| name | orbital-mechanics |
| description | Orbital mechanics — Kepler's laws, vis-viva equation, orbital elements (a, e, i, Ω, ω, ν), Hohmann transfer, bi-elliptic transfer, plane change maneuvers, delta-V budget, patched-conic interplanetary trajectories, launch window, gravity assists, atmospheric reentry, J2 perturbation, TLE format, and STK/GMAT software conventions. |
| metadata | {"priority":7,"promptSignals":{"phrases":["orbital mechanics","Hohmann transfer","delta-V budget","Kepler orbit","orbital elements","interplanetary trajectory"],"minScore":3}} |
Orbital Mechanics — Complete Skill
Kepler's Laws and Two-Body Problem
Kepler's Laws
First Law: orbit is a conic section (ellipse, parabola, or hyperbola) with central body at one focus
Second Law: radius vector sweeps equal areas in equal time → conservation of angular momentum
Third Law: T² = (4π²/μ) × a³ [T = period [s]; μ = GM_central [m³/s²]; a = semi-major axis [m]]
μ_Earth = 3.986×10¹⁴ m³/s² = 398,600 km³/s²; μ_Sun = 1.327×10²⁰ m³/s²
Period of LEO (a = 6578 km = R_Earth + 200 km):
T = 2π × √(a³/μ) = 2π × √((6.578×10⁶)³ / 3.986×10¹⁴) = 5310 s = 88.5 min
Orbital Parameters
Vis-Viva Equation
Vis-viva (energy) equation:
v² = μ × (2/r − 1/a) [v = speed at radius r; a = semi-major axis; μ = gravitational parameter]
Circular orbit: v_c = √(μ/r) [a = r for circular]
Escape velocity: v_esc = √(2μ/r) = √2 × v_c [e = 1; parabolic trajectory]
Orbital energy:
ε = −μ/(2a) [specific orbital energy [J/kg]; negative = bound orbit; zero = escape]
Angular momentum:
h = r × v_perp = √(μ × a × (1−e²)) [h = specific angular momentum [m²/s]; e = eccentricity]
Classical Orbital Elements (COE)
Six elements uniquely define orbit:
- a — semi-major axis [km]: orbit size
- e — eccentricity [dimensionless]: orbit shape (0 = circle; 0–1 = ellipse; 1 = parabola; >1 = hyperbola)
- i — inclination [°]: angle between orbital plane and equatorial plane
- Ω — RAAN (Right Ascension of Ascending Node) [°]: orbit plane orientation in space
- ω — argument of perigee [°]: orientation of ellipse in orbit plane
- ν (or M, E) — true anomaly (or mean/eccentric anomaly) [°]: position in orbit at epoch
Perigee and apogee:
r_p = a(1−e) [closest approach; perigee for Earth orbits]
r_a = a(1+e) [farthest point; apogee]
v_p = √(μ(1+e)/(a(1−e))) = √(μ(2/r_p − 1/a)) [from vis-viva at perigee]
v_a = √(μ(1−e)/(a(1+e))) [at apogee]
Orbital Maneuvers
Hohmann Transfer (Minimum Energy)
Hohmann transfer: two-impulse maneuver between coplanar circular orbits; most fuel-efficient for orbit change
Transfer ellipse: perigee = r₁ (departure); apogee = r₂ (arrival)
Semi-major axis of transfer: a_t = (r₁ + r₂) / 2
Delta-V calculations:
ΔV₁ (at departure, perigee burn):
ΔV₁ = v_t_p − v_c1 = √(μ/r₁) × (√(2r₂/(r₁+r₂)) − 1)
ΔV₂ (at arrival, apogee burn):
ΔV₂ = v_c2 − v_t_a = √(μ/r₂) × (1 − √(2r₁/(r₁+r₂)))
Total ΔV = |ΔV₁| + |ΔV₂|
Example — LEO to GEO (r₁ = 6578 km; r₂ = 42,164 km):
v_c1 = √(398600/6578) = 7.784 km/s (LEO)
v_c2 = √(398600/42164) = 3.075 km/s (GEO)
a_t = (6578 + 42164)/2 = 24,371 km
ΔV₁ = 7.784 × (√(2×42164/(6578+42164)) − 1) = 7.784 × (√(0.865) − 1... )
Actually: v_t_p = √(2×398600×42164/(6578×(6578+42164))) = √(33,588,480,000/323,561,476) = √103.81 = 10.19 km/s
ΔV₁ = 10.19 − 7.784 = 2.41 km/s
v_t_a = √(2×398600×6578/(42164×48742)) = √(5,244,776,000/2,055,714,688) = √2.551 = 1.597 km/s
ΔV₂ = 3.075 − 1.597 = 1.478 km/s
Total ΔV_Hohmann = 2.41 + 1.48 = 3.89 km/s
Transfer time:
t_transfer = π × √(a_t³/μ) = π × √(24,371³/398,600) = π × √(36,300) = π × 190.5 s × ...
t = π × √(24371000³ / 3.986×10¹⁴) = π × √(1.443×10¹¹) = π × 379,900 s/...
