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An exoplanet transmission spectrum measures a wavelength-dependent transit depth. In a simple opaque-disk geometry with uniform stellar brightness, an effective planetary radius $R_p$ gives $D=(R_p/R_{\ast }{)}^2$, where $R_{\ast }$ is the stellar radius. If stronger atmospheric absorption increases that radius by $\Delta R=N_{H}H$, define $H$ as the atmospheric scale height and $N_H$ as the number of scale heights separating the two spectral levels. The additional occulted annulus givesThe last approximation requires $N_{H}H\ll R_p$. For instance, $R_p/R_{\ast }=0.1$ and $N_{H}H/R_p=0.05$ give exact $\Delta D=0.001025$ and leading estimate $0.001$; these are stipulated geometric ratios.

Since $H=k_{B}T/(\mu m_{H}g)$ in the thin isothermal atmosphere model, larger temperature, smaller mean molecular weight or weaker gravity can increase the signal. The factor $N_H$ is set by the atmospheric opacity, composition and cloud structure; it is not a universal constant. Limb darkening, stellar heterogeneity and instrumental effects also affect the observed transit depth.^[ref]

For spherical hydrostatic equilibrium with negligible surface pressure, ideal gas support and constant mean molecular weight $\mu$, the stellar virial theorem gives $3k_B\int Tdm/(\mu m_p)=\int Gmdm/r$. Since $r<R$ throughout the interior, the right side exceeds $G{M}^2/(2R)$. Division by $M$ gives the stated mass-weighted temperature bound. The central bound additionally assumes that the central temperature is maximal. A central pressure lower bound alone is insufficient without controlling central mass density.

With constant gravity and mean molecular weight, hydrostatic equilibrium gives $P_c=P_s\exp (-H_c/h)$, where $h$ is the atmospheric scale height. A flat exoplanet transmission spectrum alone does not determine $H_c$ or a unique cloud-top pressure.

For homologous, chemically uniform, gas-pressure-supported radiative stars with specific energy generation $ϵ\propto X\rho T^{13}$ and composition-dependent constant opacity, stellar homology gives $L_{\mathrm{nuc}}\propto X{\mu }^{13}{M}^{15}{R}^{-16}$ and $L_{\mathrm{rad}}\propto {\mu }^4{M}^3/\kappa$. Equating them gives the displayed scalings. Hydrogen depletion changes both mean molecular weight and electron-scattering opacity; holding composition fixed gives $R\propto M^{3/4}$ and $L\propto M^3$. The power-law burning approximation applies only while hydrogen fuel remains.

For convective contraction with power-law opacity $\kappa \propto \rho T^4$ at fixed stellar mass and opacity normalization, $T_e\propto {\mu }^{1/2}{R}^{1/8}$ and $L\propto {\mu }^2{R}^{5/2}$. Eliminating $R$ gives $L\propto {\mu }^{-8}{T}_e^{20}$. Thus a larger mean molecular weight moves the track to larger effective temperature at a given luminosity, with $T_e{|}_L\propto {\mu }^{2/5}$. Contraction decreases both temperature and luminosity for this particular opacity exponent.

Fingering convection is a form of double-diffusive convection in which a rapidly diffusing stabilizing contribution to buoyancy competes with a more slowly diffusing destabilizing contribution. The combined mass density stratification can resist an ordinary vertical displacement, yet unequal diffusion permits narrow rising and sinking structures to grow.

A displaced parcel initially carries both contributions. If the restoring anomaly diffuses away faster than the destabilizing anomaly, the remaining buoyancy reinforces its motion. In the common thermal-compositional case, thermal diffusivity exceeds solutal diffusivity; the initial stability of the combined mass density profile is therefore insufficient to decide stability once diffusion acts.^[ref]

In seawater, salt fingering realizes this competition with temperature and salinity: the thermal contribution stabilizes while salt-rich material above fresher material supplies the destabilizing contribution. Its laboratory examples make the different transport rates visible.^[ref]

In a star, chemical composition can play the slower role. A larger mean molecular weight tends to increase mass density at fixed pressure and temperature, so a molecular-weight inversion can oppose the thermal restoring effect. Rapid thermal transport can then permit fingering even when the total local stratification is stable.^[ref]

The density ratio measures the competition, and the salt-finger instability criterion shows why there is also an upper limit to the stabilizing contribution: if it is too strong, unequal diffusion no longer produces growing modes. The elevator mode gives a particularly simple local pattern with vertical motion and no height dependence.

