re Pyro:<br />A flyby can use a relatively short, direct trajectory, like the upcoming New Horizons mission. However, an orbiter must be travelling relatively slowly relative to its target planet, or else it will use an inordinate amount of fuel to brake into orbit.<br /><br />The most fuel-efficient trajectory--as well as the slowest direct option--is a Hohmann transfer ellipse. Calculating travel times is relatively simple; for an outer planet mission, just calculate the semimajor axis for an elliptical orbit with a perihelion at the Earth's orbit and the aphelion at the target planet's (for Neptune, this would be about 15.5 AUs), and then calculate *half* the revolutionary period from Kepler's Third Law (a^3 = p^2). For a Neptune Hohmann, this would be almost 31 years, in line with the estimate you cited.<br /><br />Of course, there are more fuel-costly (but shorter) routes, and there's always the handy gravity-assist, but in any event the slow boat to Neptune would be slow indeed. Personally, the farthest I'd use a Hohmann transfer would be Uranus; the 16-year transit would be a long, long time, but at least theoretically bearable.