Okay, we can work it backwards from brightness of the Moon.
A full Moon has apparent magnitude –12.92
Apparent magnitude
I'll take sixth magnitude as the faintest stars we can see - though an observer with excellent vision at a dark site can see down to seventh magnitude.
So - a sixth magnitude star is 18.92 magnitudes fainter. Brightness ratio therefore, by the formulae for magnitudes, is 2.512^18.92.
So, the area of patch of the illuminated Moon equivalent to sixth magnitude star is
PI * (MOONS RADIUS)^2 / 2.512^18.92
Note - we can treat the moon as a disk here. It doesn't matter that it is a sphere in actuality - as - it still has pretty much uniform brightness over its surface as seen from Earth. The illuminated Moon has no limb darkening. And our answer will give the size of a patch at the centre of the Moon as seen from Earth with same brightness as a sixth magnitude star.
Call the radius of that patch r, then
PI*r^2 = PI * (MOONS RADIUS)^2 / 2.512^18.92
So, r = (MOONS RADIUS) / sqrt(2.512^18.92)
Diameters of course in same relation (multiply both sides by 2)
So it's diameter is 3474800 / sqrt(2.512^18.92)
or about 571 meters.
Its area is PI * (285.57)2 or about 256,000 square meters.
A hundred watt bulb has 2.6% efficiency Incandescent light bulb - so only puts out about 2.6 watts of light energy.
Each square meter of the Moon is illuminated with about 1000 watts - but - albedo is 0.12 - so only 120 watts is reflected back.
So, each square meter reflects back light equivalent to about 46 of the "100 watt" light bulbs. So our 571 meter diameter, 256,000 square meters area of illumination of the Moon is equivalent to about about 11 million light bulbs shining on the Moon.
If you had a concentrated city of a over 5 million people on the moon - and each of them had spill into space of about two 100 watt light bulbs per person into the sky -then you might just spot it as a sixth magnitude star in ideal conditions during lunar eclipse.
This assumes the habitats are on the surface. And also close together - if you had the combined light of a sixth magnitude star - but spread over a large area of the Moon you wouldn't see it, because of low surface brightness.
A complication though is - that on the Moon - you would cover all the habitats with several meters of regolith for shielding from cosmic radiation.
Also, I rather expect that human habitats would have few windows, and the windows all tiny, as in this Bigelow mockup.
At most you might have occasional larger ones like the Cupola on the ISS but they would be rare, surely - that's because windows are really hard to construct when you have to hold in ten tons per square meter of outwards atmospheric pressure.
Unless we develop some way of making amazingly strong transparent material (like the Star Trek "Transparent aluminum") they would surely rely on view screens to see the outside surface, or for that matter to display movies and scenes from Earth - but probably not much by way of windows as such, just one or two as special features like the Cupola.
So - chances are even if they are on the surface - that hardly any light would escape from the human habitat.
Here is a rather fine Russian video about a lunar city from 1965. Sadly I don't speak Russian so not sure what they are saying.
So - chances are even if they are on the surface - that no lights would escape from the human habitat.
Human habitats could also be underground, for instance in lunar caves.
Hadley Rille photographed by Apollo 15, probably example of a large collapsed lava tube on the Moon.
If so, then it may originally have been a huge hollow cavern. If so, what about lava tubes that are not collapsed, underground?
A study in spring 2015 found that they could be stable up to ten kilometers wide in the much weaker lunar gravity. If so, there may be caves on the Moon large enough to house entire cities, and if humans lived in them we'd see no trace of them from the Earth.
And, we wouldn't see solar panels, not when the habitats are in the lunar night - or anything reflective, as there would be nothing for them to reflect.
All in all, I don't think we have much chance of spotting human habitats in the areas of darkness on the Moon, at night, not for a long time to come, not directly.
Greenhouses are more promising. They would be large very strong spherical or hemispherical buildings - if on the surface. They'd be made of some material far stronger than e.g. the plastic of "polytunnels" as they have to withstand several tons per square meter outwards pressure - but would probably be transparent to let in the daylight during the 14 day "day".
And if used for growing vegetables would need to be illuminated with close to full daylight for growth during the lunar night - probably in 12 hour day / night cycles so as viewed from Earth, they would blink in and out of sight every 12 hours.
Of course the light would be directed downwards onto the plants. But the plants themselves absorb only part of the light that is shone on them, and reflect the rest back.
Using albedo for green grass, 0.25, then diameter needed to see it from Earth is 571/sqrt(0.25/0.12) = 395 - or about 400 meters for the diameter of the greenhouse. Area about 126,000 square meters. Or about 31 acres.
