Why is kinetic energy a constant for a satellite in a circular orbit but not for a satellite in an elliptical orbit?

A satellite (or any other celestial body) orbiting another massive body, can describe one of these types of orbits depending on its Orbital Total Mechanical Energy :

* when we talk about an open orbit in which the satellite escapes from the attraction of the planet's gravitational field. The shape of its trajectory is a parabola.

* when we talk open orbits, which are hyperbolic, being the kinetic energy greater than the Potential Energy

* when we talk about closed orbits, that is, the satellite will always be "linked" to the gravitational field of the planet and will describe an orbit that periodically repeats with a shape determined by the relationship between its kinetic and potential energy, as follows:

-Elliptical orbit: Although is constant, and are changing along the trajectory

-Circular orbit: When at all times both the kinetic energy and the potential remain constant, resulting in a total mechanical energy .  

In this context, in a circular orbit, no work is done by gravity on the satellite (or orbiting body), since it always acts perpendicular to the movement. However, in an elliptical orbit this is not fulfilled, because gravity force is not always acting perpendicular to the orbiting body.

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