How would the sudden destruction of Mars affect Earth?
If Mars was vaporized by aliens, how would it affect us? Assume that the core of Mars exploded, sending the debris flying in all directions, and that some would pass close by, but not hit, Earth. How would it affect Earth's orbit? Would it cause another planet to diverge from its orbit which would then affect ours?
Mars isn't blown up, it's vaporized, and all of the particles are sent flying outward in a sphere, so the chunks of Mars are irrelevant. it's just the absence of it's gravity you need to take into account.
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The effect of Mars on Earth is negligible. The total force between the planets is — at their closest approach — $$F_{\text{max}}=\frac{GM_{\text{Earth}}M_{\text{Mars}}}{r_{\text{min}}^2}\approx8.6\times10^{16}\text{ Newtons}$$ This produces an acceleration of . . . well, pretty much zero. Mars does not affect Earth much, and so its absence will not affect Earth much. In the long-term, there will be effects, as the Solar System is chaotic (i.e. the orbits cannot be predicted well) over a couple hundred million years. However, there aren't important resonances between Mars and Earth, and in the short-term, the effects will be minimal.
The same goes for the other planets. The only affected bodies will be Mars' two moons, Deimos and Phobos. Even so, they'll most likely continue to orbit the Sun on Mars's current orbit.
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