Lu-Tze can’t think of a funny name asks

I want to go to the west coast of America. But like, really really fast. Of all the stupid inventions of science fiction over the years (Gravity Elevators, Teleporters etc.) what’s actually possibly feasible?

None of them. I can’t deal with science fiction on this blog, which is what gravity elevators, teleporters etc. fall within the realm of. They might be possible using crazy future science, but I can’t reasonably predict that. I have to stick to the world of mundane physics, and in this world there are some nasty limiting factors on exactly how quickly you can get about the planet, the major one being atmospheric drag.

As you travel through the Earth’s atmosphere, you encounter a force caused by Newton’s third law; conservation of momentum means that as you hit air molecules in the atmosphere and move them out of the way, the air molecules impart an equal and opposite force on you that resists your motion through the atmosphere. This is air resistance, and the resisting force is the drag force. Drag force can be calculated by

Where F_D is the drag force, ρ is the density of the atmosphere/fluid through which you are moving, v is your velocity relative to the atmosphere, C_D is the drag coefficient of the vehicle you’re travelling in (it varies depending on the shape of your vehicle) and A is the cross-sectional area of your vehicle (which can be broadly thought of as the surface area of the vehicle perpendicular to the direction of motion).

It looks complicated, but since the drag coefficient is also dependent on ρ and V and A is going to be as small as you can make it we can boil it down to just saying that FD is proportional to the density of the atmosphere you’re travelling through multiplied by the square of your velocity.

It’s that square that effectively kills any hope we have of faster Earth-based travel using near-future technology. As you go faster, the drag force resisting your movement increases exponentially and the amount of thrust you get from burning fuel gets smaller and smaller. I got this graph from Wikipedia which demonstrates the problem.

Specific impulse is a measure of how fuel-efficient something is. A thing with a high specific impulse will use less fuel to create the same amount of thrust as something with a low specific impulse. You will notice that turbofan jet engines, while only being capable of relatively low velocities, have an incredibly high specific impulse because they can sustain those low velocities on very small amounts of fuel. As you get faster the specific impulse of even the experimental air-breathing engines like ramjets and scramjets decreases exponentially because the amount of atmospheric drag opposing their motion is increasing exponentially.

So, while we can go fast inside the Earth’s atmosphere it’s not very fuel-efficient to do so, and that’s the main consideration we have when we consider mass transit of humans about its surface. What about going outside the Earth’s atmosphere? That would reduce the density term in the drag equation to zero and remove atmospheric drag entirely. This permits spacecraft to achieve much higher velocities than anything could ever manage inside the atmosphere, but there’s a small snag involved. You see that the specific impulse for rockets on that graph is both very small and totally flat with respect to velocity? That’s because rockets, in order to operate at the velocities they do and in order to operate outside an atmosphere, have to cart their own oxidiser along with them wherever they go. This makes them really, really fuel-inefficient.

Still, you asked me what is possibly feasible, not what is likely to happen, and there is a very feasible method of getting to the other side of the planet very quickly. It’s the space plane concept, and the best example of that at the moment is Virgin Galactic. It might seem odd for me to say that since I’ve spent quite a lot of time on this blog tearing Virgin Galactic apart because it’s not a real spacecraft. This is because it goes up to an arbitrary line separating “the atmosphere” from “space” but doesn’t actually go into orbit, which is what really qualifies as space. However, the sub-orbital flights that Virgin Galactic makes would actually be quite handy for getting around the Earth’s surface on short notice, in much the same way that intercontinental ballistic missiles are quite handy for incinerating the capital city of an opposing superpower on the other side of the world on short notice. The US was even investigating the space plane concept fairly recently as a way of delivering a rapid reaction force of marines to anywhere on Earth within two hours. The drawback, as ever, is that sub-orbital flights are still bloody expensive despite not actually going into space. Virgin Galactic costs $200,000 per berth, which is why it’s marketed as a way for the super-rich to experience “space” rather than as a next generation transport mechanism and why I think it’s never going to go much beyond that in a commercial capacity. Maybe I’m wrong, though. Maybe in fifty years we’ll all be flitting around in Virgin Galactic space planes while the embalmed corpse of Richard Branson cackles madly in its life support pod. You never can tell.

