= Solution
\b[Third body and steady partition.] The third molecule removes excess association energy and momentum, stabilizing newly formed <ozone> rather than being consumed in the net reaction. Its decreasing abundance with altitude suppresses three-body <ozone> formation strongly. Write $x=[\mathrm O]$, $y=[\mathrm O_3]$, $a=k_1n_2n_M$, $j=J_3$ and $b=k_3$, with $n_2,n_M$ held as reservoirs. The kinetic equations are
$$
\dot x=2J_2n_2-ax+jy-bxy,\qquad
\dot y=ax-jy-bxy,\qquad
\frac{d(x+y)}{dt}=2J_2n_2-2bxy.
$$
The fast association/photolysis subsystem conserves <odd oxygen>, $x+y$, and has <eigenvalues> $0,-(a+j)$. Its partition-adjustment time is therefore
$$
\boxed{\tau_F\simeq\frac1{a+j}=\frac1{k_1n_2n_M+J_3}}.
$$
This neglects the slower production and mutual-destruction terms while finding that fast mode. On its equilibrium partition, $ax\simeq jy$, and total odd-oxygen equilibrium gives $bxy=J_2n_2$. Hence
$$
x\simeq\left(\frac{J_2j}{bk_1n_M}\right)^{1/2},\qquad
y\simeq n_2\left(\frac{J_2k_1n_M}{bj}\right)^{1/2},\qquad \frac xy\simeq\frac j{k_1n_2n_M}.
$$
With fixed oxygen mole fraction and temperature, the atomic-oxygen mixing ratio scales as $x/n_M\propto n_M^{-3/2}$ in this approximation, and thus increases upward as pressure decreases.
\b[Lifetime versus relaxation.] The distinction is formalized by <Chapman family lifetime versus relaxation time>. In the ozone-dominated regime $y\gg x$, the family abundance is approximately $y$, and its gross chemical lifetime is
$$
\tau_{O_x}\simeq\frac{y}{2bxy}=\frac1{2bx}
=\boxed{\frac12\left(\frac{k_1n_M}{J_2J_3k_3}\right)^{1/2}}.
$$
This recovers the printed $\tau_S$ as a family turnover time under its additional partition assumption. It is not generally an e-folding relaxation time. With rapid partitioning, the family equation is $\dot X=P-kX^2$. Linearizing about equilibrium gives relaxation rate $2kX_e$ but turnover rate $kX_e$, a factor of two difference. More generally the approximate family lifetime is $(x+y)/(2bxy)$, and its relaxation time is half that quantity. Neither can be replaced by $1/(2bx)$ when <atomic oxygen> dominates.
For a direct check using the complete two-species equations, the equilibrium <Jacobian matrix> is
$$
\mathsf J=\begin{pmatrix}-a-by&j-bx\\a-by&-j-bx\end{pmatrix}.
$$
The numerical high-altitude example below gives eigenmode times about $321\ \mathrm s$ and $4.0\times10^5\ \mathrm s$, whereas the printed $\tau_S$ is only $5.15\times10^3\ \mathrm s$. Thus interpreting both printed expressions as universal eigenmode time constants would be false, even though the fast partition and the ozone-dominated turnover approximations are useful.
Setting $x=y$ in the fast-partition approximation gives $a=j$, hence
$$
n_M\simeq\sqrt{\frac{J_3}{0.2k_1}},\qquad
\boxed{p\simeq k_BT\sqrt{\frac{J_3}{0.2k_1}}=12.3\ \mathrm{Pa}=0.123\ \mathrm{hPa}}.
$$
Retaining mutual destruction gives $a=j+bx$ and changes this estimate by less than one percent. The formal high-altitude limit of $\tau_F$ is $1/J_3\simeq333\ \mathrm s$: <ozone> photolysis, unlike three-body association, does not slow merely because collider density decreases. This is a partition timescale, not the lifetime of all <odd oxygen>. Extrapolating the fixed-reservoir approximation indefinitely upward would eventually cease to be self-consistent.
\b[Day and night.] At $p=0.01\ \mathrm{hPa}=1\ \mathrm{Pa}$ and $230\ \mathrm K$, $n_M=p/(k_BT)=3.15\times10^{20}\ \mathrm{m^{-3}}$ and $n_2=0.2n_M$. The exact equilibrium is conveniently obtained from
$$
k_1n_Mx^2-J_2x-\frac{J_2J_3}{k_3}=0,\qquad y=\frac{J_2n_2}{k_3x}.
$$
The physical root gives
$$
\boxed{x=9.97\times10^{16}\ \mathrm{m^{-3}},\quad y=6.38\times10^{14}\ \mathrm{m^{-3}},\quad
\chi_O=3.17\times10^{-4},\quad\chi_{O_3}=2.03\times10^{-6}}.
$$
Their ratio is $156$, consistent with the requested approximate factor $150$; the simpler fast-partition estimate gives approximately $151$. The printed fast time is $331\ \mathrm s$ at this pressure.
For <nighttime Chapman ozone>, after sunset set both photolysis rates to zero. <Ozone> initially grows by converting <atomic oxygen>, but simply putting every initial oxygen atom into one <ozone> molecule neglects the simultaneous $\mathrm O+\mathrm O_3$ sink. The night equations are
$$
\dot x=-x(a+by),\qquad \dot y=x(a-by).
$$
While sufficient <atomic oxygen> remains, the <ozone> balance tends toward
$$
\boxed{y\sim a/b,\qquad \chi_{O_3}\sim\frac{k_1n_2}{k_3}=6.36\times10^{-5}}.
$$
This is a useful long-night sunrise estimate, roughly thirty times the sunset <ozone> abundance, not an exact result for an unspecified night duration. The precise phase-plane relation is
$$
x-x_0=y-y_0+\frac{2a}b\ln\left(\frac{a-by}{a-by_0}\right)\qquad(y<a/b).
$$
Together with the night equations it supplies $y(t_{\rm night})$ for any supplied duration. For illustration only, a twelve-hour dark interval gives $\chi_{O_3}\simeq5.8\times10^{-5}$; no twelve-hour duration is stated by the paper. Even the infinite-dark-time endpoint is slightly below $a/b$, because the finite initial atomic reservoir is consumed. This conditional estimate avoids inventing a unique sunset-to-sunrise duration.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2016-chapman-ozone-night.png]
{title=Daytime Chapman partition and the duration-dependent nocturnal ozone increase for the supplied rates}
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