Why Things Orbit Rather Than Fall
The common intuition about orbits — that gravity and centrifugal force balance — is subtly wrong, and the correct picture is more interesting. A satellite in orbit is not hovering. It is falling. It falls continuously toward Earth, and it moves horizontally fast enough that by the time it has fallen a certain distance, the Earth's surface has curved away by the same amount. It is perpetually falling and perpetually missing the ground. Newton illustrated this with his cannon thought experiment: fire a cannonball horizontally from a very tall mountain. Too slow and it hits the ground. At exactly 7.9 km/s (at sea level), the curvature of its fall matches the curvature of the Earth — it orbits. At 11.2 km/s it escapes Earth's gravity entirely. The orbit is not a balance between forces; it is inertia and gravity producing a curved path that never intersects the surface.
Kepler's Three Laws: The Geometry
Working from Tycho Brahe's planetary observations in the early 1600s, Kepler deduced three empirical laws that Newton later derived from gravitational theory:
- First Law: Orbits are ellipses with the central body at one focus. Two parameters describe orbit shape: semi-major axis a and eccentricity e.
- Second Law: A line from body to satellite sweeps equal areas in equal times — so objects move fastest at periapsis (closest approach) and slowest at apoapsis.
- Third Law: T² ∝ a³ — directly gives the orbital period from the orbit size: T = 2π√(a³/μ).
The Vis-Viva Equation
The most useful single equation in astrodynamics relates orbital speed at any point to the orbit's semi-major axis and current distance from the central body:
Delta-v: The Currency of Space Travel
Moving between orbits requires a change in velocity (Δv), achieved by firing rocket engines. The most efficient two-burn transfer between circular orbits is the Hohmann transfer. For a transfer from LEO (200 km) to GEO: The total Δv budget determines propellant mass, which determines spacecraft mass, which determines which launch vehicle is needed. Minimising Δv is the fundamental optimisation of mission design. Enter semi-major axis and eccentricity. EngForge computes orbital period, periapsis/apoapsis altitudes and velocities, Δv to circularise, and draws the orbital path to scale.