## Transcribed Text

When deciding between Gibbs and Herrick-Gibbs, let a separation less than 5 indicate you should
use the latter.
Unless stated otherwise, assume that:
= 398600.4415 km³/s²
Rg 6378.1363 km
we = 7.2927 x 10-5 rad/sec
Problems
1. You are given the following three right ascension / declination observations:
Time (min) Right Ascension (deg) Declination (deg)
18.661
8.6916
10
21.547
7.9973
Note the use of minutes in the first column. Assume that, at all times, the observer has position
R = R@ i km in the GCRF We assume e = 0 and a = 42,000 km. Given the data and our
assumptions, what are the orbital elements i, n. and u (all in degrees) of the spacecraft? (Don't
forget to use angle ambiguity checks as needed!)
2. For each of the following problems, you are given three position vectors in the GCRF at times
ti - 100 (i 1) sec. For each problem, state:
The appropriate method to use (Gibbs or Herrick-Gibbs) and why.
The velocity vector V2 at t2 for the given method.
Assume that the provided vectors for a given problem are coplanar.
(a)
T1 III r(h) = 341592 + 23915 j - 417 k km
T2 III r(22) = 339792 + 241673 - 422 k km
T3 III r(tz) = 337962 + 24418 j - 426 k km
(b)
T1 = r(t1) = - 1970i 40333 - 5413 k km
72 = r(tz) = -2111î + 34213 - 5800 k km
Ta = r(tz) = 27703 - 6122 A km
1
(c)
T1 = r(t)) = 20502 150403 - 5516 k km
T2 = r(t2) = 20418i + 152637 5206 k km
Ta III r(tz) = 20330i + 15483 j - 4895 k km
3. You are given the three radar observations (range p. azimuth 8. and elevation el):
Time (sec) Range (km) Azimuth (deg) Elevation (deg)
651.343
45.000
26.411
30
865.398
45.715
17.782
60
1088.356
46.102
12.210
Note the values in the first column are in seconds! The observer has ITRF position vector
R = Rgi'. For this problem, TATHE = Rs(-o(f) where o(E) - wp t radians.
(a) What is the relative position vector Pa in the SEZ frame at each time t.?
(b) What is the rotation matrix for the station at R?
(c) What is the position vector T; at each time ti in the GCRF frame?
(d) Which IOD method would you use for this problem and why?
(e) What is the velocity vector V2 at 12? You may assume the vectors are coplanar for this
problem.
2

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%Question 1

clear;

clc;

R=6378.1363;%Earth radius ER in km

e=0;

a=42000;%in km

%w=2*pi/86164;

R_GCRF_t1=[R 0 0]';

%x=w*10*60;

%R_3=[cos(x) sin(x) 0

% -sin(x) cos(x) 0

% 0 0 1];%Alternate:angle2dcm(0,0,x,'XYZ')

%R_GCRF_t2=R_3*R_GCRF_t1;

R_GCRF_t2=[R 0 0]';

alpha1=deg2rad(18.661);

alpha2=deg2rad(21.547);

delta1=deg2rad(-8.6916);

delta2=deg2rad(-7.9973);

p_cap_1=[cos(delta1)*cos(alpha1) cos(delta1)*sin(alpha1) sin(delta1)]';

p_cap_2=[cos(delta2)*cos(alpha2) cos(delta2)*sin(alpha2) sin(delta2)]';

R_cap_1=R_GCRF_t1/R;

R_cap_2=R_GCRF_t2/R;

theta_1=pi-(acos(dot(R_cap_1,p_cap_1)));

theta_2=pi-(acos(dot(R_cap_2,p_cap_2)));

%Solving Quadratic Equation

%{p^2-[2*norm(R_GCRF_t1)*cos(theta_1)]p+(norm(R_GCRF_t1)^2-a^2)]

a_1=1;

b_1=-2*norm(R_GCRF_t1)*cos(theta_1);

c_1=(norm(R_GCRF_t1)*norm(R_GCRF_t1))-(a*a);

coef_1=[a_1 b_1 c_1];

p1=roots(coef_1);

p1=p1(p1>0);...