[Alan McDougall (OP)]

Hi,

What is the avarage temperature of the whole earth, not surface, but avarage from core to surface?

If the earth were placed far beyond the influence of the sun, how long would it take to cool off into cold solid rock?

Alan

[Don_1]

Take a trip to the UK and you'd be forgiven for thinking the earth has already been moved far from the sun.

Would it ever cool to such an extent? Or would the pressure at the core maintain it's heat?

[syhprum]

This I think is the most challenging question on this board that I have seen for some time, I will put forward my initial ideas.
Without the insulating effect of the atmosphere and oceans the surface temperature of the earth would be about 255°K and I am guessing that heat leaking up from the hot interior adds another 5°.
In the absence of the sun I estimate it would quite rapidly drop to about 8°K but then as the internal heat from the initial compaction and the radioactivity in the core dissipates it would drop to the CMBR temperature which in turn fades away as the universe expands.
I think the time scale would be in the order of hundreds of billions of years.   

[Alan McDougall (OP)]

syhprum

Thanks lets see what the others think

[Soul Surfer]

The residual radioactivity is a significant energy source and without it the earth would cool down in a few million years but the radioactivity will keep it hot for thousands of millions of years

[Alan McDougall (OP)]

Well the atmosphere would freeze in layers from hydrogen at the surface to the heaviest gas at the bottom.The huge quantity of water in the oceans would also freeze making a sort of igloo effect.

Then the almost absolute zero of space would at first freeze the crust of the earth and the billion year battle between the heat in the core and mantel trying to dissipate into the cold of the universe, entropy must win in the end even if it takes ten thousand billion years

But to get back to my first question guys, what is the present temperature of the entire earth, core mantel, crust oceans and land continents when all are mixed into into a very hot doe like cake ?  I think maybe we should imagine mixing and breaking the whole earth and doe and then hypothetically mix and then put in our thermometer

(I forgot how to spell DOE doe Doe??)

Alan

[syhprum]

I think a computation of the average temperature of the Earth could be done by dividing it into separate volumes starting with the core at say 10,000°K with the temperature falling at a uniform rate to the 300°K of the outer layers.
This would only be a first order approximation as allowance for the varying thermal capacity and conductivity between layers should be made.

Alan try the english 'dough'

[Alan McDougall (OP)]

syhprum 



Your approach is a good one but will only lead to an approximation and maybe that is the best we could hope for.

Alan try the english 'dough'

Heck just after posting I went to bed and the correct spelling popped into my silly head Oh!! I don't spend much time is the kitchen and that is my best excuse  [:I] [:I] []

[Democritus]

Most impressive question. Alan, I'll give it a shot.
To make things easier for me at least, I'm going to make some assumptions and idealise some circumstances.

Firstly, take a 2d cross section of the spherical earth (forget 'oblate asymmetrical' for a minute) thru the centre. So now we have an enclosed circle (2 dimensional) of earth stuff. This is not hard to imagine, after all a sphere is but a circle rotated thru a diameter...

Assume the circle of earth stuff is homogeneous in temperature conductivity, matter and density (which it isn't, but what the heck) and static ( which is to say forget convection effects).

Now, plug in some specs.
Very roughly the Earth's radius is 6,000km.
Very roughly the Earth's core temperature has been estimated/conjectured at from 5,000C to 7,000C, with many other estimates either side.
Very roughly the surface of the Earth is 0C, plus or minus about 50C, and that just below the surface, a 100m, a km or two it may be a tad hotter.
Very roughly, well, let's forget atmospheric/solar radiation in/terrestrial radiation out/oceanic & other influences, because we can.

So, we have a circle (or rather a 2d disc if you like, of earth stuff) of radius 6,000km with a temperature gradient in degrees C of 6,000 at the centre to 0 at the surface. (The two 6,000s are a mere coincidence of different measuring sticks and are not significant.)

Assume the temperature gradient is linear, which is to say that the temperature drops off at the same rate per kilometer from center to surface.
 
So, what and where is the average temperature of the circle? That is the question.

