## 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.

artwork by Ed Emshwillerdetail(ed note: In the future people use handwavium paragravity to terraform asteroids. They also use them as torchships. Some royal morons want to move an inhabited asteroid from one cluster to another, an action that will spark a localized war. Our heroes Captain Dhan Gopal Radhakrishnan and Engineer Knud Axel Syrup arrive in the Mercury Girl, and are captured by the royal morons. All radios have been confiscated, so no warning can be broadcast. But our heroes figure out how to make a makeshift rocket out of local materials in order to travel to another asteroid to spread the alarm. They are forced to use only locally available materials.)

The firstbeer-powered spaceshipin history rested beneath a derrick by the main cargo hatch.

It was not as impressive as Herr Syrup could have wished. Using a small traveling lift for the heavy work,he had joined four ten-ton casks of Nashornbräu end to end with a light framework. The taps had been removed from the kegs and their bungholes plugged, simple electrically-controlled Venturi valves in the plumb center being substituted. Jutting an orthogonal axes from each barrel there were also L-shaped exhaust pipes, by which it was hoped to control rotation and sideways motion. Various wires and shafts, their points of entry sealed with gunk, plunged into the barrels, ending in electric beaters (to agitate the beer. Much like shaking up a bottle of beer before opening the lid). A set of relays was intended to release each container as it was exhausted. The power for all this— it did not amount to much—came from a system of heavy-duty EXW batteries at the front end.

Ahead of those batteries was fastened a box, some two meters square and three meters long. Sheets of plastic were set in its black-painted sides by way of windows. The torso and helmet of a spacesuit jutted from the roof, removably fastened in a screw-threaded hatch cover which could be turned around. Beside it was a small stovepipe valve holding two self-closing elastic diaphragms through which tools could be pushed without undue air loss. The box had been put together out of cardboard beer cases, bolted to a light metal frame and carefully sized and gunked.

"You see,’’ Herr Syrup had explained grandly, "in dis situation, vat do ve need to go to New Vinshester?Not an atomic motor, for sure, because dere is almost neglishible gravity to overcome. Not a nice streamlined shape, because ve have no air hereabouts. Not great structural strengt’, for dere is no strain odder dan a very easy acceleration; so beer cardboard is strong enough for two, free men to sit on a box of it under Eart’ gravity. Not a fancy t’ermostatic system for so short a hop, for de sun is far avay, our own bodies make heat and losing dat heat by radiation is a slow process. If it does get too hot inside, ve can let a little vater evaporate into space though de stovepipe valve to cool us; if ve get shilly, ve can tap a little heat though a coil off de batteries.

"All ve need is air. Not even mush air, since I is sitting most of de time and you ban a Martian. A pair of oxygen cylinders should make more dan enough;ja,and ve vill need a chemical, carbon-dioxide absorber, and some desiccating stuffs so you do not get a vater vapor drunk. For comfort ve vill take along a few bottles beer and some pretzels to nibble on.

"As for de minimal boat itself, I have tested de exhaust velocity of hot, agitated beer against vacuum, and it is enough to accelerate us to a few hundred kilometers per hour, maybe t'ree hundred, if ve use a high enough mass ratio. And ve vill need a few simple navigating instruments, an ephemeris, slide rule, and so on. As a precaution, I install my bicycle in de cabin, hooked to a simple homemade g’enerator, yust a little electric motor yuggled around to be run in reverse, vit’ a rectifier. Dat vay, if de batteries get too veak ve can resharshe dem. And also a small, primitive oscillator ve can make, short range, ja, but able to run a gamut of freqvencies vit’ out exhausting de batteries, so ve can send an SOS ven ve ban qvite close to New Vinshester. Dey hear it and send a spaceship out to pick us up, and dat is dat.”

The execution of this theory had been somewhat more difficult, but Herr Syrup’s ears aboard theMercury Girlhad made him a highly skilled improviser and jackleg inventor. Now, tired, greasy, and content, he smoked a well-earned pipe as he stood admiring his creation.Partly, he waited for the electric coils which surrounded the boat and tapped the ship’s power lines, to heat the beer sufficiently;but that was very nearly complete, to the point of unsafeness. And partly he waited for the ship to reach that orbital point which would give his boat full tangential velocity toward the goal; that would be in a couple of hours.

Er … are you sure we had better not test it first?” asked Sarmishkidu uneasily.

"No, I t’ink not,” said Herr Syrup. "First, it vould take too long to fix up an extra barrel. Ve been up here a veek or more vit’out a vord to Grendel. If O’Toole gets suspicious and looks t’ rough a telescope and sees us scooting around, right avay he sends up a lifeboat full of soldiers; vich is a second reason for not making a test flight.”

"But, well, that is, suppose something goes wrong?”

"Den de spacesuit keeps me alive for several hours and you can stand vacuum about de same lengt’ of time. Emily vill be vatching us t’rough de ship’s telescope, so she can let McConnell out and he can come rescue us.”

"And what if he can’t find us? Or if we have an accident out of telescopic range from here? Space is a large volume.”

“I prefer you vould not mention dat possibility,” said Herr Syrup with a touch of hauteur.

artwork by H. R. Van Dongendetail

**A BICYCLE BUILT FOR BREW**by Poul Anderson

*(1958)*

*Using an oxygen tank as an improvised rocket to rescue a crewman*

From Destination Moon

The intelligent lady uses her brains and utilizes a soda siphon as a soda water rocket

From BEYOND MARS (1952)

## Rules of Thumb

*The Star Spear from***Tom Swift and his Rocket Ship**by Victor Appleton II, 1954

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.

5 milligee(0.05 m/s^{2}) : General rule practical minimum for ion drive, laser sail or other low thrust / long duration drive. Otherwise the poor spacecraft will takeyearsto change orbits. Unfortunately pure solar sails are lucky to do 3 milligees.

0.6 gee(5.88 m/s^{2}) : General rule average for high thrust / short duration drive. Useful for Hohmann transfer orbits, or crossing the Van Allen radiation belts before they fry the astronauts.

3.0 gee(29.43 m/s^{2}) : General rule minimum to lift off from Terra's surface into LEO.

Ken Burnside:Ignoring the ground to orbit issue for the moment, I see list consensus has found three thrust regimes.

EKLUNDIAN— thrusts are greater than solar gravitation, but not by much. Isps (specific impulse) are low enough that conserving delta v is the paramount concern. Travel time is known more or less in advance, and everyone has launch windows to observe.

HEINLEINIAN— thrusts are so significantly greater than local gravitation that orbital mechanics is meaningless. Isps are high enough in concert with these thrusts that Heinlein style "burn-flip-burn" moves are the norm; travel time reduction becomes the paramount concern.

THE FUZZY MIDDLE— thrusts are higher than local gravitation, but usually within an order of magnitude of it. When local gravitation as a function of range exceeds some percentage of thrust, it turns things into an Eklundian model. Isps are low enough that total delta v doesn't permit Heinlein-style brachistochrone orbits.

Now the questions:

1)Have I categorized this properly? Or is there a category I'm missing? Does category three need a better name, or further subdivision?

2)At what percentage of thrust does local gravitation force Eklundian style "slow spiral orbits"?

3)At what range of ISps do we get to "It's better to just burn more gas to save time" assuming point 2 is met?

GIGAWATT THRUSTER

Mini-Mag Orion Z-Pinch Fission

from boardgame High FrontierTERAWATT THRUSTER

Colliding FRC^{3}He-D Fusion

from boardgame High Frontier

RICK ROBINSON'S RESPONSE:Ken Burnside:

Ignoring the ground to orbit issue for the moment, I see list consensus has found three thrust regimes.

EKLUNDIAN— thrusts are greater than solar gravitation, but not by much. Isps are low enough that conserving delta v is the paramount concern. Travel time is known more or less in advance, and everyone has launch windows to observe.

Thrust hardly really matters in the limited Isp regime, so long as it is an appreciable fraction of a milligee — it can probably even be less than solar gravitation, so long as it isn't too much less.

Thrust above about 0.1 g allows more efficient planetary departures, saving a few km/s, but this is nearly irrelevant to interplanetary transfer orbits, so the Eklundian conditions apply.

Ken Burnside:

HEINLEINIAN— thrusts are so significantly greater than local gravitation that orbital mechanics is meaningless. ISps are high enough in concert with these thrusts that Heinlein style "burn-flip-burn" moves are the norm; travel time reduction becomes the paramount concern.I don't like the term Heinleinian, because Heinlein also (and more often) described Eklundian or fuzzy-middle travel. Call it torchship, or torchlike.

Note that acceleration of about 5-10 milligees is enough for torchlike performance,

if you have the delta v.You'll still have to spiral out from planets, but once clear of them, 10 milligees gives you 8.5 km/s per day, solar escape speed in a week (with displacement in the frame of reference less than 0.1 AU).Ken Burnside:

THE FUZZY MIDDLE— thrusts are higher than local gravitation, but usually within an order of magnitude of it. When local gravitation as a function of range exceeds some percentage of thrust, it turns things into an Eklundian model. ISps are low enough that total delta v doesn't permit Heinlein-style brachistochrone orbits.

Yes.

Note also a relationship between the inner and outer Solar System (roughly, inside and outside Jupiter). Ships that are Eklundian in the inner system can barely reach the outer system at all. Ships that are fuzzy-middle in the inner system behave nearly like torchships in the outer system — they have to coast most of the way, but nearly in a straight line.

Ken Burnside:

Now the questions:

1)Have I categorized this properly? Or is there a category I'm missing? Does category three need a better name, or further subdivision?

No, these sound about right. I'd call the fuzzy middle "transitional."

My impressionistic description. The Solar System is a vast, slowly revolving whirlpool. Eklund ships are galleys caught in it; by hard rowing they can shift themselves inward or outward to visit the whirlpool's floating, revolving islands.

Torchships are hydrofoils that zip from island to island, so fast they can effectively ignore the motion of the whirlpool, except for the movement of their destination.

