Extra Planets – How visible would they be?

At 30 AU (30 times the Earth-to-Sun distance, about 4.5 billion km) Neptune has a visible magnitude of +8. Astronomical Visual Magnitude is a strange scale – it goes up by 5 for every 100-fold *decrease* in observed brightness. Thus the brightest thing in our sky (the Sun, of course) has an apparent visual magnitude of -26.75. The absolute visual magnitudes of stars is their magnitude as if they were 10 parsecs away. As a parsec or “parallax second” is defined by the IAU as 648,000/pi = 206,264.8 AU, 10 parsecs is 2,062,648 AU, at which distance the Sun’s magnitude *increases* by +2.5 x Log10(2,062,648^2) = 31.572 to +4.82. Notice that I’ve squared the parsec distance – that’s the inverse square law at work. For example, every doubling in distance, the light intensity decreases by 2^2 = 4 fold. When light shines on a planet, then travels back to us it’s an inverse quartic law – i.e. distance to the -4 power. Twice the distance, 1/16 the intensity observed.

In the case of Neptune, if it was closer to the Sun, it would thus reflect much more light – about 900^2 = 810,000 times more if it was 1 AU from the Sun and us. That’s a *decrease* in magnitude by 14.77 to -6.77 so Neptune would be brighter than everything except the Sun and full Moon.

Planet Nine averages a distance of 700 AU from the Sun and is about Neptune size. In that case it’s 13.7 magnitude points dimmer than Neptune’s current +8 at about +22. If it’s near its aphelion (furthermost position from the Sun) at a distance of ~1,200 AU, then it’d be ~40^4 times dimmer than Neptune, with a magnitude of +24. At its closest approach to the Sun (about 270 AU) then it’d be 9^4 times dimmer at magnitude 17.5.

Earth, at Neptune’s distance, would have a magnitude of +11, while a Mars sized planet would be +12.3. At Planet Nine distance Earth’d be +24.7 in magnitude, Mars +26 . A couple of extra planets have been suggested at distances of 213 AU and 329 AU. Thus if they were Earth & Mars sized they’d be +19.5/+20.8 at the first distance, and +21.4/+22.7 at the second. Currently the Pan-STARRS1 Sky Survey is trying to capture everything that can be seen from Hawaii down to a magnitude of +22. If the possible planets are in its field of view, then it’s likely to capture them. The survey has been gathering data every night (10 terabytes a night!) since 2010, and checks the positions of everything it sees from night to night. Everything that changes position and/or brightness is flagged by the computer system as an object of interest. Eventually Pan-STARRS hopes to push down to +24 in magnitude, which *might* capture Planet Nine, if Pan-STARRS sees its part of the sky.

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