Keplerian Orbit, Escape Speed & Roche Explorer
Compute Keplerian orbital period, circular speed, escape velocity, and tidal Roche disruption radius for any system.
Under Newton's law of universal gravitation, a body orbiting primary mass M at semi-major axis a has orbital period P = 2π√(a³/GM), circular orbital speed v_circ = √(GM/r), escape speed v_esc = √2 v_circ, and fluid tidal Roche limit d_Roche = 2.44 R_M (ρ_M / ρ_m)^(1/3).
Interactive Physics Calculator
Governing Equation & Worked Benchmark
Kepler's Third Law, Vis-Viva Velocity & Fluid Roche Limit
P_yr = √[ (a_AU)³ / (M / M_☉) ] ; v_circ = 29.78 km/s · √[ (M/M_☉) / a_AU ] ; d_Roche = 2.44 R_M (ρ_M / ρ_m)^(1/3)At any point r in a circular orbit, multiplying speed by √2 ≈ 1.4142 reaches the parabolic escape threshold where total mechanical energy E = K + U = 0.
| Benchmark Step | Physical Details & Result |
|---|---|
| Scenario | Earth orbiting the Sun (M = 1.0 M_☉, a = 1.0 AU) versus star S2 at pericenter around Sgr A* (M = 4.15 × 10⁶ M_☉, r = 120 AU). |
| Calculation | Earth (1 AU): P = 1.00 yr, v_circ = 29.78 km/s, v_esc = 42.12 km/s. Star S2 around Sgr A* (r = 120 AU): v_circ = 29.78 × √(4.15×10⁶ / 120) = 5,537 km/s (and v_peri ≈ 7,650 km/s on its eccentric e = 0.885 orbit). |
| Physical Takeaway | Kepler's third law scales across 15 orders of magnitude in mass—from asteroid moonlets and planetary rings to stars orbiting supermassive black holes. |
Physical Mechanism & Interpretation
Inside the fluid Roche limit d_Roche ≈ 2.44 R_M (ρ_M / ρ_m)^(1/3), the primary's differential tidal stretching exceeds the satellite's self-gravity, dispersing moons or comets into planetary rings (like Saturn's main rings inside 140,000 km) or triggering Tidal Disruption Events around black holes.
Outside the Roche limit, a planet commands its own gravitational domain out to the Hill sphere radius R_H ≈ a (m / 3M)^(1/3).
Model Assumptions & Systematic Uncertainties
Close to compact objects (r < 100 R_s), post-Newtonian general-relativistic corrections add pericenter precession and gravitational redshift.
Related Concepts & Methods
Scientific Sources & Provenance
- NASA/JPL Solar System Dynamics & Planetary Fact Sheets (DE441 (2021/2026)) — Park, R. S., et al. 2021, AJ, 161, 105 (The JPL Planetary and Lunar Ephemerides DE440 and DE441). Archive / Reference ↗
- ESA Gaia Archive — Data Release 3 (DR3 (2022-06-13)) — Gaia Collaboration, Vallenari, A., et al. 2023, A&A, 674, A1 (Gaia Data Release 3). Archive / Reference ↗
- NASA Exoplanet Archive (2026-10-01 snapshot) — Akeson, R. L., et al. 2013, PASP, 125, 989; NASA Exoplanet Archive (IPAC/Caltech). Archive / Reference ↗