Introduction

RocketCat sez

Propellant is the crap you chuck out the exhaust pipe to make rocket thrust. It's Newton's Law of Action and Reaction, savvy? Fuel is what you burn to get the energy to chuck crap out the exhaust pipe. As I told you before they ain't the same. And any show-oaf who asks me "what about reactionless drives" is gonna get an instant RocketCat Atomic Wedgie.

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.

Example

Say the good ship Polaris has a mass of 181,000 kilograms (181 metric tons). It has super powerful Nuclear Salt Water Rockets with an exhaust velocity of 182,000 meters per second (because it is using 22% enriched uranium) and a remass flow of 71 kilograms per second. If the Polaris is floating in space with a speed of zero, how fast will it be moving if it burns its engine for ten seconds?

In ten seconds it will spit out 71 × 10 = 710 kilograms of reaction mass.

The exhaust velocity is 182,000 m/s, so the momentum of the action is 182,000 × 710 = 129,220,000 kg⋅m/s

The Polaris gets an equal but opposite reaction of 129,220,000 kg⋅m/s. The mass of the Polaris is 181,000 kg. So the velocity is 129,220,000 / 181,000 = 714 m/s, which is about 1,597 miles per hour (in the opposite direction the remass is traveling). Not bad for a ten-second burn.

THE MARTIAN WAY

(ed note: Long and Rioz are Scavengers: citizens of Mars whose job is to intercept cast-off propellant tanks, brand them, and send them to Deimos. They are listening to a broadcast from Terra. It is a propaganda piece from a nastly little politician named "Hilder" {Asimov's thinly disguised stand in for you-know-who}. Hilder is making a fake controversy about Mars wasting water as rocket propellant, which is nonsense. But Hilder needs an issue if he is going to be elected as global co-ordinator)

      Rioz took a hesitant step forward. Space was clear, so to hell with sitting and looking at a blank, green, pipless line. He said, “What's the Grounder been talking about?”
     “History of space travel mostly. Old stuff, but he's doing it well. He's giving the whole works—color cartoons, trick photography, stills from old films, everything.”

     As if to illustrate Long's remarks, the bearded figure faded out of view, and a cross-sectional view of a spaceship flitted onto the screen. Hilder's voice continued, pointing out features of interest that appeared in schematic color. The communications system of the ship outlined itself in red as he talked about it, the storerooms, the proton micropile drive, the cybernetic circuits…
     Then Hilder was back on the screen. “But this is only the travel-head of the ship. What moves it? What gets it off the Earth?”
     Everyone knew what moved a spaceship, but Hilder's voice was like a drug. He made spaceship propulsion sound like the secret of the ages, like an ultimate revelation. Even Rioz felt a slight tingling of suspense, though he had spent the greater part of his life aboard ship.
     Hilder went on. ”Scientists call it different names. They call it the Law of Action and Reaction. Sometimes they call it Newton's Third Law. Sometimes they call it Conservation of Momentum. But we don't have to call it any name. We can just use our common sense. When we swim, we push water backward and move forward ourselves. When we walk, we push back against the ground and move forward. When we fly a gyro-flivver, we push air backward and move forward.
     “Nothing can move forward unless something else moves backward. It's the old principle of ‘You can't get something for nothing.'
     “Now imagine a spaceship that weighs a hundred thousand tons lifting off Earth. To do that, something else must be moved downward. Since a spaceship is extremely heavy, a great deal of material must be moved downwards. So much material, in fact, that there is no place to keep it all aboard ship. A special compartment must be built behind the ship to hold it.”
     Again Hilder faded out and the ship returned. It shrank and a truncated cone appeared behind it. In bright yellow, words appeared within it: MATERIAL TO BE THROWN AWAY.
     “But now,” said Hilder, “the total weight of the ship is much greater. You need still more propulsion and still more.”
     The ship shrank enormously to add on another larger shell and still another immense one. The ship proper, the travel-head, was a little dot on the screen, a glowing red dot.

(ed note: i.e., The Tyranny of the Rocket Equation)

     Rioz said, “Hell, this is kindergarten stuff.”
     “Not to the people he's speaking to, Mario,” replied Long “Earth isn't Mars. There must be billions of Earth people who've never even seen a spaceship; don't know the first thing about it.”

     Hilder was saying, “When the material inside the biggest shell is used up, the shell is detached. It's thrown away, too.”
     The outermost shell came loose, wobbled about the screen.
     “Then the second one goes,” said Hilder, “and then, if the trip is a long one, the last is ejected.”
     The ship was just a red dot now, with three shells shifting and moving, lost in space.
     Hilder said, “These shells represent a hundred thousand tons of tungsten, magnesium, aluminum, and steel. They are gone forever from Earth. Mars is ringed by Scavengers, waiting along the routes of space travel, waiting for the cast-off shells, netting and branding them, saving them for Mars. Not one cent of payment reaches Earth for them. They are salvage. They belong to the ship that finds them.”

     Rioz said, “We risk our investment and our lives. If we don't pick them up, no one gets them. What loss is that to Earth?”
     “Look,” said Long, “he's been talking about nothing but the drain that Mars, Venus, and the Moon put on Earth. This is just another item of loss.”
     “They'll get their return. We're mining more iron every year.”
     “And most of it goes right back into Mars. If you can believe his figures, Earth has invested two hundred billion dollars in Mars and received back about five billion dollars' worth of iron. It's put five hundred billion dollars into the Moon and gotten back a little over twenty-five billion dollars of magnesium, titanium, and assorted light metals. It's put fifty billion dollars into Venus and gotten back nothing. And that's what the taxpayers of Earth are really interested in—tax money out; nothing in”.