In the transition are steamboats, which are able to cut steeply across the vortex, but cannot ignore it.

Ken Burnside:

2)At what percentage of thrust does local gravitation force Eklundian style "slow spiral orbits"?

Less than perhaps 5-10 milligees.

In terms of whether you have to spiral out from individual planets, or can make the more efficient slingshot burn from low orbit, I would say that the threshold is about 0.1 g.

Ken Burnside:

3) At what range of Isps do we get to "It's better to just burn more gas to save time" assuming point 2 is met?

I am going to swag this as a ship delta v of about 50-100 km/s — assuming roughly a 65 percent fuel fraction, your exhaust velocity is the same, so Isp about 5000-10,000 seconds. For fast commercial travel you probably want a lower fuel fraction, so you need a drive with upwards of 10,000 seconds of Isp.

Ken Burnside:

Ignore specifics of rockets - we've seen enough arguments over people's favorite propulsion systems for the last 5 weeks. :) I'm just looking to establish the categories for now, so we can use them to organize discussions in the future.

Keeping it general, chemfuel is strictly Eklundian in spite of its high thrust.

Nuclear-thermal (even Orion) is largely Eklundian, struggling to get ship delta v above 20 km/s or so. The advantage over chemfuel is that it does Hohmann and near-Hohmann orbits with a considerably lower fuel fraction, allowing perhaps 2x to 5x the payload.

Nuclear-electric drives like VASIMR live in the transition zone — they have supra-Eklund delta v, but sluggish acceleration around 1 milligee. In practice, commercial ships at least may still live in Eklundian space, using the higher specific impulse to further reduce fuel fraction, allowing more cargo, rather than for higher speed.

The threshold of torchlike performance is roughly Isp of 10,000 seconds combined with thrust around 5 milligees. For a 1000-ton ship this requires 2.5 gigawatts of thrust power. If the drive engine itself is 250 tons, that requires a drive power density of 10 kw/kg, comparable to a jet engine.

So, very roughly 1 gigawatt (GW, 10

^{9}W) thrust power for a moderate size ship marks the minimum torchship threshold. But "classical" torchships are about 1000x more powerful, approaching the terawatt (TW, 10^{12}W) range, allowing acceleration near 1 g or exhaust velocity near 1000 km/s.

**THRUST REGIMES AND ISP FOR INTERPLANETARY TRAVEL**

by Rick Robinson (2007)

## Mass Ratio

Since almost all rockets are giant propellant tanks with an engine on the bottom and the pilot's chair at the top, most of the rocket is propellant. A titanic metal foil balloon with tiny rocket bits stuck on with vacuum tape.

"Mass Ratio" is just a fancy way to measure how much mass is the propellant and how much is the rest of the blasted rocket.

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 propellant out the exhaust pipe. As I told you before they ain't the same.

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. 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)*.

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 / M _{e}**

or

**R = (M _{pt} / M_{e}) + 1**

where:

- R = mass ratio
*(dimensionless number)* - M = mass of rocket with full propellant tanks, the Wet Mass
*(kg)* - M
_{pt}= mass of propellant, the Propellant Mass*(kg)* - M
_{e}= mass of rocket with empty propellant tanks, the Dry Mass.**M**_{e}=M-M_{pt}*(kg)*

If the Star Spear carries 70 metric tons of propellant, and the rocket masses 40 metric tons with dry tanks, its mass ratio is (70 / 40) + 1 =

2.75. This means that for every ton of rocket and payload there is 2.75 tons of propellant. Alternatively, if the Star Spear masses 110 metric tons full of propellant and 40 metric tons empty, the mass ratio is still 110 / 40 =2.75. Note that mass ratios are generally always much higher than 1.0.

The equation you will actually use *(later)* is:

**P _{f} = 1 - (1/R)**

**R = 1 / (1-P _{f})**

where

- P
_{f}= propellant fraction, that is, percent of total rocket mass**M**that is propellant: 1.0 = 100% , 0.25 = 25%, etc.

The Star Spear's propellant fraction is 1 - (1 / 2.75) =

0.63 or 63%

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 = e ^{(Δv/Ve)}**

where

- e
^{x}= antilog base*e*or inverse of natural logarithm of x, the*"e*key on your calculator^{x}"

This section is intended to address some gaps in available information about spacecraft design in the Plausible Mid-Future (PMF), with an eye towards space warfare. It is not a summary of such information, most of which can be found at Atomic Rockets.

The largest gap in current practice comes in the preliminary design phase. A normal method used is to specify the fully-loaded mass of a vessel, and then work out the amounts required for remass (propellant), tanks, engine, and so on, and then figure out the payload

(habitat, weapons, sensors, cargo, and so on)from there.While there are times this is appropriate engineering practice

(notably if you’re launching the spacecraft from Earth and have a fixed launch mass),in the majority of casesthe payload mass should be the starting point. The following equation can be used for such calculations:

M = R * ( M_{pl}/ (1 - (P_{f}* (R-1)) - (Pi * R)) )

- M = mass of rocket with full propellant tanks, the Wet Mass
(kg)- R = mass ratio
(dimensionless number)- M
_{pl}= Payload Mass(kg)- P
_{f}= propellant fraction, that is, percent of total rocket massMthat is propellant: 1.0 = 100% , 0.25 = 25%, etc.- P
_{i}= inert fraction, that is, percent of total rocket massMthat is Inert Mass: 1.0 = 100% , 0.25 = 25%, etc.(ed note: P

_{f}is actually any mass that "scales" with the propellant mass, such as the mass of the tank. "Scale" means if the propellant mass is increased, the tank mass will also increase since you need more tankage to hold more propellant.P

_{i}is actually any mass that "scales" with the size of the spacecraft, such as such as engines or structure.M

_{pl}is actually any mass that is of fixed mass (does not scale) regardless of size of spacecraft, such as habitats, weapons, or sensors.)

## Exhaust Velocity

To find the engine's exhaust velocity, look it up in the table. Now you can skip the rest of this section.

If you ony have the engine's specific impulse, mulitiply it by 9.81 to get exhaust velocity. You

cando multiplication, can't you?Rocket scientists like to use specific impulse instead of exhaust velocity because then they can use any other units they want for the rest of the equations. I know you are not a rocket scientist or you would have hurt yourself laughing by now reading this site. Therefore I'm giving you all the equations with fixed units, because otherwise it is just one more thing to that will cause math mistakes.

*The Aldebaran designed by Dandridge Cole, 1960. The best place to watch lift-off is from an adjacent continent.*

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

**V _{e} = I_{sp} * 9.81**

where

- I
_{sp}= specific impulse*(seconds)* - V
_{e}= exhaust velocity*(m/s)* - 9.81 = acceleration due to gravity
*(m/s*^{2})

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.

MaturinTheTurtleWhen a physicist tells you something is "specific" he means that quantity is per something else. Specific impulse is

impulseper unitweightof propellant.Impulse, in a rocketry context, is thrust applied over time. One newton of thrust (metric system) applied for one second results in one newton-second of impulse.

The importance of impulse in rocketry should be pretty obvious: thrust alone is not a meaningful quantity if you're talking about get-up-and-go. Thrust tells you how hard an engine can push, but it's not until that engine pushes

for some timethat you get anywhere.But how long an engine can push for depends on how much propellant you have. If you have infinite propellant then you can keep

anyengine going for infinite time; that's obvious. But if you have afiniteamount of propellant, then how long can you make an engine go? Well,that depends on the engine.Which is where specific impulse comes in. Specific impulse is how much impulse — thrust over time — you get out of a given weight of propellant. If you have a thousand pounds of propellant and that results in your engine giving you a kilonewton of thrust for three seconds, then your engine has a specific impulse of 3000 newton-seconds per thousand pounds, or 3 newton-seconds per pound.

Except these days people tend to measure

everythingin the metric system, which results in a bit of confusion. See, both thrust and weight, in the metric system, are measured in newtons or multiples thereof. So you end up quantifying specific impulse in units of newton-seconds per newton, and then people cancel out the newtons … even though they really shouldn't, because they'redifferent kinds of newtons.Specific impulse really has units of newton (thrust)-seconds per newton (weight), but it's become traditional to just drop the newtons and call it seconds instead.

DrScrubbingtonTL;DR, specific impulse is how long an engine can hover for, while carrying its own fuel and neglecting the mass of the engine.

MaturinTheTurtlePeople keep saying that, but it's not right. Thrust is constant under given conditions but weight falls continuously, so your "the engine is hovering" thing is only true for a single instant. After that, it's accelerating steadily upward at an increasing rate (the third derivative of altitude is positive).

If you want to explain it succinctly to somebody, say that specific impulse is the amount of time it takes for a given engine to burn a weight of propellant equal to its thrust. Then tell them what it really means — thrust time per unit weight of propellant — once they point out to you that that succinct explanation is useless.

Dimensional analysis: a newton is a unit of weight or force (same thing, different points of view). Weight and force are both mass accelerated, so a newton is mass times acceleration. Integrate that over time and you have mass times acceleration times time … but acceleration is length per unit time per unit time. So that become mass-length-per-time, which is how you quantify impulse. (You will recognize these as the units of momentum; impulse and momentum are two sides of the same coin. Momentum is mass moving with a certain velocity, and impulse is thrust applied for a given time. Tomayto, tomahto.)

But if you then divide that out by mass, you end up with mass-length-per-time-per-mass, and the masses cancel leaving you just with length-per-time. That looks like a velocity, which turns out to be a very inconvenient way to quantify the specific impulse of a motor.

If you multiply the specific impulse of a motor times the conversion factor between units of weight and units of mass (which in the metric system is 9.80665 m/s/s exactly by definition; it is NOT local g) you get a quantity called the "effective exhaust velocity" which shows up in a few equations, but in practice nobody uses that quantity. Everybody just writes "Isp g

_{0}" instead.

