75 research outputs found
A Dynamical Systems Approach to Schwarzschild Null Geodesics
The null geodesics of a Schwarzschild black hole are studied from a dynamical
systems perspective. Written in terms of Kerr-Schild coordinates, the null
geodesic equation takes on the simple form of a particle moving under the
influence of a Newtonian central force with an inverse-cubic potential. We
apply a McGehee transformation to these equations, which clearly elucidates the
full phase space of solutions. All the null geodesics belong to one of four
families of invariant manifolds and their limiting cases, further characterized
by the angular momentum L of the orbit: for |L|>|L_c|, (1) the set that flow
outward from the white hole, turn around, then fall into the black hole, (2)
the set that fall inward from past null infinity, turn around outside the black
hole to continue to future null infinity, and for |L|<|L_c|, (3) the set that
flow outward from the white hole and continue to future null infinity, (4) the
set that flow inward from past null infinity and into the black hole. The
critical angular momentum Lc corresponds to the unstable circular orbit at
r=3M, and the homoclinic orbits associated with it. There are two additional
critical points of the flow at the singularity at r=0. Though the solutions of
geodesic motion and Hamiltonian flow we describe here are well known, what we
believe is a novel aspect of this work is the mapping between the two
equivalent descriptions, and the different insights each approach can give to
the problem. For example, the McGehee picture points to a particularly
interesting limiting case of the class (1) that move from the white to black
hole: in the limit as L goes to infinity, as described in Schwarzschild
coordinates, these geodesics begin at r=0, flow along t=constant lines, turn
around at r=2M, then continue to r=0. During this motion they circle in azimuth
exactly once, and complete the journey in zero affine time.Comment: 14 pages, 3 Figure
Where Did The Moon Come From?
The current standard theory of the origin of the Moon is that the Earth was
hit by a giant impactor the size of Mars causing ejection of iron poor impactor
mantle debris that coalesced to form the Moon. But where did this Mars-sized
impactor come from? Isotopic evidence suggests that it came from 1AU radius in
the solar nebula and computer simulations are consistent with it approaching
Earth on a zero-energy parabolic trajectory. But how could such a large object
form in the disk of planetesimals at 1AU without colliding with the Earth
early-on before having a chance to grow large or before its or the Earth's iron
core had formed? We propose that the giant impactor could have formed in a
stable orbit among debris at the Earth's Lagrange point (or ). We
show such a configuration is stable, even for a Mars-sized impactor. It could
grow gradually by accretion at (or ), but eventually gravitational
interactions with other growing planetesimals could kick it out into a chaotic
creeping orbit which we show would likely cause it to hit the Earth on a
zero-energy parabolic trajectory. This paper argues that this scenario is
possible and should be further studied.Comment: 64 pages, 27 figures, accepted for publication in A
On the Regularizability of the Big Bang Singularity
The singularity for the big bang state can be represented using the
generalized anisotropic Friedmann equation, resulting in a system of
differential equations in a central force field. We study the regularizability
of this singularity as a function of a parameter, the equation of state, .
We prove that for it is regularizable only for satisfying relative
prime number conditions, and for it can always be regularized. This
is done by using a McGehee transformation, usually applied in the three and
four-body problems. This transformation blows up the singularity into an
invariant manifold. The relationship of this result to other cosmological
models is briefly discussed.Comment: 22 pages, 0 figure
The classical supersymmetric Coulomb problem
After setting up a general model for supersymmetric classical mechanics in
more than one dimension we describe systems with centrally symmetric potentials
and their Poisson algebra. We then apply this information to the investigation
and solution of the supersymmetric Coulomb problem, specified by an 1/|x|
repulsive bosonic potential.Comment: 25 pages, 2 figures; reference added, some minor modification
On Optimal Two-Impulse Earth-Moon Transfers in a Four-Body Model
In this paper two-impulse Earth-Moon transfers are treated in the restricted four-body problem with the Sun, the Earth, and the Moon as primaries. The problem is formulated with mathematical means and solved through direct transcription and multiple shooting strategy. Thousands of solutions are found, which make it possible to frame known cases as special points of a more general picture. Families of solutions are defined and characterized, and their features are discussed. The methodology described in this paper is useful to perform trade-off analyses, where many solutions have to be produced and assessed
Optical 2-metrics of Schwarzschild-Tangherlini Spacetimes and the Bohlin-Arnold Duality
We consider the projection of null geodesics of the Schwarzschild-Tangherlini
metric in n+1 dimensions to the space of orbits of the static Killing vector
where the motion of a given light ray is seen to lie in a plane. The projected
curves coincide with the unparametrised geodesics of optical 2-metrics and can
be equally understood as describing the motion of a non-relativistic particle
in a central force. We consider a duality between the projected null curves for
pairs of values of n and interpret its mathematical meaning in terms of the
optical 2-metrics. The metrics are not projectively equivalent but the
correspondence can be exposed in terms of a third order differential equation.
We also explore the extension of this notion of duality to the
Reissner-Nordstrom case.Comment: 10 page
Survey of highly non-Keplerian orbits with low-thrust propulsion
Celestial mechanics has traditionally been concerned with orbital motion under the action of a conservative gravitational potential. In particular, the inverse square gravitational force due to the potential of a uniform, spherical mass leads to a family of conic section orbits, as determined by Isaac Newton, who showed that Kepler‟s laws were derivable from his theory of gravitation. While orbital motion under the action of a conservative gravitational potential leads to an array of problems with often complex and interesting solutions, the addition of non-conservative forces offers new avenues of investigation. In particular, non-conservative forces lead to a rich diversity of problems associated with the existence, stability and control of families of highly non-Keplerian orbits generated by a gravitational potential and a non-conservative force. Highly non-Keplerian orbits can potentially have a broad range of practical applications across a number of different disciplines. This review aims to summarize the combined wealth of literature concerned with the dynamics, stability and control of highly non-Keplerian orbits for various low thrust propulsion devices, and to demonstrate some of these potential applications
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