Interception of missiles in space combat

Question

How to approach this problem!

Numerically. Trying to find some closed form equation to spit things out is extremely difficult, especially in the face of how well this lends itself to numeric methods. Solving this numerically is better in every way.

I was able to get a proof of concept out at 1 am in a few minutes, and causative was able to make a program that iterated numerically with full features in not much time at all. Coming up with how big things are, how fast, how accurate, and how good your sensors are will take a lot of research if you want to postulate the future, and with that basis you can 'turn knobs' for a look and feel you'd like if you're writing a story.

Basic C# code numerically solving this

https://pastebin.com/tq3d2TEr Seems that, yes, you can rather easily, if you don't have a factor to account for missiles being hard to spot. Interesting.

Premise

I'm trying to make a formula for rough estimates to find the "time to kill" of an inbound missile with a starting range, terminal range, the missile's initial closing speed, its dodge dV, its cross section, how many hits to kill (could be 1), and the jitter of your weapons that are trying to fire at it.

For people who write Sci-Fi and like to include numbers, or just people interested by the problem space, having a way to 'turn the knobs' of "what if this is better, what if that is worse" to see what works and what doesn't would be helpful. The problem is, such an undertaking is beyond my present abilities.

The first thing that came to mind is "oh, integrate probability over time!" but I have no clue how to do this, being decades out of school. While I have formulas for likeliness of hitting a target based on jitter, range, and so on, at a fixed distance, all the dependent variables of moving through space, and integrating probability, is something I have zero experience with.

Another thing I have to clear about is that the point of this is to have multiple terms to plug in, given that these are all assumptions. Given that this is to help fiction writers, not the space force, there's no right or wrong answers, as much as "let's assume this is better than that, what shakes out?"

Finally, given that this is inherently general, if I plugged in a very low velocity but a large cross section, that is, a ship, and not a missile, this formula or something very similar could determine time-to-hit on the ship firing the missiles, and thus determine how far away they need to be to not get chewed up by anti-missile weapons themselves.

Definitions

Missile: a maneuvering, disposable vehicle that either crashes into something or carries a ranged warhead.

Laser: A weaponized pulse laser, likely with a very large aperture (100+ meters!) and UV wavelengths. These damage by ablating what they shoot. At very long ranges, due to diffraction, it doesn't drill as much as flash away a wide, flat region of what it does shoot.

Particle beam: I'm assuming the use of either neutralized or relativistic particle beams so you don't have to worry about bloom. You still have to worry about hitting anything. These cause damage by irradiating the insides of whatever they hit, and depending on the specifics of the particle beam, cause damage not unlike making a needle to pencil sized cross section of whatever they intersect with behave not unlike it was turned into det-cord. Some particle beams may well deposit their energy in a shallow cross section, too, but that isn't so important for this exercise.

Jitter: The ultimate limit of precision that your weapons and sensors have to deal with. Measured in micro to nano-radians.

Particle beams can be approximated as points. Lasers, however, not so much, since even with UV lasers and 200meter apertures, spot sizes can be in centimeters or meters over the distances I'm thinking of, which goes from .1 to 3 light seconds. So, you can have a cross section intersecting another cross section.

Formulas I've found through research

If you know your jitter J, and your range R, the diameter of the region your beam wanders over is, roughly: $D_r=2\cdot R \cdot tan(J/2)$.

For very small ranges, you can approximate with $RJ$.

If your target has a profile area A, you can approximate the chance of hitting with: $A / ( (\pi/4) * 2\cdot R \cdot tan(J/2) )$

I do not know how to account for a profile of your spot size, or for that matter, LaTeX.

Now it gets hard

How do you go about integrating this?

I'd need to include a firing rate for pulsed weapons, as much as how many of them I have, so one would have some finite number of shots over time. You'd start from the outside of the envelope (let's say 3 light seconds for now) and it would end with the missile hitting the ship trying to shoot it down, or reach the minimum range for a warhead like a reactor-pumped laser or a bomb-pumped particle beam, casaba howitzer, or whatnot.

That's where I just stare blankly at my screen and wish I did more stats back in school.

Oh, let's make it harder

Uncertainty radiuses due to jamming or other EW methods. I wouldn't ask about specific modeling, I'd want some simple polynomial term that can be used, and a cutoff for "burn through", that being your own active sensors 'burn through' the jamming. Jamming in space could be many things, such as the ship that fired the missile having a spot size big enough on their lasers they can't miss and dazzling your sensors, or the missiles being multi stage, and the n-1 stage backlighting them with radar, lidar, lasers, or whatnot.

