Space Fleet - Missile Versus Laser

Now I'd like to write a bit about the game of anti-ship missile versus laser defenses and some of the interesting information it reveals. I covered some aspects of this in Space Fleet - Lasers. This post is just going to be a rough analysis, though sometimes delving into the details, to get a feel for the new game you get when you combine missiles and optics together, looking to see if equations and tactics pops out. So as Steven Den Beste often states, don't lose the forest for the trees. We're not trying to write a grant proposal here, we're just hypothesizing on space fleet combat, and fortunately telescopes and missiles are two of my hobbies.

I'll start by noting that a missile in space must maneuver only with rocket thrust, which is unlike our usual conceptions gleaned from looking at Sidewinders and other atmospheric missiles, which are steered with guidance fins. Our basic space missile has two components, a thrust motor that gets it up to high velocity, so it can minimize its flight time through the enemy defenses, and a maneuver motor so it can track the target by moving side-to-side. Our SDI program produced some missiles with just these features, under the program called "Brilliant Pebbles".

Let's assume that once the boost engine has gotten the missile up to speed, it's already got more than sufficient closing velocity to destroy the target. At this point the boost motor is just dead weight, which would be a detriment to agility during the closing phase of the missile's attack. So we'll use the boost motor as the first stage of a two stage missile. The second stage doesn't have only a single motor, but multiple ones aimed perpendicular to the missile's flight path, so that it can jink, dodge, or track a target. Odd as it may seem, our final missile doesn't have a motor in the back propelling it forward, it has little motors that push it side-to-side. You could use a single motor in the back and steer it conventionally, but then most of your thrust is being applied along an axis where you've already got sufficient velocity for the kill, instead of along an axis that you desperately need to maneuver in to intercept the target. So that basic type of missile will be the basis for the rest of this particular post.

Now in terms of what the missile must accomplish, I'll go back my earlier equations from Space Fleet – Evasion to see what our missile has to accomplish. We have a missile traveling at velocity V_m and would like to destroy a ship with it. The ship is going to maneuver along some path with an acceleration A_s, and the ship length along this path (probably the ship's shortest dimension) is L. Our missile is probably going to have its guidance abilities destroyed at some point along the path, but if we can get it close enough and fast enough we can still guarantee a kill on the ship. If we can keep our missile, traveling at velocity V_m alive and on course to within range R_kill=sqrt(V_m²L/A_s) we guarantee a kill on the enemy ship, if the enemy ship can't directly stop the missile.

Note that the faster the missile the greater the range where we can have the missile's guidance knocked out and still guarantee that our path should result in a kill. Also note that if the ship can't accelerate the kill range goes to infinity, meaning we could in theory guarantee a hit from extreme ranges, just like NASA probles heading for non-maneuvering planets. I'm assuming a kinetic kill weapon for the simple reason that it's a highly efficient method when missiles are moving at extreme velocities. For example, in an earlier post I ran some calculations for a missile traveling at V_m=19359 m/sec = 63,515 fps = 43,300 mph = 12 miles per second. If it slammed to a stop the energy released would be 76.3 megajoules, which is equivalent to 17 kg of TNT. Just because of the impact velocity our missile potentially possesses 17 times more energy than it would if the whole thing was made out of high explosives. All they need to do is make a hit.

Laser Weapons

I posted earlier on laser weapons, and now I'd like to mention some fundamental physics that come into play. Optical systems are generally "diffraction limited", meaning that the wave nature of light puts a limit on focusing, which means there's also a limit on how finely a system can direct a beam. The reason is that below about a quarter of a wavelength electromagnetic radiation isn't further affected very much by the precision of the reflecting or refracting surface, and this is why astronomical mirrors are often only finished to "1/8 wave". They're not perfect, just good enough to where adding more surface perfection doesn't keep increasing the performance.

There may be a bizarre way around this problem using materials with a negative index of refraction, and although it sounds strange we're actually able to do just that, but I don't know if a telescope could make use of the effect. I'll ignore such potential tricks and see how a laser would perform if we were to build it like we would either now or in the near future.

