Proof for elliptical orbits

erkentrs

erkentrs

Answered question

2022-08-11

Its mentioned in several books that a satellite launched with a velocity less than the escape velocity and other than the critical velocity will follow an elliptical orbit. However I can't find a derivation of its equation of trajectory.

Answer & Explanation

Bridget Vang

Bridget Vang

Beginner2022-08-12Added 11 answers

This problem is most easily done in polar coordinates. In polar coordinates, the Newtonian gravity is given by
g = G M r 2 r ^
Applying Newton's Second Law in polar coordinates gives two differential equations: one radial and one angular:
(1) r ¨ r θ ˙ 2 = G M r 2
(2) r θ ¨ + 2 r ˙ θ ˙ = 0
Multiplying (2) by r, we observe that r 2 θ ˙ = h is a constant. This is really just a result of conservation of angular momentum, as there is no force in the angular direction. h is nothing more than the specific angular momentum of the orbiting body.
By substituting u = 1 / r into (1) and (2) and solving, it can be shown that
d 2 u d θ 2 + u = G M h 2
This can then be solved to obtain
r = h 2 G M ( 1 + e cos ( θ θ 0 ) )
where e is the eccentricity.
This is the equation we were looking for: a conic section. e = 0 gives a circle, 0 < e < 1 gives an ellipse, e = 1 gives a parabola, and e > 1 gives a hyperbola.

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