A satellite of mass m is in Keplerian orbit around a parent body of mass M >>m. The satellite has total energy E and angular momentum l, where angular momentum is measured about the parent body and the zero of potential energy is at infinity. Detnermine the semimajor axis a and the eccentricity e of the orbit, in therms of the above quantities and Newton'ss constant G (use the symbol, not the numerical value). HINT: Find the relation amonb the energy, angular momentum and orbit raduis at the apside of the orbit. Solve this relation for the apsidal radii r*, and use thed fact that the apsidal radii are r*= a(1+-e).

To determine the semimajor axis (a) and eccentricity (e) of the orbit in terms of the given quantities, total energy (E), angular momentum (l), and Newton's constant (G), we can follow the steps outlined in the hint.

1. Find the relation between energy, angular momentum, and orbit radius at the apside of the orbit:
- At the apsis (the farthest or closest point to the parent body), the satellite's kinetic energy is zero.
- The total energy (E) of the satellite is the sum of its potential energy and kinetic energy:
E = U + K
- At the apsis, the potential energy (U) is equal to zero, so:
E = K = 1/2mv^2
- The satellite's kinetic energy can also be expressed in terms of angular momentum (l) and orbit radius (r):
K = (l^2) / (2mr^2)

2. Solve the relation for the apsidal radii (r*):
- Equate the expressions for kinetic energy:
1/2mv^2 = (l^2) / (2mr^2)
- Simplify the equation by canceling out common factors:
v^2 = l^2 / (mr^2)
- Rearrange the equation to solve for orbit radius (r):
r^2 = l^2 / (m * v^2)
r = (l / v) * (1 / sqrt(m))

3. Use the fact that the apsidal radii are r* = a(1+-e):
- Plug in the expression for orbit radius (r) at the apsis into the equation:
(l / v) * (1 / sqrt(m)) = a(1+-e)
- Rearrange the equation to solve for semimajor axis (a):
a = [(l / v) * (1 / sqrt(m))] / (1 +- e)

So, the semimajor axis (a) of the orbit is given by:
a = [(l / v) * (1 / sqrt(m))] / (1 +- e)

Note: The hint suggests that the apside radii are r* = a(1+-e), which implies elliptical orbits (e > 0). If the orbit is a circle (e = 0), the equation still holds with r* being the constant radius and a = r* = l / v * (1 / sqrt(m)).