The post Physics Can Be Such A Drag. appeared first on The Scientific Gamer.

On Monday there were words about the various types of manned spacecraft we’ve flung into orbit (and beyond). Particularly discerning readers will have noticed that the vast majority of them — Vostok, Voskhod, Soyuz, Mercury, Gemini and Apollo – were designed and launched within a single decade between 1960-1970. Since then we’ve had exactly one new manned space vehicle: the Space Shuttle. Manned spaceflight has been more or less left to stagnate by national governments, but there are promising signs that the next decade may be as groundbreaking as 1960-1970. Are governments becoming interested in spaceflight again? Hardly. The US government is still dragging its feet over the design and development of the MPCV, while I’ve heard very, very little about the proposed replacement for Soyuz (on the other hand that doesn’t really need replacing since it does what it’s supposed to extremely well). No, what’s going to be exciting about the next ten years is the opening up of human spaceflight to a variety of commercial efforts.

If you know me you’d probably expect me to hate the idea of private companies going into space, but to be honest governments have ignored it for the last fifty years and there’s at least a chance that profit will provide the driving force that simple discovery could not, so I’m broadly in favour of it. Regardless: not only has development of private “spaceflight” been in progress for a decade or so what with Virgin Galactic and all, but the US government recently initiated something called the Commercial Crew and Cargo program (CCC, or C3) whereby they subsidise the development of several different private spaceflight efforts with the view to buying or leasing the most suitable vehicle once development is complete.

It probably sounds inefficient to pay private companies to develop spacecraft that you then have to pay to use, and that’s because it is. It’s very inefficient. However, going up to LEO is relatively routine these days and by subsidising private efforts NASA ensures that the private sector has the technology base to take up any slack created by them abandoning LEO. And they are abandoning it; the MPCV mentioned in the last post theoretically has the capability to do LEO, but the ultimate intention of the program is to return astronauts first to the Moon or a suitable nearby asteroid, and then eventually in a few decades to possibly put a man on Mars (I nearly said person, but you know it’s going to be a guy). As it is they have to pay the Russians to make flights up to the ISS, so why shouldn’t they give some of that money to their own commercial efforts? It makes a twisted sort of sense, if you think about it.

Anyway, there’s several different competitors vying for a slice of the C3 pie. Excluding the Mickey Mouse effort of Virgin Galactic (which is literally a plane that goes really high rather than a true spacecraft), these private spacecraft are:

Boeing’s CST-100 – This catchily-named capsule is basically a space bus with a very simple design brief: deliver anywhere from between two to seven astronauts to LEO and then return them to Earth. It’s not going to go any further than that because it doesn’t have the capacity to do so.

Sierra Nevada’s Dream Chaser – Like Boeing’s effort, designed to take up to seven astronauts to LEO and back. You may notice that it looks like a miniature space shuttle, but the Dream Chaser launches on top of a rocket booster just like a capsule does. The wings and whatnot are just to allow it to land on a runway instead of splashing down in the ocean.

SpaceX’s Dragon – The interesting one. Like CST-100 and Dream Chaser, Dragon can do LEO with no problems, but it also has the theoretical capacity to go further – perhaps even to the Moon. It’s also the only one of the three private efforts to have actually made an unmanned test flight into orbit, so it actually does work. This got them noticed by NASA and they’ve been awarded a contract to resupply the ISS, so this is the one you’ll most likely see flying in the next decade or so.

In theory, then, that’s LEO taken care of. What about going further afield? You may or may not remember some hullaballoo about five or six years ago when President Bush announced the Constellation program. This was supposed to set NASA’s manned spaceflight goals for the next quarter-century, and those goals were lofty: to return to the Moon, to establish a permanent base there, and to put a man on Mars by 2030. Even the most insufferably optimistic space scientists thought that achieving all this on such a short timescale was a little unrealistic (it could be done with enough money and political will, but NASA doesn’t have anywhere near that much in their manned spaceflight budget), and sure enough when Obama was elected one of the first things he did was cancel the Constellation program. This meant that the proposed Ares heavy rocket booster and the Altair lunar lander were both cancelled, but what didn’t get cancelled was the Orion space vehicle (similar to the Apollo command and service modules). They couldn’t cancel that because despite the shifting political winds they still needed a replacement for the shuttle – which had been pushed far beyond its projected operational lifetime – and Orion was the best they had.

