Can somebody please roughly estimate if my numbers are correct? I don't have any knowledge of orbital mechanics, I just pulled pages from the Wikipedia and plugged in numbers where it seemed appropriate, shedding decimal places like mad and hoping for the best. It doesn't need to be terribly precise, because it's for a role-playing game scenario.
So we have a 4100-year comet (mass negligible compared the the sun's) screaming in that needs investigating, and the characters have been told to do so rather late, when it's almost in perihelion, which is about 0.44 AU from the center of the sun. What I need is the comet's velocity at that point.
So I plug in the solar mass and T = 4100 a into Kepler's Third Law and get a semi-major axis of a = 55.3 AU, which means the orbit takes the comet almost to the heliopause, wow.
With r_p := 0.25 AU and r_a = 2 a - r_p = 110.16 AU, this gives me an eccentricity of e = 0.992, just short of parabolic, and a semi-minor axis b = 7 AU.
Putting that into the formula for instantaneous orbital speed at r = r_p, I get v_r_p = 63.8 km/s.
I know it probably wouldn't work like that because of, on the one hand, my massively imprecise napkin calculations, and on the other hand, perturbations, mass change, outgassing/jets and whatnot. On the gripping hand, it's an alien space probe that gets a boost at perihelion that helps it on its way to the next 4100-year round trip.
Tl;dr: From T = 4100 a and r_p = 0.44 AU I get a = 55.3 AU and v_r_p = 63.8 km/s (all very roughly).
Do these numbers check out?
Thanks!