Why does the ellipse fail to close?

The extra 3u² term in the orbit equation is a GR correction absent from Newtonian 1/r potentials. It advances the periapsis each radial period so the tip of the ellipse slowly rotates, drawing a rosette. This is spacetime curvature, not a classical inverse-cube ‘extra force’ in flat space—though the math looks similar in the u(φ) equation.

How is this related to Mercury’s 43″/century?

The weak-field limit of Δφ ≈ 6πM/[a(1 − e²)] per orbit, accumulated over a century for Mercury’s a and e, yields about 43 arcseconds after other Newtonian perturbations are removed. The simulator shows the strong-field version of that same mechanism in geometric units.

What do r = 2M and r = 6M mark?

r = 2M is the Schwarzschild event horizon; timelike circular orbits do not exist stably down to the horizon. The innermost stable circular orbit sits at r = 6M for a non-spinning hole—inside that, circular orbits are unstable. The annotations orient the rosette relative to those scales.

Schwarzschild Orbit Precession (Rosette)

Schwarzschild geodesic in the φ-form d²u/dφ² + u = 1/L² + 3u² (G = c = M = 1) integrated by RK4. The closed Newtonian ellipse is replaced by an orange precessing rosette with apsidal advance Δφ ≈ 6πM/[a(1 − e²)] per orbit — the same mechanism that produces the historic 43″/century perihelion shift of Mercury. Horizon r = 2M and ISCO r = 6M annotated.

About this model

Test-particle motion in the Schwarzschild metric is integrated in the φ-form d²u/dφ² + u = 1/L² + 3u² with G = c = M = 1 and u = 1/r, using RK4. The Newtonian closed ellipse becomes an orange precessing rosette; the apsidal advance per radial period is Δφ ≈ 6πM/[a(1 − e²)], the same GR effect behind Mercury’s historic 43″/century perihelion shift. Horizon r = 2M and ISCO r = 6M are annotated. Assumptions: geodesic motion of a test mass, Schwarzschild (non-spinning) spacetime, and geometric units—no radiation reaction. Vary orbital parameters to compare nearly Newtonian ellipses with strongly precessing rosettes near the hole.

Who it's for: Advanced undergraduate and graduate general relativity and relativistic astrophysics courses.

Key terms

Schwarzschild geodesic
perihelion precession
apsidal advance
rosette orbit
ISCO
event horizon

How it works

Schwarzschild orbit equation d²u/dφ² + u = 1/L² + 3u² integrated by RK4 (G = c = M = 1). The orange GR geodesic precesses by Δφ ≈ 6πM/[a(1−e²)] per orbit — the same mechanism that produces the famous 43″/century perihelion advance of Mercury. A gray Newtonian ellipse is overplotted to isolate the GR correction; ISCO (r = 6M) and horizon (r = 2M) are marked.

Frequently asked questions

Why does the ellipse fail to close?: The extra 3u² term in the orbit equation is a GR correction absent from Newtonian 1/r potentials. It advances the periapsis each radial period so the tip of the ellipse slowly rotates, drawing a rosette. This is spacetime curvature, not a classical inverse-cube ‘extra force’ in flat space—though the math looks similar in the u(φ) equation.
How is this related to Mercury’s 43″/century?: The weak-field limit of Δφ ≈ 6πM/[a(1 − e²)] per orbit, accumulated over a century for Mercury’s a and e, yields about 43 arcseconds after other Newtonian perturbations are removed. The simulator shows the strong-field version of that same mechanism in geometric units.
What do r = 2M and r = 6M mark?: r = 2M is the Schwarzschild event horizon; timelike circular orbits do not exist stably down to the horizon. The innermost stable circular orbit sits at r = 6M for a non-spinning hole—inside that, circular orbits are unstable. The annotations orient the rosette relative to those scales.

Continue with similar topics in this category — or all 27 in Gravity & Orbits.

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Mercury Perihelion Precession

Launch Simulator

GR Δω per orbit vs Newton; ~43″/century readout; amplified animation.

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Gravitational Wave Binary Chirp (Inspiral)

Launch Simulator

Leading-order post-Newtonian inspiral of a compact binary: f(τ) ∝ τ^(−3/8), strain h(t) ∝ M_c^(5/3) f^(2/3) / D_L. Tune component masses m₁, m₂ and luminosity distance D_L; live h(t) and f(t) traces with the orbiting bodies on the side. The chirp mass M_c = (m₁m₂)^(3/5)/(m₁+m₂)^(1/5) is the very quantity LIGO/Virgo measures from the early inspiral; the frequency freezes at the Schwarzschild ISCO.

NewUniversity / research

Three-Body Figure-Eight

Launch Simulator

Equal masses: Chenciner–Montgomery choreography in 2D (RK4, periodic orbit).

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ISCO & Photon Sphere (V_eff)

Launch Simulator

Schwarzschild effective potential V_eff(r) for massive (timelike) and photon (null) test particles in geometric units. Sliding angular momentum L collapses the stable / unstable circular pair into the innermost stable circular orbit r_ISCO = 6M (the inner edge of accretion discs); for photons the unstable photon sphere r = 3M defines the inner ring of black-hole shadow images.

NewUniversity / research

Oberth Effect

Launch Simulator

Same prograde Δv at peri vs apo on one ellipse; higher ε when burning deep.

NewSchool

Earth–Moon Barycenter Wobble

Launch Simulator

Heliocentric path of Earth’s center: barycentric ellipse plus lunar epicycle (exaggerated).

Schwarzschild Orbit Precession (Rosette)

About this model

How it works

Frequently asked questions

Related simulators

Mercury Perihelion Precession

Gravitational Wave Binary Chirp (Inspiral)

Three-Body Figure-Eight

ISCO & Photon Sphere (V_eff)

Oberth Effect

Earth–Moon Barycenter Wobble

Schwarzschild Orbit Precession (Rosette)

About this model

How it works

Frequently asked questions