[batarjal]

There a lot of talk about the idea of an Earth mass Primordial Black Hole punching through Earth and the effects of such an event. However, Planet nine is predicted to be about five Earth masses, not one.

I don't have a physics degree, but how close would Earth have to get to such an object before the Earth is within that object's Roche limit?

[semi-extrinsic]

The Roche limit depends on the density difference between the two bodies in question. For a black hole (of any size), the density is basically infinite compared to regular planets, so the Roche limit is basically zero. As in, less than one meter.

The radius of the event horizon for a black hole even the size of Jupiter is only 2.5 meters.

[CodesInChaos]

The gravity field of a spherical object only depends on its mass and not its density/radius. So for the Roche limit only the density/radius of the object at risk of breaking apart (called minor object on wikipedia) (earth) should matter, not the one of the major object (black hole).

Wikipedia gives the formula d=R_m*(2m_M/m_m)^{1/3} which matches that intuition.

[semi-extrinsic]

Ah, you're right.

So plugging in numbers you get 2.15 times the radius of Earth, which becomes 13 750 km.

For comparison, Earth-Moon distance is approx. 400 000 km.

[perl4ever]

Yes, but you don't have to be within the Roche limit to have substantial tidal effects, which could destroy habitability at a much greater distance. Just look at Io.

[taneq]

Very close, in astronomical terms. At a guess, no further than the distance from Earth to the Moon, and probably much less. I'd love to see real numbers though. :)

Edit: According to https://en.wikipedia.org/wiki/Roche_limit the Roche limit d = Rm * (2 * MM/Mm)^(1/3) for Rm = radius of the satellite, MM = mass of the primary, and Mm = mass of the secondary. So in this case d = 6380 * (2 * 5/1)^(1/3) = 13,745km. In astronomical terms, very nearly a bulls-eye.