So, we wouldn't see a small greenhouse. But if you had greenhouses the size of a very large field - of order half a kilometer in diameter or so, seems that we would see the largest greenhouses in ideal conditions if fully illuminated to equivalent to daytime sunlight.
Let's not bother with the practical details - do they go ice skating in spacesuits? Or do they have a very thick polymer covering able to hold in ten tons per square meter? Or low pressure atmosphere?
But - just suppose we have ice, high Albedo, exposed on the surface, could we see it?
If it was, say, fresh snow - or perhaps an ice rink, then to see it from the Earth as a sixth magnitude star, it needs to have a diameter of 571/sqrt(0.8/0.12) or about a 220 meters diameter patch. Or about 9.4 acres.
A standard skating rink of 40 by 60 meters is not nearly large enough to see from Earth with the equivalent of full sunlight illumination.
A larger area - say, a fully illuminated giant slalom slope for low gravity skiing in spacesuits - that could be visible from the Earth. Length 400 meters, width 30 meters, so still nowhere near big enough. 12,000 square meters. We need ten times that to be as bright as a sixth magnitude star.
So, the answer is no, we wouldn't see ice rinks or Olympic ski slopes on the Moon from the Earth either, not unless the ski slopes are much larger than on Earth.
We'd see a complete enclosed ski resort easily though. The largest ski resorts are thousands of acres. If we had a fully developed ski industry on the Moon - if low gravity skiing really took off, they would only need to cover
I've written this up now for my science20 blog as
Could You See Moon City Lights? Or A Greenhouse? Or An Ice Rink Or Ski Slope? Just For Fun
You can now get this as a kindle book along with many other of my answers, in
Simple Questions - Surprising Answers - In Astronomy
A full Moon has apparent magnitude –12.92
Apparent magnitude
I'll take sixth magnitude as the faintest stars we can see - though an observer with excellent vision at a dark site can see down to seventh magnitude.
AREA OF MOON NEAR SIDE NEEDED FOR SAME ILLUMINATION AS SIXTH MAGNITUDE STAR
So - a sixth magnitude star is 18.92 magnitudes fainter. Brightness ratio therefore, by the formulae for magnitudes, is 2.512^18.92.
So, the area of patch of the illuminated Moon equivalent to sixth magnitude star is
PI * (MOONS RADIUS)^2 / 2.512^18.92
Note - we can treat the moon as a disk here. It doesn't matter that it is a sphere in actuality - as - it still has pretty much uniform brightness over its surface as seen from Earth. The illuminated Moon has no limb darkening. And our answer will give the size of a patch at the centre of the Moon as seen from Earth with same brightness as a sixth magnitude star.
Call the radius of that patch r, then
PI*r^2 = PI * (MOONS RADIUS)^2 / 2.512^18.92
So, r = (MOONS RADIUS) / sqrt(2.512^18.92)
Diameters of course in same relation (multiply both sides by 2)
So it's diameter is 3474800 / sqrt(2.512^18.92)
or about 571 meters.
Its area is PI * (285.57)2 or about 256,000 square meters.
CONVERSION TO "100 WATT" LIGHT BULBS
A hundred watt bulb has 2.6% efficiency Incandescent light bulb - so only puts out about 2.6 watts of light energy.
Each square meter of the Moon is illuminated with about 1000 watts - but - albedo is 0.12 - so only 120 watts is reflected back.
So, each square meter reflects back light equivalent to about 46 of the "100 watt" light bulbs. So our 571 meter diameter, 256,000 square meters area of illumination of the Moon is equivalent to about about 11 million light bulbs shining on the Moon.
WOULD WE SEE A CITY?
If you had a concentrated city of a over 5 million people on the moon - and each of them had spill into space of about two 100 watt light bulbs per person into the sky -then you might just spot it as a sixth magnitude star in ideal conditions during lunar eclipse.
This assumes the habitats are on the surface. And also close together - if you had the combined light of a sixth magnitude star - but spread over a large area of the Moon you wouldn't see it, because of low surface brightness.
A complication though is - that on the Moon - you would cover all the habitats with several meters of regolith for shielding from cosmic radiation.
Also, I rather expect that human habitats would have few windows, and the windows all tiny, as in this Bigelow mockup.
At most you might have occasional larger ones like the Cupola on the ISS but they would be rare, surely - that's because windows are really hard to construct when you have to hold in ten tons per square meter of outwards atmospheric pressure.
Unless we develop some way of making amazingly strong transparent material (like the Star Trek "Transparent aluminum") they would surely rely on view screens to see the outside surface, or for that matter to display movies and scenes from Earth - but probably not much by way of windows as such, just one or two as special features like the Cupola.