So we're talking about diffraction limited optics, and the key formula is also very common to astronomy.

D_spot = 2.44 Rl / D_lens

D_spot is the diameter of the spot that contains 84% of the transmitted light.

R is the range (in meters)

l is the wavelength of light

     If the light is visible it will be from 410x10^-9 meters (violet) to 680x10^-9 meters (red).

D_lens is the diameter of the mirror

For a bit more on high power lasers try this link, which details some of our advances at cleaning up atmospheric distortion so we can retain tight focus when targeting orbiting satellites with ground-based laser weapons. This link discusses some pros and cons of chemical lasers, and also mentions that laser weapons operate at the very margins where any flaw in reflectivity results in sudden destruction of the mirror surface. Yet that the laser weapon designer knows exactly how perfect his coatings have to be and where, and can use active cooling on hot spots. The defender has none of these advantages, and the slightest flaw in his coating will result in a heated region where his coating suffers catastrophic and expanding failure. In short, mirroring your craft might protect against generalized heating but won't succeed in stopping an extremely intense, tight beam. Another use for high-powered lasers is transmitting energy, and this link points out the potential application of powering a lunar base through the lunar night by beaming laser energy up from earth to the same photovoltaic solar array that the base uses throughout the lunar day.

So, we want to stop an incoming missile with a laser, yet the missile designer, if he so chooses, can coat his missile with a reflector every bit as good as the laser's mirror, which gets us to a crucial range equation. Let's assume the laser makes a single reflection of the primary mirror (the big one) and is driving the primary mirror's reflective surface to the very limits of self-destruction. If the missile uses the same coating technology as the laser's mirror then we'd have to hit the missile with at least this much energy density to destroy the reflective coating on the missile, and until this is done 99% to 99.9% of the laser energy is just reflecting away instead of heating. And a missile is certainly still intact if its thin, delicate reflective coating is.

Now the primary mirror is coated with full knowledge of the laser's frequency, so we can have its reflective layer tuned for maximum reflectivity in a narrow band. At present, this might mean that instead of a broadband reflector at 99% reflectivity, we might have 99.8% reflectivity. These numbers will improve somewhat, because optics is a very fast paced field and massive amounts of dollars and physicists are involved in it, and we certainly would never choose a reflector that's worse than what we have available now. For a small example look here or just Google up "mirror reflectors" or "reflective coatings mirror laser".

The defender might or might not have prior knowledge of our laser's frequency, but we must assume that he does. Since both sides are going to be using the best available coating technologies, if they're at all competent, and since I'm assuming some scientific and technical information was being shared prior to hostilities, I'll say that both sides are just as good at reflective coatings.

The formula given above tells the size of the spot that contains 84% of the laser's energy. Since this means our spot is a little bit diffuse, and since we need a spot that's even more intense than what's bouncing off our primary mirror if we're to destroy a similar coating elsewhere, then we need the area of the target spot to be less than 84% of the area of the primary mirror, which means the diameter of the spot has to be less than 91.7% of the diameter of the lens. So our formula for the closest point where we can possibly blow off a technically optimal reflector becomes this.

0.917 D_lens = 2.44 Rl / D_lens

R = 0.376 D_lens² / l

For violet (410 nm) light this comes to R (km) = 917 D_lens² (meters)

or R (miles) = 14.47 D_lens² (inches).

Note that a little 10" mirror has a possibly useful anti-missile range of 145 miles. I've ground and polished an 8 inch and 10 inch mirror by hand, and it's a common size for an amateur telescope. As you get into Hubble sizes the range starts pushing out to almost 1500 miles. At a closing velocity of 12 miles per second (mach 57), you'd have about two minutes to intercept the missile. But of course it won't be alone. But in truth these ranges are going to be much smaller, because there are some very interesting optical tricks the missile designer will definitely use, but more on that a bit later.

