Orbital precession due to central-force perturbations

We calculate the perifocus precession of Keplerian orbits under the influence of arbitrary central-force perturbations. Our result is in the form of a one-dimensional integral that is straightforward to evaluate numerically. We demonstrate the effectiveness of our formula for the case of the Yukawa potential. We obtain analytic results for potentials of the form $V(r)=\alpha r^n$ and $V(r)=\alpha \ln (r/\lambda )$ in terms of the hypergeometric function ${}_{2}F_1(\frac{1}{2}-\frac{n}{2},1-\frac{n}{2};2;e^2)$, where $e$ is the eccentricity. Our results reproduce the known general relativistic ($n=-3$), constant force ($n=1$), and cosmological constant ($n=2$) precession formulas. Planetary perihelion precessions are often used to constrain the sizes of hypothetical new weak forces—our results allow for more precise, and often stronger, constraints on such proposed new forces.

Figure 1

Contour plot showing the relative precession $I(\kappa ,e)$ for the Yukawa potential as a function of the range parameter $\kappa$ and eccentricity $e$. The contour lines are at 0.75, 0.85, …, 1.55. The darkest (lightest) shading corresponds to the smallest (largest) values for the relative precession. The medium gray shown on the left side of the plot corresponds to values near one. As expected, the largest variations from one occur for large eccentricity. Surprisingly, there is a range of $\kappa$ for which the relative precession is depressed from its $e=0$ value.