What is the magnitude of the angular momentum of the asteroid at aphelion?

Express the magnitude of the angular momentum in terms of , , , , and .

=

m*V_a*R_a

The total energy and angular momentum of the asteroid at perihelion are obtained by replacing with and with in the equations just derived. Instead of considering aphelion and perihelion separately, let's consider them simultaneously by replacing and with the generic symbol , and replacing and with the generic symbol , where the subscript "e" stands for extremum (since aphelion and perihelion are the extrema of the orbit). The reason for doing this will become clear in a moment.

Find a quadratic equation of the form that relates to the known quantities , , , , and .

Express your answer in terms of , the known quantities and , and the fixed constants , , and . Construct your equation such that the coefficient of the term is 1.

0 =

R_e^2+(G*M*m/E)*R_e-0.5*L^2/(m*E)

The quadratic you have just derived,

,

has two solutions: One will correspond to , and the other will correspond to . This was the reason for using the generic symbol . Solve for using the quadratic formula and find (the larger root). Remember that must be negative, which means that the term in the quadratic equation is positive. (This is to be expected since the results should give radii, which need to be positive.)

Express the distance at aphelion in terms of , the known quantities and , and the fixed constants , , and . The results are a bit messy, so be careful when typing in your answers.

=

0.5*(-G*M*m/E+sqrt((G*M*m/E)^2+2*L^2/(m*E)))

Now solve your quadratic equation to get the perihelion distance (the smaller root).