Black hole the size of an apple

Question

I was intrigued by this question: Would touching a black hole of a small mass (the mass of an apple) cause you to spiral in and get dead? until several answerers pointed out that a black hole with the mass of an apple would just have the gravitational pull of an apple and so would not be very interesting from a gravitation perspective. (Although it could be interesting in other ways, notably Hawking radiation.)

I felt a bit stupid not thinking of that myself but also realized that subconsciously I (and perhaps the original poster of the original apple question?) was picturing a black hole with the size rather than mass of an apple. So now I'm stuck with the followup question:

What would it be like to have an apple sized black hole somewhere here on earth, and what happens when you get too close to it?

Using the formula's in the accepted answer to the other question (and some fruit statistics I found online) I figured that the apple size black hole would have a mass of roughly 4.5 earth masses, live extremely long (ca 3.5 times the age of the universe) and be extremely cold (close to absolute zero) so that the horror scenario of that answer would not occur. But does that mean that everything round the apple black hole is nice and cozy, or am I being too optimistic?

Answer 1

Very roughly, the tidal acceleration would be of order $$a_{\rm tidal} \sim 2 \left(\frac{GM_{\rm BH}}{r^3}\right)l\ ,$$ where $l$ is the length of your arm and $r$ is how far away your elbow is from the black hole.

So say you got within touching distance, with $l \sim 0.5$ m and $r \sim 0.3$ m, and with $M_{\rm BH} \sim 2\times 10^{25}$ kg, then $a_{\rm tidal} \sim 5 \times 10^{16}$ m s$^{-2}$.

In other words, you would be shredded (spaghettified) by tidal forces before you got anywhere near the black hole.

Furthermore, even if you allow $r= 6400$ km and $l \sim 13800$ km, one gets a tidal acceleration of 140 m s$^{-2}$ (i.e. "14 g"), so the Earth would be tidally disrupted too.

Answer 2

In additon to the tidal disruption described by ProfRob, there would be accretion. Basically, most of Earth will be trying to get into an apple-sized volume. An Earth mass falling under 14g over several thousand km acquires tremendous amount of kinetic energy (hundreds of megajoule per kg or more). That flow and compression will heat things up. A lot.

The result is that the super-heated plasma radiates more and more until it reaches a brightness so the radiation pressure equals the gravity, at which point the mass flow chokes. The flow rate at this point produces roughly 10% of solar luminosity. Given the high temperature the output will be in the hard gamma ray range, converting remaining as yet unaccreted matter into plasma.

The timescale for this is basically the free-fall timescale for an Earth mass, about half an hour. After an hour Earth would mostly be a rotating cloud/disk of iron plasma, irradiated by the core accretion disk.