Have we been able to see the shadow on the Moon's surface of any artificial satellites that orbit the Moon?
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Possible duplicate of Picture of equipment left on the Moon? – Hobbes Jan 20 '19 at 19:28
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No. Just like the lunar landers are too small to see from Earth, satellites shadows are too small. – Hobbes Jan 20 '19 at 19:29
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The short answer is no.
Any artificial satellites around the moon is too small and too far away to cast a shadow. The angular diameter of the sun from the moon is approximately 0.5°.
This means that a satellite has to have an equal or greater angular diameter observed from the moon than the sun. The formula for angular diameter is:
$$\delta =2\arctan({\frac{d}{2D}})$$
where $\delta$ is the angular diameter, $d$ is the diameter of the object, and $D$ is the distance from the object to the observer. Using the Lunar Reconnaissance Orbiter as an example, it has a periselene of around 20 km.
Substituting $\delta=0.5$ and $D=20 km$:
$$0.5° =2\arctan({\frac{d}{40km}})$$ $$\tan(0.25°) = \frac{d}{40km}$$ $$d = 40km \cdot \tan(0.25°) = 0.175km = 175m$$
Unless you can find a 175 meter wide artificial satellite around the moon, I'm afraid we won't be seeing shadows anytime soon.
QiLin Xue
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2@OscarLanzi Thanks, I edited it to periselene instead, which refers specifically to the moon. – QiLin Xue Jan 20 '19 at 23:03
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Great answer! This diagram (found in this answer) may also be helpful here. https://en.wikipedia.org/wiki/File:Solar_eclipse_types.svg – uhoh Jan 20 '19 at 23:19