On the perturbed harmonic oscillator and celestial mechanics

TL;DR

This paper develops a perturbation theory for the 3D isotropic harmonic oscillator (3DIHO) using the conserved Runge-Lenz tensor $\mathds{A}$, whose eigenvalues encode the orbit axes and eccentricity $\epsilon$. By averaging the time derivative of $\mathds{A}$ over the unperturbed orbit, the authors derive a practical method to compute the precession velocity $\Omega$ for perturbations that induce precession, including Larmor-type and central $r$-power forces, with closed-form results involving hypergeometric functions $_2F_1$. The work also analyzes precessionless perturbations such as Euler-type forces and velocity-dependent drags $\boldsymbol{\delta F}_n=-m\gamma_n v^{n-1}\boldsymbol{v}$, showing that linear drag preserves eccentricity while higher-order drags increase it, and that Chandrasekhar friction tends to circularize orbits. The results have direct relevance to stellar dynamics in clusters and can inform mass inferences via virial analyses, while highlighting the distinctive behavior of the 3DIHO relative to the Kepler problem under dissipative perturbations.

Abstract

We study the influence of perturbations in the three dimensional isotropic harmonic oscillator problem considering different perturbing force laws and apply our results in the context of celestial mechanics, particularly in the movement of stars in stellar clusters. We use a method based on the Runge-Lenz tensor, so that our results are valid for any eccentricity of the unperturbed orbits of the oscillator. To establish basic concepts, we start by considering two cases, namely: a Larmor and a keplerian perturbation; and show that, in both cases, the perturbed orbits will precess. After that, we consider the more general problem of a central perturbation with any power-law dependence, that also only causes precession. Then, we consider precessionless perturbations caused by an Euler force and by the non-central dragging forces of the form $\boldsymbol{δF}=-γ_nv^{n-1}\boldsymbol{v}$, where $\boldsymbol{v}$ is the velocity of the particle and $γ_n\geq0$. We demonstrate that, in the case of a linear drag $(n=1)$, the orbits eccentricities remains constant. In contrast to what occurs in the well-known Kepler problem, for $n>1$ the orbit becomes increasingly eccentric. In the case $n=-3$, where the force is interpreted as a Chandrasekhar friction, we show that the eccentricity diminishes over time. We finish this work by making a few comments about the relevance of the main results.

Figures (9)

• Figure 1: Orbit of a charged 3DIHO submitted to a perturbation given by a constant and uniform magnetic field perpendicular to the plane of the orbit. In this graph we used $a=1$, $b=0.5$, and $\Omega=0.05$ in arbitrary units.
• Figure 2: Model of a cluster consisting of a sphere of radius $r_1$ and a concentric spherical shell of radii $r_1$ and $r_2$. The mass densities of the sphere and the shell are $\rho_1$ and $\rho_2 < \rho_1$, respectively. The prescribed (unperturbed) ellipse of a given star has semiaxis of length $a$ and $b$ such that $r_1 <b<a < r_2$.
• Figure 3: Plot of $\Omega_n/k_n'$ as a function of $n$ for fixed values of the eccentricity. Note that nothing imposes $n$ to be an integer. A black line at $n=1$ separates the anti-clockwise and clockwise precessions.
• Figure 4: Orbit of the 3DIHO subjected to a perturbation of the form $\boldsymbol{\delta F}=-m\boldsymbol{\alpha}\times \boldsymbol{r}$, with $\alpha=\omega^2/2$ for 16 revolutions. The initial semiaxes are defined as $a=1$ and $b=0.5$ in arbitrary units. Note that there is no precession, but the perturbing orbit becomes more eccentric over time.
• Figure 5: Orbit of the perturbed 3DIHO under a perturbing force of the form $\boldsymbol{\delta F}=-Cv^{-3}\boldsymbol{v}$ for 32 revolutions. The initial semiaxes are defined as $a=1$ and $b=0.5$ in arbitrary units. In this case, the eccentricity decreases with time.