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 the masses M and m) and Newton's constant (G), we can follow these steps:
1. Find the relation between energy, angular momentum, and orbit radius at the apside of the orbit:
At the apsides (the points of closest and farthest distance), the satellite's speed is either maximum or minimum, resulting in kinetic energy (K) equal to half the total energy (E) and potential energy (U) being negative. So we have:
K = 1/2 m v^2 = E/2 (1)
U = -GMm / r = -E (2)
Also, the angular momentum (l) can be expressed as:
l = mvr (3)
From equations (1) and (2), isolate v and substitute it into equation (3) to get a relation between E, l, and r.
2. Solve the relation for the apsidal radii (r*):
Arrange the relation obtained in step 1 to obtain r in terms of E, l, and G.
r = l^2 / (G^2 M^2 m^2 E - 2G M m l^2) (4)
Note: We assume that the energy E is negative for a bounded orbit.
3. Use the fact that apsidal radii are r* = a(1 ± e):
Rewrite equation (4) as r* = l^2 / (G^2 M^2 m^2 E - 2G M m l^2) = a(1 ± e).
Solve this relation for a and e by equating the coefficients of r* on both sides.
By following these steps, you should be able to derive the formulas for semimajor axis (a) and eccentricity (e) of the orbit in terms of the given quantities and Newton's constant.