🛰️ Orbital Mechanics & Kepler's Laws

The Physics of Staying in Space

Orbital mechanics (astrodynamics) describes how spacecraft and natural bodies move under gravity. It rests on Newton's law of gravitation and Kepler's empirical laws, and it provides the equations engineers use to place satellites, plan transfers between orbits, and send probes across the solar system.

Kepler's Three Laws

Johannes Kepler distilled planetary motion into three laws that apply to any two-body orbit:

1. Law of ellipses: every orbit is an ellipse with the central body at one focus. A circle is just an ellipse with zero eccentricity.
2. Law of equal areas: the line joining the body and the orbiting object sweeps out equal areas in equal times. The object therefore moves fastest at perigee (closest point) and slowest at apogee (farthest point).
3. Harmonic law: the square of the orbital period is proportional to the cube of the semi-major axis, T² ∝ a³. Larger orbits take dramatically longer.

Why Orbits Don't Fall Down

A satellite in orbit is in continuous free fall. It is constantly accelerating toward Earth under gravity, but its high tangential velocity carries it sideways so quickly that the curved Earth falls away beneath it at exactly the same rate. Gravity supplies precisely the centripetal force that bends the path into a closed loop, so no propulsion is needed to stay in orbit — only to change it.

Describing an Orbit: The Orbital Elements

Six orbital elements fully specify an orbit and the spacecraft's position within it:

Element	Symbol	Describes
Semi-major axis	a	Orbit size (and, via Kepler 3, period)
Eccentricity	e	Orbit shape (0 = circle, <1 = ellipse)
Inclination	i	Tilt of the orbit plane to the equator
Right ascension of ascending node	Ω	Orientation of the orbit plane
Argument of perigee	ω	Orientation of the ellipse within its plane
True anomaly	ν	Position of the satellite along the orbit

Circular and Elliptical Orbits

For a circular orbit at radius r, gravity equals the required centripetal force, giving a simple orbital speed:

v = √(μ / r)

where μ = GM is the gravitational parameter of the central body. An elliptical orbit has a varying radius and speed: fast at perigee, slow at apogee, in accordance with Kepler's second law. The semi-major axis a is the average of the perigee and apogee radii.

The Vis-Viva Equation

The single most useful relation in orbital mechanics is the vis-viva equation, which gives the speed anywhere on any orbit:

v² = μ (2/r − 1/a)

From it you can find the velocity at perigee and apogee, check whether an orbit is bound or escaping, and — most importantly — compute the velocity changes (delta-v) required for maneuvers. It unifies circular, elliptical, parabolic, and hyperbolic trajectories in one formula.

The Hohmann Transfer

To move a spacecraft from one circular orbit to a higher one, the most propellant-efficient maneuver is the Hohmann transfer, using two burns:

1. First burn: accelerate tangentially to enter an elliptical transfer orbit whose perigee touches the starting orbit and whose apogee touches the target orbit.
2. Second burn: at apogee, accelerate again to circularize at the higher orbit.

The vis-viva equation gives the speed before and after each burn, and the delta-v is their difference. The Hohmann transfer is the textbook method for raising satellites from a low parking orbit to a higher operational orbit, trading time for fuel economy.

Common Earth Orbits

Orbit	Altitude	Period	Use
Low Earth orbit (LEO)	~160–2000 km	~90 min	ISS, imaging, constellations
Medium Earth orbit (MEO)	~2000–35,000 km	2–12 h	GPS and navigation
Geostationary (GEO)	~35,786 km	24 h	Communications, weather

Geostationary orbit is special: at exactly 35,786 km over the equator, the orbital period equals one sidereal day, so the satellite appears to hover motionless over a fixed point — perfect for communications and weather observation. LEO, by contrast, offers cheap access and high resolution at the cost of rapid ground-track motion and the need for many satellites for continuous coverage.

Escape Velocity

To break free of a body's gravity entirely — to reach a parabolic trajectory with just enough energy to coast to infinity — a spacecraft needs the escape velocity:

v_esc = √(2μ / r) = √2 × v_circular

Escape velocity is exactly √2 (about 1.41) times the circular orbital speed at the same radius. From Earth's surface it is about 11.2 km/s. Exceeding it puts the craft on a hyperbolic, unbound trajectory — the gateway to interplanetary travel.

From Kepler to Mission Design

Kepler's laws set the geometry, the vis-viva equation provides the speeds, and the Hohmann transfer and escape velocity give the recipes for maneuvers. Together they let mission designers compute exactly how much delta-v — and therefore how much propellant — a mission needs, turning the elegant geometry of orbits into the practical business of flying spacecraft.