Perihelion Precession

Perihelion Precession

A planet orbiting the Sun does not follow a perfectly closed ellipse — its orbit slowly rotates. Most of this perihelion advance is explained by the gravitational pull of other planets, but a small residual remained unexplained by Newtonian gravity for decades. For Mercury, 43 arcseconds per century were unaccounted for.

General relativity provides the explanation. The extra precession per orbit is:

$\Delta\phi = \frac{6\pi G M}{c^2 a (1 - e^2)} \quad \text{(radians per orbit)}$

where:

• $M$ = mass of the central body (Sun)
• $a$ = semi-major axis of the orbit
• $e$ = orbital eccentricity

Mercury's Parameters

Plugging these values into the GR formula gives:

$\Delta\phi \approx 5.02 \times 10^{-7} \text{ rad/orbit} \approx 0.1036 \text{ arcsec/orbit}$

Mercury completes about $414.9$ orbits per century, giving:

$\Delta\phi_{\text{century}} \approx 43.0 \text{ arcsec/century}$

This was one of the earliest triumphs of GR, confirmed with the observed anomalous precession known since 1859.

Your Task

Implement three functions. All physical constants must be defined inside each function body.

• precession_per_orbit(M, a, e) — precession in radians per orbit
• precession_arcsec_per_orbit(M, a, e) — convert to arcseconds: multiply by $(180/\pi) \times 3600$
• precession_arcsec_per_century(M, a, e, period_days) — arcseconds per century: multiply by $(100 \times 365.25 / \text{period\_days})$