Originally published July 22, 2017

Rocket Efficiency

I was always confused when I would hear people say that a rocket is “more efficient” when it has a higher exhaust velocity. This was because I tended to assume that the word, in the absence of qualifiers, referred to energy efficiency. But what they meant is fuel efficiency. A rocket with higher exhaust velocity gets more miles per gallon — or rather, more miles per hour per gallon — but it doesn’t save energy, it wastes energy. For a car, energy efficiency and fuel efficiency are the same thing, but for a rocket they’re opposites.

Why is this? Because what a rocket needs to get from its propellant is momentum, which is linearly proportional to the exhaust speed, but what it has to put into that exhaust is kinetic energy, which is proportional to the square of the speed. If you double the exhaust speed, you double the momentum of each unit of mass in it, and therefore also double the opposite velocity gained by the rocket. But the energy cost is quadrupled, so the amount of speed gain you get per unit of energy is cut in half. This means that the faster you want to be able to go, the more energy you have to waste for each increment of speed.

If a rocket expels mass at a constant rate, the momentum change per second is also constant. Momentum change per span of time is force: this is the rocket’s thrust. A given energy budget produces the most thrust when the reaction mass expelled is as heavy and slow as possible, not when it’s fast and light. The lighter the reaction mass, the more it takes the lion’s share of kinetic energy, and the less thrust you get for your energy dollar. That’s one reason why heavier solid fuels are often used for that first hard push off the launchpad, when the need for thrust is greatest.

And this is why airplanes are more energy efficient than rockets. By gulping in large volumes of air, they apply thrust by using a reaction mass which is far heavier than what they could carry in tanks. And it’s why cars are even more efficient: by exerting their thrust against the solid ground, they use the entire Earth as reaction mass, and practically all of the kinetic energy goes into the car itself.

As an aside, I can mention that helicopters hover much more easily if they use a large rotor than if they use a smaller one spinning more quickly. A big one grabs more air, which means heavier reaction mass, which means less engine power for the same lift.

The tradeoff for these terrestrial vehicles is that the faster they move, the harder they have to work in order to add any momentum to a reaction medium which is already moving backward. A change of speed starting at 200 miles an hour takes twice the work (literally — the same force over twice the distance) as adding that same increase when starting from 100 mph. Combine that with friction, and you see why cars, planes, trains, boats, and so on all have a limit on their top speed — a velocity which they are incapable of exceeding by their own power, relative to the medium against which they are applying thrust.

Rockets have no top speed, when used in a vacuum. Their limit is on how much they can change speed. The two metrics of a rocket’s performance are its thrust and its total capacity for velocity change, which is commonly called “delta V” (∆v), from the mathematical use of delta to mean “change of”. Both thrust and delta V are proportional to the exhaust velocity. The former is also proportional to the rate at which reaction mass is consumed, but the latter is not: you can have high thrust for a short time or low thrust for a longer time, and it comes out to the same total change of velocity in the end.

When planning a space mission, the supply of delta V is what you have to carefully budget for each necessary stage of the journey. Once in orbit, you need this much to escape Earth’s gravity, that much to transfer to the path of a destination object, this much more to slow down into orbit by it... adding the costs of each step of the journey tells you how capable a rocket you need to build.

Rocket fuels and engines are often ranked by their “specific impulse”. It’s the amount of momentum gained per unit of fuel. For other vehicles such as airplanes it’s a complicated empirical measurement, but for rockets it simply equals the exhaust velocity. Values range from less than one kilometer per second for black powder to more than five km/s, or mach fifteen, for the liveliest of all fuel reactions, lithium and fluorine. Most of the popular and practical chemical fuels can realistically produce speeds a bit over three km/s (mach nine). Liquid hydrogen and oxygen can go above four, but much of that advantage is negated by the huge bulk of the tanks that are required to contain the hydrogen. The effective exhaust speed in practice is always lower than the raw molecular speeds coming out of the reaction, as it’s impossible to herd all of the hot gas into moving perfectly in parallel. The standard rocket nozzle — a tapered constriction that smoothly narrows and then widens into a roughly parabolic bell — does an amazingly good job of turning extreme heat and pressure into hypersonic speed, but the gases still emerge with a lot of remaining disordered molecular motion.

The reason rocket builders want high exhaust velocity is to increase their delta V budget. Here is a map of the solar system, showing the delta V costs to move between various locations, in kilometers per second (click to embiggen):

delta-V map of the Solar System
For example, let’s say you want to land on the Moon. You start from Earth (bottom left), and use at least 9.4 km/s to get from the surface to orbit. Then you add another 2.44 to raise one axis of your orbit to geostationary altitude, but you don’t pause there, and expend another 0.68 to reach an orbit where the high end falls back to the Moon instead of the Earth. You coast a while, then at the right time hit the brakes by applying 0.14 in reverse to make sure you’re in lunar orbit. Another 0.68 of braking lowers that orbit close to the Moon’s surface, and slowing to a landing requires at least another 1.73 km/s of retro-thrust. That all adds up to 15.07 kilometers per second of velocity change for which you need to have fuel. And it doesn’t matter what alternative paths you take: no possible trajectory can get you there for less than fifteen km/s. And note that this minimum assumes some harsh and risky conditions at some points, particularly the launch and the landing, where the most efficient path is to use high G forces close to the ground.

To come back, you need another 1.73 + 0.68 + 0.14 to get back to an Earth transfer orbit, but then you can cheat: by slamming into Earth’s atmosphere with a heat shield, you can kill plenty of speed without using fuel. That’s the meaning of the red arrows pointing down. These arrows appear near any body which has an atmosphere.

What this map doesn’t indicate is the durations of these journeys — the time you have to spend coasting. These stretches can be very prolonged if minimal delta V is used, like eight months to reach Mars, or years to reach Saturn. Using additional velocity change beyond the minimum can greatly reduce the time necessary for the trip. If you’re going on a trip hundreds of millions of kilometers long, an extra five or ten kilometers per second will make a big difference in how long you’ll have to coast in the middle.