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**

**V _{e} = sqrt( ((2 * k) / (k - 1)) * ((R' * T_{c}) / M) * ( 1 - (Pe/Pc)^((k-1)/k) ) )**

where

**V**= ideal exhaust velocity_{e}*(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)***T**= Combustion chamber temperature_{c}*(Kelvin)***P**= Combustion chamber pressure_{c}*(standard for comparison is 68 atm)***P**= Pressure at nozzle exit_{e}*(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.

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 (H_{2}) 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.

(ed note: engineer Rob Davidoff gently points out that I don't know what I am talking about. It

ispossible to use low molecular weight propellant, at least a little bit)It's worth noting you can actually "spice up" the propellants with low mass exhaust products. This is how a lot of tripropellant chemical works, by adding hydrogen as much to have lower average molecular mass in the exhaust as to actually burn it for its energy release.

It's also among two or three reasons why you see a lot of hydrolox engines run fuel-rich, so there's unburnt excess H

_{2}in the exhaust.

*(2018)*

**EXHAUST VELOCITY OF FUSION ROCKETS**

Particle | Mass (unified atomic mass units) |
---|---|

n (Neutron) | 1.008665 |

p (Proton) | 1.007276 |

D (Deuteron) | 2.013553 |

T (Tritium) | 3.015500 |

^{3}He (Helium-3) | 3.014932 |

^{4}He (Helium-4) | 4.001506 |

^{11}B (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 = mc ^{2}**? 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:

**V _{el} = sqrt(2 * E_{p})**

where

- E
_{p}= fraction of fuel that is transformed into energy - V
_{el}= exhaust velocity*(percentage of the speed of light)*

Multiply V_{el} 299,792,458 to convert it into meters per second.

To see more about this check out the page about Fusion Fuels.

### 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.

Engine | Nominal Temperature | Propellant | Exhaust Velocity | Specific Impulse |
---|---|---|---|---|

Solid Core | 2,750K | Molecular Hydrogen | 8,300 m/s | 850 s |

Liquid Core | 5,250K | Atomic Hydrogen | 16,200 m/s | 1,650 s |

Gas Core | 21,000K | Atomic Hydrogen | 32,400 m/s | 3,300 s |

### Deuterium-tritium fusion rocket

Deuterium-Tritium Fusion rockets use the fusion reaction **D + T ⇒ ^{4}He + 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: E

_{p}= 0.018882 / 5.029053 =

**0.00375**.

Plugging that into our equation V_{e} = sqrt(2 * 0.00375) = 0.0866 = **8.7% c**.

### Deuterium-helium 3 fusion rocket

Deuterium-Helium^{3} Fusion rockets use the fusion reaction **D + ^{3}He ⇒ ^{4}He + 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. E

_{p}= 0.019703 / 5.028485 =

**0.00392**.

Plugging that into our equation V_{e} = sqrt(2 * 0.00392) = 0.0885 = **8.9% c**.

The **D + ^{3}He** 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 rocket*

Deuterium-deuterium Fusion rockets use the fusion reaction **D + D ⇒ T + p** or ** ^{3}He + 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. E
_{p}= 0.00108 - a Helium-3 and a neutron: 3.014932 + 1.008665 = 4.023597. 0.003509 converted into energy. E
_{p}= 0.000871

Plugging that into our equation

- V
_{e}= sqrt(2 * 0.00108) = 0.0465 =**4.7% c** - V
_{e}= sqrt(2 * 0.000871) = 0.0418 =**4.2% c**

### Hydrogen-boron thermonuclear fission rocket

Hydrogen - Boron Thermonuclear Fission rockets use the reaction **p + ^{11}B ⇒ 3 × ^{4}He**. 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. E

_{p}= 0.012068 / 12.016586 =

**0.001**.

Plugging that into our equation V_{e} = sqrt(2 * 0.001) = 0.045 = **4.5% c**.

### Watch the Heat

*Chart from***"The Atomic Rocket"**by L. R. Shepherd, Ph.D., B.Sc., A.Inst.P., & A. V. Cleaver, F.R.Ae.S., 1948. Collected in Realities of Space Travel*Magnetic Nozzle*

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

**F _{p} = (F * V_{e} ) / 2**

where

- F
_{p}= thrust power*(watts)* - F = thrust
*(newtons)* - V
_{e}= 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:

**A _{e} = (0.5 * A_{m} * A_{v}^{2}) / B**

where

- A
_{e}= particle energy*(Kelvin)* - A
_{m}= mass of particle*(g)*(**1.6733e-24**grams for monatomic hydrogen) - A
_{v}= exhaust velocity*(cm/s)* - B = Boltzmann's constant:
**1.38e-16***(erg K*^{-1}) - x
^{2}: 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:

**Q _{e} = (V_{e} / (Z * 129))^{2} * P_{w}**

where

- Q
_{e}= engine reaction chamber temperature*(Kelvin)* - V
_{e}= exhaust velocity*(m/s)* - Z = heat-pressure factor, varies by engine design, roughly from 1.4 to 2.4 or so.
- P
_{w}= 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 Langson
Field is used in the ship's drive, and as a force screen defense. The Langson 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/m^{2}. 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:

**A _{f} = sqrt[(1/E_{l}) * (1 / (4 * π))]**

**R _{c} = sqrt[H] * A_{f}**

where

A_{f}= Attunation factor. Anthony Jackson says 0.126, Luke Campbell says 0.133

E_{l}= Maximum heat load(MW/m. Anthony Jackson says 5.0, Luke Campbell says 4.5^{2})

π = pi = 3.141592...

H = reaction chamber waste heat(megawatts)

R_{c}= 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)*

Say your propulsion system has an exhaust velocity of 5.4e6 m/s and a thrust of 2.5e6 N. Now

Fso the thrust power is 6.7e12 W. So, 6.7e12 watts divided by 1.0e6 watts per megawatt gives us 6.7e6 megawatts._{p}=(F*V_{e})/2Assuming Anthony Jackson's more liberal 5.0 MW/m

Plugging this into the equation results in sqrt[6.7e6 MW] * 0.126 = drive chamber radius of 326 meters or a diameter of almost half a mile. Ouch.^{2}, this means A_{f}= 0.126

Here is how the above equation was derived. If you could care less, skip over this box.

It is based on the good old Inverse-square law.

The reaction chamber is assumed to be spherical. Obviously the larger the radius of the chamber, the more surface area it has, and the given amount of waste heat has to be spread thinner in order to cover the entire area. If you only have one pat of butter, the more slices of toast means the lesser amount of butter each slice gets.

E

_{l}is the Maximum heat load, or how many megawatts per square meter the engine can take before the blasted thing starts melting. Anthony Jackson says 5.0 MW/m^{2}.The idea is to expand the radius of the reaction chamber such that the inverse-square law attenuates the waste heat to the point where it is below the maximum heat load. Then we are golden.

The attenuation due to the inverse square law is:

ISLA = (4 * π * R_{c}^{2})where:

ISLA = attenuation due to the inverse square law

π = pi = 3.141592...

R_{c}= reaction chamber radius(meters)The heat load on the reaction chamber walls is:

C_{l}= H / ISLAwhere:

H = waste heat(megawatts)

C_{l}= heat load on chamber wall(MW/m^{2})Merging the equations:

C_{l}= H / (4 * π * R_{c}^{2})Solve for R

_{c}:C_{l}= H / (4 * π * R_{c}^{2})

4 * π * R_{c}^{2}= H / C_{l}

4 * π * R_{c}^{2}= H * (1/C_{l})

R_{c}^{2}= (H * (1/C_{l})) / (4 * π)

R_{c}^{2}= H * (1/C_{l}) * (1 / (4 * π))

R_{c}= sqrt[H] * sqrt[(1/C_{l}) * (1 / (4 * π))]

Looking at the last equation, take the right half and swap C

_{l}for E_{l}to get:

A_{f}= sqrt[(1/E_{l}) * (1 / (4 * π))]and the entire equation is where we get:

R_{c}= sqrt[H] * A_{f}which is what we were trying to derive. QED.

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.

Chart from"To The Stars"by Gordon Woodcock, (1983). Collected inIslands In The Sky, edited by Stanley Schmidt and Robert Zubrin (1996). Most of the engines on this chart are torchships.Calculating the performance of a spaceship can be complicated. But if the ship is powerful enough, we can ignore gravity fields. It is then fairly easy. The ship will accelerate to a maximum speed and then turn around and slow down at its destination. Fusion or annihilation-drive ships will probably do this. They will apply power all the time, speeding up and slowing down.(ed note: a "brachistochrone" trajectory)

In this simple case, all the important performance parameters can be expressed on a single graph. This one is drawn for the case when 90% of the starting mass is propellant. (ed note: a mass ratio of 10) Jet velocity (exhaust velocity) and starting acceleration are the graph scales. Distance for several bodies are shown. Mars varies greatly; I used 150 million kilometers. Trip times and specific power levels are also shown. "Specific power" expresses how much power the ship generates for each kilogram of its mass, that is, its total power divided by its mass. The propellant the ship will carry is not included in the mass value.

An example: Suppose your ship can produce 100 kW/kg of jet power. You wish to fly to Jupiter. Where the 100 kW/kg and Jupiter lines cross on the graph, read a jet velocity of 300,000 m/s (Isp = 30,000) and an initial acceleration of nearly 0.01g. Your trip will take about two months.

The upper area of the graph shows that high performance is needed to reach the nearest stars. Even generation ships will need, in addition to very high jet velocities, power on the order of 100 kW/kg. The space shuttle orbiter produces about 100 kW/kg with its three engines. The high power needed for starflight precludes its attainment with means such as electric propulsion.

## Delta-V

Konstantin Tsiolkovsky is

The Manand don't you forget it! Every single time you design a rocket, you will be using his brilliant rocket equation. It is thesine qua nonof rocketry, without it this entire freaking website would not exist. If you are a serious rocket geek, you should have Tsiolkovsky's portrait hanging on your wall and the rocket equation on your T-shirt.I love the smell of delta-V in the morning. Smelled like ...

trajectory.