My conjecture is that..curses, I've been called away for a few days. Hope to get back to this then, or rather hope more you all have solved it by then. 

(Just scribbling in my notebook ...I have a proof in mind but my margin is too narrow to include it here. But .707 comes also to mind for some reason...)

Just a shot
Best wishes
Democritus
       





   

 
 

[Alan McDougall (OP)]

Democritus

Very impressive effort. Strangely I have never heard this question posed in any of my physics or science community of at University . Surly little me has not asked an original question?

However, this will not stop me having a bash at it like you did

Thanks

Alan

[Alan McDougall (OP)]

What about an approach something like this?

  I have got to thinking how to solve the problem. I recently saw the documentary of the amazing huge ship the Queen Mary 2. They did extensive testing with an exactly scaled down model in a huge simulated ocean tank.

If we knew exactly the constitution of the earth from core, mantel to crust and made a model as exact as possible, maybe a billion trillion times smaller that the earth with the exact same density.

Would the weak gravity of the earth make any difference to heat flow in our model?, I think not.

We could then put our hypothetical model in a vacuum as near to absolute zero as possible. Take temperature readings from sensor probes we had placed at various depth with the model. Wait until all the heat is dissipated by the process of entropy, measure the time accurately and multiply extrapolate the time by a billion trillion and get our estimate.

Of course I admit making this exact model would prove very difficult, the flow of simulated lava, iron core, mantel. Crust and the insulting affect of the oceans and atmosphere would somehow be simulated in the model

Maybe I am sprouting a whole lot of nonsense, so guys take this suggestion and tear it apart with a better solution.

With we humans the nurse now just sticks a sensor in our ear and immediately knows the average temperature of our body

Take Care

Alan

[syhprum]

Allan

Your idea to solve the problem with a model has allready been done

http://www.myconfinedspace.com/tag/liquid-sodium/

[Alan McDougall (OP)]

syhprum,

No this was not to test how long it would take the earth to cool down to near absolute zero in a vacuum near absolute zero beyond the influence of the sun and neither was it done to determine the actual average temperature of the whole earth. That was the two questions n my thread and this experiment had nothing to do with answering my questions

It only mimics metal the core for research into the magnetic field of the earth

Nevertheless it is very interesting to me as I had no idea that anyone thought of an approach similar to my suggestion, but to resolve the problem in the way I suggested would be much more difficult, but difficult does not mean impossible

I would like others to come into the problem maybe our group minds could come up with a logical easy problem

By the way when I said extrapolate and multiply in my previous post I meant coming up with a mathematical formulae E.G. Original model volume 1 ==1 volume 2==4 volume 4= 64 extrapolating right up to where we equal the volume of the earth.

Maybe we could reduce the model volume to a sphere of 100 metres ? It would have to be larger than the test for magnetic effects as all factors would have to betaken into account in constructing a realistic true small model to mimic earth as close as possible

We can ignore density in my suggestion, as this would be take care of in our model Thus; Model Density = Earth Density.

Alan

“Researchers at the University of Maryland have constructed a 30-ton sphere that spins at more than 90 mph to generate magnetic fields. The 10-ft.-dia. sphere is filled with 13.5 tons of liquid sodium to mimic the Earth’s liquid-iron center core. A 3.3-ft.- dia. stainless-steel sphere inside the larger one counterrotates to approximate the motion of the planet’s solid iron inner core. The action of Earth’s inner liquid produces a magnetic field that makes compasses work, deflects harmful cosmic rays and protects the planet from solar wind. The field reverses every couple of hundred thousand years. By using a model instead of a computer simulation, scientists hope to determine how these reversals occur and predict the next one

[syhprum]

I know of course this device was not to solve the problem you put forward, I just wanted to make the point that not every problem is reduced to a simulation in a computer but there is a place for an analogue approach.

PS an 100m diameter sphere filled with molten Sodium would be a rather large experiment and might be difficult to fund.

