Exercises with solutions

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Exercise list 1: The two body problem Orbital Mechanics 2018/2019 Exercise list 1: The two body problem Exercise 1.1 . Characterize a Geostationary orbit. Answer: Circular, equatorial. T = 23 h 56 min. Radius r = 42164 km (altitude h = 35786 km). Exercise 1.2 . Calculate the maximum latitude and the percentage of the Earth’s surface visible from GEO. GEO view at different longitudes Answer: Max atitude φ = 81.3◦. Percentage of hemisphere 84.9%. Exercise 1.3. Two spacecraft perform a parabolic trajectory about the Earth with a perigee of 7000 km. Considering that the two s/c are, after their respective fly-by, at P1 (8000 km) and P2 (16000 km) from the centre of the Earth, calculate i. The distance among the two spacecraft. ii. Their flight path angle. iii. The escape velocity at P1 and P2 and at infinite. Answer: i. P1P2 = 13.270 km. ii. γ1 = 20.70◦,γ 2 = 48.59◦. iii. v1 = 9.9825 km/s, v2 = 7.0587 km/s. Exercise 1.4 . At a given point on a spacecraft geocentric trajectory the radius is 14600 km and the speed is 8.6 km/s and the flight path angle is 50 degrees. Show that the orbit path is hyperbolic and calculate i. h. ii. e. iii. θ. 1 Orbital Mechanics 2018/2019 Exercise list 1: The two body problem iv. rp. v. a. vi. C3. vii. Turn angle. viii. Aiming radius. Answer: h = 80.708 km 2/s. e = 1.3393. θ = 84.889◦. rp = 6986 km. a = −20590 km. C3 = 19.36 km 2/s 2. δ = 96.60◦. ∆ = 18340 km. Exercise 1.5 . The Asteroid Express spacecraft communicates its telemetry data to Earth’s ground station on Oct 4th 2017, at 6:15 pm (local civil time at ground station). The space- craft is travelling at a velocity relative to Earth of 10 km/s, directed as depicted (β = 120◦), at a distance of 10000 km from the center of the Earth. i. Characterize the geometry of the spacecraft’s orbit ( a,e…