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…
Kepler-equation Newton-Raphson iteration: M = E - e sin(E); one Newton step from initial guess E_0 = M
Kepler-equation Newton-Raphson framework (Kepler 1621; Newton 1686; Danby 1987 Celest Mech 40, 303). Setup: Kepler's equation M = E - e…
Jacobi integral in CR3BP via canonical Poisson bracket: {x, H} = dH/dp_x = Omega y + p_x
Jacobi-integral framework via canonical Poisson brackets (Jacobi 1836). Setup: in the circular restricted three-body problem (CR3BP), the…
KAM golden-ratio invariant torus: phi = (1+sqrt(5))/2 solves phi^2 - phi - 1 = 0 (most irrational frequency)
KAM golden-ratio-torus framework (Kolmogorov 1954 Dokl Akad Nauk SSSR 98, 527; Arnold 1963 Usp Mat Nauk 18, 13; Moser 1962 Nachr Akad Wiss…
Theorem: Kepler Newton step from E = M gives (M e cos M - M - e sin M) / (e cos M - 1)
Theorem (Kepler single-Newton-step canonical): starting Newton-Raphson from initial guess E_0 = M applied to f(E) = E - e sin E - M gives…
Theorem: Poisson bracket {x, H_CR3BP} = Omega y + p_x (Coriolis-shifted generalised velocity)
Theorem (CR3BP Poisson-bracket canonical): for the rotating-frame Hamiltonian H = (p_x^2 + p_y^2)/2 + Omega^2 (x^2 + y^2)/2 - Omega (x p_y…
Theorem: golden ratio phi = (1+sqrt(5))/2 satisfies phi^2 - phi - 1 = 0 (KAM most-irrational)
Theorem (golden-ratio KAM-algebraic canonical): the golden ratio phi = (1 + sqrt(5))/2 is the unique positive real root of the polynomial…
Planet formation (Safronov 1969)
V Safronov 1969 planetesimal-accretion; modern pebble-accretion Lambrechts-Johansen 2012; CAI-to-planet timescale ~10 My.
Nice model (Tsiganis 2005)
Tsiganis-Gomes-Morbidelli-Levison 2005 giant-planet migration; Late Heavy Bombardment ~3.9 Gya; modern jumping-Jupiter variant.
Habitable zone (Kasting 1993)
J Kasting-D Whitmire-R Reynolds 1993 conservative + optimistic CHZ; modern Kopparapu 2013 update; basis of biosignature target-selection.
Geodynamo (Glatzmaier-Roberts 1995)
Glatzmaier-Roberts 1995 first 3D self-consistent geodynamo simulation; reproduced field-reversals; modern multi-scale + sub-grid models.
Mass-radius (Seager 2007)
Seager 2007 + Zeng 2019 M-R relation: rocky / water-world / gas-giant; modern Bayesian retrieval + JWST observations 2024 expanding sample.
Subsurface oceans
Cassini-magnetometer + Galileo-induced-field; Europa + Enceladus + Titan + Ganymede subsurface-oceans; modern Europa-Clipper 2024 mission.
Kepler laws (1609-1619)
J Kepler 1609-1619 + Newton 1687 universal-gravitation; modern modern foundational text + 2024 ML-driven solar-system stability long-term.
Titius-Bode (1772)
J Titius 1766 + J Bode 1772 distance-law a=0.4+0.3·2^k; modern modern foundational text + statistical-significance debate +…
Safronov (1969)
V Safronov 1969 planetesimal-accretion; modern modern foundational text + Wetherill 1980 + 2024 streaming-instability + pebble-accretion…
Nice model (Tsiganis 2005)
K Tsiganis-Gomes-Morbidelli-Levison 2005 Nice-model planet-migration; modern modern foundational + LHB explanation + 2024…
Geodynamo (Glatzmaier-Roberts 1995)
G Glatzmaier-P Roberts 1995 first-3D-MHD-geodynamic; modern modern foundational text + 2024 ML-paleomagnetic Ediacaran field weakness.
Equilibrium temperature (1900s)
S Arrhenius 1896 + Hulburt 1931 T_eq=(L*(1-A)/16πσd²)^¼; modern modern foundational text + greenhouse-correction + 2024…