# Puzzler: Precisely what maximum distance from the Earth did the Apollo 13 astronauts achieve?

tl;dr:

Both this answer and this answer quote Wikipedia to state that the Apollo 13 astronauts reached 404,171 km from Earth. I'm not able to reproduce that value.

According to JPL's Horizons the distance between the center of the Earth and the center of the Moon at the beginning and end of occultation times from here are:

          Event                        UTC        Earth-Moon R (km)
-------------------------  --------------------   -----------------
Lunar occultation entered  1970-Apr-15 00:21:00       404418.652
Lunar occultation exited   1970-Apr-15 00:46:00       404423.901


Just in case I entered something wrong using the Horizons interface, I've double checked the numbers with Skyfield on DE421 and go the same thing except for about 100 meters because the times scales are slightly different due to accumulated leap seconds. Note these distances are not light time or otherwise corrected, they are just the differences in positions.

Let's call the Earth-Moon distance at the midpoint time to be 404 421.3 km, and the equatorial radii of the Earth and The Moon 6378.1 km and 1738.1 km.

Wikipedia's Apollo 13 (quoted in this answer) says:

The flight passed the far side of the Moon at an altitude of 254 kilometers (137 nautical miles) above the lunar surface, and 400,171 km (248,655 mi) from Earth, a spaceflight record marking the farthest humans have ever traveled from Earth.

Wikipedia's List of spaceflight records: Speed and altitude records (quoted in this answer) says:

Farthest humans from Earth

The Apollo 13 crew (Jim Lovell, Fred Haise, and Jack Swigert), while passing over the far side of the moon at an altitude of 254 km (158 mi) from the lunar surface, were 400,171 km (248,655 mi) from Earth. This record-breaking distance was reached at 0:21 UTC on 15 April 1970.

Question: I don't know how to combine the Horizons distances, the radii of the Earth and Moon, and the reported altitude in order to get to 400,171 km. Can someone help?

Secondary item is that the 2nd Wikipedia quote gives the maximum at 00:21 UTC but that seems to be the time of entry into occultation, not the midpoint 12.5 minutes later, that could be a clue, or just an oversight.

• @OrganicMarble max Earth altitude: 404421 - 6378 + 1738 + 254 = 400035, max distance from Earth center: 404421 + 1738 + 254 = 406413 So there's perhaps a 136 kilometer disparity with Wikipedia
– uhoh
Dec 31 '18 at 1:37
• Wow, that 1970 datestamp is annoying - my programmer brain has been trained to immediately suspect it as wrong/a bug 😂 Dec 31 '18 at 17:06
• @LightnessRacesinOrbit it's less problematic for people who started to program "way back" in the 1900's.
– uhoh
Dec 31 '18 at 17:18
• @uhoh Alan Shepard on his first mission probably won the US award for spaceflight travel the farthest from the moon.... Jan 1 '19 at 2:26
• Yeah, I meant the Mercury one. Jan 1 '19 at 2:36

## 1 Answer

Let's start with the easy bits. The distance from the center of the Moon at the far point. The max distance that the Moon was from Earth was actually at the exit point. The exact distance involves some complex geometry. Another thing to keep in mind is the Earth and Moon are not exact spheres, the radius can vary quite a bit. The Earth's actual radius can vary between 6353-6384 km depending on exactly where you are, it is less towards the poles. The Moon's similarly can vary between 1736-1738 km. Thus whatever value we give could be off as much as 33 km, depending on which point of Earth/ the Moon we are talking about.

The distance should be (dMoon_Earth)-rE+rM+dA13. Let's use the values that will give the largest number for all of these, and we get 400,061 km, which is still less then the reported value. This would assume the min distance from the Equator of Earth (Unlikely) and the equator of the Moon (Likely).

Interestingly enough, the NASA article cites the Guinness Book of World Records as the source it uses. Other sources give even higher numbers, such as 401,056 km.

My guess is that they picked the distance to a point on the Earth, but not necessarily the closest point to Earth. A point on Earth that wasn't directly in the line of closest approach would give a slightly larger value, as is seen.

• As this is a puzzler and has survived for two rotations of the Earth without explanation, educated guessing is welcome. Britannica's num is very close to what I got (404421 - 6378 + 1738 + 254 = 400035) and that's encouraging, and if you use Earth's polar radius you get exactly what they get! I think you may be very close to a solution; Wikipedia is wrong.
– uhoh
Jan 2 '19 at 2:39
• They cite NASA, who comes the Guinness book if world record. Note that the Britanica value is off by 1000 km. Jan 2 '19 at 2:49
• Oh! I was so focused on the last few digits I missed the thousands. Okay, mystery not solved... yet ;-)
– uhoh
Jan 2 '19 at 2:50
• I actually did the first time I saw it too, then looked a bit more careful... Jan 2 '19 at 3:11