A Problem of Dimensions
We’ve more-or-less sorted out who will be teaching what next term in the Department of Theoretical Physics at Maynooth University next term (starting a month from now) and I’ll be taking over the Mathematical Physics module MP110, which is basically about Mechanics with a bit of of special relativity thrown in for fun. Being in the first semester of the first year, these is the first module in Theoretical Physics students get to take here at Maynooth so it’s quite a responsibility but I’m very much looking forward to it.
I am planning to start the lectures with some things about units and dimensional analysis. Thinking about this reminded me that I posted a dimensional analysis problem (too hard for first-year students) on here a while ago which seemed to pose a challenge so I thought I would post another here for your amusement.
The period P for an elliptical orbit of semi-major axis a of a moon of mass m around a planet of mass M, depends only on the quantities a, m, M and G (Newton’s Gravitational Constant).
(a). Using dimensional analysis only, determine as completely as possible the relationship between P and these four quantities.
(b). How would the period P compare with the period P′ of a system consisting of a moon of mass 2m orbiting a planet of mass 2M in an ellipse with the same semi-major axis a?
Please submit your efforts through the comments box below.
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In the Dark 
August 21, 2019 at 6:48 pm
(a) P = f(m/M) r^(3/2) G^(-1/2) M^(-1/2), where f is an unknown function of m/M.
(b) P’/P = 2^(-1/2).
August 22, 2019 at 1:05 pm
Correct! An alternative and equally valid form would have M replaced by m in the second part and a different function of the dimensionless m/M.
August 22, 2019 at 3:27 pm
I’m not sure what happened to your first comment, by the way. It seemed to go into some sort of limbo but I didn’t get a notification that it was pending.
August 21, 2019 at 7:11 pm
G has dimensions LLL/(MTT) (where M momentarily denotes the mass unit), so the only dimensionless quantities are GMPP/(aaa) and GmPP/(aaa), and one of these has to be an arbitrary function of the other. This gives an implicit relation to be solved for P.
More cannot be said without knowledge of the functional form. In the limit m/M <<1, the function giving GMPP/(aaa) in terms of GmPP/(aaa) becomes constant, so that GMPP/(aaa) is constant and PP is proportional to aaa/m, which is one of Kepler's laws.
August 21, 2019 at 7:13 pm
I think you can say more…
August 22, 2019 at 8:55 am
Hint: m/M is also dimensionless.
August 22, 2019 at 9:39 am
That hadn’t escaped my notice but it is not an independent variable if you are already working with GmPP/(aaa) and GMPP/(aaa) as it is their ratio.Of course you can isolate P by doing it with GMPP/(aaa) as a function of m/M.
August 22, 2019 at 9:51 am
So if you do that you can answer the second part precisely….
August 22, 2019 at 2:11 pm
Here’s how to match Anton’s approach onto my solution. Let F be the function Anton refers to, specifically
F( GmPPP/(aaa) ) = GMPP/(aaa)
Define
f(u) = F(GMPPP/(aaa) u)
Then the left side of the first equation is f(m/M). Solve for P, and Bob, as they say, is your uncle.
August 22, 2019 at 4:16 pm
There’s another fun one you can do with deriving the plausibility of white dwarf stars (by considering the dimensions implied from Newtonian mechanics plus Planck’s constant), and then the existence of the Chandrasekhar limit once you add the speed of light as an additional constant to play with.
August 22, 2019 at 5:16 pm
Yes, I like doing dimensional analy. using the units operator U. Sometimes it makes teaching easier.
U(period) = U(circumference) / U(velocity)
= U(circumference) / sqrt(U(K.E.) / U(mass))
= U(circumference) / sqrt(U(P.E.) / U(mass))
since U(P.E.) = U(G)*U(M)*U(m) / U(a^2),
then, U(P) = U(a) / sqrt(U(G)*U(M)*U(m) /
(U(a^2)*U(m)))
At this point see that the U(m) cancels in the denominator. Doubling the mass of the objects changes the period by a factor of 2^-(0.5).
However, I think the rigorous calculation does not predict that the period depends only on M (i.e. referring to Bunn’s f(m/M)).
August 22, 2019 at 5:19 pm
whoops! Sorry about that, should be U(a) in the denominator of U(P.E.), not U(a^2).