Consider a planet of mass \(m\) moving at constant speed \(v\) in a circular orbit of radius \(R\) under the influence of the gravitational attraction of a star of mass \(M\).

(a) What is the planet's kinetic energy, in terms of \(m, M, G\), and \(R\) ?

(b) What is the energy of the star-planet system?

(c) Using the expression for the eccentricity obtained in Example 13.5, obtain an expression for \(E_{\text {mech }}\) for the circular orbit.

(d) Compare your answers to parts \(b\) and \(c\).

Data from Example 13.5

A satellite of mass \(m_{\text {sat }}\) is in an elliptical orbit around a star of mass \(m_{\text {star }} \gg m_{\text {sat }}\). If the mechanical energy of the starsatellite system is \(E_{\text {mech }}\) and the magnitude of the angular momentum of the satellite about the star is \(L\), what are the semimajor axis \(a\) and the eccentricity \(e\) of the satellite's orbit?