Spaceship 101

Eine Kleine Mathematik*

Hello all. In science fiction, one of the staple offerings of fertile author’s imagination is The Spaceship. The addition of one of these to a story immediately tells the reader what genre they are in, almost like how a dragon would signal you are in a fantasy novel. Spaceships, of course, run the gamut of importance to a story ranging from being a convenient conveyance from one distant point to another to that of a character in the story in its own right.

Seeing how important they can be to the fabric of the story, how the ship operates determines how hard (or well-grounded, if you prefer) your science fiction story is. Designing a spaceship then becomes a crucial part of worldbuilding. I use the term worldbuilding in the general sense as there are potentially many worlds. It is science fiction after all.

To design your trusty stellar steed, you need to know the function of the spaceship, what is it doing in your story, how do your characters need to interact with it, what does it need to provide to the story. These are not new questions at all. They are the basics for worldbuilding, it just happens the world you are currently building is rather small and confined to the shape of a spaceship. Beyond the normal worldbuilding, you also need to provide for the basics to support life, of course, as well as what is needed for performing the duty that it is supposed to fulfill. Those are not quite the normal worldbuilding questions but one can see the need to answer them. I guess they are shipbuilding questions.

Invariably, though, you need to ask the question, how does this spaceship get from point A to point B?

This is probably the biggest design decision that you will make for the ship and it potentially impacts your entire story and probably where you fall on the Mohs Scale for Science Fiction.

If you wish to have your ships flit around like old timey planes without caring about things like the Rocket Equation or inertia, by all means. That makes for a fine story but please don’t have them in the middle of a crucial moment having to break out the slipsticks and calculation how much propellent they are going to need to make the turn.

Likewise, if your ship design takes into account fuel and propellent mass separately, you probably need to run a few calculations to see if something you want done can actually happen.

It is to this later category of shipbuilding that I will offer some equations in a effort to get to the appropriate crunchiness level of hardness.

The first and most important question that you need to answer is “How much does the ship mass in kilograms? (BTW, we are using the MKS system. No Imperial units. These are not the droids you are looking for.)

There are lots of ways to estimate this ranging from gut feel like how many cars big is it to an engineering drawing with densities and volumes calculated. Important side note when designing a ship if you are striving for realism: Every gram counts. Make it as light as you can get away with.

This brings us to the first bit of math. It’s not a complicated thing. I am purposely simple calculation, not enough to actually get you to the Moon but real enough that it will satisfy story purposes. This first equation is the force equation.

F = ma

It may not seem like much but it governs the entire physical universe. Basically, it says that the amount of force (F) required to get something moving is governed by the mass (m) of an object and how fast you are trying to get it going (a). m is the mass in kilogram, from your estimate before, and a is acceleration in meters per second per second (m/s/s or m/s^2). Acceleration is how fast your velocity changes in a second. Velocity is how many meters you cover in a second. The units for F are called Newtons. I will frequently refer to them.

Say you have a slug of monocrystalline iron that you wish to shoot at your opponent using your hand-wound railgun. The slug masses 1 Kg and your gun can accelerate it at 100 g. A g is the acceleration of gravity which is 9.8 m/s/s, call it 10 m/s/s just so you can do it in your head. Plugging all this in and you get F = 1 Kg x (100 x 10 m/s/s) or 1000 Newtons. A 9mm bullet has about 350 N of force, btw. Ouch.

So, how does this work for a spaceship? Well, it goes the other way too. If you have an engine that can output 10,000 N of thrust and you mass 1000 Kg, you can accelerate at 1 g. Handy for keeping gravity effects throughout your ship because as Special Relativity says, It Don’t Matter Where the Acceleration Comes From.

So let’s supposed you’ve roughed out how much you ship masses and you have provided a sufficiently powerful engine at the business end. It dumps out oodles of Newtons of force in the backward direction. How much? Well, you can calculate it because you know the force equation and basically you are taking some amount of mass (called the propellent) and accelerating it away from the ship. According to Sir Isaac Newton (yes, that is where it gets the name), that has an equal and opposite reaction and makes you go forward, governed by the same equation.

To summarize, you want to take something small and throw it off the back of your ship really, really fast and depending how massive you are, you’ll go forward a little bit. The faster you throw stuff backward, the faster you’ll go forward.

Of course, there is a little complication. If you through something off your ship, you no longer have the same mass as before. You have less. You read earlier something I called the Rocket Equation. That’s it. The math for that is non-trivial. But what it means for your story is that your thrust will last a bit longer than just taking the amount of propellent and dividing it up by how much you use per second because you would have to taper it down to maintain the same acceleration. If you didn’t, your acceleration would increase and you probably don’t want that.

