This question is about a scientific survey probe in circular orbit around the planetary body Chronos.
Define gravitational potential energy of an object at a point in a gravitational field.
The probe has mass 1800 kg1800\text{ kg}1800 kg. The orbital radius of the probe around Chronos is 6.8×107 m6.8 \times 10^7\text{ m}6.8×107 m. The orbital period of the probe is 4.68×104 s4.68 \times 10^4\text{ s}4.68×104 s. The mass of Chronos is 8.5×1025 kg8.5 \times 10^{25}\text{ kg}8.5×1025 kg. (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)
Show that the magnitude of the gravitational potential energy of the probe is about 1.5×1011 J1.5 \times 10^{11}\text{ J}1.5×1011 J.
Show that the kinetic energy of the probe is half the magnitude of its gravitational potential energy.
Calculate the total energy of the probe.
The power source for the instrumentation on board the probe is a plutonium-238 radioisotope thermoelectric generator, which initially provides a thermal power of 450 W450\text{ W}450 W.
Plutonium-238 decays by α\alphaα-particle emission with a half-life of 88 years88\text{ years}88 years. The kinetic energy of each α\alphaα-particle is 8.9×10−13 J8.9 \times 10^{-13}\text{ J}8.9×10−13 J. (Take 1 year=3.16×107 s1\text{ year} = 3.16 \times 10^7\text{ s}1 year=3.16×107 s)
Calculate the number NNN of plutonium-238 nuclei needed to provide the power of 450 W450\text{ W}450 W initially.
Calculate the power PPP still available from the plutonium-238 source 40 years40\text{ years}40 years later.
