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The Math would add up
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this post was submitted on 15 Feb 2024
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Light doesn't have to spread wider than a target or else you wouldn't be able to have telescopes or magnifine glasses. Each panel in unison can act like a giant magnifine glass. The difference in power density would be the ratio of the distance of sun to earth squared vs (sun to earth + earth to satellite) squared which is basically negligible. Where do you get the 3km wide beam? Suns rays are almost parallel.
Yes, you are right, considering the rays emerge from a point. And yes, each panel or all panels in unison can act like a magnifying glass. However, if they focus the light on a point at the height of the satellite, they work like a magnifying glass, or telescope with a focal length of the satellite – power plant distance, so at least 300 km. Considering the angular size of the sun, this telescope would lead to an image of the sun, the size of 3 km.
No sun rays are not parallel. If you looked at the sun (don't, it will burn your eyes), would you see it as a point or a disc? As a disc. Why? Because even looking in slightly different directions, you see the sun. So the rays from the sun are not almost parallel, the rays from other stars are, they look point like.
Two interesting images for you: A solar eclipse viewed trough tree leaves: You can see the partial sun disc by using the small free points in the tree cover as pinhole cameras. Sure, the tree cover does not have lenses, but they only make the image sharper, not smaller. In this image the focal length is only the height of the trees and the image is already a few cm across. It also shows that the rays from the sun are not parallel. If they were, all rays going through the small free spots in the tree cover would end up at the same spot on the ground.
International Space Station, ISS, flying in front of the Sun: As the sun and the satellite are far away, we can assume that the angular size of the original sun and the virtual sun image are approximately the same when viewed from the power plant. Hence, this image shows how the mirrors would form an image of the sun, where only a small part of it hits the sun.
As the sun is much larger than the ISS, the angle of rays which come from the sun is much larger than the angle of rays which hit the ISS.
Unfortunately I can't open the links on my phone so it's hard to follow what you're describing. I understand the rays aren't perfectly parallel but they're pretty close to parallel. Do you mind doing the math of where you are getting 3km from I'm not really following your logic. It doesn't make sense to me that the light should suddenly spread out way more after bouncing off a mirror than if it has just continued traveling straight.
The links are actually only random images from an image search with the terms "solar eclipse through tree leaves" and "iss in front of sun".
I think you have in mind, that the rays are not parallel because they have to arrive at different positions. As you say, this is negligible, and it can even be avoided by the tilting of the mirrors. However, the rays start from different parts of the sun, and as the sun is huge, this angle is not so small.
I'll try to explain it in more detail, sorry for the wall of text, it got longer than expected. In this case, we can use simplified ray optics and ignore the wave nature of light. This means, the light of the different mirrors or even pieces of mirrors just adds to one another. An important point, even if obvious, is that each point on the mirror surface can only have one orientation. Now we "select" the orientation of this point, we orient it in a way, that it reflects the rays from the center of the sun^1^ directly on the sun.^2^ But until now, we have ignored the rays which come from the rim of the sun. These rays start at a different position, namely the sun radius (695700 km). Due to different starting position, the rays have a different angle to arrive at the power plant, arcsin(sun radius / sun-earth-distance), which is 0.27°. Now we already oriented the mirrors parts in a way, that the rays from the center of the sun are reflected onto the satellite, but the rays from the rim of the sun come at an 0.27° differing angle. If the incidence of the ray on a mirror is changed by an angle, the outbound ray is also changed by the same angle. This leads reflected rays leaving in a direction 0.27° offset from the direction to the satellite. Assuming the satellite is at a height of 300 km and directly above, it is 300 km away, the smallest realistic distance. With this angle, it leads to a miss of plant-satellite-distance * sin(angle) leading to 1.4 km. This thought is valid for all points on the rim. Similarly, the rays between the rim and the center land between the satellite and 1.4 km off target. Hence the plant projects an image of the sun onto the satellite with a radius of 1.4 km.
^1^ Well they actually do not come directly from the sun, they still come from very close to the surface, but they seem to come from the center of the sun and for rays it is not important how far they have already traveled. We can just assume the sun is a disc.
^2^ If we assume the mirror is optimally shaped, we can reflect every ray, which seems to come from the center of the sun, perfectly on the satellite. Such a mirror would be part of an ellipsoid, with focal points at the center of the sun and the center of the satellite. In practice, it would be practically indistinguishable from a paraboloid with the satellite (deviance of 1.5 µm with a guessed plant size of 1 km). This is possible as the rays through the center of the sun falling onto the plant are, as you say, almost parallel.
I get what your saying now but it would be possible if you made the mirrors with a focal point at 300km and it would be less effective at different lengths but it shouldnt change all that much since you would only need like a couple seconds with that kind of intensity.