Relativistic Kill Vehicles are a staple of space opera.  The idea is simple — impart enough kinetic energy on your projectile, and make sure that energy carries it into something or someone you don’t like.  Upon impact, the energy is shared between the projectile and the target, and much of it will go into remaking the target into a form you find more pleasing.

The interesting properties of RKVs all involve observation.  Detecting an RKV is often very difficult — since terminal guidance is virtually impossible for them, they don’t expend a lot of thrust near the target, and therefore pretty much just reflect signals rather than emitting much of their own.  Also, because they are traveling at a significant fraction of the speed of light, they have moved considerably by the time they are detected in a particular location.

Let’s take the scenario presented in yesterday’s video:  six RKV’s enter the Sol system.  Each is 1 ton traveling at 0.5c relative to Earth, their target.  At that speed, relativistic effects are noticeable, but not extreme — the observed mass of each object from locations in the solar system is about 1.15 tons, not incredibly more than the projectile at rest.  Let’s presume for the moment that Earth has the ability to detect an object the size of that projectile as far out as Neptune (~30 AU.)  Since there’s minimal radiation from the object itself, let us also assume they’re using active radar to detect it.  The fastest line of communication is a straight line, so let’s further presume that that radar station is on or near Earth.

The radar beam is pretty quick — it will bounce off the RKV as soon as it enters detection range, because it is always being sent out.  The return, though, has to come from 30 AU all the way back to Earth at 1 AU, which will take about 4 hours.  In those 4 hours, the RKV’s are closing in — when the radar return reaches each, the RKV’s are now 14.5 AU closer than they were — already inside Uranus’ orbit and only 4 hours from impacting Earth.

Because Earth is in the cross-hairs, this trend will continue until impact — the RKV will always be half the distance to Earth that the most recent returns say it is.

In the scenario depicted in the video, Earth attempts to defend itself by using Mass Drivers location on the Moon and in Australia to throw hunks of raw minerals in the path of the on-rushing RKV.  Mass Drivers are fascinating devices of almost mythical ability depending on where you hear about them, but for our purposes, they accelerate heavy objects up to orbital velocities in a controlled way.  They are essentially space guns  that shoot big rocks.

Escape velocity for the Sol System from Earth’s orbit is about 41 km/s.  There’s really no reason a Mass Driver on Earth would ever need to throw much harder than that, so I take 41 km/s to be the “muzzle velocity” of my defensive guns.  I give them each a payload size of 100 tons, which is couple of cubic meters of some reasonably heavy metal they don’t care too much about.  Each of these chunks of rock is bestowed with about 8.4 x 10^13 joules of energy.  Some of it will be lost to gravity by the time they intercept the RKV, but for now let’s use that number.

Assuming the first shot comes off the drivers at T-4 hours exactly they will have made it to about 590,000 km by the time they reach the RKV and have a chance of hitting it.  At this time, the RKV is four seconds from hitting Earth.

To save the Earth, the first ore packet would need to slow the RKV down to less than 1% of the speed of light.  Unfortunately, even if it connects with the target, it will only have a fraction of the necessary energy.  The blow would certainly change the RKV’s course, cause a different spot on Earth’s surface to be hit and reduce the final impact energy, but it wouldn’t come close to diverting it from slamming to the Earth with an energy release of about 3 gigatons.