Is there any model explaining why would a particle revolve around another one in an elliptical orbit? [duplicate]

The reason for asking this question is to understand the elliptical orbits of planets and why aren't they circular, we know that if a particle moves with an initial velocity perpendicular to the gravitational force of another particle, it orbits it in a circle as the perpendicular acceleration is constantly changing the direction of the velocity.

I recently came across Kepler's laws of planetary motion, and didn't see him explain why are the orbits elliptical.

But I did some research on it and I learnt that a particle can move around/orbit another particle in any of the four conic sections, all I want now is to compare the conditions of a circular orbit(I know them) to the ones for an elliptical orbits(that's why you're here).

If theoretically planets should orbit in circles then my guess would be that modeling the planets and their stars as particles when they aren't even perfect spheres and the effects of external gravitational fields is the reason.

marked as duplicate by Qmechanic♦Mar 3 at 18:09

• General tip: Look in the right margin for related questions. – Qmechanic Mar 3 at 16:26
• I did say that I did research on it before asking, didn't find someone explaining the model they just said it exists. – user597368 Mar 3 at 16:30
• Elliptical orbits are implied by Newton's law of gravitation. The derivation is a standard one, done in any moderately advanced classical mechanics course. – Javier Mar 3 at 16:36
• “If theoretically planets should orbit in circles...” This is an incorrect assumption. – G. Smith Mar 3 at 16:56
• We thank Kepler for finding that the orbits are elliptical in the first place. Not he but Newton explained 150 years or so later why this is so. – my2cts Mar 3 at 17:28

Yes. The model is that the two particles attract each other with a force that decreases as the inverse square of their separation. $$\mathbf{F}=m\mathbf{a}$$ then gives a differential equation for the orbit of the form
$$\frac{d^2\mathbf{r}}{dt^2}\propto-\frac{\mathbf{r}}{|\mathbf{r}|^3}$$
where $$\mathbf{r}(t)$$ is the separation vector.