Propulsion · two-body dynamics · the Solar System
A working reference for how rockets accelerate and how orbits behave — derived from first principles, with live calculators, worked examples, and the real numbers for every planet, moon and satellite regime. Hover every i for the detail.
Propulsion · 1
A rocket carries its own reaction mass and throws it backwards. Conservation of momentum — the exhaust gains backward momentum, the rocket gains an equal forward momentum — leads directly to Tsiolkovsky's 1903 result.
It comes from integrating m·dv = −v_e·dm. Because Δv depends on the logarithm of the mass ratio, gains get punishingly expensive — the tyranny of the rocket equation. To reach low Earth orbit you need ~9.4 km/s.
Propulsion · 2
The rocket equation tells you the final speed; thrust tells you how fast you get there and whether you leave the pad at all.
The pressure term is why engines have two I_sp ratings, and why I_sp = F/(ṁg₀). Lift-off requires the thrust-to-weight ratio to exceed 1.
Propulsion · 3
Because Δv grows only with the log of the mass ratio, a single tank can't reach orbit: most of its dry mass is empty structure you keep dragging along. Staging throws that structure away mid-flight, so each stage starts with a fresh, favourable mass ratio. Total Δv is simply the sum.
Each upper stage is "payload" to the stage below it, so lower stages must lift everything above. The penalty of an extra stage is more dry mass and complexity, so real rockets settle on 2–3 stages. The payload fraction to LEO is typically just 2–4%.
Orbital mechanics · 1
Once in space, a vehicle coasts under gravity alone. For one body orbiting a much larger one, the path is a conic section and the motion is governed by energy. The vis-viva equation gives the speed anywhere on any orbit.
Energy alone decides the type of orbit: ε<0 is bound (ellipse/circle), ε=0 is the escape parabola, ε>0 is an unbound hyperbola. Set up an orbit and read its full state on the right.
Orbital mechanics · 2
Empirical in 1609–1619, derived from Newton's gravity decades later, they still describe every closed orbit.
Each planet moves on an ellipse with the Sun at one focus — not a circle, not centred on the Sun.
The line from Sun to planet sweeps equal areas in equal times, so a body moves fastest at perihelion, slowest at aphelion. This is conservation of angular momentum.
The square of the period is proportional to the cube of the semi-major axis: T² ∝ a³. Bigger orbits are slower, by a precise rule.
Eight wedges, each swept in T/8 — equal areas, wildly unequal shapes.
Orbital mechanics · 3
Two numbers fix an orbit's size and shape; three fix its orientation in space; one fixes where the body is right now. Together they pin down a trajectory completely. Tap each element.
Orbital mechanics · 4
You can't steer in space; you can only change velocity. Two classic problems: raising an orbit (Hohmann transfer) and tilting it (plane change).
From a 300 km parking orbit around Earth.
A 30° change in LEO costs ~4 km/s — more than reaching the Moon from LEO. This is why rockets launch toward their target inclination, and why the Oberth effect rewards burning low and fast.
The Solar System · 1
The same equations describe all eight planets. Watch them run (inner planets race, outer ones crawl — Kepler's third law made visible), and click any one for its data and the formulas applied to it.
Orbit radii compressed for display (∝ √a) — not to scale. Speeds reflect real relative periods.
log T vs log a — slope 3/2 for every planet. Click a point.
The Solar System · 2
Where satellites live is set entirely by altitude — which fixes period and speed through Kepler's third law. Click a regime for the numbers and what it's used for.
Concentric regimes around Earth (compressed). GEO sits where the orbital period equals one sidereal day.
The Solar System · 3
Satellites obey the same laws around their planets that planets obey around the Sun — only the value of μ changes. Click any moon for its orbit data.
Quick reference
Planetary and satellite values are standard mean elements rounded for clarity; orbits are idealised two-body (no perturbations). For mission-grade work use full ephemerides (e.g. NASA JPL).