns out that this value is slightly modified with respect to the corresponding Newtonian one. In the second part of the paper, we first evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters and describe a tiny departure from the equilateral triangle. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. In other words, we deal with a theory involving quantum corrections to Einstein gravity, rather than to Newtonian gravity. By virtue of the effective-gravity correction to the long-distance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. In the latter case, for the point L1, general relativity corrects Newtonian theory by 7.61 meters, comparable, as an order of magnitude, with the lunar geodesic precession of about 3 meters per orbit. The latter is a cumulative effect accurately measured at the centimeter level through the lunar laser ranging positioning technique. Thus, it is possible to study a new laser ranging test of general relativity to measure the 7.61-meter correction to the L1 Lagrangian point, an observable never used before in the Sun-Earth-Moon system. Performing such an experiment requires controlling the propulsion to precisely reach L1, an instrumental accuracy comparable to the measurement of the lunar geodesic precession, understanding systematic effects resulting from thermal radiation and multi-body gravitational perturbations. This will then be the basis to consider a second-generation experiment to study deviations of effective field theories of gravity from general relativity in the Sun-Earth-Moon system
An Exactly Conservative Integrator for the n-Body Problem
The two-dimensional n-body problem of classical mechanics is a non-integrable
Hamiltonian system for n > 2. Traditional numerical integration algorithms,
which are polynomials in the time step, typically lead to systematic drifts in
the computed value of the total energy and angular momentum. Even symplectic
integration schemes exactly conserve only an approximate Hamiltonian. We
present an algorithm that conserves the true Hamiltonian and the total angular
momentum to machine precision. It is derived by applying conventional
discretizations in a new space obtained by transformation of the dependent
variables. We develop the method first for the restricted circular three-body
problem, then for the general two-dimensional three-body problem, and finally
for the planar n-body problem. Jacobi coordinates are used to reduce the
two-dimensional n-body problem to an (n-1)-body problem that incorporates the
constant linear momentum and center of mass constraints. For a four-body
choreography, we find that a larger time step can be used with our conservative
algorithm than with symplectic and conventional integrators.Comment: 17 pages, 3 figures; to appear in J. Phys. A.: Math. Ge
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