It is Planet X not Planet IX and they want our women! Get your tropes right. :-)
Does anyone know off hand the limit for reflected light from the Sun in terms of detection? Assuming a rockie planet is 'cold' and has a diameter 'd', at what distance is its magnitude so low that it is only detectable by star occlusion rather than reflected sunlight.
Not an astronomer, but let's try going by Wikipedia: Neptune has apparent magnitude of about 8. If you displaced it to twice as far, it'd become 16 times as dim (twice the distance, squared). The scale of apparent magnitude is a factor of 2.512 per step, so that'd be a change of about 3, to magnitude 11. The biggest ground-based telescopes can go up to around 24-26, if you dedicate significant observing time. So (26-8)/3=6 doublings: they could detect Neptune at 64 times as far out as it is, if they knew where to look. That's ~1900 AU.
But of course Neptune is bigger and likely more reflective than the target. I guess you could expect a brightness about one more magnitude lower. A quick skim of https://arxiv.org/abs/1601.05438 (the paper proposing the new planet) is difficult for me, but one concrete number they mention as a possibility is 700 AU. So, apparently, that'd be within reach of a ground-based telescope but it'd have to be a darned big state-of-the-art one.
Does anyone know off hand the limit for reflected light from the Sun in terms of detection? Assuming a rockie planet is 'cold' and has a diameter 'd', at what distance is its magnitude so low that it is only detectable by star occlusion rather than reflected sunlight.