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Is it possible for an Earth-like planet to exist where the light reflected by the moon is, not as powerful, but close in intensity to daylight?

And if so, what would be necessary for it? A moon much closer to the planet? Different chemical composition? Different moon cycle? Multiple moons??

What would be the effect of these on the projected moonlight? Would its spectrum shift towards blue or towards red?

Hesitant to tag this as hard-science as I'm not necessarily looking for exact equations for light propagation, etc...

The sun is about 400000 times brighter than the full moon. That's quite a lot.

The moon, depsite looking quite white, is actually a surprisingly dingy grey with an average albedo of about 0.12 (equivalent to damp soil). If you painted the moon a brilliant glossy white and raised its albedo to 1, it would be a little over 8 times brighter, which still leaves it 1/48000 times as bright as the sun.

(incidentally, an ideal material for the surface of your super white moon would be ice, which is a little implausible close in to the parent star but not entirely impossible)

A moon much closer to the planet? Different chemical composition? Different moon cycle? Multiple moons??

None of the above. The apparent brightness of our super-high-albedo moon is related to its size and its distance from the sun. Even if you had ten moons, and each one had four times the apparent size of the moon (so about twice the apparent angular diameter), you'd still be 1/1200th of the sun's brightness, and that's such an astonishingly unlikely and gravitationally unstable arrangement that it isn't really worth thinking about.

You'd either need to move much close to the parent star, or to substantially increase the brightness of the parent star. In either case, the apparent brightness of the sun during the daytime would be correspondingly higher, and that means that your planet is going to be roasted and won't be likely to support life (or even an atmosphere, to be honest).

If you want something almost as bright as the sun without incinerating the world, you should see about building giant orbital mirrors and have them oriented such that they reflect sunlight onto earth.

What would be the effect of these on the projected moonlight? Would its spectrum shift towards blue or towards red?

Moonlight actually has a slightly warmer colour temperature than sunlight (about 4100K vs about 50-5800K for sunlight). A perfectly colour balanced briliant white moon should therefore have a slightly cooler colour temperature than the moon (eg. more blue), which isn't what I would have expected at all.

Moonlight has the (lack of) intensity it does because the Moon's surface (bright as the full moon looks at night, against the black of space) is quite dark -- about like worn asphalt pavement, gravel with tar between the pebbles.

To make it brighter, it would need to be covered with brighter material. One fine candidate is ice; a fresh ice surface, if it's finely divided, like snow, could reflect about 5 times as much light as the regolith we see. Unfortunately, ice doesn't stay white like snow over geological time when exposed to space; it darkens and turns red. The undisturbed ice surfaces of Kuiper Belt objects and first-time comets can be darker than the rock dust that covers most of the Moon's face, and redder than a building brick, so if it's to stay bright, it'll need some mechanism to replace the surface every few thousand years or so.

Most ices will reflect nearly white -- that is, they won't change the color of the light that strikes them much. Spectrography can tell what the ice is made of, to some extent, by what light it absorbs, but the color of the reflected light will read as white to the eye.

I am not sure you understand the vast difference between the brightness of sunlight and moonlight when you ask for moonlight "almost" as bright as sunlight. And in fact the moonlight on Earth is quite adequate for many purposes, so it is possible that your story might work with moonlight no brighter than that of on Earth.

The magnitude scale for apparent brightness is a reverse logarithmic scale. The higher the magnitude number, the lower the apparent brightness of a light source. The lower the magnitude, the higher the brightness. A magnitude one magnitude lower corresponds to being 2.512 times brighter. A magnitude five magnitudes lower corresponds to being 100 times brighter.

The new moon, the Moon at its minimum brightness, has an apparent magnitude of -2.50, while the full moon, the Moon at its maximum brightness, has an apparent magnitude of -12.90, a difference of 10.4 magnitudes. A difference of only 10.00 magnitudes corresponds to a difference of 10,000 times in brightness.

The Sun, as seen in a clear sky on Earth, has an apparent magnitude of -26.74. That is a difference of 13.84 magnitudes. A difference of 13.00 magnitudes is a difference of 126,202 times the brightness, and a difference of 14 magnitudes is 317,021 times the brightness. So as seen from Earth the Sun has a few hundred thousand times the brightness of the full Moon.

https://en.wikipedia.org/wiki/Apparent_magnitude1

You might want to ask yourself exactly what you want the extra brightness of your planet's moon for in your story, and then do research to find out how much light is needed for that, and then figure out if it is possible to increase the brightness of moonlight on your planet that much.

On a clear night, you can see fairly well by starlight if you are far from man made light sources and the resulting light pollution.

I used to go out at night and walk up a hill to a grassy field and look at stars and astronomical bodies with binoculars. I didn't take a flashlight with me to light my way because I wanted my eyes to become dark adapted to see in the darkness better.

Human eyes adapt to see better in darkness after a few minutes in darkness. So amateur astronomers don't use flashlights or lanterns, or use only red artificial light, when setting up their equipment to observe the skies, because they don't want to interfere with their eyes adapting to see better in the dark.

https://en.wikipedia.org/wiki/Adaptation_(eye)2

On a cloudy night close to a big city, you can see fairly well by city lights reflected from the clouds due to man made light pollution.

On a clear moonlit night you can see fairly well without any artificial light sources.

Both history and fiction have many examples of single persons or entire armies sneaking around in the dark.

Of course if someone travels by night without artificial light sources they would probably have a higher than usual probability of tripping over something they don't notice or stepping into an unseen hole than if they traveled during the day. But if someone doesn't watch where they are going they could trip during broad daylight also.

