Calculating Lagrange 1
Posted: 11.12.2003, 21:10
I used the equations for the force due to gravity, and circular motion equations to derive the following for the distance of Lagrange 1:
(d^5) - 2a(d^4) + (a^2)(d^3) - (1-k)(a^3)(d^2) + 2(a^4)d - (a^5) = 0
Where d is the distance from the primary to Lagrange 1, a is the separation of the primary and secondary objects and k is the mass of the secondary divided by the mass of the primary.
Now this looks suspiciously similar to a binomial expansion to me, and while I am perfectly capable of using the Newton-Raphson process to find a numerical solution to the equation, I would like to know if there is an algebraic solution.
If you could outline the technique to find such a solution, then maybe I could also apply it to L2 and L3.
(d^5) - 2a(d^4) + (a^2)(d^3) - (1-k)(a^3)(d^2) + 2(a^4)d - (a^5) = 0
Where d is the distance from the primary to Lagrange 1, a is the separation of the primary and secondary objects and k is the mass of the secondary divided by the mass of the primary.
Now this looks suspiciously similar to a binomial expansion to me, and while I am perfectly capable of using the Newton-Raphson process to find a numerical solution to the equation, I would like to know if there is an algebraic solution.
If you could outline the technique to find such a solution, then maybe I could also apply it to L2 and L3.