Elliptic orbit of earth round the sun

[Apache]

Hello, I would like to ask why if the Earth rotate around the Sun in circle orbit, it must have an kinetic energy equal to negative half of potential energy it has at anytime rotating round the sun, which Ek = -Ep/2 ! If this situation cannot be achieved, our Earth will move in a new ellipse orbit round the Sun. I read this article from a Chinese website, just troubling why Ek = -Ep/2 for circle orbit to be formed. Hope can get perfect answer...

 

[D H]

First things first: The Earth does not orbit the sun in a perfectly circular orbit. In reality, nothing does: There is no such thing as a perfectly circular orbit. They are impossible to achieve in practice.

I'll ignore:

• The presence of the Moon, other planets, other stars;
• That the Sun and Earth are not perfectly spherical;
• That the Sun is losing mass because it is radiating light and spewing particles in the form of the solar wind;
• That nature does not obey Newton's law of gravity (general relativity is closer to the truth).

Each of the above makes a perfectly circular orbit not only a practical impossibility but also a theoretical impossibility. Ignoring those details, assuming the Sun and Earth are point masses, and assuming Newtonian gravity rules the universe, then circular orbits are a theoretical possibility.

However, they are still impossible to achieve in practice. For a given orbital distance between the (point mass) Sun and a (point mass) planet, there is one and only one orbital speed that yields a perfectly circular orbit, and then only if the velocity is perpendicular to the radius vector. If the velocity vector exhibits any deviation whatsoever from orthogonality or any deviation whatsoever in magnitude from the circular velocity then the planet will not be in a circular orbit. It will be in an elliptical orbit (or a parabolic orbit, or a hyperbolic orbit) instead.