Here's how you keep your guns.

To start off with, we must remember that railguns are only one sort of electric cannon.

*Rail*guns have a conducting projectile trapped between two rails. An electric current run up one rail, across the projectile and down the other creates the Lorentz Force. This tries to push the rails apart, but if they are braced, the projectile is ejected instead. The projectile can be conducting, or held within a conducting sabot.

*Coil*guns have a projectile levitated inside a series of electrical coils. These coils are switched on and off one in series, creating magnetic fields that push and pull the projectile along. The projectile can be ferromagnetic, conductive or containing a magnetic coil within itself.

Railguns are better as lower velocity, lower complexity weapons. They are less efficient due to electrical resistance and friction between the projectile/sabot and the rails, leading to lower velocities, but are rather cheap to produce, and damage to the cannon in operation can usually be fixed by simply swapping out the rails. The projectile has to be heat resistant and able to maintain contact with the rails at a wide range of velocities. The rails can easily cooled by running loops of coolant against the them.

The projectile might need some magnetic shielding too. |

Coilguns are more high-tech and higher velocity weapons. Since they can levitate the projectile and use magnetic fields to manipulate it, they eliminate friction losses. They also generally have lower electrical resistance and if superconducting, can be very efficient. However, they have many technical challenges. Their maximal velocity is limited by how fast the coils can be switched on and off. Cooling the coils themselves is more difficult. Overall, they are heavier than railguns.

Each coil needs bracing against repulsive forces. |

**So what can electric cannons do?**

Impressive amounts of damage, that's what. The main damage mechanic is kinetic. The projectile encounters no aerodynamic resistance in space, so muzzle exit velocity is equal to impact velocity.

At the projectile velocities we are used to on Earth, we get plastic deformation. This means that the forces on impact are so great, that even steel bullets bend and smush like plasticine. How far a projectile penetrates into a dense material, like steel armor, is a complex interaction between kinetic energy, projectile and target material strength, projectile shape, speed of sound within materials and so on...

Soon, we will have operational railguns on Navy warships, later, on the ground. These will enter the hypersonic regime. Even so, they are slower than Explosively Formed Projectiles found in HEAT shells and ATGMs.

Between 800m/s and 2km/s (Mach 6), we have the Newtonian model. Basically, it states that at those velocities, the kinetic energy involved dwarfs the strength of cohesion of the materials. All that matters is the length of the penetrator and the relative density of the projectile and armor.

Newton did more than mess with apples. |

However, the velocities that will likely be involved in space combat are even greater. To cross the vast distances between two spacecraft quickly, projectiles have to be shot at speeds of 20km/s, 100km/s or even a few percentages of the speed of light in some settings.

In that case, even the Newtonian model is insufficient, since projectile is vaporized into plasma upon impact. The best model for these high-energy events is hydrodynamic or 'crater' model. Densities, shapes... nothing matters except the kinetic energy involved and the vaporization energy of the target material.

The hydrodynamic model states that at those velocities, the material strength is irrelevant and you can treat the problem as one hydrodynamic jet of fluid penetrating another stationary fluid.

The equation is

- Penetration depth = Projectile Length * sqrt(Density A/ Density B)

However, even this penetration depth can be surpassed by the crater produced by projectiles of extreme kinetic energy:

The bottom of the crater is usually regarded as the penetration depth, but armor is deformed deeper down. |

- Kinetic energy = 0.5 * Projectile Mass * Velocity^2
- Crater Volume = Kinetic Energy /Crater strength of the armor

The 'crater strength' of the armor is about three times the yield strength of the material. By approximating the crater as a hemisphere, we can work out the radius, therefore depth:

- Crater Depth = (Crater Volume * 0.21) ^ 0.33

Let's say we have a 50km/s coilgun projectile trying to punch through a meter of steel armor:

50km/s coilgun

10kg projectile

Kinetic energy = 12.5GJ

Yield Strength of steel =250MPa

Crater Strength of Steel = 750MJ/m^3

Crater Volume = 16.7m^3

Crater Depth = 1.52m

Our coilgun digs through a

*minimum*of 152cm of armor per shot, and can do so at any distance. Multiple shots will overlap their craters and achieve full penetration of the armor.

The greatest benefit to kinetic weapons is their secondary damage mechanics. Impact craters create shockwaves that are supersonic within the material, shattering and fracturing armor outwards from the impact site and making further impacts penetrate further. Furthermore, shockwaves bouncing off the rear face of the armor causes spallation which have to be dealt with using Whipple shielding.

