On effective range.

Here we shall determine the effective range, meaning the range at which you can hit what you are shooting at, of a weapon in space.

I'm planning on posting updates to the blog weekly, every Sunday night, for as long as I have material to work with. This one is a bit late. If you have any questions you'd like me to tackle, post them in the comments or telegram me on Nationstates.

To business. Our starting assumptions:

• The weapon's intrinsic range is much greater than its effective range. This is valid for solid projectiles, though may not be for realistic lasers or particle beams which diverge.
• The weapon shoots one bullet at a time. We aren't considering shotgun-like approaches.
• The bullets travel in straight lines and don't have any active manoeuvring systems. We aren't considering guided munitions.
• The weapon is intrinsically precise, it won't miss a non-moving target.
• The defender can change their acceleration without delay.
• The attacker and defender both have effectively-instant FTL scanners. This is not uncommon in science-fiction.

The scenario is thus simple. The attacker fires, aiming the bullet so it would strike the middle of the defending ship. The defender immediately detects this and takes evasive action by accelerating. If the defender can completely vacate the space it was occupying when the shot was fired, the shot misses, while if the defender is still in that place then it is a hit. We shall use the reference frame of the defender, prior to their taking the evasive action. The below diagram depicts a miss, the defender having just managed to move half its own length for the shot to pass harmlessly behind it.

With the projectile travelling at velocity v and having to cover distance r, the defender has a time to evade

t = r/v

To determine how far the defender can move within that time, we can use one of the SUVAT equations for uniform acceleration situations to find that distance

s = ut + at²/2

u, the initial velocity, is zero since we chose our reference frame so that would be the case. Therefore, substituting the first equation into the second, we get

s = ar²/2v²

If

s < l/2

Then the projectile will hit. Substituting and rearranging, we get

This I refer to as the "Range Equation". If the target is closer than r, it will be hit. If not, it will be missed.

For a worked example, consider the 1 km long ship capable of 100 g of acceleration from the previous post, being targeted by a projectile moving at 0.95c. Converting to SI units

v = 2.85 x 10⁸ ms^-1
l = 1000 m
a = 981 ms^-2

r = 2.88 x 10⁸ m

So the target can be reliably hit if it's as far as 288 thousand kilometres away. For context, the average Earth-Moon distance is 385 thousand km.

The range equation has various consequences. Most obviously, smaller targets can get closer to an attacker safely than larger ones. Critically, if two ships are in one-on-one combat, both using weapons that fire their projectiles at the same speed, the smaller one can sit at a range where it will always hit its enemy while never being hit itself. Compactness is thus advantageous, something only emphasised by the previous posts's result that shows a more compact ship will also accelerate harder.

Also, the faster the projectile can be fired the better. Laser shots will of course go at light-speed, and particle beams won't be far off, so massive projectiles need to be doing relativistic speeds or have other advantages in order to compete.

Increasing acceleration while holding size the same, for example by advancing technology, has a limited benefit compared to reducing size and thus benefiting from the natural consequent acceleration increase.

Finally we can tell that the intrinsic precision needs to be pretty precise, to well under a second of arc. The modern Hubble Space Telescope is capable of pointing with a precision of 0.01 arc seconds, so this is not an especially demanding requirement.

If the first assumption is violated, and range is actually limited by the intrinsic behaviour of the weapon, the argument becomes moot. A consequence of such a situation is that large ships can now get just as close to their attacker as smaller ones. It's also possible for the range equation to apply to small targets but not to large ones, creating a two-regime situation. This would likely result in the 'borderline' area being vacated as ships would either be larger for more general capability, or smaller for being hard to hit.

If the second assumption is violated, then the geometry needs to be considered in more detail. The overall conclusion that smaller ships are harder to hit - which is the intuitive result after all - will probably remain intact though.

If the third assumption is violated the argument becomes moot. Guided munitions are likely to have much longer effective ranges. However they may have other drawbacks.

If the fourth assumption is violated then the imprecise pointing will result in probabilistic concerns, but the overall effect is likely to be similar to limited intrinsic range.

If the fifth assumption is violated, then a 'reaction time' delay must be incorporated into the equations, making them more complicated.

If the sixth assumption is violated, then the defender of course cannot actively dodge the incoming fire if it's travelling at or near light-speed. However, since the attacker's information will always be out of date, the defender can make random manoeuvres. If they are within a range half that given by the above equation, they will surely be hit (the derivation of this will come in a future post). Beyond that, the situation becomes probabilistic instead of a certain miss.