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Sure it is possible (in theory).

Moon's albedo is 12%

Photovoltaic panel efficiency is around 19% (commercially available) and can go up to 40% with more exotic technologies.

Assuming Moon gets the same amount of solar radiation as Earth, surface receives 1367 watts per square meter, 42% of which is visible light, which gives us 574 watts per square meter to play with. LEDs should beam back 69 watts. Assuming that we are using commercially available LEDs with 50% efficiency, 1 square meter should house 138 watts of LED power. This is a lot, but our bulbs will cover only a fraction of surface. The rest can be used for solar panels. Solar panels, on the other hand, will give us 229 watts per square meter.

During lunar day, panels will be baking in sunlight, converting it to electricity, which would be stored in batteries (do we have enough lithium on Earth? Hmm...) during the night, the bulbs will turn on, creating illuminated Moon face.

Also note that while solar panels can cover 100% of Moon surface, LEDs need to be installed only on the visible side, which should double the energy balance in our favor.

P.S. Calculations above assume that lunar LEDs work just like the Moon's surface, i.e. their emission is omnidirectional. Our efficiency can be improved A LOT is we are allowed to beam light only in Earth's direction.

Numbers, numbers

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  The actual Moon surface is quite dark; the albedo of the Moon is 0.136. This means that the Moon reflects only 13.5% of the sunlight it receives. Moreover, the reflection is diffuse, that is, the reflected light goes all over the place, not only towards Earth.

  
  
  

  In order to move light from the far side of the Moon to the near side in the form of electric power, we need to (1) capture the energy of the light and convert it to electric energy, (2) transfer the electric energy to the near side, and finally (3) convert the electric energy into light. The overall efficiency of the process must at least match the 13.6% achieved by the Moon rocks through reflection.

  
  
  

  Can this be done?

  
  



• The efficiency of a decent photovoltaic panel is about 20%, meaning that the panel converts 20% of the incoming light energy into electric energy.




• The efficiency of a white LED lamp is currently around 15%, but 20% efficient lamps exist; the theoretical maximum luminous efficacy of a white LED is about 40%. Let's say that the emperor's scientists have achieved the capacity to make white LEDs with 30% luminous efficacy.




• Let's put the efficiency of electric power transmission at 90%.



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  Overall this gives 20% (light-to-electricity conversion) × 90% (transmission) × 30% (electricity-to-light conversion) = 5.4% overall efficiency. This means that, at best, the artificially illuminated new moon will have about 40% of the luminosity of the full moon; in photographic terms, that's a difference of about 3.5 stops of exposure; in astronomical terms, this is a difference of one magnitude.

  
  
  

  How visible is the difference in luminosity? Here is an image showing a normally exposed full Moon and a copy with the luminosity reduced to 40%.

  
  

The photograph of the Moon on the left is exposed so that the highlights are close to the maximum value, without exceeding it. The Moon on the right is the same image, digitally manipulated to make the Moon have 40% of the luminosity. Own work.

• But what about the phases of the Moon? Won't there be a marked difference in luminosity between the naturally lit and the artificiall lit parts? Yes, there will be a one-magnitude, or 3.5 stops, difference; visible, but hey, it very much better than the current situation.




• But what about the non-uniform illumination of the photovoltaic panels? True, the Moon is spherical, and the conversion efficiency of the photovoltaic panels on the far side will vary between the theoretical maximum when the Sun is up in the sky to zero when it is on the horizon; this will bring the available power down a factor of two, and make the artificially illuminated part even darker. Actual calculation remain as an exercise for the reader; however, overall we can confidently say that we can build a decent artificial lunar illumination system for our glorious and much beloved emperor.

EDIT: It can be done, see my second edit below.

Fundamentally can't be done. This issue was parodied at XKCD. The gist of the problem is this: you can't duplicate the firepower of the sun, especially if you're using a solar-based power system that isn't 100% efficient. Even if it was. It would need to acquire 100% of the solar energy that would hit the moon during a full moon, transfer that power perfectly (100% efficiency... the engineer within is starting to weep) to LEDs, which can emit the collected power as photons with 100% perfect conversion (oh, the pain!).

Can't be done without serious power. Serious. Check out the link. Serious.

Edit:

Also, remember that where there's solar panel, there isn't LEDs. You can hide the batteries underground and put the panels on the backside of the moon (it's tidally locked), but that means you must capture and store enough power to illuminate all those buka-watt LEDs for each night. Serious.

Edit:

OK, Shadoweze has piqued my curiosity. Lunar albedo for a full moon is 0.12. Albedo is the ratio of energy received to energy reflected. The sun bathes the moon in 1kw/m. So the reflection, what we need to achieve, is 0.12kw/m.

