An angular-momentum preserving dissipative model for the point-mass N -body problem

Abstract

A simple mathematical model emulating energy dissipation due to tidal effects is proposed. In this model, forces acting between masses remove energy but preserve the total angular momentum of the system. We study the effect of such forces on a particular family of orbits, namely those corresponding to central configurations, and show that a specific dependence on the mutual distances between the bodies leads to homographic equations equivalent to those of the two-body problem with dissipation. We then describe in detail the topology of solutions of the dissipative two-body system via Poincaré compactification. Finally, we present equations averaged over Keplerian motion showing no influence of the dissipation on periapsis precession.

Figures (7)

• Figure 1: Phase space portraits for the regularized two body system (eq. \ref{['eq:two-bodies-regularized']}) for $C = \gamma = 1$. (A) The conservative case ($\alpha = 0$) showing elliptic (blue), parabolic (green), and hyperbolic (orange) orbits. (B) The system with dissipation ($\alpha = 0.5$) showing a captured orbit (blue), the weak-unstable-center manifold (red) from the equilibrium $(r_e, v_e) = (0, 2)$, and an escaping orbit (orange); the green dot marks the circular equilibrium $(r_c, v_c) = (1, 0)$. (C) Zoom of (B) near $(r_e, v_e)$; stable branches along $r=0$ are shown in blue.
• Figure 2: Phase space portraits for the compactified system (\ref{['eq:two-bodies-compact-xy']}) for $C = \gamma = 1$. (A) The system without dissipation ($\alpha = 0$) showing elliptic (blue), parabolic (green), and hyperbolic (orange) orbits. The green dot is the circular orbit equilibrium $(y_c, x_c) = (1, 0)$. (B) The system with dissipation ($\alpha = 0.5$) showing a captured orbit (blue), and an escaping orbit (orange). The curve in red represents the weak-unstable-center manifold from the degenerate equilibrium $(r_e, v_e)$ which here is a point at infinity. (C) Zoom of (B) near the degenerate equilibrium $(y_\infty, x_\infty) = (0, 0)$. The hyperbolic, attracting, and repelling sectors are denoted $S_h, S_a$, and $S_r$. The purple dashed lines $\Gamma_{u(s)}$ are local approximations to the unstable (stable) manifolds of $(y_\infty, x_\infty) = (0, 0)$.
• Figure 3: Phase space portraits for the compactified system (\ref{['eq:two-bodies-compact-wz']}) for $C = \gamma = 1$. (A) The system without dissipation ($\alpha = 0$). Dashed lines show the stable and unstable manifolds for $(w_{v\infty}, z_{v\infty}) = (0,0)$ and solid lines a few illustrative orbits. (B) The system with dissipation ($\alpha = 0.5$). The equilibrium point $(w_e, z_e) = (0, 0.5)$ divides the phase space, as indicaded by its manifolds (dotted pink lines).
• Figure 4: Sketches of the phase space topology on the Poincaré disk for the Kepler system. Green dots mark the circular equilibrium $(r_c, v_c)$. (A) Conservative case ($\alpha = 0$) showing the parabolic separatrix (red). (B) Dissipative case ($\alpha = 0.1$) showing the separatrix and two center manifolds. (C) Dissipative case ($\alpha = 0.5$) showing a new ejection center manifold and the connection of the center manifold from $(r_e, v_e)$ to $(r_c, v_c)$.
• Figure 5: Phase space portrait for the compactified system (\ref{['eq:two-bodies-compact-xy']}) for $C = \gamma = 1$ and $\alpha = 0.1$, showing that the weak-center-manifold, associated with $(r_e, v_e) = (0, 2)$, which in $(y,x)$ is a point at infinity, is located within the $S_a$ sector and connects to $(y_\infty, x_\infty) = (0, 0)$, instead of $(y_c, x_c) = (1, 0)$ as seen for $\alpha = 0.5$ (Fig. \ref{['fig:two-bodies-compact-xy']} (B) and (C)).