Put another way, if a missile's cross section is, say, a meter, your sensors won't necessarily be able to precisely know where it is exactly with the enemy spamming your sensors with nonsense to dazzle them, so there would be a multi-meter blob that you'd think it's in, at least until it gets close enough your own active sensors can get returns.

Really hard

The laser spot size getting smaller as the missile approaches. This is roughly 1.22(Distance)*(Wavelength/Aperture). Instead of flashing the surface, you can start drilling higher and higher aspect ratio holes into the target. On the other hand, if you have enough time, why not just flash the entire thing and not miss?

The Space Force is now interested

Active maneuvering? An earlier stage acting as a missile bus coming in obliquely and taking pot shots as it arcs away? I know 'path integrals' are a thing but I remember little about that part of Calc 3

And now something completely different

Is this even viable? Could making a program to actually simulate be easier?

Chat Responses

"I don't think lasers work"

General response

Based on what? While not relevant to the question, it's a matter of aperture size (mirror or lens), laser wavelength, and power. In this setting, ships are going to be big, and powerful - terawatt or hundred-gigawatt effective power will be what is trying to shoot at missiles over multiple light seconds. Many calculators are online to compute drilling speed over time, or per-pulse-train. If you want something to start with, go with an excimer laser deep in the UV (157 nm), an aperture size of 200 meters, and a power of 1TW. Yes, that's big. Yes, that's powerful! The setting I have in mind has goes big.

Mirrors

In the first place, once you reach a certain intensity, it does not matter what it's made out of. The electric field from that spot will be so intense that nothing chemical will withstand it. Electrons will be smacked off of nuclei, and then things go crazy. I believe it is a coulomb explosion. Once you get into extremely high intensities, you actually make a reflective plasma that means you don't put all of the energy in the target, but I'm not qualified to speak to it. Yes, you can make a "plasma mirror" or "plasma lens" with this effect, but you run into efficiency issues and other complexities outside of "it can work." No, this isn't really useful as an armor.

https://en.wikipedia.org//wiki/Laser_damage_threshold If you want to go into it, this briefly discusses how lasers do damage, such as dielectric and avalanche breakdown.

Prior art calculators to play with

http://panoptesv.com/SciFi/LaserDeathRay/DamageFromLaser.php This also comes with a lot of information if you want to scroll down and read it all.

Space is big

If a missile is traveling multiple light seconds while you shoot at it, unless it's juking and dodging like a space squirrel on space crack, you're going to smack it quite a few times. Not only that, but the spot size of a laser could easily be as big as the entire nose of the missile, perhaps by design.

Sensor attrition

Valid, but as some have already said, drones scattered around your side of space (and maybe a few shot at the enemy) with line of sight laser communications that are cryogenically cold are useful. While they can't be too big lest they make odd holes in the sky, or you handwave meta materials and hope they don't occult a star, this will help. Also, messing up datalink would basically require glassing the outside of the ship by getting a nuke close enough. If they've got almost an hour and a half to do so, that's not so guaranteed, now is it?

Anti missiles

Oh, indeed. You want to have screens of some sort. Drones (glorified missile buses, or, a big stage that has multiple terminal stages) out ahead of you, if you know the enemy is coming, are a great idea. The same formula I want to make would be useful to see how the enemy could shoot them down.

Answer 1

Edit: added a simulator for missile interception.

So.

