AEE 4263 HW 7

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Florida Institute of Technology *

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4263

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Mechanical Engineering

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Apr 3, 2024

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docx

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AEE 4263-02 Space Flight Mechanics Homework HW7 Fall 2023 Use the following constants: Earth gravitational parameter: μ = 398600.4418 km 3 / s 2 Earth radius: R E = 6378.137 km 1. (15 points) Given the orbital state vector below, compute the corresponding Geocentric Equatorial (GCE) to Radial-Intrack=Crosstrack (RIC) transformation matrix – i.e., C GCE RIC . Provide numeric precision to six decimal places. r = [ 5089.9839 2974.0851 3342.2275 ] km v = [ 0.473944549 6.07102533 4.67328585 ] km / s Solution: 0.751101 -0.438870 -0.493194 C GCE RIC = 0.062093 0.790702 -0.609044 0.657261 0.426830 0.621148 MATLAB: close all clear all clc %given: mu= 398600.4418; %km^3/s^2 Re= 6378.137; %km r= [5089.9839; -2974.0851; -3342.2275]; %km v= [0.473944549; 6.07102533; -4.67328585]; %km/s h= cross(r,v); R= r/norm(r); C= h/norm(h); I= cross(C,R); C_GCE_RIC= [R I C]' 1
2. (35 points) Given a circular orbit with a 350 km altitude, compute the two velocities changes ( v s ) needed to maneuver to a 35,000 km altitude using a Hohmann transfer. Compute the velocity changes in meters/second, to two decimal places. a. (10 points) v 1 = ¿ b. (10 points) v 2 = ¿ c. (5 pts) What is the transfer orbit semi-major axis in km, to 3 decimal places? d. (5 pts) What is the transfer orbit eccentricity to six decimal places? e. (5 pts) What are the flight path angles at the two v s in degrees, to two decimal places? Solution: a. v 1 = ¿ 2398.34 m/s b. v 2 = ¿ 1462.21 m/s c. a= 24053.137 km d. e= 0.720280 e. λ at v 1 = ¿ 0.00, λ at v 2 = ¿ 0.00 MATLAB close all clear all clc %given: mu= 398600.4418; %km^3/s^2 Re= 6378.137; %km height1=350; %km height3= 35000; %km r1= Re+height1; r3= Re+height3; ra2= Re+height3; rp2= Re+height1; a2= (rp2+ra2)/2 deltav1= (sqrt(mu*((2/rp2)-(1/a2))) - sqrt(mu/rp2)) *1000 %m/s deltav2= (sqrt(mu/ra2) - sqrt(mu*((2/ra2)-(1/a2)))) *1000 %m/s e2= (ra2-rp2)/(ra2+rp2) 2
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