Astronomy & Celestial Mechanics — the sky, computed An illustrated collection of essays about how the sky is worked out — orbits, distances, brightnesses and lifetimes, with every figure computed from the mechanics it describes. https://www.astronomy-celestial-mechanics.com/ 2026-08-03T12:30:35.224Z The orbit is an ellipse, and the Sun is not in the middle of it https://www.astronomy-celestial-mechanics.com/essays/the-orbit-is-an-ellipse/ 2026-08-03T12:30:35.224Z 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, which is angular momentum in disguise https://www.astronomy-celestial-mechanics.com/essays/equal-areas-in-equal-times/ 2026-08-03T12:30:35.224Z 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. The law that links period to size, and weighs everything https://www.astronomy-celestial-mechanics.com/essays/the-law-that-links-period-and-size/ 2026-08-03T12:30:35.224Z 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, and the number that picks between them https://www.astronomy-celestial-mechanics.com/essays/every-orbit-one-force-allows/ 2026-08-03T12:30:35.224Z 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. Neither body is still, and the wobble is how planets are found https://www.astronomy-celestial-mechanics.com/essays/neither-body-is-still/ 2026-08-03T12:30:35.224Z 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. The tide is a difference, which is why there are two of them https://www.astronomy-celestial-mechanics.com/essays/the-tide-is-a-difference/ 2026-08-03T12:30:35.224Z 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. Five places that keep station, in a problem with no solution https://www.astronomy-celestial-mechanics.com/essays/five-places-that-keep-station/ 2026-08-03T12:30:35.224Z 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 sphere that is not there, and why it is still the right model https://www.astronomy-celestial-mechanics.com/essays/the-sphere-that-is-not-there/ 2026-08-03T12:30:35.224Z The stars are at wildly different distances and the celestial sphere is a fiction. It is also the most useful fiction in observational astronomy, because for pointing at things, distance is exactly the information to throw away. The Sun's path, and the tilt that makes the seasons https://www.astronomy-celestial-mechanics.com/essays/the-suns-path-and-the-seasons/ 2026-08-03T12:30:35.224Z 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. Phases are not shadows, and eclipses are https://www.astronomy-celestial-mechanics.com/essays/phases-are-not-shadows/ 2026-08-03T12:30:35.224Z Half the Moon is lit at every instant of every month. The phase is which part of the lit half faces the Earth — and confusing that with a shadow is the commonest error in astronomy. The triangle that reaches the stars, and stops https://www.astronomy-celestial-mechanics.com/essays/the-triangle-that-reaches-the-stars/ 2026-08-03T12:30:35.224Z Parallax is the only distance measurement in astronomy that assumes nothing. It is also the only one with a hard ceiling, and everything beyond that ceiling rests on it. A scale that runs backwards, multiplies, and works https://www.astronomy-celestial-mechanics.com/essays/a-scale-that-runs-backwards/ 2026-08-03T12:30:35.224Z Brighter stars have smaller magnitudes, and five steps is a factor of a hundred. A scale invented by eye in the second century BC turned out to be logarithmic, because eyes are. Colour is a thermometer, and it reads across the galaxy https://www.astronomy-celestial-mechanics.com/essays/colour-is-a-thermometer/ 2026-08-03T12:30:35.224Z A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy. The diagram that sorted the stars, by plotting two things against each other https://www.astronomy-celestial-mechanics.com/essays/the-diagram-that-sorted-the-stars/ 2026-08-03T12:30:35.224Z Plot brightness against colour for a few thousand stars and they do not scatter. They fall on a narrow band with two islands off it, and explaining that structure is most of stellar astronomy. Mass decides everything, by a power of three and a half https://www.astronomy-celestial-mechanics.com/essays/mass-decides-everything/ 2026-08-03T12:30:35.224Z Two stars of the same mass are almost the same star. Double the mass and the output multiplies by eleven — which is why a modest range of masses produces a colossal range of stars. Why the biggest stars die first, and take the galaxy with them https://www.astronomy-celestial-mechanics.com/essays/why-the-biggest-stars-die-first/ 2026-08-03T12:30:35.224Z A star thirty times the Sun's mass has thirty times the fuel and burns it forty thousand times faster. It lasts a few million years, and everything heavier than iron exists because of it. The cheapest way between two orbits, and why it is so slow https://www.astronomy-celestial-mechanics.com/essays/the-cheapest-way-between-two-orbits/ 2026-08-03T12:30:35.224Z 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. The speed that does not come back, and the √2 that separates it https://www.astronomy-celestial-mechanics.com/essays/the-speed-that-does-not-come-back/ 2026-08-03T12:30:35.224Z 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. Stealing speed from a planet, which does not notice https://www.astronomy-celestial-mechanics.com/essays/stealing-speed-from-a-planet/ 2026-08-03T12:30:35.224Z 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.