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The helium mass fraction is $Y=1-X-Z$. For the two compositions, $Y_1=0.28$ and $Y_2=0.18$. The mean molecular weight is thereforeAt fixed mass and with the same prescribed opacity coefficient, part (a) gives $T_e\propto {\mu }^{1/2}{R}^{1/8}$ and $L\propto {\mu }^2{R}^{5/2}$. Eliminating radius givesThis is the composition shift of a convective pre-main-sequence track. The higher-helium mass fraction case $X_1=0.7$ is hotter at a given luminosity, lying to the left of the $X_2=0.8$ case in a Hertzsprung-Russell diagram. Its fixed-luminosity temperature ratio is $(349/324{)}^{2/5}\simeq 1.030$.

Both stars contract, so both $L$ and $T_e$ decrease in this particular opacity model. Their nearly vertical Hayashi tracks therefore run downward and slightly to the right, with arrows in that direction:

The plotted luminosity and effective temperature are normalized to arbitrary common references; no opacity normalization or absolute age was specified. The comparison holds the given opacity coefficient fixed.

These paths cease to apply when their assumptions fail. Deuterium-burning energy-generation rate can temporarily slow contraction; sustained hydrogen burning ends the purely gravitational phase. Formation of a radiative core breaks the fully convective adiabat and can lead to a Henyey track toward hotter effective temperature. Partial ionization, a changing atmospheric opacity, and electron degeneracy pressure alter the equation of state or boundary relation. Accretion luminosity, changing mass, and magnetic inhibition of convection can also modify a young star's path.

Write $u=\cos \theta$, $S=j/\kappa$ and use axial symmetry. The radiative flux is $F=2\pi {\int }_{-1}^{1}uIdu$. Integrating the radiative transfer equation over directions givesThere are no energy sources in the atmosphere, so radiative equilibrium makes $F$ constant. Consequently

For the linear angular approximation $I=A+Cu$, the radiation-field moments areThus it satisfies the Eddington closure approximation, andSince $F$ is constant, $C^{\prime }=0$. Substitution in the radiative transfer equation, using $S=A$, gives $u{A}^{\prime }=Cu$, hence $\boxed{A^{\prime }=C}$.

In this moment approximation, the inward hemispheric flux at the surface isSetting this boundary moment to zero gives $A_0=2C/3$. ThereforeCombining it with the thermal source function and $F=\sigma T_e^4$ gives the grey atmosphere temperature law:

The surface condition is an approximate angular-moment condition. If the affine intensity were interpreted as an exact nonnegative boundary radiation field, no incident radiation would require $I(0,u)=0$ for every $u<0$; a linear polynomial vanishing on that interval would have $A_0=C=0$ and no outward flux either. The derived formal profile indeed has $I(0,-1)=-F/(4\pi )<0$. This positivity limitation of a linear Eddington intensity is why one uses the Eddington surface boundary condition as a moment closure, not the affine ansatz as an exact emergent angular intensity. An outward direction-independent intensity with no inward intensity gives the same boundary moments.

For the gas-pressure-dominated atmosphere, combine $dP/dz=-\rho g$ with $d\tau /dz=-\kappa \rho$. Thus the hydrostatic equilibrium in optical depth is $dP/d\tau =g/\kappa$. Let $w=T^4$. The grey atmosphere law implies $dw/d\tau =3T_0^4/2$. With the specified power-law opacity,At the outer vacuum limit take $P=0$ and $T=T_0$. Since $\alpha ,\beta >0$, integration givesThis is the grey-atmosphere pressure with power-law opacity relation.

At the matching radius the local radiative flux is $L_r/(4\pi r^2)=\sigma T^4$. In the thin plane-parallel atmosphere approximation this equals $F=\sigma T_e^4$, so $T=T_e$ and $T^4=2T_0^4$. Multiplying the pressure relation by ${\kappa }_0{T}^{4-4\beta }/g$ yields the required surface pressure-opacity relationHere $P\kappa$ is a product, not a pressure subscript.

Use the notation $I_{\nu }(\mu ,\tau )$ throughout, with optical depth increasing inward and $0<\mu \le 1$ for an outward ray. This removes the change in argument order used in the printed surface-intensity notation. Multiplying the radiative transfer equation by its integrating factor givesIntegrating from the surface to a deeper boundary $b$ yieldsFor a semi-infinite atmosphere with no exponentially growing homogeneous term, the boundary term vanishes as $b\to \infty$. The formal solution of the radiative transfer equation is thereforeThe usual deep-atmosphere boundary condition is necessary; a first-order differential equation alone does not fix the emerging specific intensity.

For a linear source function, the zeroth and first exponential moments give $I_{\nu }(\mu ,0)=a_0+a_1\mu$. Equivalently $I_{\nu }(\mu ,0)=S_{\nu }(\tau =\mu )$, the Eddington-Barbier relation. If the source rises inward, rays near the limb sample cooler, shallower layers, producing limb darkening.

