The New Horizons spacecraft, launched in 2006, approached the quasi-planet Pluto at an altitude of 14,000 km with an orbital velocity of 14 km/s in July 2015. The spacecraft then performed a “flyby,” passing by Pluto without entering its orbit.
Suppose that the spacecraft did not perform a flyby, but instead, as shown in the Figure, the spacecraft thrusted its propellant back at point A on orbit I and entered an elliptical orbit (orbit II) with an eccentricity e = 0.8 and apogee altitude H = 14,000 km. Answer the following questions for this case. Assume that Pluto’s radius Ro is 1185 km, the gravitational parameter μp is 8.749 x 10^2 (km^3/s^2), and the velocity V on orbit I is 14 km/s. Calculate to five significant digits.
(1) Find the semi-major axis a, semi-minor axis b, and semi-focal distance c of the elliptical orbit (orbit II).
(2) Find the orbital velocity Va at the apogee of the elliptical orbit (orbit II), and find the difference in velocity ΔV between before and after the injection.
(3) At perigee B on the elliptical orbit (orbit II), a reverse thrust is made to shift to a circular orbit (orbit III) with e=0. Find the orbital altitude h and orbital velocity Vb .