A satellite of mass mmm is in a circular orbit of radius rrr around a planet of mass MMM. Show, using Newton's law of gravitation and the equations of circular motion, that the relationship between the orbital period TTT of the satellite and its orbital radius rrr is given by:
T2∝r3T^2 \propto r^3T2∝r3
An exploration probe of mass 480 kg480\text{ kg}480 kg is launched into an elliptical orbit around an exoplanet of mass 3.50×1024 kg3.50 \times 10^{24}\text{ kg}3.50×1024 kg. The probe is recorded to have an orbital period of 16.0 days16.0\text{ days}16.0 days.
The closest distance (periapsis) of the probe to the centre of the exoplanet is 2.00×107 m2.00 \times 10^7\text{ m}2.00×107 m and its furthest distance (apoapsis) is 1.00×108 m1.00 \times 10^8\text{ m}1.00×108 m.
A small inner moon of the exoplanet serves as a reference point. This moon has a mean orbital distance of 1.50×107 m1.50 \times 10^7\text{ m}1.50×107 m and an orbital period of 2.00 days2.00\text{ days}2.00 days. Use Kepler's third law to calculate the mean orbital distance of the probe from the exoplanet.
Assuming that the total mechanical energy of the probe in its elliptical orbit remains constant, calculate the change in the kinetic energy of the probe as it moves from its furthest point to its closest point.
(Take G=6.67×10−11 N m2 kg−2G = 6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}G=6.67×10−11 N m2 kg−2.)
Suggest why the total mechanical energy of the probe in its orbit around the exoplanet is not equal to the total mechanical energy it possessed during the launch phase from the surface of its home planet.