# The Wheels on the Spacecraft Go Round and Round

Spinning a ten kiloton spacecraft around is no easy feat. Even more impressive if you want to be able to turn on a dime. This article covers the issues of Attitude Control of spacecrafts.

On land, sea, or in air, rotation is easily done by pushing off the nearest medium, with anything from a tractor tread to an impeller to a rudder. In space, no such thing is possible, which means rotation can only be accomplished in one of two ways: by expelling mass via a rocket engine, or by storing rotational momentum internally. Incidentally, the second technique is only really viable in space due to the lack of any major medium, as friction would quickly degrade stored internal momentum.

The goal of these systems is to rotate enormous spacecrafts at reasonable or high speeds in combat, while being both cheap and non-massive.

The first technique is simple. Firing a thruster off center of your spacecraft will cause it to torque. Since rotation is the goal, and rotational acceleration is not, a second thruster must be fired to decelerate it at the end. And because such a thrust would send the center of mass off center, often two thrusters on opposite sides are used to start rotating, and two different opposing thrusters are fired to stop rotating.

These are called Vernier Thrusters, and see heavy use in space travel.

There are significant disadvantages to this method, however, chiefly that additional reaction mass is needed. Not only that, such thrusters usually can’t be Nuclear Thermal Rockets (NTRs), due to radiation concerns (recall that most crew modules are placed as far away from the main engine NTRs as possible). Cold gas thrusters don’t provide near enough thrust to be useful in combat, which means combustion rockets and resistojets are in.

Combustion rockets suffer from the issue that they require propellant(s) that are almost guaranteed to be different than NTR propellants, so additional propellant tanks must be added in, which takes up space and mass. This leaves resistojets as the prime method of providing torque to your spacecraft, since they can use the same propellant as your NTR.

Alternatively, instead of a system of multiple Vernier Thrusters spread out across your hull, simply putting a gimbal on your main thruster will do the trick instead. This allows your main engine to turn, producing off center thrust on your ship, rotating it. In this way, your main engines can serve the dual purpose of getting you places as well as orienting you in combat. This is one of the cheapest forms of thrust vectoring.

The primary disadvantage of it is that propellant is still expended when turning, however, no additional propellant tanks are needed as your main thrusters are doing the turning. For capital ships, the amount of delta-v spent turning is negligible, though for smaller crafts like missiles, the delta-v spent can raise a few eyebrows.

A more subtle disadvantage of gimbaled thrusters is that the size of the opening that the engines need to have can balloon significantly. This can yield much larger aft sections of the ship, and increase the targetable cross section of the spacecraft heavily.

Gimbaled thrust tends to be the cheapest and simplest solution to turning in space, and many capital ships and all drones and missiles in Children of a Dead Earth use them. Some capital ships opt for resistojet Vernier Thrusters, primarily for getting a smaller targetable cross section.

But suppose you don’t want to spend precious propellant turning? Then you need to invest in the second technique towards turning in space: exploiting Conservation of Angular Momentum.

Given a system without little or no external medium (such as space) and assuming you don’t want to expend propellant, the only way to rotate is by using conservation of angular momentum.

A quick example. Consider a pair of identical masses loosely attached to one another, floating in space. If one of the masses begins to spin in one direction, the other mass must spin in the opposite direction at an equal speed, otherwise it would violate Newton’s Laws of Motion.

This is the basic principle of operation for a Reaction Wheel, which is a flywheel with a motor attached. A flywheel is a mass with a high Moment of Inertia, or ability to resist rotational changes, along a single axis. When you spin a reaction wheel inside your ship using the motor, your spacecraft must spin in the opposite direction. Three reaction wheels must be used to get a full range of motion (one for each axis: pitch, yaw and roll). The Kepler Spacecraft uses reaction wheels.