So Orion morphed into the Multi-Purpose Crew Vehicle. This is a four-man spacecraft designed to be launched on refurbished shuttle rocket boosters, and it’s multi-purpose because it can, theoretically, do pretty much any manned mission we can think of. Want to make a trip up to LEO? No problem, just launch the capsule. How about a near earth object? Stick a service module on it to extend its flight time and send it out there. Mars? Even that’s doable if you daisy-chain several service modules together to increase power, life-support and fuel capacity. This is the great advantage of a modular capsule design: it is hella adaptable in comparison to a fixed-mass spacecraft like the shuttle. It’s a great design if they can pull it off; an unmanned test flight of the Orion is scheduled for 2014, although I really wouldn’t expect to see it flying with people inside until the 2020’s at the very earliest.

Other countries are also looking at getting into space; Japan and India both have nascent manned spaceflight programs, while the European Space Agency was somewhat put out when NASA announced they wouldn’t be accepting international participation in Orion like they did with the shuttle, thus forcing ESA to start developing their own solution pretty damn quickly; it currently consists of a jury-rigged version of the unmanned Automated Transfer Vehicle that carries supplies up and down from the ISS. All of these plans are still very much on the drawing board though, and it usually takes a spacecraft design at least a decade to actually make it off the drawing board and into space – and as the Chinese have proved, even then it can be a slow, arduous process developing the technology. No, I think that in the future LEO will belong to Soyuz and the commercial spacecraft, with the MPCV providing a longer-range option should we ever want one.

The post A Brief History Of Spaceflight, Part Two. appeared first on The Scientific Gamer.

Following on from why the Space Shuttle sucked, I’m going to do a little summary of manned spaceflight – both where it’s been, and where it’s going.

You’d have to look pretty hard to find someone who didn’t know that Yuri Gagarin was the first man in space in 1961. However, finding someone who doesn’t have a clue what the hell he was flying would be significantly easier. Gagarin made Earth orbit in what was essentially a tin can strapped to a converted ICBM. The tin can was called Vostok, and it was incredibly basic; it was a metal capsule containing the cosmonaut attached to an equipment module with RCS¹ thrusters and a small rocket engine to make the re-entry burn. Vostok amuses me because once the re-entry capsule had separated from the equipment module the cosmonaut had no way of controlling his re-entry – no thrusters, no parachutes, no nothing. Well, I suppose they did have parachutes, but only in the sense that the cosmonaut would physically have to eject from the capsule and parachute down before the capsule smashed into the ground. Still, it proves that you can do spaceflight in an incredibly basic way if you’re willing to economise.

The American Mercury program was next; this is slightly better known thanks to Freedom and Hollywood and whatnot. Alan Shepard was beaten by Gagarin’s flight by a mere month, but what the films and documentaries tend not to emphasise was that Shepard’s flight was sub-orbital – i.e. there wasn’t a huge amount separating it from a regular aeroplane flight except the altitude. It wasn’t until 1962 and the third Mercury flight that NASA finally achieved a stable orbit. On the other hand the problem lay more in the Redstone rocket boosters they were using than the Mercury spacecraft themselves; compared to Vostok it was a model of sophistication, with a conical shape for controlled re-entry (this is important because you only have to plaster your heat shield along the flat bottom of the capsule rather than shielding the whole thing as they had to do with Vostok), limited fly-by-wire maneuvering capability and a parachute for a “soft” splashdown into the ocean². Most of this was down to the direct participation of the Mercury Seven astronauts in the design process – they were aviators³ used to being in control of whatever they were flying, and so they demanded features like a bigger window and a manual re-entry control system.

