Is it possible to have a moon of a planet that is practically day on both the star facing side and the other side, if the planet itself is mostly made of highly reflective surfaces like Silver and Aluminum.
It's not possible, because there are configurations where part of the moon would not be seeing neither the star nor the planet.
For a reference, look at the diagram of the moon phases
You can see that there is almost always a portion of the moon which is both on the half away from the star and on the half away from the planet. At first quarter and last quarter that correspond to a whole quarter of the moon, at the full moon it would be a whole half moon.
This portion of the moon would therefore receive no light.
That apart, reflection from a planet can hardly be compared in terms of intensity with the light directly shining from a star, even if the planet is 100% reflective.
There are two issues (that I can think of) with this:
The orbital arrangement required is unstable, so it can't exist naturally.
Unless the planet surface has mirrors specifically arranged to reflect light towards the moon, the illumination on the dark side of the moon will be less.
There may also be an issue with the moon being too close (within the Roche limit), but calculating that is beyond me.
Short AnswerIt is close to impossible to imagine an astronomical situation where an astronomical object like a planet or a moon is in perpetual daylight all over its surface.
The particular situation you imagine, with the moon between the planet and its star, having the side facing the planet illuminated by light reflected from the planet, would not be able to keep all parts of the moon in constant daylight.
The trouble with this idea is that moons orbit planets, and planets orbit stars. Therefore, the relative positions of a moon, its planet, and its star, will constantly change.
But it is possible for a world to be in an eternal daylight all over its surface area if it is in a very weird star system, with one or more rings of stars orbiting around a giant black hole.
Part One: Does Temperature or Illumination make it Daytime?
You would expect that the moon would be tidally locked to the planet, with one side always facing away from the planet, and one side always facing the planet, but that does not have much effect on the problem if it is desired to have eternal day on both sides of the moon, instead of eternal day on only one side of the moon.
The question doesn't specify whether there is supposed to be eternal day temperature-wise, or merely eternal day illumination wise.
As well as I can guess, it would be impossible to have eternal day on both sides of the moon, if daytime temperature is required. I think it will be impossible for the moon to receive such a constant amount of radiation that it will have a constant temperature, the same on both sides and at all times. Instead the amount of radiation received will vary and any part of the moon will get hotter or colder as the amount of radiation increases and decreases.
But it would be much more likely to have eternal day on all parts of the moon if day is defined by how bright the surroundings look. The surroundings can have the same apparent brightness despite a considerable difference in the actual light levels.
That is because the human eye, and the eyes of many other animals, can adjust to greatly differing levels of illumination. How well you can see shapes and colors is not directly proportional to the amount of light which is arriving from light sources and being reflected from objects.
Part Two: How Bright is Bright Enough to count as Day?
Anyone who has been outside at night without a light and been able to see where they are going by the light of the full moon will know that they don't need a light sources as bright as the noonday sun to see where they are going.
So what is the difference between the brightness of the full moon and the brightness of the sun overhead?
Astronomers measure the brightness of objects seen in the sky with apparent magnitude. And the lower the apparent magnitude, the brighter the object appears from Earth. A star of magnitude 1 or the first magnitude is brighter than a star of magnitude 2 or the second magnitude, which is brighter than a star of magnitude 3, and so on.
And the brightest objects in the sky appear so bright that their magnitudes have negative numbers.
The full Moon can have an apparent magnitude as bright as -12.90, while the Sun has brightness of -26.74, which is 13.84 magnitudes. The magnitude scale is logarithmic, and 5 magnitudes is a difference of 100 times the brightness, and 10 magnitudes is a difference of 10,000 times the brightness of. The Sun is about 400,000 times brighter than an average full moon.
So if a light level at least equivalent to the light of a full moon is all that is necessary to turn night into day for the purpose of the story, there is a lot of room for variation in light levels while remaining day.
But you might have a higher requirement for daylight.
An illumination level of 1 lux is equal to apparent magnitude - 14.20, which should be more that 3 times the light of the full Moon.
My answer to this question discusses light levels: Moonlight bright enough to see by
Of course, if day has to be bright enough for human eyes to see color vision, the possible range in daytime light levels will be much smaller.
And if there has to be enough light for scattered light to make the sky appear blue instead of transparently showing the blackness of space, the possible range in light during the day will be even smaller.
A clear moonless night lit by starlight and airglow will have a brightness of 0.002 lux.
A quarter moon will have a brightness of 0.01 lux.
A full moon has 0.25 lux.
The sky under very dark storm clouds at midday will have about 200 lux.
Sunrise or sunset on a clear day has 400 lux.
An overcast day at midday will have 1,000 to 2,000 lux.
A shaded area illuminated by a clear blue sky at midday will have 20,000 lux.
