Let $\theta =T-T_0>0$ and ${\theta }_1=T_1-T_0$. The aperture spans the enclosure's depth $H$, so the incoming volume flux, equal to the outgoing volume flux, isOnly one of these equal exchange fluxes multiplies the temperature difference in the heat budget: incoming fluid brings $q{T}_0$, outgoing fluid removes $qT$. With constant mass density and specific heat capacity, those factors cancel and the well-mixed ventilation temperature balance isIntegrating from ${\theta }_1$ givesThe cooling is algebraic rather than exponential because the buoyancy-driven single-opening exchange flow weakens as its driving reduced gravity decreases. The formula assumes negligible wall heat capacity, no heat input, the specified aperture geometry, and a uniform interior temperature maintained by mixing.

With a low-level opening, the dense incoming cold fluid tends to spread along the floor instead of falling through the warm interior. A cold lower layer and warm upper layer develop, giving stable density stratification. The incoming fluid can short-circuit back out through the low opening after that layer occupies it, leaving warm fluid above poorly ventilated. Thus the uniform-temperature assumption is generally inappropriate: cooling becomes vertically nonuniform and substantially slower for the upper warm fluid. If mixing were artificially maintained, the same lumped budget would still apply; it is the loss of that assumption, rather than a reversal of buoyancy, that changes the physical outcome.

Optical cluster observations give galaxy positions and line-of-sight redshifts. Subtract the mean recession to obtain the line-of-sight velocity dispersion ${\sigma }_{\mathrm{los}}$. If the tracers are in approximate virial equilibrium with isotropic velocities, $⟨v^2⟩=3{\sigma }_{\mathrm{los}}^2$ and $K\simeq 3M{\sigma }_{\mathrm{los}}^2/2$. Write the gravitational energy as $U_G=-\alpha G{M}^2/R$, with $\alpha$ determined by the density profile and radius convention. The virial theorem givesFor a uniform sphere $\alpha =3/5$, this becomes $5R{\sigma }_{\mathrm{los}}^2/G$. Projected separations are commonly used to estimate a gravitational radius rather than literally assume a uniform sphere. Interlopers, substructure, anisotropy, tracer bias and a nonzero surface term affect the estimate. The inferred mass is total gravitating mass, not merely the galaxies' stellar mass.

For X-ray observations, the hot intracluster gas emits thermal bremsstrahlung and atomic lines. Spectroscopy determines its temperature, and the surface brightness, after deprojection, determines a gas-density profile. Under spherical hydrostatic equilibrium with thermal gas pressure, $d{P}_g/dr=-{\rho }_{g}GM(<r)/r^2$ and $P_g={\rho }_g{k}_{B}T/(\mu m_p)$. ConsequentlyThe order-of-magnitude virial version is $k_{B}T\sim \mu m_{p}GM/R$. X-ray gas temperature is therefore a second measure of the depth of the gravitational potential. Nonthermal pressure, mergers, clumping and temperature gradients must be considered before equating the thermal estimate to the true mass.

For dense clusters the vacuum force is small compared with their internal gravity. If retained in the hydrostatic equation, it changes $GM/r^2$ to $GM/r^2-\Lambda r/3$ and adds $\Lambda r^3/(3G)$ to the mass formula above. The optical virial relation likewise uses the separate vacuum contribution derived in part (c). Comparing optical and X-ray estimates tests the dynamical assumptions and exposes a gravitating mass much larger than the luminous stellar component.

Use the physical root of the cubic which tends to $1/2$ as $\eta \to 0$. For $0<\eta <1$, the cubic has value $1$ at $x=0$ and value $-\eta /4$ at $x=1/2$, and its derivative $6\eta x^2-(2+\eta )$ is negative throughout that interval. Thus it has exactly one root there. Implicit differentiation givesThe smallest limiting value occurs as $\eta \to 1^{-}$, since larger $\eta$ cannot have a finite turnaround. At $\eta =1$ the equation factors:The continuation of the collapsing root is thereforeThe lower endpoint is an infimum, not an attained minimum for a shell that turns around in finite time. This is the minimum virial-radius ratio with positive cosmological constant. The separate root $x=1$ at $\eta =1$ describes the limiting static situation and is not the collapse branch.

The smaller ratio does not mean vacuum repulsion makes turnaround easier. At fixed turnaround data, $U_m=-A-B$ is more negative than for $\Lambda =0$. If one tried $x=1/2$, the virialized energy would be $-A-B/2$, which is not negative enough to equal $U_m$. Extra contraction is necessary to increase the magnitude of the gravitational binding energy. Thus vacuum repulsion inhibits formation of a bound shell, while the shell that does form has a smaller idealized final-to-turnaround ratio under the stated energy and profile assumptions.

No. Assign the two free generators to $(1,0)$ and $(0,1)$ in the lexicographically ordered group ${\mathbb{Z}}^2$. The universal property gives a lattice-ordered group homomorphism onto this group: its image contains both ordinary group generators. It is not an Archimedean ordered group, sinceThus a lattice-homomorphic image can contain a nonzero positive infinitesimal even when the original free object has an Archimedean pointwise function representation.

No. The free Abelian lattice-ordered group on two generators is countable, since every element is represented by a finite term over a countable alphabet of operation symbols and the two generators. The additive group of all real-valued functions on the given half-plane is uncountable: it already contains a distinct constant function for every real number. A lattice-ordered group homomorphism that is an isomorphism would be a bijection, contradicting these cardinalities. The concrete free-group model consists of specially constrained integer-linear term functions, not all real-valued functions on a domain.