There's a whitepaper out, on putting powerful rocket-propelled missiles on the Moon. This for defence against meteors; although I should bring in the STL1 magnet against Carrington, which would require a rocket component, as long as it needs station-keeping.
The various solar-system moons all have less gravity than our Earth, so can support larger rockets than our planet can. With the notable exception of Titan they also don't have atmospheric interference. As long as Earth is inhabited, we on Earth should like defence against space rocks, either mindlessly intersecting us or, perhaps, flung this way by future hostiles - The Expanse has cited the Belt, here. And nobody is terraforming any of the airless satellites; thus, most of their surfaces will always be wasteland, allowing for even stronger (read: atomic) rockets - starting with Old Boom Boom himself.
Fair enough: I am sold. Problem: any rocket that is tank enough to pound a 100 km asteroid (or to balance, what, 500 kilotonnes of Rebco plus heat-mitigants at L1) is tank enough to plow Washington DC a new bayou. If that sounds fun to you, then consider triggering the Yellowstone Caldera or the Campi Flegrei near Naples. I know some of us talk carbon alternatives or even direct mitigation, but for the latter I should like more from the process.
So I wonder. Railguns in specific parts of the Moon that can't hit the Earth and can't be redirected to hit the Earth? The best place for that is the direct opposite side, pointing up. That gun will be strafing TLL2, admittedly. But that's a metastable site anyway and if we're worried about Apophis, we just tell the station-keepers to nap this one out for a few weeks. Of more concern is that the rock won't be in the exact sights of the railgun like, ever. That'll be true of every railgun that isn't given the flexibility to point, load, and calculate energy-requirements.
My answer is a rocket launched from the Orion site - TLL2 remains a fine place for the post-Orion rocket to hang out, so that site can be the farside as mooted earlier. Railguns can feed to it more propellant and more mass generally once it gets to L2. Then run it to Apophis. We are absolutely maximising mass and velocity for this; I foresee a high-Isp solution, saving on propellant, because what's thrust at L2.
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