Math in Space: Light-Years, Rockets, and Gravity

The twenty-two that embarrassed me

My daughter asked why we cannot simply point a plane upward and fly to the Moon. I said it was because planes cannot go that high, which was true and beside the point. So we did it properly, on the back of a receipt at the kitchen table: how much fuel would it take to put a spacecraft into orbit?

I already knew the answer is roughly twenty-two kilograms of propellant for every kilogram of rocket and payload you want to keep. I had never turned that into something she could picture. Twenty-two kilos of fuel per kilo of spacecraft meant her capsule needed roughly forty tonnes of fuel, which is about fifty thousand litres, which is more than a road tanker of petrol, sitting on the launchpad, for a passenger list of one. She said that was a lot of petrol for a short trip. It is. It is also why a spacecraft is mostly a tank with a small payload attached, and why it weighs so little at the top.

Here is what I could not explain at the table. Twenty-two is not a design choice. It falls out of arithmetic, and the shape of that arithmetic — a logarithm, of all things — is why going higher is so much harder than going up.

A light-year is a distance wearing a time's name

A light-year is the distance light covers in one year: roughly 9.46 trillion kilometres, or 9.46 × 10^12 km. Nothing about it is slow. The speed it borrows its name from is the fastest thing we know of. So the phrase is a translation error from the word "year", and it is why people assume the nearest star is four years away. Four years is how long light takes. Getting there is another question entirely. Proxima Centauri is 4.24 light-years away, and at the fastest speed a crewed vehicle has ever sustained — around 300 kilometres per second — the crossing works out to roughly 4,200 years.

That gap between four and 4,200 is the most useful thing about the unit. It is why astronomers measure in light-years for their own convenience while everyone else measures in years. And 9.46 × 10^12 km is exactly the sort of figure the big-number prefixes were invented to keep readable — and the conversion is the same discipline as any unit conversion that looks easy and then does not.

Gravity is not a wall, it is a slope

The instinct is that gravity is a force with a range, like a magnet that switches off. It is not. Gravity falls off with distance squared, and the square is everything. Move twice as far out and gravity is not half as strong, it is a quarter. One Earth radius up, the pull is a quarter of what you feel on the ground. At the Moon's distance it is about one sixth. Not zero. There is no height at which it becomes nothing.

Which sounds like it should make space easy. It does not. In low orbit, a few hundred kilometres up, gravity is roughly 88% of what you have on the ground. Astronauts float because they and the station are falling at the same rate, not because anything switched off. The difference between orbit and a sealed box on the ground is not the gravity. Both are free.

The rocket equation, and the logarithm nobody mentions

Here is the whole thing. Divide the mass of a rocket at ignition by the mass of everything still left when the fuel runs out, and call the result the mass ratio R. Take the natural logarithm of R and multiply it by the exhaust velocity — how fast the exhaust leaves the nozzle, about 3 kilometres per second for a good kerosene rocket. The result is Δv, the total change in velocity the vehicle can manage:

Δv = v_e × ln(R)

Escape from the surface of Earth needs about 11.6 kilometres per second; just reaching orbit needs around 9.4. Divide 9,400 by 3,000 and you are asking what R makes ln(R) ≈ 3.1. The answer is roughly 23. So twenty-three kilograms of fuel for every kilogram kept — which is the number I could not explain at the table, and now it is just a logarithm.

The consequence is why spaceflight is shaped like a wine glass. Because Δv grows with ln(R) and not with R, the return on propellant is measured in equal steps per factor of ten, not per kilo. A mass ratio of 10 gives about 6.9 kilometres per second, 100 gives 13.8, and 1,000 gives 20.7. Every step is the same size and every step costs ten times the fuel of the last. So doubling your Δv from 6.9 to 13.8 needs a tenfold jump in mass ratio and roughly eleven times the propellant, not twice the fuel.

There is an obvious reason for the shape. Every kilogram of fuel has to shove not only itself but all the unburned fuel behind it. Early on, each kilogram is dragging the whole tankful; by the time you burn the last tenth of the propellant you are pushing a tenth of the mass you started with, and that final burn is extraordinarily efficient. The rocket is least efficient when it is heaviest, and it is heaviest at the start. That is why it empties itself upward: the vehicle that arrives is the small fraction still left at the end.

One detail deserves attention, because exhaust velocity sits outside the logarithm. Double v_e and you double Δv outright, and since doubling Δv means squaring the mass ratio — from 23 to about 530 — this one linear term beats every clever staging scheme. Hydrogen is expensive, awkward and harder to fly than kerosene, and rockets are still built around it for exactly this reason. The mechanism behind those power laws is in our exponents and roots guide.

Orbit is falling, sideways

The last piece trips up most people, and it is a geometry problem wearing an engineering costume. To stay in orbit you do not point up. You go sideways, at around 7.7 kilometres per second, and then stop.

Gravity pulls you down continuously. Your sideways speed carries you forward. The two bend your path into a curve, and if it falls at exactly the rate the ground does, you never reach the ground. That is all orbit is: a fall that keeps missing. It is just projectile motion, and the curve is the same parabola either way, so it helps to have the parabola explanation in your back pocket.

Point the same rocket straight up at that speed and you get very high and very fast, in a direction that brings you straight back to the start. Vertical speed buys altitude. Horizontal speed buys orbit. A launch is mostly a controlled conversion of the first into the second.

What to try this week

Pick any Δv — 9.4 for orbit, 11.6 for escape — and an exhaust velocity, then solve for R and work out the kilograms of fuel per kilogram of spacecraft that implies. Do it twice, at 3 and at 4.5 kilometres per second, and put the two fuel numbers side by side. Three small calculations, and space stops being a place you admire and becomes a place you have costed.

Frequently Asked Questions