Download Spacecraft Guidance I Homework 2: Orbital Parameters and Spacecraft Position Calculation and more Assignments Mechanical Engineering in PDF only on Docsity!
MAE 180A: Spacecraft Guidance I, Summer 2009
Homework 2 Due Tuesday, July 14, in class.
Guidelines: Please turn in a neat and clean homework that gives all the formulae that you have used as well as details that are required for the grader to understand your solution. Show all work. Required plots should be generated using computer software such as Matlab or Excel. Answers should be written in the blank spaces provided in these homework sheets. Use the back of the page in case you need additional space (not recommended to use more space than provided), for which a clear indication should be written to warn the reader of the presence of text there. Vector quantities are denoted in bold letters in what follows.
Student’s Name:.......................................................... Student’s ID:.............................
Question 1 (30 pts). State in words the six fundamental orbital parameters and the geometrical meaning for each of them. For a circular orbit, provide a list with the parameters that are undefined and those that best describe the orbit geometry and spacecraft location.
Question 2 (20 pts) Select the true answer (only one) out of the choices from the list provided for each question. A complementary and brief mathematical proof of your answer on the available space would be welcome, but it is not needed in order to get full credit. 3.1 An equatorial retrograde orbit has an inclination i of a) 45◦ b) 90◦ c) 0◦ d) 180◦
3.2 The topocentric-horizon system of coordinates a) is an inertial system. b) is a non-inertial system. c) does not rotate with the Earth. d) is irrelevant for spacecraft orbital calculations.
3.3 The latitude of the Cape Canaveral launch site used by NASA is 28◦, which represents the a) geocentric latitude of the site. b) geocentric longitude of the site. c) geodetic latitude of the site. d) topocentric longitude of the site.
3.4 A sidereal day is a) longer than a mean solar day since the Earth is not a perfect sphere. b) shorter than a mean solar day. c) equal to a mean solar day. d) longer than a mean solar day because of the orbital path of the Earth around the Sun and the rotation of the Earth about its polar axis.
3.5 The longitude of the ascending node of an equatorial orbit is a) always 0◦. b) 90◦. c) undefined. d) − 90 ◦.
Problem (50 pts)
A radar site at Cape Canaveral (latitude 28. 5 ◦^ N, longitude 80. 5 ◦^ W) detected a spacecraft passing directly overhead with the following data: ρ = 0.5 DU⊕, Az = 45◦, EI = 90◦, for position relative to the radar site, and ρ˙ = 0 DU⊕/TU⊕, Az˙ = 0, EI˙ = 3 rad/TU⊕, for velocity relative to the radar site, where ρ, Az and EI are the position, and the azimuth and elevation angles relative to the radar site (see figure). Assume that the longitude of the radar site with respect to the vernal equinox direction is θ = 45◦^ at the time of observation.
Z
S
vernal−equinox direction
EI
Az
C
I
J
K
Cape Canaveral
radar station
C=
E P
P=Spacecraft
orbital path
L
Figure 1: The figure shows the latitude of the radar site L, and its longitude θ with respect to the vernal equinox direction. The {S, E, Z} system is the topocentric-horizon coordinate system, and the {I, J, K} is the geocentric-equatorial coordinate system.
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Part I (25pt) a) Determine the components of the spacecraft position vector ρ in the topocentric-horizon coordinate system.
b) Determine the components of the spacecraft velocity vector ρ˙ in the topocentric-horizon coordinate system.
c) Obtain the spacecraft position vector r from the center of the Earth - assuming that the Earth is perfectly spherical - in the topocentric-horizon coordinate system.
d) Transform the r vector into geocentric-equatorial coordinates.
k) Calculate the perigee and apogee altitudes of the orbit.
l) Obtain the orbit inclination. Is it a retrograde or direct orbit?.
m) Calculate the longitude of the ascending node, the argument of the periapsis, and the true anomaly at epoch.
n) Determine the argument of latitude at epoch, the true longitude at epoch and the longitude of the periapsis.
o) What minimum set of parameters, of those calculated above, would you choose in order to determine the orbit orientation, orbit geometry and object location?.
