## Introduction

### ROCKET ENGINES 101

If you already know about Newton's three laws of motion and how rockets work, you can skip ahead to the next section.

Spaceships have it hard because space does not have all the advantages we take for granted on Terra. Things like friction.

You want to make your automobile move? No problemo, just make sure all the wheels touch the ground. The wheels spin, they push against the friction of the road, the auto moves forwards. Easy peasy.

There ain't no road in space. There ain't no nothing in space, that's why they call it space. There is nothing with friction to push against. In space your auto will literally spin its wheels while going nowhere. As will your motor-boat and airplane uselessly spinning their propellers.

So how are you going to get your poor spaceship to move? Newton's third law, that's how.

In a physics textbook Newton's third law says for every action there is an equal and opposite reaction. In English this means things will recoil. If you fire a .577 Nitro Express bullet out of your rifle, the rifle's recoil is going to kick like a mule and dislocate your shoulder. You can see this even better if you are standing on something with little friction, like on a piece of glare ice or in a raft on the water. You will go sliding backwards on the ice or overboard the edge of the raft, with your dislocated shoulder.

But you will really move backwards fast if you are in a place with no friction. Like, say, in space.

So a rocket engine is just a way to fire some "reaction mass" (propellant) out the exhaust nozzle, so the recoil kicks the spaceship forwards. Because conventional propulsion won't work in space.

Now the action and the equal but opposite reaction are not measured in speed, they are measured in momentum. But don't panic, it is reasonably simple. Momentum is the object's mass times the velocity. Which is a fancy way of saying that an object with a tiny mass but an enormous velocity can have the same momentum as an object with a huge mass but a tiny velocity.

The practical effect is even if the mass of the propellant shooting out the engine is tiny compared to the spaceship, if the propellant is moving really fast the recoil will give the heavy space ship a substantial velocity in the other direction.

## General Rules

So the good ship Polaris has to have engine(s) and enough propellant to manage a total deltaV of 39,528 m/s. We have to make a trial spacecraft design, calculate its total deltaV capacity, and see if it equals or exceeds 39,528 m/s. If not, its time to go back to the drawing board to tweak the design a bit.

Here's how to calculate a spacecraft's total deltaV capacity. (You can find a more in-depth explanation of the following process here). In order to calculate the spacecraft's total deltaV capacity, you need to know two things: the spacecraft's Mass Ratio, and the exhaust velocity of the engine. Surprisingly, you don't need to know anything else, not even the ship's mass. I will lead you through the steps in the sections below.

Rick Robinson's Rocketpunk Manifesto blog has some important points to make. The performance of available rocket engines will affect the rate of exploration, what destinations can be reached, and the travel time.

Eric Rozier has an on-line calculator that will assist with some of these equations.

## Mass Ratio

Mass Ratio tells the percentage of the spacecraft's mass that is propellant. You generally try different values for the mass ratio until you get a deltaV that is sufficient to perform the mission that is contemplated. You want a mass ratio that is low, but you'll probably be forced to settle for a high one. As a general rule, a mass ratio greater than 4 is not economical for a merchant cargo spacecraft, mass ratio 15 is at the limits of the possible, and a mass ratio greater than 20 is probably impossible (At least without staging. But we won't go into that because no self-respecting Space Cadet wants to go into space atop a disintegrating totem pole. For purposes of illustration, the Apollo Saturn V uses staging, and had a monstrous mass ratio of 22).

When a rocket designer is given the mission deltaV and the engine specific impulse (or related exhaust velocity), the required mass ratio can be calculated. Now the rocket designer is faced with the daunting task of trying to cram all the rocket's payload, structural mass, engine mass, and everything else that is not propellant into the design's dry mass.

As a side note, propellant is also called "reaction mass" or "remass". Please note, there is a difference between propellant and fuel. Fuel is the material used by the propulsion system to generate energy. Propellant is "reaction mass", i.e., what comes shooting out the exhaust nozzle to work the magic of Newton's law of action and reaction. Only in rare cases (like chemical propulsion) are propellant and fuel the same thing. For most of these propulsion systems the fuel is uranium or plutonium and the propellant is hydrogen.

You probably won't use this equation, but the definition of mass ratio is:

R = M / Me

or

R = (Mpt / Me) + 1

where:

• R = mass ratio (dimensionless number)
• M = mass of rocket with full propellant tanks, the Wet Mass (kg)
• Mpt = mass of propellant, the Propellant Mass (kg)
• Me = mass of rocket with empty propellant tanks, the Dry Mass. Me=M-Mpt (kg)

If for some odd reason you have the mass ratio and only one of the masses, simple algebra will show you that:

M = Me * R

Me = M / R

The equation you will actually use (later) is:

Pf = 1 - (1/R)

R = 1 / (1-Pf)

where

• Pf = propellant fraction, that is, percent of total rocket mass M that is propellant: 1.0 = 100% , 0.25 = 25%, etc.

If you happen to have the rocket's delta V (or you are designing for a target delta V) and exhaust velocity, there is an equation that will allow you to calculate the required mass ratio:

R = ev/Ve)

where

• ex = antilog base e or inverse of natural logarithm of x, the "ex" key on your calculator

## Exhaust Velocity

The engine and its type determine Exhaust velocity.

Often instead of exhaust velocity your source will give you an engine's "specific impulse". This can be converted into exhaust velocity by

Ve = Isp * 9.81

where

• Isp = specific impulse (seconds)
• Ve = exhaust velocity (m/s)
• 9.81 = acceleration due to gravity (m/s2)

Generally you will find the exhaust velocity (or specific impulse) of a given propulsion system listed in some reference work. I have a table of them here.

It is possible to calculate the theoretical maximum of a given propulsion system, but it is a bit involved. I have a few notes for those who are interested, those who are not can skip to the next section. I'm only going to mention thermal type propulsion systems, non-thermal types like ion drives are even more involved.

EXHAUST VELOCITY OF THERMAL TYPE ROCKETS

Ve = sqrt( ((2 * k) / (k - 1)) * ((R' * Tc) / M) * ( 1 - (Pe/Pc)^((k-1)/k) ) )

where

• Ve = ideal exhaust velocity (m/s)
• k = specific heat ratio (hydrogen = 1.41, water = 1.33, methane = 1.32, ammonia = 1.32, carbon dioxide = 1.28, carbon monoxide = 1.40, nitrogen = 1.40, chemical rocket = 1.2)
• R' = Universal gas constant (8,314.51 N-m/kmol-K)
• M = exhaust gas average molecular weight (atomic hydrogen = 1, molecular hydrogen = 2, water = 18)
• Tc = Combustion chamber temperature (Kelvin)
• Pc = Combustion chamber pressure (standard for comparison is 68 atm)
• Pe = Pressure at nozzle exit (standard for comparison is 1 atm)

The main thing to notice is that for thermal engines, the lower the molecular weight of the propellant, the better. When you are dividing by M, you want the number you are dividing by to be as small as possible.

For combustion chamber temperatures below 5000K with hydrogen propellant, for M use the value for molecular hydrogen (2). Above 5000K the hydrogen atoms dissociate into atomic hydrogen, for M use 1.

If you do not want to calculate this for yourself, you can use Ian Mallett's Online Calculator.

In Robert Heinlein's novels, he postulated a magic way (which he never explains) of storing stabilized atomic hydrogen in propellant tanks in order to have the ultimate propellant boost. He called it "Single-H". In reality, a tankfull of atomic hydrogen would explosively recombine into molecular hydrogen quicker than you can say "Stephen Hawking". The least unreasonable way of preventing this is to make a solid mass of frozen hydrogen (H2) at liquid helium temperatures which contains 15% single-H by weight.

As an example: the chemical engines on the Space Shuttle Main Engine (SSME) have a much higher temperature than a solid core nuclear thermal rocket (NTR) (4,000K as opposed to 2,000K). But the NTR has a higher exhaust velocity because it uses low molecular weight hydrogen as propellant, instead of that high molecular weight water that comes out of the SSME. So the NTR has a theoretical maximum exhaust velocity of around 8,000 m/s while the SSME is lucky to get 4,400 m/s. Behold the power of low molecular weight propellant: the higher temperature of the SSME is no match for the NTR's lower weight propellant.

Why can't chemical engines use low molecular weight propellant? Because in chemical engines, the fuel and the propellant are one and the same, but in an NTR the fuel is the uranium and the propellant is whatever you want to use. With chemical you are stuck with whatever chemical reaction products are left over after the fuel has finished burning.

EXHAUST VELOCITY OF FUSION ROCKETS

Nuclear rocket fuel
ParticleMass
(unified atomic
mass units)
n (Neutron)1.008665
p (Proton)1.007276
D (Deuteron)2.013553
T (Tritium)3.015500
3He (Helium-3)3.014932
4He (Helium-4)4.001506
11B (Boron)11.00931

Pure fusion rockets use the reaction products themselves as reaction mass. Fusion afterburners and fusion dual-mode engines use the fusion energy (plasma thermal energy, neutron energy, and bremsstrahlung radiation energy) to heat separate reaction mass. So afterburners and dual-mode reduce the exhaust velocity in order to increase thrust.

For pure fusion rockets calculating the exhaust velocity is as follows (for afterburners or dual mode see the fusion engine entry).

