First of all, while I appreciate the extent you’re willing to go to for technical accuracy, you’re making this way too complicated. It’s all well and good for Heinlein to boast how he and his aerospace engineer wife spent three days and yards of paper calculating an orbital ascent in order to get one line of dialogue, but unless you do something utterly foolish–like have the Earth rotating the wrong direction[sup]1[/sup]–nobody without an advanced degree in aerospace engineering or astrophysics is really going to take notice (and in fact, few that fit that description will). Just SWAG[sup]2[/sup] it and go on with life.
That being said, given a distance and initial (vector) velocity, it’s a simple application of Kepler’s Laws to figure out the free unpowered orbit of a spacecraft. I say “simple” in terms of someone who regulary plays with orbital mechanics or toys around with Matlab/Scilab as a hobby[sup]3[/sup]. Even if you don’t, the calculations aren’t that hard, though you’re going to have to take into account tidal effects, the effects of the Galilean moons (we can dispense with the smaller outer moons out of hand), and of course, as you note, atmospheric effects in computing the resultant degenerate orbit. There’s no easy closed form solution to that; any generalized n-body solution, and especially one with variable forces, is going to a) have a large family of solutions, and b) be highly perturbative (i.e. very sensitive to initial and boundary conditions) and so will require interative simulation. If you want a reference, I think highly of Prussing’s Oribtal Mechanics, but seriously, anything you need to know for this calcuation you should be able to get from a basic university-level physics text.
However, if you fail to give complete detail on velocities and times, nobody is going to be able to calculate squat, and unless you make some egregious error, even the most pedantic physics grad student will give you a pass so long as the story is interesting. And since your characters are (presumably) going to suffer a hideous death in the crushing pressures of Jupiter’s inner atmosphere (the pressure-distance relationship isn’t linear, by the way; beyond the Jovian upper atmosphere very little is known about its actual composition, so feel free to make up things as you go along) then nobody’s really going to be checking up on them anyway. Rhubarb’s calculation is for an incompressable fluid (derived from Bernoulli’s Principle) but if you need a number you might as well use this as anything else. It won’t necessarily be any more accurate than pulling a number out of the air, but at least it’ll have the benefit of having a totally indefensible calcuation behind it, which is sufficient for NASA Safety Management[sup]4[/sup].
Just write your story. Run a few rule-of-thumb Kepler equations if it’ll make you more comfortable (or offer up some specifics here and you’ll have a dozen engineering/physics geeks running Matlab simulations of your scenerio thirty minutes after you hit submit) and get on with the business of describing your characters.
Stranger
[sup]1[/sup]As in the first edition of Ringworld by Larry Niven. Never mind that he totally missed the point that the Ringworld is unstable.
[sup]2[/sup]Scientific Wild Ass Guess. It’s a technical term. Trust me.
[sup]3[/sup]Guilty.
[sup]4[/sup]Seriously, don’t ask. It’s been a hard week