With such high velocity change requirements for even the slowest journeys to some destinations, and so much more needed to make the durations of the longer trips tolerable, it’s clear that making the solar system accessible to people requires increasing the delta V that our craft are capable of. How might we do that?

The straightforward way is to increase the energy yield of the fuel. The exhaust velocity is proportional to the square root of the energy content per kilogram of propellant. It is also inversely proportional to the mass of the individual exhaust molecules. This is one reason why the combination of hydrogen and oxygen yields higher exhaust velocity than other common chemical fuels: because of the low molecular weight of the water which is produced by the combustion. In fact, hydrogen-burning rockets are typically designed to run fuel-rich, including unburned hydrogen in the exhaust just to lower the average molecular weight. This reduces the energy yield, but the increase in velocity due to lowered molecular weight makes the tradeoff worthwhile.  Unfortunately, there is not really any way to make chemical fuels such as hydrogen better than they already are. The lithium-fluorine reaction is superior on paper, but it would be extremely hazardous and toxic to use, and would offer only a slim improvement in performance over fuels which are not only much safer, but also very inexpensive. Nobody could make a good case for trying it.

There is a way to increase delta-V without improving the fuel, but it’s costly. Since a rocket carries its reaction mass with it, and discards it during operation, the craft gets lighter as it runs. This means that as the tanks run down, the same thrust will produce higher acceleration, as it’s working against less inertia. Higher acceleration means greater change of velocity. You can get a big bonus in delta V if you make the rocket’s payload, and any dead weight accompanying it, very light in relation to the initial fuel supply. This is why rockets use stages: the more dead weight you can shed, the faster you can accelerate the important part that remains. If you start by burning two hundred tons of fuel, you don’t want the remaining part to waste its energy on hauling empty 200-ton-sized tanks.

In an idealized rocket, the velocity change it can achieve is given by the Tsiolkovsky equation ∆v = log(m0 ∕ mf) ve, where m0 is the rocket’s initial mass fully fueled, mf is the remaining mass after the fuel is consumed, log is the natural logarithm function, and ve is the exhaust velocity, averaged over all the gases that leave the vehicle. This equation means that you can reach a speed equal to your exhaust velocity with a mass ratio equal to Euler’s constant e (2.718... — the base of natural logarithms), meaning you have to load 1.718 tons of fuel for each ton of dry weight. And it means you can double that speed with a ratio of e² (7.389...), triple speed with e³ (20.085...), and so on. Ratios higher than 20 are quite commonplace in real rockets.

Note that when stages are used, you have to apply the equation separately to each one. Let’s take a real world example: a SpaceX Falcon 9. It has two stages and is designed for a low-orbit payload of up to 17.4 metric tons in the current Block 3 “Full Thrust” version (though in practice the limit is normally lower, as maxing it out prevents recovery of the first stage for reuse). They plan to increase that capacity to 22.8 later this year, but let’s use the lower figure. Applying the formula naively would take the mass ratio as about 33, and the answer you get with that isn’t terribly far off, so you could use it for a rough estimate. But for accuracy you need to know the weights of each stage.

The best estimates I can find say that the first stage weighs 437 metric tons full and 26 tons empty, and the second stage is 111 tons full and just 4 empty. The second stage is part of the first stage’s payload, so for the first stage, the final mass is 26+111+17 = 154 tons, and the initial mass is 411 tons more, or 565. The mass ratio is therefore 3.67, and the natural log of that is 1.30, so that’s what we multiply our exhaust velocity by. That exhaust velocity is about 3.05 km/s, so in theory (if there were no gravity or atmosphere to struggle against) the first stage can get the speed up to 3.9 km/s before separating. Once the first stage detaches, the second stage has an initial mass of 128 tons and a final mass with payload of 21.4, for a ratio of 5.98, which has 1.79 as its logarithm. The exhaust speed improves to 3.41 km/s thanks to a larger bell on the second stage engine, yielding a velocity change of 6.1 km/s. Add the two stages and you get 10.0, which after losses to gravity and the atmosphere is just enough to go into orbit.

If you lighten the payload the mass ratio improves considerably, allowing the second stage to reach higher orbits. If the payload is only two tons, the second stage’s mass ratio is now 21, yielding an extra 4.3 km/s of delta V. However, if you want to put something heavier into a very high orbit, or to escape the Earth, the best way to do it is to use some of your 17 ton capacity to add a third stage. If such a stage starts with 17 tons and ends with 3, that’s 5.9 km/s available after reaching orbit — enough to go to Mars.

That fully loaded Falcon 9, sitting on the pad, has gravity exerting a force of 5.6 meganewtons holding it down, because Earth’s surface gravity is 9.8 newtons per kilogram. The engine thrust at liftoff is 6.8 meganewtons... but that’s because the atmospheric pressure reduces the exhaust velocity to under 2.8 kilometers per second. Once it reaches an altitude with thin enough air for it to produce 3.05 km/s, the thrust increases to 7.4 meganewtons. This figure means that fuel is being expended at a rate of 2.4 metric tons per second. The kinetic energy of each kilogram at 3.05 km/s is 4.8 megajoules, meaning that the first stage’s power output is ...holy crap, 13.4 gigawatts! If you compare that to the output of an electrical power plant, that would rank it among the biggest ever built... but unlike those plants, it can only run at that rate for less than three minutes.

The second stage produces only one eighth as much thrust, and lasts six and a half minutes. The first stage produced an acceleration of around four G at its lightest; the second stage starts out with less than one G (which is mostly applied horizontally), but also climbs to around four G as it empties, depending on the payload.