Mercury and Vostok were essentially proofs of concept. We knew spaceflight could be achieved, and now it was time to do something with it. That something was the space race for the Moon. Pretty much every single thing the Americans did post-Mercury had the final goal of supporting the Apollo program and the Moon landings. By contrast, while the Soviets had the same general objective they were more concerned with space “firsts” – first EVA, first space station, first probe around another planet and so on. This is one of the reasons they went running up the blind alley of the Voskhod programme.

Voskhod wasn’t even a second generation spacecraft – not really. It was simply a Vostok capsule modified to take two cosmonauts instead of one, and it retained most of the limitations of the Vostok capsule. Improvements in the boosters used to carry them into space meant that Voskhod had a braking rocket added which allowed for a “soft” descent on land with the cosmonauts still inside, but otherwise it was basically the same spacecraft. Jury-rigging Vostok in this way meant the Soviets got two of their firsts – first multi-person spacecraft, and the first spacewalk⁴ – but since Voskhod was a technological dead end it turned out to be mostly wasted effort, with only two flights being made.

Gemini on the other hand was a pure technological testbed for Apollo. There were ten manned Gemini flights over a two year period in 1965 and 1966; not only were these testing the systems and procedures required for multi-person spaceflights, they were also testing the maneuvers that would be necessary for a Moon shot – docking two Geminis in orbit, endurance spaceflights, that sort of thing – and giving the Gemini astronauts valuable experience in these manuevers that they would later find useful on Apollo missions. Design-wise Gemini superficially resembled a larger brother of Mercury, but the internal mechanisms were very, very different. In particular where the Mercury spacecraft combined its critical systems like power and life support into the Mercury capsule itself, Gemini hived these off into a separate equipment module similar to Vostok. This laid the groundwork for the modular construction of Apollo.

Not that the Soviets were sitting around all dejected after the relative failure of the Voskhod programme. They started design work on the Soyuz capsule in 1963, with the first generation Soyuz flying in 1967. Soyuz was intended to be part of the Soviet Moonshot⁵; the idea was that it’d make up part of a modular spacecraft attached to a lunar lander in the same way as Apollo’s command/service module. Soyuz has the same sort of construction, with orbital and service modules that contain all the stuff the spacecraft needs while it’s actually in space which is then jettisoned before landing to make the re-entry module’s job easier. An unmanned Soyuz capsule even made a test flight around the Moon in 1969. You can essentially think of it as the Russian equivalent to Apollo – and while you’re thinking about that, consider that while it’s undergone several generations of redesigning the Soyuz spacecraft is still flying today ferrying astronauts up to the ISS and back, albeit in a heavily modified form. Imagine what NASA could have done had they stuck with Apollo instead of putting all their hopes in the space shuttle.

Speaking of Apollo, there’s a decent argument to be made that it represents the peak achievement in manned spacecraft design. The modular construction meant that it could be tailored to a specific mission – there’s no reason an Apollo capsule couldn’t make routine flights to LEO just as well as it flew to the Moon – and considering it had to keep astronauts alive during the most insanely ambitious and dangerous missions in the history of manned spaceflight it speaks volumes about how well it was designed that it did an astonishingly good job, with only three astronaut fatalities when Apollo 1 caught fire on the launch pad.

This is why I think Apollo is so amazing (and indirectly, why I think the space shuttle was so terrible). Here is a picture of an Apollo spacecraft.

The big cylinder with the rocket is the service module with life support, power and propulsion systems⁶. The small cone on top is the command module which contains the astronauts during flight and re-entry. The service and command modules would be launched into space together on top of a Saturn V, and then they’d dock with the lunar module in Earth orbit before switching to a trans-lunar orbit.

Everyone got that? Right. That was spacecraft design in 1969. This on the other hand is spacecraft design in 2012. It’s the proposed design for what used to be the Orion spacecraft, part of the Multi-Purpose Crew Vehicle system that’s supposed to replace the space shuttle. Note the way it looks exactly the goddamn same as Apollo. That’s how good Apollo is: when today’s NASA engineers were asked to design a next-generation spacecraft, they used Apollo as the blueprint.

