May 27, 2004
Space Fleet - Beams Beat Missiles
I ran some more rough numbers on space fleet combat to see if the defending ship could possibly have enough energy to disable a missile volley representative of what it could itself deliver. The calculations are pretty simple, didn't come out like I expected, and now instead of my original question I'm left wondering if there's even a remote possibility that a ship could carry a missile volley capable of penetrating the defenses of another ship with similar mass, as long as that ship was defended properly with lasers or some other type of beam weapon.
The answer, detailed below, is that using bullets, rail guns, or missiles is probably a losing proposition, and as you keep upping the projectile velocity the energy expended by the attacker in propelling the projectile is growing with the square of the velocity, while the energy required to vaporize the projectile remains some small constant. The faster you make your projectiles, required to make them unavoidable by the defender from longer and longer ranges, the greater the energy advantage accruing to the defender.
If you don't like boring analysis, don't bother clicking the fold.
Let's keep things simple by starting with missiles using a regular chemical fuel and defensive lasers powered by crude chemical reactions. Organic polymers have about the same heat of combustion per gram of oxygen consumed, which is 13.1 +- 0.7 kJ/g-O2. A hydrogen-oxygen reaction produces about 13.4 kJ/g-H2O, and converting the units to kilograms means we have somewhere between 13 MJ/kg of energy stored up ready for defensive use.
Suppose we could turn this fuel into electrical power at 30% to 62% efficiency and then turn the electrical power into laser power at 50% to 80% efficiency. That gives us 15% to 50% efficiency, meaning that we can burn fuel to deliver about 2 to 6.5 megajoules of laser power per kilogram of fuel consumed. Let's also allow that the defending ship has about 10% of its mass devoted to fuel for laser power. Keep in mind I wouldn't actually do things like this, and rather use nuclear or some other source, but it gives a good basis for comparison. So let's say we can deliver laser energy where the total megajoules is the same as about 0.2% to 0.65% of the mass of the ship, measured in kilograms.
One the flip side, the enemy missile is burning chemical fuel to propel the incoming warshot, and since we're talking about a kinetic kill weapon this too can be converted to megajoules of warhead kinetic energy per kilogram of missile fuel consumed. I'll use liquid hydrogen and liquid oxygen to power our missile, which gives us an engine with a specific impulse (Isp) of roughly 400. I'll pick an Isp of 408 just to make the exhaust velocity c an even 4,000 meters per second.
It starts out with some fully fueled mass, or launch weight M0, when sitting in the missile bay and burns all its onboard fuel to some final mass, M1. The ratio of the initial mass M0 to M1 is known as the mass ratio, MR, and the basic rocketry equation says that the final velocity of the rocket is given as vfinal = c * ln(MassRatio), where ln() denotes a natural logarithm. What we're concerned about is the final kinetic energy of the missile after the fuel is spent, which is given as KEmissile = 1/2 * M1 * vfinal2. The fuel consumed to get the missile up to this kinetic energy is simply the initial mass minus the final mass, M0 minus M1. The fuel consumed (doing some algebra) is Mfuel = M1*(MR-1).
Let's set the final mass to 1 kg to make the units easy, so the fuel consumed for each kg of final mass is Mkg-final = MR-1. The kinetic energy per kg is KEmissile = 1/2 * vfinal2, so KEmissile = 1/2 * (c * ln(MR))2. Divide the kinetic energy per kilogram of final missile mass by the fuel consumed to get a curve of kJ of kinetic missile mass per kg of missile fuel. This curve is pretty pathetic for low mass ratios, where you burn fuel to barely get moving, but is increasing dramatically to peak efficiency at a mass ratio of 4.92, regardless of exhaust velocity. The final velocity at his mass ratio is always 1.59 times the missile's exhaust velocity, and past that the efficiency tapers back off. Keep in mind that the rocket's engine efficiency isn't actually changing, and that these numbers are artifacts of the fixed reference frame for calculating kinetic energy, and the way mass ratio and final velocity are related. You can make the missile go as fast as you want, you're just not going to be optimally fuel efficient doing it, since most of your added fuel is just spent pushing the existing fuel along.
Now the efficiency of a missile starts out as abysmal at launch. I mean, think of the mighty thrust of a Saturn V, with fuel pouring through fuel lines you could crawl through, and then look how slowly it's kinetic energy builds up as it starts to clear the pad. It's just not going very fast in those early stages, despite the horrendous fuel consumption. Rockets are like that. It's one of the reasons we don't bother with them when a gun will do the job.