     The screen was filled, as he spoke, with diagrams of the Scavengers on the route to Mars; little, grinning caricatures of ships, reaching out wiry, tenuous arms that groped for the tumbling, empty shells, seizing and snaking them in, branding them MARS PROPERTY in glowing letters, then scaling them down to Phobos.
     Then it was Hilder again. ”They tell us eventually they will return it all to us. Eventually! Once they are a going concern! We don't know when that will be. A century from now? A thousand years? A million? 'Eventually.' Let's take them at their word. Someday they will give us back all our metals. Someday they will grow their own food, use their own power, live their own lives.
     “But one thing they can never return. Not in a hundred million years. Water!
     “Mars has only a trickle of water because it is too small. Venus has no water at all because it is too hot. The Moon has none because it is too hot and too small. So Earth must supply not only drinking water and washing water for the Spacers, water to run their industries, water for the hydroponic factories they claim to be setting up—but even water to throw away by the millions of tons.
     “What is the propulsive force that spaceships use? What is it they throw out behind so that they can accelerate forward? Once it was the gasses generated from explosives. That was very expensive. Then the proton micropile was invented—a cheap power source that could heat up any liquid until it was a gas under tremendous pressure. What is the cheapest and most plentiful liquid available? Why, water, of course.
     “Each spaceship leaves Earth carrying nearly a million tons—not pounds, tons—of water, for the sole purpose of driving it into space so that it may speed up or slow down.
     “Our ancestors burned the oil of Earth madly and wilfully. They destroyed its coal recklessly. We despise and condemn, them for that, but at least they had this excuse—they thought that when the need arose, substitutes would be found. And they were right. We have our plankton farms and our proton micro-piles.
     “But there is no substitute for water. None! There never can be. And when our descendants view the desert we will have made of Earth, what excuse will they find for us? When the droughts come and grow—.”

     Long leaned forward and turned off the set. He said, “That bothers me. The damn fool is deliberately—”

From THE MARTIAN WAY by Isaac Asimov (1952)
CANDLE 5

(ed note: Keven, Glenda, and Jacob have been stranded on a tiny asteroid orbiting Ceres by the Bad Guys. They are in a mostly stripped base, trying to figure out how to get down to Ceres using only what is available)

     Kevin prowled through the corridors of their prison. There has to be some way, he told himself. Ceres mocked him from below, less than three hundred kilometers down. It hung huge in the night sky.
     Three hundred kilometers down, and we're moving about half a kilometer a second relative to Ceres, Kevin thought. That's not very much velocity. Under a thousand miles an hour. It doesn't take much energy to get to that speed. How much gasoline does it take to accelerate a car on Earth up to a hundred miles an hour—a gallon or so? We only need ten times that, not even that much.
     There's plenty of hydrogen and oxygen. Marvelous rocket fuels if we only had a rocket. More than enough to get us down, except that the temperature of hydrogen burning in oxygen is a lot hotter than anything we have to contain in it—
     No. That's not right. The fuel cells do it. But they do it by slowing down the reaction, and they can't be turned into rocket engines.
     He remembered the early German Rocket Society experiments described by Willy Ley. The Berliners had blown up more rockets than they flew, and they were only using gasoline, not hydrogen. Liquid-fuel rockets need big hairy pumps, and Kevin didn't have any pumps.
     What did he have? Fuel cells, plenty of them, and so what? An electric-powered rocket was theoretically possible, but Kevin didn't have the faintest idea of how to build one, even if there was enough equipment around to do it with. He wasn't sure anyone had ever built one—certainly he couldn't.
     Back to first principles, he thought. The only way to change velocity in space is with a rocket. What is a rocket? A machine for throwing mass overboard. The faster the mass thrown away goes in one direction, the faster the rocket will go in the other, and the less you have to throw. All rockets are no more than a means of spewing out mass in a narrow direction. A rocket could consist of a man sitting in a bucket and throwing rocks backward.
     That might get a few feet per second velocity change, but so what? There simply wasn't enough power in human muscles—even if he did have a lot of rocks. Was there any other way to throw them? Not fast; and unless the thrown-away mass had a high velocity, the rocket wouldn't be any use. He went on through the tunnels, looking at each piece of equipment he found, trying to think of how it might be used.
     You can throw anything overboard to make a rocket. Hydrogen, for example. That's all Wayfarer's engines did, heat up hydrogen and let it go out through the rocket nozzle. We have hydrogen under pressure— Not enough. Nowhere near enough hydrogen and nowhere near enough pressure, not to get velocity changes of hundreds of miles an hour. Ditto for oxygen. Gas under compression just can't furnish enough energy. What would? Chemical energy; burning hydrogen in oxygen would do it, but it gave off too much; there was nothing to contain that reaction except the fuel cells and they did it by slowing the reaction way down and—
     And I'm back where I started, Kevin thought. Plenty of energy in the fuel cells if I could find a way to use it. Could I heat a gas with electricity? Certainly, only how—
     His eye fell on the hot-water tank in the crew quarters. An electric hot-water tank. There was a pressure gauge: forty pounds per square inch. Forty p.s.i.—He looked at the tank as if seeing it for the first time, then went running back to the others.
     "Glenda, Jacob, I've got it."

