# Apollo 11 space mission Lunar Orbit Insertion

I'm doing a basic analysis of the trip to the Moon.

I've estimated the speed as follows:

distance:       363,104,000 m
time:           4 days, 6 hours, 45 minutes =  369,900 s

average speed:  363,104,000 m / 369,900 s   ~ 982 m/s

1. How much fuel was used to decelerate the CSML at the moon.
2. How well does this speed estimate compare to the actual velocity of the CSML as it approached the Moon?
• Where does " 937 m/s" come from? Can you add a link or citation for your source? Thanks! – uhoh Apr 11 '19 at 4:08
• The kinetic energy of Apollo 11 command-service module and lander (CSML) that is propagating to the moon is calculated using the distance to the moon (363,104,000 m) and the time that the Apollo 11 space craft (CSML) propagated to the moon (4 days 6 hours and 45 minutes [364,900 seconds]), v = (distance)/(time) = (363,104,000 m)/(364,900 s) = 983 m/s – carl eto Apr 11 '19 at 22:01
• Okay I see, thanks! I've made an edit to your post to include that information, and expanded your question a bit to help it match the style of a Stack Exchange question. Usually the more details you can add to the question, the better it is received. Please have a look and see if you are comfortable with the adjustment. I think that answers will point out that the speed at arrival will be much faster than the average speed you've calculated due to acceleration under the Moon's gravity. – uhoh Apr 11 '19 at 22:22
• @OrganicMarble I just noticed that there's been an answer to both parts about 40 minutes ago, so I can't quickly delete it – uhoh Apr 11 '19 at 23:32

## 1 Answer

This is not a complete answer, but some estimates.

1. The Apollo stack needs to switch from near-parabolic orbit around the Moon to a low circular orbit (LLO), LLO speed is $$\approx1700\text{ m/s}$$, so delta-V is about half that, $$\approx900\text{ m/s}$$. The $$I_{SP}$$ of Apollo engine is about three times that, $$\approx2700\text{ m/s}$$, so the amount of fuel needed is roughly one quarter of the mass of the full stack.

2. The speed estimate is done with a simple model of constant average speed. A better model would be to calculate perigee speed of the elliptic orbit (Hohmann elliptic orbit), which would give $$\approx600\text{ m/s}$$, and subtract the Moon speed, which is $$\approx1000\text{ m/s}$$, giving $$\approx400\text{ m/s}$$ relative to the Moon when getting into the zone where Moon gravity is dominant (roughly sphere with radius 65 000 km). The speed isn't high, approximately Moon parabolic velocity, which is $$\approx2400\text{ m/s}$$ relative to the Moon by the time the spacecraft gets to the Moon surface. The LLO speed is $$\approx1700\text{ m/s}$$, so the difference which needs to get corrected to get to LLO from translunar trajectory is, as above, $$\approx900\text{ m/s}$$.