[Alan McDougall (OP)]

syhprum,

I was not thinking about using molten Sodium , I was suggesting making an exact scaled down model of the earth, Small hot iron core up to the crust. Of course you are  correct computer  modeling is the best way to go and it would be much more cost effective than creating my difficult model.

But has anyone tried to solve this problem computer or by any other method?

I think off pat that it might take billions of years for the earth to cool off to that of space beyond the influence of the sun. What is your estimate?, it would be interesting later to see how close we came

Alan

Alan   
 

PS an 100m diameter sphere filled with molten Sodium would be a rather large experiment and might be difficult to fund.

[Bored chemist]

Before anyone does the arithmetic they need to think what "average" they want.

Imagine that I have a bucket of cold hydrogen (say 10C) and a bucket of warm iron (at 30C).
The simple average is (10+30)/2 =20
But I have about 1000 times as much mass of iron as H2 (It's under pressure) so the weighted average is near 30C
Hang on, I dont have 1000 times as many molecules of hydrogen so the weighted average is nearer 10C.
Perhaps what I mean by an average is what would I get it I mixed them together, and to calculate that I need the heat capacities of the materials involved. So to answer the original question, I need to know what the earth is made of as well as how hot each layer is. Plus I need to know how density and heat capacity vary with pressure etc.

That's certainly an interesting question.

[Alan McDougall (OP)]

Well lets revisite the topic with a few other views

Research by myself seems to indicate that this topic is too unimportant for science to try to solve by expensive simulations

I had a dream, which was not all a dream
The bright sun was extinguish'd,
and the stars Did wander darkling in the eternal space,
Rayless and pathless,
and the icy earth Swung blind and blackening in the moonless air;
Morn came and went-and came, and brought no day,
And men forgot their passion in the dread of this their desolation;
And all hearts were chill'd into a selfish prayer for light.

Different parts of the Earth would freeze at different rates if the Sun were to suddenly go out. We can guess some of the rates just from considering what happens at night. Over the course of 12 hours without sunlight, temperatures tend to drop between 10 and 30 degrees Fahrenheit, depending on whether you're in a desert or near the ocean. So you might expect that things would start to freeze in only two or three days with no Sun. The ocean acts like a big reservoir of heat, since its layers can mix and bring warm material easily to the surface. Areas near the ocean would remain warmer for a while longer, but the freezing would probably start in only a few months. After the time of an arctic winter, the ocean would be filled with pack ice. Since the pack ice of the arctic ocean averages about a dozen feet thick during the winter, one might guess the oceans would freeze at about the rate of a 50 feet a year. This rate would slow as a thick blanket of insulating ice was accumulated, but the effect would be the same, the warm water would be so far down as to be useless to people on the surface unlucky enough to be looking for it. Eventually, the atmosphere would liquefy and then freeze. That would probably take a few decades to begin

It would take about a month for the heat stored in the surface of the Earth to dissipate. Since the interior of the Earth is so hot, and the temperature of the universe so cold (2.7 K), the surface temperature would probably drop to around 200 K and remain there for billions of years, maintained by the energy from the interior of the earth.

•   And the earth's surface temperatures would continue to fall, towards -400 degrees, then towards absolute zero, around -459 degrees Fahrenheit. These falling surface temperatures would begin cooling even the earth's lower crust & possibly parts of the mantle. However, at some point the earth's center temperatures(which are generated in large part by radioactive fissioning materials & its own heat of contraction) would stop the decrease of earth's outer layer temperatures. In short, the earth has a good chance that its interior would NOT freeze solid for a very very long time. The fissioning materials might provide heat for the earth's interior for billions of years, even tho the earth's surface might be hovering just above absolute zero.

Of course over countless trillions of years the earth will finally dispence all its internal energy


What do you guys think?

[syhprum]

Them is my views too as I have already expressed.

[Democritus]

Alan

Recall that my very simplistic model assumes…
An Earth:
 
A sphere.
6,000km radius.
6,000C degrees temperature at centre.
0C  at surface.
The temperature gradient is linear.
Earth stuff is homogeneous and static.
The loci of average mass are the loci of average temperature.

