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Design and Computer Simulation
June 16 – July 2, 2008
Final Project Report-Hohmann Transfer Orbit Extension
Xiaojing Wu
In order to improve the Hohmann Transfer Orbit project, our group decided to do the following: improve the GUI, add a parameter for the Karman line, and transfer from a low earth orbit to the moon. I worked on the transfer from the low earth orbit to the moon. This included researching the needed equations and information, writing the MATLAB code for this part of the program as well as discussing with my group members about the necessary components. Because there was no formula available to successfully calculate how the rocket would travel from the lower orbit directly to the moon’s orbit, I adjusted the program so that the rocket would first travel into the moon’s orbit and then catch up to the moon using the Hohmann Transfer Orbit. I also assumed that the Moon’s eccentricity was negligible and thus the orbit was circular because I could not find the information necessary to calculate a transfer of a truly elliptical orbit and because the moon’s eccentricity was very small. The first part of the code was to transfer the rocket from the low earth orbit—generally 160 to 2000 km from the earth—to the moon orbit.
First the rocket needs to change its velocity at the perigee of its current orbit, which under our current assumptions is the same all around the orbit. The new velocity needed at that point is determined by the equation:
where GM is the standard gravitational parameter, e is the eccentricity of the rocket’s elliptical path and P is the distance from the center of the earth to perigee. After this change in velocity the rocket will be traveling in the elliptical path. When the rocket reaches apogee, another velocity change is needed to slow it down. The equation for the new velocity is:
Then another velocity change is required for the rocket to stay in the moon’s orbit. The new velocity is given by:
The change in velocity needed for each thrust is simply calculated by subtracting the new velocity by the previous velocity. The time for the rocket to enter the moon’s orbit can be calculated by:
where is the radius from the earth to rocket and is the radius from the earth to moon. It is divided by two because the rocket only travel half the total distance of the ellipse. At this point the program calculates the angle between the moon and rocket and performs another Hohmann transfer like in the original program to catch up to the moon.
The Karman line is the “boundary” between outer space and earth’s atmosphere. It has an altitude of 100 km above sea level. Our program was modified to incorporate this line so that if the Hohmann transfer comes below this line, a warning will be displayed that such an orbit cannot be performed.
This project has helped me gain insight on more complex applications to computer programming. I have also become skilled in manipulating MATLAB script. The style of this project required much independent research, individual work, and group collaboration and this allowed me to experience what a real project in engineering may be like. Working with professors and learning MATLAB has also given me a glimpse at the structure of a college class. This program has broadened my views on the various applications of computer programming and modeling. The skills I have learned from this program will no doubt be of great benefit to any engineering or science career I pursue in which computer modeling may be essential.
Sources
Velocity equations
http://jwilson.coe.uga.edu/EMAT6680Fa05/Bacon/hohmanntransfers.html
Period equation
http://en.wikipedia.org/wiki/Hohmann_transfer_orbit
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