GR ε:800speed:4
🪐 Orbital Precession — What's going on?
Newton's law of gravity predicts that a planet in a two-body system
traces a perfect, closed ellipse — the same ellipse, forever.
This is exactly right for a simple sun + one planet system.
But Mercury doesn't do this. Its orbit slowly rotates.
The axis of Mercury's ellipse drifts by 575 arcseconds per century.
Most of that (532") is explained by the gravitational pull of other planets.
The remaining 43 arcseconds per century stumped astronomers for 50 years.
Einstein's general relativity explained those 43 arcseconds exactly.
It was one of the first experimental confirmations of the theory.
WHY DOES GR CAUSE PRECESSION?
In Newtonian gravity, the force falls off as exactly 1/r².
This produces a perfectly closed orbit (Bertrand's theorem).
General relativity adds a tiny extra attractive term:
F_GR = GM/r² + 3GML²/(c²r⁴)
The extra term is strongest at periapsis (closest approach), where r is
smallest. It gives the planet a slightly stronger "slingshot" than
Newtonian physics predicts, so the orbit overshoots closure by a small
angle each revolution. That small angle accumulates over centuries
into the measurable drift called perihelion precession.
THE SIMULATION
This animation integrates two planets from identical starting conditions:
BLUE — Newtonian gravity: F = GM/r² (orbit closes perfectly)
GOLD — GR gravity: F = GM/r² · (1 + ε/r²)
The ε slider exaggerates the GR correction so it's visible in seconds
rather than centuries. You can watch the gold ellipse slowly rotate
while the blue ellipse stays fixed.
The algorithm is Euler integration:
a = −F/m · r_hat (acceleration toward sun)
v += a · dt (update velocity)
pos += v · dt (update position)
Run multiple steps per frame for a stable, smooth orbit.
CONTROLS
• GR ε — size of the relativistic correction (0 = pure Newton)
• speed — integration steps per frame
• reset — return both planets to their starting positions