AEE 4263 HW 3
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AEE 4263-02 Space Flight Mechanics Name: Homework
HW3
Fall 2023
For all problems, use the following constants:
Earth gravitational parameter: μ
=
398600.4418
km
3
/
s
2
Earth radius: R
E
=
6378.137
km
1. (15 pts) Given a satellite in an elliptical Earth orbit, with semi-major axis a
=
8684.137
km
, and eccentricity e
=
0.214538
.
a)
What is the orbit’s specific angular momentum magnitude h
in km
2
/
s
?
b)
find the following parameters for true anomalies of 0, 30, 90, 120, 180, 240, 270, and 330 degrees:
i.
orbital radius r
in km
.
ii.
orbital speed v
in km
/
s
.
iii.
radial velocity component v
r
in km
/
s
.
iv.
tangential (horizontal) velocity component v
⊥
in km
/
s
.
v.
flight path angle γ
in degrees
.
c)
What are the apogee and perigee altitudes?
d)
What is the symmetry between the parameters in part b) ?
Solution:
a)
h = 57464 km^2/s
b)
i.
r(0)=6821km, r(30)=6986km, r(90)=8284km, r(120)=9280km, r(180)=10548km, r(240)=9280km, r(270)=8284km, r(330)=6986km
ii.
v(0)=8.42km/s, v(30)=8.26km/s, v(90)=7.09km/s, v(120)=6.33km/s, v(180)=5.45km/s, v(240)=6.33km/s, v(270)=7.09km/s, v(330)=8.26km/s
iii.
vr(0)=0.00km/s,
vr(30)=0.74km/s, vr(90)=1.49km/s, vr(120)=1.29km/s, vr(180)=0.00km/s, vr(240)=-1.29km/s, vr(270)=-1.49km/s, vr(330)=-0.74km/s
iv.
v
⊥
(0)=8.42km/s, v
⊥
(30)=8.23km/s, v
⊥
(90)=6.94km/s, v
⊥
(120)=6.19km/s, v
⊥
(180)=5.45km/s, v
⊥
(240)=6.19km/s, v
⊥
(270)=6.94km/s, v
⊥
(330)=8.23km/s
v.
γ
(0)=0.00deg, γ
(30)=5.17deg, γ
(90)=12.11deg, γ
(120)=11.76deg, γ
(180)=0.00deg, γ
(240)=-11.76deg, γ
(270)=-12.11deg, γ(330)=-5.17deg
c)
ha = 4169.5km, hp = 442.53km
d)
There is symmetry between 30&330deg, 90&270deg, 120&240deg, for the orbital radius, orbital
speed, and horizontal velocity. There is symmetry between 0&180deg, 30&330deg, 90&270deg, 120&240deg for the magnitude of radial velocity component and flight path but they are opposite in sign.
MATLAB:
close all clear all
clc
mu= 398600.4418; %km^3/s^2
Re=6378.137; %km
a= 8684.137; %km
e= 0.214583;
%a)
What is the orbit’s specific angular momentum magnitude
h= sqrt(mu*(1-e^2)*a)
% b) find the following parameters for true anomalies of 0, 30, 90, 120, 180,
240, 270, and 330 degrees:
% i. orbital radius r in km.
% ii. orbital speed v in km/s.
% iii. radial velocity component v_r in km/s.
% iv. tangential (horizontal) velocity component v_
⊥
in km/s.
% v. flight path angle γ in degrees.
theta= [0 30 90 120 180 240 270 330]; %deg
n=length(theta);
%i
for i=1:n
r(i)= (h^2)/(mu*(1+(e*cosd(theta(i)))));
fprintf(
'r(%g)=%0.0fkm,\t'
,theta(i),r(i))
end %ii
for i=1:n
v(i)= sqrt(mu*((2/r(i))-(1/a)));
fprintf(
'v(%g)=%0.2fkm/s,\t'
,theta(i),v(i))
end
%iii
for i=1:n
vr(i)= (mu*e*sind(theta(i)))/h;
fprintf(
'vr(%g)=%0.2fkm/s,\t'
,theta(i),vr(i))
end
%iv
for i=1:n
vperp(i)= (mu*(1+e*cosd(theta(i))))/h;
fprintf(
'v
⊥
(%g)=%0.2fkm/s,\t'
,theta(i),vperp(i))
end
%v
for i=1:n
fp(i)= atand((e*sind(theta(i))/(1+e*cosd(theta(i)))));
fprintf(
'γ(%g)=%0.2fdeg,\t'
,theta(i),fp(i))
end
%c)
What are the apogee and perigee altitudes?
ha=(h^2/(mu*(1-e)))-Re %km
hp=(h^2/(mu*(1-e)))-Re %km
%d) What is the symmetry between the parameters in part b) ?
disp(
'There is symmetry between 30&330deg, 90&270deg, 120&240deg, for the orbital'
)
disp(
'radius, orbital speed, and horizontal velocity. There is symmetry between'
)
disp(
'0&180deg, 30&330deg, 90&270deg, 120&240deg for the magnitude of radial'
)
disp(
'velocity component and flight path but they are opposite in sign.'
)
2. (5 pts) Given a spacecraft departing Earth on a hyperbolic trajectory, with a semi-major axis
a
=
10,000
km
and eccentricity e
=
1.284
, find the following:
a)
True anomaly of the asymptote θ
∞
in degrees
.
b)
Hyperbolic excess speed v
∞
in km
/
s
.
c)
Aiming radius ∆
in km
.
Solution:
a)
θ
∞
1
= 141.15deg, θ
∞
2
= 218.85deg
b)
v
∞
= 6.31km/s
c)
∆
= b = 8053.9km
MATLAB:
close all clear all
clc
mu= 398600.4418; %km^3/s^2
Re=6378.137; %km
a= 10000; %km
e= 1.284;
%a) True anomaly of the asymptote θ_∞ in degrees.
theta_inf1=acosd(-1/e) %deg
theta_inf2=360-theta_inf1 %deg
%b)
Hyperbolic excess speed v_∞ in km/s.
v_inf=sqrt(mu/a) %km/s %c) Aiming radius ∆ in km.
b= a*sqrt(e^2-1)
%km
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