# Celestial Mechanics and Orbital Period

AE-641 Problems Set No. 1 1. The orbital period of an Earth satellite is 106 min. Find the apogee altitude if the perigee altitude is 200 km. 2. Find the orbital period of a satellite if the perigee and apogee altitudes are 250 km and 300 km, respectively. 3. Find the maximum and minimum orbital speed of the Earth if the eccentricity of the Earth’s orbit around Sun is 1/60. What is the mean speed if the mean radius is 1 AU? (Sun’s Gm1=1. 327×10 11 km3/s 2. ) 4.

Given the orbital period of Mars around Sun as 687 Earth mean solar days, find the semi-major axis of Mars orbit in AU. 5. Estimate solar gravitational constant using Kepler’s third law. 6. A spacecraft in a 200 km high circular Earth orbit fires its retro-rocket, reducing speed instantly by 600 m/s. What is the speed of the spacecraft when it reaches an altitude of 100 km? (Assume zero atmospheric drag. ) 7. What is the parabolic escape velocity from a geosynchronous orbit?

What extra speed will be required for a geosynchronous satellite to escape Earth’s gravity? 8. A hyperbolic Earth departure trajectory has a perigee speed of 15 km/s at an altitude 300 km. Calculate (a) hyperbolic excess speed, (b) radius and speed when true anomaly is 100o. 9. Voyager-I’s closest approach to Saturn was at a periapsis radius of 124000 km and the hyperbolic excess speed was 7. 51 km/s. What was the angle through which the spacecraft’s velocity vector was turned by Saturn? (Saturn’s m = 37. 931×10 6 km3/s2. ) 10.

Derive expressions for the position and velocity vectors of a spacecraft in a coordinate system fixed to the orbital plane such that the unit vectors of the axes are along the eccentricity vector, e, the direction of parameter, p, and the angular momentum vector, h. Express the answers in terms of semi-major axis, a, eccentricity, e, and true anomaly, q. 11. Halley’s comet last passed perihelion on February 9, 1986. Its orbit has a semi-major axis, a=17. 9564 AU and eccentricity, e=0. 967298. Predict the date of next return (perihelion) of the comet.