Every essay
Orbits
Kepler's three laws, and the family of curves a single force allows.
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, 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.
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, 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.
Gravitation
Two bodies pulling on each other, and everything that goes wrong at three.
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.
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.
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 observed sky
Coordinates, seasons, phases and shadows — geometry seen from inside it.
The sphere that is not there, and why it is still the right model
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
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
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.
Starlight
Distance, brightness and colour: what can be measured when nothing can be visited.
The triangle that reaches the stars, and stops
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
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
A star's colour gives its surface temperature, from two brightness measurements and no other information. It is the cheapest useful measurement in astronomy.
Stars
The diagram that sorted them, and the one quantity that decides a star's whole life.
The diagram that sorted the stars, by plotting two things against each other
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
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
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.
Spaceflight
Celestial mechanics used forwards — where to burn, and what it costs.
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.
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.
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.