= Solution
\b[<virial pressure-radius curve with uniform-sphere binding energy>.] Put $c_s^2=k_BT/\mu$. Isothermality makes $\int(p/\rho)dM=c_s^2M_0$, so the <virial theorem> gives
$$
\boxed{p_0=\frac{A}{r_0^3}-\frac{B}{r_0^4},\qquad
A=\frac{3c_s^2M_0}{4\pi},\quad B=\frac{3GM_0^2}{20\pi}}.
$$
The curve crosses zero at $r_z=B/A=GM_0/(5c_s^2)$, rises to its maximum at
$$
\boxed{r_c=\frac{4B}{3A}=\frac{4GM_0}{15c_s^2}},
$$
then decreases toward zero at large radius. Since $p_0'=(4B-3Ar_0)/r_0^5$, a cloud displaced outward at fixed external <pressure> has a restoring <pressure> deficit only on the descending branch. Thus \b[the homologous, isothermal radial mode is stable for $r_0>r_c$], unstable for $r_z<r_0<r_c$, and marginal at $r_c$. The negative-pressure branch is not a physical equilibrium with a positive confining <pressure>. This conclusion uses the stipulated gravitational-energy approximation, not the exact isothermal <density> profile.
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2014/paper-1-virial-pressure.png]
{title=Original pressure-radius curve and stable versus unstable branches of the stipulated virial cloud model}
\b[<critical mass from a virial pressure turning point>.] At the turning point, $B=3Ar_c/4$; hence
$$
\boxed{p_{\max}=\frac{A}{4r_c^3}
=\frac{10125}{1024\pi}\frac{c_s^8}{G^3M_0^2}
\propto\left(\frac{T^2}{M_0}\right)^2}.
$$
Using the cloud's <mean density> $\bar\rho=3M_0/(4\pi r_0^3)$, the condition $r_0>4GM_0/(15c_s^2)$ becomes
$$
\boxed{M_0<M_{\rm crit}=\left(\frac{15c_s^2}{4G}\right)^{3/2}
\left(\frac{3}{4\pi\bar\rho}\right)^{1/2}
\propto\left(\frac{T^3}{\bar\rho}\right)^{1/2}}.
$$
Using the boundary <density> instead changes the numerical coefficient, not this scaling. The coefficient should not be identified with the exact <Bonnor-Ebert sphere> threshold because the assumed <potential energy> is different.
\b[<mass and luminosity weighting of a stellar initial mass function>.] Impose finite birth-mass cutoffs $m_l,m_u$. The stellar birth <mass> and main-sequence <luminosity> scale as
$$
M_{\rm born}=\frac{\xi_0}{0.35}(m_l^{-0.35}-m_u^{-0.35}),\qquad
L_{\rm tot}\propto\frac{\xi_0}{2.65}(m_u^{2.65}-m_l^{2.65}).
$$
For a broad <mass> interval, the low-mass end controls the first <integral> and the high-mass end the second. Per logarithmic <mass> interval the weights are respectively $m^2\xi\propto m^{-0.35}$ and $mL\xi\propto m^{2.65}$. Finite lifetimes remove massive stars from an aging population, so the <luminosity> conclusion applies while those stars remain on the <main sequence>.
\b[<exponential closed-box gas consumption>.] The gross formation rate is $\Psi=\epsilon M_g$, but the permanent locking rate is $\alpha\Psi$. The <instantaneous recycling approximation> and <mass conservation> give $\dot M_g=-\alpha\epsilon M_g$, $\dot M_*=\alpha\epsilon M_g$. Therefore
$$
\boxed{M_g=M_0e^{-\alpha\epsilon t},\qquad
M_*=M_0(1-e^{-\alpha\epsilon t}),\qquad
f_g=e^{-\alpha\epsilon t}}.
$$
Here $M_*$ means surviving locked stellar <mass> in the <galactic gas fraction>, not the cumulative gross <mass> that has passed through <star formation>.
\b[<locked-mass and returned-mass yield conventions>.] Pre-existing metals are locked at rate $\alpha Z\Psi$; the stipulated newly produced metal source is $b(1-\alpha)\Psi$. Thus
$$
\frac{d(M_gZ)}{dt}=-\alpha Z\Psi+b(1-\alpha)\Psi,
\qquad M_g\dot Z=b(1-\alpha)\Psi.
$$
For $\alpha>0$ and initially pristine gas,
$$
\boxed{Z=y\ln\frac{M_0}{M_g}=y\ln\frac1{f_g},\qquad
y=\frac{b(1-\alpha)}\alpha,\qquad
Z(t)=b(1-\alpha)\epsilon t}.
$$
The effective yield $y$ is per unit permanently locked <mass>; it is not the source parameter $b$. For $\alpha=0$, $M_g=M_0$ and the direct time equation gives $Z=b\epsilon t$. For $\alpha=1$, the explicitly stipulated metal source vanishes, so this particular enrichment model gives $Z=0$. Constant yield is a trace-enrichment approximation; the formal logarithmic divergence as gas is exhausted must not be interpreted as a physical metal fraction exceeding one.
\b[<backward extrapolation of exponential closed-box star formation>.] Denote the present <age of the universe> by $t_{\rm now}$. With the last paragraph's $\alpha=1$,
$$
\boxed{\Psi(z\simeq3)=\left(1M_\odot\,\mathrm{yr^{-1}}\right)
\exp\left[0.5\left(\frac{t_{\rm now}}{\mathrm{Gyr}}-2\right)\right]}.
$$
Adopting a present <age of the universe> $t_{\rm now}=13.8\,\mathrm{Gyr}$ for the estimate gives about \b[$365M_\odot\,\mathrm{yr^{-1}}$], requiring $7.3\times10^{11}M_\odot$ of gas at that epoch in the same model. The boxed expression makes the dependence on the adopted present age explicit. This is an extreme history for a present Milky-Way-like system, although intense high-redshift <star formation> by itself is not forbidden.
The extrapolation forces a fixed gas reservoir to decline exponentially over many <gas-depletion times>. Gas <accretion>, outflows, evolving gas <density> and star-formation efficiency, delayed recycling, and galaxy assembly break that assumption. The empirical disk-averaged relation is not a universal constant-efficiency law for all gas: the original measurements combined <Balmer series> H-alpha, atomic gas, <carbon monoxide> and <infrared> data, with tracer calibration and dust corrections entering the inference.<The Global Schmidt Law in Star Forming Galaxies by Robert C. Kennicutt Jr.>[Abstract: observed tracers, sample, and fitted index]{ref} Cosmological calculations explicitly support continuing cold-gas supply to rapidly forming high-redshift galaxies, providing a physical alternative to this closed-box backward extrapolation.<Cold streams in early massive hot haloes as the main mode of galaxy formation by Avishai Dekel and others>[Abstract: stream-fed galaxies and comparison with observed abundance]{ref}
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