So that’s a typical example of cutting edge liquid-fueled rocketry: burn 518 tons of hydrocarbons and liquid oxygen to get just 17.4 tons to low orbit, or 3.8 tons to escape Earth. People would like to do better.

Let’s say we could build a hypothetical rocket which could spit out its fuel at a speed of six kilometers per second. One way this might be done is to build an engine which pushes a nuclear fission reactor to the edge of meltdown, and then uses that to heat hydrogen gas. Because hydrogen gas is so light, the exhaust velocities are faster than those of any chemical rocket even if the temperature is no higher... and the fuel is capable of any higher temperature you want, if you have a way to keep the engine in one piece. So if you decide not to care about the health and environmental risks of such a monstrosity, it can achieve these exhaust velocities in vacuum. A riskier design could get eight km/s, even. Prototypes of such engines were actually built, back in the space race era. NASA was at one point saying that its NERVA engine was ready to be built into a Mars rocket.

Let’s say that your output power limit was still 13.4 gigawatts (a much higher figure than NASA achieved, I will note). What kind of rocket would you get? At 6 km/s exhaust velocity, the rate of fuel consumption would be 740 kilograms per second instead of 2400 as in the Falcon, and the thrust would be 4.4 meganewtons: not enough to lift off from Earth, especially because these nuclear-thermal rockets lose more than half of their ve when operated in the atmosphere. But out in space, things get better... except not really, because your reactor adds many tons to the ship’s dry mass, losing in mass ratio much of what you gained in exhaust velocity. It might pay off if the entire vessel is built on a large enough scale, but this unknown extra bulk means that I can’t estimate what the delta-V would be. I doubt the improvement would be major. No wonder that, despite very active research in the past, and some who to this day advocate bringing the idea back, nobody ever actually flew one of these. I hope they never do.

Let’s try a gentler option: an ion engine. This uses electricity to ionize cold gas (usually xenon) and accelerate it electrostatically or magnetically, producing exhaust velocities up to 90 km/s. Such engines are in use today on many probes and satellites. They scale down very nicely for small applications. Could we use them for a big one? This would require a large source of electric power, and there is presently just no possible way to generate 13.4 gigawatts of juice on a lightweight spacecraft. Let’s try for 13.4 megawatts — and amount that’s within reach for big solar arrays, or portable nuclear generators. (The Russians say they are currently working on a spaceship with nuclear-powered thrust, presumably via ion engines, which will be able to make repeated trips to destinations such as Mars.) This 13.4 megawatt figure is still far beyond today’s ion thrusters, which barely crack a tenth of a megawatt — even the Russian proposal is only one megawatt — but the engines themselves should not be difficult to scale up. And speaking of scaling up, xenon will be way too rare and pricey to use in large tonnages, but krypton makes a pretty good substitute (SpaceX uses it in their Starlink swarm), and fortunately there are ion engine variants which can work on highly abundant gases such as argon or nitrogen or ammonia or even hydrogen, and others that work on vaporized metals. One new type coming to the market uses iodine.

(There are also intermediate steps which have exhaust velocities between those of nuclear and ion engines, such as “arcjets” which heat gases by shooting electric sparks through them, but except for rare niche uses these have no advantage over ion engines.)

Taking the upper end of the exhaust velocity range, and adding 20% more power consumption for the ionization step, with 13.4 megawatts we get a gas consumption rate of just 2.7 grams per second, and a thrust of 245 newtons. If the power supply weighed fifty tons, which it might need to in order to yield all those megawatts, then even with tanks near empty it could only manage to pull an acceleration of around 1/2000 of a G. It would be helpless against even the faintest wisp of atmosphere, as indeed is the case with the little engines used today. But with tanks full... well, with a mass ratio of only four you’d have a spaceship capable of over 130 km/s of delta V, an amount with which you could visit every interesting destination in the solar system without refueling, though it would take many years.

If we magically grant this ship 13.4 gigawatts with no extra weight, it could make this grand tour more quickly, but you’d have to be careful not to waste fuel, as you could easily go through your 130 km/s in a day or two. To really get around the solar system faster, you’d want to bump the mass ratio up again, adding tanks until you had like 300 km/s of delta V.

How many megawatts per ton might be achievable? With solar panels, it’s now becoming possible to get the mass of the cells below fifty grams for a square meter, which might yield up to 300 watts. That means a 13.4 megawatt array might weigh a lot less than the fifty tons mentioned above. Make it flimsy enough and the entire array, which would be well over 200 meters across, might weigh under three tons. If we actually achieved such an extreme power-to-weight ratio, the ceiling on our acceleration might exceed 1/100 G... but I bet the wires alone would weigh more than that, let alone the structural scaffolding.

If instead of solar cells we use a fission reactor, similar to the powerplant of a nuclear submarine or aircraft carrier, we might get 13.4 megawatts out of something like 25 tons of equipment. That would only yield more like 1/1000 of a G as a ceiling, but it would be usable far from the sun. The delta V would be the same as with solar. NASA is currently working on a small fission power plant: they figure it’ll weigh a bit under a quarter ton and supply a hundredth of a megawatt. A bigger model would improve the ratio, but there would be two disadvantages if used in a crewed spacecraft: one is the risk of cancer and three-headed grandchildren if anything leaks, and the other is that people would make fun of you for flying a steam-powered spaceship, because as yet we do not have any better way to extract large amounts of electricity from fission power than to boil water with it and spin turbines with the steam.

Is there any way to use nuclear power to get exhaust speeds comparable to an ion engine, without the power limitations? Maybe there is, and it’s absolutely nuts. It would be so dirty that it should probably never be used anywhere in Earth’s magnetic field, let alone anywhere near the atmosphere. And it would be so challenging to build that to get it to work without melting itself or exploding its fuel tank would be an epic engineering achievement. But if it worked, it would have an exhaust velocity approaching that of an ion engine, and the thrust of a large chemical rocket, at the same time. And it would have to be built on a large scale: imagine something the size of a Starship, or even the size of a Saturn V, but with far more power output.

