Astronomy & Celestial Mechanics
the sky, computed
Theme

Small perturbations, large consequences

The corrections that were supposed to be negligible: Neptune found in the residuals, Mercury's extra arcseconds, and orbits that are stable only for a while.
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.

 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.

 05101520-40-200204060hour (local solar time)June solstice — 61° at noon, 16.5 h of dayequinox — 38° at noon, 12.0 h of dayDecember solstice — 15° at noon, 7.5 h of daybelow the horizon The observed sky

The Sun's path, and the tilt that makes the seasons

Summer is not when the Earth is closest to the Sun — that happens in January. It is when the Sun climbs higher and stays up longer, and both come from a 23.4° tilt.

 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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