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.

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