The equation of Binet in classical and relativistic orbital mechanics

Abstract

Binet's equation provides a direct way to obtain the geometric shape of orbits in a central force field. It is well known that in Newtonian gravitation Binet's equation leads to all the conic curves as solutions for an inverse-square force. In this work, we show how Binet's equation arises from the horizontal and vertical infinitesimal displacements of a body in free fall and in inertial motion. This derivation uses elementary concepts of infinitesimal calculus. Second, we derive the relativistic version of Binet's equation for the Schwarzschild-(anti-)de Sitter metric. This derivation, which is novel, directly relates the coordinates involved in Binet's equation without the need to introduce potentials or the use of Killing vectors. Finally, we tackle some controversies related to the role of the cosmological constant in the trajectory of photons in a Schwarzschild-(anti-)de Sitter or even in Reissner-Nordström-(anti-)de Sitter spacetimes.

Figures (2)

• Figure 1: Representation of the instantaneous position of the reduced mass $\mu$ in a Cartesian coordinate system $(x,y)$ centred at the attracting centre. The vertical $y$ axis is aligned with the direction from the attracting centre to $\mu$ at this instant. The horizontal $x$ axis is perpendicular to $y$. The initial velocity components $v_{x}$ and $v_{y}$ represent the inertial motion along $x$ and $y$, respectively. The dotted curves represent some possible orbits of $\mu$ around the attracting centre, depending on the initial conditions.
• Figure 2: A massive particle follows a trajectory (solid curve) in curved spacetime determined by the metric $g_{\mu\nu}$. At each point of the trajectory, a local Lorentz frame (dashed arrows) can be constructed, in which the metric is locally flat. The tangent vector to the trajectory, $\vec{v}$, is parallel transported along the curve, meaning that its covariant derivative is zero. This parallel transport ensures that the particle's motion is consistent with the curvature of spacetime.