This is not intended to be a math question, so much as an understanding of how the current technology and techniques would view this problem.
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1$\begingroup$ Welcome to the site. This may get a better answer on Astronomy, but let's see. $\endgroup$– GdDJan 16 at 18:36
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3$\begingroup$ Satellites currently in geostationary orbit are a lot smaller than 1 mile across (more on the order of 10's of feet or meters) and are visible to the unaided eye under the right conditions. Something 1 mile cube should be easy to see by everyone even when it is black. $\endgroup$– JohnHoltzJan 16 at 19:36
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1$\begingroup$ @GdD This would most likely by off-topic on Astronomy.SE, since it does not appear to be about a natural object or an astronomical device. $\endgroup$– PM 2RingJan 16 at 23:48
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$\begingroup$ @JohnHoltz do you have a reference for this? I thought that all satellites visible to the naked eye were in very low orbits (e.g. starlink or the ISS) $\endgroup$– Oscar SmithJan 17 at 16:25
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1$\begingroup$ If I was to answer as to how to spot it, could you clarify what you mean by black. are we talking ultimate non-reflection here? or just normal matte-black colouring. Does it reflect laser and lidar? because if not the only way to spot it would be to spot what it blocks out (starlight, sunlight, planets, satellites passing behind it.) $\endgroup$– Anti Elon GuyJan 19 at 2:06
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3 Answers
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The initial detection would almost certainly be a ground based telescope that could no longer see a particular star (because there is a giant cube in the way, cf. occultation). From that point, once a bunch of people looked they would be able to triangulate to discover that there was an unexpected something and would be able to figure out roughly the size and orbit. At this point a geostationary satellite could be moved over to where the object was to get a direct image.
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1$\begingroup$ It's in GEO, which means it's moving at about 3 km/sec, and so would cover a star for about a half of a second. I think it would not be detected by occultation. A black object would get toasty pretty quickly and glow like crazy in thermal infrared, so I think one has to wait for a ground-based or space-based thermal infrared telescope to pick it up by accident. $\endgroup$– uhohJan 17 at 7:08
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$\begingroup$ even though it's moving at 3km/sec as long as it has a low inclination, that doesn't matter. There will be times in it's orbit where it will be almost perfectly still relative to decent chunks of the earth's surface. $\endgroup$ Jan 17 at 16:21
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5$\begingroup$ ...but not with respect to the stars, which if you point in a geostationary direction, are moving at 15 arcsec/s. A 1 mile cube at GEO is 7.8 arcsec wide, so we arrive once again at the same result and conclusion (but now in your rotating frame) that it "would cover a star for about a half of a second." So certainly if one were viewing enough GEO satellite tracking telescope footage pointed in a direction that included the black cube, one just might see a little break in the trail of a star and get curious. $\endgroup$– uhohJan 17 at 23:14
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$\begingroup$ It won't be detected at first but sooner or later someone trying to figure out the glitch in their instruments will find it. $\endgroup$ Jan 18 at 3:34
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1$\begingroup$ @uhoh It would likely be discovered by a technician at one of the satellite operators investigating a mysterious loss of uplink. A giant cube getting in the way between your ground station and your GEO slot is bound to cause trouble. $\endgroup$– TooTeaJan 18 at 19:33
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Remember, the moon is black, or at least almost as dark as fresh asphalt (the moon reflects about 7% of incoming light, compared to 5% for asphalt). It's just brightly lit against a truly dark background when we see it. So, if the cube were only as black as asphalt, I suspect it would be naked-eye visible, looking like a star that never rises nor sets but just stays at the same spot in the sky.
Now, if you mean Vantablack black, then you'd have to wait for occultation or IR visibility.
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1$\begingroup$ a freshly polished specularly reflective cube in space will also look black until one of its faces is oriented such that it reflects a light source in your direction. Of course over time, it will get roughened/pitted and gain a diffuse component. $\endgroup$– uhohJan 18 at 22:58
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1$\begingroup$ The only way that cube would be naked eye visible would be as a point source of reflected light. You aren't going to resolve any details or see a shape. Human eye resolution from ISS to Earth is about 100m, from a distance of about 400km. GEO is 36,000 km away... $\endgroup$ Jan 21 at 20:42
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$\begingroup$ I never meant to suggest otherwise. But I guess it's worth making that clear. $\endgroup$ Jan 22 at 3:39
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Once each day, the cube measuring 9.5 arc seconds, would pass across the sun once. With all the filters needed to view the sun through a telescope in place, you would have 2.16 mins to spot it every day.
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$\begingroup$ Including assumptions about the orbit, and please mention just how small the footprint would be on Earth - it will only pass in front of the Sun for a fairly small band of latitudes (if at all). $\endgroup$– uhohFeb 1 at 5:27
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$\begingroup$ Depends on the geosynchronous orbit. If the orbit is geostationary, it only passes across the Sun for about two months a year, and then only for sufficiently low latitudes. $\endgroup$– MarkFeb 2 at 3:35