• Missile is moving towards you at speed $v_M$.
• You have a weapon (laser or particle beam) that can shoot out an attack at speed $v_a$. $v_a$ is much faster than $v_M$.
• The attack can be modeled as a cone with angle $\theta$, accounting for jitter and dispersion. $\theta$ is assumed fairly small.
• The attack cone gets slightly larger because of the missile's frontal radius $r_M$. This will make it not quite a true cone; the effective radius of the attack at distance $x$ will be $r_a(x) = x \tan(\theta) + r_M$. If this sort-of cone passes over the perfect center of the missile, the attack can do damage; if not, it can't.
• You have a function $\mathrm{DAM1}(x)$ which tells you the average amount of damage that the weapon will do, if the cone of the attack touches the missile when the missile is at distance x. When you have done 100 damage cumulatively, the missile is destroyed.
• Your weapon can fire $f$ times per second.
• You predict that if you shoot the weapon now, the attack will meet (or miss) the missile when the missile is at distance $x$ from you.
• The missile will maneuver sideways randomly to dodge potential attacks. So when the missile is at distance $x$, you only know it will pass through a circle of uncertainty with radius $r_U$.
• $r_U$ increases based on the missile's maneuvering acceleration $a_U$, and based on the delay $d$ before your attack arrives. The delay includes speed of light delay ($d_1$) for you to see where the missile used to be, processing delay ($d_2$) to predict where it will be, the time to physically aim and fire your weapon ($d_3$), and the time for the attack to arrive at the missile ($d_4$).
• $r_U$ also increases based on any uncertainty in your calculation of the missile's last known position, accounting for jamming or decoys or whatever else the missile might try to do. Call this uncertainty factor $r_{US}$ for Uncertainty of Sensors.
• We can combine the sources of uncertainty like this: $r_U = \sqrt{r_{US}^2 + 1/4 a_U^2 (d_1 + d_2 + d_3 + d_4)^4}$
• Your attack cone at distance x consists of a circle with radius $r_a = x \tan(\theta) + r_M$.
• You aim the attack so that your attack circle $r_a$ will land anywhere, at random, within the missile's circle of uncertainty $r_U$. I think normally we'd expect that $r_U$ is larger than $r_a$. (If not, then the attack circle would just always hit).

So you have this circle of uncertainty $r_M$ for the missile, and you have another circle $r_a$ for the attack. The expected damage from a single attack, accounting for the missile's position uncertainty, is therefore $\mathrm{DAM}(x) = \mathrm{DAM1}(x) \cdot (\pi r_a^2 / \pi r_U^2) = \mathrm{DAM1}(x) \cdot r_a^2 / r_U^2$. Unless $r_a > r_U$, in which case the expected damage is just $\mathrm{DAM}(x) = \mathrm{DAM1}(x)$.

Now we can also talk about the damage per second. Damage per second when your attacks are hitting at distance $x$, is $f \cdot \mathrm{DAM}(x)$.

So, how long does it take to shoot down the missile? We want to measure from when your sensors first spot it, at distance $x_0$ and time $0$. At time $t$, the missile will be at distance $x_0 - v_M t$. So, average damage per second as a function of time is $f \cdot \mathrm{DAM}(x0 - v_M t)$.

Total damage by time $T$ is $\mathrm{TDAM}(T) = \int_0^T f \cdot \mathrm{DAM}(x0 - v_M t) dt$

The missile is destroyed when $100 = \mathrm{TDAM}(T)$.

Now, we can't do this integral without knowing $\mathrm{DAM1}(x)$. For a laser, we can suppose damage falls off as the square of distance; $\mathrm{DAM1}(x) = k/x^2$. This would be assuming the laser beam at distance $x$ is substantially wider than the missile. For a particle beam with no dispersion, $\mathrm{DAM1}(x)$ is a constant $k$; if the beam hits the missile, it doesn't matter what the range is. For a particle beam with dispersion theta due to jitter, $\mathrm{DAM1}(x)$ is a constant $k_1$ when the range is close enough to ignore the dispersion. This could be used as a rough estimate when $x \tan(\theta) < r_M$. When the particle beam range is large enough that the jitter is substantially larger than the missile, $\mathrm{DAM1}(x) = k_1 \cdot (\mathit{area of missile}) / (\mathit{area of jitter}) = k_1 \cdot r_M^2 / (x \tan(\theta))^2$. This could be used as a rough estimate when $x \tan(\theta) \geq r_M$.

Now we can do the integral. Or we can ask Sage Cell to do the integral, because that's easier.

Let's use Sage Cell to run an example. There is a mosquito flying initially 10 meters away (x0 = 10). The mosquito is 5mm radius (r_M = 0.005). We aim a pulsing laser at it that shoots ten times a second (f = 10). The laser spreads out to 2cm radius over the ten meters (theta is about 0.002). The speed of light delay to see the mosquito can be ignored (d1 = 0), the processing delay is 200ms (d2 = 0.2), the aiming delay is 100ms (d3 = 0.1), and the time for the laser to hit the mosquito can be ignored (d4 = 0). The mosquito flies slightly erratically so that in that 300ms it might wobble 5cm left, right, up, or down. This means its effective acceleration a_U = 1.111 m/s^2. We are also having trouble seeing the mosquito, to within 2cm, which means r_US = 0.02. We have to settle for aiming the beam somewhere in that slightly more than 5cm radius circle where the mosquito might be, and hoping it hits. The mosquito is slowly flying towards us (v_M = 1). And, it would take 10 of those laser pulses to kill the mosquito at 10m, from which it follows that k/100 = 10, or k = 10000.