Frontal Area of Missiles

Against a solitary enemy ship we might first go with a cylindrical missile, long and thin, to keep the frontal area minimal. If we can keep the nose aimed directly at the targeted ship then the defense lasers are striking the minimal area on the missile. Further, we're probably at the longest range where the laser could just manage to blow off that super reflective coating on our nose, which means the laser's beam size is still pretty large. As calculated previously, the laser probably can't damage our nose reflector till its beam size is less than 91% of its mirror diameter. If we're talking about a 100 inch diameter mirror means that we're at the point where 84% of the laser energy is in a 91 inch diameter circle. If our missile is only 8 inches in diameter then the area being heated by the laser only represents 0.77% of the central beam area. Note that we have a transmitted beam 91 inches in diameter, but only an 8 inch object there to catch the light.

Considering that the 91 inch central beam is only 84% of the total beam energy, this means that only 0.65% of our beam energy is even hitting the missile. That means the beam is 99.35% wasted. As a missile designer trying to penetrate their defenses, I really like that number. They defending ship would surely love to focus the beam tighter and reduce that value, but the diffraction limit on their optics means there's nothing they can do. But also note that if I'd sent in a monster 10 foot diameter super missile then that same beam would put nearly 100% of its energy on target. Small targets at extreme range are not an efficient way for a defending ship to expend its power. If we wait till this small missile has traveled three fourths of the way to the ship then our beam diameter is only 22.75 inches, and 10.4% of our beam energy is hitting it, a sixteen-fold improvement.

This effect creates another range equation that takes into account the target size. For targets that look circular we get a range beyond which the laser beam widens to the point where the beam is fatter than the target, and past which the delivered energy goes down with an inverse square law. I'll denote this range as the energy efficient range, R_ee.

R_ee = D_missile D_lens / 2.44l

For violet (410 nm) light this comes to R (km) = 1000 D_missile D_lens (meters)

and R_ee (miles) = 0.401 D_missile D_lens (inches)

Since the energy efficiency of the defenses drops with an inverse square law past this range, the offensive-missile designer can reduce the frontal area of his missiles to both shrink the defenders energy efficient defensive range or to force him to fight less efficiently at the longer ranges. These equations aren't completely exact, since they're based on that 84% number from the diffraction equations, but the actual values aren't off by much.

Missile Heating

So, we've blown off the super-reflective surface protecting the nose of the incoming missile. Now we need to apply heat to disable, melt, and even vaporize it. Obviously if the missile is has nothing between the reflective nose and the fuel it will immediately explode, since the 0.1 to 1% of absorbed laser intensity, which just blew off the nose reflector, is really going to chew up a fuel tank or guidance system with energy absorptions that are 50 times higher or more. Additionally, the reflectivity of metals drops under high intensity illumination as they melt and vaporize, and will probably be reflecting roughly half the energy instead of 99% of it. So we need to stick some thermal "armor" behind that reflector. As a first attempt, I'll suggest a heavy high-temperature metal, though this may not be nearly a good a choice as some type of high-temperature ceramic.

We're going to go through a couple iterations of missile improvements and see where we end up. First off, suppose I put a slug of uranium up at the nose. I'm using uranium mostly because it makes an incredibly good penetrator in the event we actually hit the enemy ship. Here are some really rough stats on 1 kg of uranium. The final numbers are exact since specific heat varies a bit with temperature, but they're good enough for our purposes.

molecular weight 238.029 - density 18.9 g/cm^3 - 1 kg = 4.2011 moles
Specific heat 0.12 kJ/kgK
Melts at 1405K (So it takes 168 kJ/kg to raise the temperature of 1 kg from 5K to 1405K)
Boils at 4203K (So it takes 336 kJ/kg to raise the temperature from 1405K to 4203K)
Heat of fusion 8.520 kJ/mol (So it takes 35.79 kJ to melt 1 kg)
Heat of vaporization 477.0 kJ/mole (So it takes 2004 kJ to vaporize 1 kg)

