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 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. 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 over 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. This is one reason why heavier solid fuels are often used for that first hard push off the launchpad, when the need for thrust is greatest.

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 using the Earth as reaction mass, 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. 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, at least not in 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 for the liveliest of all fuel reactions, lithium and fluorine. Most of the popular and practical chemical fuels can realistically produce speeds in the ballpark of three km/s. 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 stuff. 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 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
What this map doesn’t indicate is the durations of these journeys, which 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?

There is a way to do it 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. 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 cubic meters of fuel, you don’t want the remaining forty cubic meters to waste its energy on hauling empty 200m³ tanks.

In an idealized rocket, the velocity change it can achieve is given by the Tsiolkovsky equation v = ln(m0mf) ve, where m0 is the rocket’s initial mass fully fueled, mf is the remaining mass after the fuel is consumed, ln 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.

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 “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 438 metric tons full and 27 tons empty, and the second stage is 116 tons full and 4.5 empty, thanks to the lightness of carbon fiber composites. The second stage is part of the first stage’s payload, so for the first stage, the final mass is 27+116+17 = 160 tons, and the initial mass is 411 tons more, or 571. The mass ratio is therefore 3.57, and the natural log of that is 1.27, so that’s what we multiply our exhaust velocity by. That exhaust velocity is about 3.1 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 133 tons and a final mass with payload of 22, for a ratio of 6.05, which has 1.80 as its logarithm. The exhaust speed improves to 3.4 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 enough to go into orbit at about 8 km/s.

If you lighten the payload the mass ratio improves considerably, allowing the second stage to reach higher orbits. If the payload is only 1.5 tons, the second stage’s mass ratio is now 20, yielding an extra 4.1 km/s of delta V. However, if you want 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 another 5.9 km/s — 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 3.1 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 3.1 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 520 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. If you decide not to care about the health and environmental risks of such a monstrosity, it could achieve such 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, the rate of fuel consumption would be 740 kg per second instead of 2400, 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. No wonder that, despite very active research in the past, and some who to this day advocate bringing the idea back, we can be thankful that nobody ever actually flew one of these.

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 racks of nuclear thermoelectric generators.

Taking the upper end of the exhaust velocity range, and adding 20% more power consumption for the ionization step, we get a gas consumption rate of just 2.4 grams per second, and a thrust of 225 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 1/3000 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 fly by every planet 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 faster, you’d need to bump the mass ratio up again, until you had like 300 km/s of delta V.

What if we upgraded 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, and improved designs might mean that the exhaust from a fusion motor could eventually be a lot hotter than that. And with such a reactor, should we ever manage to build one, an output of 13.4 gigawatts is not at all unreasonable.

Using the 600 km/s exhaust speed, this gives you a fuel consumption rate of 75 grams per second, for a thrust of 45 kilonewtons. If the ship weighed 100 metric tons empty, 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 from Jupiter to Saturn (perhaps a billion kilometers depending on orbital positions) in just a month.

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 time-consuming and expensive.

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(ln(m0mf)) 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... I guess 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 Galactic’s 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. Because at that point, a rocket is moving about as fast as an airliner, only vertically instead of horizontally. 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. 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 risks not too terrible. 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 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 there’s no limit. 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 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). 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 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 about seven 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 loop around 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 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 door to your microwave oven does. This would allow the sail to be even 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.

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. 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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