Elliptical orbit in Newtonian gravity

Question

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?

Answer 1

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.