How many hours of daylight can a planet in an elliptical orbit receive?

We have a very specific issue. My friends and I have been working on a world-building passion project for the past few years. One core theme of the planet is that it is almost entirely snowy, has no axial tilt, and experiences endless night, save 3 months every year, when the sun rises and sets as it would on Earth. Because we are far out of our depth when it comes to astronomy, we have assumed that this world's physics simply differ from ours somehow. We've established that the planet is travelling in an elliptical orbit about its sun, and sunlight is only able to reach the 4 moons of the planet and not the planet itself (except during this three month-long period, which occurs when the planet is closest to the sun.)

Obviously, we have encountered an issue with this: we do not know how many hours of daylight each topographical "section"/hemisphere of the planet experiences during these 3 months, which is integral for knowing the agricultural advancements of different nations. We theorised that the sunlight durations vary across different parts of the planet (hence the distinction), but we have no solid idea of how feasible what we could establish is (as this world's physics cannot differ too much from ours), or what the most optimal establishments regarding the odd physics of this world are to ensure the world's surface population is able to survive.

So, my question is, how much sun would a planet in an elliptical orbit receive?

A few clarifications about this planet to note are that:

What you want is sadly not possible. To answer your primary question the whole planet would receive 12 hours of day light and 12 hours of night on an ongoing basis. The following provide some further explanations which you might find useful.

Axial tilt is the predominant cause of seasons on Earth, but not the only cause. Earth’s slightly elliptical orbit means that the Earth is always closer to the Sun during the southern summer and further away during the southern winter. This is part of the reason why the southern summers are a little hotter and winters are a little colder than in the north (all other things being equal).

So, a world with a much more pronounced elliptical orbit would still show seasonality. But the seasons would be planet wide. Even on a planet with no axial tilt it will be warmer at the equator and much colder at the poles because the surfaces of the planet at high and low latitudes are inclined to the Sun light. It is possible to have a planet that is very cold just by virtue of its sun being a little dimmer or the planets proximity to its sun being a little further away.

It would be possible for a planet to have seasonality due to a pronounced elliptical orbit, but it is not possible within gravitational physics far as I know it, to (effectively) turn off or hide the sun for 9 months of the year. The best you can expect is for the sun to be much less bright in winter.

Perhaps the sun might be blotted out by other means such as vulcanism or cloud cover? Although I don’t know how that might be triggered on a global seasonal basis, and it would not be strictly “night” just dark.

Without an axial tilt the whole planet would always experience 12 hours of day and 12 hours of night with a few exceptions such as hilly terrain causing shadow especially at the poles. Additionally, From the exact poles on a uniform world the sun would circle the horizon in a perpetual state of dawn/dusk half risen/half set. Heading towards the equator the sun would appear progressively higher in the sky at noon until at the equator the sun would pass directly overhead. So your world would still have polar and equatorial climates, but they would be the same all year (no arctic circle or 6 months of darkness etc).

Illuminating the moons and not the planet itself is also problematic. And on any Earth sized planet 4 moons of any significant size would in all likelihood not be gravitationally stable. This would be especially true if they were all in the same orbit as you describe.

It might work for a very short time, but disturbances caused by (for example) the variation in mountain masses on the planet itself and the variable forces during the planets elliptical orbit would rapidly cause the moons orbits to vary and as soon as the moons were no longer equally spaced moon-moon gravitational forces would lead to chaos, a total derangement of the moons orbits and ejection or catastrophic collisions.

(sorry)

Binary system with a Red Dwarf star and a hot Jupiter = Flare Star

Let's craft a star system that changes its luminosity on a recurring time scale. To do it, we'll need a red dwarf star like Bernard's Star, but well add a hot jupiter in an elliptical orbit around a common barycenter. It would look like this:



Binary Star System

Here is Bernard's Star next to jupiter for reference:



Bernard's Star

Now we create a mechanism to cause your star to increase and decrease in luminosity periodically. This can be (theoretically) done by creating a flare period:

Most flare stars are dim red dwarfs, although recent research indicates that less massive brown dwarfs might also be capable of flaring.

Flaring is theorized to be caused by disrupting the star's magnetic field by another body (in our case, a hot Jupiter in an elliptical orbit):

but it is understood that these flares are induced by a companion star in a binary system which causes the magnetic field to become tangled.

