Linearized averaged resonant equations and their solution for dust particles. (arXiv:1506.09033v5 [astro-ph.EP] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Pastor_P/0/1/0/all/0/1">Pavol Pastor</a>
The averaged resonant equations of motion for the planar circular restricted
three-body problem are solved on the linearization basis taking into account
also non-gravitational effects. The averaged resonant equations are derived
from Lagrange’s planetary equations with additional Gauss’s terms caused by the
non-gravitational effects. The time depending solution has the standard form
with exponential, quadratic, linear and constant terms. The existence of a
rotational symmetry in the action of the non-gravitational effects around the
star determines the order of a characteristic equation of the linearized
system. In the symmetrical case (order 3) the considered non-gravitational
effects are the stellar electromagnetic radiation and the radial stellar wind
(stellar radiation). In the asymmetrical case (order 4) the stellar radiation
and interstellar gas flow are considered. It is investigated how well the
linearization solution describes real solution obtained from an equation of
motion by a comparison of the resonant libration frequency found analytically
and numerically. It is found that from initial values of the evolving orbital
parameters in the averaged phase space (semimajor axis, eccentricity, longitude
of pericenter, and resonant angular variable) the linearization frequency
depends most sensitively on the initial value of the resonant angular variable.
For small libration amplitudes of the resonant angular variable the best match
of the real libration frequency and the linearization frequency is located
approximately at the solution of the resonant condition ($da / dt$ $=$ 0). If
the initial averaged conditions are chosen close to the solution of resonant
condition, then the linearization frequency for practically all simple
oscillatory evolutions matches the real libration frequency and the
linearization solution very well approximates the real evolution.
The averaged resonant equations of motion for the planar circular restricted
three-body problem are solved on the linearization basis taking into account
also non-gravitational effects. The averaged resonant equations are derived
from Lagrange’s planetary equations with additional Gauss’s terms caused by the
non-gravitational effects. The time depending solution has the standard form
with exponential, quadratic, linear and constant terms. The existence of a
rotational symmetry in the action of the non-gravitational effects around the
star determines the order of a characteristic equation of the linearized
system. In the symmetrical case (order 3) the considered non-gravitational
effects are the stellar electromagnetic radiation and the radial stellar wind
(stellar radiation). In the asymmetrical case (order 4) the stellar radiation
and interstellar gas flow are considered. It is investigated how well the
linearization solution describes real solution obtained from an equation of
motion by a comparison of the resonant libration frequency found analytically
and numerically. It is found that from initial values of the evolving orbital
parameters in the averaged phase space (semimajor axis, eccentricity, longitude
of pericenter, and resonant angular variable) the linearization frequency
depends most sensitively on the initial value of the resonant angular variable.
For small libration amplitudes of the resonant angular variable the best match
of the real libration frequency and the linearization frequency is located
approximately at the solution of the resonant condition ($da / dt$ $=$ 0). If
the initial averaged conditions are chosen close to the solution of resonant
condition, then the linearization frequency for practically all simple
oscillatory evolutions matches the real libration frequency and the
linearization solution very well approximates the real evolution.
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