It takes 203 kJ/kg to melt 1 kg
It takes 2340 additional kJ/kg to vaporize 1 kg of liquid U

Our 8 inch diameter missile could have a tip that's of course 8 inches across and 6.4 inches thick, which would weigh 100 kg. If it was 50% absorptive of laser energy, it would take about 40 mega joules to melt it, and about 500 mega joules to vaporize it, neglecting that fact that boiling hot uranium will also be radiating some heat back into space. Now some of you may not exactly recall what a Joule is, so I'll mention that it's simply a Watt-second. A 100 Watt light bulb uses 100 Joules per second, so our 40 mega joules could come from a 0.1 second burst from a 400 megawatt laser, or a 10 second burst from a 4 megawatt laser. Considering that the target will be radiating some of the applied energy away as heat (the target will be glowing) it's a bit more efficient to use the bigger laser for a shorter time.

Now obviously, a heavier missile is going to take more energy to melt, disable, or vaporize, and here we get into the frontal area effect again. If I make missile heavier by fattening it, say going to a 16" diameter missile with the same 6.4 inch thick nose, and arbitrarily treat the nose as the payload for rocket equation purposes, then to maintain the same mass ratio (and thus achieve the same speeds), the entire missile becomes four times heavier. But also note that beyond the energy efficient defensive range, R_ee, this fat missile is getting heated up by the defending laser four times as efficiently because it's got four times the frontal area. So the new fat boy missile is no better able to survive long enough to get inside R_ee than the old missile, yet the attacker can only carry a fourth as many of them for the same payload weight. That's a dramatic step backwards. The attacker is better off using four of the smaller missiles, or even better, just make his missile longer and thinner, to provide more frontal area that has to be melted prior to killing the missile.

Now also note that as long as you guarantee that your attacker can't disable your missile before it crosses the "sure kill" range, given earlier as R_kill=sqrt(V_m²L/A_s), the question is simply whether the defensive lasers can completely vaporize your missile, because it's going to hit the attacking ship if he doesn't, unless his point defenses (guns, metal plates, or armor) can stop it.

So the design of missile versus laser goes round and round as we figure out that we need more insulation between the nose and the guidance, rocket fuel, and engines, because melting uranium is really freaking hot. You can plug through the basic rocketry equations to try and optimize your missile velocity versus warhead mass, and go through many other design exercises, but keep in mind we're still only talking about missiles optimized against a solitary target.

Conic Noses, or "Nose Cones"

Of course sometime in the first three days someone will point out the benefits of using a long cone for the nose of the missile, which obviously is just a nose-cone. Normally we use these to reduce drag, but in this case "drag" is irrelevant and we'll use it to better resist the defense lasers.

The energy being absorbed by the nose cone is just a function of the frontal area and the energy in the laser beam, while the actual surface area receiving this energy can be made vastly larger by having the beam strike it obliquely, meaning across the area of a long, thin cone or other shape. The energy striking the nose is just the energy density in the beam multiplied by the frontal area, or pi times the square of the missile radius. The area of our cone is pi * r * sqrt (r²+h²), where h is the length of the nose cone. Not only is the laser beam impinging less efficiently across the surface, probably allowing the initial reflector to stay intact much longer, but as this new nose heats up to glowing it has a much greater surface area to radiate the heat away. It's simply a more efficient design.

Grazing Angle Nose Cones

As the idea of continually lengthening the nose cone to radiate heat away more and more efficiently is pursued, eventually becoming a thin, rolled piece of metal, a startling thing happens. It stops absorbing laser energy altogether, instead reflecting 100.00% of the incident beam.

The reason for this is that the uranium on our nose has what's called a grazing angle, or critical angle, below which all of the light reflects. This only occurs at extremely shallow angles, however, so our missile is now looking more like a long, thin needle or the blade of a rapier. Grazing angle is also dependent on the wavelength of light, as a formula in the above link shows, so the warship could try and move into the UV and X-ray portion of the spectrum to try and take out the warhead. But wouldn't you know it, uranium also happens to be potentially the best grazing-angle reflector up into the X-ray band, at over 100 times the frequency of violet light.