Let's let the attacking missiles have a mass ratio of 4.92, a final velocity of 6373 meters per second, which give it a kinetic energy of 20.3 megajoules per kilogram of final mass. Considering the fuel consumed, that gives us 5.18 MJ/kg of kinetic energy per kg of fuel consumed, which is compares favorable with the defenders 2 to 6.5 MJ/kg of laser energy. From an energy per mass standpoint, using this particular fuel combination, that's as good as it gets for the attacker, and I'm even considering that the entire missile constitutes the payload, so fuel pumps and valves are figured in as part of my final mass. I'm also not allowing any extra fuel or mass for maneuvering, so this missile would be simple to dodge. In reality things would look much worse for the missile.
So now the defender has to disable this incoming missile with laser energy. Supposing that the warhead was uranium then it can be vaporized with just about 5 megajoules of laser output per kilogram, allowing for about 50% reflectivity of the vaporizing uranium under high-intensity laser light. So the defender has to expend anywhere from 1 to 2 kg of fuel to vaporize the missile that took almost 4 kg of fuel to deliver.
Yet this missile isn't really going very fast. Suppose it has a 10" diameter missile and 100 kg mass, and is being targeted by a 100" diameter laser the defender can keep his beam efficiently on it (84% of the emitted light striking the missile) at a range of 401 miles, giving us 1 minute and 40 seconds to vaporize it. We need 500 megajoules to completely vaporize it, which we could deliver with 5 megawatts of emitted power maintained for 100 seconds flight time, and since we're already working to put a megawatt class laser on an AC-130 gunship this probably isn't going to be very hard. So the missile is not only defeated but vaporized, at a cost of 77 to 250 kilograms of fuel for the defender, but launched at a cost of 392 kilograms of fuel for the attacker. As mass expenditure goes, which includes the expended mass of the missile, the attacker lost 492 kilograms and one very expensive piece of hardware.
If the attacker launched a full volley of these "energy efficient" missiles, amounting to 10% of the mass of his ship, then these missiles' final mass is about 2% of his ship mass. If both ships weighed 1000 metric tons then the warheads weigh at most 20 metric tonnes and could be vaporized by turning 15 to 50 tonnes of fuel into laser energy, which is just 1.5 to 5% of the mass of the defending ship. Yet these missiles are allowing the defending ship almost 2 minutes of firing time on them, which gives the defender plenty of time to use a far more energy efficient defense.
We've been trying to vaporize the incoming warshot, which requires about 2.5 MJ/kg for uranium, but I'm assuming it's 50% reflective. If instead we merely melt the incoming missile then we're assured it can't track or steer, since machines don't work very well after they melt to a blob. We can melt uranium by expending only 0.168 MJ/kg, even if it maintains 75% reflectivity under our laser pulses this would still require only 0.672 MJ/kg, or about 7 times less energy than vaporizing it. So now the volley that cost the attacker 10% of his ship mass only cost the defender 0.2% to 0.7% of his own ship's mass, an energy advantage of 14 to 1 up to 50 to 1 in favor of the defender. Of course the defender still has to void the molten blob of missile, but even if the missile is as small as 10" in diameter the defender only needs to move a ship length in 100 seconds. If our 1000 ton ship is about 150 feet in diameter then it needs to accelerate at 0.009 m/sec, or about 1/1000th of a G, reaching a final velocity of about 1 meter per second relative to its previous. So our 1000 ton ship had to burn about 228 kg of fuel, or 0.023% of its mass to dodge the shot. That disappears into the round off error and the defender has defeated the expensive attack at almost no cost.
So the attacker will switch to something like beryllium for his warshot, since instead of the 0.17 MJ/kg it takes to melt uranium, beryllium takes about 2.8 MJ/kg. This makes up much of the energy deficit, but beryllium is only about 10% as dense as uranium and the energy efficient laser range doubles. The attacker wants to minimize that range by using smaller warheads, while the defender wants to increase it with larger aperture mirrors for his lasers. So the defender has to work 10 times harder to melt the warhead but has twice as long to do it, and his job of dodging is even easier, assuming he neutralizes the missile twice as far out.
I'll spare you the details, but the warhead is likely going to be beryllium, boron, graphite, or some ceramic. It will probably be designed to ablate like an Apollo era heat shield instead of just heat up, because vaporizing a material generally takes about 10 times as much energy as just melting or boiling it. By making the laser have to vaporize its way from the front of the missile to its back, the attacker is forcing the defender to expend more energy while assuring that the attacking missiles penetrate deeper into the defenses.
