Doubt about the elliptic orbits of the planets

[LCSphysicist]

Homework Statement

I was thinking, if it were ask to me why is the orbit of planets elliptic...

Relevant Equations

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I was thinking to myself, if it were ask to me why is the orbit of planets elliptic, how to justify?:

"It is easy to show, first, that the force is central, we have the motion in a plane, we can also show the energy and angular momentum is conversed without mention to power function of the force."
"Now, we can show that the only forces that could result in bound orbits, whatever be the particle, is the f = -kr^n, n equals one or minus two"
"But, n equals one is such unlikely, the only one n possible is equals minus two"
"So, we can now integrate the orbit equation, and get the equation of conics"

"Here is the thing, since we could say that the orbit is closed (or, at least, approximately), the only possible path is circular or elliptic"
"It's sufficient one statement to close the discussion: Find out if the orbital speed of the planet is constant or not, because L = rxmv,for circular orbits, |r| remains constant and perpendicular to the velocity tangential, but L is constant, so V would be constant."
"But the speed with respect to the stars, or the center of mass of universe or whatever frame inertial, varies with the time"
"Only the elliptical orbit remained"
"So is elliptical"

The bould phrases i just know to justify with collected dates.

IS this a reasonable justification?

 

[etotheipi]

You might find it interesting to have a look at the justification for Bertrand's Theorem.