I'm writing a system where the moon is always full from the perspective of the Earth-type planet--it does not go through phases. Short of the moon itself being luminous, what would it take for this to be naturally possible?
A special case known as Earthshine is a nice example:
Earthshine is visible earthlight reflected from the Moon's night side. It is also known as the Moon's ashen glow or as "the new Moon with the old Moon in her arm".
Source for image and text: same link as above
Now to get this by natural means, the planet and moon in question might need much higher albedos than Earth and Moon. A human standing on the planet by day would also be blinded and possibly scorched.
But! As an act of engineering, this might be safer. With about a couple billion 1MW laser pointers spread over a circle spanning 120⁰ of both latitude and longitude on the surface of an Earth-sized planet you just might make it. The calculations, and the whole process to get there, were figured out by this site's favourite nerd god, Randall Munroe, in this What If article.
As far as I know, it is pretty close to impossible to have a planet with a single moon that always appears full as seen from the planet. Thus I suggest the next best thing would be a ring of so many moons that at least one would always be in the full phase.
Part One of Four: A Moon with an Orbital Period of One Year.
I can imagine a moon that has an orbital period around the planet exactly as long as the planet's orbital period around the star in the system. So the moon should stay in the same posiiton relative to the angle between the planet and the star all the time, and have the same phase all the time, possibly a full phase.
Here is a link to an orbital period of planet calculator.
I give the "sun" the mass of the Earth and the "planet" the mass of the Moon" and then put in different semimajor axises to calculate the orbital period.
At 500,000 kilometers the orbital period is be 0.110802 Earth years, at 750,000 kilometers the orbital period is 0.203557 Earth years.
At 1,000,000 kilometers the orbital period is 0.313396 Earth years, at 2,000,000 kilometers the period is 0.886418 Earth years, at 2,100,000 kilometers the period is 0.953723 Earth years, at 2,167,400 kilometers the period is 1.00000 Earth years.
The Hill sphere of an astronomical body is the region in which it dominates the attraction of satellites. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.1
The Hill sphere for Earth thus extends out to about 1.5 million km (0.01 AU).
The Hill sphere is only an approximation, and other forces (such as radiation pressure or the Yarkovsky effect) can eventually perturb an object out of the sphere. This third object should also be of small enough mass that it introduces no additional complications through its own gravity. Detailed numerical calculations show that orbits at or just within the Hill sphere are not stable in the long term; it appears that stable satellite orbits exist only inside 1/2 to 1/3 of the Hill radius
So the true region of stability for Earth satellites should only extend to about 500,000 to 750,000 kilometers, far closer than the 2,167,400 kilometers necessary for a moon to have an orbit 1 year long.
In fact I have read that the orbital period of a planet around its star needs to be at least 9 times as long as the orbital period of a moon around the planet for the moon to have a stable orbit.
The longest possible length of a satellite’s day compatible with Hill stability has been shown to be about P∗p/9, P∗p being the planet’s orbital period about the star (Kipping 2009a)
Which is this paper:
A moon with an orbital period of 1/9 (0.111111) year around Earth would be orbiting at a distance of about 500,928 kilometers.
Part Two: A Ring of Many Moons.
Thus I suggest an alternative, a planet with a ring of moons at the same distance, where always at least one or two of the moons appears full.
Possibly the planet could have a circle of moons of the same size equally spaced in the same orbit around the planet.
That would be very unlikely to have happened naturally, so it would probably have been an orbital situation created by an advanced civilization. The advanced civilization would put tiny moons equally spaced along the circular orbit and gradually bring in more and more material to build them up, carefully making certain that all the moons increased im mass at the same rate.
According to this blog post:
A ring of 7 to 42 objects of equal mass equally spaced along an orbit around a larger body would be stable over long periods of time. And it gives as source a scientific paper whose abstract is at this link:
So you could theoretically have a star system where there are 7 to 42 moons of equal mass orbiting a habitable planet at the same distance.
With 42 moons sharing the orbit, they would be 8.5714285 degrees apart. If one moon was directly opposite to the Sun and thus full, the 2 nearest moons to it would be 8.5714285 degrees ahead of it and 8.5714285 degrees behind it in orbit. And they should also look full to the naked eye of someone on the planet. If two moons were equally space ahead of and behind the direction exactly opposite the Sun, they would be 4.2857142 degrees being exactly full. And they would look full to someone looking at them without a telescope. And possibly some of the other moons close enough to that direction would also look full.
With 42 equally spaced moons, at least two or three would always look full at one time to people looking at them without telescopes, and possibly others would seem full enough.
And if you reduce the number of moons from 42 to a lower number, at first you would still have several moons looking full at the same time, and it would probably take a serious reduction in the number of moons to have a situation where there was not always one or two moons at a time that looked full.
And that is not the same thing as a planet with one moon which is always full as seen from the planet. But it is a planet where at least one full moon is always seen from the night hemisphere.
And once the ring of coorbital moons is created, it woun't need the constant applcation of immense amounts of power to keep a single moon in the L2 point of the planet suggested in other answers. And it would not need the constant application of immense amounts of power to keep it orbiting the planet with an orbital period of one year and not wandering away into space, escaping from the planet.
Part Three of Four: Why Not Many Moons in Different Orbits?
What about many different moons at many different distances from the planet? Earth has only one moon, and Venus and Mercury have no moons, but the small planet Mars has two tiny moons. The tiny dwarf planet Pluto has 5 known moons. The giant planets Neptune, Uranus, Saturn and Jupiter have 14, 27, 83, and 80 known moons resepectively.
