# 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.

It is not obvious, but the general solution to this equation producing a bound orbit (where the separation stays finite) can be shown to be an ellipse. A circular orbit is just an elliptical orbit with no eccentricity. The initial conditions determine whether the orbit is circular or elliptical.

It is not true that you necessarily get a circular orbit if the initial velocity is perpendicular to the force. (In an elliptical orbit, the velocity is perpendicular at perihelion and aphelion.) You also have to have just the right amount of speed. So circular orbits are unusual, while elliptical orbits are typical.

The idea that orbits “should” be circular is unjustified. Circular orbits have simpler math that doesn’t require calculus, so you just learn about them earlier. You could argue that the orbit should be a circle because circles are more beautiful than ellipses, but this would be an example of purely aesthetic arguments leading physicists astray, which occasionally happens.