Originally Posted by

**Gusbob** The temperature at r distance from centre is $\displaystyle T = a^2 - r^2 $

So when a small length ds is moved to from the centre, it will vary in proportion to the temperature T.

The new length $\displaystyle dr \propto T ds \rightarrow dr = k(a^2 - r^2)ds $ for some constant k.

But if you think about it, when you're at the centre w/ r = 0, you'll get $\displaystyle dr = ka^2 ds $ which is true iff $\displaystyle k = \frac{1}{ a^2 }$

Therefore $\displaystyle dr = \frac{a^2-r^2}{a^2} ~ds $

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A more intuitive way to think of it is using the ratio of lengths (L).

We have a ratio $\displaystyle \frac{dr}{ds} $ .

Because the lengths themselves are proportional to the specific temperature at the radius they are at,

$\displaystyle \frac{dr}{ds} = \frac{T_r}{T_s} = \frac{a^2 - r_2}{a^2 - r_1}$

where $\displaystyle r_1, r_2$ are the distance from center the lengths dr and ds are at.

At $\displaystyle L = ds, r = r_1 = 0, T_r = a^2$

At $\displaystyle L = dr, r = r_2, T_s = a^2 - r_2^2$

So we have $\displaystyle \frac{dr}{ds} = \frac{a^2-r_2^2} {a^2} \Rightarrow dr = \frac{a^2-r^2}{a^2} ds $

Yes. That's another way of saying this is the value of all the sums of dr. Because dr changes with distance from radius, we sum all the small values of dr at each distance to make the radius. If we make dr sufficiently small, as with integration, it will be a good estimation for the radius.