Aaaaanyway. After the Moon landing everyone kind of lost interest in space flight as a national pursuit, and the design of new spacecraft stagnated. The US launched the last Apollo in 1972 and haven’t made a manned spaceflight that didn’t involve the space shuttle since.  The Soviets – and then the Russians – continually refined the design of Soyuz and made great advances for the long-term human habitation of space with their Mir space station, but they too are somewhat stuck in the past as far as manned spaceflight is concerned. China is the only other nation with an operational manned spaceflight program; while people sneer at their Shenzhou capsules as barely being at the Gemini level of technology, you have to remember what I said about the MPCV – NASA is still using the Apollo design after running down the blind alley of the shuttle for the last thirty-five years. It’s the launch rate that concerns me, with the Chinese managing just three manned missions in nine years. They’re not going to make any progress if they don’t physically launch the things into space with taikonauts inside.

So spaceflight has been rather static for the last forty years. It’s only now starting to take off again (har) and the major driving force behind that has come from a rather surprising source: the private sector. More on that on Thursday.

1. RCS stands for reaction control system; if you want to make a small adjustment to the trajectory of something flying through space the best way to do it is to squirt a small amount of pressurised gas in the opposite direction and let Newton’s third law do the rest. That bit in WALL-E with the fire extinguisher? That was scientifically accurate.
2. American capsules all splashed down into the ocean. Soviet capsules all landed in Siberia or the deserts of Kazakhstan. Why is this? It’s a more a divergence in the cultural design philosophy than anything else — both methods are equally valid in their fulfilment of the main requirement for a spacecraft landing site: that it be really, really, really hard to miss.
3. Flying ability wasn’t really necessary for astronauts, except in so far as it demonstrated their ability to handle very complex machines. However what was useful was their complete self-confidence and conviction that they could handle anything, which was a trait common to combat pilots.
4. Which nearly went comically awry when Alexei Leonov realised his air-filled spacesuit — which he’d clambered into in the pressurised cabin of the spacecraft — had inflated in the vacuum of space to the point where he could no longer bend his joints to get back inside the airlock. His problems didn’t end there, either; Voskhod 2 overshot its landing site by some 400km and the two cosmonauts ended up spending a very chilly night in the Ural mountains before they could be recovered.
5. The Soviet moonshot didn’t fail because their spacecraft were bad. It failed because they couldn’t get their heavy lift rocket launch vehicles working properly before the Americans landed, thus cementing my opinion that Rockets Suck.

6. Putting all the critical systems in one place does have a drawback, as the Apollo 13 disaster demonstrated. An exploding oxygen tank damaged the service module, and because the astronauts didn’t know how bad the damage was they couldn’t trust any of the systems contained on board – including, critically, the large propulsion engine that they’d normally be using to adjust their orbital trajectory. If they hadn’t had a convenient set of backups in the form of the lunar module life support systems and descent engine, the Apollo 13 astronauts would have been screwed.

The post A Brief History Of Spaceflight, Part One. appeared first on The Scientific Gamer.

Warning: post contains moderately difficult algebra along with hints of calculus. But don’t worry, it’s not that bad.

Rockets are kind of sucky for getting into space. Right now, though, they’re all we’ve got.

Rockets work by the classic principle of Newton’s Third Law: every action has an equal and opposite reaction. Push against a wall, and the wall also pushes against you with the same force. Friction between your feet and the ground stops you from moving backwards in response to this force, but if there wasn’t any friction – if you were standing on a big sheet of ice or something – then if you thumped the wall hard enough you would start to slide away from it. In a true Newtonian environment like space, even a simple action such as throwing a wrench away from you can nevertheless produce an appreciable impulse on you in the opposite direction to which you threw it, and that’s where you’d end up drifting.

This is basically how rockets work, except instead of a wrench they’re using thousands of kilograms of rocket fuel shot out of the back of the rocket at several kilometres per second. This creates an equal and opposite force which pushes the rocket upwards – thrust, in other words. When they’re considering how much fuel to put in a rocket, rocket scientists don’t think “Well, we want the rocket to reach such and such an altitude so we need this much fuel,” because that would be needlessly complex. Instead they think in terms of something called delta-V, or ΔV, which is the change in the velocity of the spacecraft that will be caused by burning X amount of rocket fuel. If your rocket’s delta-V is roughly equal to the Earth’s escape velocity, then congratulations! You’re on your way out of Earth orbit.