Any type of nuclear or high-explosive warhead would likely be abandoned, since explosives just mean your weapon has a relatively low-temperature thermal self-destruct mechanism built in, and the defender happens to be using a thermal attack on the missile. The only advantage I see for a nuclear weapon is that when the defending laser is focused on it and fires, the warhead could self-detonate and hopefully fry the defending laser, which happens to be focused right on the warhead. Still, it seems rather wasteful to trade one nuke for each defending laser on a ship that would probably have dozens.
But let's get back to the basic energy problem. Our missiles are already probably far to slow to even have a chance of being effective, and if the attacker increases the velocity using conventional fuels then the ratio of kinetic energy to fuel consumed starts to drop. For example, using the same fuel we'd have to use a mass ratio of 26 to 1 instead of 4.9 to 1 just to double in inbound missile velocity. The attacker is now forced to burn up far more mass as fuel, so for a fixed initial mass his volley is one fifth has heavy and thus five times easier for the defender to melt or vaporize, while the defender merely needs to fire on the missiles in half the time, yet only has to deliver a fifth as much energy. His required power levels have actually dropped. So the attacker comes up with a more efficient missile, say switching to nuclear thermal to provide twice the final velocity with the same warhead mass. Yet the defender just has to double his output power to deliver the same amount of energy in half the time, and his advantage remains.
As the attacker tries to further stress the missile defenses he'll continue to move to higher and higher velocities, trying to get the missiles so fast that they can't be dodged even if they're disabled at a fairly long range. However, the energy required to vaporize a given warhead mass is constant, whereas the kinetic energy of the warhead, which has to be supplied by the attacker, keeps going up with the square of the missile velocity.
For example, when you get uranium up to around mach 6.5 it has enough kinetic energy to vaporize itself. Of course this vaporization would require hitting something, and this would require the warhead to come slamming to a stop. Titanium doesn't cross this threshold till mach 13.6, and graphite till about mach 35. Below these velocities it takes more energy to vaporize the warhead than it did to accelerate it, assuming a 100% efficient propulsion method. Past these velocities it's physically impossible for the attacker to have expended less energy launching the projectile than will be required to vaporize it.
Now based on what we know about dodging, it takes a very fast dumb shot to be able to hit the defending ship. Yet as the defending lasers grow in size and thus efficient range, the shots have to come faster and faster to constitute an unavoidable hit, and even more so to cross the defensive range of the lasers so fast that the lasers simply can't deliver sufficient joules in the available time to vaporize the warhead. As this game is playing out the attacker keeps spending vaster and vaster amounts more energy getting his warheads up to speed than the defender is expending to destroy them, and you quickly cross a point where the missile or any other physical projectile is simply a failing technique to employ.
For example, if I speed up my missiles by a factor of 10, their energy expenditure went up by a factor of 100, yet the energy spent to destroy them is unchanged. It's a losing game.
However, their may be regions of this game where you can try to weight the attacker with enough advantages to just make it pay off, and never discount human cleverness, but the defender could double his mirror size to double his effective range, requiring the attacker to double his missile velocity and thus either vastly slash the mass in his attack or vastly increase the energy in his attack. I don't think certain possible advantages of the missiles are sustainable against the long term trends.
I wish my analysis hadn't gone this way, because I really love missile design, but it looks like beam weapons are the way to go, since their "projectile" is massless and not subject to the very trends that say high speed missiles aren't a good primary weapon for fleet attack. If the result is that projectiles aren't able to penetrate laser defenses, then they're relegated to roles such as eliminating stragglers after a fleet action, using grazing angle nosecones that lasers can't easily cope with, or hitting ships whose defensive lasers have been otherwise disabled. We may end up playing a game of energy beam weapon versus energy beam weapon, which will probably more resemble WW-I naval thinking where you need to slightly outrange and outgun the enemy to sweep to an overwhelming victory.
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If you use a beaded ship design you don't even need spend mass to move out of the way of a missile. You just move the bead out of the way and why you may want to use chemicals to fuel the beam weapons. They can have a much higher energy peak and when they are spend you can use them as "fuel" for ion engine.
Posted by: carl at May 27, 2004 9:31:04 PM
Interesting. My concern would be not energy but power. If I have 10% of my ship mass as missles, it seems likely I can launch them all more-or-less at once.
The defender may have a 5-1 energy advantage, but to defend himself with it that requries that his lasers have 20% the power output of my missles; I'd want to see at least rough estimates for laser/generator/battery mass (in terms of KW/Kg) to verify that's plausible.
Or to put it another way, if each laser can destroy 5 missles in an incoming volley, a laser had better mass less than 5 missles.
Of course, the real issue is cost, not mass, but given the need for drive systems etc. mass is probably a reasonable proxy.
