Nghi said:
I do have one relevant question, though: can you use conservation of energy to answer this question? Because I was browsing through the threads, and I found one where the guy used 1/2mv^2 - GMm/r = 1/2mv^2 - GMm/r to try and find his answer. I tried doing that for this problem, and I got a radically different (but still wrong) answer.
OK, this took some time to track down the problem. Yes, in principle, you should get the same result using the conservation of mechanical energy equation.
We ran into a similar difficulty a few weeks ago on another celestial mechanics problem of this sort. What it illustrates is the danger of providing data for a highly elliptical orbit to insufficient precision. If you use conservation of mechanical energy and make the calculations for energy per unit mass (just drop the little m) with the data you are given, you get this (GM = 1.32733·10^20 for the Sun):
using perihelion data --
r_p = 8.823·10^10 m
v = 54,600 m/sec
E/m = -13,817,600 J/m
2a (major axis) = 9.606·10^12 m
r_a (aphelion distance) = 9.518·10^12 m
using aphelion data --
r_a = 5.902·10^12 m
v = 816.2 m/sec
E/m = -22,156,404 J/m
2a = 5.991·10^12 m
r_p (perihelion distance) = 8.87·10^10 m
We get the major axis of the elliptical orbit using the vis-viva equation (I don't know if you had it in your course -- it's either in your book or look on Wiki, for instance). The value of E/m is equal to -GM/2a , where 2a is the distance from the perihelion point to the aphelion point (the "length" of the ellipse) and 2a = r_{p} + r_{a}.
So from this, at first glance, it looks like two different orbits and that your instructor goofed, with the results using the aphelion information being consistent with the problem's distances and the perihelion velocity
not being consistent. But if you use the conservation of angular momentum as you did for the problem and apply it to your aphelion velocity (which looks reliable at present), you get a perihelion velocity of
(5.902·10^12 / 8.823·10^10) x 816.2 m/sec = 66.893 · 816.2 m/sec = 54,598 m/sec ,
which is awfully close to what he used.
So what gives? The answer is that for a highly elliptical orbit, the values of the aphelion distance and aphelion speed are
extremely sensitive to the value of the perihelion speed. When you make up a problem like this, you must either avoid using an orbit with a huge difference between the perihelion and aphelion distances or else given the values to at least four or five significant figures. (This is just the trouble the other student ran into on their problem last month. Depending on how they rounded off the numbers they used, they got considerable discrepancies in their answers.)
You're welcome to pass these calculations on to your instructor, if they haven't discovered the difficulty already. Obviously, they're going to have to allow a fair amount of leeway in grading everyone's work...