Remember Einstein's famous e = mc2? For our thermal calculations, we will use the percentage of the fuel mass that is transformed into energy for E. This will make m into 1, and turn the equation into:

Vel = sqrt(2 * Ep)

where

• Ep = fraction of fuel that is transformed into energy
• Vel = exhaust velocity (percentage of the speed of light)

Multiply Vel 299,792,458 to convert it into meters per second.

### Nuclear fission thermal rocket

The higher the temperature, the higher the exhaust velocity. Unfortunately, at some point the temperature is so high that the reactor would melt. That is why the nominal temperature for the solid core reactor is only 2,750K.

Liquid core and gas core nuclear thermal rockets are where the reactor is normally molten or gaseous in order to have a higher exhaust velocity.

Nuclear Thermal Rocket
EngineNominal
Temperature
PropellantExhaust
Velocity
Specific
Impulse
Solid Core2,750KMolecular Hydrogen8,300 m/s850 s
Liquid Core5,250KAtomic Hydrogen16,200 m/s1,650 s
Gas Core21,000KAtomic Hydrogen32,400 m/s3,300 s

### Deuterium-tritium fusion rocket

Deuterium-Tritium Fusion rockets use the fusion reaction D + T ⇒ 4He + n. If you add up the mass of the particles you start with, and subtract the mass of the particles you end with, you can easily calculate the mass that was converted into energy. In this case, we start with one Deuteron with a mass of 2.013553 and one atom of Tritium with a mass of 3.015500, giving us a starting mass of 5.029053. We end with one atom of Helium-4 with a mass of 4.001506 and one neutron with a mass of 1.008665, giving us an ending mass of 5.010171. Subtracting the two, we discover that a mass of 0.018882 has been coverted into energy. We convert that into the fraction of fuel that is transformed into energy by dividing it by the starting mass: Ep = 0.018882 / 5.029053 = 0.00375.

Plugging that into our equation Ve = sqrt(2 * 0.00375) = 0.0866 = 8.7% c.

### Deuterium-helium 3 fusion rocket

Deuterium-Helium3 Fusion rockets use the fusion reaction D + 3He ⇒ 4He + p. Start with one Deuteron with a mass of 2.013553 and one atom of Helium 3 with a mass of 3.014932, giving us a starting mass of 5.028485. We end with one atom of Helium-4 with a mass of 4.001506 and one proton with a mass of 1.007276, giving us an ending mass of 5.008782. Subtracting the two, we discover that a mass of 0.019703 has been coverted into energy. Ep = 0.019703 / 5.028485 = 0.00392.

Plugging that into our equation Ve = sqrt(2 * 0.00392) = 0.0885 = 8.9% c.

The D + 3He reaction is of particular interest for rocket propulsion, since all the products are charged particles. This means the they can be directed by a magnetic field exhaust nozzle, instead of spraying everywhere as deadly radiation.

Unfortunately, if you want to minimize the amount of x-rays emitted, you have to choke the reaction down to 100 keV per particle, resulting in a pathetic exhaust velocity of 2.5% c (7,600,000 m/s).

### Deuterium-deuterium fusion rocket

Deuterium-deuterium Fusion rockets use the fusion reaction D + D ⇒ T + p or 3He + n. Start with two Deuteron with a mass of 2.013553 for a starting mass of 4.027106.

We end with either

• a Triton and a proton: 3.015500 + 1.007276 = 4.022776. 0.00433 converted into energy. Ep = 0.00108
• a Helium-3 and a neutron: 3.014932 + 1.008665 = 4.023597. 0.003509 converted into energy. Ep = 0.000871

Plugging that into our equation

• Ve = sqrt(2 * 0.00108) = 0.0465 = 4.7% c
• Ve = sqrt(2 * 0.000871) = 0.0418 = 4.2% c

### Hydrogen-boron thermonuclear fission rocket

Hydrogen - Boron Thermonuclear Fission rockets use the reaction p + 11B ⇒ 3 × 4He. Start with one Proton with a mass of 1.007276 and one atom of Boron with a mass of 11.00931, giving us a starting mass of 12.016586. We end with three atoms of Helium-4, each with a mass of 4.001506, giving us an ending mass of 12.004518. Subtracting the two, we discover that a mass of 0.012068 has been coverted into energy. Ep = 0.012068 / 12.016586 = 0.001.

Plugging that into our equation Ve = sqrt(2 * 0.001) = 0.045 = 4.5% c.

### Watch the Heat

From my limited understanding, the basic problem with increasing exhaust velocity is how to keep the engine from vaporizing.

Fp = (F * Ve ) / 2

where

• Fp = thrust power (watts)
• F = thrust (newtons)
• Ve = exhaust velocity (m/s)

The problem is that at high enough values for exhaust velocity and thrust, the amount of watts in the jet is too much. "Too much" is defined as: if only a fractional percentage of those watts are lost as waste heat, the spacecraft glows blue-white and evaporates. The size of the dangerous fractional percent depends on heat protection technology. There is a limit to how much heat that current technology can deal with, without a technological break-through.

Jerry Pournelle says (in his classic A STEP FARTHER OUT) that an exhaust velocity of 288,000 m/s corresponds to a temperature of 5 million Kelvin.

As an exceedingly rough approximation:

Ae = (0.5 * Am * Av2) / B

where

• Ae = particle energy (Kelvin)
• Am = mass of particle (g) (1.6733e-24 grams for monatomic hydrogen)
• Av = exhaust velocity (cm/s)
• B = Boltzmann's constant: 1.38e-16 (erg K-1)
• x2: square of x, that is x * x

(note that the above equation is using centimeters per second, not meters per second)

A slightly less rough approximation:

Qe = (Ve / (Z * 129))2 * Pw

where

• Qe = engine reaction chamber temperature (Kelvin)
• Ve = exhaust velocity (m/s)
• Z = heat-pressure factor, varies by engine design, roughly from 1.4 to 2.4 or so.
• Pw = mean molecular weight of propellant, 1 for atomic hydrogen, 2 for molecular hydrogen

The interiors of stars are 5 million Kelvin, but few other things are. How do you contain temperatures of that magnitude? If the gadget is something that can be mounted on a ship smaller than the Queen Mary, it has other implications. It is an obvious defense against hydrogen bombs, for starters.

Larry Niven postulates something like this in his "Known Space" series, the crystal-zinc tube makes a science-fictional force field which reflects all energy. Niven does not explore the implications of this. However, Niven and Pournelle do explore the implications in THE MOTE IN GOD'S EYE. The Langston Field is used in the ship's drive, and as a force screen defense. The Langston field absorbs energy, and can re-radiate it. As a defense it sucks up hostile laser beams and nuclear detonations. As a drive, it sucks up and contains the energy of a fusion reaction, and re-radiates the energy as the equivalent of a photon drive exhaust.

(And please remember the difference between "temperature" and "heat". A spark from the fire has a much higher temperature than a pot of boiling water, yet a spark won't hurt your hand at all while the boiling water can give you second degree burns. The spark has less heat, which in this context is the thrust power in watts.)

Reaction Chamber Size

If one has no science-fictional force fields, as a general rule the maximum heat load allowed on the drive assembly is around 5 MW/m2. This is the theoretical ultimate, for an actual propulsion system it will probably be quite a bit less. For a back of the envelope calculation:

Af = sqrt[(1/El) * (1 / (4 * π))]

Rc = sqrt[H] * Af

where

Af = Attunation factor. Anthony Jackson says 0.126, Luke Campbell says 0.133
El = Maximum heat load (MW/m2). Anthony Jackson says 5.0, Luke Campbell says 4.5
π = pi = 3.141592...
H = reaction chamber waste heat (megawatts)
Rc = reaction chamber radius (meters)
sqrt[x] = square root of x

As a first approximation, for most propulsion systems one can get away with using the thrust power for H. But see magnetic nozzle waste heat below.

Science-fictional technologies can cut the value of H to a percentage of thrust power by somehow preventing the waste heat from getting to the chamber walls (e.g., Larry Niven's technobabble crystal-zinc tubes lined with magic force fields).

Only use this equation if H is above 4,000 MW (4 GW) or so, and if the propulsion system is a thermal type (i.e., fission, fusion, or antimatter). It does not work on electrostatic or electromagnetic propulsion systems.

(this equation courtesy of Anthony Jackson and Luke Campbell)

Playing with these figures will show that enclosing a thermal torch drive inside a reaction chamber made of matter appears to be a dead end. Unless you think a drive chamber a half mile in diameter is reasonable.

Therefore, the main strategy is to try and direct the drive energy with magnetic fields instead of metal walls. The magnetic field is created by an open metal framework ("magnetic nozzle"). The metal framework lets the heat escape instead of trying to stop the heat to the detriment of the metal reaction chamber. The magnetic field cannot be vaporized since it is composed of energy instead of matter. Note this is different from an ion drive, where the exhaust is being accelerated by electromagnetic or electrostatic fields. In this case, the exhaust is being accelerated by thermal, fusion, or antimatter reactions; the magnetic fields are being used to contain and direct the exhaust.

Magnetic nozzles are used in some fusion and antimatter propulsion systems.

With these propulsion systems, H is not equal to thrust power. It is instead equal to the fraction of thrust power that is being wasted. In other words the reaction energy that cannot be contained and directed by the magnetic nozzle. Which usually boils down to neutrons, x-rays, and any other reaction products that are not charged particles.