Classic scene from Star Trek IV "The Voyage Home" about Transparent Aluminum - sadly we don't have anything like this to build our windows of our space habitats yet.
So - chances are even if they are on the surface - that hardly any light would escape from the human habitat.
Here is a rather fine Russian video about a lunar city from 1965. Sadly I don't speak Russian so not sure what they are saying.
So - chances are even if they are on the surface - that no lights would escape from the human habitat.
WOULD WE SEE GREENHOUSES?
Human habitats could also be underground, for instance in lunar caves.
Hadley Rille photographed by Apollo 15, probably example of a large collapsed lava tube on the Moon.
If so, then it may originally have been a huge hollow cavern. If so, what about lava tubes that are not collapsed, underground?
A study in spring 2015 found that they could be stable up to ten kilometers wide in the much weaker lunar gravity. If so, there may be caves on the Moon large enough to house entire cities, and if humans lived in them we'd see no trace of them from the Earth.
And, we wouldn't see solar panels, not when the habitats are in the lunar night - or anything reflective, as there would be nothing for them to reflect.
All in all, I don't think we have much chance of spotting human habitats in the areas of darkness on the Moon, at night, not for a long time to come, not directly.
WOULD WE SEE GREENHOUSES?
Greenhouses are more promising. They would be large very strong spherical or hemispherical buildings - if on the surface. They'd be made of some material far stronger than e.g. the plastic of "polytunnels" as they have to withstand several tons per square meter outwards pressure - but would probably be transparent to let in the daylight during the 14 day "day".
And if used for growing vegetables would need to be illuminated with close to full daylight for growth during the lunar night - probably in 12 hour day / night cycles so as viewed from Earth, they would blink in and out of sight every 12 hours.
Of course the light would be directed downwards onto the plants. But the plants themselves absorb only part of the light that is shone on them, and reflect the rest back.
Using albedo for green grass, 0.25, then diameter needed to see it from Earth is 571/sqrt(0.25/0.12) = 395 - or about 400 meters for the diameter of the greenhouse. Area about 126,000 square meters. Or about 31 acres.
So, we wouldn't see a small greenhouse. But if you had greenhouses the size of a very large field - of order half a kilometer in diameter or so, seems that we would see the largest greenhouses in ideal conditions if fully illuminated to equivalent to daytime sunlight.
A very large field, perhaps half a kilometer in diameter or more, if fully lit up with artificial light equivalent to the sun, on the Moon, would be as bright as a sixth magnitude star. This is a photo I took myself, so copyright Robert Walker.That is - unless they optimized the light for vegetation so that it supplies almost no green light, then the plants might look much darker.
It is of the corner of a field in the Scottish Borders - not sure of its acerage.
WHAT ABOUT AN ICE SKATING RINK OR SKI SLOPE?
Let's not bother with the practical details - do they go ice skating in spacesuits? Or do they have a very thick polymer covering able to hold in ten tons per square meter? Or low pressure atmosphere?
But - just suppose we have ice, high Albedo, exposed on the surface, could we see it?
If it was, say, fresh snow - or perhaps an ice rink, then to see it from the Earth as a sixth magnitude star, it needs to have a diameter of 571/sqrt(0.8/0.12) or about a 220 meters diameter patch. Or about 9.4 acres.
A standard skating rink of 40 by 60 meters is not nearly large enough to see from Earth with the equivalent of full sunlight illumination.
A larger area - say, a fully illuminated giant slalom slope for low gravity skiing in spacesuits - that could be visible from the Earth. Length 400 meters, width 30 meters, so still nowhere near big enough. 12,000 square meters. We need ten times that to be as bright as a sixth magnitude star.
Olympic Super-G - sadly you couldn't see a Super G ski slope on the Moon - unless lunar slalom ski slopes are ten times wider or longer than on Earth.Also that assumes they are lit at similar levels to full sunlight. But they would probably be lit at much lower light levels than that, at night.
So, the answer is no, we wouldn't see ice rinks or Olympic ski slopes on the Moon from the Earth either, not unless the ski slopes are much larger than on Earth.
We'd see a complete enclosed ski resort easily though. The largest ski resorts are thousands of acres. If we had a fully developed ski industry on the Moon - if low gravity skiing really took off, they would only need to cover
I've written this up now for my science20 blog as
Could You See Moon City Lights? Or A Greenhouse? Or An Ice Rink Or Ski Slope? Just For Fun
You can now get this as a kindle book along with many other of my answers, in
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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