The Two-Body Problem
The foundation of orbital mechanics is the idealized two-body problem: a small object (satellite) in the gravitational field of a much larger body (Earth). Under this assumption, the only force acting on the satellite is Earth's gravity, and the motion can be described analytically.
Newton's law of gravitation gives us the equation of motion:
r = position vector from Earth's center to satelliter̈ = second time derivative (acceleration)μ = GM = gravitational parameter = 398,600.4418 km³/s²r = |r| = scalar distance from Earth's center
This differential equation has analytical solutions that trace out conic sections — circles, ellipses, parabolas, or hyperbolas — depending on the satellite's total energy. Satellites in stable orbit follow ellipses.
In reality, many forces act on a satellite (atmospheric drag, solar radiation, Moon's gravity). The two-body problem ignores all of these. It gives us a clean analytical solution — a perfect baseline that perturbation theory then corrects. See Chapter 03 for perturbations.
Kepler's Three Laws
Johannes Kepler (1609–1619) empirically derived three laws from Tycho Brahe's planetary observations. These laws emerge naturally from the two-body problem and remain central to modern astrodynamics.
First Law — Elliptical Orbits
The orbit of a satellite around Earth is an ellipse with Earth's center at one focus. This means the distance between the satellite and Earth varies continuously — minimum at perigee, maximum at apogee.
r = orbital radius at true anomaly θp = a(1 − e²) = semi-latus rectuma = semi-major axise = eccentricity (0 = circle, 0–1 = ellipse)θ = true anomaly (angle from perigee)
Second Law — Equal Areas
A satellite sweeps out equal areas in equal times. This is conservation of angular momentum in disguise: a satellite moves faster near perigee (lower altitude) and slower near apogee (higher altitude).
h = specific angular momentum vector (constant throughout orbit)Third Law — Period Relation
The square of the orbital period is proportional to the cube of the semi-major axis. This is why GPS satellites at ~20,200 km orbit once per ~12 hours, while the ISS at ~420 km orbits once per ~92 minutes.
T = orbital period (seconds)a = semi-major axis (km)μ = 398,600.4418 km³/s²
| Object | Altitude (km) | Semi-major axis (km) | Period | Velocity (km/s) |
|---|---|---|---|---|
| ISS | ~420 | 6,791 | 92 min | 7.66 |
| Starlink | ~550 | 6,921 | 95.5 min | 7.60 |
| GPS | ~20,200 | 26,571 | 11h 58m | 3.87 |
| GEO (Clarke Belt) | 35,786 | 42,164 | 23h 56m | 3.07 |
The Vis-Viva Equation
The vis-viva equation is perhaps the single most useful result in orbital mechanics. It relates a satellite's speed at any point in its orbit to its distance from Earth and the orbit's semi-major axis — through conservation of energy.
v = orbital speed at radius r (km/s)μ = gravitational parameter (km³/s²)r = current distance from Earth's center (km)a = semi-major axis of the orbit (km)
For a circular orbit, r = a everywhere, giving v = √(μ/r). This is why lower satellites move faster — they're in a deeper gravitational well. A 1 m/s increase in speed at ISS altitude raises the opposite side of the orbit by ~1.75 km.
The vis-viva equation underlies all delta-v calculations in the maneuver planning module. When a conjunction is detected, the Clohessy-Wiltshire model computes the minimum Δv needed — and vis-viva tells us how that translates to an altitude change.
