# In the absence of a heat source how long would it take for the earth to freeze solid?

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Let’s try a thought experiment; of course you can use mathematics if you want to also.

Let’s remove the earth and place it in another universe that is absolutely void and lacking energy of any type, absolute cold at absolute zero.

How long do you think it would take the entire earth from atmosphere, oceans, crust right down to the very core to freeze solid (This might not be the right expression) or put another way, reach the now almost the absolute zero of the empty universe it now inhabits?

Alan

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Let's try a thought experiment; of course you can use mathematics if you want to also.

Let's remove the earth and place it in another universe that is absolutely void and lacking energy of any type, absolute cold at absolute zero.

How long do you think it would take the entire earth from atmosphere, oceans, crust right down to the very core to freeze solid (This might not be the right expression) or put another way, reach the now almost the absolute zero of the empty universe it now inhabits?

Alan

Absolute zero is unreachable (third law of thermodynamics) in our universe and I do not know how dissipation would work in an hypothetical universe without third law. Do you?

Average temperature of outer space is about 3 kelvin. If you move the Earth to a place without Sun or similar source of heat. It would achieve about 10-100 kelvins in 102 days or something as that. Earth would not achieve 3 kelvin due to internal heat. Of course in geological scales of time, volcanoes and other geological stuff would finally stop and then Earth would achieve equilibrium with outer space.

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Absolute zero is unreachable (third law of thermodynamics) in our universe and I do not know how dissipation would work in an hypothetical universe without third law. Do you?

Average temperature of outer space is about 3 kelvin. If you move the Earth to a place without Sun or similar source of heat. It would achieve about 10-100 kelvins in 102 days or something as that. Earth would not achieve 3 kelvin due to internal heat. Of course in geological scales of time, volcanoes and other geological stuff would finally stop and then Earth would achieve equilibrium with outer space.

Thanks for the reply but it does not really answer my question, I know absolute zero is unreachable, thus I said in my post 'Almost Absolute Zero", please read it again.

In the theoretical totally void universe in which I put the earth , absolute zero is possible, of course "until you put our earth into it", which then will become its only heat source. Given infinite time the earth in this universe the avarage temperater of the earth would and must approach as close as the 3rd law of thermodynamics allows but in countless trillions of earth years. But it must in the end freeze solid or reach equilibrium in the proposed hypothetical universe.

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Thanks for the reply but it does not really answer my question

I think that your question was "In the absence of a heat source how long would it take for the earth to freeze solid?". Under the assumptions and limitations stated in my post my answer is

in 102 days or something as that.

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It would achieve about 10-100 kelvins in 102 days or something as that.

Only for the surface, but I think it'd take a whole lot longer than that.

Temperatures would plunge fastest over continents with freezing conditions in only a few days temps at the coast would be fairly stable until the oceans froze over. This could take a month or so but once a thin layer of ice has formed, the process will accelerate again as the oceans heat reservoir is cut off.

As the earth cools down and all water and CO2 precipitates out of the atmosphere, the rate of cooling will again slow down. Not only because the radiative losses depend on T4 but because the major source of heat will be deep geological heat. simple conduction through the crust. It will be enough to stop the atmosphere precipitating as it is sufficient to maintain a ~90K surface temperature (although the atmosphere may be able to keep it warmer than this.)

The geological heat is primarily from radioactivity K-40 is an important isotope. it has a half life of 1.25 Ga (billion years) so complete cooling is going to take longer than that. After about a billion years, then you might see the atmosphere start precipitating leaving only trace quantities of hydrogen and helium as a tenuous atmosphere (it won't escape like it currently does as it will be too cold) Infact, the longer we go on the thicker the atmosphere is likely to become again. as more and more isotopes decay, the helium will eventually diffuse to the surface. even if we leave it long enough that everything has decayed and all geological temperatures have reached <1K we will still have a thin helium atmosphere and helium oceans. not likely to be rivers though as everything will be at a standstill.

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Only for the surface, but I think it'd take a whole lot longer than that.

Yes, you are right. I omitted to quote this part:

Of course in geological scales of time, volcanoes and other geological stuff would finally stop and then Earth would achieve equilibrium with outer space.
Edited by juanrga
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Only for the surface, but I think it'd take a whole lot longer than that.

Temperatures would plunge fastest over continents with freezing conditions in only a few days temps at the coast would be fairly stable until the oceans froze over. This could take a month or so but once a thin layer of ice has formed, the process will accelerate again as the oceans heat reservoir is cut off.