How is this possible? You start with enriched uranium or plutonium, and you combine that into salts to make it water soluble. You then concentrate those salts into water until they’re thick enough that it will go critical and explode if you don’t store it in tanks full of neutron-absorbing baffles. Then all you have to do is squirt this satanic solution into a reaction chamber where it’s got some neutron enhancement instead of neutron absorbtion. The result is a nuclear explosion that instantly heats the water to hundreds of thousands of degrees... and if you pump in more juice fast enough, the explosion becomes continuous without the chain reaction creeping back up the fuel pipe. Or so you hope. It’s called a Nuclear Salt Water Rocket.

It should go without saying that such an engine would be insanely dangerous, as would its fuel at every stage of production and transportation. It would also be extremely challenging to make a reaction chamber that could withstand the heat, even with layers of cold fluid doing their best to protect it from direct exposure. But maybe someday, such a thing could be built. If so, everyone around it would be acutely aware of its potential for misuse as a terror weapon, which might make it extremely unpopular. Even the old Orion concept — a ship that propels itself with repeated detonations of full-sized nuclear bombs — looks safe and sane compared to this.

There are many variations on this concept — many ideas for how to get thrust out of miniature nuclear explosions. Some use dry powder or pellets of nuclear fuel, some use pulsed explosions instead of a continuous blast, some enrich the mix with deuterium and tritium for extra yield, and some use magnetic fields to guide the blast and protect the engine’s solid surfaces. Some of these might be more feasible and less terrifying than the salt-water engine... but they would still be difficult and dangerous and dirty. I will not try to calculate any performance values for these, as too many figures are unknown.

But, speaking of deuterium and tritium, what if we upgraded our power source from fission to fusion? At high enough temperatures, the ²H or ³H isotopes of hydrogen, known as deuterium and tritium respectively, can be merged together to produce helium, yielding enormous energy. If such a reaction can be made self-sustaining, it produces plasma with nuclei having speeds of at least 600 km/s, which is way beyond current ion engines. And improved designs might mean that the exhaust from a fusion motor could eventually be a lot hotter than that. A more advanced engine might confine all the ions except the hottest few percent, and if it becomes possible to use the helium nuclei coming directly out of the moment of fusion as the exhaust... the speed of those could be over 15,000 km/s, depending on which nuclear reactions are used. And power won’t be a problem: should we ever manage to build a fusion reactor, an output of 13.4 gigawatts is not at all unreasonable. In fact, it may be that one reason we have had so much difficulty getting a fusion reactor to work successfully here on Earth is that we’ve been thinking too small; such reactors might be more stable and efficient if built at an enormous size. And yet such an engine would have almost none of the danger of the salt-water monstrosity, as the fuel is extremely difficult to detonate outside of the engine. (Even detonating it inside the engine, where it’s supposed to happen, is very difficult, which is why we can’t presume that this kind of engine will necessarily ever be feasible.) And there is no need for the hot plasma to touch a solid nozzle. It’s all handled by magnetic fields. This means that if you could get such an engine to work at all, it might be robust and reliable, with little maintenance needed. All in all, it’s by far the most attractive option for nuclear propulsion, on paper.

Using the 600 km/s exhaust speed, the 13.4 gigawatt power level gives you a fuel consumption rate of just 75 grams per second, for a thrust of 45 kilonewtons. If the ship weighed 100 metric tons empty (I have no idea whether that weight is a reasonable guess or not, but it might well be very overoptimistic), that would pull less than 1/20 G, but with a mass ratio of four you’d have 830 km/s of delta V: enough to go to Saturn (well over a billion kilometers) in about ten weeks. And if you found a way to really ramp it up, you could get a comfortable one G of acceleration by expending 1.65 kg of fuel per second, with an output power of 295 gigawatts. That would cut the time to Saturn down to six or seven weeks. And if those figures still aren’t quick enough, just add more fuel tanks and bump up the mass ratio.

If you want to envision what a mature interplanetary civilization might look like in five hundred years, that sort of fusion rocket is probably our best guess for how they’d get around. Even for them, travel around the outer solar system might remain unavoidably time-consuming and expensive.

What if fusion is too difficult, and the salt water option and its ilk remain unusable? In that case, we may be stuck with ion engines, or some other electromagnetic system of similar efficiency. With these devices, building the engine and supplying it with propellant is the easy part; the hard part is providing the electric power it needs. In the inner solar system, we could get along with solar arrays, and improve their power-to-weight ratio to some degree, but in the outer system we might still have to lug around a fission power plant. It’s hard to conceive of any power supply which is better than solar or fission, yet more attainable than fusion.

But if we did solve the power supply problem, that opens up an attractive possibility. It’s being worked on now. It’s called VASIMR, for Variable Specific Impulse Magnetoplasma Rocket. The basic idea is simply to heat your exhaust gases in a microwave oven. It can be tuned for high thrust or for high delta V just by varying the oven temperature. They expect to be able to make plasma temperatures of a million degrees Kelvin or so, which would mean an exhaust velocity of up to 50 km/s, and more may be attainable in the future. So it’s comparable to an ion engine in efficiency, but can be scaled up to much higher levels of brute power, and can also produce bursts of extra thrust by using more propellant at lower temperatures. It might even be usable in the upper atmosphere, sucking in air and heating it in bulk, if you have the power for it. The propellant it’s mainly been developed for is argon, but in a pinch it might be able to use just about anything.

Ion and VASIMR engines might have a lot of value in the inner solar system, where solar power and argon are inexpensive. Even for such a near-term mission as schlepping cargo from low Earth orbit to the Lunar Orbital Platform-Gateway, ion-powered tugs might move more stuff at less cost than the conventional rockets which NASA is planning to use.

