Answer to Question #186981 in Classical Mechanics for Aqib Siddique

Question

For an elliptical orbit in gravity with energy E and angular momentum L, (a) find an expression for the semi-major axis (in terms of m, M, G, E, and L) and (b) for the eccentricity. This is also how the book operates: in our language, the equation for u = 1/r is 

For numbers 3 & 4, it is easier to use the differential equation for the orbit that comes from analysis of the FORCES, not the ENERGETICS. This is also how the book operates: in our language, the equation for u = 1/r is  [The book’s equation contains the parameter ℓ which is just L/m, a constant; NOT the latus rectum, but related to it since both depend on for the inverse-square force. The orbits in 3 & 4 have no definable latus rectum].

 

Answer 1

The eccentricity of the orbit

$$\epsilon=\sqrt{1+\frac{2EL^2}{\mu (GmM)^2}}$$

$$a=\frac{L^2}{\mu GmM}$$