Not a great representation of metal armor, but displays fracturing and shockwave rebound. |

So we've confirmed that electric cannons are deadly effective at digging through enormous volumes of armor, and thanks to crater ejecta, shockwave fracturing, plastic deformation and spallation, are more devastating joule per joule compared to lasers.

**Why are they shunned by hard scifi?**

The main reason is Time to Target.

Imagine two spacecraft, one equipped with a laser and the other with a railgun. The railgun has to stay outside of the laser's effective range unless it want to be reduced to scrap.

As we've seen in the Laser Problem posts, the effective range of lasers can be tens of thousands, hundreds of thousands or even millions of kilometers. The railgun's projectiles must cross this distance.

- Time to Target = Distance / Projectile Velocity

At 10km/s, a railgun projectile takes three hours and 46 minutes to reach its target from 100000km away. It would need more than a day to cross a million km.

We quickly see that even the fastest projectiles would take hours, or even days, to cross those distances. In that time, they can be destroyed by laser defenses, or the target craft can simply manoeuvre out of the way.

Another problem is their acceleration per meter. Most railguns have a limit in the heat the projectile can survive and the forces the bracing can support, and coilguns are limited by how fast the switching is and how much momentum they can impact per switch. These translate into a maximal increase in projectile velocity per meter length of the cannon.

In the 50km/s railgun example above, the cannon needs to impart 12.5GJ of kinetic energy to the projectile. If it has capacitors that can discharge at a rate of 100kW/kg (super-supercapacitors by today's standards), and is expected to have about 400kg/m length (according to DARPA's EMM project estimates) then it can have the following design:

50km/s railgun

20 tons of 100kW/kg capacitor: 2GW discharge rate

Total acceleration time: 6.25 seconds

Acceleration rate: 8000m/s^2 or about 815G

Railgun length: 156km

Railgun mass: 156000*0.4 = 62400 tons

Total mass: 62420 tons

This mass is ridiculous, as is the length.

The solution is to massively increase the acceleration, so that the length is shortened.

50km/s railgun 100000G acceleration

Acceleration time: 50000/(100000*9.81) = 0.051 seconds

Energy transfer rate: 245GW

Capacitors needed: 245/0.000001 = 2450000kg or 2450 tons

Railgun length: 1.27km

Railgun mass: 510 tons

Total mass: 2960 tons

In either case, the mass dedicated to the railgun is still very high, and the extreme length imposes design restrictions. Improving your 'technological' numbers, such as mass per meter or capacitor discharge rate, also helps competing weapon systems (supercapacitors help create pulsed lasers that are even more effective than continuous lasers).

In comparison, a laser-armed opponent using lightweight weaponry can outrange and generally out-perform railgun-equipped spaceships. At 1000000km, the railgun rounds will take 5.6 hours to hit their targets. At 100000km, it still takes over half an hour.

In the next post, we'll discuss ways to improve the usefulness of electric cannons, and even make them competitive against weapon systems such as lasers or missiles. After all, variety is what makes combat interesting!

I noticed a few technical errors.

ReplyDeleteFirst, Newton's approximation of the impact depth is not really applicable. At around 2 km/s, you are in the realm where both material strength and fluid flow are important, and you at the very least need a one-dimensional integration (using the Tate equations, for example). At higher speeds (such as once you reach the 10 km/s mentioned), the material strength is irrelevant and you can treat the problem as one hydrodynamic jet of fluid penetrating another stationary fluid. In this case, the penetration of the jet will be D = L sqrt(A/B) (using the terminology used here, where D is the penetration depth, L is the length of the projectile, A is the mass density of the projectile, and B is the mass density of the target).

Second, when you have an explosion, the volume of the crater is E/Y, where E is the energy released by the explosion (kinetic energy, in this case), and Y is the crater strength of the material, usually about 3 times the yield strength. It is not the energy to vaporize the material - you can move stuff out of the way and scatter it around the landscape with significantly less energy than it takes to vaporize it.

First of all, quite honored to have Luke Campbell himself comment here.

DeleteSecond: I'll make those corrections as soon as possible.

Third: Where does the boundary between the crater and hydrodynamic model occur?

There really isn't a sharp boundary. A jet, or just a hypervelocity impactor, will produce enough material blasting sideways to make an expansion cavity. For solid projectiles, which slow down, you get sort of a carrot-shaped hole, where near the entry point the projectile is blasting out a wide cavity and near the tip of the hole it isn't going fast enough to blow out much of a cavity at all. Fully hydrodynamic jets don't have this property (but Munroe-effect jets used in armor piercing munitions show similar shaped holes despite the fact that they are well into the hydrodynamic regime - likely because different parts of the jet are moving at different speeds and the faster parts hit first). As the speed increases, bigger cavities are blown out, until the size of the cavity is much larger than the penetration distance. For a quick estimate, you can just compute the crater radius in the usual way, and if it is larger than the penetration depth, use that instead.