VASIMR may be more exciting than most of its competitors, but without tons of power it is just one more of a wide variety of potential designs for electromagnetic rocket engines. There are endless variations of ion or plasma thruster designs, and they all share roughly similar capabilities and constraints. Some offer the possibility of raising the exhaust velocity into the hundreds of kilometers per second, but that just makes them even more power hungry.  They are all far better than chemical fuel in delta V, but far short of fusion or fission power in thrust. But even if a thousandth of a gee is all they ever deliver, one or another of these ion or plasma drives will probably be the dominant form of interplanetary propulsion for much of the next century, unless fusion rocketry is attained more easily than most expect. Personally, I’m optimistic: I think it’s possible that a fusion rocket might turn out to be easier than building a fusion power plant on the ground, just as the first successful harnessing of a fission reactor was not in a powerhouse, but in the engine room of the USS Nautilus.

If fusion remains difficult, there might be a way to get extremely high exhaust velocity from fission reactions without insane risk. When radioactive atoms fission, the fragments fly out at enormous speed, but they don’t usually get far because they’re inside solid material. But what if the fuel were in the form of a fine dust, or bonded to a microscopically thin membrane, or to an aerogel? Many of the nuclear fragments would escape at high speed. Being ions, they could be funnelled out an exhaust bell with electromagnetic fields. The exhaust velocity would be comparable to that from fusion, though the power would be miniscule by comparison. It would be difficult to coax more than a wisp of thrust from such an engine, but its power to weight ratio might be far better than that of solar cells or conventional reactors. Such an engine might be useful in the outer solar system. And we could get even higher exhaust velocity, though even lower power, by using isotopes that undergo ordinary alpha decay. This is the approach proposed for initial experiments.

60 or 600 km/s, or even 6000, is useless for interstellar travel. What would it take to visit other solar systems in reasonable time? You’d need relativistic exhaust velocities. If you accelerated ions or something to 150,000 km/s, half of lightspeed, then your ship could also reach a relativistic speed. The endpoint of that progression would be to use light itself as reaction mass: to propel yourself forward by shining a lamp out the back. This would be the ultimate in fuel efficiency, and in energy inefficiency.

Let’s imagine a ship powered by the total conversion of matter and antimatter into energy. Its exhaust is the gamma radiation formed by the mutual annihilation of particles and antiparticles, somehow forced into a tight beam. Let’s say the craft has a mass ratio of ten: a hundred tons of ship with 900 tons of matter and antimatter to burn. Let’s say it can pull a hundredth of a G just before it runs out, so the thrust is 10 kilonewtons.

The power needed for a light beam exerting 10 kN of force is three trillion watts. That’s enough to power an entire first-world nation. To put it another way, that’s a rate of three Hiroshima bombs per minute. And like those bombs, that exhaust could be extremely dangerous and destructive for anything in its path.

The rate of matter/antimatter consumption would be 33 milligrams per second, or 2.9 kg per day. At that rate it would take 850 years to empty the tanks!

We need to move things along more quickly. Let’s consume 3.3 grams per second, producing one meganewton of thrust, and one G of acceleration when nearly empty. Power is now 300 terawatts, which is about fifteen times the combined activity of all present-day civilization. What speed can this ship now reach?

The formulae we used previously are now incorrect due to the relativistic effects. The equation for velocity change with light as exhaust is ∆v = tanh(log(m0 ∕ mf)) c, where c is lightspeed and tanh is the hyperbolic tangent function, which approaches a value of 1 as its input rises toward infinity. For our example with a mass ratio of 10, the result is 98% of lightspeed! At that pace the ship’s onboard rate of time would be dilated by a factor of five, significantly shortening the subjective length of the journey once it gets up to speed.

But we’ll probably want to stop at the other end. To do that we’ll need to use half of our ∆v to speed up and half to slow back down. Each half then uses a mass ratio of √10, or 3.16, so our peak velocity in midflight is 81% of lightspeed. The time dilation factor for that is only 1.73, so it won’t shorten the subjective duration by all that much. Using fuel at 3.3 grams per second (those are shipboard seconds), it would take 6.6 subjective years to speed up and 2.1 to slow down again. That’s an awfully long time to spend cooped up in the confines of a 100 ton spacecraft, but it’s not inconceivable.

How much distance would be covered during the acceleration? I haven’t got a formula for that, but I’ll try to fake it... something in the ballpark of five lightyears covered during the speedup phase, and one and a half during slowdown. So to reach the nearby sunlike star 82 G Eridani, which appears to have multiple rocky planets, you’d have to coast for about 13 lightyears before braking, which would add about 9 subjective years of thumb twiddling, for a total of about 17 years onboard for the whole trip. It’s doable, on paper... congratulations, we have achieved interstellar travel.

But the waste! The energy packed into 900 tons of matter/antimatter could benefit the lives of billions of people for hundreds of years. That’s something like eighty times the total energy that human industry has ever used or produced across the entire history of civilization. Is there a better way?

Maybe we can make some kind of spacecraft which acts more like a plane or a car than like a rocket. In the Earth’s atmosphere, at least, we might literally be able to fly like a plane. Why not do the liftoff with jet engines instead of rockets? Some small launch systems are in fact using this approach, by dropping the rocket from an airplane instead of making it lift itself off the ground: examples include Orbital ATK’s Pegasus launcher and Virgin’s LauncherOne, as well as their suborbital tourist rides. It’s certainly true that the most wasteful part of a rocket’s flight is that first minute after launch, when it’s moving slowest. And as mentioned near the beginning of this article, a given amount of thrust requires much less fuel in a jet engine than in a rocket.