So, what is the average temperature of the Earth?
Well, we have to tunnel with our thermometer deeply within the Earth to where there is as much stuff beneath us as there is above. This is not halfway to the centre at the 3,000km mark.

At the 3,000km point there is a lot more stuff above us than there is below because the volume of a sphere increases in proportion to the cube of its radius.

Abandoned Model
I’m abandoning my circle/disc model as its area/volume increases in proportion to the square of the radius. But if we were to pursue that model we would similarly find that at the halfway mark at 3,000km there would be lot more area of the circle outside that 3,000km radius than there is within from there towards the centre.

In fact, for any circle, to find an inner radius that halves its area, we have to multiply the outer radius (as it were) by the inverse of the square root of two. (1/sqr2) Which is about 0.707   Interestingly, many Newtonian telescope mirrors are supported at points about .7 radii from the mirror’s centre, so as to balance the mirror’s weight or mass there to avoid deformation.

So in the circle model the average temperature of the Earth would be located at a point 0.707radius unit along a radius from centre to circumference.
Or at the 0.707 x 6,000km = 4,242km distance from the centre.
(Hmm… a couple of 42s…spooky… where are you Douglas Addams?)
 Or 1758km from and within the circumference.

Given assumptions above, and a happy coincidence of units, and a rise in temperature of 1C/km tunnelled below the surface our thermometer would measure a temperature there of 1,758C

. 
But a sphere is different.
The volume of a sphere is calculated by multiplying the cube of its radius by four thirds. V= (4/3) r^3

From which we can calculate that the radius of a sphere with a volume of 1 cubic unit is 0.6208 units in length.
And that the radius of a sphere with a volume of ½ a cubic unit is 0.4927 units in length.

So for any sphere, the ratio of these lengths (0.4927divided by 0.6208 =  0.7937) is the ratio of the radius of an inner sphere (as it were) to that of an outer sphere (ditto), sharing a common centre, so that the inner volume is half that of the total volume.

In other words to divide a sphere using an inner radius so that there is as much stuff inside as there is outside that inner radius then that inner radius shall have a value of
0.7937 of the total radius.

So, in the spherical model the loci of average mass of the Earth would be located at a point .7937 radius units from centre to surface.
Or at the .7937 x 6,000km = 4762km distance from the centre.
Or 1,238km below the surface.

Given assumptions above, and a happy coincidence of units, and a rise in temperature of 1C/km tunnelled from the surface our thermometer would measure a temperature there of 1,758C.

Average Temperature, a Prediction
I predict that terra-tunnellers millennia hence shall measure the average temperature of the Earth where the loci of average mass intersects the temperature gradient. I predict that the temperature shall be measured as 1,758C (plus or minus 2C accounting for thermometer error), at a uniform depth of 1,758km beneath the surface of the Earth, provided that no one lights a campfire on the surface above a measuring location.

Disappointment
I had hoped with this exercise to finally put to rest one of the great unsolved mysteries of science. Sadly I find that on reflection this exercise is of no comfort there. So we must wait for one far greater than me to emerge and confirm whether the Earth is round, or whether it is flat. Sigh.
 []

Conclusion
Hope this helps Alan. My best shot.
Best wishes

Democritus

[Alan McDougall (OP)]

Democritus,

Indeed a fine effort at that, [8D] [] I will think about it and return if needed. Another method I thought about was to use a computer model and turn the earth inside out, and work from there

Alan

Average Temperature, a Prediction
I predict that terra-tunnellers millennia hence shall measure the average temperature of the Earth where the loci of average mass intersects the temperature gradient. I predict that the temperature shall be measured as 1,758C (plus or minus 2C accounting for thermometer error), at a uniform depth of 1,758km beneath the surface of the Earth, provided that no one lights a campfire on the surface above a measuring location.

Disappointment
I had hoped with this exercise to finally put to rest one of the great unsolved mysteries of science. Sadly I find that on reflection this exercise is of no comfort there. So we must wait for one far greater than me to emerge and confirm whether the Earth is round, or whether it is flat. Sigh

.