1. How do we know that a rock is from Mars, especially when its composition is different from what we've found on Mars to date.
Isotope ratios and certain element ratios. These depend on the history of a planetary body, and you can rule out every planet / asteroid but Mars. I always liked the conclusion in this paper [sciencedirect.com]
There seems little likelihood that the SNCs are not from Mars. If they were from another planetary body, it would have to be substantially identical to Mars
Of course, there is no such other Mars in the solar system.
The existence and composition of little atmospheric inclusions (i.e., tiny little bits of Martian air trapped in the rock) were another convincing piece of evidence for the Mars meteorites, as was the evidence of alteration by water.
2. How do rocks leave Mars' gravity well in the first place? Are they shrapnel from Mars being hit by big meteorites?
In a way. Suppose you have a big meteor hit (the size of the one that formed the Baringer Meteor Crater, or bigger). The meteor drills into the body and goes beneath the surface. At some point, it is stopped, and it dumps its kinetic energy into the body of the planet (i.e., for big impacts the meteorite explodes at depth). The shock wave is roughly spherical, and so the part directed upwards lifts up the surface above where the meteorite hit. Most of this material is lifted not much more than the depth of the explosion, forming the characteristic lip of the crater, and typically turning the layers in the rock upside down at the lip. Some of this material can be accelerated to much higher velocities, however, forming (for example) the rays of the new craters on the Moon. If the meteorite is really big, some of the surface material is accelerated to escape velocity and away it goes. After a little while (a few dozen to no more than a million years), some material will hit another planet. Mars and Earth have been trading material like this for the life of the solar system.
The really amazing thing is that some of the material ejected is not treated too roughly. Spores and seeds etc. could definitely survive the trip.
Source: http://rss.slashdot.org/~r/Slashdot/slashdotScience/~3/oTUELNKKjAc/story01.htm
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