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Take $z$ to increase inward, so the positive coefficient $A$ describes an inward-increasing temperature. Constant gravity and hydrostatic equilibrium give $dP/dz=\rho g$, henceMatching the radiative temperature gradient to the adiabatic temperature gradient gives the formal local boundary relationThe same result follows from radiative diffusion: for constant upward thermal flux $F$ and Rosseland mean opacity $\kappa$, $A=3\kappa F/(16{\sigma }_{\mathrm{SB}})$.

There is an important limitation to treating $A$ as constant over the entire radiative layer. Integration from an irradiated outer boundary givesFor a diatomic ideal gas, ${\nabla }_{\mathrm{ad}}=2/7>1/4$, so this profile cannot actually reach a radiative-convective boundary. Formally, imposing a constant ${\nabla }_{\mathrm{ad}}$ would givewhich is positive only for $0<{\nabla }_{\mathrm{ad}}<1/4$. A finite boundary for a normal molecular atmosphere requires additional opacity, flux, or thermodynamic variation. The local matching formula is usable near a real boundary, but constant $A$ is not a complete global model of it. This is the convective stability of a constant-opacity irradiated atmosphere.

For the intended order-of-magnitude scaling, suppose the local values of $g/A$ and ${\nabla }_{\mathrm{ad}}$ are comparable for Jupiter and a hot Jupiter, and assume $T_b$ scales with planetary equilibrium temperature. Equal absorbed-flux factors around the same stellar luminosity give $T_b\propto a^{-1/2}$. Taking $a_J=5.2\mathrm{AU}$ and a representative close-in orbit $a_h=0.05\mathrm{AU}$ yieldsThis illustrates how irradiation can push a boundary much deeper. It is a conditional estimate calibrated from the supplied $0.1\mathrm{bar}$ reference, not a self-consistent prediction of the globally constant-$A$ model. Different intrinsic cooling flux, opacity, gravity, or atmospheric metallicity of a giant planet can substantially alter it.

Four mechanisms that can produce disequilibrium chemistry in an exoplanet atmosphere are:

The exoplanet transport and photochemical examples follow kinetic atmosphere calculations and models including horizontal transport; the condensate example is examined in cold-trap calculations.

Use the standard local chemical quench level approximation: carbon monoxide-rich gas from the hot deep region is transported upward, and conversion becomes slower as the gas enters the cooler layers. Assume that neither a faster loss process nor strong compositional fractionation removes carbon monoxide above the quench level. For an effective mixing length $L$, the eddy mixing time is ${\tau }_{\mathrm{mix}}\simeq L^2/K_{zz}$, so outrunning conversion near $1\mathrm{bar}$ requiresThe usual order-of-magnitude choice is $L\simeq h_b$, the local atmospheric scale height at $T_b=1400\mathrm{K}$. Equal planetary mass and radius give the same gravity as Jupiter; with the same mean molecular weight, the atmospheric scale height scales linearly with temperature. To use the supplied $30\mathrm{km}$ reference, additionally adopt a representative Jovian reference temperature $T_J=150\mathrm{K}$. ThenUsing $k_{B}T/(\mu m_{H}g)$ directly with $\mu \simeq 2.3$ and Jovian $g\simeq 25\mathrm{m}{\mathrm{s}}^{-2}$ gives $h_b\simeq 200\mathrm{km}$ and $K_{zz}\gtrsim 4\times 10^5{\mathrm{m}}^2{\mathrm{s}}^{-1}$, the same order of magnitude. The supplied Jovian height is approximate and does not specify its reference temperature. Inserting $30\mathrm{km}$ unchanged for the hot gas would instead give $9\times 10^3{\mathrm{m}}^2{\mathrm{s}}^{-1}$ and neglect this temperature scaling.

The pressure separation is relevant to a stronger, whole-column transport estimate. With constant gravity and mean molecular weight, the profile givesIf one additionally requires diffusion through the entire column within $10^5\mathrm{s}$, a sufficient conservative condition is $K_{zz}\gtrsim \Delta z^2/{\tau }_{\mathrm{chem}}\sim 10^7{\mathrm{m}}^2{\mathrm{s}}^{-1}$. It is not a necessary local quench condition: the chemical relaxation time is expected to become much longer in the cooler gas, so a longer total transit time can still preserve carbon monoxide.