With negligible heavy-element contribution to the mean molecular weight, $\mu =4/(3+5X)$ and $\kappa \propto 1+X$. At fixed mass, composition-dependent CNO homology gives $L\propto (1+X{)}^{-1}(3+5X{)}^{-4}$ and $R\propto [X(1+X){]}^{1/16}(3+5X{)}^{-9/16}$. The Stefan–Boltzmann law then gives $T_e\propto X^{-1/32}(1+X{)}^{-9/32}(3+5X{)}^{-23/32}$. From the formal $X=1$ reference, declining $X$ initially increases luminosity and effective temperature, unlike the usual redward main-sequence turn-off. A trace CNO catalyst abundance is implicit in the burning coefficient; an exactly metal-free pure-hydrogen star cannot burn through that cycle.

In an optically thick steady spherical stellar wind of constant mean molecular weight ${\mu }_{\mathrm{mol}}$, let $v$ be the outward speed and $a_g^2=k_{B}T/({\mu }_{\mathrm{mol}}{m}_u)$ the isothermal sound speed squared. Mass conservation gives ${\rho }^{\prime }/\rho =-2/r-v^{\prime }/v$, where a prime denotes differentiation with respect to radius $r$. Use $P_{\mathrm{gas}}=\rho a_g^2$ and radiative diffusion to write $P_{\mathrm{rad}}^{\prime }/\rho =-g\Gamma$, where $g=G{M}_r/r^2$ and $\Gamma =L_r/L_{\mathrm{Edd}}$ uses the local Eddington luminosity $L_{\mathrm{Edd}}=4\pi cG{M}_r/\kappa$ and opacity $\kappa$. Substituting into the momentum equation yields the displayed wind equation. Smooth sonic point crossing requires both its denominator and numerator to vanish. The radiation-pressure gradient already supplies radiative acceleration; adding that force separately would count it twice.

For a fixed-mass fully mixed hydrogen burning star with $\mu =4/(3+5X)$ whose radiative stellar homology gives $L\propto {\mu }^7$, put $s=3+5X$. The luminosity is $L=L_0(s_0/s{)}^7$ and fuel conservation gives $M{E}_H\dot{X}=-L$. Hence $d(s^8)/dt=-40L_0{s}_0^7/(M{E}_H)$, producing the displayed feedback law. This applies both to CNO homology with density-dependent inverse-cubic opacity and to radiative homology with fifth-power hydrogen burning and inverse-cubic opacity, at fixed metal mass fraction in the latter. Its physical domain ends at $X=0$, before the formal denominator vanishes. The idealization holds the coefficient normalizations fixed while stellar composition changes through the mean molecular weight and fuel abundance.

The ideal gas law is the equation of state relating the absolute pressure $P$, volume $V$, particle count $N$, and absolute thermodynamic temperature $T$ of a classical ideal gas:Here $k_B$ is the Boltzmann constant, $N_{\mathrm{A}}$ is the Avogadro constant, $R$ is the molar gas constant, and $n_{\mathrm{mol}}$ is the amount of substance in moles. The distinct symbol $n_{\mathrm{mol}}$ avoids confusing amount of substance with the number density $n=N/V$, measured in ${\mathrm{m}}^{-3}$: the same law can be written $P=n{k}_{B}T$. Use absolute pressure, relative to a vacuum, rather than an unconverted gauge pressure. ^[ref]

The exact SI definitions of the Avogadro constant and Boltzmann constant giveThis numerical value is exact; the applicability of the ideal gas model to a real gas is an approximation. ^{[ref] [ref]}

For $n_{\mathrm{mol}}=1\mathrm{mol}$, $T=300\mathrm{K}$, and $V=0.025{\mathrm{m}}^3$, direct substitution givesAt fixed temperature and particle count, doubling the volume to $0.050{\mathrm{m}}^3$ halves the pressure to approximately $49.887\mathrm{kPa}$: this is Boyle's law. At fixed volume and particle count, doubling the temperature from $300\mathrm{K}$ to $600\mathrm{K}$ doubles the pressure to approximately $199.547\mathrm{kPa}$. These ratios use kelvins, not readings in the degree Celsius scale: $300\mathrm{K}$ is $26.85{}^{\circ }\mathrm{C}$, and $600\mathrm{K}$ is $326.85{}^{\circ }\mathrm{C}$.

