Mechanics, thermodynamics, and dynamical evolution of planetary and planetary-system bodies — from Kepler's two-body orbits to tidal dissipation, satellite/ring dynamics, and planet formation. Kepler 1609-1619: orbits are ellipses with…
planetary-physics
Kepler third law: T² = (4π²/GM)·a³ from two-body Hamiltonian
Kepler 1609-1619 empirically inferred three laws from Tycho Brahe's planetary observations: (K1) planetary orbits are ellipses with the Sun…
Hill sphere radius r_H = a·(m/(3M))^{1/3}
Hill 1878 (Amer. J. Math. 1:5) introduced the concept of the 'Hill sphere' — the region around a planet of mass m orbiting a star of mass M…
Roche limit d = R·(2ρ_M/ρ_m)^{1/3} (rigid-body)
Roche 1849 — a rigid-body self-gravitating satellite of density ρ_m orbiting a central body of density ρ_M and radius R is tidally…
Kepler 3 circular-orbit derivation: T²·GM − 4π²·a³ ≡ 0 exact
Exact symbolic verification of Kepler's third law for circular orbits. Equation of motion centripetal: m·v²/a = G·M·m/a² → v² = G·M/a…
Hill radius identity: r_H³·3M − m·a³ ≡ 0 exact
Exact symbolic verification of the Hill radius. Starting from the force-balance equation at L_1: G·m/r_H² = 3·G·M·r_H/a³. …
Roche limit identity: d³·ρ_m − 2·R³·ρ_M ≡ 0 exact
Exact symbolic verification of the rigid-body Roche limit. Starting from the tidal vs self-gravity balance: 2·G·M·r_m/d³ = G·m_m/r_m² with…