# HW6_sol (1).pdf - AE 202 Aerospace Flight Mechanics...

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AE 202: Aerospace Flight Mechanics Homework Assignment 6 - Solutions Spring 2020 Guidelines • Follow all requirements for assignment submission as outlined in the syllabus. Submissions not conforming to the requirements may be penalized up to 10%. • Assignments should be submitted electronically as a single file in PDF format via the course website on Compass2g. Multiple files, and those that are not in PDF, will not be graded. • Show all work and clearly indicate final answers. • The following constants may be useful: Gravitational parameter, Earth ( μ Earth ): 3 . 986 × 10 5 km 3 /s 2 Gravitational parameter, Sun ( μ Sun ): 1 . 327 × 10 11 km 3 /s 2 Radius, Earth: 6,371 km Problems 1. [10 points] Write a computer program to convert between orbital elements and position and velocity vectors in the ECI frame. You can check your program using the following orbit data (at Earth): Parameter Orbit 1 Orbit 2 a , km 7371.0 17248.3 e , nd 0.1 0.5965 i , deg 28.5 6.0126 ω , deg 191 354.68 Ω , deg 273 0 ν 0 , deg 10 5.3247 r , km [-2413.9, 6083.8, -1136.0] [6971, 0, 0] v , km/s [ -6.5461, -3.1365, -3.6385] [0.3315, 9.4942, 1] Use your program to compute (at Earth): (a) r and v if a = 6871 . 0 km 1
e = 0 . 07 i = 5 . 2 deg ω = 37 . 8 deg Ω = 89 . 0 deg ν 0 = 213 . 4 deg (b) orbital elements ( a , e , i , ω , Ω , ν 0 ) if r = [4589 . 7 , 4980 . 4 , 2997 . 5] km v = [1 . 259 , 4 . 613 , 6 . 208] km/s Include your computer program/script as text in your assignment. 2. [10 points] Download the data file trv.csv, which contains position and velocity vector data as a function of time for a spacecraft in orbit around the Earth. Using your code from problem 1, create plots of the six orbital elements (one element per plot, use km and deg as appropriate) as a function of time using the data. Discuss the spacecraft trajectory–what is going on here? 3. [15 points] A spacecraft is in an elliptical orbit around the Earth with a = 1 . 8 × 10 7 m and e = 0 . 4 . (a) How much time is required for the spacecraft to travel from ν = 30 deg to ν = 185 deg, in hours? (b) What will the true anomaly angle be 1 hour after the spacecraft passes through periapsis, in deg? 4. [5 points] A spacecraft is launched into a circular parking orbit at Earth with an altitude of 300 km. The spacecraft operators wish to transfer the spacecraft to a geosynchronous orbit (circular, altitude of 35,786 km) at the same inclination. (a) What is the minimum Δ V required for this transfer, in m/s? (b) What is the time of flight associated with this transfer, in hours? 2
Solutions 1. (a) First, calculate semilatus rectum p and scalar position r . Then, calculate r and v in perifocal (PQW) frame. p = a (1 - e 2 ) p = 6837 . 332 km r = p 1 + e cos ν r = 7261 . 701 km r = ( r cos ν ) P + ( r sin ν ) Q r PQW = [ - 6062 . 416 , - 3997 . 427 , 0] km v = r μ p ( - sin ν ) P + ( e + cos ν ) Q v PQW = [4 . 203 , - 5 . 839 , 0] km / s Next, need to rotate position and velocity vectors to an Earth-centered Inertial (ECI) frame.
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