Evasion
Space is big, and space battles could plausibly occur between ships located thousands or even millions of km apart. The first defense is not to get hit, and being very far from the enemy makes that easier. You may have several seconds or minutes to get out of the way.
To know how much time is available to evade, we need to know how fast a projectile you're being shot with. Hypervelocity projectiles on Earth, such as the Navy railgun, might have a muzzle velocity of up to 5 km/s. This is actually not very fast by solar system standards. Meteoroids strike Earth typically at 20 km/s. The relative speed between your ship and your enemy's ship is likely higher than the muzzle velocity of the enemy's railgun relative to his ship.
Sometimes you hear about the Casaba-Howitzer idea for producing hypervelocity projectiles. The notion you sometimes hear is that you could set off a small nuke and have it propel a solid bullet at 100 km/s. However, more realistically, the initially solid bullet would be vaporized by the nuke into a plume of material that spreads out at 100 km/s in a cone. The effect if the plume hits your ship would be very different from a solid projectile, and less damaging; it would dump energy broadly across your hull, heating it up and vaporizing the outer layers of hull. It wouldn't punch a hole. So we won't worry about Casaba-Howitzer any further here. It's not a solid projectile and it would take different measures to defend against.
So, realistically, we might say you are faced with only 30 km/s projectiles, with most of that speed being due to the relative velocities between your ship and the enemy's ship, and not due to the muzzle velocity of the enemy's railgun. This means that if you are 1000 km apart when the enemy shoots, you have around 30 seconds to evade. (If you're a million km apart, you have nine hours to evade - the enemy would likely not even bother with kinetic projectiles at such ranges!)
Evasion can be divided into "eyes-open" evasion, and "random walk" evasion. You would use "eyes-open" evasion if you know the trajectory of the enemy's projectile so you can get out of the specific target area. This could potentially be done using a radar telescope. You can certainly use radar to detect that a projectile is incoming (although the enemy can also cover their projectile in angular radar-reflective surfaces like a stealth bomber), but determining specifically where it is aimed down to a few meters is a much bigger ask, although possible in theory.
For random walk evasion, you thrust in random directions at specific intervals, to maximize the enemy's uncertainty in your future position. This is what bombers did in WW2 to avoid flak. This is generally a good idea to do any time you are in hostile space, because you won't always be aware of the specific projectile or bomb before it hits you, if it is cold, small, and stealthy to radar.
A ship that is good at evasion should:
- Have a small cross-section presented to the enemy, and be capable of thrusting sideways while presenting the small cross-section
- Have a high thrust-to-mass ratio, not weighed down by too much heavy armor or weapons
- Have good "eyes" (telescopes) and "reflexes" (computers) for eyes-open evasion when possible
Armor
A 1.2 cm aluminum sphere impacted an aluminum plate at 6.8 km/s, leaving a crater 5.3 cm deep:
The amount of armor you need certainly depends on the size of projectile you're being shot with. There is a trade-off between the weight of armor, which lets you tank bigger hits, and the maneuverability of your craft, which lets you avoid hits entirely.
When a projectile hits your armor, it has two effects that can be considered separately. The first effect is that it penetrates some amount based on its density and length. The second effect is that it dumps its energy into the armor, causing an explosion and crater.
As Starfish Prime helpfully mentioned, at such high speeds both the projectile and the armor act like fluids; the molecular bonds are too weak compared to the force of impact to be very important. I'll go ahead and steal his formula for penetration depth:
$$P_d = \ell \sqrt{\rho_j \over \rho_t}$$
where $P_d$ is the depth of penetration, $\ell$ is the length of the penetrator, $\rho_j$ is the density of the penetrator (with j-for-jet) and $\rho_t$ is the density of the target.
Note: this depth of penetration is before the crater is formed. The crater will be usually several times deeper than this formula gives. More on that later. But we do want to minimize this initial penetration depth according to this formula, because it affects the crater depth too.