For instance, D-T (deuterium-tritium) fusion produces 80% of its energy in the form of uncharged neutrons and 20% in the form of charged particles. The charged particles are directed as thrust by the magnetic nozzle, so they are not counted as wasted energy. The pesky neutrons cannot be so directed, so they do count as wasted energy. Therefore in this case H is equal to 0.8 * thrust power.

Magnetic nozzles are gone into with more detail here in the Torchship section.

And don't forget the Kzinti Lesson.

## Delta-V

The main number of interest is deltaV. This means "change of velocity" and is usually measured in meters per second (m/s) or kilometers per second (km/s). A spacecraft's maximum deltaV can be though of as how fast it will wind up traveling at if it keeps thrusting in one direction until the propellant tanks run dry.

If that means nothing to you, don't worry. The important thing is that a "mission" can be rated according to how much deltaV is required. For instance: lift off from Terra, Hohmann orbit to Mars, and Mars landing, is a mission which would take a deltaV of about 18,290 m/s. If the spacecraft has equal or more deltaV capacity than the mission, it is capable of performing that mission. The sum of all the deltaV requirements in a mission is called the deltaV budget.

This is why it makes sense to describe a ship's performance in terms of its total deltaV capacity, instead of its "range" or some other factor equally silly and meaningless. In Michael McCollum's classic Antares Dawn, when the captain asks the helmsman how much propellant they have, the helmsman replies that they have only 2200 kps (kilometers per second) left in the tanks.

Astronautical Engineer Lauren Potterat suggests replace delta-V or Δv with "yeet" in professional contexts. Personally I'm all for it.

To calculate the spacecraft's total DeltaV you can thank Konstantin Tsiolkovsky and the awsome Tsiolkovsky rocket equation. Sir Arthur C. Clarke called the most important equation in the whole of rocketry.

Anyway, the equation is:

Δv = Ve * ln[ M / Me ]

Δv = Ve * ln[R]

where

• Δv = ship's total deltaV capability (m/s)
• Ve = exhaust velocity of propulsion system (m/s). Remember if you only have specific impulse (Isp), you can calculate Ve with Ve = Isp * 9.81
• M = mass of rocket with full propellant tanks (kg)
• Me = mass of rocket with empty propellant tanks (kg)
• R = ship's mass ratio
• ln[x] = natural logarithm of x, the "ln" key on your calculator

The inverse of the deltaV equation sometimes comes in handy.

R = ev/Ve)

where

• ex = antilog base e or inverse of natural logarithm of x, the "ex" key on your calculator

In rocket design, you generally start with the deltaV needed for a given mission. The above equation will then tell you the mass ratio required, which gives you the mass budget your rocket design has to fit into.

As a matter of interest, if the mass ratio R equals e (that is, 2.71828...) the ship's total deltaV is exactly equal to the exhaust velocity. Depressingly, increasing the deltaV makes the mass ratios go up exponentially. If the deltaV is twice the exhaust velocity, the mass ratio has to be 7.4 or e2. If the deltaV is three times the exhaust velocity, the mass ratio has to be 20 or e3.

In the real world, multi-stage rockets use a low exhaust velocity/high thrust engine for the lower stages and high exhaust velocity/low thrust engines in the upper stages.

### Delta-V Implications

There is a very important consequence of the delta V equation that might not be obvious at first glance. What it boils down to is that if the delta V requirements for the mission is less than or about equal to the exhaust velocity, the mass ratio is modest and large payloads are possible. But if the delta V requirements are larger than the exhaust velocity, the mass ratio rapidly becomes ridiculously expensive and only tiny payloads are allowed. Most of the ship will be propellant tanks.

If the engine has a variable exhaust velocity (if it can shift gears) the general rule is the maximum economic mass ratio is about 4.0. If the engine has a fixed exhaust velocity, the maximum economic mass ratio is about 4.95.

The implication is that for a mass ratio of 4.0 (variable exhaust velocity), the delta V requirement for the mission cannot be larger than about 1.39 times the engine's exhaust velocity (i.e., ln[4.0]). This is because Δv / Ve = ln[R]

For a mass ratio of 4.95 (fixed exhaust velocity), the delta V requirement for the mission cannot be larger than about 1.5 times the engine's exhaust velocity (i.e., ln[4.95]).

Refer to the chart above to see how quickly the mass ratio can spiral out of control. Divide delta V by exhaust velocity and find the result on the bottom scale. Move up to the green line. Move to the left to see the required mass ratio. For instance, if the delta V requirement is 105,000 m/s, and you are using Gas Core rockets with an exhaust velocity of 35,000 m/s, the ratio is 3. Find 3 on the bottom scale, move up to the green line, then move to the left to discover that the required mass ratio is a whopping 20!

I personally did not notice the above implication until I read about it in Jon Zeigler and James Cambias' book GURPS: Space.

Turning it around, this means for a once you choose a variable exhaust velocity propulsion system, you will know that it will not be able to do a mission with a delta V requirement over Ve * 1.39, not if you want to keep the mass ratio below 4.0

And once you choose a fixed exhaust velocity propulsion system, you will know that it will not be able to do a mission with a delta V requirement over Ve * 1.5, not if you want to keep the mass ratio below 4.0

Turning it around again, if you have chosen the mission, once you know the mission delta V you can calculate the optimal exhaust velocity for your variable exhaust velocity propulsion system: Ve = Δv * 0.72 (where 0.72 = 1/ln[4.0]).

And once you know the mission delta V you can calculate the optimal exhaust velocity for your fixed exhaust velocity propulsion system: Ve = Δv * 0.63.

Why is there an optimum value? If the exhaust velocity is too high, you are wasting energy in the form of high-velocity exhaust. If the exhaust velocity is too low, you are wasting energy by accelerating vast amounts of as-yet unused propellent. Dr. Geoffrey A. Landis says that this optimization is somewhat tedious to prove mathematically, you have to use calculus to maximize the value of kinetic energy of payload as a function of exhaust velocity. You have to iteratively solve the equation 0.5 = x * (1 - e-1/x). If you are interested WolframAlpha has a calculator for that function here.

If you are using gas-core or plasma core antimatter engines (or other engine where the fuel mass is microscopic compared to the propellant mass) there are some unexpected implications.

### Alternate Delta-V Equations

If you are using a Laser Thermal engine or a Solar Moth engine the equation is slightly different.

Δv = sqrt((2 * Bp * Bε) / mDot) * ln[R]

R = ev/sqrt((2 * Bp * Bε) / mDot)

where

• Bp = Beam power (watts) of either laser beam or solar energy collected
• Bε = efficiency with which engine converts beam power into exhaust kinetic energy (0.0 to 1.0)

Basically the exhaust velocity Ve is equal to sqrt((2 * Bp * Bε) / mDot)

If you are using a beam-core antimatter engine the equation is hideously different. Just the beam-core antimatter, the standard delta V equation does apply to solid-core, gas-core, and plasma-core antimatter engines.

First off the whole matter-into-energy process invalidates the assumption that the matter in the system at the start is the same as the matter in the system at the end. Secondly the exhaust particles are commonly moving near the speed of light, so relativistic effects changes the particle's rest mass.

This increases the mass ratio required for a given delta V. It also forces the simple delta V equation to turn into a monster:

I'm not even going to try and explain it, much less try to use it. If you want more details, refer to the Wikipedia article.

## Shifting Gears

VASIMR
Thrust Power5,800,000 w
High Gear
Exhaust velocity294,000 m/s
Thrust40 n
Medium Gear
Exhaust velocity147,000 m/s
Thrust80 n
Low Gear
Exhaust velocity29,000 m/s
Thrust400 n
LANTR
NERVA mode
Exhaust velocity9,221 m/s
Thrust67,000 n
LOX mode
Exhaust velocity6,347 m/s
Thrust184,000 n

Certain propulsion systems can "shift gears" much like an automobile. Basically they can trade thrust for exhaust velocity (specific impulse) and vice versa. There are four main reasons this can be a useful feature.

What does this mean? Thrust affects acceleration, or how fast the ship can increase speed or reduce speed. Exhaust velocity (specific impulse) affects the ship's gas mileage, or how fast it uses propellant. So you can shift the rocket into low gear if you want to burn rubber, but at the cost of the rocket guzzling propellant. Or shift it to high gear when you want to make the propellant last as long as possible, and you can live with the rocket accelerating at a snail-like pace.

Shifting gears is analogous to an aircraft activating an afterburner, or a drag racing car turning on their nitro injector.

Example spacecraft and engines include:

Many engines (such as LANTR) can change gears by simply injecting a heavy cold propellant into the hot exhaust (LANTR uses hydrogen for propellant and oxygen for gear-shifting propellant). Usually the cold propellant is a different compound than the hot propellant. The addition of a new tank of gear-shifting propellant does increase the total propellant mass, the ship's mass ratio, and the ship's delta V (but cuts into your payload mass).

Other engines such as VASIMR can change gears by altering internal operations (the amount electromagnetic propellant heating and levels of propellant mass flow). VASIMR engines do not need a second type of propellant, so there is no change to mass ratio or delta V. But the propellant mass flow (propellant consumption) rises to ugly levels.

A crude form is the Santarius Fusion Rocket. It is a fusion engine with three different operating modes. All three modes share the fusion reactor as the power source, but use the power in three different ways. I guess a given fusion rocket design can be built with one, two, or all three modes. Those with more than one can change gears.