Posted by: mike earl at May 28, 2004 10:50:40 AM
Very interesting assessment, but I think you make some assumptions that aren't neccessarily valid:
1) You assume the attacker needs to supply all the kinetic energy of the missile. In many scenarios, the attacking and defending ships will already be approaching each other with velocities on the order of 10's of km/sec (normal orbital velocity). That totally changes the energy equation.
2) You assume that you can put a 100" laser beam on a 10" target at 400 miles. Assuming a maximum allowable error of 45", your aiming system needs pointing accuracy of 0.37 arcseconds. That is _very_ hard, especially from possibly accelerating and/or rotating spaceship. If the missile is heading for a different ship, the job gets even tougher - you need to maintain sub-arcsecond pointing while slewing at perhaps one degree per second. Even if the missile is coming "straight down your throat", it will still be thrusting from side to side as it maneuvers for a hit - your laser needs to track that.
3) You assume that detecting and tracking the missile is limited only by the resolution of your optics. The real task is limited by several other factors: (a) separating missiles from background objects such as stars. (b) missiles are fast (a few minutes flight time) and telescopes are slow (seconds or more per image) (c) space is big, and the field of view of high resolution telescopes is small. (d) missiles will be designed for stealth - they will fire thrusters in short bursts only, they will be painted flat black, and they will be kept as cool as possible.
The folks doing sky surveys for asteriods that might someday hit the Earth talk in terms of decades to locate all objects in the sky over 500m in diameter. Granted, many of the objects they are seeking are much further away than missiles during space combat - but to assume that you can spot one meter diameter missiles in seconds (or even minutes) seems wildly optimistic.
Posted by: John at May 28, 2004 11:55:23 AM
I share your concern about tracking, but I don't think it's insurmountable. I suspect the answer would be to use optical (IR?) sensors to detect/identify missles and a separate active radar system for targeting. Aiming would be hard but not obviously infeasible.
The relative velocities point is interesting, especially with regard to defense of fixed locations (imagine a mass driver on a moon throwing gravel into likely approach windows...). It would also matter if ships could use some kind of nuclear drive that wasn't feasible for missles; otherwise you could just put the fuel into the missles directly.
One relevent issue is what exactly space warships are used for besides destroying each other: destroying transports? Ground bombardment (strategic or tacticial)? This would affect what kind of engagements would tend to be fought.
Posted by: mike earl at May 28, 2004 12:16:48 PM
Good point Mike,
When I started out I was thinking that the only way to penetrate the defenses would be by having the volley come in a dense enough wave to overwhelm the maximum power output of the lasers, since if we had a nuclear reactor powering them their long term energy output is enormous, but their maximum power output could still be swamped.
The lasers can probably deliver extremely intense pulses if your power system can supply or store up the required energy. Currently the record for laser power is beyond megawatts, gigawatts, and terawatts, and into the petawatt class. However the pulse durations we're using in that range is very brief, so the total energy per pulse is less than a kilojoule.
However, my chemical fuel vs. chemical power source already seems to have an advantage for the laser, and as we try to increase the incoming missile velocity to make a more intense "pulse" of warheads, the attacker starts having to expend larger and larger amounts of energy and mass, while the defender just has to figure out how to deliver his fixed energy in a more intense pulse. So let's look at some solutions to the problem of power storage.
As for storage, lithium-ion batteries are already at 280 kJ/kg, but I really doubt anyone will consider batteries.
Fuel cells lack the energy density we need, but here's a fuel cell PDF.
A composite flywheel can get you into the 500 kJ/kg class, and on paper a carbon fibre flywheel should be good to 1000 kJ/kg. That's a better energy density than we need to melt most warheads on a weight for weight basis, even ignoring the mass lost in launching the missile. And current composite fly wheel systems are small and you can buy them off the shelf, plus you can strip off their vaccuum pumps and just port them to empty space, so maybe we can get a discount.
However, a much better alternative is to use superconducting magnetic energy storage, were you store energy in a coil of superconducting wire. Strange as the idea sounds this is what's keeping the lights on in northern Wisconsin and Mississippi. With sub-millisecond response times and the ability to discharge in less than a second they might be idea for space fleet use. They are currently available into the gigawatt class, and it's a very new an immature technology where massive improvements can be expected. Since by nature they're a DC technology, you could probably hook them directly to laser diodes with only very small conversion losses.
With technologies like these available, which will be continuously improving, it's likely that our power limits may be in the actual lasers instead, in terms of reflector area. But we'll just keep having to go through the math.
