I am in space. What is the maximum distance that I could be from Neptune and see it with the naked eye. You can choose the optimal position. Also, assume that I have 20/20 vision.
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4$\begingroup$ What is the point of the question? Are to doing any calculation or is it a worldbuilding question? $\endgroup$– Nilay GhoshApr 12 at 1:57
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$\begingroup$ Curiosity @NilayGhosh $\endgroup$– Starship - On StrikeApr 14 at 10:33
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1 Answer
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Fortunately, Neptune is in an almost circular orbit that is almost in the same plane as that of the Earth. The NASA planetary fact sheet has Neptune at an apparent magnitude of 7.7 when it is $4.35\times 10^9$ km from Earth and the Sun, Earth and Neptune are in line.
If you were to move closer to Neptune along that line, it would grow brighter (but still appearing as a point source to the naked eye) as the inverse square of the distance to Neptune. Thus $$7.7 - m = 5\log_{10}\left(\frac{4.35\times 10^9}{d}\right)\ ,$$ where $d$ is the distance you are looking for (in km) and $m$ is the apparent magnitude you think you can see with the naked eye - typically, $m=6$ for good vision in a dark sky.
This equation can be rearranged as $$ d = 4.35\times 10^9\times 10^{(m-7.7)/5}\ {\rm km}. $$
e.g., for $m= 6$, $d = 1.99\times 10^9$ km.
Possibly you can do a bit better than $m=6$ out of the Earth's atmosphere. For $m=6.5$, $d = 2.50\times 10^9$ km.
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$\begingroup$ That means Uranus comes close enough if it gets into proper alignment, so you might see Neptune from hovering above Uranus or a Uranian moon. $\endgroup$ Apr 14 at 23:20