Another crude form is the Hybrid BNTR/EP. This is a bimodal nuclear thermal rocket with the electrical power output hooked up to an ion thruster. Meaning the nuclear engine can produce either thrust or electrical power for the bolted-on ion drive. In this case changing gears is more like turning off one engine and turning on another. Naturally the electical power produced has less energy than the nuclear rocket thrust power, since the power conversion equipment is nowhere near 100% efficient.

Remember that the thrust power is equal to the exhaust velocity times thrust, divided by two. Usually when a drive changes gears the thrust and velocity change, but the thrust power stays the same. But not always (e.g., LANTR).

The point is if the thrust power stays the same, you can use that equation to calculate the changes in thrust and exhaust velocity.

The side effect is the propellant consumption (or "mDot") kilograms of propellant expended per second of engine burn. mDot is equal to thrust divide by exhaust velocity. In other words, if you shift gears so that the thrust increases, your propellant consumption will increase as well.

Fp = ( Ve * F) / 2

F = (Fp * 2) / Ve

Ve = (Fp * 2) / F

mDot = F / Ve

(you won't need these following equations unless you are reverse-engineering)

F = mDot * Ve

Ve = F / mDot

Ve = sqrt[(Fp *2) / mDot]

mDot = (Fp *2) / Ve2

where:

Fp = Thrust Power (watts) should be a constant for a given engine
F = Thrust (Newtons)
Ve = Exhaust Velocity (m/sec) = specific impulse * 9.81
mDot = Propellant Mass Flow (kg/sec) sum of both kinds of propellant
sqrt[x] = square root of x

High and Low Gear

By analogy with the terminology for automobile gear ratios, low thrust/high exhaust velocity is called high gear, and high thrust/low exhaust velocity is called low gear. You put your automobile into low gear when you are trying to pull something heavy, trading speed for pulling force. You put your automobile into high gear when you are flying down the highway, trading pulling force for speed.

Dr. Stuhlinger notes that low gear mode allows fast human transport vessels with short trip times while high gear mode allows cargo vessels with large payload ratios. He compares these to sports cars and trucks, respectively.

Why would you want to change gears? Four main reasons are:

1. Optimizing the exhaust velocity to the mission delta V
2. Engine has super-fantastic exhaust velocity but thrust is ludicrously tiny
3. Using the same engine for lift-off and interplanetary flight
4. Using the same engine for interplanetary flight and dodging hostile weapons fire

[1] Remember that given the delta V requirements for a mission, the optimal exhaust velocity is Ve = Δv * 0.72. By changing gears, you can throttle the exhaust velocity to the optimal value.

[2] Engines like fission-fragment rockets have ultra-fantastic exhaust velocity / specific impulse but the thrust is so low as to be worthless. As a general rule the acceleration should be at least 5 milligees (0.05 m/s2) or the ship will take years to change orbits. By shifting to low gear the exhaust velocity drops from ultra-fantastic to just fantastic, but the thrust rises to something worthwhile.

[3] Interplanetary flight is mostly indifferent to thrust, but lift-off and landing have to deal with the gravity tax. When sitting on the launch pad at Terra, Terran gravity imposes 9.81 m/s of delta V downward, per second (1 g). This is the gravity tax. If a spacecraft on the pad does not have enough thrust to accelerate more than the gravity tax, it is going nowhere. Every second the ship needs enough thrust to make enough acceleration to pay the tax plus the acceleration needed to lift into orbit.

Rick Robinson had a sample torchship with a high gear acceleration of 0.3 g (ship mass 1,000 metric tons, thrust of 3,000 kN, exhaust velocity of 300 km/s). This was not enough to pay the gravity tax. So in order to lift off, it would shift to low gear. This had an acceleration of 1.5 g (thrust 14,700 kN, exhaust velocity 50 km/s) which is enough to pay the tax and get into orbit. The drawback is that in low gear the torchship has a total delta V of only 40 km/s, but in high gear it has 200 km/s.

[4] The first rule of spacecraft combat is: Don't get hit. By dodging around with evasive maneuvers you complicate the enemy's targeting solution (i.e., make yourself much harder to hit). And the higher the thrust the better you can dodge.

## Pulsed Mode

Many rocket engines are what we call "power limited". It would be great if their power could be increased, which generally increases the thrust and/or the specific impulse. But all to often there comes a point where the waste energy from the reaction will destroy the engine.

The classic example is the nuclear thermal rocket. Feed propellant into a hot nuclear reactor, then send the now-hot propellant out the exhaust nozzle to create thrust. The hotter the reactor, the higher the specific impulse. Problem is that if the reactor gets hotter than 3200° K or so, the poor reactor melts. The molten core goes shooting out the exhaust like radioactive lava diarrhea and things go downhill from there. Just below the disaster point the rocket can crank out a specific impulse of 1,200 seconds or so. Admittedly this is better than the 450 seconds that chemical engines max out at. But still disappointing.

Engineers try to do an end run around this by designing nuclear thermal engines where the reactor is already molten or even gaseous. But these compound the problem of engine design something horrible.

Then some rocket designers had an idea.

What if you pulsed the nuclear reaction?

That is, you could run the reaction at a temperature far over the melting point of the reactor, but only for a fraction of a second. Not long enough for the engine to actually start melting. Then you let the engine cool off. After that it is time for the next pulse. The idea is to make sure that the average power never overloads the system. The technical term is something like "transient overpower".

By cleverly choosing the pulse parameters, one could get spurts of propellant emerging at a specific impulse far greater than a mere 1,200 seconds but without the engine melting. The result was the Pulsed Solid-core NTR.

Because engineers cannot leave well enough alone, they tried pulsing a gas-core reactor as well. Pulsing was supposed to avoid a gaseous reactor altogether, but in this case they were more pulsing the neutron flux than they were the temperature in an attempt to burn up all the uranium before it escaped out the exhaust.

Engines that use the pulsing trick include:

You can find automatic pulsing in the TRIGA reactor. It has what is known as a prompt negative fuel temperature coefficient of reactivity. Which means as the temperature of the reactor core rises, the nuclear reaction level rapidly decreases.

Sometimes in science fiction they take this to extremes. Instead of a pulse which allows the machine to run intermittently a bit overpower, instead they feed the machine several orders of magnitude as much power as it is rated for. In the split second before the machine explodes, it hopefully produces an output pulse which is several orders of magnitude larger than usual.

## Camera Flash

Certain propulsion systems (particularly inertial confinement fusion) require electricity in "jolts"; that is, very very powerful but mercifully brief in duration.

Electronic flash units on cameras had a similar problem. The flashtube requires a flash pulse of 250 to 5,000 volts but of brief duration.

Since AA batteries can only manage a few volts, you can see they are woefully inadequate to the task. The solution is to use a photoflash capacitor. The AA batteries can take a few seconds to fill the capacitor to the brim with electrical power. Then the capacitor can shoot all the stored power through the flashtube in a fraction of a second.

This is why after your camera's flash has gone off, you'll hear a whine for a few seconds while the batteries charge up the capacitor. When the capacitor is loaded, the orange "ready light" comes on to indicate another flash is ready to fire.

Anyway, IC fusion and related engines also use huge capacitor banks for the same reason as camera electronic flash units.

Typically the capacitor banks get their initial charge from a little one-lung fission nuclear reactor. A few days before a scheduled engine burn in the ship's mission, the nuclear reactor will chug away and gradually fill up the capacitor banks. This takes a day or two.

At the start of the burn, the capacitor banks dump all their power into the fusion engine in a single brief pulse. The engine will use this to ignite a pellet of fusion fuel, creating a small fusion explosion. Most of the fusion energy is used for thrust, rebounding from a magnetic field. But cleverly, some of the energy is skimmed off and used to instantly re-charge the capacitor banks. The nuclear reactor is shut down since it should not be needed for the remaining duration of the engine burn. The fusion engine will continue zapping pellets of fusion fuel at a rate of about one to one-hundred fusion explosions per second, recharging the capacitors by skimming the energy. This stops once the required amount of delta-V is imparted to the spacecraft.

The "skimming off" of some of the fusion energy is usually done by tapping the magnetic field. Remember the fusion explosion slams into the magnetic field, moving the field, which moves the magnetic field coils, which move the ship's thrust frame, which propels the ship. Since the basis of an electrical power generator is moving a magnetic field through conducting cables, it is relatively easy to convert some of the energy of the slamed magnetic field into electricity.

At least one of the engine designs use belt-and-suspenders logic, that is, redundancy.

What if the engine misfires? If you drain the capacitor into the engines but it fails to produce a fusion explosion, there will be no fusion energy to skim off for capacitor recharge. It will take two days to charge up the capacitor banks with the one-lung nuclear reactor, which will throw off the ship's trajectory. Scheduled engine burns have to be precisely timed or you might miss your destination altogether, dooming the crew.

The belt-and-suspender engines have not one but two capacitor banks. If the engine misfires, you still have a spare charged bank. Hopefully the engine does not misfire again. You can temporarily skim off double the amount of fusion energy to recharge both banks.

Engines that use the pulsing trick include:

## Propellant-less Rockets

There are a couple of utterly bizarre propulsion systems that do not use propellant, at least not propellant that is composed of matter. The problem is with these weirdos is the mass ratio and delta-V equations don't work with them.

More to the point, these propulsion systems are not subject to The Tyranny of the Rocket Equation.

Please understand I am not talking about engines like the Laser Thermal or Solar Moth. They use matter propellant, it is just that their power source is located at some distance from the actual spacecraft.