Posted by: George Turner at May 28, 2004 5:35:46 PM
Good points too. However I'm seperating out the act of firing the missiles from the equivalent of a high speed firing pass where the kinetic energy is supplied by the ships themselves, since the defenders could see the enemy ships accelerating and use a countermove, even matching the delta V of the attacking ships in the same direction, which is a standard French Navy technique for not engaging English fleets.
If you had kinetic weapons in a reverse orbit it's likely that the defending fleet wouldn't move themselves into harms way in the first place, although this may be unavoidable. However, they could also keep to the high ground an possible take shots at the orbiting missile launchers until the threat is neutralized, while making sure the kinetic kill weapons have to expend energy climbing up out of the gravity well to successfully attack. Plus, orbital velocity weapons are going to be fixed in their closing velocities.
As for pointing accuracy, we were achieving such feats back in the 1980's with a 747 firing on Sidewinders where everything was also subject to turbulence and vibration, so it's probably not an insurmountable obstacle.
However, you are correct in noting that if the defending ship gets hit with anything the optics are going to ring like a bell for a while, unless of course they're mounted in electromagnetic bearings, which might not be a bad idea.
But in terms of the difficulty of laser tracking, this is stuff we're already doing now.
You assume that detecting and tracking the missile is limited only by the resolution of your optics. The real task is limited by several other factors: (a) separating missiles from background objects such as stars.
But your onboard computers already know where all of those are, including variable stars and small planetary bodies. You've also got a network of optical systems up there keeping watch on everything. It's very likely that your tracking scopes are not even on your ship, and in space there's little limit on scope size. You could even go with aerogel telescopes a hundred feet in diameter, making use of the world's best thermal insulator with the best strength-to-weight ratio to support a very thin optical surface.
(b) missiles are fast (a few minutes flight time) and telescopes are slow (seconds or more per image)
But the missiles have to come from a missile launcher, and there's probably no point in the previous ten years where you weren't looking at the enemie's $50 billion dollar piece of hardware. There's no way the missile can accelerate without leaving a trail of very hot gas behind it, which makes a plume. So you just have a more sophisticated version of the missile threat warnings that our fighter planes already employ. Further, as long as the missile is firing you can use the plume to provide updates on its projected flight path, and unless it fires thrusts again it has to lie on that flight path, so you know exactly where to look. The only way the launching ship can counteract this is by using lasers to try and blind your optics, but you might have hundreds of different satellites in your network.
(c) space is big, and the field of view of high resolution telescopes is small. (d) missiles will be designed for stealth - they will fire thrusters in short bursts only, they will be painted flat black, and they will be kept as cool as possible
But here we get into another problem. You already are watching the enemy ships, just as they are watching you, so you don't have to scan the whole sky in real-time. The missile left a major exhaust plume when it launched, so you know where to look, and if it can only use cold-gas thrusters to maneuver then its maneuver envelope is extremely limited. If you even suspect that hostilities are imminent you can just keep your fleet slowly accelerating in random directions, just like anti-wolf pack zig-zags, and such a missile volley would miss.
Posted by: George Turner at May 28, 2004 6:02:04 PM
You could use magnetic propulsion for that. It is not really fast but is fast enough for zig-zags. Especially if you hide the center of mass
Posted by: carl at May 28, 2004 7:02:32 PM
That might be an interesting technique, Carl, since our fleet doesn't necessarily have to expend propulsion mass as long as the center of gravity of the fleet is unaccelerated. Of course doing this may either be a bit difficult, inefficient, or involve guys pulling a hawser with a capstan.
I'll keep calculating on some missiles, powerplants and whatnot to see if I can flesh all this out with some harder numbers.
Posted by: George Turner at May 28, 2004 8:50:32 PM
'as long as the center of gravity of the fleet is unaccelerated'
I have a half-formed idea about pairs of ships, or binary ships, or ships and rocks being tethered in pairs and spinning to create artificial gravity and/or variable closing velocities (launch missles while moving toward opponents, then cut the cable as you're moving away).
I haven't quite worked the numbers but I'm still thinking the drive may dominate. A propulsion system capable of pushing across interplanetary distances in reasonable time is pretty darned impressive. If we can sustain acceleration of 2Gs, I suspect you can pretty much just run away from any chemically powered missle given maybe 3-5 minutes warning - it just doesn't have the delta-v to catch you. Assuming your drive is aligned properly...
Posted by: mike earl at May 29, 2004 10:10:16 PM
interplanetary distances can be done with ease wih much lower accelerations. You need a runtime of a few hours at 2G to have a traveltime of a few months to travel to the inner 6 planets. So a .1G engine will need to burn for a few days to accomplice that. A .1G engine is in all likelyhood much more efficient than a 2G
Posted by: carl at May 30, 2004 1:30:01 AM