exponential law of cooling

[kowalskil@alpha.montclair.edu]

On 12/8/96 Jack Uretsky wrote:

> The cooling of a radiating (black) body of temperature T, immersed in a
> radiation field of black-body temperature T_0, is given by the equation:
> 	dT/dt = -k[T^4 -(T_0)^4]
> where k is some constant.
>
> Show that if T is much larger than T_0, then the temperature decrease is 
> proportional to the time and the proportionality constant is -k(T_i)^4.  

Let me try. I assume that initially t=0 and T=T_i. If T>>T_0 then 
dT/dt=-k*T^4  or  dT/(T^4)=-k*dt. I integrate and get 

        (1/T_i^3 - 1/T^3) = -3*k*t  or  1/T^3 = (1/T_i^3) + 3*k*t

Suppose that T_0 is close to zero, k=1/10^9 and T_i=1000. 
Then the numerical results (all in some arbitrary units) are

      t =   0    1    5    9    30   100   300  1000
      T = 1000  630  397  329  222   149   103   69

Did I make a mistake somewhere? 
Which "temperature decrease" should be proportional to t?

              Ludwik Kowalski