Gravity N-Body and Black Hole Sandbox - 2D Orbit Lab: Precession, Light Bending and Figure-Eight

Presets

Live readouts

Graphs

Experiments

Orbit: raise p slowly and watch the periapsis advance shrink like 1/p; at p below 6 + 2e the orbit plunges. Light: sweep b through 5.196 M, the critical impact parameter. N-body: switch the step to coarse and see the energy error grow, then to fine.

Numerical checks

running...

How to use

Pick a preset or the experiment selector. In the orbit view the orange track is the general-relativistic orbit and the dashed blue ellipse is what Newton predicts for the same p and e: the gap is the periapsis advance. In the light view, rays come from the left; red ones are captured. In the N-body view, drag on the scene to launch a new body (the drag length sets its velocity). Space pauses, R resets.

The science

Units. G = c = 1 and the black-hole mass M = 1, so lengths are in M (the horizon is at the Schwarzschild radius r_s = 2M) and times in M as well.

Massive orbits. With u = 1/r and L the specific angular momentum, a bound geodesic obeys d2u/dphi2 + u = M/L^2 + 3 M u^2 The last term does not exist in Newton's theory; it makes the ellipse rotate. The orbit is parametrised by the semi-latus rectum p and eccentricity e, r = p/(1 + e cos chi), with L^2 = p^2 M / (p - 3M - e^2 M), E^2 = ((p - 2M)^2 - 4 e^2 M^2) / (p (p - 3M - e^2 M)) Stable bound orbits need p > 6M + 2eM; the innermost stable circular orbit (ISCO) is r = 6M. The periapsis advances per orbit by delta phi ~ 6 pi M / p for p >> M (Mercury: 43 arcseconds per century). The animation advances in coordinate time using dphi/dt = L (1 - 2M/r) / (E r^2); proper time follows dtau/dt = (1 - 2M/r)/E.

Effective potential. Radial motion is a particle in V(r) = (1 - 2M/r)(1 + L^2/r^2) with energy level E^2: turning points are where the curve crosses the line. The potential has a maximum that disappears at the ISCO.

Light. For null rays d2u/dphi2 + u = 3 M u^2. Far away the deflection is 4M/b (twice the Newtonian value, confirmed by the 1919 eclipse); rays with impact parameter below b_c = sqrt(27) M = 5.196 M are captured and the photon sphere sits at r = 3M.

N-body. The sandbox integrates Newtonian gravity with softening eps using the kick-drift-kick leapfrog, which is symplectic: energy error oscillates instead of growing. The figure-eight is the Chenciner-Montgomery solution with period 6.3259.

Honest simplifications. Orbits are test particles (no gravitational waves, no spin); the N-body part is Newtonian and 2D; the fixed-step leapfrog is not adaptive, so close encounters in the random cluster lose accuracy. The 3D page uses a different, adaptive integrator.

Checks run in the page. Periapsis advance against 6 pi M/p; radial-oscillation frequency of a near-circular orbit against sqrt(1 - 6M/r); light deflection against 4M/b with its second-order term; the capture threshold against sqrt(27) M; figure-eight period closure and energy conservation.