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Robert Walker
Theoretically, I'd say, it's possible, unlike the other answers here, so long as we are given enough time - which is pretty likely.

Not too likely a new planet will come sailing through the solar system from interstellar space. And if it did, the chance it would hit the Earth first time around would be almost infinitesimal.

So it would need to do many swings by of the sun before it hits the Earth. If from interstellar space or the Oort cloud, first need to pass close by Jupiter to get deflected into an 0rbit in the inner solar system (otherwise, it could swing close by the sun, but would then just fly out again to outer space or to the Oort cloud after that one pass by, never to be seen again, or at least not for probably tens of thousands of years).

Then that orbit gradually adjusted until it becomes a near Earth asteroid frequently crossing the Earth's orbit. Then it would have a high probability of hitting the Earth - incidentally an object as big as this would also have a chance of changing orbits of other planets the inner solar system too - you can't just add a planet to our solar system without making some changes in the orbits of the other planets.

So - would expect probably a few thousand years at the very least to do something about it. In that case, then  it could be done, though it would require huge outlay of energy. And note this is just with existing technology - during those thousand years surely we'd develop new technology as well to handle situations like this.

What you'd do is use many gravitational slingshots

In that way you get energy multiplication. You aren't getting the energy for free, but you are using slingshots to convert some of the energy e.g. of Jupiter';s orbit about the sun into extra delta v for your comets.

You can send comets and asteroids and trojans, - and have cascading effects using smaller objects to move larger objects and those to move even larger objects.

Result is that Jupiter's year gets slightly faster, and you end up sending heavy objects towards the object with far more energy than you would expect with the rocket power available to you.

I don't know about detailed calculations - but ideas like this have been discussed for mega-engineering projects like impacting icy objects into Mars and Venus.

Perhaps therefore not totally impossible but hard.

Also depends a lot on how far in advance you know about it. If you know about it a million years before the event, far more practical.

And - there is a chance of this happening - but we'd probably have several hundred million years or so of warning so with mega technology we'd have if still technological a few hundred million years from now - if still a long lasting stable technological civilization by then - may be relatively easy.

The object is Mercury, Nowhere near 50% of Earth's mass - but it is over 5% of Earth's mass. Far larger than anything else likely to impact us in the future.
Planetary Fact Sheet - Ratio to Earth

There's a tiny probability of it hitting Earth long into the future
Stability of the Solar System

If so, we have a chance and with such long warning, not such a mega engineering challenge as all that - but would need to have a sustained peaceful civilization able to do long projects, at least, I imagine, centuries or thousands of years long projects, perhaps longer.

If you need to do it really quickly, only a few years warning - of something as big as Mercury - not sure if it is possible with present day technology - though who knows what might be available then.

It would probably take some decades, and perhaps more on the scale of centuries just to set everything up to start the cascade of gravitational slingshots needed to change its trajectory if you used that approach.

There were many impacts like this in the early solar system as the planet embryos hit each other and coalesced to form our solar system. But since then is very rare and you can see from impact history of other planets and the Moon - a few really big impacts but they all come from the first few hundred million years of our solar system. Later impacts not as large as that.

The best dates we have are for the Lunar Maria - and the most recent of those are over a billion years old
Lunar mare

So doesn't seem there have been any major impacts - e.g. big enough to create a mark on the Moon visible to naked eye from the Earth - for a billion years or so - and those of course are far smaller impactors than the ones you are thinking about.

Theoretically we have the larger KBOs and large comets in the Oort belt which could be this large or larger. But you can see from the cratering history - that if they do every get diverted into inner solar system - is so rare and so unlikely to hit anything that we probably don't need to lose any sleep on that - Mercury probably more of a risk than those. Though can't say an absolute no. Just very very unlikely.

About the Author

Robert Walker

Robert Walker

Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.
Studied at Wolfson College, Oxford
Lives in Isle of Mull
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