Astronomy
Trojan swarms
Swarms integrated about the triangular points in the rotating frame: tadpoles at small amplitude, horseshoes past the critical one, dispersal once the planet exceeds Routh’s limit.
The model and its limits
The planar circular restricted problem: the planet on a circle, no Saturn, no inclination, no eccentricity of the planet. The linear readings (period, frequency, growth) are ring 2’s formulas; the integrated swarms are the Tier A run.
Equations used in the module
Equations ν/n = √((1 − √(1 − 27μ(1−μ)))/2); φ = ±π/3 + A cos(νt); stability: 27μ(1-μ) < 1 integrated: a = −(1−μ) r/r³ − μ (r − r_J)/|r − r_J|³ − μ r_J; C_J = x² + y² + 2(1−μ)/r₁ + 2μ/r₂ − v²
