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 v_final = 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 KE_missile = 1/2 * M₁ * v_final². 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 M_fuel = 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 M_kg-final = MR-1. The kinetic energy per kg is KE_missile = 1/2 * v_final², so KE_missile = 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.