Astronomy & Celestial Mechanics
the sky, computed
Theme

Everything is a conic

One force law, four curves, and the discovery that a comet and a planet differ only in how fast they were going.
empty focusabae = 0.6arperiapsisapoapsis Orbits

The orbit is an ellipse, and the Sun is not in the middle of it

Kepler's first law is usually drawn wrong. The interesting content is not the ellipse — it is the focus, and the fact that one of the two is empty.

 common periapsiscircleellipse, e = 0.5ellipse, e = 0.9parabola, e = 1hyperbola, e = 1.5e < 1 returnse ≥ 1 never does Orbits

Every orbit one force allows, and the number that picks between them

Circle, ellipse, parabola, hyperbola. A single inverse-square force permits exactly these four, and one number decides which — including whether the body ever comes back.

 burn 1: +0.202burn 2: +0.158total 0.360 — and no way to spend lesscoast: half an ellipse, 1.21 of an inner-orbit yearspeeds in units of the inner circular speed Spaceflight

The cheapest way between two orbits, and why it is so slow

Two burns and a long coast is the least fuel that will move a spacecraft between two circular orbits. It is also, for anything beyond the Moon, an unreasonably long wait.

 2468101200.511.5distance, in body radiiescape speedcircular speedthe surfacea high orbitgeostationarythe gap is always a factor of √2 Spaceflight

The speed that does not come back, and the √2 that separates it

Escape speed is exactly the square root of two times circular speed, at every distance from every body. Being in orbit is already 71% of the way to leaving.

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