     "Sure it works." Kevin grinned. "Steam at forty p.s.i. will come out fast. About a kilometer a second."
     "I believe you," Glenda said. "But it sounds silly. Steam rockets?"
     Kevin shrugged. "It is silly. There are a lot more efficient systems. But this will work—"
     "In a low g field," Jacob said. "You will not have much thrust. Of course you won't need much."
     "I'm sure it works," Kevin said. "Now all we have to do is build it." He made himself sound confident; he knew how much room for error there was in his figures. "Look, it takes nine hundred and eighty calories to turn a gram of water into steam. We heat that steam up another thirty or forty degrees and let it out. The energy is moving molecules. We know the molecular weight of water, so we can figure the number of molecules in a gram and—"

     They disconnected the hot-water tank and drilled holes in it. Several turns of heating wire went through the holes, then they sealed them in epoxy. At one end of the tank they drilled a large hole and threaded a pipe into it, threaded a large valve onto the pipe, and welded a makeshift rocket nozzle beyond that.
     When it was done they tethered the tank and filled it with water, then connected a fuel cell to the heating leads. "Here goes," Kevin said. He threw the switch to start the heaters.
     Slowly the water inside heated, then began to boil. The pressure shown on the gauge began to rise. In half an hour they had forty-five pounds of pressure. "All right, let's try it," Kevin said.
     Glenda turned the valve to let out steam. A jet of steam and water shot out across the surface of the moonlet. Ice crystals formed in space and slowly settled to the rocket surface. The jet reached far away from them, well off the moonlet itself. The tank pulled against its tether lines, stretching the rope.
     "It works!" Kevin shouted. "Damn it, we're going to make it!" He shut off the electricity. "Let's get her finished."

     It didn't look like a spaceship. It didn't even resemble a scooter, crude as those were. It looked like a hot-water tank with fuel cells bolted onto it. For controls it had vanes set crosswise in the exhaust stream, spring-loaded to center, with two tillers, one for each vane; a valve to control steam flow; and switches to connect the fuel cells to the heaters. Nothing else.
     The tank itself was fuzzy: They'd sprayed it with Styrofoam, building it up in layers until they had nearly a foot of insulation. There were straps on opposite sides of the tank to hold two passengers on.
     The tank held nearly a hundred gallons of water. Kevin calculated that they had more than enough energy to boil it all in their two fuel cells, and they would only need sixty gallons to get to Ceres. The number was so small that he ran it four times, but it was correct.
     The strangest part was the stability system: a pair of wheels taken from a mining cart and set up in front of the water tank. Electric motors rotated the wheels in opposite directions.

     The total mass of Galahad with full water tank was just under 550 kilograms.

     It took only a gentle effort to push the steam rocket away from the moonlet, but the cartwheel-gyros resisted any effort to turn it. Finally they got it oriented properly in space. Then they climbed aboard.
     "Full head of steam," Kevin said. "Almost fifty pounds. Ready?"
     "Ready—"
     He twisted the steam valve. At first both steam and water were expelled from the tank, but as they began to accelerate, the water settled and the exhaust valve let out only steam. C-2 dropped away. They missed it. It was a prison, but a safe one; now they had only their makeshift steam rocket.
     Galahad showed a tendency to tumble, but with the gyros resisting, they were able to control it with the steering vanes. A plume of steam shot from the tank, rapidly crystallizing into ice fog that engulfed them.
     "Damn. That's going to make it hard to see," Kevin said. "Nothing we can do about it." He peered down toward Ceres. It didn't seem any closer. Jacob's farewell faded in their headsets.
     Norsedal's calculations had shown that twenty minutes' thrust should be enough to cancel all their orbital velocity. It would use up just about half their fuel. Once Galahad was stopped dead in orbit above Ceres, they would fall toward the asteroid, and they would have half their steam left to counteract that.
     The trouble was that Jacob couldn't calculate how high above Ceres they would be when the twenty minutes were finished. As they lost velocity, they would lose altitude, and their orbit would no longer be a smooth circle, but an ellipse intersecting Ceres—somewhere. At the end of twenty minutes Kevin cut the power off. He was pleased that they still had thirty pounds of steam pressure.

     "Yes, but that's what the numbers say."
     "All right."
     And a year ago I was working equations in school, Kevin thought. Numbers to crunch and write down for examinations. Now they're something to stake your life on.

From EXILES TO GLORY by Jerry Pournelle (1977).
CANDLES 4

(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 first beer-powered spaceship in 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 the Mercury Girl had 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.

From A BICYCLE BUILT FOR BREW by Poul Anderson (1958)

Rules of Thumb

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.

ACCELERATION GENERAL RULE

5 milligee (0.05 m/s2) : General rule practical minimum for ion drive, laser sail or other low thrust / long duration drive. Otherwise the poor spacecraft will take years to change orbits. Unfortunately pure solar sails are lucky to do 3 milligees.

0.6 gee (5.88 m/s2) : 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/s2) : General rule minimum to lift off from Terra's surface into LEO.

From Ken Burnside
THRUST REGIMES AND ISP FOR INTERPLANETARY TRAVEL

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?


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, 109W) thrust power for a moderate size ship marks the minimum torchship threshold. But "classical" torchships are about 1000x more powerful, approaching the terawatt (TW, 1012W) range, allowing acceleration near 1 g or exhaust velocity near 1000 km/s.

Mass Ratio

RocketCat sez

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 / Me

or

R = (Mpt / Me) + 1

where:

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

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:

Pf = 1 - (1/R)

R = 1 / (1-Pf)

where

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

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 = ev/Ve)

where

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

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 cases the payload mass should be the starting point. The following equation can be used for such calculations:

M = R * ( Mpl / (1 - (Pf * (R-1)) - (Pi * R)) )

  • M = mass of rocket with full propellant tanks, the Wet Mass (kg)
  • R = mass ratio (dimensionless number)
  • Mpl = Payload Mass (kg)
  • Pf = propellant fraction, that is, percent of total rocket mass M that is propellant: 1.0 = 100% , 0.25 = 25%, etc.
  • Pi = inert fraction, that is, percent of total rocket mass M that is Inert Mass: 1.0 = 100% , 0.25 = 25%, etc.

