**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.

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**Conclusion**Hope this helps Alan. My best shot.

Best wishes

Democritus