The light of stars, planets, and even the full Moon, is not intense enough for most people to read by. Even the light of the full Moon is not intense enough to see colors, except that objects may look faintly blueish.

If you really want the moonlight on you planet to be "almost" as bright as daylight, then you do have a problem designing a different astronomical set up allowing the moonlight to be almost as bright as daylight, because on Earth daylight is hundreds of thousands of times as bright as moonlight.

To get moonlight a thousand times more intense than moonlight on Earth, you might have a moon that occupies a thousand times the area of the sky as seen from the planet as the Moon has seen from the Earth. The square root of 1,000 is 31.622776. The Moon has an angular diameter in Earth's sky of about 29 to 34 arc minutes, so if your fictional planet's moon has an angular diameter of about 916.4 to 1,074 arc minutes, or 15.273 to 17.9 arc degrees, it will have 1,000 times the angular area of the Moon.

If your fictional moon is at the same distance as Earth's Moon, it can have 31.622 times the angular diameter of the Moon if it has 31.622 times the actual physical diameter of the Moon. That would make the fictional moon several times the diameter of any Earth-like planet, so if the planet is supposed to be Earth-like and thus have an Earth-like size the "moon" in your story will actually be a large planet orbited by an Earth-like moon.

Or the moon in your story could be the same size as the Moon but orbit the planet 31.622 times as close as the Moon orbits the Earth. Other things being equal, that will make it appear to be 1,000 times as bright as the Moon seems in Earth's sky. It should actually be more than 1,000 times as bright since the moon will be closer to the planet and it's reflected light will be more concentrated when it hits the planet.

The Moon has an average distance of about 384,402 kilometers or 238,856 miles from Earth. Divided by 31.622 that makes about 12,156.157 kilometers or 7,553.4754 miles, which would be really close to Earth.

I believe that a moon that orbits an Earth-like planet that closely would actually be slowing spiraling in toward the planet and would break up into rubble or collide with the planet within a few million more years.

Or you could make the moon both larger than the Moon, and also much closer to the planet than the Moon, so that the two factors combined give the moon an angular diameter 31.622 times that of the Moon to make it 1,000 times brighter than the Moon, while still being far enough away to not be spiraling in to its doom.

You could also make the surface material of the moon in your story more reflective than the surface of the Moon. The Moon has a rather dark, dull surface and only reflects a small percentage of the light that hits it. So your fictional moon could be more reflective than the Moon.
Maybe your Earth-like planet has several large and close moons orbiting it in different orbits.

And maybe your planet could have a ring of large moons around it at a fairly close distance.

Recent calculations indicate that it is possible for many equally spaced objects of equal mass to share the same orbit, so a few dozen large moons sharing the same orbit around a planet would not be physically impossible, though such an arrangement would be extremely improbable.

https://planetplanet.net/2017/05/03/the-ultimate-engineered-solar-system/3

So you could make an astronomical arrangement where your planet has moonlight a few thousand times as bright as moonlight on Earth.

But sunlight on Earth might still be tens or hundreds of times as bright as the moonlight on your planet, even if you make the moonlight on your planet a few thousand times as bright as moonlight on Earth.

In my opinion, making your "planet" actually a giant, Earth-sized moon of a giant planet may be the way to get the other astronomical body as large as possible in the sky of your world, and thus reflect as much light as possible on to that world.

And if you decide that is the case you should look up other questions and answers in this site about stories set on the moons of giant planets.

But of course the astronomical set up necessary for your story depends on exactly what you want more moonlight for in your story, and thus how much brighter the moonlight needs to be.

A Moon could supply a lot of light given the correct conditions. A much bigger moon in a much closer orbit would work, although such a situation might best be described as a double planet rather than a planet and a moon. Such a moon might well fill a large portion of the sky and even when only half lit by the sun would still be very bright especially if its surface where composed of highly reflective materials.

I should point out that there are a number of issues with this type of arrangement. Although the roche limit for similarly sized bodies would allow a close approach such an arrangement is unlikely to be very stable and would produce a lot of strange gravitational anomalies.

Multiple Moons might sound attractive but in close proximity to a planet multiple large moons would probably be highly unstable leading to a collision.

Salt deposits are more stable, and they are white when powdered.

The bright spots on Ceres are hydrated magnesium salts and brine deposits. I don't know if hydrated salts can retain their water content on our moon, but salts like sea water salt are white by nature and have an albedo much higher than that of Regolith.

https://en.m.wikipedia.org/wiki/Bright_spots_on_Ceres

Your setup may involve a shower of meteors made-up of frozen brine hitting the moon and changing its color.

Bright enough to see by can mean many things. And the human eye can adapt to a very wide range of brightness values.

For example, the brightest noon sun can be 120,000 lux, but a very cloudy overcast day can be as low as 200 lux. Most people barely notice the difference because our pupils expand and contract to keep the perceived brightness roughly the same.

A full moon is about .25 lux, but when fully adapted it can be possible to read by moonlight. Interior light generally is between 100 and 250 lux (the latter for a bright office, for example). So if you can deal with the amount of light in a classroom or office, you only need to improve your moonlight by a factor of 400-1000.

If your people are aliens, you could also leave the moonlight about the same and simply give them slightly bigger eyes, or their eyes could have a tapetum lucidum structure at the back (that reflective surface that makes dogs and cats eyes glow when you shine a light in them), which greatly enhances their low light performance.