A Momentum Wheel is a reaction wheel which is constantly spinning at a very high speed. If a momentum wheel is spinning inside a spacecraft, simply braking it can cause a huge change in rotational momentum, causing a significant torque on your ship. Six momentum wheels are needed instead of three, two for each axis in opposite directions.

Momentum Wheels are often used for spin stabilization, as the huge amount of stored rotational energy will resist external torques. The Hubble Space Telescope uses Momentum Wheels.

Control Moment Gyroscope (CMG) is a single momentum wheel on a dual-axis gyroscope. By rotating the momentum wheel about the two gimbal axis, the angular momentum balance of the spacecraft can be altered at a whim. A single CMG can rotate a spacecraft along any axis by simply rotating the gyroscope to be in line with that axis. They are the most expensive and complex of the above systems, and are used on the ISS.

These systems promise the ability to rotate your spacecraft without expending propellant. Not only that, they require no exposed external systems, so they can’t be damaged unless the main bulkhead armor is penetrated. Sounds like a win-win, right?

Unfortunately, their effectiveness is very poor for combat operations. Modern CMGs are often used to very slowly change orientations over the course of minutes and hours. I originally had never intended to implement Vernier Thrusters or Gimbaled Thrusters in game, and was only going to use Momentum Wheels and CMGs. That rapidly fell apart when I did the math on them.

Consider the above example with the two masses on a string. Because they are identical, the spins will be equal and opposite. But if one mass has twice the moment of inertia, it will spin at half the (reverse) speed as the other. The greater the moment of inertia, the slower the spin.

If one of these masses is the spacecraft and the other is the reaction wheel, you want the spacecraft to have a lower moment of inertia. Thus, if your spacecraft has a moment of inertia 100 times that of your reaction wheel, it will spin 100 times slower than that reaction wheel.

Moment of Inertia is proportional to mass and the square of distance from the rotation axis. Roughly speaking then, very voluminous and very massive objects will have the greatest moment of inertia.

Spacecrafts by nature will have a greater volume than your reaction wheels, since they envelope these wheels. And they are guaranteed to have a greater mass than your reaction wheels, unless you are okay with an abysmal mass ratio (less than 2). Thus, you are guaranteed a moment of inertia far tinier than your spacecraft’s moment of inertia. And as I discovered, even spinning your wheels at enormous speeds yields rotations that take minutes or even hours.

In short, these techniques were not viable for combat rotations, barring some sort of future technology.

This leaves thrusters as the only viable method of spinning about in combat. How fast can they spin?

Because thrusters affect acceleration rather than velocity, the answer is that it varies. For instance, the time to spin 90 degrees is not going to be twice the time it takes to spin 45 degrees. And it varies based on which axis of rotation is used.

A simple metric is the Full Turnabout Time, which is the time it takes to spin 180 degrees about the slowest axis. This is essentially the “slowest” possible turning time for the ship, and most turns will be much faster, a fraction of this time.

For medium sized capital ships with gimbaled thrusters in game, 20-30 seconds is a common value. Capital ships with vernier thrusters tend in the 10-20 second range, as do small sized capital ships. Very large capital ships can take up to a minute to do a full turnabout. Gimbaled drones and missiles tend to take 5 seconds or less for a full turnabout, with some being able to do a 180 in under a second.

Much faster turnabouts are possible by simply adding more and more vernier thrusters or gimbaled thrusters. However, this is often fast enough to deal with the rapidly changing nature of space combat. It is rare for a capital ships to ever need to flip a 180. Most of their turns are much smaller angle shifts, small dodges and broadsides.

# Gasping for Fumes

Finally we take a look at which propellant we should use for our rocket motor here. As mentioned in earlier posts, the two prime candidates for near future warship propulsion are the combustion rocket, and solid core Nuclear Thermal Rocket (NTR).

