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**Physics, Astronomy & Cosmology / Re: What is the average temperature of the earth?**

« **on:**11/10/2008 22:19:25 »

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

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