Dynamics of a small planetoid in Newtonian gravity field of Lagrangian configuration of three primaries

Abstract

Novel method for semi-analytical solving of equations of a trapped dynamics for a planetoid m₄ close to the plane of mutual motion of main bodies around each other (in case of a special type of Bi-Elliptic Restricted 4-Bodies Problem) is presented. We consider here three primaries m₁, m₂, m₃ orbiting around their center of mass on elliptic orbits which are permanently forming Lagrangian configuration of an equilateral triangle. Our aim is to obtain approximate coordinates of quasi-planar trajectory of the infinitesimal planetoid m₄, when the primaries have masses equal to 1/3 (not stable configuration of the Lagrange solution). Results are as follows: (1) equations for coordinates $\{ \overline{x},\;\,\overline{y}\}$ are described by system of coupled second-order ODEs with respect to true anomaly f and (2) expression for $\overline{z}$ stems from solving second-order Riccati ordinary differential equation that determines the quasi-periodical oscillations of planetoid m₄ not far from invariant plane $\{ \overline{x},\overline{y},\,0\}$.

Appendix

We have provided here in the current research the numerical calculating for appropriate semi-analytical solutions of the first and second Eq. of system (11). The Runge–Kutta scheme of the fourth order (step 0.001 proceeding from initial values) has been used for calculating the data. Also, eccentricity e = 0.0167 has been chosen for calculations (e.g., as in “Sun–Earth” system) for modeling the same binary mutual motions for various pairs of primaries m₁, m₂, m₃. Graphical results of numerical calculation are depicted in Figs. 2, 3, 4, 5 and 6, with the initial data presented below:

1) $\overline{x}_{\,0} = 0.001,\;(\dot{\overline{x}})_{\,0} = - 0.4,\;\overline{y}_{\,0} = - 0.2,\;(\dot{\overline{y}})_{\,0} = - 0.3$

Fig. 2

Numerical solution for $\overline{x}\,(f)$, depicted on ordinate axis

Fig. 3

Numerical solution for $\overline{y}(f)$, depicted on ordinate axis

Fig. 4

Numerical solution for distance $\overline{r}_{{{\kern 1pt} 1}}^{{}} (f)$, depicted on ordinate axis

Fig. 5

Numerical solution for distance $\overline{r}_{{{\kern 1pt} 2}}^{{}} (f)$, depicted on ordinate axis

Fig. 6

Numerical solution for distance $\overline{r}_{{{\kern 1pt} 3}}^{{}} (f)$, depicted on ordinate axis

Meanwhile, it was numerically obtained for the dynamics of infinitesimal planetoid m₄ (see Figs. 2, 3, 4, 5 and 6) that this planetoid should be moving not far from primaries m₁, m₂, m₃ ({$\overline{r}_{{{\kern 1pt} 1}}^{{}}$,$\overline{r}_{{{\kern 1pt} 2}}^{{}}$,$\overline{r}_{{{\kern 1pt} 3}}^{{}}$} < 1.3) up to the meaning of true anomaly f ≅ 14.5 or more than 2 full turns of the first primary around the common center of masses. It is worth noting that dynamics of components of the numerical solution is checked to be quasi-stable (at least, up to the value of true anomaly f = 50).

We should note that additional numerical experiments regarding solving Eq. (8) (with already known numerical solutions for coordinates $\{ \overline{x},\overline{y}\}$) have brought reasonable results which can indeed be regarded as quasi-periodical oscillations of a planetoid in close vicinity of plane $\{ \overline{x},\overline{y},\,0\}$, see Fig. 7

Fig. 7

Numerical solution for $\overline{z}(f)$, depicted on ordinate axis

Thus, trajectories have the quasi-stable dynamics (for the chosen initial conditions, including those $\{ \overline{z}_{\,0} = - \,0.2,\;(\dot{\overline{z}})_{\,0} = - \,0.3\}$ for coordinate $(\overline{z})$) without sudden jumping of the solutions.

Also, works [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53] should be mentioned as a part of novel methods used in celestial mechanics applications related to the current research.