(ed note: Pf 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.

Pi is actually any mass that "scales" with the size of the spacecraft, such as such as engines or structure.

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

by Byron Coffey

Exhaust Velocity

RocketCat sez

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 can do 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 engine and its type determine Exhaust velocity.

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

Ve = Isp * 9.81

where

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

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

WHAT EXACTLY DOES SPECIFIC IMPULSE MEAN?

MaturinTheTurtle

When a physicist tells you something is "specific" he means that quantity is per something else. Specific impulse is impulse per unit weight of 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 time that 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 any engine going for infinite time; that's obvious. But if you have a finite amount 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 everything in 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're different 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.


DrScrubbington

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


MaturinTheTurtle

People 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 g0" instead.

From What exactly does specific impulse mean? comment by MaturinTheTurtle (2015)

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


EXHAUST VELOCITY OF THERMAL TYPE ROCKETS

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

where

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

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

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

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


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

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

(ed note: engineer Rob Davidoff gently points out that I don't know what I am talking about. It is possible 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 H2 in the exhaust.

Rob Davidoff (2018)

EXHAUST VELOCITY OF FUSION ROCKETS

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

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

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

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

Vel = sqrt(2 * Ep)

where

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

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

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.

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

Deuterium-tritium fusion rocket

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

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

Deuterium-helium 3 fusion rocket

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

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

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

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

Deuterium-deuterium fusion rocket

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

We end with either

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

Plugging that into our equation

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

Hydrogen-boron thermonuclear fission rocket

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

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

Watch the Heat

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

Fp = (F * Ve ) / 2

where

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

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

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


As an exceedingly rough approximation:

Ae = (0.5 * Am * Av2) / B

where

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

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


A slightly less rough approximation:

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

where

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

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

Larry Niven postulates something like this in his "Known Space" series, the crystal-zinc tube makes a science-fictional force field which reflects all energy. Niven does not explore the implications of this. However, Niven and Pournelle do explore the implications in THE MOTE IN GOD'S EYE. The 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/m2. This is the theoretical ultimate, for an actual propulsion system it will probably be quite a bit less. For a back of the envelope calculation:

Af = sqrt[(1/El) * (1 / (4 * π))]

Rc = sqrt[H] * Af

where

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

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

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

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

(this equation courtesy of Anthony Jackson and Luke Campbell)

Example

Say your propulsion system has an exhaust velocity of 5.4e6 m/s and a thrust of 2.5e6 N. Now Fp=(F*Ve)/2 so the thrust power is 6.7e12 W. So, 6.7e12 watts divided by 1.0e6 watts per megawatt gives us 6.7e6 megawatts.

Assuming Anthony Jackson's more liberal 5.0 MW/m2, this means Af = 0.126

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.

Equation Derivation

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.

El 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/m2.

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 * Rc2)

where:

ISLA = attenuation due to the inverse square law
π = pi = 3.141592...
Rc = reaction chamber radius (meters)

The heat load on the reaction chamber walls is:

Cl = H / ISLA

where:

H = waste heat (megawatts)
Cl = heat load on chamber wall (MW/m2)

Merging the equations:

Cl = H / (4 * π * Rc2)

Solve for Rc:

Cl = H / (4 * π * Rc2)
4 * π * Rc2 = H / Cl
4 * π * Rc2 = H * (1/Cl)
Rc2 = (H * (1/Cl)) / (4 * π)
Rc2 = H * (1/Cl) * (1 / (4 * π))
Rc = sqrt[H] * sqrt[(1/Cl) * (1 / (4 * π))]

Looking at the last equation, take the right half and swap Cl for El to get:

Af = sqrt[(1/El) * (1 / (4 * π))]

and the entire equation is where we get:

Rc = sqrt[H] * Af

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.

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.

Gordon Woodcock

Delta-V

RocketCat sez

Konstantin Tsiolkovsky is The Man and don't you forget it! Every single time you design a rocket, you will be using his brilliant rocket equation. It is the sine qua non of 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.

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.

Anyway, the equation is:

Δv = Ve * ln[ M / Me ]

Δv = Ve * ln[R]

where

  • Δv = ship's total deltaV capability (m/s)
  • Ve = exhaust velocity of propulsion system (m/s)
  • M = mass of rocket with full propellant tanks (kg)
  • Me = mass of rocket with empty propellant tanks (kg)
  • R = ship's mass ratio
  • ln[x] = natural logarithm of x, the "ln" key on your calculator
Example

Suppose that the Polaris has a 1st 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 = ev/Ve)

where

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

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

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

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

THE TYRANNY OF THE ROCKET EQUATION

      Tyranny is a human trait that we sometimes project onto Nature. This projection is a form of rationalization, perhaps a means to cope with matters that we cannot control. Such is the case when we invent machines to free us from the bounds of Earth, affecting our escape into space. If we want to expand into the solar system, this tyranny must somehow be deposed.

     Rockets are momentum machines. They spew gas out of a nozzle at high velocity causing the nozzle and the rocket attached to it to move in the opposite direction. Isaac Newton correctly defined the mathematics for this exchange of momentum in 1687. Conservation of momentum applied to a rocket was first done by Russian visionary and scientist Konstantin Tsiolkovsky in 1903. All our rockets are governed by Tsiolkovsky’s rocket equation.
     The rocket equation contains three variables. Given any two of these, the third becomes cast in stone. Hope, wishing, or tantrums cannot alter this result. Although a momentum balance, these variables can be cast as energies. They are the energy expenditure against gravity (often called delta V or the change in rocket velocity), the energy available in your rocket propellant (often called exhaust velocity or specific impulse), and the propellant mass fraction (how much propellant you need compared to the total rocket mass).