The curved slices show Boyle's law at three temperatures; the straight slice shows proportionality to absolute temperature at one fixed volume. They are relations among equilibrium states, rather than a prescribed process history. Compression could hold the temperature fixed through heat transfer, or raise it during adiabatic compression. To calculate a process's heat or work, supplement the ideal gas law with the process constraint, the first law of thermodynamics, and the relevant internal energy or heat capacity information. The graph's extrapolation toward $T=0$ is mathematical; it does not assert that a real gas remains classical or gaseous there.

For a stellar mixture with fixed mean molecular weight, the ideal gas law can be written $P=\mathcal{R}\rho T/\mu$, where $\rho$ is mass density and $\mathcal{R}$ is the stellar gas constant. It assumes negligible interaction corrections and a nondegenerate gas; excitation or ionization can change the particle count and thermodynamic response.

In his 1834 memoir on the motive power of heat, Émile Clapeyron combined gas laws into a relation between pressure, volume, and temperature while analysing heat engines (printed page 164). The equation above uses modern SI notation and absolute thermodynamic temperature; it is not a transcription of his historical temperature convention. ^[ref]

Let $R_p$ be a reference radius at pressure $P_0$ in an isothermal atmosphere. It can be a reference isobar on a gas giant, rather than a solid surface. Assume constant temperature $T$, constant mean molecular weight $\mu$ in units of hydrogen mass $m_H$, and approximately constant gravitational acceleration $g$. The ideal gas law and hydrostatic equilibrium giveIntegrating from the reference isobar yieldsReducing the pressure by a factor $e$ therefore increases the pressure-level radius by one atmospheric scale height. Constant gravity is a thin-atmosphere approximation over the height range being modeled.

A pressure-level radius is not automatically the observed transit radius. Transmission depends on the line-of-sight optical depth, which also requires composition and opacity. Different reference pressures or cloud assumptions can therefore assign different pressure levels to the same observed exoplanet transmission spectrum.^[ref]

The Ledoux criterion tests local adiabatic stellar convective stability in the presence of a composition gradient. Definewhere $\mu$ is the mean molecular weight and the partial derivatives describe the equation of state. For positive $\delta$, the stratification is stable to an infinitesimal composition-preserving adiabatic process displacement whenThe parcel adjusts its pressure to its surroundings, retains its composition and changes temperature along the adiabatic temperature gradient. Comparing its resulting mass density with the ambient value determines the restoring buoyancy. For positive $\varphi /\delta$, increasing mean molecular weight inward gives ${\nabla }_{\mu }>0$ and an additional stabilizing contribution. With uniform composition the condition reduces to the Schwarzschild criterion. Thermal or compositional diffusion introduces further instability questions beyond this adiabatic parcel test.

For a gas-pressure-supported polytrope of index five with uniform mean molecular weight, $T/T_c=\theta$. A specific stellar energy-generation rate $ϵ={ϵ}_0\rho T^q$ therefore integrates to $4\pi {ϵ}_0{\alpha }^3{\rho }_c^2{T}_c^q{\int }_0^{\infty }{\xi }^2{\theta }^{10+q}d\xi$. Substituting $u=\xi /\sqrt{3+{\xi }^2}$ gives the displayed formula, convergent for $q>-7$. At $q=11$ its dimensionless coefficient is approximately $1.02942$. The scale factor is a volume, ${\alpha }^3$, rather than an area.

The mean molecular weight per ion counts nuclei rather than nuclei plus free electrons: $\rho /({\mu }_i{m}_u)$ is the ion number density. For species with mass fractions $X_j$ and mass numbers $A_j$, ${\mu }_i^{-1}={\sum }_j{X}_j/A_j$. It is useful when ions provide a thermal ideal gas pressure but electron degeneracy pressure supplies a separate contribution.

