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Rearranging the outgoing radial null geodesics of the Vaidya metric equation along $r=r_0(v)$ givesFor the prescribed parametrization by $w$, the chain rule yieldsSince $n>1$, $v/w=1+2\beta w^{n-1}\to 1$ as $w\to 0^{+}$. The asymptotic equivalence $w\sim v$ therefore impliesMore precisely $w=v+O(v^n)$ and $M(v)=(\beta /2)v^n[1+O(v^{n-1})]$, using Big O notation. The positivity of $dv/dw$ justifies treating $w$ as a single-valued function of $v$ locally.

First,For $n\ge 1$,Thus $({\chi }_n)$ is an asymptotic sequence.

Termwise asymptotic equivalence fails for every $n\ge 1$, becauseFor example, as $x\to 0$, the positive scale ${\varphi }_n(x)=|x{|}^n$ gives ${\chi }_n(x)=|x{|}^{n-1/2}$ for $n\ge 1$. This illustrates the geometric-mean refinement of an asymptotic sequence.

For $E\to \infty$, the outer part of the action integral for a linear-to-square-root potential dominates:The WKB quantization condition therefore givesTaking the power $2/5$ yields the asymptotic equivalenceHence

The approximation in part v requires the dimensionless stagnant depthto remain small. Since $0<\theta \le 1$ makes the numerator order one, this condition isUsing the solution from part v, the equivalent time range isor, at the level of asymptotic equivalence,The cooling becomes appreciable on the shorter scale $\tau =O({\mathcal{F}}^{-1})$, so these ranges overlap widely when $\mathcal{F}\gg 1$.

The Rayleigh number based on the convecting depth isThe definition of $\mathcal{F}$ implicit in part iii givesIt follows thatIf $1-\eta =O(1)$ and $\theta \gg {\mathcal{F}}^{-3/5}$, then $\mathrm{Ra}\gg O({\mathcal{F}}^{9/5})$. The thermal convection therefore remains strongly supercritical throughout the range in which the thin stagnant-layer approximation is valid.