The usual quench estimate is $K_{zz}$ of order $10^6{\mathrm{m}}^2{\mathrm{s}}^{-1}$ under the stated scale-height assumptions. A unique bound for survival to $10^{-3}\mathrm{bar}$ cannot be inferred from one reaction time without assumptions about its variation, the mixing length and upper-atmospheric losses.

At 10 parsecs, $0.01$ arcseconds corresponds to $a=0.10$ astronomical units. Take a solar-radius star with $T_{\star }=5772\mathrm{K}$, zero Bond albedo, full day-night heat redistribution, negligible internal heating, and a hydrogen-helium atmosphere with mean particle mass $2.3m_H$. The planetary equilibrium temperature isFor Jupiter mass and radius, $g\simeq 24.8\mathrm{m}{\mathrm{s}}^{-2}$. Its atmospheric scale height isAssume a strong band spans $N_H=5$ atmospheric scale heights and saturates in the annulus model for transmission spectroscopy. The atmospheric spectral-feature amplitude isFor a five-standard-deviation detection, the uncertainty of the measured differential contrast must satisfyThis estimate scales linearly with the assumed feature height; one atmospheric scale height would require about $7.5\mathrm{ppm}$. The question gives no opacity or abundance from which to fix $N_H$, so an atmospheric detection threshold is necessarily assumption-dependent.

For thermal emission at $10\mu \mathrm{m}$, assume the planet and star emit as blackbodies. The thermal eclipse depth from the Planck law isThus the uncertainty required for a five-standard-deviation exoplanet secondary eclipse detection isThese are uncertainties of the final transit or eclipse contrasts, including the uncertainty of their reference levels. The distance affects photon counts and observing time but cancels from the flux ratios. An opaque exactly isothermal atmosphere emits a featureless blackbody spectrum: an eclipse detects its thermal light, while identifying atmospheric composition requires spectral features and a nonisothermal structure or other diagnostics.

Assume all three planets retain approximately their original hydrogen-helium bulk composition. Stellar encounters change orbital energy and irradiation, not automatically the planet's elemental abundances. Their atmospheric structures can become approximately stationary long before their interiors finish cooling.

For the unperturbed Jupiter-like planet at $a\simeq 5.2\mathrm{au}$, the zero-albedo globally averaged planetary equilibrium temperature is about $122\mathrm{K}$. Stellar light heats the outer atmosphere, while internal cooling supports a deeper temperature gradient and convection. At cooler pressures, chemical equilibrium favors methane and ammonia; condensate clouds can include ammonia at high levels and water deeper down. Disequilibrium chemistry in an exoplanet atmosphere can preserve carbon monoxide or other species from deeper layers.

For the inward-migrated planet at $a\simeq 0.052\mathrm{au}$, the irradiation-only planetary equilibrium temperature is ten times higher, about $1220\mathrm{K}$, because $T_{\mathrm{eq}}\propto a^{-1/2}$. Its irradiated planetary atmosphere has a heated radiative exterior, potentially strong day-night differences, and a deep radiative-convective boundary. At suitable pressures, chemical equilibrium increasingly favors carbon monoxide over methane; water remains important, while alkali absorption and high-temperature condensates can matter. An atmospheric thermal inversion depends on absorbers, clouds, and irradiation and is not guaranteed simply by migration.

The ejected object is a rogue planet. Its irradiation-based planetary equilibrium temperature becomes very small, but its actual emitting temperature is set primarily by internal cooling and Kelvin-Helmholtz contraction. After only several million years it can remain warm and self-luminous, with deep convection and an outward-cooling radiative atmosphere. Its photospheric chemistry and clouds depend on that cooling temperature; they cannot be inferred from the absence of a host star alone. None of these cases fixes an exact pressure-temperature profile without a cooling model, opacities, and atmospheric composition.

For a spherical planet, combine the hydrostatic pressure support equation with $dm/dr=4\pi r^2\rho$:Neglect surface pressure. Since $r(m)\le R$, the hydrostatic lower bound on planetary central pressure isFor a physically usual mass density decreasing outward, the mean interior density exceeds the global mean. Hence $r(m)\le R(m/M{)}^{1/3}$, giving the sharper minimum within this class,Uniform mass density attains the sharper bound; direct integration with $m(r)=M{r}^3/R^3$ gives $P_c=3G{M}^2/(8\pi R^4)$. Real planets are centrally concentrated through compression and dense cores, so their central pressure is higher.