The amount of fuel you need to accelerate a given payload to a certain delta-V is dependent on how heavy that payload is. A bigger payload means more fuel. And as mentioned in the post on satellite orbits, it’s not enough just to add more fuel to the rocket, because now you’ve just added a whole bunch of extra kilograms that also needed to be lifted into orbit. So you need more fuel to lift the fuel, and then more fuel to lift the fuel to lift the fuel, and the whole thing would get dreadfully complicated if it weren’t for something called the Tsiolkovsky rocket equation. Because I’m trying to re-teach myself some basic physics I’m going to derive it from first principles. Don’t worry if you can’t or don’t want to follow what I’m doing, just skip to the end.

WARNING. WARNING. AWFUL CALCULUS STARTS HERE. SKIP FORWARD IF YOU’RE ALLERGIC TO MATHS.

Start with the principle of conservation of momentum.  Momentum is mass times velocity. Consider the state of the rocket at two different times: t, which for our purposes is the rocket sitting on the launch pad just before blast off, and t + dt, which is a time dt after that when the rocket has burned off all of its fuel.

(Note: dt is part of the Leibniz notation of calculus. The d doesn’t stand for a specific quantity, but instead basically means “change in”. So dt is the change in time, dm is the change in mass, and so on. They can be combined, too, so dm/dt would be the rate of change in mass over time. For a real world example, dV/dt is the rate of change in velocity over time, which is equivalent to acceleration a. Here, t + dt is similar to saying something like “D-Day +17”, where the +17 is equivalent to + dt.)

The rocket will start with a mass M(t), and will eject a quantity of exhaust gas as it burns its fuel which will have a total mass dm (because the mass of the rocket will change by this much). Due to conversation of momentum, the momentum of the payload after the fuel burn  — which for our purposes we will treat as everything on the rocket that isn’t the fuel (i.e. M(t) – dm) – will be equal and opposite to the momentum of the exhaust gas dm. We can model these separately to find out their respective momentums at the times t and t + dt, but to do this we need to make some assumptions about the velocity of the payload and the fuel.

At time t, the fuel is unburnt and flying along with the payload, so it and the payload have the same velocity V(t). At time t + dt, the fuel will have been converted into exhaust gas travelling at a constant velocity U (since for the purposes of simplicity we’ll say a single fuel type will burn at a constant rate). The payload will be travelling at the same velocity V(t) plus some positive change in velocity dV provided by burning the fuel, making its total velocity V(t) + dV. The fuel will be travelling in the opposite direction with velocity U, but it had a velocity of V(t) to start with, so its true velocity with respect to the payload will be V(t) – U. (The U is negative because the fuel is travelling in the opposite direction to V(t)).

Right, now we can get on with the interesting stuff. The momentum of the payload and the fuel can be expressed as:

To find out the change in momentum over time dt, we have to subtract the intial momentum at time t from the final momentum at time t + dt.

Because we’re treating the rocket as a closed system with no external forces, conservation of momentum means the sum of the change in momentum of the rocket and the payload will be zero. In other words, adding together the terms we’ve derived for ΔP_payload and ΔP_fuel should equal zero.

We’re trying to find the rocket’s change in velocity – its delta-V, or dV – so we rearrange this in terms of dV.

And now comes the part where I’ll probably lose even the more dedicated amongst you. The mass of the burned fuel dm will be equivalent to the change in mass of the rocket M(t), or -dM(t), so we can substitute that in for dm.

Then we do a wonderful, magical thing called integration. Say you have a line on a graph which is described by the equation y = x. Integrating that equation will give you the area underneath the line y = x. This is bloody complicated and there’s a whole bunch of different rules for doing it. For our simple example, the integral of y = x will be x²/2. Don’t understand? Don’t worry, I don’t either. The one thing you should grasp about integrals is that since most lines described by set equations will be technically infinite, we need to identify the bit we actually want to find the area under and integrate between upper and lower limits that define that part of the line.

