= Solution
New oceanic <lithosphere> is hot and mechanically weak. As it cools, the depth over which rock can sustain elastic <stress> on the loading timescale generally increases, so \b[<effective elastic thickness> tends to increase with plate age at the time of loading]. It is this mechanical state when the load is emplaced, together with subsequent relaxation, that matters; the present age alone is insufficient for an ancient load.
The <half-space cooling model> illustrates the thermal dependence. With age $\tau$, surface <temperature> $T_s$, initial interior <temperature> $T_m$ and constant <thermal diffusivity> $\kappa$,
$$
T(z,\tau)=T_s+(T_m-T_s)
\operatorname{erf}\left(\frac{z}{2\sqrt{\kappa\tau}}\right).
$$
A chosen <temperature> $T_*$ lies at
$$
\boxed{z_*=2\sqrt{\kappa\tau}\,
\operatorname{erf}^{-1}\left(\frac{T_*-T_s}{T_m-T_s}\right)
\ \propto\sqrt\tau.}
$$
If the base of the mechanically strong layer approximately follows such an isotherm, $T_e$ initially follows a similar age trend. Cooling models constrained by ocean-floor depth and heat flow provide the thermal motivation; their fitted thermal thickness must not be mistaken for measured <elastic thickness>.<An analysis of the variation of ocean floor bathymetry and heat flow with age by Barry Parsons and John G. Sclater>[1977, pages 803–806 and discussion of half-space versus finite-plate cooling]{ref}
In practice $T_e$ is an effective bending thickness rather than the depth of one universal <temperature>. Stress-dependent yielding, <grain size>, water content, composition, loading duration and reheating can alter it. Finite-plate cooling and basal heat input also limit indefinite square-root growth at old ages. Nor should $T_e$ be identified with either the thickness of the <crust> or the full seismically defined <lithosphere>.
\Image[/past-exam-of-the-physics-of-the-earth-as-a-planet-course-of-the-university-of-cambridge/2015/paper-1-plate-cooling.png]
{title=Cooling deepens a fixed isotherm as the square root of age in the half-space model}
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