## Post-mortem body cooling in variable environments

In a recent post I noted that the standard post-mortem body cooling method used to estimate time of death (TOD) does not take into account a varying environmental temperature (i.e. it assumes a constant ambient temperature when applying Newton’s Law of Cooling).

Consider the differential equation

$\frac{dy}{dx} = -ky$.

Suppose we take the difference between the body temperature and the ambient temperature, $T-T_\alpha$, as being $y$ and the time, $t$, and being $x$. Then

$\frac{d}{dt} (T-T_\alpha ) = \frac{dT}{dt} - \frac{dT_\alpha }{dt} = -k(T-T_\alpha )$.

Since, as I noted, the ambient temperature is usually considered constant, the $\frac{dT_\alpha }{dt}$
term is eliminated and the usual solution is given.

So I figured I’d take some actual data from the NOAA/NWS website, throw it into Excel, and get a rough estimate of the function $T_\alpha (t)$. The 24-hour data I chose from Sanford, Maine (nearest data to my house) fit a sixth-order polynomial with an $R^2$ value of 0.977. Since I then had a pretty good function for $T_\alpha (t)$ I simply plugged it (and $\frac{dT_\alpha }{dt}$) in to the above to obtain a first-order non-homogenous differential equation of the form

$\frac{dy}{dx} + y = f(x)$.

Note that there are $k$‘s in there but not intuitively distributed. In terms of the variables in this situation, we have

$\frac{dT}{dt} + kT = kT_\alpha + \frac{dT_\alpha }{dt}$.

In any case, I then plotted this for a 24-hour period assuming the body started out at 98.6º. Unfortunately, the data for $T(t)$ is just nuts. In short, as one of my students noted, it is too strongly coupled to the ambient temperature for this instance. The question is: what’s wrong with the above and what is the correct way to handle this? I really think the solution, if found, will demonstrate that in some instances treating the ambient temperature as a constant results in an incorrect TOD.