*Konstantin Tsiolkovsky, the father of modern rocketry. "The Earth is the cradle of humanity, but mankind cannot stay in the cradle forever."*

Finally it is time to calculate the spacecraft's total DeltaV. For this, 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.

*View from the International Space Station. Notice whose picture they have on the wall. The other photo is Yuri Gagarin, first man in space.*

Anyway, the equation is:

**Δ _{v} = V_{e} * ln[ M / M_{e} ]**

**Δ _{v} = V_{e} * ln[R]**

where

**Δ**= ship's total deltaV capability_{v}*(m/s)***V**= exhaust velocity of propulsion system_{e}*(m/s)***M**= mass of rocket with full propellant tanks*(kg)***M**= mass of rocket with empty propellant tanks_{e}*(kg)***R**= ship's mass ratio**ln[**x**]**= natural logarithm of x, the*"ln"*key on your calculator

Suppose that the

has a 1Polaris^{st}generation Gaseous Core Fission drive. Exhaust velocity of 35,000 m/s(see table in engine list).Let's try a mass ratio of 2

(50% propellant). 35,000 * ln[2] =24,260 m/s. Not good enough, we need 39,528 m/s.Let's try a mass ratio of 3.1

(68% propellant). 35,000 * ln[3.1] =39,600 m/s. That'll do.

The inverse of the deltaV equation sometimes comes in handy.

**R = e ^{(Δv/Ve)}**

where

- e
^{x}= antilog base*e*or inverse of natural logarithm of x, the*"e*key on your calculator^{x}"

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 e^{2}. If the deltaV is three times the exhaust velocity, the mass ratio has to be 20 or e^{3}.

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 **Δ _{v} / V_{e} = 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]).

*High mass ratios mean ridiculously high propellant loads.*

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 V _{e} * 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 V _{e} * 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: V_{e} = Δ_{v} * 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: V_{e} = Δ_{v} * 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.

On ProjectRho, there are a few nebulous statements about "optimal" exhaust velocities existing for a fixed, given mission Δv. In-particular, regarding the optimization, there is this:

"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 [the] payload as a function of exhaust velocity. You have to iteratively solve the equation 0.5 = x * (1 - exp(-1/x))."This confused me. For one thing, if we take "optimal" in the usual sense taken by armchair rocket scientists—that is, lowest propellant mass—we see that the statement is completely bogus. Recall the Tsiolkovsky rocket equation:

Δv = Vₑ ln( m₀ / m₁ )Given a mission Δv, increasing Vₑ will make m₀ closer to m₁ (that is, reduce the propellant expended). There is, in-fact, no optimal value! Vₑ should be increased as far as you possibly can given the limits of your technology (in-practice, the speed of light; the rocket equation is a Newtonian approximation).

But there is another sense of "optimal": the lowest total energy used to accelerate all propellant for the mission, and this time it turns out there is an optimal value. The way to think of this is that, while accelerating your propellant to a ludicrous speed allows you to use as little of it as you like, you'll use more energy to do so (kinetic energy is, to first order, quadratic in speed). Conversely, accelerating your propellant to a low speed requires little energy, but the loss of specific impulse drives the mass ratio up higher, requiring a bigger rocket, more propellant, and more energy in the end. The optimal (lowest-energy) solution is somewhere in the middle.

Finding this optimum value is somewhat tricky. Here's my derivation. Note that it breaks down (in several ways) for relativistic speeds.

First, you rearrange the rocket equation for initial mass m₀:

m₀ = m₁ exp( Δv / Vₑ )Then, you figure out how much energy you spend accelerating the exhaust. This is just the total mass of the propellant and the exit velocity subbed into the standard kinetic energy equation (it's this simple because the energy required is expended in an instantaneously co-moving reference frame):

E = ½ (m₀ - m₁) Vₑ²Then you substitute the first into the second to get a formula for the energy required to accelerate the payload mass m₁ as a function of (fixed) Δv and (variable) Vₑ:

E(Vₑ) = ½ m₁ ( exp(Δv/Vₑ) - 1 ) Vₑ²If you plot this function for Vₑ>0 (choose some sensible value for Δv), you'll see a curve that swoops down from infinity, then back up to infinity more-slowly. The lowest point of this chart is the lowest energy we can expend for that Δv. To find that minimum mathematically, we apply a basic calculus trick, first differentiating:

d E(Vₑ) / d x = m₁ Vₑ ( exp(Δv/Vₑ) - 1 ) - ½ Δv m₁ exp(Δv/Vₑ)Then setting to zero and solving for Vₑ:

We're kindof stuck here because of the form the equation takes. Note that this is very nearly the equation that Landis presents (his formula is probably intended to½ Δv m₁ exp(Δv/Vₑ) = m₁ Vₑ ( exp(Δv/Vₑ) - 1 )

½ Δv exp(Δv/Vₑ) = Vₑ exp(Δv/Vₑ) - Vₑ

½ Δv = Vₑ - Vₑ / exp(Δv/Vₑ)

½ Δv = Vₑ (1 - exp(-Δv/Vₑ))express the form of the equation, rather than the exact instance thereof). Landis suggests solving it iteratively, but using the (admittedly less-common) productlog function W, we get:

Vₑ = Δv / (2 + W(-2/e²))

≈ 0.627500 ΔvSo, in the idealized, Newtonian case, the least energy is used when the exit velocity is about 63% the total mission Δv.

Getting back to the question of "optimal", how useful is this?

The energy expended by chemical rockets to accelerate their propellant comes from the reaction of the fuels that form the propellant in the first place. It may not even be possible to produce a Vₑ as high as desired, given the comparatively poor ISPs of chemical fuels. Therefore, energy doesn't really enter into the calculation except as the fuel choice (which was probably already pre-specified, particularly if the rocket is already built, and is often limited by available technology anyway). Mass is what matters for a chemical rocket, because lower mass means less fuel is required, and given the high mass ratios, this is a major expense.

For interstellar drives, again the formula is not super-useful. The travel time is ridiculous, and a major design constraint is reducing it, even at a hefty cost in mass and especially energy. Of the two, mass tends to be the limiting factor, with mass ratios becoming enormous to satiate long burns at high ISPs and/or high thrust. Power, meanwhile, can be stored efficiently in nuclear materials, or beamed remotely, as in some concepts.

Probably the most-useful application is near-future variable-ISP drives, such as ion or MPD engines. Due to sociopolitical issues, nuclear power in space is for now a hard sell, severely constraining power budgets. At the same time, the ISP is adjustable while still high enough that, regardless, the mass ratio ends up reasonable.

*(2018)*

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.

To those rocket engineers inured to the inevitable rise in vehicle mass ratio with increasing mission difficulty, antimatter rockets provide relief. The mass ratio of an antimatter rocket for any mission is always less than 4.9:1 [Shepherd, 1952], and cost-optimized mass ratios are as low as 2:1 [Forward, 1985]. In an antimatter rocket, the source of the propulsion energy is separate from the reaction fluid. Thus, the rocket's total initial mass consists of the vehicle's empty mass, the reaction fluid's mass, and the energy source's mass, half of which is the mass of the antimatter. According to the standard rocket equation, the mass ratio is now (assuming

m»_{r}m)_{e}

equation 11.13where

Δ= change in vehicle velocity (m/s)_{v}

v= rocket exhaust velocity (m/s)_{e}

m= initial mass of the vehicle (kg)_{i}

m= final mass of the vehicle (kg)_{f}

m= empty mass of the vehicle (kg)_{v}

m= mass of the reaction fluid (kg)_{r}

m= mass of the energy source (kg)_{e}The kinetic energy (

K.E.) in the expellant at exhaust velocity (v) comes from converting the fuel's rest-mass energy into thrust with an energy efficiency (_{e}η):_{e}

equation 11.14where

K.E. = kinetic energy (kg·m^{2}/s^{2})

c = speed of light (3 × 10^{8}m/s)Solving Eq. (11.14) for the reaction mass (

m), substituting into Eq. (11.13), and solving for the energy source's mass (_{r}m) produces_{e}

equation 11.15We can find the minimum antimatter required to do a mission with a given Δ

. We set the derivative of Eq. (11.15) with respect to the exhaust velocity_{v}vequal to zero, and solving (numerically) for the exhaust velocity:_{e}

equation 11.16Substituting Eq. (11.16) into Eq. (11.13), we find that, because the optimal exhaust velocity is proportional to the mission Δ

, the vehicle mass ratio is a constant:_{v}

equation 11.17The reaction mass (

m) is 3.9 times the vehicle mass (_{r}m), while the antimatter fuel mass is negligible. Amazingly enough, this constant mass ratio is independent of the efficiency (_{v}η) with which the antimatter energy is converted into kinetic energy of the exhaust. (If the antimatter engine has low efficiency, we will need more antimatter to heat the reaction mass to the best exhaust velocity. The amount of reaction mass needed remains constant.) If we can develop antimatter engines that can handle jets with the very high exhaust velocities Eq. (11.16) implies, this constant mass ratio holds for all conceivable missions in the solar system. It starts to deviate significantly only for interstellar missions in which the mission Δ_{e}approaches the speed of light [Cassenti, 1984]._{v}

(ed note: Translation: to compensate for poor efficiency of antimatter energy converted into kinetic energy you do not need more reaction mass, you just need a few more milligrams of antimatter. Assuming the engine can resist being vaporized by the higher temperatures that come with the higher exhaust velocities.)We can obtain the amount of antimatter needed for a specific mission by substituting Eq. (11.16) into Eq. (11.15) to get the mass of the energy source (

m). The antimatter needed is just half of this mass. We find it to be a function of the square of the mission velocity (Δ_{e}) (essentially the mission energy), the empty vehicle's mass (_{v}m), and the conversion efficiency (_{v}η):_{e}

equation 11.18The amount of antimatter calculated from Eq. (11.18) is typically measured in milligrams. Thus, no matter what the mission, the vehicle uses 3.9 tons of reaction mass for every ton of vehicle and an insignificant amount (by mass, not cost) of antimatter. Depending on the relative cost of antimatter and reaction mass after they have been boosted into space, missions trying to lower costs may use more antimatter than that given by Eq. (11.18) to heat the reaction mass to a higher exhaust velocity. If so, they would need less reaction mass to reach the same mission velocity. Such cost-optimized vehicles could have mass ratios closer to 2 than 4.9 [Forward, 1985].