Some reminders. Combustion rockets tend to achieve up to 5 km/s of exhaust velocity, and NTRs achieve almost twice that at their best. At the same time, NTRs suffer from lower thrust generally. NTRs in Children of a Dead Earth generally achieve around 3000 K temperatures in their reactor, limited by the materials that make up the core. Combustion rockets can achieve greater temperatures, but it’s wholly dependent upon the reaction used and the stoichiometric mixture ratio of the propellants (if using a multi-propellant reaction).

A popular combustion rocket is the LOX/LH2 engine, which uses liquid oxygen and liquid hydrogen to achieve almost 3000 K as well (assuming a 1:1 mixture ratio). This is the main engine of the space shuttle orbiter, and I’ll refer it to primarily when discussing combustion rockets, though later we’ll explore its limitations, and switch to different reactions.

Looking at the energy densities of nuclear energy versus chemical energy, one comes to the realization that nuclear power is roughly 600,000 more energy dense than hydrogen. So how on earth is a hydrogen combustion rocket even remotely comparable to a nuclear rocket?

After all, remember the rocket power equation:

$P=\frac{1}{2} T v_e$

Where $T$ is the Thrust, $v_e$ is the exhaust velocity, and $P$ is the power. If you increase the power by 600,000, either the thrust or the exhaust velocity must also increase by 600,000.

The trouble comes with releasing that power all at once. We have the ability to do so: it’s called a nuclear bomb. However, releasing it in a way that we can control is difficult, and must be done with a nuclear reactor. (This is one reason why the theoretical Nuclear Salt Water Rocket is so powerful: it tries to unlock that 600,000x power factor and still control it.)

A nuclear reactor’s rate of energy release can be seen through how high the temperature of the core can get. This means that between a NTR with a chamber temperature of 3000 K and a combustion rocket with a chamber temperature of 3000K, if they have identical mass flow rates, they must have identical rocket power.

Mass flow rate is how fast you can feed propellant into the rocket, which is governed by the turbopump injector you use to feed the rocket. The flow rate increases with pump size and with pump speed, and in general, is the same between an NTR and a combustion rocket.

This means, given an NTR and a combustion rocket of similar sizes and similar temperatures, the total power is roughly the same. And if we assume the exhaust velocity of the NTR is roughly twice that of the combustion rocket, the thrust of the NTR must be roughly half that of the combustion rocket. By extension, if the NTR has the same exhaust velocity as the combustion rocket, then the thrust must be the same.

A more direct way to see this is to look at the rocket thrust equation:

$T=\dot{m} v_e$

Where $T$ is the thrust, $\dot{m}$ is the mass flow rate, and $v_e$ is the exhaust velocity. It’s obvious from this that given a constant mass flow rate, exhaust velocity and thrust are inversely proportional. On the other hand, in order to increase your rocket’s thrust, you simply need to increase the mass flow rate by using a bigger turbopump.

Essentially, this means the biggest advantage of NTRs, their high exhaust velocity, is the root cause of their lower thrust. Additionally, NTRs which do not have this advantage, the high exhaust velocities, have comparable thrust as combustion rockets!

In Children of a Dead Earth, Methane is the primary propellant used, because it achieves slightly better exhaust velocities than the best combustion rockets, which means in terms of thrust, it’s only slightly worse than combustion rockets. Decane and Water are also other NTR propellants that see heavy use.

But at that point, is there a purpose to using NTRs at all? If we only use NTRs that yield roughly similar stats to combustion rockets, why not just go with combustion rockets altogether? After all, combustion rockets are cheaper, don’t spew neutron radiation, and are somewhat less massive.

The trouble with combustion rockets, particularly the LOX/LH2 rocket, is the propellants. As mentioned in the previous post (Slosh Baffles), each propellant tank has an ultimate mass ratio ceiling. Roughly speaking, higher density propellants have higher allowed mass ratios. Given standard tank materials, water has an excellent mass ratio ceiling (in the hundreds), while hydrogen has an awful mass ratio limit (< 10 generally).

