newtonian mechanics - Elliptical orbit of revolution of earth

Well known Kepler laws state that earth revolve around sun in an elliptical path with sun at one of the focii. My question is rather simple. Why so?

I mean for an equilibrium of earth that is it is not accelerated towards or away from sun its gravitational force must be balanced by centrifugal force (pseudo force as frame corresponding to earth)

But following an elliptical path it distance from sun won't be same and hence the gravitational force and hence how is this?

I suspect may be it is due to presence of other celestial bodies but that seems too vague and escaping from the truth

Also maximum of derivations in gravitation are done by assuming circular path of earth around sun. (atleast at my level)

Please give me the real physics behind it?

Answer

What you are missing is that Earth speed is varying along its orbit, contrary to a circular one.

More precisely, at any point of the orbit, the acceleration of the Earth has a component tangent to the orbit, and a component perpendicular to the orbit. For a circular orbit, the former is always zero, and the latter is therefore exactly equal to the gravitational force, i.e. a centripetal acceleration. That was in an inertial frame, so now if we take a frame rotating along with the Earth, that's the picture you had in mind: the centrifugal force, which is opposite to the centripetal acceleration in the inertial frame, compensates the gravitational force.

But for an elliptic orbit, the component of the acceleration tangent to the orbit is only zero at the apogee and perigee. Since the gravitational force is equal to the sum of the tangent and normal component of the acceleration, we can say that part of the gravitational force bends the trajectory toward the Sun, and part accelerate the Earth along the trajectory (or decelerate it, depending on the position on the orbit).