var('x x0 t k theta r_M d1 d2 d3 d4 a_U r_US v_M f T')
x0 = 10 # m
k = 10000 # m^2
theta = 0.002 # radians
v_M = 1 # m/s
r_M = 0.005 # m
r_US = 0.2 # m
a_U = 1.111 # m/s^2
f = 10 # 1/s
d1 = 0 # s
d2 = 0.2 # s
d3 = 0.1 # s
d4 = 0 # s
r_a(x) = x * tan(theta) + r_M # meters
r_U(x) = sqrt(r_US^2 + 0.25 * a_U^2 * (d1 + d2 + d3 + d4)^4) # m + m/s^2 * s^2 = meters
DAM1(x) = k/x^2 # laser wider than the missile. units of "damage" (k is m^2. m^2/m^2 = dimensionless function of meters)
DAM(x) = DAM1(x) * r_a(x)^2 / r_U(x)^2 # dimensionless * m^2/m^2 = dimensionless function of meters
TDAM_indefinite(t) = integral(f * DAM(x0 - v_M * t), t) # f has units 1/s so integrand has units of 1/s. integrate over time cancels the seconds. result is dimensionless.
TDAM(T) = TDAM_indefinite(T) - TDAM_indefinite(0) # between time 0 and time T, how much damage has been done?
plot(TDAM(T), T, 0, 6) # plot damage as a function of time from 0 to 10
#find_root(TDAM(T) - TDAM(0) == 100, 0, 9) # solve for when 100 damage has been dealt

The result is here. You can see from the plot that the laser will kill the mosquito in just under 6 seconds.

I've also made a full simulator here. It can be run in the browser.

Answer 2

You can add uncertainties like perpendicular vectors. If your thrown rock has a standard deviation of 3m, and your target position has a standard deviation of 4m, then the total accuracy from the two combined will be a standard deviation of 5m. Or just think of an average scatter, rather than standard deviations: the result is the same.

Let's start with trying to hit a ship with an impacting piece of tungsten (say). Now this is tricky - most missile attacks do not try to ram the ship, but to blow up close enough to it - but this is a place to start. If the ship is travelling under its own momentum or uniformly accelerating, you can predict with some precision where it should be. Think of the Voyager probes passing Jupiter, and people worrying about a missing 0.01s.

If you ship has power, and it is aware of a threat, it might put a slight weave in its route. This would waste a small amount of deltaV, but it might mean you could not predict its position from a week away to within 100m.

It therefore follows that your impactor should have some ability to fine-tune its aim. It could have camera sights on the front, and a subtle thruster so it can track the ship. This is good when you are a long way off, but it becomes increasingly useless the closer you get.

If your average error is 100m, then you can blow up your impactor at the last moment. If the shrapnel cloud is about 100m in radius, you are probably going to hit it with something. If your impactor relative velocity is large, a small hit may well do enough.

Can the ship launch a defence? A thin tungsten rod with a pointy end would be very difficult to spot in space, seen end-on from the target ship. It might look for the boost phase when the impactor is given most of its momentum. It might catch some course-correction, though it is hard to see how if the impactor uses an ion drive. The ship's best defence would be to have a wiggly and unpredictable course. It all feels like WW2 ships and submarines, doesn't it?

I have not tried to deal with lasers and particle-beam weapons. The ship will have a hard time dodging, but there may be practical, lightweight defences that could thwart these.

Answer 3

Frame Challenge

Based on my comments and discussion with BMF - I don't think that Lasers for long range point defence are a good option.

Now, I'm assuming for the moment that the type of Missile we are talking about in Space is a Ship-killing missile. Whilst the hazardous conditions of Space means that something that would be a mere annoyance on Earth could be catastrophic in Space - I'm going to presume that for such an advanced civilization (that is having Space Lasers and Space Missiles and Space Combat) - we have made our ships survivable enough to be on an equivalent basis for modern Warships.

That is, something around the size/weight of the P700 Granit Anti-Ship missile (probably larger - but you get my point).

So, from the Question, we have a max engagement distance of 3 light seconds (800,000 Km)

This means first and foremost that an sensors detecting the launch at minimum have a 3 second delay.

