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This answer notes that the Apollo 13 crew holds the record for the distance that humans have traveled beyond Earth.

Had the mission exactly followed the flight plan, would they still have been the farthest crewed mission from Earth? (This includes the accident -- which occurred on April 13, 1970 -- not happening.)

I am looking for a solid answer (calculations or references that have actually done the calculations), and not just speculation.

This question is part of a series honoring the 50th anniversary of Apollo 13, "NASA's finest hour".

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You've hit on a really interesting question. To answer this, I'm going to look at JPL Horizons, using the center of the Earth and the center of the Moon as the distances provided. I'm going to look at each of the Apollo missions, with the time that they were orbiting the Moon, showing the max distance, with 10 minute increments included. All distances in kms, just copying the line from Horizons. This will not give a perfect answer, but it should be close enough, as from orbit to orbit the distance to the Moon doesn't change much, and the perigee on the far side is pretty much always the same, a bit under 100 km. If there was a VERY close distance, I'd look more in to it, but...

 * Apollo  8- 1968-Dec-25 06:10     3.8172776659E+05 km
 * Apollo 10- 1969-May-21 20:40     4.0438516167E+05 km
 * Apollo 11- 1969-Jul-19 17:20     3.9451125435E+05 km
 * Apollo 12- 1969-Nov-21 20:50     3.8770443152E+05 km
 * Apollo 13- 1970-Apr-15 00:21     4.0445729295E+05 km
 * Apollo 14- 1971-Feb-07 01:40     3.9520555839E+05 km
 * Apollo 15- 1971-Jul-29 20:00     4.0305615160E+05 km
 * Apollo 16- 1972-Apr-25 02:20     4.0264314992E+05 km

So during the Apollo 13 flyby, the Moon was actually the furthest from Earth compared to any other orbit during the Apollo missions. No need to take in to account that it flew further from the Moon than any other mission, in and of itself it was further.

Typical settings I used are as follows:

!$$SOF
COMMAND= '301'
CENTER= '500@399'
MAKE_EPHEM= 'YES'
TABLE_TYPE= 'OBSERVER'
START_TIME= '1972-04-19 20:20'
STOP_TIME= '1972-04-25 02:20'
STEP_SIZE= '10 m'
CAL_FORMAT= 'CAL'
TIME_DIGITS= 'MINUTES'
ANG_FORMAT= 'HMS'
OUT_UNITS= 'KM-S'
RANGE_UNITS= 'KM'
APPARENT= 'AIRLESS'
SUPPRESS_RANGE_RATE= 'NO'
SKIP_DAYLT= 'NO'
EXTRA_PREC= 'NO'
R_T_S_ONLY= 'NO'
REF_SYSTEM= 'J2000'
CSV_FORMAT= 'NO'
OBJ_DATA= 'YES'
QUANTITIES= '20'
!$$EOF
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    $\begingroup$ Sorry I am a noob. What units are these? $\endgroup$
    – Gabriel Fair
    Commented Apr 14, 2020 at 14:17
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    $\begingroup$ You can certainly do to the second, so this isn't really required, yes. None of the missions crossed a boundary, so just simply putting in the time of orbital insertion and the time of leaving orbit would be enough. But it's close enough to answer the question, and was half of the number of queries, so I was content with that. $\endgroup$
    – PearsonArtPhoto ♦
    Commented Apr 14, 2020 at 16:34
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    $\begingroup$ You can't ignore the actual orbits. When Apollo 10 orbited the Moon, the Moon was only 72 km closer to the Earth than it was for Apollo 13, and Apollo 10 had a more-eccentric Lunar parking orbit than any of the later missions -- at a 317 km apoapsis, the additional altitude more than covers the difference. You'll need to look into which way that orbit was pointing. $\endgroup$
    – Mark
    Commented Apr 14, 2020 at 21:20
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    $\begingroup$ @StarMan, that needs to be in the answer, then, preferably with a source. $\endgroup$
    – Mark
    Commented Apr 14, 2020 at 22:51
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    $\begingroup$ There's still a piece that might be obvious to you but not obvious to us. The moon was furthest during Apollo 13, but I don't know the magnitudes enough to know if the space ship flight plan is negligible. If I'm not mistaken the moon was 72 km further away from Earth during Apollo 13 than it was during Apollo 10. How much did the flight plans very in distance. $\endgroup$
    – Jetpack
    Commented Apr 16, 2020 at 15:59
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I believe the answer is yes, but just barely.