And I am most certainly not talking about Reactionless Drives (keep your voice down! If RocketCat hears you it is Atomic Wedgie time). Those crack-pot drives allegedly do not use Newton's Third Law at all, and are only taken seriously by those who think the law of conservation of momentum is more what you'd call a 'guideline' than actual law.

Propellant-less rockets include:

• Photon Sails and Laser Sails: which create thrust by bouncing photons (which are energy, not matter) off a mirrored sail. The photons typically come from the Sun or a remote laser station. Do not confuse them with solar moth or laser thermal. Both are powered by the Sun or a remote laser station, but the sails bounce photons with mirrors while the others use photons to heat up material propellant.
• Photon Drives: the propellant is a beam of photons. Basically the engine is a honking huge laser.
• Tachyon Drive: the exhaust is a beam of tachyons, which are technically matter. The point is that the engine does not have a tank full of tachyons at the start, the tachyons are created out of energy as needed. So the mass ratio equation does not work since the propellant mass at the start of the mission is zero.

## Rocket Engine Components

Rocket engines use Newton's Third Law to generate thrust. The action of sending propellant out the rocket nozzle causes the reaction we call thrust.

As RocketCat so brusquely put it: "Propellant is the crap you chuck out the exhaust pipe to make rocket thrust. Fuel is what you burn to get the energy to chuck crap out the exhaust pipe."

1. The first component of the engine is the fuel that is burnt to generate energy
2. The second component is the generator which burns the fuel.
3. The third component is the propellant or reaction mass.
4. The fourth component, the energy conversion system uses the energy from the burnt fuel to make the propellant move at high velocity.
5. The high speed propellant stream is sent through the fifth component the exhaust deflection system in order to direct the stream in the appropriate direction

There are some cases where the fuel and the propellant are one and the same. Examples include chemical rockets and fusion drives that use the fusion products as reaction mass.

After going through the list of existing and experimental rocket propulsion systems, I've compiled lists of the various components. These lists are not complete, I'm not a rocket scientist so I might have forgotten a few. And some of the classifications might be incorrect.

But again, until a real rocket scientist decides to do it right, I'll be forced to do the best I can.

### Fuel

This is the fuel that is burnt in order to generate energy. Remember that fuel and propellant are two different things.

Antimatter
βPositrons
pAntiprotons
HAntihydrogen
Chemical Liquid
CH4/O2Liquid Methane / Liquid OxygenPoor performance, but the stuff can be stored almost indefinitely in space, unlike other liquid fuels. It is also available on some outer moons.
H2/F2Liquid Hydrogen / Liquid FluorinePretty close to the maximum possible performance out of a chemical rocket. A pity that fluorine is insanely dangerous and will burn up pretty much anything. Let's just say that on Dr. Derek Lowe's list of things he will not work with Fluorine is near the top of the chart.
H1/O2Single-H / Liquid OxygenFree Radical Hydrogen (atomic hydrogen) has about five times the performance of molecular hydrogen. It is a pity the stuff wants to implode back into molecular hydrogen at the slightest provocation.
H2/O2Liquid Hydrogen / Liquid OxygenAlmost as good performance as H2/F2, but without the nasty fluorine.
RP-1/O2RP-1 / Liquid OxygenRP-1 is highly refined kerosene. This is NASA's favorite fuel. Almost as good performance as H2/O2, but without liquid hydrogen's strict cryogenic requirements and lamentable lack of density.
UDMH/NTODimethylhydrazine
+ Nitrogen Tetroxide
MMH/NTOMonomethylhydrazine
+ Nitrogen Textroide
Chemical Solid
Al/APAluminum / Ammonium PerchlorateSolid ammonium perchlorate composite propellant (APCP). Powdered aluminum fuel is mixed with ammonium perchlorate oxidizer in a rubbery binder. Burn rate catalysts are also added to control the burn rate. APCP was used in the Space Shuttle solid-rocket boosters.
Chemical Hybrid
Al/O2Aluminum / Liquid OxygenFinely sintered aluminum dust is sprayed with liquid oxygen. The aluminum has the storability advantage shared by chemical solid fuel rockets, and the liquid oxygen gives the throttle and turn-off capabilities of chemical liquid rockets. The specific impulse is poor, but the raw materials are availabled by in-situ resource utilization (i.e., on Luna and the asteroids, the raw material is in the dirt!).
Metastable
Met-HMetallic HydrogenHydrogen squeezed until it turns into a metallic soid, then somehow convinced not to explode into gas until needed.
He*Metastable He*Helium in a long-lived excited state
He IV-AMetastable He IV-AHelium in a long-lived excited state
Electrical Power
10 MWeTen megawatts of electrical inputMany drives are "fueled" by electricity. They typically use solar photovoltaic array or fission reactors. Example: Ion drive.
External
Ext Plas-BeamExternal Plasma BeamA fixed installation such as space station sends a beam of plasma to the spacecraft. Example: MagBeam.
Ext LaserExternal LaserA fixed installation such as space station sends a laser beam to the spacecraft. Example: Laser Thermal and Laser Sail.
Ext KineticKinetic PelletsA fixed installation such as space station sends a stream of kinetic pellets to the spacecraft. Some use the kinetic energy of the pellets, but many us the momentum of the pellets. Also includes concepts like Kare's Sailbeam.
Sol MagSolar MagnetismSpacecraft utilizes the environmental solar magnetic field for propulsion. Example: M2P2.
Sol PhotonSolar PhotonsSpacecraft utilizes the environmental sunlight for propulsion. Example: Photon Sail.
Sol WindSolar WindSpacecraft utilizes the environmental solar wind for propulsion. Example: E-Sail.
Fission
245CmCurium-245
6LiLithium-6
239PuPlutonium-239
233UUranium-233
235UUranium-235
UBr4Uranium-235 Tetrabromide
UF6Uranium-235 Hexafluoride
FIGeneric Fissionable
Fusion
4xHProton - Proton
D-DDeuterium - Deuterium
D-TDeuterium - Tritium
H-BHydrogen - Boron
H-FeHydrogen - Iron
H-6LiHydrogen - Lithium-6
H-7LiHydrogen - Lithium-7
3He-DHelium-3 - Deuterium
3He-3HeHelium-3 - Helium-3
FUGeneric Fusion Fuel
D-T + 6Li-nDeuterium - Tritium fusion
+ Lithium-6 fission

### Generator

This is the generator that consumes the fuel and burns it into energy. The energy output can be thermal, electric, high-speed subatomic particles, or other forms. The energy will be used to accelerate the propellant.