The problem with an airplane is that you’re literally only helping that first minute — maybe just the first thirty seconds. Because at the one minute point, an orbital rocket is moving about as fast as an airliner, but vertically instead of horizontally, which airliners can’t do. At a minute and a half it’s getting to the speed of a fighter jet, and before two minutes it’s exceeding the speed of an SR-71 Blackbird, the fastest thing we’ve ever pushed through the air with jets, and we’re about out of air to feed a jet with. We’re way beyond the capabilities of existing airplane engines, and we’ve still gained only one km/s of speed, less than one eighth of the way to orbit. Unless the plane itself is rocket-powered (which is a reasonable way to make a first stage reusable), the economic improvements of lifting rockets with planes are marginal.

But it sure would be nice to get one’s oxidizer and reaction mass for free from the air instead of carrying them onboard, and get some free lift from wings. Is there a kind of jet which can do rocketlike speeds? Perhaps there is: a supersonic combustion ramjet, or scramjet for short. Current experiments (which are proving difficult, with many failures) are getting these things up to about two km/s, and there are hopes to eventually reach around five km/s, which would get you more than halfway to orbit. Rocket launches might then take the form of extremely rapid horizontal flight in the fringes of the upper atmosphere, before cutting loose a pure rocket to climb to orbit, while the scramjet craft returns to the ground. But this scheme may be unworkable due to the high weight and low thrust of foreseeable scramjets, and the air drag at those speeds may negate any fuel savings.

But there is a company trying to build something like this right now. A company called Reaction Engines Ltd say they’ve cracked the problem and built an engine which can function as a superfast jet in the atmosphere, and then as a rocket by adding liquid oxygen. Their motor is called SABRE, for Synergistic Air-Breathing Rocket Engine. Their trick is that they rapidly refrigerate the incoming air, and then pump it into a rocket-style combustion chamber. Apparently this gives them much higher thrust than a scramjet, and it has no trouble starting up when stationary at sea level. If this really works, they will have achieved an orbital spaceplane which needs no staging, and access to the Space Station and so on should get much cheaper indeed. But it sounds like the speeds it would reach by air-breathing would be well under 2 km/s, so it might still need a buttload of conventional rocket fuel.

You can’t drive a car in space... but how about an elevator car? If we put a heavy asteroid into geosynchronous orbit, and lowered a super-strong cable from there to the Earth’s surface, we could raise a ton of payload to that orbit with only about 60 billion joules of energy, as opposed to the 600 billion or so that a rocket would expend, because you put the energy into lifting only the payload, not into accelerating a bunch of hot exhaust.

What are we asking of this super-strong cable? Well, the bottom end has to be strong enough to lift the payload, but has to be incredibly light so as not to break the top end, which has to be incredibly strong. The whole thing would be incredibly heavy because of its enormous length. Can real materials pull it off? Maybe. Today’s plastic-impregnated carbon composites aren’t there, but if we could make something closer to the strength of pure graphene on a large scale, it might just be feasible.

There’s a material under development which consists of extra-large carbon nanotubes with intentional flaws and pores and wrinkles. It’s thought that the stuff might be woven into a sort of fabric, and people are talking about actually making clothes out of it. It might be rather stretchy to make an elevator cable out of, but in theory, you could hang a one ton weight off a thread of this stuff which weighs only 400 grams per kilometer. Of course, you can’t get away with building a permanent structure out of a material that’s at the limit of its strength, so to make even a faint pretense of safety margin, we’d better start with a thickness of one kilogram per kilometer per ton capacity. The first thousand kilometers would be able to hoist your payload well beyond any atmosphere, but that strand would add another ton of weight to the ton of payload, meaning that the upper parts now have to be twice as thick as the lower part, which in turn adds yet further weight. The next thousand km would have to start out at around triple thickness, and it gets worse from there... but to balance that out, the force of gravity starts to ease off as you go up. At 2600 km altitude, gravity is half what it is on the surface, and at 6400 km it’s a quarter. Plus, as you get higher you’re also gaining partial orbital speed. Once you’re a quarter of the way to synchronous orbit, about 10,000 km up, the strain on the cable is increasing much more slowly. You can now make the cable thick enough to support all the weight below, without adding too much further weight penalty to the part of the cable that’s still above you.

Nevertheless, the tension on the cable still increases as you go up, meaning that the outer part has to be very heavy and strong indeed, which suggests that the whole thing might be as massive as the Great Wall of China. It would be quite an undertaking to build... and if it broke and fell, the scale of catastrophe on the ground would be huge. The lower part of it might whip-crack across an entire continent before the remaining piece achieves a halfway stable lower orbit.

There are a whole list of other ways we might supplant rockets for the task of lifting loads off of Earth’s surface. At the most easily achievable end, we should be able to build an air-breathing rocket: the British “Skylon” project aims to use the SABRE engine to build a plane that can go from runway to orbit to runway without dropping any stages. Or, more ambitiously, we could power a rocket by blasting its underside with a big laser from the ground, instead of having to store all of its lift energy in the fuel tanks. Or we could try to build huge mega-structures that can reach above the atmosphere; they’d be almost inconceivably enormous and difficult to make, but they’d still be less extreme than a geosynchronous elevator. Some variations of this idea include:

The Skyhook might be the easiest to build, but if scaled up to a large size it would be very dangerous to have flying overhead. If it’s big, any failure to keep its orbit maintained could threaten large areas of land near the equator, and the threat would be quite real as each use would tend to pull its orbit downward. Fortunately, if we keep it on the small side and ask it to handle only reasonably light payloads, it might actually be doable with foreseeable technology, with reasonable risks. I may do a separate article on that topic later.

These all might save a lot of energy or money in launching stuff away from Earth, or other planets, but that isn’t really the question of interest here. What we’re here to address is the travel through the open space between destinations — the task for which some form of rocket still looks like just about the only option.