     The energy expenditure against gravity is specified by where you want to go. For human exploration, there are only a handful places we can realistically consider at this time. The most likely candidates are: from the surface of Earth to Earth orbit, Earth orbit to surface of the Moon, Earth orbit to surface of Mars, Earth orbit to cis-lunar space (the region between the Earth and the Moon, including a variety of locations such as Lagrange points, geostationary orbit, and more). Of course there are permutations to these routes but they are the most likely ones considering our current state of technology.
     In planning an expedition into space, we first must select where we want to go. The energy expenditure against gravity is then specified by the starting and ending points of our journey. As humans, we are powerless to change this number. We simply have to accept its consequences. I like to think of this as the travel cost.

     Next we need to choose the type of rocket propellant, thus specifying the available energy. Currently, all our human rated rocket engines use chemical reactions (combustion of a fuel and oxidizer) to produce the energy. There are limits to the quantity of energy that can be extracted from chemistry and thus bounds placed outside of human control on the energy we can pack into a rocket. Some of the most energetic chemical reactions known are chosen for rocket propulsion (e.g. like hydrogen-oxygen combustion) and thus, the second variable is now specified. Again, we simply have to accept the limit to what chemistry can offer (unless we choose other energy sources, such as nuclear). I like to think of this selection as what you have to pay for the travel cost.

     With these two variables set, the rocket mass fraction is now dictated by the rocket equation. We must build our rocket within this mass fraction or it will not reach its destination. This also applies to existing rockets when new uses are contemplated. There is very little we can do to alter this result. With some clever engineering we might be able to shave a few percentage points off the fraction, but the basic result is set by the gravitational environment of our solar system (choice of where we want to go) and the chemistry of the energetic bonds of our selected chemical components (choice of propellant).
     It is constructive to put a few numbers together to illustrate the grip that simple momentum balance places upon our rockets. Here the approximate cost in energy has been given in terms of velocity (kilometers per second, km/s), a common ploy engineers use to simplify the discussion. These numbers assume ideal conditions such as no losses for atmospheric drag or combustion but are close enough for the sake of this illustration.

DestinationEnergy Cost
(km/s)
Surface of Earth to Earth orbit8
Earth orbit to cis-lunar locations: Lagrange points
3.5
Earth orbit to cis-lunar locations: Low Lunar orbit4.1
Earth orbit to near-Earth asteroids> 4
Earth orbit to surface Moon6
Earth orbit to surface Mars8

     From this simple table, a few conclusions can be drawn. Travelling from the surface of Earth to Earth orbit is one of the most energy intensive steps of going anywhere else. This first step, about 400 kilometers away from Earth, requires half of the total energy needed to go to the surface of Mars ("halfway to anywhere"). Destinations between the Earth and the Moon are only a fraction of that required to simply get into Earth orbit. The cost of this first step is due to the magnitude of Earth’s gravity. And physics dictates that paying a penny less than the full cost will result in Earth repossessing your spacecraft in a not so gentle way. The giant leap for mankind is not the first step on the Moon, but in attaining Earth orbit.

     Listed next are the major categories for our chemical rocket propellants and their energy content used for payment of the gravitational cost of travel. These are selected from propellants with an operational history in manned spacecraft. “Hypergols” are contact-ignited propellants, used in the Lunar Module ascent stage to simplify the engine design and methane-oxygen has not been used in space to date, but is under consideration for future human missions to the Moon and Mars. The first law of thermodynamics was used to convert the energy of combustion into an equivalent exhaust velocity so that these units of payment are consistent with the costs shown above.

PropellantPayment Energy
(km/s)
Solid Rocket3.0
Kerosene-Oxygen3.1
Hypergols3.2
Earth orbit to near-Earth asteroids:3.4
Methane-Oxygen4.5

     Hydrogen-oxygen is the most energetic chemical reaction known for use in a human rated rocket. Chemistry is unable to give us any more. In the 1970’s, an experimental nuclear thermal rocket engine gave an energy equivalent of 8.3 km/s. This engine used a nuclear reactor as the source of energy and hydrogen as the propellant.
     Since the giant leap for mankind is the first step off of Earth, our illustration of the rocket equation uses earth orbit as the destination with the cost of 8 kilometers per second. To pay for this cost, each of the chemical propellants above are used with the rocket equation which results in the following mass fractions (given as percent of the total rocket mass):

PropellantRocket Percent Propellant
for Earth Orbit
Solid Rocket96%
Kerosene-Oxygen94%
Hypergols93%
Methane-Oxygen90%
Hydrogen-Oxygen83%

     These are ideal numbers free from losses due to atmospheric drag, incomplete combustion, and other factors that reduce the efficiencies of a rocket. Such losses make these numbers even worse (moving the mass fraction closer to a rocket being 100% propellant). However, clever engineering constructs such as rocket staging, multiple kinds of propellants (1st stage solids or kerosene, upper stages hydrogen), and gravitational lean (converts radial velocity into tangential) can help compensate. When making a rocket that is near 90% propellant (which means it is only 10% rocket), small gains through engineering are literally worth more than their equivalent weight in gold.

