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This is how Stellarium depicts the positions of the Earth and Sun 24 hours before the solar eclipse (from Lunar perspective) on Jan. 13, 2131

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A close up of the Earth on that day (Jan 12) shows no indication of any crescent Earth visible.

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On Jan. 11, there is the barest of indications of a crescent Earth visible.enter image description here

On Jan. 10, more of a crescent Earth is depicted. enter image description here

Is there anyway of telling the accuracies of these depictions? Is there reason to think the human eye could see something of the Earth or its atmosphere on January 12, or would it appear entirely black?

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While a black disk might not be entirely accurate, it is not a too bad approximation. Better would be using orbital night-time images of Earth.

There are images in the NASA archive which show Earth at night, the programme is titled "The Black Marble" which depicts Earth at night. There's a video which covers 22 days. The actual image depends on atmospheric conditions and on which side Earth shows - you might see some lights of civilisation. Similar to new moon conditions where you can also see features on the moon, an observer on the Moon would see some features on the Earth's surface. Additionally, maybe some of the refracted light of the atmosphere or airglow or aurorae is visible - but with the atmosphere being less than 1% of Earth radius, that's not much in comparison to the disk of Earth in the observer's FOV; yet it might be enough to be detectable, depending on sensitivity of the observer.

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