Binet equation

Binet’s equation rewrites a central-force orbit using $u(\theta )=1/r$. If $\ell =m{r}^2\dot{\theta }$ is angular momentum and the potential is $V(r)$, then

For an attractive central force of magnitude $c/r^5$, a circle of radius $R$ through the force center has polar equation $r=2R\cos \theta$. With specific angular momentum $h$, substitution into the Binet equation gives $c=8m{h}^2{R}^2$. Consequently $R$ is inversely proportional to the magnitude of the angular momentum, and the angular-time integral gives $T=2\pi R^2/|h|\propto R^3$. This is the finite travel time along the punctured circle from one approach to the singular center to the next; the force law alone does not specify continuation through the collision.

For radial acceleration $-A/r^2-B/r^3$, conserved specific angular momentum $l$ and $u=1/r$ reduce the Binet equation to the displayed linear equation. If $l^2>B$, its angular oscillation frequency is $\sqrt{1-B/l^2}$; attraction proportional to $r^{-3}$ changes the apsidal angle. A zero of $u$ ends the physical positive-radius branch and corresponds to escape, not passage to a negative radius. A spatial self-intersection need not be a periodic return because velocity may differ.

For $u=u_c[1+e\cos \nu (\psi -{\psi }_0)]$ with $\nu >0$ and $e\ge 1$, the positive-radius scattering branch has $|\nu (\psi -{\psi }_0)|<\arccos (-1/e)\le \pi$. Thus the principal complex argument in the displayed expression is continuous throughout that branch, unlike at an apocentre of a bound orbit. Its derivative is $\nu$, so the angle modulo $2\pi$, or the associated unit apsidal vector in the orbital plane, is a stationary integral of motion fixing the periapsis direction. Together with energy and three components of specific angular momentum, it gives five regular scattering labels even for irrational $\nu$. Periodicity is therefore necessary for five orbit labels in recurrent bound motion, not in general for escaping trajectories.

In the gravitational potential $-\mu (1/r+a/r^2)$, a regular noncircular bound orbit requires $L^2>2\mu a$ and has $u=\mu (L^2-2\mu a{)}^{-1}[1+e\cos \nu (\psi -{\psi }_0)]$ with $0<e<1$. Each radial period advances the angle by $2\pi /\nu$. The full position and velocity return after finitely many radial periods precisely when $\nu$ is rational. Irrational $\nu$ gives a dense rosette on the two-frequency invariant torus. Rationality permits a fifth orbit label fixing the apsidal phase on that resonant angular momentum family, but does not give global superintegrability for all nearby angular momenta. When $a=0$, every regular bound Kepler orbit is closed and the Laplace-Runge-Lenz vector supplies the globally defined additional integral.

For radial acceleration $-\mu /r^2+C/r^3$, the Binet equation gives $u^{\prime \prime }+(1+C/h^2)u=\mu /h^2$, where $h$ is specific angular momentum. A regular bound orbit has $h^2+C>0$, $a=-\mu /(2\mathcal{E})$ and $e^2=1+2\mathcal{E}(h^2+C)/{\mu }^2$. Its radius is $a(1-e^2)/[1+e\cos (\alpha \theta )]$, with $\alpha =\sqrt{1+C/h^2}$. The apsidal precession per radial period is $2\pi ({\alpha }^{-1}-1)$, which reduces to the displayed expression for a small perturbation. Negative $C$ produces prograde precession; positive $C$ produces retrograde precession. The dimensionless coefficient is $C/(\mu a)$, since $C$ has units of length to the fourth power divided by time squared.

For radial acceleration $-GM(1+\gamma /r)/r^2$, conserved specific angular momentum $h$ gives an angular radial frequency $\alpha =\sqrt{1-GM\gamma /h^2}$ through the Binet equation. Successive pericentres differ by $2\pi /\alpha$, so a small perturbation produces apsidal precession $\Delta \varpi \simeq \pi GM\gamma /h^2$. Writing the unperturbed semilatus rectum as $p=h^2/(GM)$ gives the displayed inverse relation. For a nearly circular orbit of period $P$, use $p\simeq [GM(P/(2\pi ){)}^2{]}^{1/3}$. Positive $\gamma$ means prograde precession; $\gamma$ is a length. A measured rate per century must first be converted to an angle per orbit.

Constant areal velocity gives constant specific angular momentum $l=r^2\dot{\theta }$, hence zero transverse acceleration and a central force. For a focus-based ellipse $u=1/r=(1+\epsilon \cos \theta )/A$, one has $u_{\theta \theta }+u=1/A$. The Binet equation then gives the displayed radial acceleration, proving an attractive inverse-square law. An orbit's shape alone does not specify its acceleration without a time law.

An attractive inverse-square force together with radial drag $-k\dot{r}/r^2$ preserves angular momentum $h=r^2\dot{\theta }$ but modifies the Binet equation for $u=1/r$ by a first derivative. For $k/|h|<2$ and $h\ne 0$, the reciprocal radius is a decaying oscillation about $p/h^2$. Forward angular distance $\mathrm{sgn}(h)(\theta -{\theta }_0)$ makes its damping sign explicit for either orbital orientation.

A bounded trajectory in the damped Binet equation with nonzero angular momentum asymptotically becomes circular. Energy loss is nonnegative in magnitude and vanishes only for zero radial velocity. The centrifugal effective potential energy excludes collision; bounded radius forces indefinitely increasing forward angular distance, so the reciprocal-radius oscillation decays. Escaping trajectories are excluded by the boundedness hypothesis.

ForBinet’s equation isIts circular orbits solve $K=u(1-r_{0}u{)}^2$; the branch $u<1/(3r_0)$ is stable and the branch $u>1/(3r_0)$ is unstable.