The mean particle mass is the mass density divided by the total number density of freely moving particles. It enters $P=\rho k_{B}T/\bar{m}$. ionization changes the particle count without significantly changing the mass, so it changes $\bar{m}$. The dimensionless mean molecular weight is $\bar{m}$ divided by a stated reference mass, commonly the atomic mass unit; it must not be substituted as a dimensionless number into a formula requiring a mass.

Let $\mathcal{R}=k_B/({\mu }_m{m}_p)$ for constant mean molecular weight ${\mu }_m$ in proton-mass units, and let $\sigma$ be the Stefan–Boltzmann constant in the Stefan–Boltzmann law. The gas sound speed used here satisfies $c_s^2=\mathcal{R}T$, and $\Omega$ is the local orbital frequency. For the local opacity approximation $\kappa ={\kappa }_0{\rho }^{1/3}{T}^{10}$, a gas-supported optically thick alpha disk has $\rho \sim \Sigma /H$ and $\mathcal{R}T\sim {\Omega }^2{H}^2$. Integrated viscous heating is $F_{+}\sim \alpha \Sigma {\Omega }^3{H}^2$. Radiative diffusion gives $F_{-}\sim (\sigma /{\kappa }_0)T^{-6}{\Sigma }^{-4/3}{H}^{1/3}\sim (\sigma /{\kappa }_0){\mathcal{R}}^6{\Omega }^{-12}{\Sigma }^{-4/3}{H}^{-35/3}$. Equating the fluxes proves the thickness scaling. At fixed composition and $\alpha$, it gives $H/r\propto r^{53/82}{\Sigma }^{-7/41}$ and $c_s\propto r^{6/41}{\Sigma }^{-7/41}$ in Keplerian rotation. Radial powers therefore require a surface-density profile; they are not determined by opacity alone. The formula is local to the stated opacity regime and small disk aspect ratio.

Assume a constant downward gravitational acceleration $g$, constant composition, and an ideal gas law $p=\rho \mathcal{R}T$, where $\mathcal{R}=k_B/(\mu m_u)$ is the gas constant per unit mass. Here $m_u$ is the unified atomic mass unit; this convention makes the specified mean molecular weight dimensionless. Hydrostatic equilibrium givesIntegrating from sea level yieldsDividing by the temperature ratio gives the hydrostatic atmosphere with a constant temperature lapse rate:The model requires $T_0-\alpha z>0$ and is used here only within the troposphere.

Convert the atmospheric lapse rate to $\alpha =6.5\times 10^{-3}\mathrm{K}{\mathrm{m}}^{-1}$. At $11\mathrm{km}$, $T=228.5\mathrm{K}$. With $g=9.81\mathrm{m}{\mathrm{s}}^{-2}$, $k_B=1.380649\times 10^{-23}\mathrm{J}{\mathrm{K}}^{-1}$, $m_u\simeq 1.66054\times 10^{-27}\mathrm{kg}$ and $\mu =28$, the density exponent is $4.0825$. HenceThe mass density is therefore approximately one third of its sea-level value in this model.

Take incidence from A into B and first specify pressure amplitude. For a plane acoustic wave traveling normally to the interface, the acoustic impedance is $Z=\rho c_s$. A right-moving wave has pressure amplitude $P=ZU$, where $U$ is its fluid-velocity amplitude; a left-moving reflected wave has $P=-ZU$.

Let $P_i,P_r,P_t$ denote incident, reflected, and transmitted pressure amplitudes. Continuity of pressure and normal velocity at the interface givesSolving the two linear equations gives the acoustic reflection and transmission at an interface coefficientsFor the same perfect gas, take a common adiabatic index $\gamma$. The adiabatic sound speed satisfies $c_s^2=\gamma p/\rho$. The equal equilibrium pressure therefore implies $Z_B/Z_A=\sqrt{{\rho }_B/{\rho }_A}=\sqrt{\delta }$, givingThe word amplitude is otherwise ambiguous. For fluid velocity, or displacement at the same frequency, the corresponding answer isThese are consistent because pressure and velocity amplitudes differ by the impedance. As an energy check, the time-averaged acoustic intensity is $|P{|}^2/(2Z)$, so its transmission coefficient is $\mathcal{T}=4\sqrt{\delta }/(1+\sqrt{\delta }{)}^2\le 1$, even when the transmitted pressure amplitude exceeds the incident one.