t = 0.5 × T_transfer = π × √(24,371³/398,600) ≈ 18,925 s ≈ 5.26 hours
Bi-Elliptic Transfer
Bi-elliptic vs. Hohmann:
Bi-elliptic uses 3 burns; more fuel for small/medium ΔR; less fuel if r₂/r₁ > ~11.94
For GEO from LEO: bi-elliptic with aphelion at 500,000 km saves ΔV vs. direct Hohmann
Plane Change
Pure plane change (inclination change):
ΔV_plane = 2 × v × sin(Δi/2) [v = circular velocity at maneuver point; Δi = inclination change [rad]]
Very expensive! At LEO (v = 7.7 km/s): 28.5° plane change: ΔV = 2×7.7×sin(14.25°) = 3.8 km/s
Principle: always combine plane change with speed change (do plane change at apogee of transfer where v is lowest)
Delta-V Budget
Mission ΔV Components
Complete ΔV budget for LEO-GEO satellite:
| Maneuver | ΔV [km/s] |
|---|
| Gravity losses (launch) | ~1.5 |
| Drag losses (launch) | ~0.1 |
| LEO injection | 0 (included above) |
| LEO to GTO (super-synchronous) | ~2.44 |
| GTO to GEO (apogee kick) | ~1.48 |
| Station-keeping (15 yr) | ~0.5 |
| Disposal to graveyard | ~0.01 |
| Total mission | ~5.5–6.5 |
Tsiolkovsky rocket equation:
ΔV = I_sp × g₀ × ln(m₀/m_f) [ΔV [m/s]; I_sp = specific impulse [s]; g₀ = 9.81 m/s²; m₀ = initial mass; m_f = final mass (after burn)]
Propellant mass fraction: m_prop/m₀ = 1 − exp(−ΔV/(I_sp × g₀))
Example:
ΔV = 1000 m/s; I_sp = 300 s (bipropellant)
m_prop/m₀ = 1 − exp(−1000/(300×9.81)) = 1 − exp(−0.340) = 1 − 0.711 = 0.289 = 28.9% of wet mass
Interplanetary Trajectories
Patched Conic Approximation
Sphere of Influence (SOI):
r_SOI = a_planet × (m_planet/m_sun)^(2/5) [radius of planet's gravitational dominance]
Earth SOI: 924,000 km; Mars SOI: 577,000 km
Interplanetary Hohmann:
Launch planet departure: hyperbolic escape from planet SOI; v_∞ = excess hyperbolic velocity
Arrival planet: hyperbolic capture at v_∞; ΔV to capture into orbit
Earth to Mars (opposition-class):
Phase angle at launch: ≈ 44.4° ahead of Mars
Transfer time: ≈ 8.5 months (180° transfer)
ΔV from Earth: ΔV₁ ≈ 3.6 km/s (heliocentric) + Earth escape
Synodic period (launch window repeat): T_syn = 1/(1/T_Earth − 1/T_Mars) = 1/(1/1 − 1/1.881) = 2.135 years
Gravity Assist (Flyby)
Gravity assist: spacecraft gains (or loses) energy relative to Sun without fuel burn
ΔV_assist = 2 × v_∞ × sin(δ/2) [δ = turn angle; v_∞ = hyperbolic excess speed]
Vinf entry ≠ Vinf exit (gravity focus); energy gained from planet's orbital motion
Example: Voyager 2 used Jupiter, Saturn, Uranus, Neptune gravity assists; achieved solar escape velocity
Earth Orbit Perturbations
J2 Perturbation (Oblateness)
J2 effect (Earth's equatorial bulge):
RAAN precession rate: dΩ/dt = −(3/2) × n × J₂ × (R_E/a)² × cos(i) / (1−e²)² [rad/s]
J₂ = 1.08263×10⁻³; R_E = 6378.1 km; n = √(μ/a³)
Sun-synchronous orbit: choose i and a so dΩ/dt = 360°/year (Earth's orbital rate about Sun)
RAAN advances ~0.9856°/day → orbital plane stays fixed relative to Sun → constant illumination angle
For a = 700 km, i = 98.2° (slightly retrograde) → sun-synchronous
TLE Format
Two-Line Element Set
TLE (Two-Line Elements): standard format for satellite tracking
Line 1: NORAD ID, classification, epoch, ballistic coefficient, drag, ephemeris type
Line 2: inclination, RAAN, eccentricity (implied decimal), arg perigee, mean anomaly, mean motion (rev/day)
Mean motion to semi-major axis:
n [rev/day] → n [rad/s] = n × 2π / 86400
a = (μ/n²)^(1/3)
Standards and References
| Standard | Scope |
|---|
| NASA-STD-3001 | Human factors for space vehicles |
| SMAD (Space Mission Analysis and Design) | Standard textbook / reference |
| CCSDS 502.0-B-2 | Orbit data messages (OEM format) |
| AIAA S-096 | Coordinate systems for space missions |
| AGI STK | Satellite Tool Kit (industry-standard software) |
| GMAT | Open-source trajectory optimization (NASA) |
Output
Provide: mission definition (departure: orbit/body; target: orbit/body; mission epoch), orbital elements (a [km]; e; i [°]; period T [min]; r_p [km]; r_a [km]; v at perigee and apogee from vis-viva [km/s]), maneuver sequence (Hohmann/bi-elliptic/plane change; each ΔV [km/s]; direction: prograde/retrograde/normal; burn duration given I_sp and thrust), ΔV budget (table: each maneuver ΔV [km/s]; total mission ΔV [km/s]; stationkeeping [km/s/year]), propellant mass (Tsiolkovsky: m_prop/m₀ = 1−exp(−ΔV/(I_sp×g₀)); I_sp [s]; dry mass assumption; wet mass), transfer time (Hohmann: t = π√(a_t³/μ) [hours]; patched-conic interplanetary: [months]), perturbations (J2 RAAN drift [°/day]; sun-synchronous check if applicable; atmospheric drag: ballistic coefficient; orbit decay [km/day]), and applicable standard (CCSDS 502.0 for orbit data; AGI STK for analysis tool; SMAD for systems-level trades).
