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On Earth, half the planet is illuminated at any time (let's ignore eclipses). Axial tilt lets day lengths vary, but over the course of a year, every location is illuminated half the time.

It's easy to make a planet where, over a year, everywhere is illuminated more than half the time. Use a binary star.

But is there a stable solar system that satisfies the more restrictive requirement that the planet is always more than half illuminated?

If it orbits one star of a binary, there will be a point in its orbit where the other star passes behind the one the planet orbits. If it orbits both stars, there will likewise be a point where all three are in a line.

edit: To make it clear, I'm not talking about atmospheric diffraction (Venus has some weird stuff going on there). Solutions must work for a vacuum world.

edit: It must be daylight providing meaningful illumination, not just technically visible. Let's say that more than half the planet must always be illuminated to a level at least 2-3% of the brightest illumination it receives. This rules out a binary setup using L4/L5.

edit: A planet where the same >50% receives light all the time isn't acceptable. That meets the "more than 50% illuminated" part, but not the "day longer than night" part since it doesn't have a (solar) day. I also described this goal as "over a year, everywhere is illuminated more than half the time".

You can have a binary star, and the planet in a Trojan position in the same orbit as the smaller of the two stars.

Basically, the two stars describe one side of an equilateral triangle, and the planet occupies the third vertex, in either L4 or L5 position.

One such configuration is presented here (figure 2, on the right).

Wikipedia gives stability requirements as "m1 > 100 m2 > 10,000 m3" so you'd need a large F-star as m1, and a small red-yellow dwarf as m2. This also requires a large orbital radius for habitability.

This configuration is not long lived enough for life to originate on the world, though. For that you'd need a smaller, colder, longer-lived main star (for example a Sun-type G star), and then you'd need a very small brown dwarf which, at a distance of some 8-10 light-minutes, wouldn't probably supply much of a daylight.

But if you don't require habitability, this would get you six hours of main daylight, six hours of "reinforced" daylight, six hours of "secondary" daylight and six hours of "night" every 24-hour day.

When you have a large geostationary moon that reflects enough sunlight to extend the day it should be possible. To increase the day time significantly.

Earth Example:

Formulas:

• 
  $alpha_z$ = $frac{v^2}{r}$ which is the same as: $alpha_z = frac{4pi^2}{T^2}$
  


• $F = Gfrac{m_1 m_2}{r^2}$

$alpha_z$ is the centripetal acceleration it has to be the same as $F$ from the second formula which is the gravitational force.

T is the time it takes for your object to complete one orbit in the case of earth about 24 hours or 86400 seconds.

$r$ is the distance between the two objects.

$m_1$ and $m_2$ are the masses of your moon and planet.

$G$ is the gravitational constant.

$v$ is the speed of your orbiting object relative to in this case earth.

In our case we get about 0,0000000052885 for $alpha_z$

For the second formula you leave the mass of the moon out because it cancels out with the first formula where you originally also have to adjust for the mass of the moon but you use it in both formulas so you can ignore it completely.

If we plug this value for F in the second formula and switch the equation around to give us r we get $r^2 = Gfrac{m_1 m_2}{F}$ and a value of 7.537137 × 10^22 we have to take the square root of it as it is $r^2$ and so we land at 2.74538468 × 10^11 meters which tells us that in our case it would not work because the moon would be to far away to reflect enough light onto your planet. So you would have to tweak your system if you want this solution to work.

The things you could tweak to make it possible are:

1. You could make the moon lighter


2. Let the Planet rotate much faster around it self


3. decrease the mass of the planet

Also you could place the planet between two stars in such a way that both stars pull on the planet with the same force which leads to a cancellation of gravitational forces but you would have constant day and no seasons. Hope this helps.

The star is usually much larger than the planet, and there is diffraction, so any planet is always more than half illuminated. ;-) Significantly more than half, no, imo.

Except ... if you put the orbit of the planet perpendicular to the plane in which the two binaries circumvent each other. It'll probably be tricky to stabilise such an orbit, which needs to be synced up to make sure the three objects really never line up.

An astrophysicist could probably tell us if there is a "natural" resonance which would drive a planet into such an orbit, and keep it there.

Yes. Admittedly an engineered system answer:

Central object: A black hole. This must be rotating in the plane of the system so the deadly jets never get anywhere near the planet.

Around it, a ring of 6 (or more) stellar objects.

Case A: The planet orbits between the black hole and the stars. In this case if all the stars are burning you have perpetual sun, if you want night most of them must be dead (white dwarf or neutron star.)

Case B: The planet orbits outside the ring of stars. At this point you have more than half light but it's not perpetual.

So long as the black hole is sufficiently more massive than the stars (I don't know the required ratio) this is stable.

Imagine a planet with an advanced civilization. It has launched many mirrors into space, to illuminate (part) of the planets dark side. You can tweak the amount of mirrors and increase the average length of the day to fit your story.

Pyramid planet.

With one light source and a spherical planet, I could not think of a way to illuminate more than half. It's a sphere thing. But if you can use shapes other than a sphere it is easy. The (tidally locked) pyramid planet keeps its apex point at its sun, and each of the triangular faces stays in the light. You could have it rotate around the axis down through the apex. The square side stays in the dark.

Other tidally locked elongated shapes would also keep their elongated faces in the light and the base in the dark.

OK. Tidal locking not allowed. I will borrow my answer from

Why is my Dark World so dark?

This world is a disc, turning on its axis. It stays with its edge
facing its sun. On the ground, the sun is always moving along the
horizon, never setting, never rising, never stopping. Sunlight is
always redshifted and oblique. Shadows are long.

There is sunrise and sunset on the edges of the disc. The edge is a minuscule fraction of the disc.