It has been proposed that the mechanism for this is similar to that of the RS CVn variables in that the flares are being induced by a companion, namely an unseen Jupiter-like planet in a close orbit.3

Flare Star

With some amount of speculation (a lot) we can say that during the period of time when the red dwarf and Jupiter size planet are closest, the star would undergo a continuous flare period. If we adjust the orbits to match your requirements, the star can flare for 3 months and dim again for the remainder of the time.

Now, let's put your planet on a circular orbit around the binary system outside the goldy locks zone of a red dwarf. However, during the flare period, the habitable zone of the star extends and the planet receives a useful amount of sunlight.

We have now approached your requirements with a few caveats:

A traditional Dyson sphere as commonly presented is when you make a sphere around the star the size of a planet's orbit and people live on it, with the Sun at the middle keeping everyone warm. Very expensive solution to a very simple problem, especially when you price the magic gravity generators to keep people walking around on it. We couldn't afford this.

Our precursor situation launched a vast constellation of statites, using solar sails to stay rooted to specific spots forming a sphere around the star. They had very, very thin reflectors to collect and focus light for power, Archimedes-like lens warfare, terraforming outer planets... I don't know what, who knows the minds of the gods? Point is, they completely surrounded their star in half-nanometer-thick mirrors or collectors so that no light got out except in a planned way. And their mighty homeworld, with perfectly restored ancestral ecology, revolved around their star in an orbit that they circularized perfectly just to make navigation easier, which, naturally, was just inside the statite sphere (allowing some space for the Hill radius), so that they could gaze lovingly on the light of their ancestral sun. The planet was surrounded by what sounds like a "Kemplerer rosette" [sic] of moons, a favorite of Larry Niven's but not a naturally occurring formation; it requires active maintenance in a complex orbital environment which is apparently still available. People would zip all around the system on little graphene cables that would link and unlink to the statites in all directions, getting anywhere they wanted with fast acceleration and almost no expenditure of energy.

Alas, something fouled up their golden age. A very large impactor, the size of the Moon or more, knocked their homeworld out of its orbit, and destroyed every trace of their life and technology. If I had to take a wild guess, I'd say that wasn't an accident. (Use a less drastic disaster if you'd like to retain some artifacts) The statites were decimated, but they were also automated, and they gradually scavenged the planetary debris and rebuilt themselves just as before. Fortunately the propulsion system on the moons, needing to maintain that unstable Kemplerer rosette, were highly overdesigned to maintain stable orbits around the planet "no matter what", with the help of a Dyson sphere's worth of power.

The planet now orbits elliptically, with a perihelion inside the statite sphere, but moving far beyond the sphere for most of the year due to the energy it was given during the impact. The statites leave holes in their spherical formation where they calculate it would be too difficult to maneuver out of the way of the planet, so they don't collide every time it passes through the sphere. So if you're on the planet can see the star when you're near it, but not when you're outside the statite sphere.

The moons of the planet are always illuminated. This is out of basic courtesy: the formation doesn't block starlight to any moon, spacecraft, or built environment outside the sphere wherever it goes; they just tilt their mirrors and scurry out of the way.

But the planet was a special case. The statites were programmed to avoid being knocked out of orbit by the planet, to hold formation, perhaps to shield it from the view of prying eyes outside the system... the programmers never contemplated the situation where the planet would be outside the statite sphere. I mean, that would be ridiculous, right? So for this one world the sphere stays solidly in place, blocking all light to the planet even though it gives the moons a different consideration. (It is conceivable that by some imprecision of the programming that Bailey's beads might be visible around a moon as it eclipses the sun, since the moon is illuminated)

Now for the hours of daylight: the planet is in an elliptical orbit, which could potentially have perihelion at any time of the year. When the planet is illuminated, day and night are normal. If it has an axial tilt, therefore, it could be closest during Southern hemisphere summer like Earth, or Northern hemisphere summer, or even around the equinox when the tilt doesn't matter much anyway. This gives you options to allow any amount of day or night you want during this time, on some particular continent. But it will still be 12 hours day and 12 hours night on average if you consider the entire planet as a whole. (This is even true if it rotates synchronously, in which case you're averaging 24 and 0)

The remaining detail I can't cover is There is a period of pure darkness lasting 5 hours during a full "day"/night cycle in a given region, except of course during the 3 month period. I thought it was dark that whole time?? Anyway, I am reasonably confident this system is flexible enough to fit this if I can figure out what it means.