Put your observatory here: Although a single planet of given mass m in orbit abo
ID: 2078834 • Letter: P
Question
Put your observatory here: Although a single planet of given mass m in orbit about let's say, a star of mass M, with a known orbital radius, can have only one period of its orbit for that given radius, if a third object (much smaller mass) were placed at certain special points within this system (i.e. the Lagrange points), then this third object (with two forces of gravity acting on it from the star and then the planet) can have the same period of orbit as the planet has the star. Find the radius of this third object from the star such that this could occur. One position would be in between the star and the planet and another position would be outside of the planet and star; both of those positions would be along a line joining the star and planet at any given instant. Find these two points. There are other Lagrange points but that's not this problem.
Explanation / Answer
Given, mass of star = M
mass of planet = m
orbit of planet around star = R
now, for the planet and star system
if linear velocity of planet in orbit is v
then v = 2*pi*R/T [ T is orbital time period]
and, GMm/R^2 = mv^2/R
GM/R = v^2 = 4*pi^2*R^2/T^2
T^2 = 4*pi^2*R^3/GM
T = 2pi*Rsqroot(R/GM)
now if an object of mass m' is on the line joining the star and the planet, there can be two cases
1. its between the star and the planet
in this case, let its linear velocity be v'
and its distance from the star be r
then v' = 2*pi*r/T [ same time period ]
but from force balance
GMm'/r^2 = Gmm'/(R - r)^2 + m'*v'^2/r
GM/r^2 = Gm/(R - r)^2 + 4*pi*pi*r*r/r*T*T
GM/r^2 = Gm/(R - r)^2 + 4*pi*pi*r*G*M/4*pi^2*R^3
M/r^2 = m/(R - r)^2 + r*M/R^3
M(R - r)^2 *R^3 = m*R^3*(r^2) + r^3*M*(R - r)^2
M(R^2 + r^2 - 2Rr) *R^3 = m*R^3*(r^2) + r^3*M*(R^2 + r^2 - 2Rr)
MR^5 + Mr^2*R^3 - 2MR^4*r = mR^3*r^2 + MR^2*r^3 + Mr^5 - 2MRr^4
the solution of this equattion of 5 degree gives the value of r
similiarly
the other distnce when the planet is in between body and star is the solution of this equation of order 5 in r
M/r^2 + m/(R + r)^2 = r*M/R^3
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