A research group is examining the energetic cost of placing a scientific satellite of negligible mass into a low circular orbit about Earth. The radius of Earth is RE=6.37×106m and the standard gravitational parameter of Earth is μE=3.99×1014m3⋅s−2. The desired orbit is 300km above Earth’s surface, and the group wishes to compare the orbital binding energy to typical chemical energy densities of 5×106J⋅kg−1.
(a) Derive an expression for the total mechanical energy per unit mass of a satellite in a circular orbit of radius r around Earth, expressing your result only in terms of μE and r. Provide clear reasoning for every step in your derivation.
(b) Using the expression from part (a), calculate the binding energy per unit mass for the orbit 300km above Earth’s surface. Report a numerical value with appropriate units.
(c) Determine the ratio of the magnitude of the orbital binding energy per unit mass you found in part (b) to the stated chemical energy density. Comment on whether ordinary chemical reactions could supply enough energy to place a payload of mass 1.0kg into this orbit if the conversion from chemical energy to orbital energy were 100% efficient.
(d) Interpret physically what it would mean if the total mechanical energy of the satellite were zero instead of the negative value obtained in part (b).