The low mass ratio of antimatter rockets enables missions which are impossible using any other propulsion technique. For example, a reusable antimatter-powered vehicle using a single-stage-to-orbit has been designed [Pecchioli, 1988] with a dry mass of 11.3 tons, payload of 2.2 tons, and 22.5 tons of propellant, for a lift-off mass of 36 tons (mass ratio 2.7:1). This vehicle can put 2.2 tons of payload into GEO and bring back a similar 2.2 tons while using 10 milligrams of antimatter. Moving 5 tons of payload from low-Earth orbit to low Martian orbit with an 18-ton vehicle (mass ratio 3.6:1) requires only 4 milligrams of antimatter.

Antimatter rockets are a form of nuclear rocket. Although they do not emit many neutrons, they do emit large numbers of gamma rays and so require precautions concerning proper shielding and stand-off distance.

[Forward, 1985] Forward, Robert L., Brice N. Cassenti, and David Miller. 1985. Cost Comparison of Chemical and Antihydrogen Propulsion Systems for High AV Missions. AIAA Paper 85-1455, AIAA/SAE/ASME/ASEE 21st Joint Propulsion Conference, 8-10 July 1985, Monterey, California.

[Pecchioli, 1988] Pecchioli, M. and G. Vulpetti. 1988. A Multi-Megawatt Antimatter Engine Design Concept for Earth-Space and Interplanetary Unmanned Flights. Paper 88-264 presented at the 39th Congress of the International Astronautical Federation, Bangalore, India 8-15 October 1988.

[Shepherd, 1952] Shepherd, L. R. 1952. Interstellar Flight. Journal of the British Interplanetary Society. 11:149-167.

artwork by Vincent Di Fate“Of course the parry isn't

necessary," he said. “It represents a tradition from the earliest days of planetary exploration. The ships at that time all used chemical rockets—”“Not nuclear?” Jan asked. “They had nuclear energy, you know, even back then.”

“They did, but they’d had bad experiences with it and a lot of people were still scared. So they used chemical rockets.”

“But the effects of chemical rockets on the atmosphere and ionosphere are a lot worse than nuclear. Didn’t they know—"

“The ships used chemical rockets. That’s not totally true, because there were already a few ion drives; but they provided such low accelerations that they were useless for passenger shipping. You can guess what it was like. Everybody was short of delta-vee for everything. They would scrounge, beg, or borrow as much momentum transfer as they could lay their hands on, but space travel was still marginal, all touch-and-go. The first ships to reach Jupiter didn’t have enough fuel to slow into orbit around the planet. If they didn’t do something different, they would arrive, swing past, and shoot away in some other direction. The answer—the only possible answer at the time—was to skim through Jupiter's upper atmosphere and use air-braking for velocity-shedding.

“The theory was simple and fully understood for more than a century. Doing it, and getting it exactly right, was another matter. The

Arbkenazywent in too deep and never came out. TheCelandineerred in the other direction. It skipped in, skipped out, and left the Jovian system completely."His voice had gradually slowed and deepened. Jan squeezed the little roll of fat at his waist. “You’re supposed to be telling me about some big patty we’ll be having, not zoning out on me. Are you drifting off?"

“I am not. I'm thinking how much easier we have it than the original explorers. The

Celandinecrew members were tough, and braver than you can believe. I've heard their recordings. They sent back data on the Jupiter magnetosphere until they were on the last drips of oxygen, then they all signed off as casually as if they were going out together for an early dinner. A dip into the Jovian atmosphere used to be a life-or-death proposition. Now it's just a game. Jupiter's atmospheric depth profile is mapped to six figures. The atmospheric swingby is a tradition and a good excuse for a party, but it is absolutely and totally unnecessary.”"Like crossing the line.” She saw Paul’s forehead wrinkle. “In the old days of Earth-sailing ships, crossing the equator was a bit dodgy. The region around the equator was called the Doldrums, where the winds would fall away to nothing for days or weeks at a time. The ship would sit becalmed, in extreme heat, with no one aboard knowing if they would live long enough to catch a saving wind. Then steamships came along, and crossing the equator offered no special danger. But a ceremony called ‘Crossing the Line’ lived on. There were high jinks on board the cruise ships; parties and, ritual shaving—not just of people's heads, either—and silly ceremonies involving King Neptune.”

“It’s King Jove on the Jupiter flyby, but the rest of it sounds much the same." Paul turned to look at Jan. “Look, I know it sounds stupid and it really is stupid, but as first officer I’m stuck with it. You don't have to go along.”

“Are you kidding? Paul, there's no way I'd miss this. If I had been there in the old days crossing the equator, I’d have been whooping it up like nobody's business. My question is, can you as first officer take part in all the fun, or is it considered too undignified?"

“Define ‘too undignified.' I suppose there are limits, but they're pretty broad. On the last Jupiter atmospheric flyby, two months ago, the chief engineer dressed himself in a baboon suit. He had cut a piece out of the back. His a** was bare, and painted blue, and he said he was selling kisses. But I didn’t hear of any takers."

“Captain Kondo permitted this?" Jan had trouble imagining the captain, short, stocky, and immensely dignified, participating in the brawl that Paul was describing—or even allowing it.

“Captain Kondo remained in his quarters throughout the party. He does that on every Jupiter swingby. His view is that what he does not see, he is not obliged to report."

**DARK AS DAY**by Charles Sheffield

*(2002)*

### Alternate Delta-V Equations

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

**Δ _{v} = sqrt((2 * Bp * Bε) / mDot) * ln[R]**

**R = e ^{(Δv/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** V _{e}** 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 Power | 5,800,000 w |

High Gear | |

Exhaust velocity | 294,000 m/s |

Thrust | 40 n |

Medium Gear | |

Exhaust velocity | 147,000 m/s |

Thrust | 80 n |

Low Gear | |

Exhaust velocity | 29,000 m/s |

Thrust | 400 n |

LANTR | |

NERVA mode | |

Exhaust velocity | 9,221 m/s |

Thrust | 67,000 n |

LOX mode | |

Exhaust velocity | 6,347 m/s |

Thrust | 184,000 n |

Certain propulsion systems can "shift gears" much like an automobile. Basically they can trade thrust for exhaust velocity (specific impulse) and vice versa.

Example spacecraft and engines include:

- Afterburner Fission-Fragment Mars DRA Ship
- Bimodal Hybrid NTR NEP
- Ehricke Fusion Ship
- LANTR
- LANTR LTV
- Pulsed Fission Fusion engine
- Pulsed Solid-core NTR
- Santarius Fusion Rocket
- VASIMR

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.

**F _{p} = ( V_{e} * F) / 2**

**F = (F _{p} * 2) / V_{e}**

**V _{e} = (F_{p} * 2) / F**

**mDot = F / V _{e}**

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

**F = mDot * V _{e}**

**V _{e} = F / mDot**

**V _{e} = sqrt[(F_{p} *2) / mDot]**

**mDot = (F _{p} *2) / V_{e}^{2}**

where:

F_{p}= Thrust Power (watts) should be aconstantfor a given engine

F = Thrust (Newtons)

V_{e}= 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

The current figure for the VASIMR's thrust power is 5.8 megawatts (5,800,000 watts). If its exhaust velocity is set to 294,000 m/s (specific impulse of 30,000 seconds), what would the thrust be?

F = (F_{p}* 2) / V_{e}

F = (5,800,000 * 2) / 294,000

F = 11,600,000 / 294,000

F =40 NewtonsWhat if you set the thrust to 400 Newtons, what would the exhaust velocity be? Remember for a given engine the thrust power is a

constant, it is still 5,800,000 wattsV_{e}= (F_{p}* 2) / F

V_{e}= (5,800,000 * 2) / 400

V_{e}= 11,600,000 / 400

V_{e}=29,000 m/s

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:

- Optimizing the exhaust velocity to the mission delta V
- Engine has super-fantastic exhaust velocity but thrust is ludicrously tiny
- Using the same engine for lift-off and interplanetary flight
- 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 **V _{e} = Δ_{v} * 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/s^{2}) 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.

Rick Robinson:Those performance stats (for the Project Daedalus) are certainly torchlike, and in fact

an exhaust velocity of 10,000 km/s is wasteful for nearly all Solar System travel — on most routes you just don't have time to reach more than a few hundred km/s.Using STL starship technology on interplanetary routes is like using a jet plane to get around town.

Jean Remy:There's no such thing as going somewhere "too fast". At least in terms of military strategy you'll want the ability to get somewhere faster than anyone else can, and damn the price at the pump. It is more costly to arrive at a battle late (and for want of a horse)

Rick Robinson:Oh, I have nothing against speed! A better way to put it is that STL starships are geared all wrong for insystem travel, like driving city streets in 5th gear.

Luke Campbell:Consider a 1,000 ton spacecraft with a 10,000 km/s exhaust velocity and an acceleration of 0.722 m/s/s. For a 1 AU trip at constant acceleration, flipping at the midpoint, it will take 10.5 days and consume 66 tons of propellant/fuel.

Now let's add extra mass into the exhaust stream, so that the spacecraft uses propellant at 16 times the rate but expells it at 1/4 the exhaust velocity (

thus keeping the same power). This brings the acceleration up to 2.89 m/s/s. We will accelerate for 1/10 the distance, drift for 8/10 the distance, and then decelerate for 1/10 the distance. The trip now takes 7 days and uses 240 tons of propellant, of which only 14 tons is fuel.Bulk inert (non-fuel) propellant is probably cheap (water or hydrogen). Fuel is probably expensive (He-3 and D).

