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So this was only supposed to be a fun little easter egg in my sci-fi novel for any eagle-eyed reader who decided to run the numbers, but I've driven myself quite literally - to distraction for the last two hours or more trying to calculate it, and I'm still getting inconsistent results. So I'm turning it over to someone who's (hopefully) an even bigger nerd about this than I am. No pressure or anything - I just hope someone finds it an enjoyable challenge.

If you want to compare your results with mine:

17 February 2173, 23:44 (UTC)

Sunrise I am defining as the time at which the first rays of sunlight strike the top of the tallest building in Tranquility Base which, at the time my book is set, is the 3000-ft-tall "Go For Landing" monument - built at the spot where Buzz Aldrin said those words in 1969, when the lunar module was 3000 feet above ground level. According to this, the location of such a monument would be 23.5625 degrees east (and 0.6889 degrees north). I've already calculated from the monument's height and the Moon's equatorial radius that the horizon would be 56.37 km away (1.85812 degrees), so we are looking for the time at which the sun is exactly on the horizon when viewed from a point on the ground at 25.42062 degrees east (assuming an uninterrupted line of sight between this point and the top of the monument - although 100 imaginary internet dollars to anyone who takes local topography into account).

I don't think the numbers above are wrong, it's just that when I plug them into this calculator, I get different results (it can vary by an hour or more) depending on which date I use as the reference time, which really shouldn't happen.

I feel like there should be a trivially easy way to calculate this, but lunar libration and the varying length of a lunar month throws several large spanners in the works.

Also, the fact that I'm bothering to calculate this on 11th of November means I've probably already failed Nanowrimo.

Jynto
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    In this answer I show how the Sun's elevation above the horizon can be obtained for a given location on the moon. I checked and Horizons will accept the year 2173. for this situation. Horizons uses well-developed ephemerides for Solar system objects and for detailed motion of the Moons rotation and axial motion. There is good information available on this because the Moon's exact distance has been tracked by the timing laser pulses for decades. – uhoh Nov 11 '20 at 22:22
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    You won't be able to tell exactly to the second because you won't be able to tell how many leap seconds will be added to UTC between now and then. –  Nov 12 '20 at 21:43

3 Answers3

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I threw it into STK (fair warning, I haven't validated the STK ephemerides in any way shape or form, and 2173 is pretty far out there, and it looks like the SPICE kernels built into STK are only good through 2048) and came up with sunrise occurring at February 7, 2173 at 6:59:32 UTC. Maybe someone with better knowledge of STK can figure out how to update the ephemerides for the appropriate years.

EDIT: All that said, other sources I'm looking at as a rough cross-check (e.g., dates for lunar phases) look closer to agreeing with a Feb 17-ish date. If someone who knows STK better than I do wants to put in a better SPICE kernel for those years and redo this calculation, I'd upvote it.

Tristan
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From JPL Horizons, the time is 2173-Feb-18 00:35:24, accurate to its resolution of 0.5 seconds. Leap seconds will ruin this by about 82 seconds (27 leap seconds in 50 years, 153 years from now), as was mentioned in the comments, although I suspect the largest uncertainty is our long term knowledge of the spin of the Moon, which can vary a bit. With an estimated leap seconds, one could change this to 00:36:46. JPL Horizons uses the latest SPK files and such, so it should be as accurate as possible. The exact set of parameters I used is as follows:

!$$SOF
COMMAND= '10'
CENTER= 'coord@301'
COORD_TYPE= 'GEODETIC'
SITE_COORD= '23.5625,0.6889,0.9144'
MAKE_EPHEM= 'YES'
TABLE_TYPE= 'OBSERVER'
START_TIME= '2173-02-18 00:35'
STOP_TIME= '2173-02-18 00:36'
STEP_SIZE= '120'
CAL_FORMAT= 'CAL'
TIME_DIGITS= 'MINUTES'
ANG_FORMAT= 'HMS'
OUT_UNITS= 'KM-S'
RANGE_UNITS= 'AU'
APPARENT= 'AIRLESS'
SUPPRESS_RANGE_RATE= 'NO'
SKIP_DAYLT= 'NO'
EXTRA_PREC= 'NO'
R_T_S_ONLY= 'NO'
REF_SYSTEM= 'J2000'
CSV_FORMAT= 'NO'
OBJ_DATA= 'YES'
QUANTITIES= '1,9,20,23,24'
!$$EOF

Short of monumental work, I would say that's close enough. Note that I used the first time a * appeared, which is the indication of daylight.

PearsonArtPhoto
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So far, I've managed to work out with a pen and paper that the sunrise time on the Moon would be the time on 17-18 February at which point the angle is exactly 64.57938 degrees (90 minus 25.42062) between the longitudes of the subsolar and sublunar points here on Earth (again, I'm hoping libration doesn't complicate this calculation). But frustratingly I can't find any data for subsolar and sublunar points with this much accuracy.

Jynto
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