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What effects would occur if the earth's core goes cold? Would the planet stay liveable after this happens?

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Once the core has solidifed it can no longer generate a magnetic field. There may be some frozen in field, but I would guess that the strength of the magnetic field with decrease as more and more of the core freezes, so any residual field is likely to be small. The core of Mars is thought to be frozen and it's magnetic field is negligable.

Anyhow, the main problem would be that without the magnetosphere to protect it the atmosphere will gradually be stripped by the solar wind and we'll be left with nothing to breathe. In addition, without an atmosphere the surface will be subject to wide temperature swings; just as the Moon is in fact.

So no, by any reasonable definition of the word liveable once the core has frozen the Earth will not be liveable. I wouldn't worry for a while yet though.

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It is not necessary for the Earth's core to freeze for the geomagnetic field to collapse. The geomagnetic field is a chaotic phenomena and has reversed polarity many times in the Earth's history. The question is why are we so fortunate to have a persistent geomagnetic field at all? I believe Mars has a molten core, but a much weaker magnetic field than Earth. Mars lacks a natural satellite as large as Earth's Moon. What role has our Moon played in creating a persistent geomagnetic field? There's a worthy geophysical question! –  Mark Rovetta Mar 2 '13 at 2:02

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