Astronomy & Celestial Mechanics
the sky, computed
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Two bodies, and no more

The two-body problem is solved exactly and the three-body problem is not solved at all. Almost everything since has been about living with that.
fast hereslow here16 equal intervals of time, one full orbit 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.

 barycentrethe line joining them always passes through it 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.

 toward the sourcethe near side is pulled harder than the centrethe far side is pulled less — and so falls behindnot to scale: the source is far outside this frame Gravitation

The tide is a difference, which is why there are two of them

The Moon pulls the ocean toward it. That explains one bulge. The second one, on the far side, is the whole of the physics — and it comes from subtracting.

 L₁L₂L₃L₄L₅L₄ and L₅ are exactlyequilateral with both bodiesthe collinear three areroots of a quintic Gravitation

Five places that keep station, in a problem with no solution

Three bodies under gravity cannot be solved. Restrict the problem slightly and five exact answers fall out anyway — three of them roots of a quintic, two of them perfect equilateral triangles.

 the planet's velocityinout70°speed before: 0.505speed after: 0.661gained 0.157, and burned nothingthe circle: constant speed in the planet's frame Spaceflight

Stealing speed from a planet, which does not notice

A flyby cannot change a spacecraft's speed relative to the planet. It changes its direction — and adding the planet's own motion back turns that into free velocity.

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