Suppose molecular hydrogen is fully dissociated into atoms across a steady planar shock, with the same far-upstream and far-downstream temperature and negligible binding-energy mass correction. The upstream isothermal sound speed obeys $a_1^2=k_{B}T/(2m_H)$, while $a_2^2=2a_1^2$. Conservation of mass gives $\chi ={\rho }_2/{\rho }_1=u_1/u_2$. Momentum conservation then gives $u_1^2+a_1^2=u_1^2/\chi +2a_1^2\chi$, producing the displayed quadratic with ${\mathcal{M}}_1=u_1/a_1$.

The high root approaches ${\mathcal{M}}_1^2/2$ at large upstream Mach number. Its real compressive domain is ${\mathcal{M}}_1\ge 1+\sqrt{2}$. At the endpoint the downstream flow is sonic; above it the high root is subsonic downstream. Completion of dissociation and maintenance of equal temperatures additionally require a consistent chemical and radiative energy budget. The two flux equations alone do not establish that physical realizability.

For a classical monatomic ideal gas, direct volume integration gives $\Delta ={\Lambda }^{-3N}(\beta P{)}^{-N-1}$. Consequently the continuous-volume ensemble has $⟨H⟩=(5N/2+1)k_{B}T$. Defining a dimensionless thermodynamic function using $\tilde{\Delta }=\beta P\Delta$ instead gives $-{\partial }_{\beta }\ln \tilde{\Delta }=(5N/2)k_{B}T$. The normalized volume distribution is identical, because the prefactor cancels in probabilities. Its temperature derivative nevertheless contributes $-k_{B}T$ to the derived enthalpy and $-k_B$ to its heat capacity. The two conventions agree at leading extensive order, but cannot be mixed in an exact finite-$N$ fluctuation identity.

For a regular closed planar curve, maximize signed enclosed area at fixed length using the variational integrand $\mathcal{L}=(x\dot{y}-y\dot{x})/2+\lambda \sqrt{{\dot{x}}^2+{\dot{y}}^2}$. Write the nonzero speed as $v$. The Euler-Lagrange equations integrate to $\lambda \dot{x}/v=y+c_1$ and $\lambda \dot{y}/v=-x+c_2$. A zero multiplier would force a zero velocity, so squaring and adding yields the displayed circle. The argument uses only regularity and is independent of the speed used to trace the curve. A positively oriented simple traversal of prescribed length $L$ encloses area $L^2/(4\pi )$; the planar isoperimetric inequality supplies the separate global maximality statement.

When collisions thermalize an optically thin two-level atom, the upper fraction is $f_u=g{e}^{-\theta /T}/(1+g{e}^{-\theta /T})$ and the emitted power per atom is $A_{21}\Delta E{f}_u$. Its logarithmic temperature slope is $(\theta /T)(1-f_u)>0$. Radiative cooling still occurs, but collision frequency no longer sets the emitted power as in the low-density limit.

At fixed pressure and fixed composition, density varies as $T^{-1}$. With temperature-independent heating per particle, low-density cooling per particle scales as $T^{-3/2}{e}^{-\theta /T}$ and is restoring for $T<2\theta /3$. For constant heating per volume instead, the cooling scales as $T^{-5/2}{e}^{-\theta /T}$ and the threshold is $2\theta /5$. The heating normalization and perturbation constraint must be specified.

At the maximum of a cooling curve with constant heating set to that maximum, the linear restoring term vanishes. Because cooling falls on both sides of the maximum, the equilibrium is not generally stable to perturbations of both signs even though a first-order eigenvalue is zero.

The stability boundary in isochoric thermal stability of a collisionally excited line assumes that the abundance of the emitting ion stays fixed during the temperature perturbation. At fixed relevant densities, the cooling factor $T^{-1/2}{e}^{-\theta /T}$ then has logarithmic slope $-1/2+\theta /T$.

If the ion fraction $x(T)$ adjusts, the cooling instead contains that additional factor and its slope gains $d\ln x/d\ln T$. Changes in the electron abundance or heating law introduce further terms. Thus the quoted $T=2\theta$ boundary belongs to the frozen-abundance model; comparing ionization and thermal relaxation times determines whether that assumption fits a particular perturbation.

Take fully ionized hydrogen with electron and proton number densities both $n$, fixed local heat input $\Gamma$ per unit volume, and volumetric cooling $3n^2{\alpha }_B(T)k_{B}T/2$. The ideal gas law is $P=2n{k}_{B}T$. At fixed pressure, net cooling becomes $\mathcal{L}=3P^2{\alpha }_B(T)/(8k_{B}T)-\Gamma$. At an equilibrium $T_0$, differentiation gives the displayed derivative.

The linearized thermal equation is ${\rho }_0{c}_p\dot{\delta T}=-{\mathcal{L}}_T\delta T$, so equilibrium is stable exactly when the logarithmic slope of ${\alpha }_B$ exceeds one. A constant or decreasing coefficient gives instability. A different heating prescription must be differentiated too; constant incident radiation does not generally imply a density-independent local heating rate.

At thermal equilibrium, let $\mathcal{L}(\rho ,T)$ be net cooling per unit mass, including a specified heating law. A gas perturbation that stays at constant pressure obeys $c_{p}d\delta T/dt=-(\partial \mathcal{L}/\partial T{)}_p\delta T$. It grows when this derivative is negative. The isobaric approximation requires pressure communication faster than thermal evolution; a volumetric cooling function instead introduces a background mass density in the denominator. Thermal conduction alters the short-wavelength criterion.

Steady mass conservation, inviscid momentum balance and $dp=c_s^{2}d\rho$ give $d\ln A=(M^2-1)d\ln u$, where $M=u/c_s$. A smooth sonic crossing needs an area extremum; a minimum-area throat supports an accelerating subsonic-to-supersonic branch. A throat is a necessary geometric condition, not a sufficient specification of operating pressures.

For a calorically perfect gas on a steady isentropic nozzle branch,Here $A_{\ast }$ is the sonic area. An area ratio greater than one generally admits a subsonic and a supersonic root; boundary conditions choose the physical branch. Numerical inversion must retain that branch information.

A converging-diverging nozzle that can accelerate a compressible flow from subsonic to supersonic speed through a sonic throat. The nozzle area-velocity relation requires acceleration in a converging subsonic section and a diverging supersonic section. Reservoir and downstream pressures determine whether this transonic branch operates; the shape alone does not ensure choked flow.

For steady polytropic flow with $p=K{\rho }^{\gamma }$, $K>0$, $\gamma >1$, a sonic section has $u_{\ast }=c_{\ast }=(\gamma K{\rho }_{\ast }^{\gamma -1}{)}^{1/2}$. Substitution in mass conservation $\dot{M}=A_{\ast }{\rho }_{\ast }{u}_{\ast }$ gives the displayed density. These conditions fix the sonic state for a prescribed flux and throat area, subject to compatible reservoir and downstream boundary conditions.