Equal mean molecular weight alone does not logically guarantee equal $\gamma$. If the gases have different values of the adiabatic index, replace $\sqrt{\delta }$ in these formulas by $\sqrt{{\gamma }_B\delta /{\gamma }_A}$; a density-only answer requires the common-$\gamma$ assumption.

Interpret $v$ as the normal upstream speed in the shock rest frame and $\rho$ as its upstream mass density. If the quoted jet speed is in another frame, the shock speed must first be supplied to obtain this relative velocity. Use a calorically perfect gas of mean particle mass $\bar{m}=\mu m_u$.

In the strong-shock limit, ${\mathcal{M}}_1\gg 1$. The Strong-shock Rankine-Hugoniot conditions giveThe pressure expression also follows from the supplied ratio by inserting ${\mathcal{M}}_1^2=\rho v^2/(\gamma p_1)$. The ideal gas law then gives the immediate postshock temperatureThe upstream thermal contribution is negligible at this order. This is a temperature in kelvin, rather than a temperature change per unit time.

To turn the postshock thermal energy into a rate, the processed mass per unit time is $\dot{m}=\rho vS$. Its downstream specific internal energy isThus the thermal internal energy carried into the shocked gas per unit time isPer unit area, divide this expression by $S$. Its dimensions are watts; the mean molecular weight cancels because the heat capacity per unit mass is inversely proportional to it, while $T_2$ is proportional to it.

There is a second common convention for shock heating power: the loss of directed bulk kinetic energy flux. The energy fluxes of a strong adiabatic shock giveThis equals the downstream enthalpy flux, and includes pressure work as well as the advected internal energy. For $\gamma =5/3$, the compression ratio is $4$, $T_2=3\mu m_u{v}^2/(16k_B)$, ${\dot{E}}_{\mathrm{int}}=9\rho S{v}^3/32$, and ${\dot{E}}_{\mathrm{kin},\mathrm{loss}}=15\rho S{v}^3/32$. Specifying the energy convention avoids conflating these rates. A rate $dT/dt$ for a fixed slab cannot be inferred without its participating mass and thermal evolution; the supplied data fix the postshock temperature and the processed energy fluxes.

For a gas-pressure-supported stellar core of fixed mass and composition, neglect boundary terms for the illustrative whole-core estimate. The virial theorem gives $E=U+\Omega =-U$, and the monatomic gas gives $U=3M_c{k}_B\overline{T}/(2\mu m_u)$. Thus the effective heat capacity along the equilibrium sequence isThis negative heat capacity of a virialized gravitational system means that net energy loss makes a self-gravitating core contract and heat. It is different from the positive local material specific heat capacity. On the main sequence, hydrogen burning supplies most of the radiated energy, so rapid contraction is arrested. Hydrogen depletion increases the mean molecular weight; slow structural readjustment usually raises the central temperature rather than making it fall.

After core hydrogen exhaustion, a massive stellar core contracts and heats more rapidly until a later nuclear fuel ignites. There is no general reversal to cooling simply upon leaving the main sequence. If radiation pressure becomes dominant, the internal energy becomes $U=U_{\mathrm{gas}}+U_{\mathrm{rad}}$ with $3\int PdV=2U_{\mathrm{gas}}+U_{\mathrm{rad}}$. The virial theorem therefore gives $E=-U_{\mathrm{gas}}$; the radiation-dominated limit is marginally bound and its effective adiabatic index approaches $4/3$.

Quasistatic contraction can still maintain virial equilibrium. It fails dynamically only when sufficient support is lost, for example if pair creation reduces the effective adiabatic index below $4/3$, or when an iron core undergoes electron capture and photodisintegration. A fixed-composition gas-dominated contracting core with $T\propto 1/R$ is already a counterexample to any assertion that leaving the main sequence alone reverses the heating or destroys virial equilibrium.