Math! In! Space!

I have ran into a few “hard science fiction” story tellers or advisers who seem to think they know how space works. They don’t. Some of these story tellers and advisers are even credentialed enough to make one think that they should know what they are talking about. Yet I, an amateur in the space science arena, know enough physics to realize they are blatantly wrong. To hopefully provide better advice for those out there looking to create better science based science fiction this week’s worldbuild wednesday is going to focus on the basics of space travel as we currently know it and the math that makes it work. Along with a few notable equations and my story teller shortcuts that help me sort the good from the bad.

Starting out there is the concept of an orbit. I believe orbiting has been described as falling and missing the ground. Which isn’t exactly wrong. An orbit has 3 important factors apoapsis, periapsis, and angle. Apoapsis is the farthest point from the body being orbited. Periapsis is the closest point to the body being orbited. Angle is a bit more complicated. First off it generally ranges from 0 to 180 between 0 and 89 is prograde, which means the orbit goes in the same direction the orbited body rotates, assuming it does, between 91 and 180 it is retrograde which is opposite of the direction the orbited body rotates assuming it does. At 90 degree the prograde, retrograde comment is largely irrelevant as the orbital plane is perpendicular to the orbital plane of the orbited body.

For those who want a complete description there is another angle that describes where the apogee is ‘pointed’ compared to the sun however most of the time that won’t mater.

There are three types of orbits, circular where the apoapsis and periapsis are the same, elliptical where apoapsis and periapsis are different and hyperbolic. Circular and elliptical orbits can be stable orbits. Hyperbolic orbits lead to the orbiting object escaping from the orbited body’s gravity well. This can viewed by eccentricity. Eccentricity of 0 is circular, between 0+ and 1-, or anything above zero and below one, is an elliptical orbit and 1 and above is a hyperbolic ‘orbit’. I would say it’s not an orbit but a flyby although that is more semantics on my part.

Now that we have the way to describe and a base understanding of an orbit, how do spaceships navigate between orbits? In short add or remove energy. Any stable orbit trades kinetic and potential energy back and forth. The amount of total energy will be equal to some range of numbers that is determined by the size of the planet. Bigger the planet the bigger the total energy will be. If you fall under this range you will end up crashing into the planet, going over the range will send you out into space. Everything else will be a “stable” orbit. I put stable in quotes because there are cases where things such as an atmosphere will cause you to slow down. I’ll be talking about that in a moment.

For those knowledgeable with the physics will know that the equation can be simplified to:

Meaning that changing your velocity is much more impactful than changing your altitude. This also highlights the fact that closer orbits are faster, which is a mix of Kepler’s Third and the Conservation of Angular Momentum. Meaning if you’re writing a race using physics, closer to the thing the faster the speed. Thus at periapsis velocity will be maxed out as it’s the minimum altitude, and apoapsis is the slowest.

<photo of KSP showing speeds at top and bottom>

This alone gives some interesting levers to pull for story telling. For example it may be advantageous for the opening salvo of the fleet engagement to be a meeting one with the characters charging in on a retrograde orbit and diving to meet the enemy at periapsis. Due to the planet’s size the closing speed is at a over ten thousand kilometers a second. Meaning even the defensive chaff clouds will be enough to shred even the battleship’s armor. This will be a joust of the highest order.

Or for those who may have less interplanetary wars in their writing. A group needs to go back, yet they are on the ‘upswing’ and are closing in on apoptosis. Yet they need to get back down as fast as possible. Do they wait and adjust at apoptosis and save fuel or burn hard now to keep from going that high up?

This then leads to the other two important concepts, ΔV and Specific Impulse. ΔV or Delta-V as is usually said is the amount the rocket can change it’s velocity in that equation above. Since there is functionally no friction in space how much you can adjust that velocity is how you measure how far you can go, ignoring gravity assists for the moment. Meaning if the ship needs to adjust where it’s going it will accelerate, to change the velocity number, and use up some of the ΔV it has access to.

This can be used as a great limiter, do the characters have the fuel they need to do this now or do they have to wait for a more opportune time, can they afford to wait? Likewise if one thinks about what Delta-V means, maybe there’s a way to throw stuff overboard to lighten the ship and get more out of what’s left of the fuel. Which certainly wouldn’t cause any future issues.

Specific Impulse is often related to Delta-V as it is the fuel efficiency to the fuel. While it is a bit more complicated than that it is a functional simplification. However more importantly is the relationship between specific impulse and thrust.

The general trend is that more thrust means worse specific impulse. The reason tends to revolve around the volumes of ejection mass and desired power output. For a given amount of power, you’d need an amount of mass tossed from the back of the rocket. More mass means a larger engine to keep the efficiency, which becomes unwieldy or runs into the classic rocket problem: More thrust burns more fuel which needs more thrust. The astute will see the circular problem.

Since more thrust will burn more fuel, you will need to bring more fuel. By bringing more fuel you’ve reduced the advantage of having that more thrust. Thus requiring more thrust for the added fuel.

This is where my storyteller axiom of rocket engines comes into play: The more force the more fuel burned. This axiom generally can be used in two ways. The first is speed costs money. This can be generally ignored for most stories, however having an expensive fuel or a large amount of fuel required to get from A to B quickly is a good way to help add another wrench into the plans of the characters. The second is that the characters will never be able to get what they want when they want it.

A good example of the latter would be trying to reverse course. To reverse an orbit, takes a lot of fuel, and generally a lot of time. Thus they have a choice, continue on or maybe make it back. If they attempt to go back they might not be able to fully get back, since returning to that stable orbit would take a lot of fuel, which they may not have anymore. In some extreme cases it would be faster to launch a rescue craft from the destination rather than turning around.

These are a few of the things to think about when looking to the stars. I’ll be back next week. Until then thanks for reading.