     Real mass fractions from real rockets include the effect of many engineering details. However, these machines at root are the result of the simple application of Tsiolkovsky’s rocket equation. The ideal results presented here are not far removed from actual rockets. The Saturn V rocket on the launch pad was 85% propellant by mass. It had three stages; the first using kerosene-oxygen and the second and third stages using hydrogen-oxygen. The Space Shuttle was also 85% propellant by mass, using a blend of solids and hydrogen-oxygen for the first stage and hydrogen-oxygen for second. The Soyuz rocket is 91% propellant by mass and uses kerosene-oxygen in all of its three stages. There is an advantage to using hydrogen-oxygen as a high performance propellant; however, it is technically more complex. Kerosene offers less performance but gives a simpler, robust, and easier to fabricate rocket. These numbers represent the best that our engineering can do when working against Earth’s gravity and the energy from chemical bonds.

     What are the engineering implications of fabricating a rocket that is 85% propellant and 15% rocket? The rocket must have engines, tanks, and plumbing. It needs a structure, a backbone to support all this and it must survive the highly dynamic environment of launch (there is fire, shake, and force at work.) The rocket must be able to fly in the atmosphere as well as the vacuum of space. Wings are of no use in space; small rocket thrusters are used to control attitude. Then there are people with their pinky flesh and their required life support machinery. Life support equipment is complex, problematic, and heavy. You can’t roll down the windows if the cabin gets a bit stale. If you want to return to Earth (and most crews do), there has to be structure to protect the crew through a fiery entry and then provide a soft landing. Wings are heavy but allow soft landings at well equipped airfields. Parachutes are light, giving a big splash finale. The Soyuz goes thump, roll, roll, roll; aptly described by one of my colleagues as a series of explosions followed by a car wreck. And finally, you want to bring some payload – equipment with which to do something other than just be in space. “Because it is there” (or possibly because it is not there, depending on your definition of a vacuum) is fitting for the first time but subsequent missions need a stronger justification. Missions into space to do meaningful exploration require bringing significant payload.

     Real payload fractions from real rockets are rather disappointing. The Saturn V payload to Earth orbit was about 4% of its total mass at liftoff. The Space Shuttle was only about 1%. Both the Saturn V and Space Shuttle placed about 120 metric tons into Earth orbit. However, the reusable part of the Space Shuttle was 100 metric tons, so its deliverable payload was reduced to about 20 tons.
     It is instructive to compare rocket mass fractions to those of other everyday Earth vehicles. Here, the approximate numbers for propellant (or fuel when air is used as the oxidizer) are given to illustrate the general categories of mass fractions:

VehiclePercent Propellant
(fuel)
Large Ship3%
Pickup Truck3%
Car4%
Locomotive7%
Fighter Jet30%
Cargo Jet40%
Rocket85%

     The percent propellant has huge implications on the ease of fabrication and robustness in achieving the engineering design (and cost). If a vehicle is less than 10% propellant, it is typically made from billets of steel. Changes to its structure are readily done without engineering analysis; you simple weld on another hunk of steel to reinforce the frame according to what your intuition might say. I can easily overload my ¾ ton pickup by a factor of two. It might be moving slowly but it is hauling the load.

     Once the vehicles become airborne, the engineering becomes more serious. Light weight structures made of aluminum, magnesium, titanium, epoxy-graphite composites are the norm. To alter the structure takes significant engineering; one does not simply weld on another chunk to your airframe if you want to live (or drill a hole through some convenient section). These vehicles cannot operate far from their designed limits; overloading an airplane by a factor of two results in disaster. Even though these vehicles are 30 to 40% propellant (60 to 70% structure and payload), there is room for engineering to comfortably operate thus there is a robust, safe, and cost effective aviation industry.

     Rockets at 85% propellant and 15% structure and payload are on the extreme edge of our engineering ability to even fabricate (and to pay for!). They require constant engineering to keep flying. The seemingly smallest modifications require monumental analysis and testing of prototypes in vacuum chambers, shaker tables, and sometimes test launches in desert regions. Typical margins in structural design are 40%. Often, testing and analysis are only taken to 10% above the designed limit. For a Space Shuttle launch, 3 g’s are the designed limit of acceleration. The stack has been certified (meaning tested to the point that we know it will keep working) to 3.3 g’s. This operation has a 10% envelope for error. Imagine driving your car at 60 mph and then drifting to 66 mph, only to have your car self-destruct. This is life riding rockets, compliments of the rocket equation.
     Here are a few other interesting examples from container engineering to further illustrate the extreme nature of rocket design:

Other ContainersPercent Useful Contents
Soda Can94%
Shuttle External Tank96%
Molotov Cocktail52%

     The common soda can, a marvel of mass production, is 94% soda and 6% can by mass. Compare that to the external tank for the Space Shuttle at 96% propellant and thus, 4% structure. The external tank, big enough inside to hold a barn dance, contains cryogenic fluids at 20 degrees above absolute zero (0 Kelvin), pressurized to 60 pounds per square inch, (for a tank this size, such pressure represents a huge amount of stored energy) and can withstand 3gs while pumping out propellant at 1.5 metric tons per second. The level of engineering knowledge behind such a device in our time is every bit as amazing and cutting-edge as the construction of the pyramids was for their time.
     A veteran astronaut who has been to the Moon once told me, “Sitting on top of a rocket is like sitting on top of a Molotov cocktail”. I took his comment to heart by first weighing a bottle of wine, emptying the bottle, and weighing it again. Simple engineering analysis allowed me to estimate and compensate for the density difference between wine and gasoline (which, for this particular vintage, I am sure was not much different). A Molotov cocktail was measured to be 52% propellant. So sitting on top of a rocket is more dangerous than sitting on a bottle of gasoline!

