Circularization in the damped Kepler problem (Q6592241)

The major planets in the Solar System (Mercury being an exception) have almost circular orbits. This has been attributed to the effect of some resistance force opposite to the velocity. In Cartesian coordinates, this can be described by the damped Kepler problem in the plane. The authors focus on the question: ``Under what conditions on \(\Delta\) (see eq. (1.1)) can we say that the orbits described by equations (1.1) in terms of the astronomical coordinates, have a tendency to become circular as they get close to the Sun''? Their approach is based on dynamical systems theory, using blowup and desingularization as main technical tools. Since the eccentricity e is an astronomical coordinate, the notion of circularization occurs as $e(t)->0$ for $t->\omega$, where $\omega$ is the time of collision ($\omega<=$infinity). In the present paper, they analyze the circularization problem using the modern machinery of dynamical systems, and they prove that circularization only occurs for: $-3<\gamma:=\alpha+2\beta-3<0$, ($\alpha,\beta,\gamma$ parameters) and for $(\alpha,\beta)$ not equal to $(0,0)$. The details of the blowup transformations, used for the proof of Theorem 3.1, depend upon $\alpha$ and $\beta$. The authors divide the analysis into three main cases: $\gamma=\alpha+2\beta-3>0$, $\gamma<0$, and $gamma=0$. The above quantity separating circularization $(-3<\gamma<0)$ where $e(t)->0$ as $u(t)->0$, from cases $(\gamma>0)$ where $e(t)->1$ as $u(t)->0$, both on open sets of initial conditions. For $\gamma=0$ relates to a certain scaling symmetry and in this case the system can be reduced to a planar system. The authors, also, describe different properties of the solutions, including finite time blowup and the limit of the eccentricity vector. They found that circularization $e(t)->0$ implies finite time blowup of solutions, and they believe that their approach can be used to describe unbounded solutions, and they conclude the paper with a discussion section.