Comparison Of Apollo 13 And Translunar Injection

Apollo 13 and Translunar Injection: Using the Hohmann Transfer Orbit to Calculate the Change in Velocity Needed to Get to the Moon

My specific area of focus is the translunar injection maneuver that propelled Apollo 13 on a trajectory towards the Moon’s sphere of influence. This part of the mission was crucial to the overall success of the mission. In order to reach the moon, the velocity of the spacecraft had to be increased just the right amount in order to exit a parking orbit around the Earth and reach the Moon. It is the calculation of this velocity that I will be focusing in on in my section.

The Apollo 13 mission was intended to be the third manned flight to the Moon. The mission began with much fanfare on April 11, 1970, at 19:13
Translunar injection is accomplished by a translunar burn which increases the velocity of the spacecraft and moves it farther away from the Earth. On the Apollo 13 mission, translunar injection was accomplished by a 350.85 second-long firing of the S-IVB. This burn enabled the spacecraft to exit a parking orbit around the Earth and head on a path towards the Moon. The key to the success of this maneuver was the S-IVB burn and resultant velocity change of the spacecraft. If the resultant velocity was too low, the spacecraft would continue to orbit the Earth and if it was too high the Apollo 13 would miss its destination. I will be computing this necessary change in velocity to exit parking orbit and reach the Moon’s sphere of influence (SOI) using a Hohmann transfer
A Hohmann transfer orbit is an elliptical orbit used to switch from one circular orbit to another on the same plane. The Hohmann transfer is based upon two main concepts: conservation of angular momentum and conservation of total mechanical energy. It is these two concepts and their corresponding equations that allow one to evaluate the velocity change needed to head on the Hohmann transfer orbit. However, the Hohmann transfer model does have several assumptions that limit its usage. The basic assumptions are as follows: only two instantaneous changes in velocity are performed throughout the transfer, the two orbits are circular and coplanar, and Newtonian gravity. For the Apollo 13 translunar injection, not all of these assumptions hold true. For example, the parking orbit of Apollo 13 prior to the S-IVB burn was not exactly circular. It was actually elliptical with a perigee of 99.3 nautical miles and an apogee of 100.3 nautical miles. However, the perigee and apogee are so close and value, that the orbit was nearly circular. Additionally, the change in the velocity of the spacecraft during the translunar injection burn was not instantaneous and occurred over a period of 350.85 seconds and two midcourse corrections were performed on the way to the Moon (more than two velocity changes present). Nonetheless, while these assumptions do not all hold true with Apollo 13’s exact conditions, it can still be