+Timothy Gowers +Philippe Beaudoin
Your (and others') argument is basically what Mark Eichenlaub answered. Excellent! +Satyr Icon
Yes, Mars does end up on the opposite side of the Sun! The semi-major axis for Mars's orbit is about 1.5 AU, which is why the distribution for Mars goes out to about 2.5 AU, which is the farthest apart the Earth and Mars can get. +Kwan Lowe
if you know how JPL's HORIZONS system works, you recognize that my response and Jesse's responses are actually the same approach ;)
It should be possible to do this problem analytically, taking into account the different planes and eccentricities of the orbits of Mercury, Venus, Earth, and Mars, in terms of elliptic integrals. But since I said "in terms of elliptic integrals", I did not want to do the problem analytically!