THERMAL
output is thermal energy
Thermal-Fission
Solid CoreFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core melts.
Liquid CoreFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from molten into vapor.
Vapor CoreFission or antimatter powered device to thermally heat propellant.
This is generally a worthless design with the performance of a liquid core, but which gives vital experience in designing a gas core reactor.
Gas Core Closed-CycleFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. Fissionables are in a second loop to prevent them from escaping into the exhaust plume.
Gas Core Vortex ConfinedFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. An attempt is made to prevent fissionables from escaping into the exhaust by tailoring a vortex in the chamber.
Gas Core MHD ChokeFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. An attempt is made to prevent fissionables from escaping into the exhaust by MHD fields.
Gas Core Open-CycleFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. Fissionables escape into the exhaust with no constraint.
Plasma CoreFission, fusion or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from ionized plasma to subatomic particles.
Pulse UnitBasically a tiny self-contained bomb. It does not require external energy such as zeta-pinch fields nor beams of antiprotons to explode. Generally it is a sort of shaped charge designed to vaporize a slab of propellant and direct it at a pusher plate. Usually either a fission or fusion device.
Ultracold Neutron CatalyzedUltracold neutrons are a way to induce tiny sub-critical masses of fissionable fuel to explode with nuclear fission without needing a critical mass or neutron reflectors.
Antimatter CatalyzedThis technique uses beams of antiprotons to catalyze tiny bits of fusion fuel or tiny sub-critical masses of fission fuel to undergo nuclear reactions without requiring huge magnetic fields, banks of laser beams, critical masses or neutron reflectors.
Zeta-PinchZeta-Pinch is a technique to use large electrical currents to generate large crushing magnetic fields. Said fields can be use to squeeze tiny subcritical masses of fission fuel into criticality, or fusion fuel plasmas into fusion reactions.
Fission-fragment HeatingFission fragments from fissionables undergoing nuclear decay heat the propellant, typically liquid hydrogen.
Thermal-Fusion
Electrostatic ConfinementFusion fuel is squeezed into reacting by electrostatic fields. Example Polywell Fusor.
Inertial Confinement LaserFusion fuel is squeezed into reacting by an encircling barrage of laser beams. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Inertial Confinement Particle BeamFusion fuel is squeezed into reacting by an encircling barrage of particle beams. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Open-field Magnetic Confinement (linear mirror)Fusion fuel is squeezed into reacting by a linear magnetic bottle. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Closed-field Magnetic Confinement (toroidal)Fusion fuel is squeezed into reacting by a toroidal magnetic tokamak. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Magneto-Inertial ConfinementFusion fuel is squeezed into reacting by a magnetically crushed metal propellant foil ring. Propellant foil is heated thermally.
Plasma CoreFission, fusion or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from ionized plasma to subatomic particles.
Muon CatalyzedThis technique uses beams of muons to catalyze induce tiny bits of fusion fuel to undergo nuclear fusion without requiring huge magnetic fields or banks of laser beams.
Pulse UnitBasically a tiny self-contained bomb. It does not require external energy such as zeta-pinch fields nor beams of antiprotons to explode. Generally it is a sort of shaped charge designed to vaporize a slab of propellant and direct it at a pusher plate. Usually either a fission or fusion device.
Antimatter CatalyzedThis technique uses beams of antiprotons to catalyze tiny bits of fusion fuel or tiny sub-critical masses of fission fuel to undergo nuclear reactions without requiring huge magnetic fields, banks of laser beams, critical masses or neutron reflectors.
Zeta-PinchZeta-Pinch is a technique to use large electrical currents to generate large crushing magnetic fields. Said fields can be use to squeeze tiny subcritical masses of fission fuel into criticality, or fusion fuel plasmas into fusion reactions.
Thermal-Antimatter
Solid CoreFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core melts.
Liquid CoreFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from molten into vapor.
Vapor CoreFission or antimatter powered device to thermally heat propellant.
This is generally a worthless design with the performance of a liquid core, but which gives vital experience in designing a gas core reactor.
Gas Core Closed-CycleFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. Fissionables are in a second loop to prevent them from escaping into the exhaust plume.
Gas Core Vortex ConfinedFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. An attempt is made to prevent fissionables from escaping into the exhaust by tailoring a vortex in the chamber.
Gas Core MHD ChokeFission or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. An attempt is made to prevent fissionables from escaping into the exhaust by MHD fields.
Gas Core Open-CycleFission, fusion or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from vapor into ionized plasma. Fissionables escape into the exhaust with no constraint.
Plasma CoreFission, fusion or antimatter powered device to thermally heat propellant.
Upper limit of temperature is where the core changes from ionized plasma to subatomic particles.
REACTION PRODUCTS
output is reaction products accelerated by chemical, fission, fusion, or antimatter reaction
Combustion ChamberFor chemical fuels, a chamber where the chemicals react or "burn."
Fission-fragment PropellantFission fragments from fissionables undergoing nuclear are used as the propellant
Electrostatic ConfinementFusion fuel is squeezed into reacting by electrostatic fields. Example Polywell Fusor.
Inertial Confinement LaserFusion fuel is squeezed into reacting by an encircling barrage of laser beams. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Inertial Confinement Particle BeamFusion fuel is squeezed into reacting by an encircling barrage of particle beams. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Open-field Magnetic Confinement (linear mirror)Fusion fuel is squeezed into reacting by a linear magnetic bottle. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Closed-field Magnetic Confinement (toroidal)Fusion fuel is squeezed into reacting by a toroidal magnetic tokamak. Reaction products can be the propellant, or the reaction can heat separate propellant thermally.
Beam CoreAntimatter powered device where the reaction products are the propellant.
No upper limit to temperature.
ELECTRICAL
output is electricity
Nuclear Power Reactor (electric)Fission powered device used to supply electrical energy to the propellant accelerator.
Fusion Power Reactor (electric)Fusion powered device used to supply electrical energy to the propellant accelerator.
Photovoltaic arraySolar powered device used to supply electrical energy to the propellant accelerator.
OTHER
Collector MirrorA device for gathering external energy, such as external plasma beams, external laser beams, and solar photons.
NoneSome designs have no "reactor", most external fuels (such as laser beams) fall into this category.

### Propellant

The propellant or reaction mass is what is expelled from the rocket engine at high velocity in order to create thrust via Newton's Third Law.

Thermal
CH4Methane
COCarbon Monoxide
CO2Carbon Dioxide
H1Single-HFree radical hydrogen. For thermal acceleration single-H has superior performance to molecular hydrogen. A pity that the stuff explosively converts back to molecular hydrogen with no provocation.
H2Liquid HydrogenMolecular hydrogen. The thermal propellant of choice.
H2OWater
Seeded-HSeeded HydrogenTransparent hydrogen is poor at intercepting thermal radiation and heating up. It will heat up much more readily if you seed it with something opaque like tungsten dust.
N2Nitrogen
NH3Ammonia
O2Liquid Oxygen
Electrical
ArArgon
BiBismuth
CdCadmiumEasy to ionize, but erodes the grid. Popular until they figured out how to efficiently ionize Xenon.
CLColloidSometimes used in ion and other electrostatic drives.
CsCesiumEasy to ionize, but erodes the grid. Popular until they figured out how to efficiently ionize Xenon.
HeHelium
IIodine
KrKrypton
HgMercuryEasy to ionize, but erodes the grid. Popular until they figured out how to efficiently ionize Xenon.
MgMagnesium
XeXenonCurrently popular in ion drives, since it does not erode the grid. It took a while to figure out how to efficiently ionize the stuff.
ZnZinc
Other
CnGraphiteFor ablative laser drives and fusion pulse ablative nozzles.
DUDepleted Uranium
LiLithium
RPReaction ProductsWhere the propellant is the product of the chemical, fission, fusion, or antimatter reaction; instead a separate propellant heated by the reaction.
RKRegolithGeneral term for dirt readily available on the surface of moons and asteroids. Usually if the accelerator can use regolith, it can use anything made out of matter that can be chopped up small enough to fit into the buckets. Raw sewage, worn-out clothing, dead bodies, belly-button lint, used kitty-litter, whatever.
SiCSilicon CarbidePopular in ablative nozzles.
WTungstenFor Orion drive pulse units
γPhotonsRays of light. Generally only used in photon drives.

### Energy Conversion System

The Energy Conversion System is the mechanism that consumes energy from the generator and uses it to accelerate the propellant to high velocities.

Input: External Power.
Thermal
Arc HeaterInput: Electricity. Propellant is accelerated electrothermally by an electrical arc.
Collector Mirror HeaterPropellant is thermally accelerated by heat from sunlight or laser beams focused by a collector mirror type reactor.
Resistance HeaterInput: Electricity. Propellant is accelerated electrothermally by an electrical resistance heater.
Microwave HeaterInput: Electricity. Propellant is accelerated electrothermally by microwaves.
Reaction HeaterInput: Thermal. Propellant is thermally accelerated by heat from the chemical, fission, fusion, or antimatter reaction.
Electrical
ElectromagneticInput: Electricity. Propellant is accelerated electromagnetically (plasma drives)
ElectrostaticInput: Electricity. Propellant is accelerated electrostatically (ion drives).
Other
AnnihilationInput: Antimatter Reaction. Propellant is the subatomic particles formed by a matter-antimatter reaction.
Fission-FragmentInput: Nuclear Fission. Propellant is split atoms flying from a nuclear fission event. May be antimatter catalyzed.
NoneIncludes Reaction Product type generators.

### Exhaust Deflection System

The Exhaust Deflection System directs the stream of high speed propellant in order to move the spacecraft in the desired direction. They are mostly classified by what sort of propellant they act upon, and whether or not it focuses the exhaust.

Note that while there are magnetic nozzles there are no electrostatic nozzles. These would be used strictly by Ion drives. Ion (electrostatic energy conversion system) do not have any nozzles at all. More precisely, the "nozzle" is part of the energy conversion system that accelerates the ions.

Nozzle Thermally Hard Standard garden variety rocket nozzle. It acts on gas pressure and focuses the exhaust. Nozzle resists exhaust heat by being constructed of high-temperature alloys. Standard garden variety rocket nozzle. It acts on gas pressure and focuses the exhaust. Nozzle resists exhaust heat by a coolant system, generally using cold propellant. Rocket nozzle that is a magnetic field, since the exhaust is far to hot to be handled by a physical nozzle composed of matter. It acts magnetically on charged particles and plasma, and focuses the exhaust. Note that most ion (electrostatic energy conversion system) and plasma (Electromagnetic energy conversion system) do not have a magnetic nozzle, or any nozzle at all. The nozzle is a hemisphere with a thick layer of solid propellant. Tiny fission, fusion, or animatter explosions detonated at hemisphere center vaporize a layer of propellant which rushes out the open mouth. Examples: ACMF, Positron Ablative, D-D Fusion Inertial. A large plate of solid propellant is impacted by kinetic pellets, ablative laser beams and ablative electron beams. Impacts create shallow craters with propellant being ejected perpendicular to the plate's surface. Propellant flow is more directional than a conventional pusher plate, but less than a nozzle. A huge armored plate attached to the spacecraft by shock absorbers. Generally used with Orion nuclear pulse drives or Medusa nuclear pulse. It acts on gas pressure but does not focus the exhaust. a large sail that acts magnetically on charged particles and plasma. Does not focus the exhaust. Basically the magnetic equivalent of a conventional pusher plate. a large sail that reflects photons. Can focust the exhaust if desired. a large sail that absorbs or scatters photons or other particles. Typically glows hot due to absorbed power. a large sail that acts on charged particles via electrostatic force, does not focus the exhaust.

"Payload" is the the mass of the valuable stuff the rocket is transporting. Basically it is the reason the rocket exists. The Apollo programs Saturn V's payload was the Apollo mission: the Command module, the Service module, and the Lunar module. The payload of a cargo transport rocket is the cargo. The payload of a rocket warship is the weapons, the crew, and the habitat module.

The "payload fraction" is the fraction of the entire wet mass of the rocket which is the payload. Typically this is depressingly small, especially if you are using chemical propulsion. NASA's Saturn V had a payload fraction of 3.9%. NASA's retired space shuttle had a payload fraction of 1.4%. SpaceX's Falcon 9 has a payload fraction of 2.6%. Arianespace's Ariane 5 has a payload fraction of 2.1%.