When using a bipropellant (like LOX/LH2), this mass ratio limit is primarily governed by the worst propellant. So in the case of LOX/LH2, the mass ratio limit is extremely low, because hydrogen’s mass ratio limit is low. Compare that to a Water NTR. A Water NTR will achieve comparable exhaust velocities and thrusts, but water is very high density compared to hydrogen, allowing much higher mass ratios.

On top of this, high density propellants allow your ships to be much smaller, making them much harder to hit in combat, and as indicted in earlier posts, minimizing your targetable surface area is critical.

Much more dense chemical propellants are needed to compete with the mass ratio limits. At this point, we have to discard the assumption that we are using the LOX/LH2 reaction. However, when looking into different chemical reactions, one finds that more dense chemical propellants tend to yield much higher exhaust molar mass.

In thermal rockets, the exhaust velocity is based primarily upon the temperature and the molar mass. Chemical reactions that have competitive or better mass ratio limits tend to yield somewhat lower exhaust velocities.

On the flip side, combustion rockets with certain reactions (particularly those involving fluorine) can achieve greater chamber temperatures than NTRs using clever cooling techniques not viable for NTRs. This means the total power of these rockets exceeds that of solid core NTRs. However, they tend to have low exhaust velocities once again, which means the power manifests as much higher thrusts.

Finally, what about the costs of propellants? Unlike just about every other equation in Children of a Dead Earth, determining the cost of something has no hard and fast rules. As a result, propellant costs are estimated primarily based on solar abundance, and on ease of extraction from common celestial bodies. In this way, common NTR propellants tend to be quite cheap. Combustion rockets with high density propellants end up being much more expensive comparatively.

So where does this leave us?

If you want thrust, thrust, thrust, you should go with combustion rockets with high density propellants. Find a reaction with a high chamber temperature and a low exhaust velocity. The high density propellants might comparatively pricey against NTR propellants, though. This sort of drive is generally what most drones and smaller capital ships in Children of a Dead Earth use.

If you want high thrust but still want a reasonable amount of delta-v, NTRs tend to win out with certain propellants like Methane or Decane. This is what ended up going on most large capital ships. These drives tend to be the good-at-everything, excel-at-nothing choice.

And if you want middling thrust and an even higher delta-v, go all the way and grab a Hydrogen Deuteride NTR. Very few ships ended up falling into this use case, though.

Finally, if you want cheap, go for a monopropellant combustion rocket. Good thrust, awful exhaust velocity, but cheaper than dirt. This is what most small, disposable missiles use in game.

And of course, if thrust is totally irrelevant to you, maybe go for an ion thruster. Only a real option if you’re making a non-combat ship and don’t ever plan to dodge. And if you are okay with taking years to get anywhere.

One final note: It may surprise some readers to find that Hydrogen Deuteride ($HD$) NTRs performs better (9.1 km/s) than pure Hydrogen NTRs (9.0 km/s), especially considering that Hydrogen ($H_2$) has a lower molar mass than Hydrogen Deuteride. This surprised me when I saw it as well.

As it turns out, the Gibbs free energy of formation of monatomic Deuterium is lower than that of monatomic Hydrogen, which yields a much lower dissociation temperature. At 3000 K, $H_2$ dissociation is less than 1%, while $HD$ dissociation is nearly 100%, yielding higher exhaust velocities. As a result, $HD$ is both denser (has a higher mass ratio limit) than $H_2$ and has a higher exhaust velocity, making it better in nearly every way. The only real advantage is that $H_2$ is slightly cheaper than $HD$.

And there you have an analysis of the major near future rocket engines that would see use in space warfare.

In the end, however, I am eager to see what sort of rockets that the players of Children of a Dead Earth can come up. Everything from the propellants to the stoichiometric mixture ratio, to the dimensions and shape of the rocket nozzle, to the turbopump injector attributes are editable in game. There are likely plenty of unexplored designs here that may beat out the designs I’ve made.