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Some (generous) assumptions:

  • We have a spaceship that can reach a reasonable fraction of light speed.
  • The ship is able to withstand the high energies of matter impacting at that speed.

Given the amount of matter in inter-stellar space, at high speed, would it encounter enough of it and frequently enough that an aerodynamic shape would significantly reduce its drag (and thus save fuel)?

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For the sorts of vehicles we're used to, like cars and aeroplanes, there are two contributions to drag. There's the drag caused by turbulence, and the drag caused by the effort of pushing the air out of the way. The streamlining in cars and aeroplanes is designed to reduce the drag due to turbulence. The effort of pushing the air out of the way is basically down to the cross sectional area of whatever is pushing it's way through the air.

Turbulence requires energy transfer between gas molecules, so you can't get turbulence on length scales shorter than the mean free path of the gas molecules. The Wikipedia article on mean free paths helpfully lists values of the mean free path for the sort of gas densities you get in space. The gas density is very variable, ranging from $10^6$ molecules per cm$^3$ in nebulae to (much) less than one molecule per cm$^3$ in intergalactic space, bt if we take the value of $10^4$ in the table on Wikipedia the mean free path is 100,000km. So unless your spaceship is very big indeed we can ignore drag due to turbulence.

A sidenote: turbulence is extremely important in nebulae, and a quick glance at any of the Hubble pictures of nebulae shows turbulent motion. However the length scale of the turblence is of the order of light years, so it's nothing to worry a spaceship.

So your spaceship designer doesn't have to worry about the sort of streamlining used in aeroplanes, but what about the drag due to hitting gas molecules? Let's start with a non-relativistic calculation, say at 0.5c, and use the density of $10^4$ I mentioned above, and let's suppose that the gas is atomic hydrogen. If the mass per cubic metre is $\rho$ and you're travelling at a speed $v$ m/sec then the mass you hit per second is:

$$ m = \rho v $$

Suppose when you hit the gas molecules you accelerate them to match your speed, then the rate of change of momentum is this mass times your speed, $v$, and the rate of change of momentum is just the force so:

$$ F = \rho v^2 $$

A density of $10^4$ atoms/cm$^3$ is $10^8$ per m$^3$ or about $1.7 \times 10^{-19}$kg and 0.5c is $1.5 \times 10^8$m/sec so $F$ is about 0.004N per square metre.

So unless your spaceship is very big the drag from hitting atoms is insignificant as well, so not only do you not worry about streamlining, you don't have to worry about the cross section either. However so far I've only talked about non-relativistic speeds, and at relativistic speeds you get two effects:

  1. the gas density goes up due to Lorentz contraction
  2. the relativistic mass of the hydrogen atoms goes up so it gets increasingly harder to accelerate them to match your speed

These two effects add a factor of $\gamma^2$ to the equation for the force:

$$ F = \rho v^2 \gamma^2 $$

so if you take v = 0.999c then you get $F$ is about 7.5N/m$^2$, which is still pretty small. However $\gamma$ increases without limit as you approach the speed of light so eventually the drag will be enough to stop you accelerating any more.

Incidentally, if you have a friendly university library to hand have a look at Powell, C. (1975) Heating and Drag at Relativistic Speeds. J. British Interplanetary Soc., 28, 546-552. Annoyingly, I have Googled in vain for an online copy.