As a matter of interest, if the mass ratio R equals e (that is, 2.71828...) the ship's total deltaV is exactly equal to the exhaust velocity. Depressingly, increasing the deltaV makes the mass ratios go up exponentially. If the deltaV is twice the exhaust velocity, the mass ratio has to be 7.4 or e2. If the deltaV is three times the exhaust velocity, the mass ratio has to be 20 or e3.

These numbers are absolute, Mother Nature doesn't allow fudging. If your ship has a mass ratio of X and an exhaust velocity of Y, it will have a deltaV of Z. If the mass ratio is decreased due to the extra mass of, say, a stowaway, the deltaV goes down. If it goes down below what is needed for the mission, this signs the death warrant for everybody on board. Period. For details see the movie Destination Moon, or the short story "The Cold Equations" by Tom Godwin.

Now, remember that the percentage of the rocket mass that is taken up by propellant is:

Pf = 1 - (1/R)

This means that the percentage of the rocket mass that is not taken up by propellant is:

Pe = 1 / R

where

• Pe = percentage of rocket mass not take up by propellant

In other words, the rocket's dry mass expressed as a percentage of the rocket's wet mass. Substituting the equation for R we get:

Pe = 1 / ev/Ve)

Pe is for the percentage of mass taken up by the propulsion system, the ship's structure, the payload, and anything else (like the crew). But hopefully most of Pe is payload, at least if this is a cargo ship. So given the ship's Δv capacity and the propulsion systems Ve, you can get a ballpark estimate of the ship's payload capacity.

This graph is the same as the previous one, only the vertical axis has be re-labeled to show how rapidly your payload shrinks (the other graph was labeled to show how rapidly the amount of propellant grows, which is more or less the same thing). See how steep the curve is? That is an example of what they call "rising exponentially", which is science-speak for "gets expensive real quick". The graph was drawn with the equation R = ev/Ve). See how v/Ve) is raised next to the e? That's what is called an exponent, its what makes the curve rise exponentially. This is why you want the delta-V to be as low as possible and the exhaust velocity to be as high as possible.

So what it is saying in English is that as the delta-V cost for the mission rises, the amount of allowed payload rapidly dwindles to zero. And using a rocket engine with a higher exhaust velocity will help. You lower delta-V by choosing more modest missions and/or using orbital propellant depots. You raise the exhaust velocity by using a more sophisticated engine.

## Delta-V Nomogram

To get some rough ballpark estimates on ship Delta-V, you can use my handy-dandy DeltaV nomogram. Download it, print it out, and grab a ruler or straightedge. You can also purchase an 11" x 17" poster of this nomogram at . Standard disclaimer: I constructed this nomogram but I am not a rocket scientist. There may be errors. Use at your own risk.

Say we needed a deltaV of 36,584 m/s for the Polaris, that's in between the 30 km/s and the 40 km/s tick marks on the DeltaV scale, just a bit above the mark for 35 km/s. The 1st gen Gas Core drive has an exhaust velocity of 35,000 m/s, this is at the 35 km/s tick mark on the Exhaust Velocity scale (thoughtfully labeled "NTR-GAS-Open (H2)").

Now, lay the straightedge between the NTR-GAS-Open tick mark on the Exhaust Velocity scale and the "2" tick mark on the Mass Ratio scale. Note that it crosses the DeltaV scale at about 24 km/s, which is way below the target deltaV of 36,584 m/s.

But if you lay the straightedge between the NTR-GAS-Open tick mark and the "3" tick mark, you see it crosses the DeltaV scale above the target deltaV, so you know that a mass ratio of 3 will suffice.

The scale is a bit crude, so you cannot really read it with more accuracy than the closest 5 km/s. You'll have to do the math to get the exact figure. But the power of the nomogram is that it allows one to play with various parameters just by moving the straightedge. Once you find the parameters you like, then you actually do the math once. Without the nomogram you have to do the math every single time you make a guess.

As with all nomograms of this type, given any two known parameters, it will tell you the value of the unknown parameter (for example, if you had the mass ratio and the deltaV, it would tell you the required exhaust velocity).

Note that the Exhaust Velocity scale is ruled in meters per second on one side and in Specific Impulse on the other, because they are two ways of measuring the same thing. In the same way, the Mass Ratio scale is ruled in mass ratio on one side, and in "percentage of ship mass which is propellant" on the other.

## Delta-V Graph

I also have a version of the DeltaV nomogram in graph form. It is not quite as easy to use. Vertical lines are the exhaust velocities of various rocket engines. Horizontal lines are the delta V requirements for various missions. Diagonal lines are the mass ratios required for the intersection of a exhaust velocity and a delta V.

## Brachistochrone Transit Time Nomogram

For calculating torchship Brachistochrone Transit Times, you can use my handy-dandy Transit Time Nomogram. Be warned, this only does torchship Brachistochrone trajectories, it cannot calculate Hohmann transfers or anything else.

Let's say that our spacecraft is 1.5 ktons (1.5 kilo-tons or 1500 metric tons). It has a single Gas-Core Nuclear Thermal Rocket engine (NTR-GAS MAX) and has a (totally ridiculous) mass ratio of 20.

The mission is to travel a distance of 0.4 AU (about the distance between the Sun and the planet Mercury). Using a constant boost brachistochrone trajectory, how long will it take this particular ship take to travel that distance?

First you calculate the spacecraft's total delta-V. Ideally this should be on the transit nomogram, but the blasted thing was getting crowded enough as it is. This calculation is on a separate nomogram found here. Lay the start of the straight-edge on Exhaust velocity scale at "NTR-GAS MAX". Have end of edge on Mass Ratio scale at 20. The total delta-V comes out to 300 kilometers per second, where the edge crosses the Delta-V scale.

Step one is to calculate the ship's maximum acceleration.

Examine the transit nomogram. On the Ship Mass scale, locate the 1.5 kton tick mark. On the Engine Type scale, locate the NTR-GAS MAX tick mark. Lay a straight-edge across the two tick marks and examine where the edge crosses the Acceleration scale. Congratulations, you've just calculated the ship's maximum acceleration: 2 meters per second per second (m/s2).

For your convenience, the acceleration scale is also labeled with the minimum lift off values for various planets. Meaning that if the ship's maximum acceleration is less than the lift off value for the planet it is sitting on, it ain't goin' nowhere. If it tries to lift-off it is just going to vibrate on the launch pad while the exhaust burns a hole in the ground.

So we know our ship has a maximum acceleration of 2 m/s2 and a maximum DeltaV of 300 km/s. As long as we stay under both of those limits we will be fine.

Step two is to calculate the travel time, which is the what we've been trying to figure out all along.

On the Acceleration scale, locate the 2 m/s2 tick mark. On the Destination Distance scale, locate the 0.4 AU tick mark. Lay a straight-edge on the two tick marks and examine where it intersects the Transit time scale. It says that the trip will take just a bit under four days.

But wait! Check where the edge crosses the Total DeltaV scale. Uh oh, it says almost 750 km/s, and our ship can only do 300 km/s before its propellant tanks run dry. Our ship cannot do this trajectory. Remember that we have to stay under both an acceleration of 2 m/s2 and under a deltaV of 300 km/s

#### Visualizing a Range of Solutions

The key to solving this dilemma is to remember that 2 m/s2 is the ship's maximum acceleration. Nothing is preventing us from throttling the engine acceleration down a bit in order to lower the DeltaV cost below its limit.

This is where a nomogram is superior to a calculator, in that you can visualize a range of solutions.

Pivot the straight-edge on the 0.4 AU tick mark (meaning, stick an imaginary pin into the 0.4 AU mark and rotate the straight-edge around it). Pivot the edge counterclockwise until it crosses the 300 km/s tick on the Total DeltaV scale. This will make the acceleration below the 2 m/s2 and have the DeltaV right at the 300 km/s limit. The ship is capable of performing this trajectory.

Now you can read the other mission values: 0.4 m/s2 acceleration and a trip time of a bit over a week. Yes, the trip time is an extra three days or so, but at least the ship can manage the trip. We will assume that the ship has enough life-support to keep the crew alive for a week or so, but that's a totally separate problem.

This pivoting technique can be used on other variables. For instance, if the time limit absolutely had to be four days, you'd stick the pin there. Rotating to keep under the acceleration and deltaV limits will tell you the maximum distance the ship can do in four days.

## Slide Rules

### RAND Rocket Performance Calculator

Back in the old days, you can spend \$15 and get the RAND Rocket Performance Calculator, which is a circular slide rule for deltaV calculations. Its a pity it doesn't do metric, and the upper limit of Isp that it will handle is disappointing. But it does give one an intuitive feel for these calculations. It is way out of print now, but below I give instructions on how to make your own, Do-It-Yourself style.

### Aristo 80123 aka "Martin Space Rule"

This is an interesting rocketry slide rule. It was made in 1962 by the Aristo company for Martin Marietta. It can do most calculations you can perform with an average run-of-the-mill slide rule, but it has extra scales that allow calculating spacecraft specific parameters. Some of the calculations relate to designing a spacecraft, the rest relate to astrogation. Specifically it can do calculations in four space technology categories:

1. Booster Design
2. Exterior Ballistics
3. Orbital Mechanics
4. Interplanetary travel

A PDF of the operating manual is available here (click on link labeled 102746940-05-01-acc.pdf).