     Another less recognized side effect of the rocket equation is the sensitivity of completing the rocket burn to obtaining your goal. To illustrate this, I will use some numbers from my Shuttle flight, STS 126 in November 2008. Our target velocity at main engine cut off was 7824 m/s (25819 ft/s). If our engines shut down at 7806 m/s (25760 ft/s), only 18 m/s (59 ft/s) shy of the target value, we would make an orbit but not our designated target orbit. We would not be able to rendezvous with space station and would lose our mission objective. Like being two pennies short of a ten dollar purchase, this is only 0.2% less than the price of admission into space. In this case, we do have some options. We could burn our orbital maneuvering propellant and make up this difference. If we were 3% shy of our target, 7596 (25067 ft/s) we would not have sufficient orbital maneuvering propellant and we would not make any orbit. We would be forced into a trans-Atlantic abort, falling back to Earth and landing in Spain. This final 3% of our required velocity comes during the last 8 seconds of our burn. For astronauts and bull riders, 8 seconds is a long time.

     If the radius of our planet were larger, there could be a point at which an Earth escaping rocket could not be built. Let us assume that building a rocket at 96% propellant (4% rocket), currently the limit for just the Shuttle External Tank, is the practical limit for launch vehicle engineering. Let us also choose hydrogen-oxygen, the most energetic chemical propellant known and currently capable of use in a human rated rocket engine. By plugging these numbers into the rocket equation, we can transform the calculated escape velocity into its equivalent planetary radius. That radius would be about 9680 kilometers (Earth is 6670 km). If our planet was 50% larger in diameter, we would not be able to venture into space, at least using rockets for transport.

     Revolting against tyranny is a recurring human trait and perhaps we will figure some way to depose the rocket equation and venture away from our planet in a significant way. I am referring to exploration with continuous human presence with the first step like Antarctic-type bases (which support several thousand people) and eventually leading to colonization, a template comparable to the expansion of western civilization across the globe during the 17th and 18th centuries. To call yourself a sea-faring nation in that time meant that you could set sail on a variety of missions in a number of different types of vessels to a myriad of destinations whenever you wanted. We have a long way to go before anyone can claim to be a space-faring nation.
     The giant leap for mankind is not the first step on the Moon but attaining Earth orbit. If we want to break the tyranny of the rocket equation, new paradigms of operating and new technology will be needed. If we keep to our rockets, they must become as routine, safe, and affordable as airplanes. One of the most rudimentary and basic skills to master is to learn how to use raw materials from sources outside the Earth. Our nearest planetary neighbor, the Moon is close, useful, and interesting. Extracting and producing useful products from the raw materials of the Moon would relieve us from the need to drag everything required in space from the bottom of Earth’s deep gravity well, significantly altering the consequences of the rocket equation more in our favor. The discovery of some new physical principle could break the tyranny and allow Earth escape outside the governance of the rocket paradigm.
     The need for new places to live and resources to use will eventually beckon humanity off this planet. Having access to space removes the lid from the Petri dish of Earth. And we all know what eventually happens if the lid is not removed.

Delta-V Implications

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

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

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

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

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

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


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

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


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


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

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

EXPLAINING OPTIMAL EXHAUST VELOCITY

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) Vₑ²

Then 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 ) Vₑ²

If 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:

½ 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))
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 to

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 Δv

So, 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.


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.

ANTIMATTER ROCKET EQUATION

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 mr » me)

where

Δv = change in vehicle velocity (m/s)
ve = rocket exhaust velocity (m/s)
mi = initial mass of the vehicle (kg)
mf = final mass of the vehicle (kg)
mv = empty mass of the vehicle (kg)
mr = mass of the reaction fluid (kg)
me = mass of the energy source (kg)

The kinetic energy (K.E.) in the expellant at exhaust velocity (ve) comes from converting the fuel's rest-mass energy into thrust with an energy efficiency (ηe):

where

K.E. = kinetic energy (kg·m2/s2)
c = speed of light (3 × 108 m/s)

Solving Eq. (11.14) for the reaction mass (mr), substituting into Eq. (11.13), and solving for the energy source's mass (me) produces

We can find the minimum antimatter required to do a mission with a given Δv. We set the derivative of Eq. (11.15) with respect to the exhaust velocity ve equal to zero, and solving (numerically) for the exhaust velocity:

Substituting Eq. (11.16) into Eq. (11.13), we find that, because the optimal exhaust velocity is proportional to the mission Δv, the vehicle mass ratio is a constant:

The reaction mass (mr) is 3.9 times the vehicle mass (mv), while the antimatter fuel mass is negligible. Amazingly enough, this constant mass ratio is independent of the efficiency (ηe) 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 Δv approaches the speed of light [Cassenti, 1984].

(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 (me). The antimatter needed is just half of this mass. We find it to be a function of the square of the mission velocity (Δv) (essentially the mission energy), the empty vehicle's mass (mv), and the conversion efficiency (ηe):

The 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.
CROSSING THE LINE

“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 Arbkenazy went in too deep and never came out. The Celandine erred 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 Celandine crew 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."

From 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.

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

R = ev/sqrt((2 * Bp * Bε) / mDot)

where

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

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


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

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

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

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

Shifting Gears

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

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

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

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

Example spacecraft and engines include:


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

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

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

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


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

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

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

Fp = ( Ve * F) / 2

F = (Fp * 2) / Ve

Ve = (Fp * 2) / F

mDot = F / Ve

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

F = mDot * Ve

Ve = F / mDot

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

mDot = (Fp *2) / Ve2

where:

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

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 = (Fp * 2) / Ve
F = (5,800,000 * 2) / 294,000
F = 11,600,000 / 294,000
F = 40 Newtons

What 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 watts

Ve = (Fp * 2) / F
Ve = (5,800,000 * 2) / 400
Ve = 11,600,000 / 400
Ve = 29,000 m/s

High and Low Gear

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

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

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

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

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


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


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

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


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

LOW GEAR / HIGH GEAR

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 is low 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 Δv * 0.72.)

