Circularization in the Damped Kepler Problem

In this paper, we revisit the damped Kepler problem within a general family of nonlinear damping forces with magnitude \(\delta \vert u\vert^{\beta }\vert \dot u\vert^{\alpha+1}\), depending on three parameters, \(\delta \gt 0,\alpha \ge 0\), and \(\beta \ge 0\), and address the general question of circularization whereby orbits tend to become more circular as they approach the sun. Our approach is based on dynamical systems theory, using blowup and desingularization as our main technical tools. We find that \(\gamma =\alpha+2\beta -3\) is an important quantity, with the special case \(\gamma=0\) separating circularization (\(-3\lt \gamma \lt 0\)) where the eccentricity converges to zero, i.e., \(e(t)\rightarrow 0\) as \(u(t)\rightarrow 0\), from cases (\(\gamma \gt 0\)) where \(e(t)\rightarrow 1\) as \(u(t)\rightarrow 0\), both on open sets of initial conditions. We find that circularization for \(-3\lt \gamma \lt 0\) occurs due to asymptotic stability of a zero-Hopf equilibrium point (i.e., the eigenvalues are \(\pm i \omega,0\)) of a three-dimensional reduced problem (which is analytic in the blowup coordinates). The attraction is therefore not hyperbolic and in particular not covered by standard dynamical systems theory. Instead we use recent results on normal forms of the zero-Hopf to locally bring the system into a form where the stability can be addressed directly. The case \(\gamma=0\) relates to a certain scaling symmetry (that is also present in the undamped Kepler problem) and in this case the system can be reduced to a planar system. We find that the eccentricity limits to \(4\delta^2\in (0,1)\) on an open subset of initial conditions for (\(0\lt \delta \lt \frac 12\) and \(\gamma=0\)). We also describe different properties of the solutions, including finite time blowup and the limit of the eccentricity vector. Interestingly, we find that circularization \(e(t)\rightarrow 0\) implies finite time blowup of solutions. We believe that our approach can be used to describe unbounded solutions.