Bright sunlight is 111,000 lux.
The brightest sunlight is 120,000 lux.
So if the brightest day on the moon is no brighter than 120,000 lux, the brightest sunlight on Earth, the darkest day might be about 400 lux, equal to a clear sunrise or sunset on Earth, and still be considered day. So you might be able to accept differences of up to 300 times in the levels of light received from different stars in your star system.
If two stars have the same luminosity, and star A is 17.320 times as far away as star B, star A will have an apparent brightness only 1/300, or 0.0033333, that of Star B. 17.320 is the square root of 300.
But fortunately stars vary greatly in their luminosities. The most luminous stars have millions of times the luminosity of the least luminous stars.
So the planet and moon in your story could be very close to a very dim star, and a very bright star that was 1,000,000 times as luminous could be 1,000 times as far as the closer star and still give the planet and moon the same amount of light.
Combining that with a possible variation of hundreds of times in the luminosity acceptable as daylight, and you can have the planet and moon receive daylight from different stars in different directions at the same time, and all parts of the moon can be in constant daylight.
If the brighter star was 1,000,000 times as luminous as the dimmer star, and was 17,320 times as far away, it would be give the moon 0.003333 times the light of the dimmer star, which would still be enough for daylight.
Part Three: Stars for a Habitable Planet or Moon
But I suspect that in your story you might want the planet to be habitable, and have intelligent lifeforms who send expeditions to the dead and lifeless that has eternal daylight in every location. Or you might want the planet to be a lifeless giant planet, and the moon to be a giant, Earth sized, moon, which is habitable for human visitors and/or for native intelligent beings.
So I suspect that you want at least one planet or moon in your star system to be habitable for organisms which need an oxygen rich atmosphere. And that will greatly limit the types of stars and their distances from each other.
The planet Earth was habitable for liquid water using organisms in general for billions of years before it became habitable for humans. Lifeforms on Earth gradually produced the oxygen rich atmosphere needed by humans and by all multicelled animals such as other intelligent beings.
Stephen H. Dole, in Habitable planets for Man, 1964, p. 63, estimated that a planet would have to exist with fairly steady temperatures from its star, for at least 2 or 2 billion years before developing an oxygen rich atmosphere.
That means all the stars in a system with a habitable planet, which contribute significant amounts of light and/or heat to that habitable world, have to have been shining with fairly steady luminosity for at least 2 or 3 billion years by the time the planet develops a breathable atmosphere.
And different spectral classes of stars shine steadily as main sequence stars for different lengths of time before their luminosities change drastically.
On page 68 Dole decided that the most luminous and massive stars that would have fairly steady luminosity for at least 3 billion years would have a mass 1.4 times the mass of the Sun, and would be spectral class 52V stars.
This question: How would the characteristics of a habitable planet change with stars of different spectral types? has a very useful answer by User177107.
The semi-major axis of Earth's orbit is defined as one Astronomical Unit, or AU.
The table of different types of star in user177107's answer, gives a lot of data about each type of star in the table, including what I call the Earth Equivalent Distance or EED, the distance at which a planet would have to orbit to receive the same amount of radiation from its star as Earth gets at a distance of 1 AU from the Sun.
According to that table a spectral class F2V star has a mass of 1.44 solar masses and a luminosity of 5.001 solar luminosities, and an EED of 2.236 AU, where a planet's year would be 1,018.01 Earth days long.
And the least massive star in the table, a M8V class star, has a mass of 0.082 solar mass, a luminosity of 0.00043 solar luminosity, and an EED of 0.0207 AU, where a planet's year would be 3.82 Earth days long.
So a F2V star would be about 11,630 times as luminous as a M8V star, and so would appear as bright as an M8V star if it was 107.87247 times as far from the planet and the moon as the M8V star was, and thus at a distance of 2.2323 AU. And the F2V star would appear 1/300th, or 0.003333, times as bright as the M8V star when it was 1,868.4059 times as far away as the M8V star, or 38.676002 AU.
Part Four: How Many Other Stars in the System?
This maximum possible distance between the planet and moon and the F2V star allows for several other stars to be within the system at intermediate distances.
The moon would orbit its planet, which would orbit the M8V star at a distance of 0.0207 AU, in a non-circumbinary or S-Type orbit around that one star, and not around any other stars in the system.
In non-circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.4
So the nearest that any other stars in the system could get to the M8V star would be 5 times the semi-major axis of the planet's orbit around the M8V star. So the nearest star (or pair of stars) would have to get no nearer than 0.1035 AU to the M8V star.
Assuming that the next nearest star (or pair of stars) in the system would have to get no nearer than 5 times that distance, it would have to get no nearer than 0.5175 AU.
With the same ratio the third nearest star (or pair of stars) would have to get no nearer than 2.5875 AU.
