34 research outputs found
Nonlinear stability analysis of the Emden-Fowler equation
In this paper we qualitatively study radial solutions of the semilinear
elliptic equation with and on the
positive real line, called the Emden-Fowler or Lane-Emden equation. This
equation is of great importance in Newtonian astrophysics and the constant
is called the polytropic index. By introducing a set of new variables, the
Emden-Fowler equation can be written as an autonomous system of two ordinary
differential equations which can be analyzed using linear and nonlinear
stability analysis. We perform the study of stability by using linear stability
analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the
Lyapunov function method. Depending on the values of these different
methods yield different results. We identify a parameter range for where
all three methods imply stability.Comment: 12 pages; new reference added; 3 new references added; fully revised
versio
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
All solutions of the n = 5 Lane-Emden equation
All real solutions of the Lane-Emden equation for n = 5 are obtained in terms
of Jacobian and Weierstrass elliptic functions. A new family of solutions is
found. It is expressed by remarkably simple formulae involving Jacobian
elliptic functions only. The general properties and discrete scaling symmetries
of these new solutions are discussed. We also comment on their possible
applications.Comment: To appear in Journal of Mathematical Physic
(In)finiteness of Spherically Symmetric Static Perfect Fluids
This work is concerned with the finiteness problem for static, spherically
symmetric perfect fluids in both Newtonian Gravity and General Relativity. We
derive criteria on the barotropic equation of state guaranteeing that the
corresponding perfect fluid solutions possess finite/infinite extent. In the
Newtonian case, for the large class of monotonic equations of state, and in
General Relativity we improve earlier results
Dynamical stability of infinite homogeneous self-gravitating systems: application of the Nyquist method
We complete classical investigations concerning the dynamical stability of an
infinite homogeneous gaseous medium described by the Euler-Poisson system or an
infinite homogeneous stellar system described by the Vlasov-Poisson system
(Jeans problem). To determine the stability of an infinite homogeneous stellar
system with respect to a perturbation of wavenumber k, we apply the Nyquist
method. We first consider the case of single-humped distributions and show
that, for infinite homogeneous systems, the onset of instability is the same in
a stellar system and in the corresponding barotropic gas, contrary to the case
of inhomogeneous systems. We show that this result is true for any symmetric
single-humped velocity distribution, not only for the Maxwellian. If we
specialize on isothermal and polytropic distributions, analytical expressions
for the growth rate, damping rate and pulsation period of the perturbation can
be given. Then, we consider the Vlasov stability of symmetric and asymmetric
double-humped distributions (two-stream stellar systems) and determine the
stability diagrams depending on the degree of asymmetry. We compare these
results with the Euler stability of two self-gravitating gaseous streams.
Finally, we determine the corresponding stability diagrams in the case of
plasmas and compare the results with self-gravitating systems
Librations with Mass Transfer in the Sun-Jupiter System
Trojan-type motion is analytically and numerically studied under mass transfer between the primaries with conservation of their total orbital angular momentum. We prove theoretically and numerically our new result that angular libration widths change as m^1/4, (m - Jupiter mass) if they are throughout smaller than about 60 arcd. Numerical examples show that for initial libration widths larger than about 60 arcd, the Trojan is ultimately driven out of the libration domain, becoming an ordinary asteroid, if Jupiter's transferred mass increases by a factor less than about two. Certain processes occurring in our solar system and in extrasolar planetary systems lead to a decrease of the Trojan's libration amplitude, while other processes lead to an increase, respectively