Interactive Math & Science Lessons
Almost nothing in astronomy is measured directly. Distances, masses and temperatures are inferred from brightness, colour and timing, and that chain of inference is the actual subject.
Short sequences rather than single tools. Each step hands the next one something specific, so they are worth doing in order.
L2: school
ASTRONothing about a star can be worked out from how bright it looks until you know how far away it is. That is the order these four go in, and it is the order astronomy went in — each rung measured against the one below it, all the way from a surveyor’s triangle to the edge of the observable universe.
L3: undergraduate
ASTROAn orbit is fixed by its size and its shape, and almost everything else follows. These five tools go from the ellipse to the fuel: where a spacecraft is, how fast it is going there, how long the trip takes, which trip is cheapest, and what that costs in propellant.
The Sun pulls the Moon twice as hard as the Earth does
The Earth pulls the Moon with 1.98 × 10²⁰ newtons. The Sun pulls it with 4.35 × 10²⁰. The Sun wins by more than two to one, the Moon stays with us anyway, and the reason is not the one most people reach for.
The night sky should be as bright as the Sun. Step outside and check.
If the universe were infinite and eternal, every direction you look would eventually hit a star, and the night sky would glare like the surface of one. It is dark instead, and the darkness is evidence about the age of everything.
Nobody has ever weighed the Sun. We timed something going round it instead.
There is one way to measure a mass you cannot touch: watch something orbit it, and time the orbit. The same equation weighs planets, stars and whole galaxies. Dark matter turned up where its answer stopped matching the sky.
A quasar at z = 3 is receding at 1.46 times the speed of light
Cosmological redshift is not a Doppler shift, and the difference stops being academic above z ≈ 1.5, where the recession speed exceeds c without anything moving through space or breaking relativity.
To catch the spacecraft ahead of you, slow down
Firing your engine towards a target in the same orbit sends you further behind it. The only way to catch up is to brake, drop into a lower and faster orbit, and come back up in front.
Most of a rocket is fuel, and better engineering will not change that
The rocket equation is logarithmic in the wrong direction: speed goes up like the logarithm of the mass you carry. Reaching orbit costs more than everything after it.
