# Smallest Black hole nessesary to devour Earth?

While looking into black holes I can across the idea of hawking radiation, and the fact that microscopic black holes would near instantaneously evaporate before doing any damage. However larger black holes have a lower temperature, and so would last longer before evaporating.

What is the minimum mass a black hole have to have in order for the rate of mass inflow to outpace the evaporation rate of the black hole, if it spontaneously appeared in a terrestrial environment. Does it make a significant difference whether it magically appears in the atmosphere, at ground level, or at the core of the earth? Are there any other factors that would have a significant impact on the required mass?

• Doesn't the Wikipedia page have an equation that says something along the lines of $dM/dt={\rm something}$? Wouldn't that "something" be the minimum mass accretion rate necessary to "outpace" the evaporation? – Kyle Kanos Mar 26 '15 at 2:21
• Closely related: physics.stackexchange.com/q/2743 – Kyle Kanos Mar 26 '15 at 2:25

For they layman (like me), you can run some numbers here:

http://xaonon.dyndns.org/hawking/

a 1 earth mass black-hole (about the size of a ping-pong ball) would radiate so little energy that it would easily destroy the earth, even from a low orbit. It might take some time to devour it, but no question, it would destroy and eat the earth.

a black hole about 1/2 the mass of the moon (this would have the size of a small grain of sand, about 1/10th of a MM in diameter), this would be about temperature stable in the backround radiation of the universe, it would add as much energy as it would radiate off, but on earth, it would see much higher temperatures and would add mass, and in time, devour the earth.

But we'd see the effects long before then. A black hole, 1/2 the mass of the moon would exert 3 G forces at 300 KM, so that that means is it would effectively tear apart an Arizona sized hole, 300 KM deep, wherever it was - now it wouldn't eat that matter right away, but it would pull it apart. Deeper in the earth the effect would be smaller, but the hole would move around with relative ease, basically tearing apart the earth as it moved.

If we look at something more manageable, say, G-force would be 1 G and 1 meter, so it would only be very locally disruptive, like if that fell on your house it would tear a hole through it, but could leave most of the house standing. A 1.5E11 KG black hole, about the mass of 50 empire state buildings. A black hole like this would radiate significant heat, temperature nearly a trillion Kelvin (link above), but passing through the earth it might absorb about as much as it gave off - that's kind of a ballpark guess. Something around that size, in the range of a 150 billion KG. A black hole that size would have a life of about 8 billion years in empty space, and it might be able to eat the earth if it was on the surface/in the core.

I think, somewhere roughly in that range. It's worth pointing out that a black hole that size, it it was in space, it would likely just fly through the earth, not got caught in the earth's gravity and the damage would be far less.

Also, they don't think primordial black holes exist. They've given up the search.

http://www.nature.com/news/search-for-primordial-black-holes-called-off-1.14551

Finally, where it appeared wouldn't matter as much as what it's orbit was. Inside the earth it would interact with more matter than in the atmosphere. Theoretically a black hole in the atmosphere but at orbital speed, could stay in a stable atmospheric orbit for some time. In orbit, far enough away from the earth so it wouldn't absorb earth matter, there would still be potential tidal issues. A moon mass black hole in orbit, 10 times closer than the moon (24,000 miles) would have 100 times the tidal effects than the moon currently has - the oceans would rise and fall about a couple hundred feet with each orbit. At 24,000 miles it would be close to a geosynchronous speed, so you'd only see the high tide every 5 days or so, but that much tidal force might make the earth close to unlivable. Earthquakes and Weather changes.

Hope my layman's answer isn't too wordy. Interesting question.