Could we apply any sort of elevator-like process to interplanetary travel? If someone were to invent a “tractor beam”, as in Star Trek — a sort of directional magnet that pulls distant objects toward you, or pushes them away — then you could. You could push and pull against various planets and other heavy bodies, and use them as reaction mass, performing a sort of planetary parkour. But there does not appear to be any loophole in physics that would allow such a beam to exist. We might as well imagine warp drive. And I can’t think of any other way you could use the planets to react against while not in contact with them, aside from the kind of “gravitational slingshot” moves we are using already, where a craft picks up some of a planet’s orbital speed by making a flyby over its evening side, or slows itself down by passing over the morning side.

But there is an idea which has been proposed on the interstellar scale — one that would resemble an airplane more than a train or an elevator car or a parkourist. It’s called the Bussard Ramjet. The idea is to make a ship which uses an enormous magnetic funnel to scoop up the extremely thin interstellar hydrogen gas, and pack it into a fusion engine, giving you a fuel supply that you pick up as you go, so it never runs out. Some science fiction types have claimed that such a rocket would have no top speed, but since it is flying through a resistive medium to pick up its reaction mass, this is clearly false.

Could this work? Hard to say, but I doubt it. I am very dubious that this intangible funnel could actually be made. If it could, it would only work on ionized hydrogen, not neutral gas. There is good news and bad news about that, as most of the very tenuous gas between solar systems is neutral hydrogen, but solar systems tend to contain mostly the ionized kind because of solar wind. The boundary between the two is called the “heliopause”, and we managed to locate ours when Voyager 1 passed through it in 2012, at a distance from the sun four times that of Neptune. Within that zone, the scoop part of a Bussard ramjet may be possible, and if you hit another such zone at the other end of an interstellar flight, maybe you could slow down again.

But there’s another problem: to get any good out of the hydrogen you gather, you’d mostly have to fuse the plain common ¹H isotope, rather than deuterium or tritium. On the one hand, this is what the sun does, so you know it yields plenty of energy... but on the other hand, there’s a reason why the sun lasts for billions of years instead of instantly going up in a bang. Fusing plain hydrogen is an extremely slow process. Unlike a deuterium or tritium or ³He reaction, it can’t just reshuffle some protons and neutrons into new nuclear arrangements: it has to make its own neutrons by a process which absorbs an electron and emits a neutrino, which is something that nature prefers to do in the opposite direction (neutron decay), and even then it’s reluctant. Some analyses of the Bussard proposal have concluded that the fusion reaction could not give back even a millionth of the energy put into compressing the fuel. So unless someone comes up with a very surprising trick, I think that this idea will probably never be viable.

So trains and elevators and jets probably won’t work for the longer distances... but there is still one type of craft that might: a sailboat. I just mentioned solar wind, didn’t I? If you make a big enough and thin enough sail, you can propel a spaceship with it, and furthermore, you can use some of the tacking techniques of a real sailboat to maneuver in various directions. It wouldn’t be able to do anything that requires a keel, but in compensation, it could move inward toward the sun just by killing some of its orbital velocity. What is the solar wind? It consists of hot ions moving away from the sun at speeds of from 250 to 750 kilometers per second. Nobody knows how the sun’s lower atmosphere is managing to produce such speeds, but it probably has something to do with the sun’s intense magnetic fields.

Even better, the sail can pick up thrust from the sun’s light as well as from its hot gases, and in fact the thrust available from light is the dominant component. You get the maximum transfer of momentum if the sail has a reflective mirror finish. Going straight out from the sun, and reflecting the light straight back to it, you get twice the momentum that you would if you just absorbed the light. At locations near Earth, the force exerted by sunlight on a perfect mirror is nine newtons per square kilometer. Such a sail would need a great many square kilometers to be effective, but this is quite feasible. All you need is a very thin sheet of plastic with metal deposited on it. Established plans tend to assume a sheet of mylar or kapton at a thickness of around two microns. It would be very fragile, but plenty sturdy enough to catch sunlight. More recent plans are turning toward graphene composite substances for greater strength, which would allow the layer to be thinner.

How much would a square kilometer of this stuff weigh? About 10 tons for kapton, 7 tons for mylar, or 3 tons for recent carbon fiber materials. It’s been claimed that foamed carbon composites might be able to create a semirigid sail material weighing a fraction of a ton per square kilometer, but that is unproven. With any of these materials, the hope is that if a meteorite pokes a hole in it, the rest of the sail would be unaffected.

Let’s say we get the mass per square kilometer down to one metric ton. Nine newtons would give that a maximum acceleration of 1/1100 of a G, with no payload at all. If you had, say, 100 tons of sail and 25 tons of support and payload, that would be about 1/1350 G at the most. It would be less if you’re using it diagonally for tacking, and less still if you’re going out where the sun is dimmer. That sure isn’t much push... but it would never stop. The supply is limitless.

It would take sixteen days to gain just one kilometer per second, but with patience you could pick up as many km/s as you need, at almost no cost in onboard energy. Your reaction mass is the light. It does have a weight: by E = mc² the amount that hits your sail over that sixteen days has an aggregate mass of four kilograms.

That sail would be eleven or twelve kilometers across, if its shape is roughly circular. Turning it to steer would be difficult; it might have to be divided into smaller sections to be controllable.

Solar sails have actually been used. The first was a Japanese probe named IKAROS, launched in 2010. It was basically just a proof-of-concept to show that solar sail maneuvering could work. Its sail was a square only 14 meters across. They showed they could steer it by using liquid crystals to selectively darken parts of the sail. They’re working on a second probe which would have a 50 meter sail plus an ion engine, which they intend to send out toward the Trojan asteroids in Jupiter’s orbit. NASA is building a tiny probe called the Near Earth Asteroid Scout, which will have a sail of the same 14 meter size as IKAROS, and weigh only about 12 kg. It would attempt to study at least one near-Earth asteroid. It would be launched as a piggyback item on the first flight of the new Space Launch System with the Orion capsule, which will orbit the moon.