DeleteI did some calculations after your first comment (checking whether water ice could be a suitable impactor) and pretty much reached the same crater/jet conclusion.

DeleteI did however, have some difficulty trying to model the effectiveness of Whipple shielding. For example, how many plates of 1mm thick steel at 1m separation are needed to stop various projectiles?

First treat the problem as a hydrodynamic rod incident on the Whipple plate. You know the penetration (1mm), now solve for the length of rod that gets splattered.

DeleteNext, you have the energy of the collision - in the rest frame of the rod, it is impacted by a 1 mm disk of steel with a radius equal to the rod's radius at the impact velocity. Find the kinetic energy of this disk. Now find the radius of the crater that is excavated in the rod from the collision energy release. If this is larger than the splattered length found above, use it.

"The projectile encounters no aerodynamic resistance in space, so muzzle exit velocity is equal to impact velocity."

ReplyDeleteThat is patently wrong, of course.

Imagine 2 vessels closing at speed, firing projectiles at each other. The impact velocity will (obviously) be the sum of the muzzle velocity and the relative closing velocity, which can be quite a bit higher that the muzzle velocity itself.

Now imagine both vessels moving away from each other (say, after passing each other at that high speed approach) and shooting parting shots. (Obviously) the impact velocity will again be the sum of the muzzle velocity and the relative closing velocity (which is negative here), so the impact velocity is smaller than the muzzle velocity.

It gets more complicated! What if the the target changes it's speed between the launch of the projectile and it's impact? Not maneuvering (or at least trying to control the range) may be suboptimal. So the impact velocity is the muzzle velocity and the relative closing velocity of the firer *at the time of firing* and the target *at the time of impact* ...

Note: "closing velocity", since relative velocity can have sideways components, which means you may have not be able to shoot straight at the target, but add a sideways component and lead the target, too.

Note 2: It's self-evident that your projectiles need to have at least some course correcting capabilities. At minutes of flight time a slight puff from the RCS bending the target's course ever so slightly downwards will cause your projectiles to miss by miles. Depending on the engagement range you may even need to have rudimentary autonomous target seeking for your projectiles due to lightspeed lag issues --- and you might want to stay far enough that your random course changes and the lightspeed lag protect you from most laser shots, if your opponent has that engagement range.

Note 3: There is little reason to have lasers that can do damage beyond their ability to acquire, steadily point to and track the target, if you can do reasonable damage against targets at that range you have enough laser power --- the mass saved can be used for more weapon systems and their needs, armor, increased operation time, Delta V or simply be lighter, cheaper *and* more maneuverable, thereby forcing enemies to come closer to be able to hit you --- which means that less maneuverable enemies can be shot at with success before they can return the compliments, forcing the other side to use more mass for maneuverability instead of weapons and stuff, or cutting mass themselves.

It isn't wrong so much as outside the scope of the discussion. Unless this is extremely advanced technology with incredible accelerations, spacecraft would only add or remove about a few hundred m/s to their velocity relative to the firer, and in any case, it is an insignificant change compared to the closing velocity of the projectile.

DeleteNote 2: Yes, this was the conclusion that was reached, but it is setting-specific. If the setting is very near future with spacecraft shooting each other in low orbit, the projectiles might not need to be guided at all due to the short distances.

Lightspeed lag issues occur at 300000km or more, and much beyond that for it to be significant relative to the target's ability to change course. That is also setting-specific, and as I don't want to negatively impact any author's creativity, I try to only talk of general truths as much as possible.

Note 3: Laser ranges... are kind of like artillery on the battlefield. If they are cut off from the rest of the army, they can only really shoot in direct fire mode. If they are able to coordinate with a scout team, they can target positions up to their maximal range of 30-50km away. Lasers are the same. If you have an advance spotter or a refocusing mirror nearer the target, your effective range goes up a lot.

Another point is that every effort you make to increase the effectiveness of your laser in 'direct' fire mode, such as burning through armor twice as quickly at 100000km, increases your maximal 'assisted' range for free, such as burning out sensors at 2000000km instead of 1000000km.

Also, the logic of laser weaponry follows that of the tank gun : a single weapon with maximized efficiency and priority when it comes to assigning weapon mass fractions. A single large laser allows you to shoot further, deal more damage, and increase your defences against projectiles and missiles. The logical consequences of this mentality are the reason why 'hard' scifi space war is so boring - every spaceship is trying to maximize range at the cost of every other variable.

Updated the article.

ReplyDelete