## Transcribed Text

Unless stated otherwise, assume that:
RG 432288 km
J2 0.0010826267
Rg = 6378.1363 km
Ja -0.0000025327
"@ - 398600.4415 km³/s²
Cp 2.0
Po = 1.3271244 x 1011 km3/s²
CR 1.5
we = 7.2921158553 x 10-5 rad/s
A/m 0.01 m2/kg
Problems
1. Consider the case where the Sun has position
T@G = 149,597,870 i km
and the spacecraft has orbital elements
a 20,000 km, e 0.0, i 90°, n = 0°.
For this problem, assume the Earth creates a cylindrical shadow. For the sake of this problem, you
may assume two-body dynamics over one orbit. What is the mean anomaly M when you enter
the Earth's shadow? What about when you exit? What is the percent of the orbit spent in the
Earth's shadow?
2. Consider the case where the Sun has position
Too = 149,597,870 i km.
For each of the satellite position vectors below. compute the fraction of the Sun visible 7 using the
umbra/penumbra model. If required for a given problem, state the area of overlap A generated
in your calculation of 7. For the sake of simplicity. you may assume T@ = o km.
(a) T = 7000î 61003 km
(b) r = 1000î + 4695 i + 4316 k km
3. The Target spacecraft has the Cartesian state
T = 1.638 - 0.199 + 1.130 k v = + 0.570k DU/TU.
As is generally done in relative dynamics, let the RSW frame be defined by the Target spacecraft.
The Chaser has relative position and velocity
p=0.1R-0.1S+0.2W DU, p = -0,05 R + 0.04 s - 0.1 w DU/TU.
What is the position of the Chaser in the i-j-k (inertial) frame?
1
4. [Software Problem] Write a function that computes aser when given the appropriate inputs.
including the location of the Sun. For this implementation ignore the Earth's shadow. (Not
realistic, but let's keep it simple!) Compute aser (in km/s²) for a spacecraft at
T
- 2781.000i+5318.000-5629.000k km -
with a Sun position
T@@ = 2379260,000 + 148079334.000 - 1009936.000 km/s.
Provide the answer in your write-up.
5. [Software Problem] To this point. you have propagated an orbit using the following forces:
The two-body equation
J2 and Ja
Atmospheric drag
Third-body perturbations due to the Sun
Now. we will combine them and add SRP. Hence, the acceleration model in your propagator is:
a
a, + aja + @TB from Sun + @Drag + @SRP
Use your Sun algorithm to also get the location of the Sun over time for the SRP model. Still
ignore the Earth's shadow for the sake of this problem. Note that the Sun algorithm must be
evaluated inside of your propagation function! Also, we highly recommend that your propagator
use time in seconds, and you convert that to fractions of a day since the start epoch inside of
your propagation function For 24 hours, propagate the satellite with initial state
a 6800 km, e 0.05, i = 71%, n = 300%, w 78°, V = 0°.
The epoch (start) time is JD - 2458200.5 UTC.
(a) What are the position and velocity vectors at the final time? Provide them in your write-up
in km and km/s.
(b) Using your method of choice, propagate the trajectory over 24 hours using only the relative
two-body equation Plot the differences in T¿. Tj. and "k over time (three subplots on the
same figure) when using the two different dynamics models
(c) What is the magnitude of the position error (in km) after 24 hours if you assumed the Earth
is a perfect sphere and no other perturbations?
2

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clear;clc;

% Initialisation

h = 1;

i=1;

sma=6800; ecc=0.05; inc=71;omega=300;w=78;ta=0;

[x, y, z,xdot,ydot,zdot]=kep2cart(sma, ecc, inc,omega, w,ta);

X=[x, y, z,xdot,ydot,zdot];

for t = 1:1:86400

k1 = Twobody(t,X);

k2 = Twobody(t+(h/2),X+(k1*h)/2);

k3 =Twobody(t+(h/2),X+(k2*h)/2);

k4 = Twobody(t+h,X+k3*h);

X = X + (h/6)*(k1+2*k2+2*k3+k4);

% Storage

Two_body_sv(i,:) = X';

Two_body_t(i,:) = t;

i=i+1;

end

sma=6800; ecc=0.05; inc=71;omega=300;w=78;ta=0;

[x, y, z,xdot,ydot,zdot]=kep2cart(sma, ecc, inc,omega, w,ta);

X=[x, y, z,xdot,ydot,zdot];

j=1;

for t1 = 1:1:86400

k1 = ceqm1(t1,X);

k2 = ceqm1(t1+(h/2),X+(k1*h)/2);

k3 = ceqm1(t1+(h/2),X+(k2*h)/2);

k4 = ceqm1(t1+h,X+k3*h);

X = X + (h/6)*(k1+2*k2+2*k3+k4);

% Storage

Storage_X(j,:) = X';

Storage_t(j,:) = t1;

j=j+1;

end

[Row Col]=size(Storage_X);...