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I'm building a Dyson sphere - a real one, like Freeman Dyson originally proposed, made of swarms of solar collectors in independent orbits that fully surround the star, something like this:

Dyson Swarm

Each collector/habitat will be a truncated triangle, approximately two million kilometers wide by one million high. The inner face of some of the collectors will be inhabited, while others - those further in, exposed to stellar radiation at all times - will be entirely automated.

I'm designing a society to live on one of the habitable plates of the swarm. Being further out, their plate spends most of its time shadowed by other plates, and is only occasionally exposed to the sun. The people of the plate have lost a lot of their advanced knowledge, and they see the star as their life-giving god, the plates closer to it as their heavens, the plates further away as hells.

What I'm trying to work out is exactly how often would they get starlight at different levels of the swarm? How many levels have to be 'above' them for them to get, say 25% sun? Is this something that can be easily calculated, or if not is there an engine or similar I could use to find a result?

Edited to Add:

Detail I neglected to include originally - the closest plates to the star are about 0.5 AU out, and there's around 2 million kilometers separation between each layer of plates.

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    $\begingroup$ Note that all plates closer to the star will always be dark, probably 99-100% black, which makes me think more of Hell. The plates farther from the star will sometimes be lit by the star (though I'm not sure how much, depending on how dark & light-absorbing the surfaces are), which makes me think more of Heaven. $\endgroup$ – BrettFromLA Nov 30 '16 at 22:00
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    $\begingroup$ As @BrettFromLA noted, there will be a lot of light shining back and forth between the collectors. At some point you'll stop adding more collectors, because they begin to get too little light. It'll depend on their frequency-dependent efficiency and reflectivity,etc and on pure economics. I dont't think there can be a simple equation for that. $\endgroup$ – Karl Nov 30 '16 at 22:51
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    $\begingroup$ @Catalyst they are not rings, they are just orbiting triangles $\endgroup$ – MolbOrg Nov 30 '16 at 22:54
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    $\begingroup$ Why to build solar collector that would be dark most of the time? Why to settle it? If for industry, you can't let people know population lose knowledge. You need a reason for that. And plates closer would be dark because you see the side farther away from the sun. $\endgroup$ – Mołot Dec 1 '16 at 6:16
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    $\begingroup$ @Werrf The plates closer to the sun would appear dark because they are "above" you. Imagine you have a big opaque solar panel. If it's on the ground, it's lit by the sun, and you can see it pretty easily. If you hold it up over your head, it's blocking out the sun and there's no sun hitting the underside of it, and since you are only seeing the underside of it, it's dark! $\endgroup$ – BrettFromLA Dec 1 '16 at 18:26
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Just do surface area questions.

For any given shell, calculate the area of all of the plates within that shell, and calculate the total surface area of a sphere at that distance from the star. Dividing those will give you the fraction of the time one can expect plates in that shell to shadow shells further out.

This should be a more than reasonable estimate. It is plausible that some pairs of distances exhibit harmonics that result in different behaviors, but the basic probabilistic approach should be more than sufficient.

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  • $\begingroup$ As an addition, these probability functions will be useful for determining how much solar radiation reaches an outer plate, but that doesn't always mean that the outer plate will be in all sun or all shade. This simple calculation does not differentiate between a plate receiving full sun 50% of the time + full darkness 50% of the time and a plate which receives 50% sunlight all the time. If you care about that, you'd need many more details, such as the size of the sun, size of the plates, distances you built at, etc. $\endgroup$ – Cort Ammon Nov 30 '16 at 23:41
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Honestly, this kinda depends on a lot of factors: Size of the plates, distance from the star, orbital period, to name a very few.

The suns surface is 12,000 times bigger than Earths, meaning if each plate had the same surface area as the entire planet, you'd need 12,000 of them to cover it.

If you have the plates set 1 AU out, that number gets exponentially bigger.
Even at 1 earth surface area, each plate is going to be insignificant, and even a string of them isn't going to block much light unless the layers are pretty close together.

so a big question is, does having the actual orbital mechanics matter, vs just saying that the cloud of plates blocks out most of the light?

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  • $\begingroup$ For my purposes at the moment, just saying the cloud blocks x amount of light is enough - I'd like my population to believe that they're in, say, a world of level 15, and if they live well they'll die and ascend to level 10, if they live poorly they'll fall to level 20...etc. The closest collectors are 0.5 AU out, there's around 2 Mkm separation between each 'level'. $\endgroup$ – Werrf Dec 1 '16 at 1:43
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Some napkin math: Assume there are n habitats each with area $A_i, 0<i \le n $ and distance from the center of the sun $R_i$, which are infinitely thin, with totally random orbits at distinct distances.

Surface area of the sphere at distance r is defined by the shell $4πr^2$, ($r$ is distance from surface + 695700 km)

Then habitat $i$ is occluded by $\prod_{k=0}^{i-1} 1-(A_k/4πR_k^2)$

If $A_k = 2e12$ for all k, and the nearest habitat is 2m km from the center of the sun, and the rest are distributed at fixed intervals of 5000 km then see wolfram alpha.

$A_k$ could also be the total surface area of all habitats at a certain distance from the sun. Of course a more sophisticated orbital scheme could be devised (this setup implies a chaotic strobe light effect) but I'd expect any race capable of engineering a dyson swarm not to use natural sunlight and instead collect the radiation and use it in a more controlled way. Inner layers radiate waste heat which is usable as energy for outer layers.

I'd also not expect them to actually be instantiated as planet-optimized life. If that is the goal, a ring-world esque construct (or swarm with 1 shell at earth-like distance from the sun) makes more sense. Wide variation in distance gives you a wide variation in temperatures.

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As AndyD273 stated, there are many factors to consider.

Having elements shield each others is very inefficient, that's why most Dyson swarms feature statites, they do not cast shadows onto each others and can collect light from the entire surface of the star.

If you want to stick with orbiting elements then you must design orbits that lie on different "planes" each perpendicular to at least other two, like this.

In this way it doesn't really matter how distant your habitat is from the star or how large the elements are, the number of yearly "nights" is equal to twice the number of orbits that lie closer to the star than yours. So if your habitat lies on the second closer orbit from the star they will experience two very short nights every "year" (orbit); if they are on the third closer orbit they will have 4 nights, on the fourth 6 nights and so on. Nights are equally sparsed throughout the year, so if a year is 200 days and your habitat is on the third orbit they will experience 4 nights a year each every 50 days. The nights all last the same time and are probably very short; their duration depends on orbital velocity.

Consider that if you want artificial gravity you will need to spin your habitat unless it is massive enough to have its own gravity (this would create additional day/night cycles). Statite elements, on the other hand, don't need to spin to have gravity even if they are not massive enough, but you need to live on the dark side.

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It is enough just one plate of the same size which at an orbit closer to the sun to block those people from sunlight completely.

To dim light to 25% of possible light, sphere below them should be about 75% covered by other plates.

Truncated triangle, let say a Rectangle 1 by 2 million km, it is about 101'250 of such rectangles below to cover 75% of 1 a.u. sphere.

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