I plan on using flat mirror projection, among other methods, to view Tuesday's partial eclipse. How do I calculate the distance my flat mirror has to be from my screen in order for it to cast a sharp, bright Sun image rather than its own shape or a blurred image?
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2$\begingroup$ The "throw" distance is ~100 times the diameter of the image. Your mirror should have an opaque mask over it with a small hole in it (~5 mm diameter). You really should test it beforehand, at approximately the same time of day as the eclipse. You will probably want to adjust the mirror angle over the course of the eclipse. $\endgroup$– PM 2RingOct 20, 2022 at 20:26
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$\begingroup$ This is a follow-on question to astronomy.stackexchange.com/q/50772/16685 $\endgroup$– PM 2RingOct 21, 2022 at 17:19
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1$\begingroup$ @PM 2Ring So the distance I need is about 100 times the size of the image I wanna get? Would that be the case without covering the mirror as well? $\endgroup$– איתי מרלובOct 21, 2022 at 18:01
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One time I put a "pinhole" in some paper, it was roughly 5 or 10 mm and roughly round (I think I just pushed a pen through it) placed the paper over a flat household mirror, inclined the mirror so that it reflected the Sun through an open window and into a darkened area indoors where the resulting image was projected on to a white wall.
To calculate the expected performance I'll suggest using the technique of similar triangles (see also Similarity system of triangles for fun) and the small angle approximation and take the limit of the distance of the Sun being infinite.
If the angular width of the sun $\theta$ is say 0.5 degrees, that's roughly 0.01 radians. That means that the width of the sun's disk at a distance $L$ from a pinhole or any small aperture (transmission or reflection) will be the product $L \theta$. If $L$ is 10 meters, then the height of the Sun's disk will be about 10 cm.
To get enough light through your aperture, it needs to be big enough. I recommend you have a series of apertures of different sizes available or some manipulable material like a simple sheet of paper where you can poke various size holes in it until you get satisfactory brightness.
Since rays from the Sun are essentially parallel, "fuzziness" of the image will simply be the aperture size $w$, independent of distance.
So for a 0.5 cm aperture at 10 meters, you have a 10 cm solar disk which is "fuzzy" by 0.5 cm.
Solar eclipse viewing via apodized mirror projection:
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2$\begingroup$ Using this, how far away from the screen should a 5-7 centimeters flat mirror be in order to provide a sharp image? $\endgroup$ Oct 21, 2022 at 17:19
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1$\begingroup$ @איתימרלוב As uhoh's last paragraph says, the fuzziness depends on the aperture size. So if you don't cover the mirror, the image will be quite fuzzy, and you will need to use a huge distance to compensate for that. $\endgroup$– PM 2RingOct 21, 2022 at 18:36
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$\begingroup$ @איתימרלוב if you add a specific mirror to your question, I can update the answer to address it. That sounds too big unless you have a really long distance, and of course someone to keep moving the mirror. I'd recommend that you just give it a try on a normal sunny day and see what happens - that will likely be an "Aha!" experience, and you'l start using a mask over the mirror (paper with a hole in it) to improve the image sharpness at the cost of a dimmer image. $\endgroup$– uhohOct 21, 2022 at 20:28
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$\begingroup$ @איתימרלוב You'll also notice that you'll probably want to shade the screen from direct sunlight. Reflecting indoors through a door or window works best. $\endgroup$– uhohOct 21, 2022 at 20:28
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2$\begingroup$ The mirror I currently have is 5 centimeters in diameter. I tried it with a distance of 10 meters and beyond and it seemed amazing, but of course I couldn't really tell from this far away from the casted image. I made some covers with varying apertures to test, but I need this to work at school so I'm not sure. $\endgroup$ Oct 21, 2022 at 20:46