Firstly we have the initial missles speed - you've not stated it, but let's use some numbers for reference - Mach 100 (100 times the speed of sound) gives around 120,000 Kph - so at max range, that's 8 hours to get to target.

Sounds a bit silly? Actually... No.

A modern Tomahawk cruise missile has a flight time of at least 2 hours to get to target. So 8 hours isn't unreasonable for the flight time of a ship-killing missile.

I'm also going to assume that our Space Farers have the concept of layered defence for their ships.

I'm also going to presume that the first few layers of the Survivability Onion have been breached (Don't be there, Don't be seen, Don't be identified, Don't be targeted, Don't be fired upon).

First layer of defence - anti-missile Missiles:

Smaller and lighter than an Anti-Spaceship missile - these can travel faster (say Mach 200) and so they intercept the enemy missile around 500,000 Km away. Initial targetting is done from the Ship, but once it gets within sub lightsecond range, on-board sensors take over.

These would be fired in a cluster and would have a directional fragmentation warhead - Directional so minimal debris doesn't come back to us and so the most is directed towards the incoming threat. Think of it like a giant shotgun - throw enough poo at the wall, something is going to stick.

Also - if there are any Mirrored surfaces or other features to reduce Laser damage, these get damaged and or blasted off.

Next Layer of Defence - Decoys:

If the Missile is still inflight and tracking then we can deploy a Decoy - a small craft with various tricks to 'fool' sensors into thinking it's the target. It moves towards the missile at high speed (say Mach 50) so it will take a while to get there, but the idea is when it's there - the return signal is stronger than our vessel (which is still over 500,000 Km away).

Why not go with a Decoy first? Hard Kill is better, Decoys have a higher cost to them and should they fall into the enemies hands - they will have detailed knowledge of how fake signals are generated and will be able to stop their missiles from getting fooled.

Mid-range Defence - Lasers!

So, now that both of our outer layers have failed, and the missile is within 1 Light second, now it's time for the Lasers to shine (Pun fully intended) - our hope at this point if the Missile is still inbound that there has been some damage done to it during it's flight.

Why within a second? Well because of the distances there is still a 2 second delay (Signal coming from the missile to the ship *and then the laser travelling from the ship to the object) - the Laser array would have to target all possible vectors within a 2 second 'cone':

Imagine a car travelling on a big open bit of tarmac. We know the speed and we know the maximum G-Force it can generate in turning, braking and accelerating (we'll call it 1 G in all possible directions for the sake of argument) - we now have a cone of possibility in a 3D space where the object might be in 2 seconds time. We also know the rough size of the object (say 10 metres long) - so we target every point, 9 metres apart (or similar) within that grid.

As I'm sure you can imagine, the amount of distance we are talking about in that time period is huge - hence why these are a closer-in option to make it practicle to target every point. You could add in some statistical analysis of 'most likely'.

Hopefully the Missile's shielding is damaged and the Lasers inflict critical damage or in forcing evasive moves, the Missile destroys itself due to damage already sustained.

Short Range Defence - Flak

Flak - Similar to our Anti-Missile, but more primitive - we are now within fractions of a Light Second (less than 100,000 km) - we would want to through out a wall of very dense material at very high speeds. This is close range only because not only is it a bad idea to have lots of material blasted into space but with increasing distance, the space between each projectile is increased and the chance to hit is lowered. On the plus side though - this type of system is relatively cheap so you can saturate an area.

Note - when I say 'cheap' I'm not talking money per-se - I'm talking time and resources

Final Layer of Defence - Sacrificial plating

So, now we are panicking, the Missile is mere minutes away, all our other countermeasures have failed - the final option is to send a big chunk of the ship (either something that has been designed to do this or a non-vital part of the ship) - directly into the path of the Missile.

The idea being to prematurely detonate the Missile against a piece of the ship, far enough away from the ship that the main hull doesn't get compromised. Similar to how Spaced Armor/Explosive Reactive armor works on Main Battle Tanks.

We've accepted that the Missile is going to go Bang - we are now on the 'Don't be Penetrated, Don't be killed' part of the Survivability Onion.

Feel free to tweak as needed, add in your numbers - and make some adjustments etc. But the above, for a Military Space Vessel, exepecting to encounter some form of Anti-Ship missile is what seems most logical:

Guided smart projectiles, Decoys, Lasers, Flak, Sacrificial material