The second option gets you there faster and cheaper.(ed note: see the mathematical details of Luke Campbell's example below)

In Rick's analogy, high exhaust velocity, low thrust, low propellant flow corresponds to

high gear. Low exhaust velocity, high thrust, high propellant flow islow gear. In this case, a lower gear than the default "interstellar" Daedelus thrust parameters is preferable.

Rick Robinson:'Gearing' is highly desirable even if the drive won't produce surface lift thrust from any significant body.

Each deep space mission also has its own optimum balance of acceleration and delta v, favoring an adjustable drive.(ed note: given the mission delta V, the optimal exhaust velocity is

Δ.)_{v}* 0.72

**ON TORCHSHIPS**comments (2010)

Luke Campbell: Consider a 1,000 ton spacecraft with a 10,000 km/s exhaust velocity and an acceleration of 0.722 m/s/s.GIVEN:

M(spacecraft wet mass)=1×10^{6}kg(1,000 tons)

V_{e}(exhaust velocity)=1×10^{7}m/s(10,000 km/s)

A(instantaneous acceleration)0.722 m/s^{2}

IMPLIED:

Isp(specific impulse) = Ve / g_{0}=1×10^{6}sec

F(thrust) = F = M * A =722,000 N

F_{p}(thrust power) = (F * V_{e}) / 2 =3.61×10^{12}watts(3.61 terawatts)

mDot(propellant mass flow) = F / V_{e}=0.0722 kg/s

mDot_{f}(fusion fuel mass flow) = mDot =0.0722 kg/s(because with pure fusion engines the fuel is also the mass)

Luke Campbell: For a 1 AU trip at constant acceleration, flipping at the midpoint, it will take 10.5 days and consume 66 tons of propellant/fuel.GIVEN:

D(trip distance)=1.496×10^{11}m(1 AU)

Trajectory=Brachistochrone(constant acceleration flipping at endpoint)

IMPLIED:

T(transit time) = 2 * sqrt[ D/A ] =910,389 seconds(10.5 days)

T_{b}(duration of burn) = T (because brachistochrone) =910,389 seconds

M_{pb}(mass of propellant burnt in current burn) = mDot * T_{b}=65,500 kg(66 tons)

Luke Campbell: Now let's add extra mass into the exhaust stream (implying that above is specifying a pure fusion ship), so that the spacecraft uses propellant at 16 times the rate but expells it at 1/4 the exhaust velocity. This brings the acceleration up to 2.89 m/s/s.GIVEN:

V_{eg}(gearshifted exhaust velocity) = V_{e}/ 4 =2,500,000 m/s

IMPLIED:

F_{g}(gearshifted thrust) = (F_{p}* 2) / V_{eg}=2,888,000 N(1/4 exhaust velocity,note F)_{p}is still 3.61×10^{12}watts!

mDot_{g}(gearshifted propellant mass flow) = F_{g}/ V_{eg}=1.1552 kg/s(propellant at 16 times the rate)

A_{g}(gearshifted acceleration) = F_{g}/ M =2.89 m/s^{2}

Luke Campbell: We will accelerate for 1/10 the distance, drift for 8/10 the distance, and then decelerate for 1/10 the distance. The trip now takes 7 days and uses 240 tons of propellant, of which only 14 tons is fuel.GIVEN:

D_{0.1}= D * 0.1 =1.5×10^{10}m(1/10 the distance)

D_{0.8}= D * 0.8 =1.2×10^{11}m(8/10 the distance)

IMPLIED:

T_{a0.1}(time to accelerate 1/10 distance) = sqrt[(D_{0.1}* 2) / A_{g}] =101,885 seconds(1.2 days)

T_{d0.1}(time to deccelerate 1/10 distance) = T_{a0.1}=101,885 seconds(1.2 days, takes just as long to slow down to stop as to speed up)

M_{pba0.1}(mass of propellant burnt accelerating 1/10 distance) = mDot_{g}* T_{0.1}=120,000 kg(120 tons)

M_{pbd0.1}(mass of propellant burnt decelerating 1/10 distance) = M_{pba0.1}=120,000 kg(120 tons)

R_{0.8}(rate of speed during drift) = A_{g}* T_{a0.1}=294,000 m/s(ship speed at end of acceleration period)

T_{0.8}(duration of drift) = D_{0.8}/ R_{0.8}=408,000 seconds(4.7 days)

T_{g}(Total gearshifted time) = T_{a0.1}+ T_{0.8}+ T_{d0.1}=611,770 seconds(7 days)

M_{pbg}(total mass gearshifted propellant burnt) = M_{pba0.1}+ M_{pbd0.1}=240,000 kg(240 tons)

M_{fbg}(total mass fuel burnt) = mDot_{f}* (T_{a0.1}+ T_{d0.1}) =14,000 kg(14 tons)

## 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."

- The first component of the engine is the
**fuel**that is burnt to generate energy - The second component is the
**generator**which burns the fuel. - The third component is the
**propellant or reaction mass**. - The fourth component, the
**energy conversion system**uses the energy from the burnt fuel to make the propellant move at high velocity. - 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 | ||
---|---|---|

β | Positrons | |

p | Antiprotons | |

H | Antihydrogen | |

Chemical Liquid | ||

CH_{4}/O_{2} | Liquid Methane / Liquid Oxygen | Poor performance, but the stuff can be stored almost indefinitely in space, unlike other liquid fuels. It is also available on some outer moons. |

H_{2}/F_{2} | Liquid Hydrogen / Liquid Fluorine | Pretty 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. |

H_{1}/O_{2} | Single-H / Liquid Oxygen | Free 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. |

H_{2}/O_{2} | Liquid Hydrogen / Liquid Oxygen | Almost as good performance as H_{2}/F_{2}, but without the nasty fluorine. |

RP-1/O_{2} | RP-1 / Liquid Oxygen | RP-1 is highly refined kerosene. This is NASA's favorite fuel. Almost as good performance as H_{2}/O_{2}, but without liquid hydrogen's strict cryogenic requirements and lamentable lack of density. |

UDMH/NTO | Dimethylhydrazine + Nitrogen Tetroxide | |

MMH/NTO | Monomethylhydrazine + Nitrogen Textroide | |

Chemical Solid | ||

Al/AP | Aluminum / Ammonium Perchlorate | Solid 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/O_{2} | Aluminum / Liquid Oxygen | Finely 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-H | Metallic Hydrogen | Hydrogen 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-A | Metastable He IV-A | Helium in a long-lived excited state |

Electrical Power | ||

10 MWe | Ten megawatts of electrical input | Many drives are "fueled" by electricity. They typically use solar photovoltaic array or fission reactors. Example: Ion drive. |

External | ||

Ext Plas-Beam | External Plasma Beam | A fixed installation such as space station sends a beam of plasma to the spacecraft. Example: MagBeam. |

Ext Laser | External Laser | A fixed installation such as space station sends a laser beam to the spacecraft. Example: Laser Thermal and Laser Sail. |

Ext Kinetic | Kinetic Pellets | A 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 Mag | Solar Magnetism | Spacecraft utilizes the environmental solar magnetic field for propulsion. Example: M2P2. |

Sol Photon | Solar Photons | Spacecraft utilizes the environmental sunlight for propulsion. Example: Photon Sail. |

Sol Wind | Solar Wind | Spacecraft utilizes the environmental solar wind for propulsion. Example: E-Sail. |

Fission | ||

^{245}Cm | Curium-245 | |

^{6}Li | Lithium-6 | |

^{239}Pu | Plutonium-239 | |

^{233}U | Uranium-233 | |

^{235}U | Uranium-235 | |

UBr_{4} | Uranium-235 Tetrabromide | |

UF_{6} | Uranium-235 Hexafluoride | |

FI | Generic Fissionable | |

Fusion | ||

4xH | Proton - Proton | |

D-D | Deuterium - Deuterium | |

D-T | Deuterium - Tritium | |

H-B | Hydrogen - Boron | |

H-Fe | Hydrogen - Iron | |

H-^{6}Li | Hydrogen - Lithium-6 | |

H-^{7}Li | Hydrogen - Lithium-7 | |

^{3}He-D | Helium-3 - Deuterium | |

^{3}He-^{3}He | Helium-3 - Helium-3 | |

FU | Generic Fusion Fuel | |

D-T + ^{6}Li-n | Deuterium - 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 Core | Fission or antimatter powered device to thermally heat propellant. Upper limit of temperature is where the core melts. | |

Liquid Core | Fission or antimatter powered device to thermally heat propellant. Upper limit of temperature is where the core changes from molten into vapor. | |

Vapor Core | Fission 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-Cycle | Fission 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 Confined | Fission 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 Choke | Fission 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-Cycle | Fission 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 Core | Fission, 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 Unit | Basically 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 Catalyzed | Ultracold 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 Catalyzed | This 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-Pinch | Zeta-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 Heating | Fission fragments from fissionables undergoing nuclear decay heat the propellant, typically liquid hydrogen. | |

Thermal-Fusion | ||

Electrostatic Confinement | Fusion fuel is squeezed into reacting by electrostatic fields. Example Polywell Fusor. | |

Inertial Confinement Laser | Fusion 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 Beam | Fusion 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 Confinement | Fusion fuel is squeezed into reacting by a magnetically crushed metal propellant foil ring. Propellant foil is heated thermally. | |

Plasma Core | Fission, 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 Catalyzed | This 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 Unit | Basically 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 Catalyzed | This 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-Pinch | Zeta-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 Core | Fission or antimatter powered device to thermally heat propellant. Upper limit of temperature is where the core melts. | |

Liquid Core | Fission or antimatter powered device to thermally heat propellant. Upper limit of temperature is where the core changes from molten into vapor. | |