Solar sails may seem very slow, but some argue that they are actually capable of interstellar missions. Let’s look at that. If you deployed a sail near Earth and just kept going straight outward, how much speed would you pick up? Time for a bit of calculus... or failing that, a numeric integration.

Acceleration for this scenario is 0.0024 m/s² at Earth’s distance from the sun, and decreases with the inverse square of the outward distance. (If velocities get very large it’ll also decrease with redshift, but I doubt we’ll need to include that.) And that distance is the double integral of that acceleration, with the velocity as the single integral.... The numbers I get say that the final velocity seems to be approaching a limit of about 47 km/s, which is reasonable for interplanetary travel but useless for interstellar. About 17 km/s of that would be lost to the Sun’s gravity — that figure is just the ∆v available from the force on the sail. It would take more than a year to reach the orbit of Saturn, by which point you’re only about 3 km/s short of the maximum possible speed. Neptune’s orbit would be most of another three years, in which time you might get within about 1 km/s of maximum speed.

You can increase that speed by starting closer to the sun, but then the technical demands on the sail get tougher. If you start at one third the Earth’s distance from the sun, you can get more than 80 km/s of outward ∆v, but you have to be able to handle nine times the heat. What you gain in light pressure you might lose in how thin you can make the sail. Either way, you’re not going anywhere near fast enough to worry about redshift, or even drag from overtaking the solar wind. 80 km/s is about twice the speed we gave to the Voyager probes, and after gravitational loss it could reach the nearest star in something like twenty thousand years.

But for interstellar travel, we could give this type of craft a helping hand... such as a giant laser. One side effect of pushing our sail along with a huge laser beam would be that anyone already present in the target solar system would be able to see us coming.

A laser big enough to push a 125 ton spaceship is beyond imagining at this time, but what about a 12.5 kilogram probe? What if it weighed only 125 grams? There are now proposals on the table to launch such tiny probes. And what if the launch beam was not visible light, but microwaves? The sail could consist of a mesh with holes as loose as a window screen, and still act like a mirror at those longer wavelengths, just as the perforated door liner on your microwave oven does, if the mesh conducts electricity. This would allow the sail to be far lighter.

Would such a laser setup save any energy, relative to an antimatter rocket with the same size payload? Not much, even optimistically, and there is lots of opportunity for waste and loss, such as part of the beam inevitably spilling around the edges of the mirror. It could well be even more wasteful, but at least it wouldn’t have to be portable. And it is something we could start building foreseeably soon, if we can figure out how to make a really tiny probe capable of sending back a message we can hear.

There’s another type of sail we could use: the electric sail. This one actually does catch the solar wind, rather than the sunlight. The thrust available from that wind is even smaller, but in compensation, the sail doesn’t need to consist of more than a few strands of very thin wire. By splaying out a loose web of conductive wires, and charging them with a high positive voltage, the whole net exerts a repulsive force on the positively charged ions in the solar wind, slowing or stopping them even in the gaps between the wires, which would be tens of meters wide. As a result, the sail needs barely any mass. It would need a power supply: the same wires which repel ions will attract electrons, and the ship will need to continually collect and expel unwanted negative charge with a device similar to the electron gun in a cathode ray tube. Steering could be accomplished without even having to reel in any wires, simply by varying the voltage in different parts of the web. And unlike a light sail, it could reverse thrust, and suck itself toward the sun by using a negative voltage, though not for very long unless it was also equipped with an ion engine as a way to get rid of positive charge. To keep the web flat against the force of the wind, the whole thing would probably have to spin. The innermost parts of the wires would be thicker and heavier, to handle the tension.

Like a real sailboat, and unlike a light-based solar sail, the piloting of such a craft would have to adapt to random changes in the intensity of the wind. Sunspots, solar storms, and so on affect the density and speed of the outbound ions.

What kind of acceleration might such a craft be capable of? The mass of the sail per square kilometer of effective area would be at most in the tens of kilograms, and can probably be pushed down to single-digit kilograms. This is hundreds of times lighter than a solar sail. Unfortunately, the amount of momentum available from the ions of the solar wind is hundreds of times weaker than that of sunlight, so the net acceleration might be no better.

The first experiments with such propulsion are just beginning. If it works, some say it could be useful further from the sun than a light-reflecting sail is. I’ve seen a claim that the dropoff in the wind’s intensity is slower than the inverse-square falloff that applies to light, though I don’t understand how this can be true. Does the solar wind preferentially hug the ecliptic plane? Whether that’s true or not, the electric sail would be of no use at all in interstellar space, because once the solar wind reaches the heliopause, it loses all its momentum in a sort of barely-detectable shock wave, and then blends into the cold, electrically neutral gas of interstellar space. The sail would also be of little or no use inside a planetary magnetosphere, such as in Earth orbit or among Jupiter’s moons.

A magnetic sail may also be possible. In this variant, an enormous hoop of superconducting wire is deployed, and it creates a magnetic field which deflects ions. Performance might be not as good as that of an electric sail, but it should require less sustaining power, and with care it might be able to do useful maneuvering near planets by pushing or pulling on their native magnetic fields. The wire would have to be stronger and therefore heavier than that used in electric sails, as the current loop would create outward tension by self-repulsion. But this would mean you don’t have to spin it.

So aside from whatever use we can make of sails, once we get away from Earth we’re stuck with rockets, and the further or faster a rocket needs to go, the more energy it will waste even on shorter trips. It looks like interplanetary travel will never be quick and cheap, and interstellar travel will probably remain a highly challenging undertaking no matter what.

In spite of this, I do think our civilization will eventually spread through the galaxy, if it is not already occupied out there. It might proceed at a very slow pace. In a time when our posthuman descendants are more or less immortal, and wise beyond our comprehension, they might not mind taking a thousand years on a journey. In the end, the secret to interstellar travel may simply be patience.

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