Newton's Law of Gravitation
F = GMm/r²
Orbital Velocity (Circular)
v = √(GM/r)
Orbital Period
T = 2π√(r³/GM) (Kepler's 3rd Law)
Escape Velocity
v_esc = √(2GM/r) = √2 × v_orbital
Total Orbital Energy
E = -GMm/(2a) (a = semi-major axis)
Binding energy = -E = GMm/(2a)
Elliptical Orbit
r_periapsis = a(1-e) · r_apoapsis = a(1+e)
v_peri = √[GM(1+e)/a(1-e)] · v_apo = √[GM(1-e)/a(1+e)]
Kepler's Laws
- Planets orbit in ellipses with the Sun at one focus
- A line from Sun to planet sweeps equal areas in equal times
- T² ∝ a³ (period squared proportional to semi-major axis cubed)
Vis-Viva Equation
v² = GM(2/r - 1/a)
Mission Outcomes
- v < v_orbital: Object crashes or enters sub-orbit
- v = v_orbital: Perfect circular orbit
- v_orbital < v < v_escape: Elliptical orbit
- v = v_escape: Parabolic escape trajectory
- v > v_escape: Hyperbolic trajectory