Vapor Core | Fission 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-Cycle | Fission 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 Confined | Fission 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 Choke | Fission 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-Cycle | Fission, 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 Core | Fission, 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 Chamber | For chemical fuels, a chamber where the chemicals react or "burn." | |

Fission-fragment Propellant | Fission fragments from fissionables undergoing nuclear are used as the propellant | |

Electrostatic Confinement | Fusion fuel is squeezed into reacting by electrostatic fields. Example Polywell Fusor. | |

Inertial Confinement Laser | Fusion 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 Beam | Fusion 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 Core | Antimatter 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 array | Solar powered device used to supply electrical energy to the propellant accelerator. | |

OTHER | ||

Collector Mirror | A device for gathering external energy, such as external plasma beams, external laser beams, and solar photons. | |

None | Some 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 | ||
---|---|---|

CH_{4} | Methane | |

CO | Carbon Monoxide | |

CO_{2} | Carbon Dioxide | |

H_{1} | Single-H | Free 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. |

H_{2} | Liquid Hydrogen | Molecular hydrogen. The thermal propellant of choice. |

H_{2}O | Water | |

Seeded-H | Seeded Hydrogen | Transparent 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. |

N_{2} | Nitrogen | |

NH_{3} | Ammonia | |

O_{2} | Liquid Oxygen | |

Electrical | ||

Ar | Argon | |

Bi | Bismuth | |

Cd | Cadmium | Easy to ionize, but erodes the grid. Popular until they figured out how to efficiently ionize Xenon. |

CL | Colloid | Sometimes used in ion and other electrostatic drives. |

Cs | Cesium | Easy to ionize, but erodes the grid. Popular until they figured out how to efficiently ionize Xenon. |

He | Helium | |

I | Iodine | |

Kr | Krypton | |

Hg | Mercury | Easy to ionize, but erodes the grid. Popular until they figured out how to efficiently ionize Xenon. |

Mg | Magnesium | |

Xe | Xenon | Currently popular in ion drives, since it does not erode the grid. It took a while to figure out how to efficiently ionize the stuff. |

Zn | Zinc | |

Other | ||

Cn | Graphite | For ablative laser drives and fusion pulse ablative nozzles. |

DU | Depleted Uranium | |

Li | Lithium | |

Pb | Lead | |

RP | Reaction Products | Where the propellant is the product of the chemical, fission, fusion, or antimatter reaction; instead a separate propellant heated by the reaction. |

RK | Regolith | General 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. |

SiC | Silicon Carbide | Popular in ablative nozzles. |

W | Tungsten | For Orion drive pulse units |

γ | Photons | Rays 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 Heater | Input: Electricity. Propellant is accelerated electrothermally by an electrical arc. |

Collector Mirror Heater | Propellant is thermally accelerated by heat from sunlight or laser beams focused by a collector mirror type reactor. |

Resistance Heater | Input: Electricity. Propellant is accelerated electrothermally by an electrical resistance heater. |

Microwave Heater | Input: Electricity. Propellant is accelerated electrothermally by microwaves. |

Reaction Heater | Input: Thermal. Propellant is thermally accelerated by heat from the chemical, fission, fusion, or antimatter reaction. |

Electrical | |

Electromagnetic | Input: Electricity. Propellant is accelerated electromagnetically (plasma drives) |

Electrostatic | Input: Electricity. Propellant is accelerated electrostatically (ion drives). |

Other | |

Annihilation | Input: Antimatter Reaction. Propellant is the subatomic particles formed by a matter-antimatter reaction. |

Fission-Fragment | Input: Nuclear Fission. Propellant is split atoms flying from a nuclear fission event. May be antimatter catalyzed. |

None | Includes 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. |
---|---|

Nozzle Regeneratively Cooled | 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. |

Nozzle Magnetic | 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. |

Nozzle Ablative | 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. |

Pusher Plate Ablative | 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. |

Pusher Plate | 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. |

Magnetic Loop | 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. |

Reflective Light Sail | a large sail that reflects photons. Can focust the exhaust if desired. |

Grey Sail | a large sail that absorbs or scatters photons or other particles. Typically glows hot due to absorbed power. |

E-Sail | a large sail that acts on charged particles via electrostatic force, does not focus the exhaust. |

None |

## Payload

Payload is the

loadthat the spacecraft owner is beingpaidto haul. Yeah, kind of like the cargo. Except the blasted cargo can be a crew of astronauts, a warship's weapon turrets, a pre-fab lunar colony, the spacecraft's built-in crew quarters, or anything else that is not propellant or ship structure.With the Polaris, our payload is Tom Corbett and his buddies, the Polaris habitat module, the life-support system, the avionics, the command deck, the astrogation deck, the engineering deck, the space boats, and the atomic missile armaments.

"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 e^{2}. If the deltaV is three times the exhaust velocity, the mass ratio has to be 20 or e^{3}.

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.

*In***Destination Moon**, they expended too much propellant while landing...*...and had to frantically reduce the structural mass of the ship.*

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

**P _{f} = 1 - (1/R)**

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

**P _{e} = 1 / R**

where

- P
_{e}= 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:

**P _{e} = 1 / e^{(Δv/Ve)} **

P_{e} 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 P_{e} is payload, at least if this is a cargo ship. So given the ship's Δ_{v} capacity and the propulsion systems V_{e}, 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 = e ^{(Δv/Ve)}**. See how

**(Δ**is raised next to the

_{v}/V_{e})**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.

artwork by Alan Magee(ed note: The good ship Johannes Kepler is about midway on a 92 day journey to Mars colony when a meteor punctures the ship. Unfortunately the idiot captain was holding a meeting of all the officers in the control room, so they are all dead now. The only officer left is Lieutenant Donald Chase, who is actually the ship's medic. However, by the chain of command he is officially the captain.

They struggle through a variety of disasters, most recent of which was a solar proton storm. Now they have to somehow contact Mars Central because they are off-course, the astrogator is in the morgue, and a passenger named Ugalde who is a mathematician is not quite up to calculating a correction. Eventually they make radio contact with Mars through the solar interference by using morse code.)

It took time, a lot of time, because the communication was so complex. Don typed a message into the computer, explaining what had happened, and this was recorded on tape as a series of dots and dashes. Another tape was prepared of up-to-date stellar observations which were recorded along with the earlier data. The computer on Mars would process these and determine the course corrections that would be needed. Time passed, and with each second they moved further from their proper course.

They waited again and, instead of the course corrections, they received a request for the amount of reaction mass that remained in their tanks. This was sent back as quickly as possible and there were minutes of silence as they waited for the answer, for the corrections that would get them back into the proper orbit for Mars. The message finally came.

‘Hello Big Joe,’ the voice rasped and, although the man speaking tried to sound happy, there was an undertone of worry in his voice. ‘We are not saying that this is the final answer, the figures are being re-run, and something will be done. But the truth is … well … you have been in an incorrect orbit for too long a time. It appears that, with the reaction mass you have remaining … there is not enough to make a course correction for Mars. Your ship is on an unchangeable orbit into outer space.’

‘What is this reaction mass that Mars Central is so worried about?’ he asked. ‘I hate to act stupid, but medical studies leave little time for reading about anything else. I thought this ship was powered by atomic engines?’

‘It is, sir, but we still need reaction mass. A rocket moves not by pushing against anything, but by throwing something away. Whatever is thrown away is called reaction mass. In chemical rockets it is burning gas. The gas goes in one direction, the rocket goes in the other. The more you throw away, the more reaction you get and the faster you go. You also get more reaction by throwing something awayfaster. That is what we do. Our reaction mass is made up of finely divided particles of silicon. It’s made from steel plant slag, vaporized in a vacuum, so the particles are microscopic. These particles are accelerated by the engines to an incredible speed. That’s what gives us our push.’ (nowadays we know that liquid hydrogen is a superior reaction mass to finely-divided silicon)

Don nodded. ‘Seems simple enough — at least in theory. So, although we have unlimited power from the atomic engines, we don’t have enough reaction mass for the course change required?’

‘Right, sir. Normally we carry more than enough mass for our needs, because the course corrections are made as early as possible. The more the ship gets away from the right orbit, the more mass is needed to get us back. We’ve waited a little too long this time.’

Don refused to give in to the feeling of gloom that swept the control-room.

‘Can’t we use something else for reaction mass?’ he asked.

Kurikka shook his head. ‘I’m afraid not. Nothing is small enough to get through the injectors. And the engines are designed to run with this kind of reaction mass only.’ He turned away and, for the very first time, Don saw that the rock-like chief petty officer was feeling defeat. ‘I’m afraid there is nothing we can do.’

‘Wecan’tgive up!’ Don insisted. ‘If we can’t change the orbit to the correct one, we can certainly alter it as much as possible, get it closer to the correct one.’

‘Maybe we can, Captain, but it won’t help. With all our mass used to change course we won’t have enough for deceleration.’

‘Well at least we’ll be closer to Mars. There must be other ships there that can match orbits with us and take everyone off. Let’s ask Mars Central about it.’

The answer was infuriatingly slow in coming, and not very hopeful.

‘We are running all the possibilities through the computer here, but there is nothing positive yet. There are no deepspacers here who can aid you, and the surface to satellite ferries don’t have the range to reach you, even with your correct orbit. Don’t give up hope, we are still working on the problem.’

‘Great lot of good that does us,’ Sparks muttered. ‘You’re not in our shoes.’

‘I am afraid I must disagree with Chief Kurikka and say that his last statement is wrong,’ Ugalde said. He had been standing in a daze of concentration for a long time, and did not realize that the Chief’s ‘last’ statement had been spoken almost fifteen minutes earlier. ‘There is something we can do. I have examined the situation from all sides and, if you will permit me to point out, you are looking at only part of the problem. This is because you have stated the question wrong.’ He began to pace back and forth.

