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Let's say an alien race drops a microscopic 72135 metric ton black hole into the center of the earth. It's Schwarzchild radius is too small (only 0.0001071103 femtometers) so it won't actually be able to interact with any particles inside the earth and it won't accrete any mass.

Instead it should falls straight through the earth, ping-ponging back and forth before reaching a critical mass and exploding in a burst of hawking radiation exactly 1 year later. However this wouldn't be enough to blow up the Earth because the binding energy of the earth is $2.24 \times 10^{32}$ J and during the final second of the black holes life it will produce only $2 \times 10^{22}$ J.

Edit: From my own calculation it seems impossible to actually blow up the Earth with a black hole, so I'm changing this question to "liquefy."

How big of a microscopic black hole would you have to drop inside the Earth for it to have enough energy to liquefy the surface from energy dumped into the interior. Let's assume it reaches the critical last second while it's deep inside the Earth (not passing back and forth at the crust level) It dumps so much energy into the interior that the surface of the planet erupts in magma and the crust is liquefied.

Edit: This is the calculator I'm using: http://xaonon.dyndns.org/hawking/

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    $\begingroup$ Related: physics.stackexchange.com/q/424929/2451 , physics.stackexchange.com/q/305536/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Jul 14, 2019 at 7:09
  • $\begingroup$ Ask an Astronomer (Cornell University) says that a billion ton BH would never eat the earth. $\endgroup$
    – Keith McClary
    Jul 15, 2019 at 5:15
  • $\begingroup$ By my calculation, you would need 2.5e12 tons of BH to dump 2.24e32 Joules into the earth. Unfortunately that would take 4e12 years to happen and we would all die of boredom first. Maybe the real solution is to dump a series of BH into the earth. (or change the problem from blow up to liquefy) $\endgroup$
    – SurpriseDog
    Jul 15, 2019 at 5:18
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    $\begingroup$ Earth already is largely liquid. Dobyoi mean liquefy its crust? $\endgroup$
    – my2cts
    Jul 16, 2019 at 6:19
  • $\begingroup$ @my2cts Yes. How large would a micro black hole have to be to liquefy the crust from an explosion within the Earth. $\endgroup$
    – SurpriseDog
    Jul 16, 2019 at 17:24

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Hawking radiation is the same for every black-hole in the last second of its existence regardless of initial mass. The energy released prior to this last second is minuscule to what comes afterwards. The integration of power over a human time-scale will not be more than the binding energy of the earth and so a single black-hole is not capable of destroying the earth if below interaction size. https://en.wikipedia.org/wiki/Hawking_radiation

If you are trying to destroy the earth via black-hole, a. the black-hole must be a radius where absorption of matter through interaction is faster than matter released through radiation so as to eventually create a world destroying accretion disk, b. the black-hole must have many little friends all timed to evaporate in a period of time short enough that heat cannot just be dissipated away; you will need around 10,000,000,000 of the little suckers to explode at once.

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  • $\begingroup$ 100% of a black hole's mass is converted to hawking radiation. If we start out with a black hole of 10,000 metric tons, then 97% of it's mass will be converted into energy in that final day before it reaches the critical second at 228 tons. All of that energy goes into heating the earth. $\endgroup$
    – SurpriseDog
    Jul 15, 2019 at 5:12
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    $\begingroup$ @Benjamin Yes 100% gets converted. But the rate of conversion is dependent only on it's current mass. And so if you make it a bit bigger, it will take a very very long time to radiate away it's mass until it's small enough for the "explosion". And that explosion will be the same size. In the mean time, radiating its mass to get down to that size will occur so slowly that the earth can absorb that additional heat and radiate it away safely. $\endgroup$
    – Shufflepants
    Jul 15, 2019 at 15:18
  • $\begingroup$ @Shufflepants A 10,000 metric ton black hole has a lifetime of approximately 1 day so yes it will dump 97% of it's energy into the Earth before reaching the critical last second. (Not a "very very long time") $\endgroup$
    – SurpriseDog
    Jul 15, 2019 at 16:40
  • $\begingroup$ @Benjamin Right, but that 10,000 ton BH only dumps 9x10^23 joules into the earth which is no where near the binding energy of the earth. And making it bigger will not help because making it bigger will only delay the time it takes to reach that critical state, and almost all mass energy expelled will be expelled very slowly up to that point. $\endgroup$
    – Shufflepants
    Jul 15, 2019 at 17:35
  • $\begingroup$ @Shufflepants I realize now that it's impossible, so I edited the question to "liquefy the crust" $\endgroup$
    – SurpriseDog
    Jul 17, 2019 at 18:14

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