Using $M_{\oplus }=5.97\times 10^{24}\mathrm{kg}$, $R_{\oplus }=6.37\times 10^6\mathrm{m}$, $M_J=1.90\times 10^{27}\mathrm{kg}$, and $R_J=7.15\times 10^7\mathrm{m}$, the universal bounds are approximately $5.75\times 10^{10}\mathrm{Pa}$ for Earth and $3.66\times 10^{11}\mathrm{Pa}$ for Jupiter. The uniform-density estimates are respectively $1.72\times 10^{11}\mathrm{Pa}$ and $1.10\times 10^{12}\mathrm{Pa}$, or $1.72$ and $11.0$ megabars. Either comparison gives a Jupiter minimum about 6.4 times the Earth minimum.

At $10\mathrm{au}$ from a four-solar-luminosity star, the incident flux is proportional to $4/10^2=1/25$, approximately the same as for Jupiter at $5\mathrm{au}$ around the Sun. Thus Jupiter supplies a possible old comparison at roughly the same mass and irradiation.

Assume the radius law holds from the young epoch to an old age of $4.5\mathrm{Gyr}$, that the irradiation history can be represented by the same fixed value, and that the present old radius is $R_J$. If $t$ is approximated by stellar age, thenIf $t$ instead starts at completion of formation, the model's formation delay of up to $3\mathrm{Myr}$ leaves the young planet with thermal age between $2$ and $5\mathrm{Myr}$. Taking the old thermal age as approximately $4500\mathrm{Myr}$ then gives $\alpha \simeq 0.090$–$0.102$.

Thus an exponent of order 0.1 is a reasonable conditional estimate. A single radius at one age cannot determine both the exponent and normalization of a power law; the old Jupiter comparison and the assumptions about formation time and irradiation are essential.

Let $L_J$ be Jupiter's current total emitted luminosity. Under the stipulated split, its intrinsic luminosity after removal of sunlight is approximately $L_J/2$. If the accessible energy reservoir down to $T_{\mathrm{cold}}$ is $E_J$, the constant-luminosity estimate gives $\tau \simeq E_J/(L_J/2)$.

Assume the more massive giant has approximately the same radius as Jupiter, a comparable structure factor, and an accessible cooling reservoir scaling as $G{M}^2/R$, as in the Kelvin-Helmholtz cooling time. Assume also that the final cold state contributes negligibly to that reservoir and that the giant radiates at the constant stipulated luminosity $L_J$. ThenThereforeThis is a characteristic scaling under the stated assumptions, not a detailed evolutionary age. A different mass-radius relation, a different accessible internal-energy fraction, or time-dependent luminosity changes the result; the mass and luminosity alone do not uniquely determine a cooling time.

Assume a hydrogen-helium perfect gas with mean particle mass $2.3m_H$, Jupiter gravity $g\simeq 24.8\mathrm{m}{\mathrm{s}}^{-2}$, and mixing over one atmospheric scale height. At $T=1000\mathrm{K}$,The eddy mixing time is ${\tau }_{\mathrm{mix}}\simeq H^2/K_{zz}$. A chemical quench level at one bar requires ${\tau }_{\mathrm{mix}}\lesssim {\tau }_{\mathrm{chem}}=10^5\mathrm{s}$, hence the vertical eddy diffusion coefficient must be of orderThe number depends quadratically on the assumed mixing length; using a fraction of $H$ reduces it accordingly.

Above the chemical quench level, neglect photochemistry, condensation, and molecular diffusion. The quenched atmospheric mixing ratio $f_A=n_A/n$ is approximately constant, while the number density isThus the abundance fraction is frozen, but the absolute number density falls with pressure and altitude. Examples are carbon monoxide–methane quenching, through $\mathrm{CO}+3{\mathrm{H}}_2\rightleftharpoons {\mathrm{CH}}_4+{\mathrm{H}}_2\mathrm{O}$, and nitrogen–ammonia quenching, through ${\mathrm{N}}_2+3{\mathrm{H}}_2\rightleftharpoons 2{\mathrm{NH}}_3$.