We integrate the left hand side of the equation with respect to V between the limits V_o and V_final, the initial velocity of the rocket and the final velocity of the rocket respectively. This gives

Simple, right? Unfortunately we then have to integrate the right hand side with respect to M between the limits of M_payload and M_o, the mass of the payload and the original mass of the payload plus the fuel.

That’s substantially gnarlier. Fortunately there’s a neat way to simplify logarithms by smooshing them together, so

Since the final velocity minus the original velocity is the change in velocity, or delta-V, then

IF YOU ARE SKIPPING ALL THE HORRIBLE CALCULUS START READING AGAIN HERE.

It’s a lot of effort to go through for such a little thing, but the final product of all that work is Tsiolkovsky’s rocket equation. From it you can easily see that the delta-V you’ll get out of a given rocket will depend on the velocity of the exhaust U and the logarithm of the ratio of the total mass of the rocket divided by the payload mass. However, since I said rocket scientists usually think in terms of “How much fuel do I need to reach a given delta-V?” rather than “What delta-V do I want to reach today?”, a more useful form of the equation is

Everything on the right side of the equation is stuff the rocket scientist should know in advance – the exhaust velocity U, the desired delta-V and the mass of the payload. And if you subtract the mass of the payload from the total mass of the rocket M_o, what you have is the mass of the fuel. This can therefore be used to quickly and easily calculate what amount of fuel you’ll need to boost a given payload to a given delta-V. Just for the lols we can make a graph of how the fuel-to-payload ratio changes for that given delta-V depending on what sort of fuel you use. For a desired delta-V of 9.4 km s^-1 (the lower limit required for low Earth orbit)

Obviously there’s a law of diminishing returns in effect here, but we can see from the graph that in order for your rocket to even be feasible you need a fuel with an exhaust velocity of at least 4-5 km s^-1. Happily there are fuel mixtures which exist which are capable of this, and some go even higher, but there’s a limit to how big you can make U. Remember the Nedelin catastrophe I referenced last week? That neatly demonstrates that high-power rocket fuels are very, very volatile and prone to exploding, and so you don’t really want a U that’s too big. It’s essentially fixed at 4-6 km s-1. Assuming that you’re not willing to reduce the mass of your payload at all, is there any way we can further increase the efficiency of a rocket?

Well, I wouldn’t have asked the question if there wasn’t an answer. There is. It’s called rocket staging. The idea is that once you have burned X amount of fuel, whatever you were using to keep that X amount of fuel in is useless dead weight. Lifting it along with the rest of the spacecraft is inefficient; it’s better to just cut it loose entirely if you can. Rocket staging is basically a matter of perching a small rocket on top of a larger, more beefy rocket. The beefy rocket fires first and uses all of its fuel, and then a series of explosive bolts cut it loose from the small rocket which then starts burning its engines. We can work out how efficient this is by using the rocket equation for each successive stage.

Assume that your rocket requires 1% of storage mass for every 8% of fuel mass. For a one stage-rocket with a payload that is 1% of the total mass, this allows for a fuel mass of 88%. Your achievable delta-V will therefore be

Compare this with two rockets stacked on top of each other, using the same fuel/storage ratios. The first rocket has a fuel mass of 80% and a storage mass of 10%, with the second rocket making up the last 10% of the mass. The first stage will boost the second stage to a delta-V of

So this first stage has less delta-V than our big one-stage rocket. However, once it’s finished burning it’s cut loose and the second rocket fires as a standalone entity with

i.e. exactly the same as the first rocket since the fuel/storage mass ratio is identical. This is added to the delta-V from the first stage to give a total delta-V of 3.22 U – half as much again as a one-stage rocket carrying the same payload with the same total mass. Depending on how far you want to go and how complex you want to make your rocket you can add even more stages; the Saturn V which sent the Apollo astronauts to the Moon was a five-stage monster.

So! Hopefully this has at least been useful for crystallising in your minds the reason why we use multi-stage rockets. It’s certainly been helpful for me; I learned this stuff back in 2004 so it was good to spend an afternoon deriving the Tsiolkovsky rocket equation from first principles. Good. Yes.

*eye twitches*

The post You Have Discovered Rocketry. appeared first on The Scientific Gamer.