The full-moon lunar surface is 10¹³ m². That means we need to generate 1.2E12 watts or 1.2 terrawatts. The most efficient solar panel in 2018 has a 22.2% efficiency. That means for every kw of solar energy we'll actually have only .222 kw to work with. That's twice-ish what we need, so far so good.

Average lunar light is about 0.015 foot-candles or about 0.0019 lumens per m² for 0.016 lumens-per-watt of lunar emittance. 2018's most efficient LED is 105 lumens-per-watt. Good news! We don't need to cover every inch of the earth-side of the lunar surface!

Better news is that for each pass of the moon in front of the sun, there isn't a commensurate "full-moon" pass for the earth. The moon varies from a new moon (100% LED use) to a full moon (0% LED use). I'm absolutely wrong with the assertion I'm about to make, but to keep this from becoming a full dissertation, let's assume we only need to store 50% of the power needed to hold a full moon all the time and there's enough space between the LEDs that we can use detectors to shut off the LEDs we don't need during each phase of the moon. (And I'm ignoring the fact that we only need to turn the LEDs on when the emporer is in the night cycle. Who cares what the peons see, right?)

OK, I'm convinced. Shadowzee's right. It can be done. It might need enough battery mass to shift the moon's orbit... but it can be done.

Why is the XKCD no longer relevant? It's emitted light from the earth reflecting off the moon. That takes a ton more power, and I'd ignored it.

I wouldn't touch the actual moon, it's 384,400km away, it seems much cheaper & within reach to make your own "equivalent" using an array of satellites (maybe only 320km - 600km away) with very bright lights, maybe in a geosynchronous orbit (approximately 35,786 km away) to keep them at least visible every night, using each one like a single pixel in a very large "display screen".

Either individual satellites just close enough together to look right, or tethered together with cables or filaments, or an extremely large single array or framework of bright lights.

So you end up with a virtual LED display in space. Perfect for displaying messages, or depending on the density of lights, even any picture or video.

With unlimited resources & energy, powering them should be within reach today with either solar power & batteries, or nuclear, or maybe even a Tesla-esque wireless power transmission from earth. Taking the "unlimited resources" more literally, then the pixel-satellites could even be brighter than the actual moon, so a "sky television" could even be seen during the day.

Some satellites are already visible from the earth now (and I'm pretty sure they don't even have any purpose-built lights aimed at the earth). Here's an image of some from How to See and Photograph Geosynchronous Satellites:

Just imagine a few million of them, tied together in a giant "screen" array, with unlimited energy for bright colours, and you've got your emperor's face, and propaganda, and a moving zooming or even exploding image of the moon, or Mars, or Jupiter, or anything really.

Here's a hopefully poor example using 160 computer keyboards (each with maybe 100 led lights), but it should give an idea of what's possible with even just 160,000 lights (from here, video here or directly on YouTube):

As already stated - lighting up the surface of the Moon with LED's to mimic a full one can't be done with just solar panels and LED's.

"Possible" other options?

• Nuclear power to provide the electricity - @Gary Walker pointed out the amount of Thorium on the moon to use for fuel. Benefit of not being limited by incoming sunlight


• Earth based generators beaming power to the moon via microwaves?


• Network of giant mirror satellites to reflect sunlight onto the moon when it's behind earth


• During non-full moons, fly a massive plane/drone equipped with a giant LED screen between the Emperor's location and the moon, to mimic the appearance of a full moon. Added bonus of being able to display messages

This is impossible for a multitude of reasons.

First of all, the moon doesn't really receive enough sunlight to power LEDs covering half of its surface. The amount of energy a single square inch of modern solar panel produces in an hour is only about 0.1 watts, whereas the amount of energy required to power a single modern LED for an hour is 6 watts. That means that an 8"x8" solar panel would be required to constantly power an LED that takes up a fraction of a square inch of space. Area-wise, the bright side of the moon just isn't big enough to power its opposite half.

Second, there is the issue of transporting the power you get to the LEDs. Every cable that we have loses power for every foot that it travels, which is a natural limit on how far power can be transported. Even with fiber optic -- the most efficient path we could use with 2018 technology -- the power loss experienced over the hundreds of miles of the moon's diameter would be too great for anything to reach the other end.

Much cheaper option:

Every evening, send a large LED panel into the upper troposphere, e.g., with balloons and haul it down every morning. Since the panel is a lot closer to the ground, it can affordably be a lot smaller in cross section than the moon.

Plus side:

You can then rig the LED panel like a normal TV or monitor.

Minus side:

• It will always remain in the same place.


• There will normally be two "moons" in the sky.