Mutual Defense With Two Ships

So now our missiles are pretty much unstoppable by light emitted directly from the target, which has a profound effect on our fleet tactics. It means solitary ships are virtually defenseless if all they can rely on is lasers. Even our earlier simple cylindrical missile was very much more vulnerable from the side, indicating a clear need to have defending ships cover each other with interlocking fields of fire, and now that we have our new conical super missile the need has become a requirement. We simply have to have at least two ships, or at least two lasers, widely seperated and covering each other, since neither is capable of self-defense, only defense of the companion ship.

Against this arrangement the first thing an attacker will do is try to launch an attack long a line intersecting both defending ships, where neither ship is off angle enough to mount any defense. Yet the incoming missiles are coming from far away, and the ships can circle each other to always "get in formation" as long as they've sensed the incoming missiles at a sufficient range. And if we keep our ships moderately close together, you can picture them as being at the center of a circle, whereby the line intersecting them can be rotated around just by having the one ship move some small distance relative to the other, whereas the attack missile, much further away, would have to move around the much larger perimeter of the much larger circle marking its range. You can imagine the two ships as swordsmen surrounded by enemies, circling each other to avoid presenting an opening.

Multiple attacking ships might try to attack down various axes, but to guarantee success they need to attack from a complete hemisphere, since a line through two ships always has an intersection in any arbitrary hemisphere centered on one ship or the other. The line makes two intersections with a full sphere, but we only need the one angle to guarantee the correct tactical position. But so far using two ships for mutual protection has made sure the "magic" bullet is only magic if you can use a vast swarm of them. And then of course, even if you do launch a swarm, the defending ships will try and eliminate a zone of incoming missiles, then change their mutual positions to put their ship-to-ship axis along the zone they just cleared out, so they can work on the rest of the swarm. But if the two ships are widely separated and the flight times of the missiles is very small, they might not succeed. And if the attacker eliminates one of the defending ships the other is a freebie, because of the "magic" bullet we've developed to take out lone ships.

Missile That Defeats Two Ships

Then we introduce a new innovation, a missile that's almost a disk, or very flat cone, kind of like a squashed rice hat. It's not only taking advantage of grazing angles along a single line to its front, but along an entire plane. Suddenly all it has to do is keep level with the two defending ships so that they're both firing right on its rim. Now we have a magic bullet against pairs of ships, and the defenders have to add a third ship to their formations. That still leaves open the game of the three circling swordsmen, but if you add a forth ship you can arrange them in a tetrahedron and close off all possibility of a magic, unstoppable missile. We've just put the "fleet" in space fleet tactics.

Beyond this we'll have to look at how much energy the ships can dump into an incoming missile volley, because on this will hinge the next question. Obviously if the ships can't stop a single missile, either killing it far enough out for a dodge or vaporizing it directly, then they can't hope to stop a volley, and if they can't stop a single missile just through purely thermal means then they ships of course lose, and we don't have a space fleet, at least not for long. So let's assume that our ships can stop a single missile using purely energy weapons. Yet each missile has to be targeted and fired upon, which consumes a great deal of energy, and the ships only can produce energy at a finite rate. If we attack with one missile followed by another, knowing that the defending ships can stop single threats, then the attacker is losing because he's shedding mass (and money) to no effect against an adversary that's expending neither.

Yet each fired missile requires the defender to expend energy to destroy it, and two missiles take twice the energy to destroy. Following this up, you can fire a volley that's going to require more energy to stop than the defender can produce and direct during the brief time those missiles are moving through the defended zone. So we know we can swamp his laser defenses given sufficient missiles. The question is whether the mass and cost of our missiles exceeds the mass of his ships, in which case we'd lose the game, if it was a battle of attrition.

But calculating all this is going to take a bit of math, and I'm sure I've probably exhausted your patience for now. So I'll just leave off here.