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Edit:
For clarification of what I mean with "static in space above the Lunar south pole " please see this answer from @Tom Spilker or his answer to the question Orbits that allows observation of a polar region as an example.

When near the south pole of our Moon a permanent human community is established to search for and produce water in the permanently shadowed regions there, it would be nice to have some light from above !

Would it be possible to instal a framework of mirrors in space above such a human settlement so there would not have to be worked in constant darkness ?

Within the framework the mirrors could turn independent from each other and could function as solar sails as well to withstand the gravity of the Moon.

Of course, software would be needed to anticipate the constantly changing position of the framework by changing the angle of the different mirrors independently.

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    $\begingroup$ Theoretically, you could have a moon rotation synchronous satellite that reflects sunlight onto the moon's surface. Of course, mirrors can't reflect more light that they receive, so the total mirror surface area would have to be at least 10% as large as the village, probably more. $\endgroup$
    – user7073
    Commented Nov 26, 2019 at 15:43
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    $\begingroup$ So... you're proposing an object that floats above the surface, in a completely impossible orbit; or am I missing something here? If it's not in orbit, and it's "static" then it needs propulsion to counteract the moons gravity. You cannot have an orbit that "stays above the pole" 100% of the time. The whole statement "static in space" doesn't really make sense. $\endgroup$ Commented Nov 26, 2019 at 16:32
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    $\begingroup$ Can you actually position a statite over the moon, though? $\endgroup$ Commented Nov 26, 2019 at 16:48
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    $\begingroup$ @starfishPrime Not over a pole while reflecting light to a specific position and maintaining attitude to counter-act gravity... Maybe the question should've been, "is a statite over the moon possible? What about over the poles?". I feel like that would've been more answerable than this is in its current state. This adds a bunch of extra layers of complexity to an already complex problem. $\endgroup$ Commented Nov 26, 2019 at 16:48
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    $\begingroup$ Software wouldn't help the problem. You cannot get the delta-v you need, in the direction you need to counter-act gravity AND reflect the light back to a specific location on the surface in quantities that are needed for illumination at the poles. Especially seeing as this sail would need to be absolutely massive. $\endgroup$ Commented Nov 26, 2019 at 17:05

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Answer: No, it ain’t gonna work.

Pole-sitter orbits for the inner planets (using a combination of light sail pressure and low-thrust propulsion) have been modeled.

enter image description here

https://www.researchgate.net/publication/260082614_Trajectory_and_Spacecraft_Design_for_a_Pole-Sitter_Mission/figures?lo=1

However, they produce high orbits which are many planetary radii in altitude.

Molniya orbits would also answer your need to have perpetual clear sightlines to both the sun and lunar pole, but once again the high orbit is many lunar diameters in altitude.

The high altitude of these orbits presents the problem of a very large projection area on the Lunar surface and therefor a very dim light. This is because any mirror which is significantly smaller than the sun will function as a pinhole lens, projecting an image of the sun the same angular diameter as the Sun itself. If your proposed mirror is at an altitude of 10,000km, the projected image on the surface of the moon will be 90 km in diameter. Even if you had an optically perfect mirror 0.9km in diameter, the light intensity on the lunar surface would be 1/10,000 of daylight.

It’s tempting (with handwaving) to propose a slightly concave mirror to concentrate the reflected sunlight in a smaller region. However, the 2nd law of thermodynamics forbids optics which form an image with a smaller angular diameter than the object being imaged (unless there is a large loss of intensity). If it were possible to create a smaller image, it could have a higher black body temperature than the source and thereby “transfer heat from a cooler to a hotter”

It's easier to give the workers headlamps.

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  • $\begingroup$ I'm not quite following the 2nd law prohibition. I'm thinking of those parabolic fire starters for camping. Placed close to an object it creates a concentrated area of high radiation. If the parabolic fire starter was placed several feet away I would think it would create a larger area of radiation equal to being in direct sunlight. I'm just saying that's intuitively how it seems like it would work, maybe the physics says differently. Also they wouldn't need the same candlepower as the Sun which is overkill as it requires our eyes to dim it down. Probably need just sports stadium candlepower. $\endgroup$ Commented Oct 16, 2023 at 9:26
  • $\begingroup$ @StevePemberton ... An analogy is the difference between heat and temperature. Solar collector furnaces can produce almost limitless heat. Want more heat? Add more mirrors. But the furnace can never exceed the temperature of the radiating body (the surface of the Sun). That would be violation of the 2nd Law. If it were possible to produce an image of the Sun with a smaller angular diameter than the Sun, the temperature of a solar furnace could exceed the surface temperature of the Sun, so the 2nd Law would be broken. $\endgroup$
    – Woody
    Commented Oct 16, 2023 at 14:08
  • $\begingroup$ @StevePemberton ... A small image can be produced with non-imaging optics (as opposed to traditional geometric optics) but this attenuates the light beam, so the 2nd Law is upheld. (This is one reason lasers are so useful: they produce light beams much more highly collimated than can be produced by collimating an incandescent source.) $\endgroup$
    – Woody
    Commented Oct 16, 2023 at 14:08
  • $\begingroup$ Very convincing answer for me, especially the projecting image of the Sun on the Moon of 90 km diameter. $\endgroup$
    – Cornelis
    Commented Oct 17, 2023 at 8:13
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The answer is no, you cannot have a static satellite. It would require high delta-V continually thrusting the mirror upwards. Solar sails do not have high delta-V. But that is okay - if all you want is polar moonbase illumination, that problem is already solved:

  • Solution 1 -

Have a network of satellites with mirrors, and use them in turn as they pass near the poles - software could do this fairly easily.

  • Solution 2 -

Build mirrors on polar mountains - there are some that are continually in sunlight. This solves the entire problem at much lower cost, no fiddly orbits to worry about, and you could climb up and fix issues if there are problems.

  • Solution 3 -

Floodlights powered by solar panels, as suggested by Magic Octopus Urn

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  • $\begingroup$ Solution 3 - flood lights powered by solar energy. $\endgroup$ Commented Nov 26, 2019 at 18:48
  • $\begingroup$ I like that @magic - I'll edit that in $\endgroup$
    – Rory Alsop
    Commented Nov 26, 2019 at 19:14
  • $\begingroup$ @MagicOctopusUrn Taking Shoemaker as an example, isn't even the rim of this 51 km wide crater permanently shadowed ? Floodlights would be the best solution i think, but the electric cables could be very long. $\endgroup$
    – Cornelis
    Commented Nov 26, 2019 at 20:02
  • $\begingroup$ I don't think you're going to be illuminating a 51km diameter crater using any methodology. $\endgroup$ Commented Nov 26, 2019 at 20:04
  • $\begingroup$ The disadvantage of Solution 2 is that in general , the mountains would be far from the moonbase, so the light from the mirror would be almost horizontal. $\endgroup$
    – Cornelis
    Commented Nov 26, 2019 at 20:06
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There isn't a problem with sunlight supply at the south pole. It's actually the opposite situation. One of the biggest attractions to the lunar south pole (after water) is the fact that there are areas there that are in almost constant sunlight. From the rim of Shackleton crater, the sun skims the horizon to a complete 360˚ as the lunar day progresses. It's quite feasible to have constant solar power for a polar settlement by placing cell arrays in the right places and linking them into a network.

If a base is deep into a crater (say to run an ice harvest operation) they could save on cable by beaming microwave power from the rim.

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    $\begingroup$ the close proximity between always in shadow regions + thermal mass of the moon as a cold source, and always in sunlight hot source could be the spot for continuous stirling operation $\endgroup$
    – user19132
    Commented Oct 15, 2023 at 16:32

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