Classical Orbital Elements
Six numbers fully describe any Keplerian orbit. These are the Classical Orbital Elements (COEs) — a compact parameterization used in TLE sets and almost every orbital database.
| Symbol | Element | Description | Range |
|---|---|---|---|
| a | Semi-major axis | Half the long axis of the ellipse. Determines orbit size and period. | 0 → ∞ km |
| e | Eccentricity | Shape of orbit. 0 = circle, 0–1 = ellipse, 1 = parabola (escape). | 0 → <1 |
| i | Inclination | Tilt of orbit plane relative to Earth's equatorial plane. | 0° – 180° |
| Ω | RAAN | Right Ascension of Ascending Node. Rotates orbit plane around polar axis. | 0° – 360° |
| ω | Argument of perigee | Angle from ascending node to closest approach point. | 0° – 360° |
| ν or M | True / Mean anomaly | Current position in orbit. True = actual angle; Mean = time-averaged. | 0° – 360° |
Converting between mean anomaly M and true anomaly ν requires solving Kepler's Equation — a transcendental equation typically solved iteratively:
M = mean anomaly (linear in time: M = n·t, n = mean motion)E = eccentric anomaly (solved iteratively via Newton-Raphson)e = eccentricity
Two-Line Element Sets (TLEs)
A TLE is the standard format used by NORAD and CelesTrak to distribute orbital data for tracked space objects. Each TLE encodes the six orbital elements plus perturbation coefficients in exactly 69 characters per line.
1 25544U 98067A 24001.50000000 .00003456 00000-0 63041-4 0 9992
2 25544 51.6416 247.4627 0006703 130.5360 325.0288 15.50377579431937
Line 2: Inclination · RAAN · Eccentricity (assumed decimal) · Arg of Perigee · Mean Anomaly · Mean Motion (rev/day) · Rev number
TLE accuracy degrades over time as unmodeled perturbations accumulate. A fresh LEO TLE is typically accurate to ~1 km; after 7 days it may be off by 10+ km. This is why VectraSpace refreshes TLEs every 6 hours from CelesTrak and Space-Track.
SGP4 / SDP4 Propagation
The Simplified General Perturbations 4 (SGP4) model is the standard algorithm for propagating TLE sets forward in time. It analytically approximates the most significant orbital perturbations — Earth's oblateness (J₂, J₃, J₄), atmospheric drag, and solar/lunar effects (SDP4 for deep-space orbits).
SGP4 takes a TLE and a time offset Δt, and returns an ECI position and velocity vector. The computation is fast — thousands of satellites can be propagated per second on modern hardware — making it ideal for VectraSpace's vectorized batch processing.
VectraSpace uses the Skyfield Python library's SGP4 implementation, propagating position arrays over 12–72 hour windows at 1-minute resolution. NumPy batching allows all satellites in a regime to be processed simultaneously, achieving 50× speedup over sequential loops.
SGP4 is a mean element theory — it models average perturbations, not instantaneous forces. For high-precision conjunction analysis (Pc < 10⁻⁶), higher-fidelity numerical propagators with real atmospheric density models are required. VectraSpace's results should be treated as screening-level estimates, not operationally certified predictions.
Reference Frames
Orbital calculations require a clear choice of coordinate system. VectraSpace uses two primary frames:
ECI — Earth-Centered Inertial
Origin at Earth's center. X-axis points to the vernal equinox; Z-axis to the celestial north pole. Does not rotate with Earth. Satellite positions and velocities are expressed in ECI for propagation calculations.
RTN — Radial-Transverse-Normal (Hill Frame)
A local coordinate frame co-moving with the reference satellite: R (radial, toward/away from Earth), T (transverse, along-track), N (normal, out-of-plane). Delta-v maneuver vectors are expressed in RTN.
Circular Orbital Velocity
For a circular orbit, the satellite's speed is constant and determined entirely by altitude. This is the regime most LEO satellites operate in:
v_c = circular velocity (km/s)R_E = Earth's mean radius = 6,371 kmh = altitude above surface (km)
At ISS altitude (420 km): v ≈ 7.66 km/s. At GEO (35,786 km): v ≈ 3.07 km/s. Two LEO satellites in crossing orbits can have a relative velocity of up to 15+ km/s — equivalent to a small car moving 54,000 km/h. A 1 cm aluminum sphere at this speed carries the kinetic energy of a hand grenade.