Transfer Orbit
From time to time I like to post nice physics problems on here. Here is a quickie that I used to use in my first-year Astrophysical Concepts course which has now been discontinued, so I don’t need to keep it to myself it any longer.
A simple way to travel from one planet in the solar system to another is to inject a spacecraft into an elliptical transfer orbit, like the one shown by the dashed curve, which is described by Kepler’s Laws in the same way that the planetary orbits (solid curves) are.
Kepler’s Third Law states that the period of an elliptical orbit is given by 
In the Dark 
November 2, 2011 at 5:13 pm
I didn’t give it its proper name because people would just look it up on the internet rather than working it out.
And for those of you who have emailed, no you do not need to know either G or the mass of the Sun!
November 2, 2011 at 5:37 pm
I am assuming then that this transfer is occuring when Mars is at its furthest distance from the Earth.
In that case it is 0.7 years for the transfer, but at closest approach it is only 0.0625 years or 23 days.
November 2, 2011 at 6:37 pm
I’m totally confused by your second answer. Would the orbit in the second case actually go around the sun?
November 2, 2011 at 6:24 pm
This exact questions in the second year IB Astrophysical Dynamics example sheet which I supervise, nice to see it’s been around a while!
****** EXTRA CREDIT******
b) Instead suppose we travel to Jupiter. Suppose that at aphelion it collides elastically with Jupiter in such a way that its velocity reverses direction. Show that it has enough speed now to escape
from the solar system.
c) If you make realistic assumptions does it still escape?
November 2, 2011 at 6:36 pm
It was of course a first-year question here at Cardiff, compared with being in the second year at Cambridge, thus proving the superior intellectual level of the education provided to our students compared to Cambridge.
November 2, 2011 at 7:55 pm
perhaps this is the solution:
i assume they’re still doing natural sciences degrees in cambridge – in which case ~25% of their first year is spent doing physics and ~50% of the second year: so in terms of content at mid-point through the second year in cambridge they would be about mid-point through the first year of a physics-only degree.
November 3, 2011 at 5:26 am
Answer to (b) – it doesn’t escape the Solar System – it would just now be in an elliptical orbit going the other way (clockwise).
Answer to (c) – no.
How much extra credit is this worth?
November 3, 2011 at 5:53 am
HA! Chris, do you mean its velocity in orbit or its velocity with respect to Jupiter? In the latter case, ok!
November 3, 2011 at 8:03 am
It’s always a bit disconcerting when I post what I think is a simple question and the majority of answers given are wrong!
November 3, 2011 at 12:01 pm
Does it mean that you have a very wide readership – lots of non-astronomers, or many readers are doing it before their first cup of coffee
November 3, 2011 at 11:36 am
Have to confess, I initially answered with the period, not the travel time, but hey, what’s a factor of two in astronomy anyway!
November 3, 2011 at 4:11 pm
Our version of this question also asks how long they should stay on Mars before starting the return journey. It is first-year in Manchester.
November 4, 2011 at 9:24 am
If Manchester is where you are returning to, surely the answer is forever.
November 3, 2011 at 4:56 pm
For completeness, I should add that this is part (a) of the problem sheet I took it from: subsequent questions ask about the velocity needed to get into the orbit, etc, but I just picked the first part for here because (a) it’s “easy” and (b) it doesn’t require working out GM explicitly.
November 4, 2011 at 2:55 pm
[…] the little orbital dynamics question I posted a couple of days ago, which seems to have attracted quite a number of responses, also reminded me of something that […]