Lagrange Points Calculation Work

[Samuelriesterer]

(Sorry text is hard to read, please see attached document for an easier read)

I am having trouble with #6, I'm not sure if what I have going on is entirely correct. Also #7 is a little confusing.

Problem Statement & work done:

For an object in orbit around a second, there are five LaGrange points where a small mass will orbit the second body with the same period as the first due to the combined gravity of the two objects. The masses at these LaGrange points are effectively remain in fixed positions relative to the first body and they all orbit the second body as if they were attached to a frame.

Three of these points lie on the line passing through the two bodies. For these points, we can use Newton's Law of gravity to get the force on a small mass at L1 from each of the two bodies. The sum of these two forces is the net force which can be represented as the centripetal force.

1 - Use Newton's law of gravity and centripetal force to get an expression for the period of mass m in orbit about mass M distance R away. (geometry!)

GMm/R^2 = mV^2/R → T = sqrt[(4pi^2R^3)/(GM)]

2 - If a small mass at L1 is to have the same period as mass m, write an expression for its orbital speed in terms of that period and the orbital radius (R - r1).

v = (2pi(R-r1))/sqrt[(4pi^2R^3)/(GM)]

3 - Write the expression for the total force on a small mass u at L1 exerted by the large mass M distance R-r1 away and the smaller mass m distance r1 away in the opposite direction.

F_net = GMu/(R-r1)^2 – Gmu/r1^2

4 - Set the net force equal to u times the centripetal acceleration using the speed from [2].

GMu/(R-r1)^2 – Gmu/r1^2 = uv^2/(R-r1)

5 - Substitute for T2 from [1] and cancel all common factors.

GMu/(R-r1)^2 – Gmu/r1^2 = (uGM(R-r1))/R^3

6 - Clear all fractions by multiplying all terms by the denominators and then divide by M.

(GMu(r1)^2)/(R-r1)^2 – (Gmu(R-r1)^2)/((r1)^2(R-r1)^2) = (uGM(R-r1))/R^3 →
(GMu(r1)^2-Gmu(R-r1)^2)/((r1^2)(R-r1)^2) = (uGM(R-r1))/R^3 →
R^3(GMu(r1)^2 – Gmu(R-r1)^2) = GMu(r1)^2(R-r1)^3

7 - Next, determine the power to which radii or combinations of them are raised
in each term. (For example, (R+r)2r3 would be counted as the power 5.) Then divide
every term by R to that power and rewrite the expressions using p = r/R as shown
below:

(r(R-r)^2)/R^3 – p(1+p)^2

You should get something equivalent to:

(m/M)(1+p)^2 = (1+p)^3 p^2 - p^2

 

[haruspex]

This treats M as fixed in space. The usual treatment allows that m and M orbit each other around a common mass centre.
Leaving that aside, I see no way to get 1+p terms. Looks like there's been a confusion between L1 and L2. The example given

(r(R-r)^2)/R^3 – p(1+p)^2

should read $(r(R-r)^2)/R^3 = p(1-p)^2$.
For L2, the sign of the left hand side of eqn 4 is also switched. I believe this accounts for all the differences between the target answer and your own answer.

 

[Samuelriesterer]

Thanks for the reply. You are right, that is the equation, I had a typo on the minus sign. And L2 is the same with reversed signs. Please take a look at my upload file, I have worked out the problem but am really unsure about the results. Thanks so much!