How to prove that the orbit of celestial bodies is elliptical?

If the orbit of a celestial body is circular, that is, it moves in a uniform circle. That is to say, gravity completely acts as a centripetal force, and there is no acceleration tangent to the orbit of the celestial body (the speed of the celestial body is constant), which is the characteristic of circular motion.

However, the orbits of actual celestial bodies are mostly elliptical, which means that gravity has a component that acts on the tangent direction of the orbit, such as the earth.

On earth, if you look carefully, you will find that the sliding speed of stars in the sky is different in winter and summer. Because the earth's orbit is elliptical, it is close to the sun in summer and runs slower than in winter. You can use the constellation near Polaris as reference coordinates.

Finding this will prove that the orbit is elliptical. Because of the speed difference of celestial bodies in the same closed orbit, it is proved that gravity does act on celestial bodies, that is, there is a tangential component.