Question

4. Consider a planet in orbit about a star of mass M. Let the x-y plane correspond to the plane of the orbit, with the star at the origin. In the limit in which the stellar mass greatly exceeds the planetary mass, the planet's radial equation of motion is GM h2 p2 + 73 where h = rė is the planet's constant angular momentum per unit mass, and G is the universal gravitational constant. Here, x = r cos @ and y = r sin 8. Equation (??) has an obvious solution. Namely, r=re where h2 rc = GM This solution corresponds to a circular orbit of radius re. Consider a nearly circular orbit such that r(t) = re + ort), where 8r/re < 1. Demonstrate that, to first order in Irl/rc. v=- (39) or, where is the period of the circular orbit. Hence, deduce that a nearly circular orbit satisfies MO) = rete cos (27-to) where e and to are arbitrary constants. (Of course, e «r.) What is the shape of the modified orbit? (Hint: It is not an ellipse.)

Answer 1

Given perturbation in the circular orbit r(t) = rc + Sr(t), where Sr/rc << 1 Putting r(t) in equation 1 we have GM h2 (re+ Sr(t))2 (re + Sr(t))3 Taking re out of the brackets we get GMh2 Sr= (1+Sr(t)/r.)2 + m2(1 + Sr(t)/r.)3 — Šr = GN (1 +5r(t)/r)2+1 + r(/.)* doing binomial approximation upto first order

Sr=- 2 (1 - 25r(t)/re)+(1 - 35r(t)/rc) a in conta com ara el curs + GM GM G ST = - +- 28r 13 3-738r(t) Now we substitute the value of rein this equation, the first and third term cancels both equal to 14 with opposite signs

Finally we have here ,

Sr(t) = - 26r(t)

Now it is given that h = r, but for rc, 0=w, i.e. frequency of the orbit T = 2/w = 2772/h = 21h/G?M? Clearly, then, örſt) =-(ZH)2 Sr(t)

This equation has the solution of the form y(x)=Acos(kx)+Bsin(kx) where, k=2*pi /T Hence for some initial condition it can be easily written as ;

27 Sr(t) = e cos("** (t – to)) Hence, complete solution will be the sum of r(t) and perturbation term .., r(t) = re + e cos(2p (t – to)]

The modified shape will still be like a circle but along with sinusoidal variations over that circle.