I was studying gravity and from my previous knowledge of circular motion. i know centrifugal forces balance the force so in gravity sun pull giving centripetal force and and inertia gives centrifugal force. so where the case of elliptical orbits arise?


The solution that classical orbits are conic curvees comes up when you solve the conservation of energy:

$½ mv^2 - \frac{GMm}{r^2}=E_{(const)}$.

Or equivalently

$ \frac{1}{2} m \dot{r}^2 + \frac{1}{2}m(r^2 \dot{\theta}^2)- \frac{GMm}{r^2}=E_{(const)}$.

You join the terms 2 & 3 into a single term called "effective potential energy". This is a differential equation that, after some manipulation, drives us to the solution

$r(\theta)=\frac{l}{1\pm e\cdot \cos(\theta-\theta_0)}$

Where $e$ is the eccentricity of the orbit. Mathematicians show that this is the equation of a conic centered on its focus.


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