Orbital energy of ellipse and hyperbolic trajectory

AI Thread Summary
To find the orbital energy of an elliptical or hyperbolic trajectory, calculate the sum of kinetic energy and gravitational potential energy at any point along the path. For an elliptical orbit, the orbital energy can be determined using the formula -GM/(2a), where G is the gravitational constant, M is the mass of the central body, and a is the semi-major axis. In contrast, the orbital energy for a hyperbolic trajectory is given by GM/(2a). Proving these formulas involves understanding the relationships between kinetic and potential energy in gravitational fields. Understanding these concepts is crucial for analyzing orbital mechanics.
IPhO' 2008
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Please tell me how to find the orbital energy of ellipse and hyperbolic trajectory.
Thank you.
 
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In both cases, it is the sum of the kinetic energy and gravitational potential at any point of the trajectory.

Alternatively, if you know the semi-major axis(a), it can be found by:

-\frac{GM}{2a}
for an ellipse

and

\frac{GM}{2a} for a hyperbola.
 
Please show me how to prove it.
 

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