Current, real-world spacecraft do not usually do this. There are a few with ion engines and they have been sent to some asteroids and comets but they are the exception. NASA and other space agencies send stuff to other planets using what’s called a Hohmann transfer orbit. That type of maneuver is one that uses the least amount of propellent, handy because you don’t have very much to begin with. Every gram counts. The basics behind it is that you “throw” something via a specific burn toward where something else will be to “catch” it in its gravity well. As you can imagine, things have to line up a particular way for this to happen. In the case of Earth and Mars, this only occurs every 26 months.

Of course, you don’t want your fabulous spaceship to be limited by such concerns. You have sufficient fuel to burn to make your propellent go fast. Yes, those are two separate things, not the case with chemical rockets where fuel = propellent. But you can’t use chemical rockets to get around the planets anyway. They don’t have enough bang for the buck.

Assuming you have propellent to spare, you have what is called a torch ship. It doesn’t need a high acceleration, it just needs to be constant. Constant acceleration is magical. With it, you end up going very fast. Which gets us to the next bit of math:

V = at

Acceleration, a, you know from before. V, your velocity, is how long, t in seconds, it is on. A day has 84600 seconds in it. If you could accelerate at 1 m/s/s, at the end of the day, you would be travelling at 84600 m/s. Call it 85 km per second to keep math easy. Voyager 1 is one of the fastest spacecraft ever made. It is travelling at 17 km per second. As I said, constant acceleration rules.

With constant acceleration, you don’t worry yourself with planetary alignments for those old, slow, Hohmann orbits. You just point yourself at your target and turn your engine on.

There is one small catch. If you have your engine on the whole time you are pointed at your destination, you’re going to be building up speed. There’s a good chance you are going to fly right by the place you wanted to get to, or heaven forbid, smack into it. Very inconvenient.

The solution is to slow down. You get to the mid-point of your journey, turn the engine off, flip over so the engine is now pointed in the direction of travel (remember, inertia keeps you going and there is no friction in space to slow you down) and turn your engine on. If you hear the term, flip and burn, that’s what they mean. Also, skew turn means the same.

Technically, if you were doing this for real, you would be following what’s called a Brachistochrone Curve and doing the flip at the minima but, again, fearsome math so just straight line it and flip in the middle.

So, how do you figure out how long it is going to take to get where you are going?

You start by getting the total distance in meters and dividing it by 2 giving you D. That’s your midpoint. We are going to assume that you were at rest relative to your departure point and you wish to be at rest when you arrive at your destination. In actuality, you’re not but we are going to ignore that because for your story, you aren’t calculating and reporting the exact seconds. If you are, then serious math time. To calculate how much time you need to get there, under constant acceleration, we need another equation.

D = vt + 1/2 at^2

This is the magic equation that can solve all your travel problems across a wide regime. Let’s take a look at it.

D is the distance to travel, previously the midpoint of the journey. Quantity v is whatever initial velocity you had to begin with. Remember we said we would be at rest which translates to zero (0) and zero times anything is zero so you can ignore this term for now. Which leaves us with 1/2at^2. The ½ comes from how the original equation of motion is derived, a you know for your spaceship and t^2 is from how you need to apply an acceleration to get distance. Acceleration is m/s/s or m/s^2 so if you multiple it by t^2, you are left with meters. You need to rearrange the terms a bit and you are left with

t=

You finally have the time needed for your torch ship to get to the half way point. Time to get to your destination from the half way point is the same because you intend to be at rest relative to the planet of your choice when you get there so double t and you have total trip time. Now you know how much time your intrepid adventurers have to kill onboard. Make sure you have enough oxygen, water and food for the trip. Don’t forget, every gram counts.

Since you have the time, you can do all sorts of things now, like calculate your maximum speed and how much propellent you are going to need based on rate of consumption. In fact, you can do all sorts of things now, including figuring out how far the planet is going to move in the interim so you can apply windage. You don’t want to miss it, of course.

These calculations are just the tip of the iceberg, of course. There are many more things you can do with this simple math. For instance, say you can’t have thrust on the whole time because you don’t have enough propellent. Ok, figure out what half the propellent is and how long a burn it would give you. Calculate how much distance you would cover, double it (decelerating at the far end, remember?) and subtract from the total distance. Calculate what your velocity would be at the end of the burn and divide the remaining distance by that velocity. Now you know how long it takes you to get there. Pack a lunch. Distances in space are usually really big.

I hope you have enjoyed this little tutorial on spaceships and how they travel from one place to another. There are many good resources on the web for these sorts of things.** They can take you much further in what is needed to make as realistic a spaceship as you need for your story. Please don’t forget that. Your spaceship needs to be as realistic as your story needs because not all science fiction stories are hard science.

Good luck and good calculating. Ad Astera.

* Yes, I know that’s not how you would say that but I thought it would be humorous as I like Mozart.

** Projectrho.com is a really good resource that I recommend. Be warned, it is a deep rabbit hole.