As the earth cools down and all water and CO2 precipitates out of the atmosphere, the rate of cooling will again slow down. Not only because the radiative losses depend on T4 but because the major source of heat will be deep geological heat. simple conduction through the crust. It will be enough to stop the atmosphere precipitating as it is sufficient to maintain a ~90K surface temperature (although the atmosphere may be able to keep it warmer than this.)

The geological heat is primarily from radioactivity K-40 is an important isotope. it has a half life of 1.25 Ga (billion years) so complete cooling is going to take longer than that. After about a billion years, then you might see the atmosphere start precipitating leaving only trace quantities of hydrogen and helium as a tenuous atmosphere (it won't escape like it currently does as it will be too cold) Infact, the longer we go on the thicker the atmosphere is likely to become again. as more and more isotopes decay, the helium will eventually diffuse to the surface. even if we leave it long enough that everything has decayed and all geological temperatures have reached <1K we will still have a thin helium atmosphere and helium oceans. not likely to be rivers though as everything will be at a standstill.

We are close to agreements lets just guess

Total freeze of entire earth right down to inner core.

A million years?

A hundred million years?

A billion years?

A trillion years?

Unimaginably longer? trillions of trillions of years?........"my guess"

Alan

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It depends on exactly how close you want the earth to get to equilibrium. and if effects such as proton decay exist.

well, the longest known halflife is Tellurium-128 with 2.2septillion years (2.2*1024 years) so to get rid of all of it your looking at something like 70 septillion years before all heat sources disappear although proton decay is estimated to be ~1030 years and could keep the earth at around 1K

you could be looking at something truely massive like ~1035 because with proton decay, you'll get beta decays of the excess neutrons. the earth would just evaporate away in a cloud of gamma rays and then it wouldn't have a temperature.

but really, it all depends on how close to absolute zero is 'close enough'

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The geological heat is primarily from radioactivity K-40 is an important isotope. it has a half life of 1.25 Ga (billion years) so complete cooling is going to take longer than that.

As the earth approaches absolute zero, will that have any impact on radioactivity? Or does radioactivity remain constant regardless of temperature?

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As the earth approaches absolute zero, will that have any impact on radioactivity? Or does radioactivity remain constant regardless of temperature?

Theoretically it would exist a temperature dependence, but it has to be extremely small. Current measurements in a span 20--300 K show not variation up to the 0.1%

http://prc.aps.org/abstract/PRC/v80/i4/e045501

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That particular variation is only expected for one type of radioactivity- electron capture.

For other types the rate wouldn't be affected, even down to absolute zero (though getting it hot enough would influence the rate).

Fundamentally, since the earth is warm, it is a source of heat and the original question is poorly defined.

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Let's try a thought experiment; of course you can use mathematics if you want to also.

Let's remove the earth and place it in another universe that is absolutely void and lacking energy of any type, absolute cold at absolute zero.

How long do you think it would take the entire earth from atmosphere, oceans, crust right down to the very core to freeze solid (This might not be the right expression) or put another way, reach the now almost the absolute zero of the empty universe it now inhabits?

Alan

It could possibly take not too long... space is very cold, and heat tends to leave a system quite quickly... The Earth's core is very hot. Because of this, this is a very hard question to answer without employing the relevant equations describing a thermodynamical runaway of heat. I don't know of any equations which have properly tackled this, but I do not doubt there have been attempts.

For a system to be efficiently hot, which runs into an environment which is cold, is what the 2nd law of thermodynamics is all about. Space is very cold... which is about -270 celsius. (If my memory serves)

(Got my Kelvin and celsius mixed up there... been a while since I read up on the temperature of space)

What can be quite fascinating, is that some of the coldest environments in the universe, can be found in your local lab

Edited by Aethelwulf
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Classically, it would never reach absolute zero.

The colder it got the more slowly it would cool.

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Yes, I should have pointed out, an entire system will never reach absolute zero... not even a macroscopic one. No entire system, being made of smaller constituents can completely freeze that the constituents have zero kinetic energy. If it did, a system would never thaw out.

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Yes, I should have pointed out, an entire system will never reach absolute zero... not even a macroscopic one. No entire system, being made of smaller constituents can completely freeze that the constituents have zero kinetic energy. If it did, a system would never thaw out.

I know this and pointed it out in one of my replies, I said reach "equalibrium" or "almost absolute zero" so that remains the question. How long do you guys think it will take?

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Let’s try a thought experiment; of course you can use mathematics if you want to also.

Let’s remove the earth and place it in another universe that is absolutely void and lacking energy of any type, absolute cold at absolute zero.

