Rodica Roman

REGULARIZATION OF THE CIRCULAR RESTRICTED THREE-BODY PROBLEM USING "SIMILAR" COORDINATE SYSTEMS

The regularization of a new problem, namely the three-body problem, using ‘similar’ coordinate system is proposed. For this purpose we use the relation of ‘similarity’, which has been introduced as an equivalence relation in a previous paper (see Roman in Astrophys. Space Sci. doi:10.1007/s10509-011-0747-1, 2011). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The ‘similar’ polar angle’s definition is introduced, in order to analyze the regularization’s geometrical transformation. The effect of Levi-Civita’s transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.

‘Similar’ coordinate systems and the Roche geometry: application

A new equivalence relation, named relation of ‘similarity’ is defined and applied in the restricted three-body problem. Using this relation, a new class of trajectories (named ‘similar’ trajectories) are obtained; they have the theoretical role to give us new details in the restricted three-body problem. The ‘similar’ coordinate systems allow us in addition to obtain a unitary and an elegant demonstration of some analytical relations in the Roche geometry. As an example, some analytical relations published by Seidov (in Astrophys. J. 603:283, 2004) are demonstrated.

A family of zero-velocity curves in the restricted three-body problem

The equilibrium points and the curves of zero-velocity (Roche varieties) are analyzed in the frame of the regularized circular restricted three-body problem. The coordinate transformation is done with Levi-Civita generalized method, using polynomial functions of n degree. In the parametric plane, five families of equilibrium points are identified:

The dumb‐bell’s restricted, photogravitational, circular three‐body problem

L_{i}^{1}, L_{i}^{2}, \ldots, L_{i}^{n}

Study of astrophysics at the “Babeş‐Bolyai” University

,

Light curve shape variability of RZ Cephei. Preliminary considerarions

i\in\{ 1,2,\ldots,5 \}, n \in\mathbb{N}^{*}

Neutron star models with Douchin‐Haensel equation of state

. These families of points correspond to the five equilibrium points in the physical plane L1,L2,…,L5. The zero-velocity curves from the physical plane are transformed in Roche varieties in the parametric plane. The properties of these varieties are analyzed and the Roche varieties for n∈{1,2,…,6} are plotted. The equation of the asymptotic variety is obtained and its shape is analyzed. The slope of the Roche variety in

The Restricted Three -Body Problem. Comments on the `Spatial' Equilibrium Points

L_{1}^{1}

RT And: Multiperiodic Orbital Period Variability, Resonance and the Hypothesis of the Light-Time Effect .

point is obtained. For n=1 the slope obtained by Plavec and Kratochvil (1964) in the physical plane was found.

Long-Term Orbital Period Variation of RT Andromedae

We study the dumb‐bell’s planar motion in the frame of the photogravitational restricted three body problem. The main topic of this paper is the connection between the translation and the spin motion of the dumb‐bell, under the action of a photogravitational field generated by a binary system. The dumb‐bell’s equations of motion in the orbital plane are established, first using an inertial reference system, and then a rotating one. A prime integral of Jacobi type is found. Then are analyzed the equipotential surfaces and the equilibrium points. A geometrical feature of equilibrium points is established.