With the same ratio the fourth nearest star (or pair of stars) would have to get no nearer than 12.9375 AU.
And with that ratio the fifth nearest star (or pair of stars) would have to be at least 64.68 AU from the M8V star, which is farther than the maximum possible distance for the F2V star to produce daylight on the planet and moon.
So that means there could be as many as five to nine stars, the M8V star, plus four other and brighter stars (or possibly pairs of stars) in the star system.
Assume that the nearest star (or pair of stars) to the M8V star gets no closer than 10 times the orbit of the planet, it would get no closer than 0.207 AU to the M8V star.
With the same distance ratio of 10 times the distance, the second closest star (or pair of stars) would orbit at 2.07 AU from the M8V star.
With the same distance ratio of 10 times the distance, the third closest star (or pair of stars) would orbit at 20.7 AU from the M8V star.
With the same distance ratio of 10 times the distance, the fourth closest star (or pair of stars) would orbit at 207 AU from the M8V star - which would be far beyond the distance at which a F2V star could be 0.003333 times as bright as the M8V star.
So that means there could be as many as four to seven stars, the M8V star, plus three other and brighter stars (or possibly pairs of stars) in the star system.
Part Five: Will Tidally Locked Planets Be Habitable?
But there is a further problem. A planet orbiting close enough to a M8V star, or even one a lot more luminous than an M8V star, to be warm enough for life, would become tidally locked to the star, with one side constantly facing the star, and the other side constantly facing away from the star. All the atmosphere and water on the planet might freeze on the extremely cold night side of the planet, leaving the planet airless, waterless, and lifeless.
There is some controversy whether a tidally locked planet in the habitable zone of a red dwarf star could be habitable or have all its water and air frozen on the night side, so that is uncertain at the moment.
According to Stephen H. Dole, in Habitable Panets for Man, 1964, page 71, if a star has less than 0.88 of the Sun's mass its habitable zone (which Dole calls its ecosphere) will not be complete since planets in the inner parts of the habitable zone will be tidally locked to the star. And if a star has less than 0.72 times the mass of the Sun, all of the star's planets in the habitable zone will be close enough to be tidally locked.
Part Six: Two Choices
So if you want the planet in your planet-moon system to be habitable, you have two choices.
You can let the planet be tidally locked and hope it can be habitable anyway, and thus use a red dwarf star as dim as you want.
Or you can use a more massive and luminous star, putting the planet at a larger distance from the star and thus reducing the relative distance to a star capable of being 0.003333 as bright or brighter, and thus reducing the number of stars you can put in your star sysyem to help illuminate all sides of the planet and moon at the same time.
Part Seven: A System where the Planet is not Tidally Locked
So you would be limited to using a class K1V or larger star for the nearest star to your planet and moon. The closest star on the table I mentioned above is a K2V star with a mass of 0.78 solar masses, a luminosity of 0.337 solar luminosity, an EED of 0.58 AU, and a year of 182.93 Earth Days.
So an F2V star would be 14.839 times as luminous as a K2V star, and would appear as bright as the K2V star at 3.852 times the distance, or 2.234 AU. The F2V star would appear to be 0.00333 times as luminous as the K2V star at a distance of 66.718 times the planet's orbit around the K8V star, and thus at a distance of 38.696 AU.
Assuming that the next nearest star (or pair of stars) would be at 5 to 10 times the planet's orbit around the K2V star, it would orbit no closer than 2.9 to 5.8 AU to the K2V star.
Assuming that the second nearest star (or pair of stars) would be at 5 to 10 times the first star's orbit, it would orbit no closer than 14.5 to 58 AU to the K2V star.
Assuming that the third nearest star (or pair of stars) would be at 5 to 10 times the second star's orbit, it would orbit no closer than 72.5 to 580 AU to the K2V star, and thus be far beyond the 38.659 AU limit.
Thus there could be one or two other single stars or pairs of stars in the star system in addition to the K2V star, making a total of two to five stars in the system.
So it would still be possible to design a multiple star system trying to get stars to illuminate the planet and moon from every direction at once, but it would be a bit more difficult.
Part Eight: A Habitable Moon?
But on the other hand, if you want the moon in your story to be habitable, and not the planet, you can use much dimmer stars and have much greater luminosity differences in your star system.
If a giant and uninhabitable planet orbits within the habitable zone of a small dim star, that planet will quickly become tidally locked. That will make the planet uninhabitable, but if it is a giant planet, it's uninhabitable anyway and that won't matter. There has been much speculation about the possibility of giant moons orbiting giant planets that orbit in the habitable zones of small, dim star. Those giant moons would become tidally locked, but to the planets, and not to the stars.