How long do you think it would take the entire earth from atmosphere, oceans, crust right down to the very core to freeze solid (This might not be the right expression) or put another way, reach the now almost the absolute zero of the empty universe it now inhabits?

Alan

That's a very complicated problem to solve. It depends on the thermal properties of the various kinds of matter which the earth is made up and how that matter is distibuted It's possible to solve it but IU sure don't know how to do it. One needs to be skilled in thermodynamics to solve a problem like that and thermodynamics was always my worst subject.

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I was curious about this question so I attempted to google it. It's amazing how many times this has been asked across many question answer forums and science forums. So far i have not found any realistic estimates, most simply say billions or trillions of years or say it will never happen because the sun will consume the earth far before it could happen...

Evidently a very difficult question to even estimate much less answer exactly...

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Nobody can give an exact answer, because such answer does not exist. Decay and other processes are not deterministic processes, You cannot say that all nuclei will decay exactly in a time span t. Moreover, there are uncertainties regarding the amount of each material in the Earth, uncertainties regarding the different transport coefficients involved. We are dealing with a complex thermodynamic process where you cannot use a single equation and extrapolate. As absolute zero is unreachable, the OP asked for "Almost Absolute Zero", but it depends on what one means by "almost" 0.01 K? 0.0001 K? 0.000000000000001 K? The time needed to achieve the first temperature is very different than the time needed to achieve the latter.

There are many more complexities. What about fluctuations? When dealing with a macro system at 300 K during a relatively small span of time (small when compared to geological scales), people usually avoids fluctuations and only considers the average behaviour (average temperature). But near zero quantum fluctuations become more important (recall that the classical limit on partition functions can be obtained by taking the infinite temperature limit) and we also know that the probability for a giant fluctuation to happen is a function of time. In a very very large span of time it is likely that a giant fluctuation will happen invalidating any 'prediction' made using dynamical equations as the equation for the decay of a nucleus

$\frac{dN}{dt}= -k N$

The best answer that I gave was about 102 days for the surface and geological scale of time for total planet (but uncertainty for this latter case is so high that I did not write an specific figure).

Edited by juanrga
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I suppose the question summarizes as to evaluate how long it takes for all heat on Earth to dissipate in the form of radiation (no convection, no conduction, no mass transfer).

Since there is no mass transfer, my guess for the answer is: never.

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How can the core of Earth ever lose all its' heat. Doesn't the pressure of gravity constantly create heat?

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How can the core of Earth ever lose all its' heat. Doesn't the pressure of gravity constantly create heat?

Heat is energy in transit and the amount of energy in a finite volume is finite.

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How can the core of Earth ever lose all its' heat. Doesn't the pressure of gravity constantly create heat?

Note: The term you're looking for is thermal energy, not heat.

The thermal energy due to pressure would eventualy dissipate. The thermal energy due to pressure is from the process of compression. That's a finite process.

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How can the core of Earth ever lose all its' heat. Doesn't the pressure of gravity constantly create heat?

This is a common misconception. It will only get hotter if it is constantly moved to a greater state of compression. if you stop the compression, say by the core material being as squashed as possible by the mass above it, then there is no more energy going into the system to make it hotter. if the system is not thermally isolated (the core of the earth isn't) then the thermal energy will dissipate away.

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Thanks for the reply but it does not really answer my question, I know absolute zero is unreachable, thus I said in my post 'Almost Absolute Zero", please read it again.

In the theoretical totally void universe in which I put the earth , absolute zero is possible, of course "until you put our earth into it", which then will become its only heat source. Given infinite time the earth in this universe the avarage temperater of the earth would and must approach as close as the 3rd law of thermodynamics allows but in countless trillions of earth years. But it must in the end freeze solid or reach equilibrium in the proposed hypothetical universe.

Depends on how close to equilibrium you will accept.

Or...

A background radiation will be created given long enough if the Universe is small enough (closed system). If this Universe is contracting at some point exact equilibrium could happen.

Edited by J.C.MacSwell
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This is a common misconception. It will only get hotter if it is constantly moved to a greater state of compression. if you stop the compression, say by the core material being as squashed as possible by the mass above it, then there is no more energy going into the system to make it hotter. if the system is not thermally isolated (the core of the earth isn't) then the thermal energy will dissipate away.

Interesting. That means that if Earth got thrown away from the Sun and froze in deep space, and if someone returned to the frozen Earth, and they tunnelled into it all the way to the core, there would be no heat anywhere, including the center of the core.

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