From ON TORCHSHIPS comments (2010)
THE MATH BEHIND LUKE CAMPBELL'S EXAMPLE

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) = 1×106 kg (1,000 tons)
Ve (exhaust velocity) = 1×107 m/s (10,000 km/s)
A (instantaneous acceleration) 0.722 m/s2

IMPLIED:
Isp (specific impulse) = Ve / g0 = 1×106 sec
F (thrust) = F = M * A = 722,000 N
Fp (thrust power) = (F * Ve ) / 2 = 3.61×1012 watts (3.61 terawatts)
mDot (propellant mass flow) = F / Ve = 0.0722 kg/s
mDotf (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) = 1.496×1011 m (1 AU)
Trajectory = Brachistochrone (constant acceleration flipping at endpoint)

IMPLIED:
T (transit time) = 2 * sqrt[ D/A ] = 910,389 seconds (10.5 days)
Tb (duration of burn) = T (because brachistochrone) = 910,389 seconds
Mpb (mass of propellant burnt in current burn) = mDot * Tb = 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:
Veg (gearshifted exhaust velocity) = Ve / 4 = 2,500,000 m/s

IMPLIED:
Fg (gearshifted thrust) = (Fp * 2) / Veg = 2,888,000 N (1/4 exhaust velocity, note Fp is still 3.61×1012 watts!)
mDotg (gearshifted propellant mass flow) = Fg / Veg = 1.1552 kg/s (propellant at 16 times the rate)
Ag (gearshifted acceleration) = Fg / M = 2.89 m/s2

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:
D0.1 = D * 0.1 = 1.5×1010 m (1/10 the distance)
D0.8 = D * 0.8 = 1.2×1011 m (8/10 the distance)

IMPLIED:
Ta0.1 (time to accelerate 1/10 distance) = sqrt[(D0.1 * 2) / Ag] = 101,885 seconds (1.2 days)
Td0.1 (time to deccelerate 1/10 distance) = Ta0.1 = 101,885 seconds (1.2 days, takes just as long to slow down to stop as to speed up)
Mpba0.1 (mass of propellant burnt accelerating 1/10 distance) = mDotg * T0.1 = 120,000 kg (120 tons)
Mpbd0.1 (mass of propellant burnt decelerating 1/10 distance) = Mpba0.1 = 120,000 kg (120 tons)
R0.8 (rate of speed during drift) = Ag * Ta0.1 = 294,000 m/s (ship speed at end of acceleration period)
T0.8 (duration of drift) = D0.8 / R0.8 = 408,000 seconds (4.7 days)
Tg (Total gearshifted time) = Ta0.1 + T0.8 + Td0.1 = 611,770 seconds (7 days)
Mpbg (total mass gearshifted propellant burnt) = Mpba0.1 + Mpbd0.1 = 240,000 kg (240 tons)
Mfbg (total mass fuel burnt) = mDotf * (Ta0.1 + Td0.1) = 14,000 kg (14 tons)

Pulsed Mode

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

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

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


Then some rocket designers had an idea.

What if you pulsed the nuclear reaction?

That is, you could run the reaction at a temperature far over the melting point of the reactor, but only for a fraction of a second. Not long enough for the engine to actually start melting. Then you let the engine cool off. After that it is time for the next pulse.

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

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

Engines that use the pulsing trick include:

Camera Flash

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


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

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

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


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

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

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

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

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

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

Engines that use the pulsing trick include:

Propellant-less Rockets

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

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


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

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

Propellant-less rockets include:

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

Rocket Engine Components

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

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

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

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


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

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

Fuel

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

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

Generator

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

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

Propellant

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

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

Energy Conversion System

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

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

Exhaust Deflection System

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

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

Nozzle Thermally HardStandard 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 CooledStandard 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 MagneticRocket 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 AblativeThe 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 AblativeA 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 PlateA 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 Loopa 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 Saila large sail that reflects photons. Can focust the exhaust if desired.
Grey Saila large sail that absorbs or scatters photons or other particles. Typically glows hot due to absorbed power.
E-Saila large sail that acts on charged particles via electrostatic force, does not focus the exhaust.
None

Payload

RocketCat sez

Payload is the load that the spacecraft owner is being paid to 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 e2. If the deltaV is three times the exhaust velocity, the mass ratio has to be 20 or e3.

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

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

Pf = 1 - (1/R)

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

Pe = 1 / R

where

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

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

Pe = 1 / ev/Ve)

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

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

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

SPACESHIP MEDIC

(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 away faster. 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.’
     ‘We can’t give 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! ’

From SPACESHIP MEDIC by Harry Harrison (1970)

Handy Aids

RocketCat sez

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.

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 1st gen Gas Core drive has an exhaust velocity of 35,000 m/s, this is at the 35 km/s tick mark on the Exhaust Velocity scale (thoughtfully labeled "NTR-GAS-Open (H2)"). Now, lay the straightedge between the NTR-GAS-Open tick mark on the Exhaust Velocity scale and the "2" tick mark on the Mass Ratio scale. Note that it crosses the DeltaV scale at about 24 km/s, which is way below the target deltaV of 36,584 m/s.

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

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

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

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

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 Isp 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)

GENERAL ELECTRIC SPACE PROPULSION CALCULATOR

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.

Atomic Rockets notices

This week's featured addition is Pulsed Closed-Cycle Gas Core NTR

This week's featured addition is Gas Core MHD Driven Rotation

This week's featured addition is Human Outer Planet Exploration (VASIMR)

Atomic Rockets

Support Atomic Rockets

Support Atomic Rockets on Patreon