The distance from the Earth to the moon varies significantly over time, from 356,400 to 406,700 km. I plugged the dates of orbital entry and departure for each of the lunar Apollo missions (8, 10-17) into pyephem to find the ranges of lunar distance.

At Apollo 13's flyby, the moon was one day past apogee and about 404,418 km from the Earth; the spacecraft was that distance, plus the moon's radius, plus about 250km altitude, away from the center of the Earth; if it had entered orbit as planned, it would have been more like 100-110 km altitude, and since the moon was past apogee it would have only gotten closer from there.

Apollo 10 was the closest contender; that mission entered lunar orbit two days past a significantly higher apogee, when the moon was about 404,344 km from the Earth -- only 74 km short of the distance at the time of 13's approach! At two days past apogee, the distance would also be decreasing after entry.

I am assuming that the nominal trajectories relative to the moon of these two missions would have been very similar, not enough to make up the 74km difference, so 13 would still have taken the record.

Apollo 15 also entered a couple of days after apogee (maximum Earth-moon distance 403,008 km), and 16 departed a couple of days before apogee (max distance 402,611 km).

Pyephem is very simple to use; just a skim of the quick reference doc was all I needed to whip up this program:

import ephem

# km per astronomical unit
AU = 149597800

apollos = [
("Apollo 8 ","1968/12/24 9:59:20","1968/12/25 6:10:17"),
("Apollo 10","1969/5/21 20:44:54","1969/5/24 10:25:38"),
("Apollo 11","1969/7/19 17:21:50","1969/7/22 4:55:42"),
("Apollo 12","1969/11/18 3:47:23","1969/11/21 20:49:16"),
("Apollo 13","1970/4/15 00:21:00","1970/4/15 00:21:00"),
("Apollo 14","1971/2/4 06:59:42","1971/2/7 01:39:04"),
("Apollo 15","1971/7/29 20:05:46","1971/8/4 21:22:45"),
("Apollo 16","1972/4/19 20:22:27","1972/4/25 02:15:33"),
("Apollo 17","1972/12/10 19:47:22","1972/12/16 23:35:09"),
]

for name,date,enddate in apollos:
    m = ephem.Moon()
    m.compute( date, epoch = "1950") 
    startdistance = m.earth_distance*AU
    m.compute( enddate, epoch = "1950") 
    enddistance = m.earth_distance*AU

    print ("%9s: %d-%d" % (name, startdistance, enddistance))

Which yields the following table:

Apollo 8 : 376372-381691
Apollo 10: 404344-396084
Apollo 11: 394473-382464
Apollo 12: 375882-387685
Apollo 13: 404418-404418
Apollo 14: 384469-395180
Apollo 15: 403008-374865
Apollo 16: 379697-402611
Apollo 17: 394279-363688
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  • $\begingroup$ I'm trying to figure out if these numbers and the numbers from PearsonArtPhoto's answer are consistent. Do you know where PyEphem gets its data from? $\endgroup$
    – Ludo
    Commented Apr 14, 2020 at 15:35
  • $\begingroup$ I believe PyEphem contains its own databases; I don't know where those are sourced. $\endgroup$
    – Russell Borogove
    Commented Apr 14, 2020 at 16:18
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    $\begingroup$ As I noted on PearsonArtPhoto's answer, you can't assume that the orbits were similar, because they weren't. $\endgroup$
    – Mark
    Commented Apr 14, 2020 at 21:27
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    $\begingroup$ @Ludo this is about solar system bodies; don't (yet) know about spacecraft. PyEphem under the hood - how does it calculate position of planets? PyEphem's maintainer is active in SE and SO so I'll ask about it and link here. $\endgroup$
    – uhoh
    Commented Apr 17, 2020 at 0:48

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