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

Question

tl;dr:

Both

• this answer to Which astronaut travelled farthest from Earth?
• this answer to How far have astronauts been in space?

quote Wikipedia to state that the Apollo 13 astronauts reached 400,171 km from Earth.

I'm not able to reproduce that value.

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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.

Answer 1

The Wikipedia article on Apollo 13 disagrees with itself. In the main text of the article it says:

At pericynthion, Apollo 13 set the record (per the Guinness Book of World Records), which still stands, for the highest absolute altitude attained by a crewed spacecraft: 400,171 kilometers (248,655 mi) from Earth at 7:21 pm EST, April 14 (00:21:00 UTC April 15).[115][note 4]

However, note 4 says:

A reconstruction of the trajectory by astrodynamicist Daniel Adamo in 2009 records the furthest distance as 400,046 kilometers (248,577 mi) at 7:34 pm EST (00:34:13 UTC).

I hacked my plotting script in this answer to plot the altitude of a satellite. Here's a plot of the Moon over the time that Apollo 13 was occulted by it (0:21:35 to 0:46:19 GMT). This plot uses TDB, which was 40.46 seconds ahead of GMT / UTC.

So the altitude of the centre of the Moon above the surface of the Earth at 1970-Apr-15 0:35 TDB was ~398046.1 km. If we add that to the lunar equatorial radius of 1738.1 km and the Apollo 13 lunar altitude of 254 km we get a total of 400038.2 km, which is within 8 km of Daniel Adamo's value.

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Here's a table of all the quantities of interest calculated by the script at 0:35 April 15 TDB.

Quantity	Value
Distance	404421.52 km
Moon altitude	398045.98 km
Earth radius	6375.53 km
Declination	20.37°
Orbit speed	966.6 m/s
Earth speed	435.8 m/s
Ground speed	577.7 m/s

• Distance is the centre to centre distance between the Earth and the Moon.
• Moon altitude is the distance from the centre of the Moon to the sublunar point (on the surface of the Earth).
• Earth radius is the distance from the sublunar point to the centre of the Earth ellipsoid.
• Declination is the declination of the Moon, which is equal to the geocentric latitude of the sublunar point (not the usual geodetic latitude).
• Orbit speed is the Moon's orbit speed relative to the centre of the Earth.
• Earth speed is the Earth's rotation speed at the sublunar point.
• Ground speed is the speed of the sublunar point relative to the ground.

Answer 2

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.

Answer 3

Screenshot of the Apollo 13 trajectory's orbital elements around the Moon, published in NASA's article Apollo 13 Moon View Using LRO Data.
Here we see that the shortest distance of the Apollo crew to the Moon's centre (pericynthion) was 1988.8 km on April 15, 1970, 00:33:57 UT.
(According to Wikipedia a reconstruction of the trajectory by Daniel Adamo records the furthest distance at 00:34:13 UTC)

Above is a screenshot of the ephemeris (R.A. and DEC) of the Moon produced by Horizons Web Application also showing the distance from Earth to the Moon (delta).
It shows that on April 15,1970 00:34:00 UT that distance was 404460 km.

The screenshot also shows that at that time the declination of the Moon was about 20.3⁰ so the shortest distance to the Moon on Earth's surface happened at 20.3⁰ latitude.
From a diagram in Wikipedia's article about Earth's radius we can deduce that at 20.3⁰ latitude the geocentric radius is about 6376 km.

Conclusion: the distance from Earth to Apollo 13 was 1989 + 404460 - 6376 = 400073 km.

On April 15,1970, the right ascension (RA) of the Sun and the Moon were 1 h, 33 min. and 8 h, 53 min. respectively.
So for the prime meridian of Greenwich on April 15, 00:00:00 UT the RA would have been 13 h, 33 min. and thus at 00:34 UT, with the Apollo 13 right behind the Moon, the Greenwich meridian RA was 14 h, 07 min.
The difference between this RA and that of the Moon is 5 h, 14 min. and this means that the sublunar point at April 15, 00:34 UT was at 20.3⁰ N, 78.5⁰ W., near the southwest coast of Cuba.