We can suppose that the enemy is shooting you with an iron projectile, because he's using a railgun. Iron has a density of 7.8 g/cm^3.
Now, in space mass is king; the more your spacecraft masses, the slower it can accelerate and the more fuel you need. So you want the most protection for the least mass. Are you better off with dense armor, or an equal mass of less-dense armor? Based on this formula, surprisingly, to minimize penetration you are better off with less-dense armor!
Let's do an example. The enemy projectile is 1m long and your armor is also 1m of iron. The density of projectile and armor are the same, so the penetration depth is 1m, and the projectile gets through.
On the other hand, you could use 2.9 meters of aluminum armor for the same mass as 1m of iron armor. With aluminum, the penetration depth is 1m * sqrt(2.9) = 1.7 meters. The projectile only gets a little over half-way through!
Whipple shields are used in practice currently, and a scaled-up Whipple shield is probably a better option than solid steel or aluminum. Current Whipple shields are designed against micrometeorites (up to 1 cm size), but the same principles will apply to larger projectiles if the Whipple shield was just bigger and thicker. The principle is that the projectile hits a thin layer of armor, which it penetrates completely but disperses as it does so. It is allowed to spread out further over a gap of empty space before the next layer of armor. Then when it hits the next layer of armor, it may penetrate that too, but it is dispersed even more. After a few layers it is dispersed enough that it no longer penetrates.
Note that a Whipple shield, with all the empty space between the plates, has a low overall density. That tracks, when you consider the penetration depth equation.
That's all to minimize penetration. The other factor to worry about is the energy of impact, and the crater. The crater volume is proportional to the impact energy. Intuitively, that's because to make a crater that size you have to break a lot of the molecular bonds in that volume of armor, which takes energy per bond broken. This means the crater depth is proportional to the cube root of the impact energy.
A 100 kg projectile at 30 km/s has the energy in about 10 tons of TNT. To minimize the crater that makes, you want strong materials that take a lot of energy to vaporize. Whipple shields are made of steel, which fits the bill. You want a (giant) Whipple shield made of several thick layers of steel with gaps in between them.
If you plan on getting hit more than once, and survive the first hit, you may want to weld new plates over the holes in the armor. However, the enemy might not bother shooting you with a weapon that doesn't penetrate your armor on the first hit. It's less efficient to hit you with ten smaller rounds that don't penetrate but just damage your armor, than to hit you with one round 10x the size that goes through and destroys your ship.
Thus, we might say that what the armor really does, rather than actually protect you if you're hit, is make the enemy pack a bigger gun and heavier ammunition, which slows him down, reduces the number of times he can shoot, and reduces the other armaments he can carry. Actually getting hit with a solid object from the enemy is probably game over.
Your idea of a water tank is interesting for the purpose of healing the armor. It is not as dense as steel, which makes it favorable for the penetration depth, and if the enemy blows a hole in water, the hole just fills up on its own. However, water takes much much less energy to vaporize than steel. The first good hit and much of your water tank will be steam. Also, water lacks any tensile strength, which does matter.
Active interception
The idea here is that you send out your own projectile to meet the projectile a few km in front of your ship. The two projectiles annihilate each other. Your ship may be hit with some chunks from the explosion, but it would be much less damaging than if the enemy projectile directly hit your armor.
Two basic ways to do this. One is, you track the incoming projectile and when it's close enough to be sure of its trajectory, you shoot it with your own railgun. This is tricky but it has the advantage that the momentum of the incoming projectile can be partially canceled by your own, so less of the incoming projectile proceeds to hit you.
The other way is, you have a rocket drone loitering a few km in front of your ship. The drone masses about as much as the incoming projectile, and maneuvers to be exactly in front of it so the projectile hits the drone. This may be easier to accomplish than shooting the incoming projectile with your railgun, but the net momentum from the collision is still towards your ship so you may be hit with more debris, though it will at least be spread out more.
