What it does
Near a black hole, gravity no longer follows Newton’s rules. Orbits do not close: each pass the closest point (the periapsis) moves forward, sometimes by more than a full turn. Inside 6M (where M = GM/c², half the Schwarzschild radius) no circular orbit is stable: this innermost stable circular orbit (ISCO) sets the inner edge of accretion discs. At 3M, the photon sphere, light itself can circle the hole, and rays aimed closer than 3√3 M fall in, which sets the size of the black hole’s shadow.
This plotter uses the exact equations for a non-spinning (Schwarzschild) black hole:
- the key radii for your mass in kilometres, and the ISCO orbital period;
- for a particle with chosen closest and farthest distances: the periapsis advance per orbit (exact, compared with the weak-field formula used for Mercury), energy and angular momentum per unit mass, the orbit drawn for several turns, and the effective potential;
- for a light ray with impact parameter b: whether it is captured, how much it is bent, and its closest approach, drawn among a fan of other rays.
How to use it
- Type the mass in solar masses (10 for a stellar black hole, 4 × 10⁶ for the Milky Way’s centre).
- Choose a massive particle and type the closest and farthest distances r_p and r_a in units of M. Equal values give a circular orbit. Choose how many turns to draw.
- Or choose a light ray and type the impact parameter b in units of M.
- Read the results and the table of key radii; download the path as CSV or the report.
Worked example
Take a made-up 10-solar-mass black hole. Its horizon is at 29.53 km, the photon sphere at 44.3 km and the ISCO at 88.6 km, where an orbit takes 4.548 ms as seen from far away. A particle moving between 10M and 30M (148 and 443 km) has E = 0.976315 and L = 4.3759 M c, and its periapsis advances 105.8° per orbit, far more than the weak-field formula’s 72°. The drawing shows the rosette this makes.
A light ray with b = 6M passes 4.453M (65.76 km) from the centre and is bent by 98.51°. Make b smaller than 5.196M and it is captured.
Tips
- Bring r_p close to 4M with a large r_a to see “zoom-whirl” orbits that circle several times near the hole before swinging out.
- Set r_p = r_a = 6 for the ISCO itself: the periapsis advance becomes huge because radial oscillations there take forever.
- With b just above 5.196M, the light ray loops around the photon sphere before escaping.
FAQ
Does it work for the Sun or Earth? Yes, outside the body: it is the exact field of any round, non-spinning mass. Use large r_p and r_a (Mercury’s orbit is about 4 × 10⁷ M) and the advance agrees with the weak-field formula.
What about spinning black holes? Spin (the Kerr metric) moves the ISCO inwards for prograde orbits and outwards for retrograde ones. It is not included.
Why use M as the unit? Every Schwarzschild black hole looks the same in units of M; only the kilometres and seconds scale with mass.
Where the numbers come from
- Schwarzschild geodesics, effective potential, circular orbits, ISCO at 6M, photon sphere at 3M, critical impact parameter 3√3 M, periapsis advance and light deflection: standard results of general relativity, for example Hartle, J. B., Gravity: An Introduction to Einstein's General Relativity (2003), chapters 9 and 15, and Misner, Thorne and Wheeler, Gravitation (1973).
- Periapsis advance in closed form with the complete elliptic integral K (the cubic-root form of the orbit equation), checked by fibub against SciPy solve_ivp integrations; light deflection checked against SciPy quad. GM☉/c² = 1.4766 km is the IAU nominal value, rounded.
- Example masses and orbits are made up.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.