But moons in different orbits cannot have orbits too close together or their gravitational interactions will tend to destabilize their orbits. So the outermost moons of a planet would have to be many times as distant as the innermost moons, and thus many times as large to appear as large as seen from the surface of the planet. And most of the moons in the solar system are less than a few tens of kilometers wide and would appear as mere dots of light unless they were very close to the planet.
Furthermore, a moon would not appear to be full from the surface of the planet unless its direction as seen from the planet is close to 180 degrees from the direction to the star. And arranging the orbits of a bunch of moons with different orbits so that at least one was always within a few degrees of opposite to the star and thus appeared full would be very difficult. Sooner or later all the moons would be too far from the direction opposite to the star to appear full, and thus there would not be a full moon seen on the planet.
So a ring of many moons equally spaced in the same orbit around the planet is the only way to get a planet where there would always be at least one full looking moon at night.
Part Four: Artifically lighting the Moon.
The disadvantage of having many moons in a ring around the planet, so that at least one or two will always appear full, is that some of the other moons will also be visible all the time and will appear less than full.
The only way to get each and every moon, or the single moon that orbits the planet, to appear full all the time and never gibbeous, half, crescent, or new, is with artifical lighting of the moon.
So the answers at this question:
Always a full Moon for the Emperor - Can this be achieved with solar panels and LEDs?
Show some ways to keep a moon from every appearing less than full.
It's possible, but extremely unlikely
In order for a full Moon to be constantly visible from Earth, all of the following must occur at all times:
1.) The Moon must be situated oppositely from the Sun at all times, so it's strictly visible from the night side of the Earth.
2.) The Moon's revolution must be directed opposite from the Eeath's rotation. Observing from the North pole, the Earth rotates counterclockwise, which means the Moon must be turning clockwise.
3.) The Moon's revolution speed must be marginally faster than the Earth's rotation, so it always remains dead-center in the Earth's night zone. This math is far too complex for me to calculate so I won't bother, if anyone feels like doing so go right ahead.
4.) Although the Moon must remain dead-center in the earth's night zone horizontally, one would be forced to choose whether it would be visible exclusively from the Northern or the Southern hemisphere. If the Moon were to be situated dead-center vertically as well, it would be eclipsed by the Earth.
PS: All of this would be far smoother to accomplish and calculate if the Earth would be tidally locked, but that requires it to be more than 50 times closer to the Sun than it currently is (much closer than Venus currently is) which would make life on it impossible (unless the Sun blew up and turned into a very special case of White Dwarf with a tolerable but close life zone)
It is not possible, at least not naturally.
The only place where the Moon is full is if it's beyond Earth's orbit.
To keep it in that place, you need something that actively keeps the Moon there - there's no stable orbit that you can use.
The best place to put the Moon would be Lagrange point L2 - the energy needed to keep the Moon in place would be minimal there.
However, you'd have to invest a humongous amount of energy to get the Moon there first, as it won't form there on its own.
And you'd need some active mechanism that reacts to perturbations and offsets them; while L2 requires just a minimum amount of energy to keep your position, given the huge mass of the Moon you'd need some alien tech pretty far beyond what humanity has today.
I.e. there's no natural way to have such a Moon, sorry.
The good thing about L2 is that it's beyond Earth's core shadow, i.e. you have enough light refracting in the atmosphere that the Moon gets illuminated despite being straight behind Earth.
It will be substantially less bright due to the greater distance (current Moon orbit: 370.000 km, L2 distance from Earth: 1.5 million km, roughly a factor of 4, meaning it's 16 times fainter). You would need a larger and a better-reflecting (higher-albedo) Moon go get our full-Moon illumination. Not too big of a problem because Moon size does not affect the dynamics too much as long as the Moon's mass stays below ca. 1/25th of Earth's mass.
You live on the Sun.
The full side of a moon points at the Sun, so your planet is always lined up right with the Sun. Why? Because your star has a massive Sunspot - a highly evolved descendant of the Common Sunspots that mark our star, which is intelligent and curious about its environment and wants to protect its local planets from the forces of stellar evolution. The Sunspot, being dark, and sheathed in intense magnetic fields under its voluntary control, shields the planet it captured, which bobs upon its surface, from the vast majority of the radiation of the star, so that only the lower 2/3 were melted and ruined, and the top surface, now flattened by the changes, faces outward. The planet does not fall into the star in part because the Sunspot induces powerful magnetic levitation effects, though I suspect a more extraordinary explanation would be needed if I took a glance at the math. The star itself, being highly evolved after all, has already begun to expand to its giant phase, so that its gravity is quite low and its heat a little less intense anyway. Your people do see the Sun - but they see only a part of it, carefully regulated prominences formed as ball lightning that replicate the ancestral appearance and warmth of the star from the planet's orbit before it began its expansion.
- Note the Moon is always naturally full. You have to think like this if you ever want to write a product label. :)
Your moon is made of mostly transparent material.
even if just a small beam of light is shining on(in to) your moon, it will get internally scattered and exit the moon from all sides.
It will not always be equally bright, for example when your earth type planet is shading the moon, it will appear more dimly lit, but it will still have light shining out of it from all sides, and thus appearing full.
If your moon is clean enough it could even act as a lens and have all kinds of interesting effects.. (making it too clean of a material would eliminate internal scattering however, keeping it from looking full)