Thus those hypothetical exomoons would have days of alternating light and dark which would be equal to their orbital periods around the giant planets. Those orbital periods and days would be between roughly 1 Earth day to about 15 Earth days long. And that would be a lot shorter than the eternal day and eternal night on the two hemispheres of a tidally locked planet.
So if those hypothetical exomoons are large enough and otherwise suitable, they could be habitable. Thus you could have a habitable moon orbiting a planet orbiting a class M red dwarf star.
But there would still be a limit to how dim the red dwarf could be.
A moon must orbit its planet with a period less than one ninth as long as the planet's orbit around their star, or the planet's orbit around the star must be more than nine times as long as the moon's orbit around the planet, for the moon's orbit to have long term stability.
So if a giant, habitable moon's orbit around a giant planet must be about 1 to 15 Earth days long, the planet's orbit around its star must be at least about 9 to 135 Earth days long.
Dole thought that the day of a habitable world would have to be less than 96 hours or 4 Earth days long. In that case the orbit of a habitable moon around its planet would have to be 1 to 4 Earth days long, and the orbital period of the planet around its star would have to be at least 9 to 36 Earth days long.
Dole also considered the case of world which was tidally locked to a companion world instead of its star, and decided such world could be habitable around stars with masses that were as low as 0.35 solar mass. The solar tides on planets in the habitable zones of stars of lower mass would be so high and strong that they should erode the continents and destroy all dry land.
An M5V star would have a mass 0.16 that of the Sun, a luminosity 0.00229 of the Sun, an EED of 0.0547 Au, and a year Earth days long. An M2V star would have a mass 0.44 that of the Sun, a luminosity 0.0268 that of the sun, an EED of 0.163 AU, and an orbital period 36.51 Earth days long.
So I guess that the smallest star which could have a habitable moon orbiting a planet in the habitable zone might be just a little less than and M2V star, and its EED would be just under 0.163 AU.
An F2V star would have 186.60 times the luminosity of a M2V star, and so would have a brightness equal to a M2V star at a distance 13.66 times as far, or about 2.2266 AU, and it would have an apparent brightness 0.00333 times as bright at a distance 236.598 times as far as the M2V star, or 38.565 AU.
Part Nine: The Fatal Flaw.
So now it should be comparatively easy for someone to design a star system with the stars placed so that every part of your planet and moon is always illuminated with greater or less brightness by one or more stars. Right?
Notice that I kept mentioning the orbits of moons, planets, and stars.
And of course, moons, planets, and stars move in their orbits. So if at one moment the objects in the solar system are arranged so that all of the the moon is always in the light of one or more stars and always in daylight, the objects will eventually move into positions which are different enough that only part of the moon is illuminated at any one time.
It is possible to devise an arrangement of orbits where all parts of the moon have daylight continuously for weeks. It is also possible, though harder, to make that condition last for months. And it gets harder and harder and harder to make that condition last for years, decades, centuries, or millennia.
The original question doesn't specify how long the condition of daylight on all parts of the moon is supposed to last. So I can only assume that it is desired to last for billions and billions of years. Which of course is fantastically more difficult to make happen than to make it last for mere millennia.
Part Ten: A Solution at Last!
There is a science fiction story, "Nightfall" (1941) by Isaac Asimov, which was expanded into a novel Nightfall by Asimov and Robert Silverburg in 1990, and made into movies in 1988 and 2000. It is set on a planet Kalgash, where night falls only once time every 2,000 years.
Sean Rayomond's PlanetPlanet blog has a section Real Life science Fiction Worlds, where Raymond tries to devise scientifically plausible versions of various science fiction worlds.
That includes the post: "Real-life Sci-fi world 11: Kalgash, a planet in permanent daytime (from Asimov's Nightfall)"
In that post Raymond finds out that if he uses the six star star system as described in the story, he can't get the eternal daytime required.
So in the next post Raymond changes the arrangement of the Kalgash system, trying to make a system which will produce eternal day all over Kalgash.
Raymond eventually came up with his best version of the Kalgash system, a system centered on a giant black with many times the mass of any star, and a ring of stars circling around the black hole. Or in some variations the planet orbits between an inner and an outer ring of stars.
And that seems to be the best way (because it is the only way I ever heard of that actually works) to design a star system where a planet or moon has eternal daylight all over its surface.
Perhaps the closest you could get is to put the "moon" in an orbit around the L4 or L5 Lagrange point. That would still leave about 60 degrees of the surface in shadow though.
Another idea... You could have a ringed planet where the ring material is highly reflective, with the moon inside the inner radius of the rings. A bit of googling tells me this is possible. Worst case, the moon is in shadow of the planet, and a portion of the rings near the moon are in shadow, but there should still be illuminated ring material visible from all points on the moon's surface.