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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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    $\begingroup$ It would not affect us in the slightest way, a medium bright "star" would go out. $\endgroup$ – Oldcat Jan 27 '16 at 1:32
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    $\begingroup$ Elon Musk will be pissed. $\endgroup$ – Thucydides Jan 27 '16 at 1:48
  • $\begingroup$ @Thucydides And a lot of people at NASA, ESA, etc. will have to find new projects. $\endgroup$ – 2012rcampion Jan 27 '16 at 3:55
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    $\begingroup$ NASA would try to find out how Mars got randomly vaporized by aliens :3 $\endgroup$ – AMACB Jan 27 '16 at 4:25
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    $\begingroup$ NASA will try to find out. Elon Musk will find out. $\endgroup$ – nigel222 Jan 27 '16 at 21:13
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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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    $\begingroup$ For the curious, the acceleration is about $1.5\cdot 10^{-8}~\text{m}\cdot\text{s}^{-2}$ $\endgroup$ – 2012rcampion Jan 27 '16 at 3:40
  • $\begingroup$ @2012rcampion That's helpful. I also found out that I managed to plug in the numbers incorrectly for the force calculation, so thank you for that. $\endgroup$ – HDE 226868 Jan 27 '16 at 3:43
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    $\begingroup$ Spiral out? They'll be in an orbit slightly different from Mars' original orbit, having a slight excess or lack of kenetic energy and being a few thousand miles displaced. That will give them an eccentric ellipse that returns to the original position: no spirals anywhere. $\endgroup$ – JDługosz Jan 27 '16 at 13:25
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    $\begingroup$ Worth pointing out that orbits of the smaller planets are long term chaotic in the mathematical sense. At present Earths orbit is pretty stable for as long as the best observations allow us to predict: about 200M years. Remove Mars and you get a new "Butterfly effect". It might doom Earth a few million years in the future... $\endgroup$ – nigel222 Jan 27 '16 at 21:20
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    $\begingroup$ @trichoplax good point. I had not considered motivation. Maybe the aliens were just passing through but noticed Earth was close to the inner edge of the Goldilocks zone and that the immature intelligent race was causing dangerous global warming. So they remove Mars at precisely the point where that will cause Earth to gradually adopt a slightly wider orbit. Random probably good deed done, they go on their way. $\endgroup$ – nigel222 Feb 1 '16 at 9:27
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As was noted, in the short term there will be no visible effect, but in the (very) long term the solar system will probably change, since now the planet orbits are the result of a balance between the various gravitational forces.

If you wipe out Mars, you alter this balance and the solar system will need to find a new balance and this can affect Earth, giving her a new orbit, not necessarily in the habitable zone. But here we are speaking of million years time frames.

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