‘The problem is to alter our orbit to the correct one, not to find more mass. Stated this way the problem becomes clear and the answer is obvious.’

‘Not to me,’ Kurikka said, speaking for all of them.

Ugalde smiled.‘If we cannot get more reaction mass, then we must get less mass for our present quantity of reaction mass to work against.’

Don smiled back.‘Of course! That’s it! We will just have to lighten ship.’

‘It is important that everything that is jettisoned be weighed first,’ Ugalde warned. ‘This will be needed in the computations.And the faster it is done the better our chances will be ! ’

‘We start right now,’ Don said, pulling over a notepad and electric stylo. ‘I want to list everything that is not essential to the operation of the ship and the lives of everyone aboard. Suggestions?’

‘The passengers’ luggage of course,’ Ugalde said. ‘They should keep what they are wearing and the rest will be discarded.’

The purser moaned. ‘I can see the lawsuits already.’

‘I’m sure that the company is insured,’ Don said, making a note. ‘Their luggage or their lives — that is really not much of a choice. They can keep their valuables and personal items, but anything that can be replaced has to go. You’d better have them all assembled in the main dining-hall in fifteen minutes. I’ll come up and tell them myself.’

Jonquet nodded and left. Don turned to the others.

‘The dining—tables, chairs, dishes, most of the kitchen equipment,’ Kurikka said, counting oif the items on his fingers. ‘All the frozen meat and refrigerated food. We can live off the dehydrated emergency rations which use recycled water.’

‘Good thinking. Who’s next?’

Once they began to concentrate on it, it was amazing the number of items that they found. Carpets and decorations and banisters on the stairs, furniture, fittings and spare parts. The list grew and Don checked off the items. There was one obvious — and heavy — item missing. ‘The cargo,’ he said, ‘what about that?’

Kurikka shook his head. ‘I only wish we could. There is heavy machinery, bales of clothing, a lot of items that we could do without. But all the cargo is container loaded for the most part, and sealed into place against the G stresses. The shuttle rockets have the special extensible power sockets to reach down past the containers to free them, but we don’t have the equipment. I suppose we could jury-rig something to get the containers out, but it would take a couple of days at least.’

‘Which is far too long for us. The cargo stays — but everything else that can go,goes! ’

**SPACESHIP MEDIC**by Harry Harrison (1970)

## Handy Aids

Everything old is new again. AFAIK there ain't a smartphone app for this, and doing it longhand is a drag. So check out this 1900's tech called a Nomogram. Sneer at it if you like, it actually has some advantages over spreadsheets and online calculators. Consider it to be steampunk, because it is. I'm sure Robert Heinlein used nomograms.

*Nomogram for the Solar System. Download here.**Example of nomogram use.**Rocket Performance Calculator*

To get some rough ballpark estimates, you can use my handy-dandy DeltaV nomogram
*(more about nomograms)*. 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 1^{st} 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.

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

For fun, 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 I_{sp} that it will handle is disappointing. But it does give one an intuitive feel for these calculations. *(Alas, it appears that this is now out of print)*

The General Electric Space Propulsion Calculator was manufactured by the GE Flight propulsion Laboratory. Front side calculates Thrust, Thrust Power, Propellant Mass Flow, Specific Impulse, and Exhaust Velocity. The flip side calculates Escape velocity, Orbital velocity, Period of revolution, and Gravitational pull for the major planets and moons of the solar system. Images are from the Slide Rule Museum. If anybody has any more information about this slide rule, please contact the webmaster.

## Popular Conceptions

These illustrations are from a 1963 Russian magazine called "Техника молодежи" magazine ("Technology Youth"), as shown in Pavel Popelskii's Science Illustration blog. They are more a popularization for children than they are a rigorous technical document, but they are interesting. I do not speak or read Russian, but I discovered that Google Translate is my friend. Any awkward phrasing is the fault of Google translate.

"Isotopic motor" (a.k.a. fission sail.)

- 1.
The radioactive isotope - a source of alpha particles.- 2.
Absorber of alpha particles, which protects the equipment from the particles emitted in a random direction.- 3.
Alpha particles.This engine is labeled an "isotopic motor", but nowadays is called a fission sail. Radioactive material has its radiation absorbed on all sides except in the desired thrust direction. Great specific impulse, but the thrust is microscopic.

"Reactor", possibly a radioisotope thermoelectric generator.

- 4.
Reactor.- 5.
Vacuum diode - a source of electrical current, working on the principle of thermionic emission.- 6.
Neutron reflector to their concentration in the reaction zone.As near as I can figure, the spherical object labeled "reactor" is actually a type Radioisotope Thermoelectric Generator. I say this because the section labeled "5" appears to be a thermocouple. The spacecraft appears to be a generalized Nuclear-Electric rocket. The unspecified engine would be some kind of electrical propulsion, like ion or plasma.

Fission reactor and magnetohydrodynamic generator.

- 6.
Neutron reflector to their concentration in the reaction zone.- 7.
Solenoid to produce a magnetic field.- 8.
Capacitor divider that separates the uranium from the hydrogen.- 9.
Hydrogen plasma, fed into accelerators.- 10.
Electrodes for the removal of the electric current created by the movement of plasma through a magnetic field.- 11.
The direction of electric current.- 12.
Zone of fission.This uses a magnetohydrodynamic (MHD) generator to harvest electricity from the uranium-hydrogen plasma. The fissioning uranium ionizes the hydrogen. The ionized stream can conduct electricity. It is shot through a magnetic field (created by a solenoid), where it induces an electrical current in the side plates. The stream then enters the "divider" where the uranium is separated from the hydrogen. The unspent uranium is sent back to the reaction chamber. The hydrogen is sent to some kind of Electromagnetic accelerator which is powered by the electricity from the MHD generator.

I have no idea if this will acually work, or if it was discredited decades ago. Up until now I had only seen MHD harvesting of electricity associated with nuclear fusion reactions, not nuclear fission.

The "reactor" is actually the reaction chamber (12). The "motor" is the Electromagnetic accelerator. "Working mass" is another name for "reaction mass", "working fluid", or "propellant". The shadow shield is up near the nose, though generally it is more efficient to put it right on top of the reactor. The "vernier motor" is an attitude jet.

These are the atomic rockets, as tipped off by the Russian word for "uranium". All of these are nuclear thermal rockets or NTR. As near as I can figure:

- 13.
Nozzle.- 14.
Uranium-graphite reactor core.Looks like every NERVA diagram I've ever seen. A nuclear reactor where the coolant flows directly to the exhaust.

- 6.
Neutron reflector to their concentration in the reaction zone.- 12.
Zone of fission.Uranium is just spraying into the reaction chamber along with the propellant. Easiest to engineer, but lots of expensive un-burnt uranium escapes out the exhaust. This angers the owner's accountants and the picketing anti-nuclear activists.

- 6.
Neutron reflector to their concentration in the reaction zone.- 12.
Zone of fission.- 15.
Openings for supply of hydrogen in the tangential and the walls of the cylindrical chamber.Uranium is injected tanjentally, to make a spiral flow around the long axis. Hopefully this forces the uranium to loiter in the reaction chamber longer, reducing the amount of un-burnt uranium that escapes.

- 6.
Neutron reflector to their concentration in the reaction zone.- 8.
Capacitor divider that separates the uranium from the hydrogen.- 12.
Zone of fission.I have never seen this one before.

By doing some research I stumbled over a paper on Russian gas-core design. There is a "recirculation intake" just before the exhaust nozzle that tries to catch the uranium before it escapes. The uranium is liquifed then pumped back to the top of the reaction chamber. Frankly I do not understand why the hot fissioning uranium does not instantly vaporize the intake scoop.

But that is not this design. In this one, the fissioning uranium is jetted in the contrary direction to the hydrogen propellant. It is captured at the top, the hydrogen is filtered out, and sent back to the bottom to be injected again. The author calls it a "coaxial gas reactor", but this is not the same thing as the coaxial-flow NTR.

- 6.
Neutron reflector to their concentration in the reaction zone.- 16.
Molten uranium carbide.- 17.
Porous wall through which the hydrogen leak.The uranium is liquid, and the reaction chamber is spun on the long axis to keep the uranium in the chamber by centrifugal force. Note the tiny arrow indicating the spin, it's a dead giveaway.

This is from a 1960 issue of Technology Youth magazine.

Fourth rocket engine (Fig. IV) works in a peculiar thermo-mechanical cycle. Part of the energy of the reactor is used to drive the pump, which feeds into the reactor core liquid working medium, where it vaporizes and heated at high pressure.

The resulting hot gas is pumped into a separate high-pressure chamber, which, through valve 11 communicates with tube shocks. At the other end of the shock tube we find structed diffuser serves to concentrate the energy of the shock wave, and the valve 12, connecting tube with a nozzle rocket. Duty cycle engine is as follows: pump 5 takes the working fluid from the reservoir and high-pressure pumps it through a reactor, where it evaporates and is heated to about 2500° C — and then injected into the high-pressure chamber. Shock tube at this point is still filled with gas of low pressure left over from the previous cycle. Then the valve 11 to quickly open, compressed gas, bursting into the pipe instantaneously compresses and heats the gas in the tube, causing the appearance in it of a strong shock wave.

The highest compression is achieved in the lower stream of the diffuser. Then, valve 11 closes and valve 12 opens and gas at high speed coming out of the nozzle. When the temperature of exhaust gas will decrease by 3-4 times compared with the maximum temperature reached in the shock tube, valve 12 closes and valve 13 opens, and by a pump 5 remains a shock tube fed into a radiator where it cools. This cycle is continuously repeated, creating "clusters" of high-temperature gas flowing from a nozzle at high speed.

The rocket marked IV appears to be using a system of 'shock tubes', heating a working fluid then pulsing it out underpressure. However, this seems like it would be less efficient then simply operating the engine directly as an NTR, so i have my doubts. As Rob Davidoff pointed out, there is no addition of further "work" after the propellant is heated in the reactor.