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Theoretical Planet Nine, proposed by Caltech's Mike Brown and Konstantin Batygin, is said to have a 15-20 thousand year orbit. Approximately how bright would the sun appear from Planet Nine aphelion and perihelion?

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Between $1/40,000$ and $1/8,000,000$ of the brightness as seen from Earth, depending on what the actual orbit would turn out to be, and where the planet is in its $15,000$ year orbit period.

Brightness drops as $\dfrac{1}{r^2}$ with distance from the light source. Earth is at $1~\textrm{AU}.$ The theoretical planet is at $200~\textrm{AU}$ when it's closest to the sun, and up to $2800~\textrm{AU}$ at the point furthest from the sun at the upper end of the estimated orbit.

So e.g. $1/200^2 = 1/40,000$ of the brightness (luminance) as seen from Earth.

For how a human would experience it, we can convert to exposure value as used in photography: The difference in exposure value (photographic 'stops') is $\log 2$ of the luminance ratio, so we would have $15$ to $23$ stops less light than on Earth. Sunny noon on Earth is $15~\textrm{EV}.$

So the brightness at noon on the planet surface would be:

  • At $200~\textrm{AU},$ planet orbit is closest to the sun: About $0~\textrm{EV},$ roughly the same as a dimly lit interior

  • $400~\textrm{AU},$ lower bound on semi-major axis: $-2 ~\textrm{EV},$ similar to a landscape lit by the full moon

  • $1500~\textrm{AU},$ upper bound on semi-major axis: $-6~\textrm{EV},$ similar to landscape lit by a quarter moon

  • $2800~\textrm{AU},$ upper bound on aphelion (the point on orbit most distant from the sun): $-8~\textrm{EV}.$ This would be dark, but you would probably still see enough to avoid running into things.

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One other important brightness milestone is 700AU. That's when the sun would appear as faint as the full moon. With dark adapted eyes it's still barely possible to see colors under the full moon, although they appear very washed out. – Dan Neely Jan 21 at 13:50
Thinking a little more, At its brightest Venus is 1 billion times dimmer than the Sun. In really dark locations it's apparently bright enough to cast a shadow; so you'd definitely be able to see something at 2800 AU. – Dan Neely Jan 21 at 13:56
@DanNeely Yes, at 2800 AU with dark adapted eyes, you should be able to see just fine for navigation purposes. But no colors, and no reading. Already at 200 AU we're into mesopic vision with reduced color perception. A bit beyond 2000 AU we should be purely monochromatic. (Barring mistakes in my calculations.) – j-g-faustus Jan 21 at 14:25

If it exists, then the orbit has a perihelion of maybe 300 au and an aphelion of perhaps 2700 au.

You can then just scale from the brightness $m=-26.7$ of the Sun at the Earth. It would be between 12.4 and 17.1 magnitudes fainter. So still much brighter than the next brightest star in the sky.

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+1 because I think it's more useful to compare the brightness to distant stars as seen from Earth. I see that this would make the Sun still much brighter than Venus appears on Earth ($M\approx-4.6$, according to Wikipedia). – Warrick Jan 21 at 12:49

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