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  • $\begingroup$ so extremely fast spaceships should be needle-shaped to minimize cross section? $\endgroup$ – endolith Jul 17 '13 at 14:33
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    $\begingroup$ Unless you're travelling very, very close to the speed of light the shape of the spaceship makes little difference. $\endgroup$ – John Rennie Jul 17 '13 at 14:40
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    $\begingroup$ Very interesting answer -- you mention the gas density increasing due to Lorentz contraction. Is it possible that the density increase (and hence the mean free path decrease) makes the turbulent length scales small enough to matter? Or does the Knudsen number stay constant regardless of velocity? $\endgroup$ – tpg2114 Dec 31 '13 at 1:29
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    $\begingroup$ Sorry, but based on my understanding you have a number of things wrong. Firstly, the breakdown into "drag caused by the effort of pushing air out of the way" and "drag caused by turbulence" is wrong. Yes, we have form drag, but the other component of drag (skin friction) would happen regardless of whether we have turbulence (and in fact would be much worse in many cases if you didn't have turbulence). Secondly, aerodynamic design addresses all kinds of drag, and in fact form drag is one of the primary issues it seeks to address, so saying "streamlining is to address turbulence" is wrong. $\endgroup$ – Asad Saeeduddin Dec 21 '15 at 21:33
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    $\begingroup$ @AsadSaeeduddin: well yes, but when answering questions we have to judge at what level to pitch the answer, and in any case turbulence isn't relevant here because the mean free path is too long. A detailed discussion of turbulent drag would be a diversion. Please feel free to post your own answer if you feel mine doesn't adequately address the question. $\endgroup$ – John Rennie Dec 22 '15 at 6:11
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At that speed aerodynamic shape would be unimportant since almost all particles would penetrate the hall. So more important would be total cross-sectional area perpendicular to velocity (how many particles per second collide with ship). So I assume that ship should have torpedo like shape to reduce total cross-sectional area.

More probable is use of some kind of electromagnetic shielding (to protect crew) but in such case aerodynamics of ship is also unimportant.

Don't forget that dimension in direction of movement is compress in such speed according to Special Relativity so aerodynamic shape becomes less effective (harder to achieve).

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Yes, this shape would be good, but not for aerodynamic reasons. As the others have commented, there's not much matter in your path. However, the matter that is in your way really hits your hull hard.

At that point, look at tank designs. Since WW2, tanks have sloped armor: armor which is at a angle to incoming shells. This increases the effective armor thickness by 1/cos(θ). An aerodynamic shape of a spaceship achieves a similar effect.

So, essentially your second assumption is dependent on this shape.

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Given the amount of matter in inter-stellar space, at high speed, would it encounter enough of it and frequently enough that an aerodynamic shape would significantly reduce its drag (and thus save fuel)?

No, in fact, you can see why even at more mundane speeds.

The drag that you're typically familiar with on cars and planes is happening at speeds well below the speed of sound. Remember what "speed of sound" actually is, it's the speed at which one air molecule can bump into the next to pass a wave through the air.

Why is that important? Because when you're going slowly, that means that when you, a plane say, impacts the air, that molecule bumps into another one in front of it, and so on... causing the entire air mass in the area to sort of "get out of the way". But it can only do that so much, so if you try to push a pie plate through the air one way the air can't move sideways fast enough and just piles up and you get lots of drag. But turn it sideways and now most of it can get out of the way in time and now you have a Frisby.

If you look at this at a macroscopic level, the result is a series of "streamlines" that the air follows under a given set of conditions. If you design your aircraft to follow those lines, you minimize the number of parts of the aircraft that impact with the air, and thereby reduce drag. So, for instance, you might find that narrowing the fuselage just so causes the air to separate ever so slightly behind the door (for instance) and thus have less drag on the rear fuselage. Until we had fast computers this was as much an art as science, but now we solve everything by simulating the crap out of it.

Ok, now what's that got to do with anything? Well when you start getting up near the speed of sound, all of this goes out of the window.

At supersonic speeds, the air molecules literally cannot get out of the way before you hit them. So every single molecule in front of you hits you. The key to lowering drag at supersonic speeds is to simply reduce your cross section as much as possible, which is why things like the F-104 and Concorde look like darts. Streamlining works very differently now, and it's perhaps not even accurate to call it streamlining.

There is an effect even at supersonic speeds that comes into play, and that's shock waves. These do indeed travel faster than the speed of sound, but they don't really "move" the air as much as just shake it. At a simple level, the shock waves have lower speed behind them, so the trick is to put something out in front of the aircraft and try to keep as much of the rest behind it. That's why supersonic aircraft have sharp noses, they're trying to generate a shock wave they can "hide behind".

Ok, now what does any of this have to do with your question? Well, you're talking about a vehicle moving at speed way beyond any sort of inter-molecular interactions. So the basic idea of streamlining is just not going to work, and the idea of using shock waves won't either because there's just not enough particles to create one.

So then the answer pops out - the key is to reduce cross-section, and that's basically it.

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