AE 202: Aerospace Flight MechanicsHomework Assignment 6 - SolutionsSpring 2020Guidelines• Follow all requirements for assignment submission as outlined in the syllabus. Submissionsnot conforming to the requirements may be penalized up to 10%.• Assignments should be submitted electronically as a single file in PDF format via the coursewebsite 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 (μEarth):3.986×105km3/s2–Gravitational parameter, Sun (μSun):1.327×1011km3/s2–Radius, Earth: 6,371 kmProblems1. [10 points] Write a computer program to convert between orbital elements and position andvelocity vectors in the ECI frame.You can check your program using the following orbit data (at Earth):ParameterOrbit 1Orbit 2a, km7371.017248.3e, nd0.10.5965i, deg28.56.0126ω, deg191354.68Ω, deg2730ν0, deg105.3247r, 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)randvifa= 6871.0km1
e= 0.07i= 5.2degω= 37.8degΩ = 89.0degν0= 213.4deg(b) orbital elements (a,e,i,ω,Ω,ν0) ifr= [4589.7,4980.4,2997.5]kmv= [1.259,4.613,6.208]km/sInclude 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 asa 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 witha= 1.8×107m ande= 0.4.(a) How much time is required for the spacecraft to travel fromν= 30deg toν= 185deg,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 of300 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 minimumΔVrequired for this transfer, in m/s?(b) What is the time of flight associated with this transfer, in hours?2
Solutions1.(a) First, calculate semilatus rectumpand scalar positionr. Then, calculaterandvinperifocal (PQW) frame.p=a(1-e2)p= 6837.332 kmr=p1 +ecosνr= 7261.701 kmr= (rcosν)P+ (rsinν)QrPQW= [-6062.416,-3997.427,0] kmv=rμp(-sinν)P+ (e+ cosν)QvPQW= [4.203,-5.839,0] km/sNext, need to rotate position and velocity vectors to an Earth-centered Inertial (ECI)frame.