#### Scales

Front Scale Upper:

• λ: ratio of the initial weight of a stage at launch to its final weight at burnout
• K4: ratio of the (n-3rd) stage weight to the payload weight
• K3: ratio of the (n-2nd) stage weight to the payload weight
• K2: ratio of the (n-1st) stage weight to the payload weight
• K1: ratio of the nth stage weight to the payload weight

Front Scale Slide:

• Isp: engine overall specific impulse (sec)
• K0: numerically equal to K'
• %Wpr: percentage of the propellant loaded that remains in the stage at burnout
• ↓MF indicator: Cursor hairline is moved onto the indicator in order to read where it is pointing on the MF scale below
• C: conventional slide rule "C" scale

Front Scale Lower:

• D: conventional slide rule "D" scale
• K': equal to λ / (λ-1)
• %Wd: Stage dry weight divided by the total stage weight (payload excluded) Ratio is expressed as a percentage
• MF: Propellant mass fraction of a stage {scale is collinear with %Wd scale}

Back Scale (no slide, uses hairline)

• (ε)ecc.: eccentricity of the orbit
• Va: velocity at apogee (103fps)
• ha: altitude at apogee (103st.mi)
• hm: mean altitude of the orbit (103st.mi)
• τ: orbital period (hr)
• V1: velocity at perigee (103fps)
• V2: velocity at burnout of booster {for exterior ballistics calculation} OR circular orbit velocity {for orbital mechanics calculation} (103fps)
• Ri: range from burnout at low altitude to impact on Earth's surface (103st.mi downrange)
• γBo: flight path angle at burnout (degrees from horizontal)
• TF: time of flight from burnout to impact (minutes)
• Ha: maximum altitude of flight (st.mi)
• hc: altitude of circular orbit (103st.mi)

Gutter: Planet, Escape Velocity (fps), Radius(st. mi.)

Slide Back:

• V3 (Impact Landing): burnout velocity required to leave Earth and coast to aphelion or the orbit of the target planet (103fps) Assumes Hohmann transfer
• V3 (Soft Landing): (103fps) required velocity to leave Earth and coast to the target planet, and to counteract that planet's gravitational attraction on landing {scale is collinear with V3 Impact Landing scale} Assumes Hohmann transfer
• Time of Travel: time to coast from Earth to interplanetary aphelion (years). Applicable only to the outer planets.
• Vcirc: velocity of circular planetary orbit around the sun (103fps) {scale is collinear with Time of Travel scale}
• (Ra/Re): aphelion distance divided by the Earth's mean orbital radius around the sun (A.U.). Applicable only to the outer planets.
• (R/Re): radius of orbit around the sun divided by radius of Earth's orbit from sun (A.U.) {scale is collinear with (Ra/Re) scale}

#### What calculations can it do?

Like any standard slide rule, the C and D scales can be used to multiply and divide.

Booster Design

Propellant Mass Fraction: %Wpr (percent weight propellant remaining), %Wd (percent weight dry), and MF (propellant mass fraction of stage) are related, so if any two are known the third can be determined. The ↓MF indicator is used with the cursor.

Example: if %Wd = 12.5%, %Wpr = 7.5%, calculate MF.

• Put cursor hairline over 12.5 on %Wd scale
• Holding cursor still, move slide until 7.5 on %Wpr is at the hairline
• Holding slide still, move cursor hairline to ↓MF indicator
• Observe where hairline crosses MF scale: at 0.809, which is the answer

Multi-stage booster performance and stage optimization can also be calculated by using the λ, K4, K3, K2, K1, Isp, K0, %Wpr, K', and %Wd scales. See instruction manual for details.

Exterior Ballistics

On the Back Scales, there is no slide, all the scales are fixed. One uses the cursor hairline as sort of a lookup table.

If either the burnout velocity V2 or range from burnout Ri is known, the hairline is set to that value. Then you can read off values for γBo, TF, and Ha. Plus the unknown of either V2 or Ri

Orbital Mechanics

This also uses the back scale, so again it is a lookup table. These are for calculations of objects orbiting Earth. The scales used for orbital mechanics lookups are (ε)ecc., Va, ha, hm, τ, V1, and hc. See instruction manual for details.

Interplanetary travel

Removal of the slide reveals in the gutter a small table of planetary values.

For interplanetary calculations, the reverse side of the slide is used, the one with the V3 scale. Only scales on the slide are used, so again it is sort of a lookup table. Scales used are V3 (Impact Landing), V3 (Soft Landing), Time of Travel, Vcirc, (Ra/Re), and (R/Re). All orbital transfers are assumed to be Hohmann type.

The main difference between the two V3 scales is the values the planetary symbols are pointed at, i.e., in the Impact Landing scale the Moon symbol indicates 35.5 fps, while on the Soft Landing scale the Moon symbol indicates about 43 fps. The range and spacing of the two scales is different as well.

### Custom Designed Slide Rule

If you are really into antique instruments, it is possible to design your very own custom slide rule optimized to perform one equation. Warning: it takes a lot of math skills. You can find instructions here, on my Nomogram site.

Arthur Harrill has made a nifty Excel Spreadsheet that calculates the total deltaV and other parameters of your rocket.

### Delta-V Slide Rule

Gather round grandchildren, and hear about primitive days of yore, back when dinosaurs roamed the Earth.

Back then, computers that had a thousandth the computing power found in your average modern-day smart phone were hulking monsters that filled several rooms, and were full of vacuum tubes and relays that lured moths to their death. For day-to-day calculations engineers used slide rules. Some were even circular in shape instead of looking like a ruler.

### RAND's Slide Rule

In 1958 the RAND Corporation came out with a circular slide rule to help estimate mass ratios, called the Rocket Performance Computer. It is set up to calculate the Δv = Ve * ln[ R ] equation. Meaning if you have any two of the three variables, the slide rule will reveal the value of the third.

The outer scale (labeled Delta-V Scale in diagram above) is of course the delta-V. It is set by rotating the top disk so that the Delta-V Cursor points at the required delta-V.

The next innermost scale (labeled Specific Impulse Scale) is the specific impulse or Isp. It is set by rotating the middle disk so that the Specific Impulse Cursor points at the required specific impulse. If you only have the Exhaust Velocity (Ve), you can calculate the specific impulse with Isp = Ve / 9.81

The innermost scale (labeled Mass Ratio Scale) is the Mass Ratio. It is set by rotating the top disk so that the Mass Ratio Cursor points at the required mass ratio.

The rule had special marks on the Delta-V Scale for delta-Vs required to reach various planets, so the rule was intended for space exploration scientists. However it also had delta-Vs required for Intermediate-Range Ballistic Missiles (IRBMs) and Intercontinental Ballistic Missiles (ICBMs), so it was intended for the military as well.

The Specific Impulse scale had a few special marks for commonly used chemical fuels. If anyone was trying to make a more modern version of the slide rule, they could add marks for various other more advanced propulsion systems. You could also increase the Specific Impulse scale to handle something more than 2,000 seconds. Another good idea would be to make the units metric instead of Imperial units.

The advantage of using a slide rule instead of an app is the ability to visualize a range of solutions.

In the far-fetched chance that anybody actually wants to know how the blasted thing works, it is much like a conventional slide rule. The three scales are divided logarithmically. The C scale is the Specific Impulse scale, the Mass Ratio Cursor is much like a slide rule's cursor, and the D Scale is the Delta-V scale. Plus the thing is bent into a circle like any other circular slide rule, to make it more compact.

### DIY

Nowadays instead of a slide rule one would use Mathematica or a smartphone app, but the RAND computer would be a nice educational toy for a very small child. Assuming you could get them to put down their blasted smartphone for five minutes. Unfortunately the slide rule went out of print before most of you were born, and mint-condition rules go for about \$500 on eBay.

However, I just happened to have one. No, you can't have it. Not the physical one at any rate.

Decades ago I read about it in Jerry Pournelle's A STEP FARTHER OUT, and managed to purchase one from RAND for the princely sum of US\$15. Before most of you were born.

Anyway, because I have an unnatural fondness for antique calculating instruments, I decided to scan it for you. Now I did not actually dismantle my slide rule because the thing is too valuable. Instead I made lots of scans with the top face in various positions to infer what the bottom looked like. The composite was a mess, so I manually traced over it to make a clean version.

Here are the files you'll need:

The PDF instruction manual explains how to use the blasted thing.

The Top, Middle, and two Bottom Levels should be printed at 400 dpi.

The top level has to be transparent, so you'll need to use a printer that can print on transparent acetate. When finished printing, cut around the border.

The middle level can be printed on card-stock, or on paper then glued to cardboard for more durability. Cut around the border, and cut out the two trapezoids labeled "CUT OUT".

The bottom level Obverse is the required side. The flip side, the Reverse, just has operating instructions, it is optional. Print both of them out and glue to the opposite sides of card stock or cardboard.

Now you have to assemble them so they can turn. Traditionally this is done by using a brass fastener but these do not allow smooth turning. What you need to do is to find a craft store that caters to scrapbook crafters and get some eyelets, the appropriate eyelet punch and an eyelet anvil. You want a punch that is tool used with a hammer, not the kind integral to a something like pliers. The slide rule is about two inches in radius, pliers are not deep enough.

## Atomic Rockets notices

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