You send a probe to orbit Mercury at 192 km above the surface. What orbital velocity (in km/s) is needed to keep it in orbit? The mass of Mercury is 3.30 * 10^23 kg, and the radius of Mercury is 2.44 * 10^3 km. What is the ratio of the time it takes a signal from Earth to reach Mercury (d = 57.9 x 10^6 km) to the time it would take to reach the Moon (d = 384,400 km)? If your signal is at 15 cm, what is the wavelength shift (in cm) at this orbital velocity? (Assume the probe is at a point in its orbit in which it is moving directly away from the Earth.)
Part 1 of 4: The orbital velocity is just the circular velocity, where the distance is the distance above the surface plus the radius of Mercury: GMercury = 6.32 * 10^6 km^3/s^2, r = 2.63 * 10^6 km.
Part 2 of 4: Since the signal is traveling at the speed of light, the time is directly related to the distance from the probe.
Where we have two expressions: t_Mercury = d_Mercury / c and t_Moon = d_Moon / c
Taking their ratios gives: t_Mercury / t_Moon = d_Mercury / d_Moon