| MadSci Network: Physics |
Your instinct to be uncomfortable using centripetal acceleration in this problem is a good one. The formula you derived is only valid for circular orbits. The relevant formula for elliptical orbits where the eccentricity isn't zero (which follows from Kepler's third law) is
v = sqrt(G * M * ( 2/r - 1/a)),
where a is the semimajor axis (you can observe how this formula simplifies for a circular orbit, r = a).
Solve this for M,
M = v^2 / (G * (2/r - 1/a))
and try with both sets of your givens,
a = (24,100 + 22,500) / 2 = 23,300E3 m
G = 6.67E-11 m^3/kg/s^2
v1 = 4280 m/s, r1 = 22,500E3 m, M = 5.974E24 kg
v2 = 3990 m/s, r2 = 24,100E3 m, M = 5.957E24 kg
The two values only differ by about 0.3%, probably because the velocities and positions as given don't correspond perfectly to a possible orbit.
An eccentric orbit can be regularized to an equivalent circular orbit. You can effectively do this by using your formula for a circular orbit, calculating the mass at both apogee and perigee, and then calculating the mean of the two values. You'll arrive at the same mass as we got using the more complete formula for non-circular elliptical orbits.
Some good references on elliptical orbits can be found at
en.wikipedia.org/wiki/Elliptic_orbit and
scienceworld.wolfram.com/physics/Orbit.html
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