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.

• It would not affect us in the slightest way, a medium bright "star" would go out. Jan 27, 2016 at 1:32
• Elon Musk will be pissed. Jan 27, 2016 at 1:48
• @Thucydides And a lot of people at NASA, ESA, etc. will have to find new projects. Jan 27, 2016 at 3:55
• NASA would try to find out how Mars got randomly vaporized by aliens :3 Jan 27, 2016 at 4:25
• NASA will try to find out. Elon Musk will find out. Jan 27, 2016 at 21:13

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.

• For the curious, the acceleration is about $1.5\cdot 10^{-8}~\text{m}\cdot\text{s}^{-2}$ Jan 27, 2016 at 3:40
• @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. Jan 27, 2016 at 3:43
• 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. Jan 27, 2016 at 13:25
• 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... Jan 27, 2016 at 21:20
• @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. Feb 1, 2016 at 9:27

In the short and medium term, it will cause: absolutely nothing.

In the long term, astronomically speaking, it will result in a slight increase in the number of asteroids coming close to and eventually hitting Earth. Mars does play a (very minor) role in herding the asteroid belt into its current extent. The extreme vast majority of this is handled by Jupiter, of course.

There will be long-term effects on both the exact shape of Earth's orbital ellipse and on the Earth's rotational axis. Both of these are affected by the other planets, but virtually all influence (99%++) is linked to Jupiter and Saturn, not Mars because even though it is closer, the planet is just such a weeny compared to the Giants of the solar system.

p.s. Astronomical "long-term" means not tens or hundreds of thousands of years, but millions. Many of them.

Other answers covered the negligible direct physical effects well.

I'd like to mention the effect on humanity. This was alluded jokingly in the question's comments, but is a very serious matter.

If a planet suddenly disappears, we'll want to know why, and particularly if something similar may happen to Earth. Expect new and revitalized space programs surpassing even Apollo in ambition and resources.

If we find out about the aliens, much further work will be directed towards contacting them and staying on their good side.

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.