Astronomy & Celestial Mechanics
the sky, computed
Generator

The ellipse-orbit generator

An orbit at eccentricity 0.6
An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.

An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00.

7 essays call ellipse-orbit. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Every figure listed here is the same construction drawn at different numbers, so a correction to one is a correction to all of them.

An orbit at eccentricity 0.6. An orbit of eccentricity 0.6. The primary sits at a focus, offset from the centre by 0.6 of the semi-major axis, and the closest and furthest points differ by a factor of 4.00. 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.

 Equal areas in equal times, at eccentricity 0.65. Positions of an orbiting body at equal intervals of time, obtained by solving Kepler's equation. The two shaded sectors span the same interval and enclose the same area — a long thin one at the far end, a short fat one at the close approach. Orbits

Equal areas in equal times, which is angular momentum in disguise

Kepler's second law is a statement about the area a radius line sweeps. It looks like an odd thing to have noticed, and it turns out to be a conservation law arriving eighty years early.

 Period against size for the planets, around the Sun. Orbital period against semi-major axis on logarithmic axes. The line has slope exactly three-halves — the harmonic law — and the measured bodies sit on it. Orbits

The law that links period to size, and weighs everything

Kepler found that the square of the period goes as the cube of the orbit. Newton found the constant of proportionality, and that constant is a mass — which is how every mass in astronomy has been obtained since.

 Every orbit one force allows. Circle, ellipse, parabola and hyperbola, all sharing a focus and a closest approach. The eccentricity alone decides which one a body is on, and whether it returns. 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.

 An orbit at i = 42°, Ω = 35°, ω = 55°. The three orientation elements. The orbit is generated in its own plane and rotated by the standard sequence, so the inclination, the node line and the argument of periapsis are the angles that produced the drawn curve rather than labels applied to it. Orbits

Six numbers that fix an orbit for all time, and the sixth is the awkward one

Five of the orbital elements describe a curve that never changes. The sixth says where on it the body is, and it is the only one that has to keep being measured.

 Two bodies at a mass ratio of 3 to 1. Both bodies orbit their common centre of mass, on similar ellipses whose sizes are in inverse proportion to the masses — here 3 to 1, so the heavier body's path is 3 times smaller. Gravitation

Neither body is still, and the wobble is how planets are found

A planet does not orbit its star. Both orbit a point between them, and the star's share of that motion is small, measurable, and the reason thousands of planets are known.

 Circular and escape speed. Orbital and escape speed against distance, in units of the circular speed at the surface. The escape curve is the circular one multiplied by the square root of two, at every distance without exception. 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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