For the finite polynomial, put $u=t/\mu$ and integrate each term. Repeated integration by parts gives ${\int }_0^{\infty }{u}^j{e}^{-u}du=j!$, so the polynomial source function moments for emergent intensity giveThe limiting grazing-ray specific intensity is $a_0$. If the displayed polynomial is only a local Taylor approximation to a more general source, this formula applies to that approximation and a source-function remainder also contributes; a finite local expansion is not automatically an exact description at every optical depth.

For a constant source in an overlying layer of optical thickness $\tau$, the radial ray has the exact transfer relationExpanding the exponentials gives the constant source transfer through a thin layer:This finite-layer result does not require the specific intensity below the layer to equal its source function. The sign of $S_{\nu }-I_{\nu }(1,\tau )$ already distinguishes emission from absorption.

An absorption line forms when a bound-bound transition increases opacity at selected frequencies. In a photosphere whose temperature decreases outward, optical depth of order one at the line frequency occurs higher than at a nearby continuum frequency. In an absorption-dominated region in local thermodynamic equilibrium, the source is the Planck function; the cooler line-forming layer then emits less specific intensity than the deeper continuum-forming layer. The result is a dark spectral feature. Scattering and departures from LTE modify the source, so this temperature-gradient picture is a useful mechanism rather than a universal identity for every line.

Two distinct situations can produce emission lines. First, a heated chromosphere has an outward temperature rise, so an optically thick line can sample gas hotter than the continuum photosphere beneath it. Secondly, a hot optically thin circumstellar envelope or stellar wind emits line photons through collisional excitation or recombination, with no comparably bright background along all rays; its added line flux can exceed the underlying continuum. Winds of hot luminous stars can also give a mixture of emission and absorption across a Doppler-broadened line profile. These examples explain the change in specific intensity through either an enhanced source function or additional emitting material.

The high luminosity-to-mass ratio makes radiative acceleration and atmospheric departures from the simple approximations important. For an illustrative fully ionized hydrogen-rich composition with electron-scattering opacity ${\kappa }_e\simeq 0.34\mathrm{c}{\mathrm{m}}^2{\mathrm{g}}^{-1}$, the stellar electron-scattering Eddington factor isThus electron scattering alone removes roughly half of the effective surface gravity. It is below the Eddington luminosity, so electron scattering alone does not prove that a hydrostatic atmosphere is impossible. Line opacity can provide substantially more radiative acceleration and can drive a stellar wind.

For a plane-parallel atmosphere, the geometrical condition is that the line-forming region be thin compared with the radius. In a simple isothermal photospheric estimate, the atmospheric scale height isThis is plane-parallel validity from pressure scale height. The mass and luminosity alone do not specify $R$ or the atmospheric temperature, so they cannot decide the geometry uniquely. For example, if $T_e=3\times 10^4\mathrm{K}$ and $\mu \simeq 0.62$, the luminosity relation gives $R\simeq 37R_{\odot }$ and $H/R\simeq 3\times 10^{-3}$. Plane-parallel geometry could then be adequate for deep weak photospheric lines. It would still fail for lines formed over an extended wind, and a cooler, more extended supergiant needs a separate geometrical assessment. This numerical example is conditional, not an additional datum about the stated star.

LTE requires collisions and local thermal processes to establish the relevant excitation and ionization populations at the local gas temperature. In the dilute outer layers of a luminous hot star, radiative transition rates can dominate collisions, and the radiation field arriving from other depths is not the local Planck function. A non-LTE stellar atmosphere should therefore use statistical equilibrium with radiative and collisional rates coupled to transfer. Deep layers may approach LTE, but surface ionization balances and abundance-sensitive spectral lines are particularly susceptible to departures. An LTE abundance can be systematically wrong even when a plane-parallel approximation is geometrically good.

A static atmosphere requires velocities to be negligible where the diagnostic lines form. Massive luminous stars commonly have line-driven outflows, so static models cannot represent wind-formed profiles or their mass density and velocity gradients. A hydrostatic approximation can still be useful below the sonic region if the chosen lines form there, provided radiative acceleration is included in the effective gravity. Large outflows call for spherical moving-atmosphere models rather than merely a changed abundance in a static model.

The three assumptions cannot be accepted on the quoted mass and luminosity alone. LTE is especially doubtful in the outer layers; wind-sensitive abundance diagnostics require non-LTE moving models, and geometry must be checked through atmospheric extension and line-formation depth. Reliable stellar chemical abundances should be supported by observed wind signatures, temperature and gravity constraints, and agreement between several suitable ionization stages or weak photospheric lines.

In stellar homology, the dimensionless radial profiles are the same after scaling radius, enclosed mass, pressure, temperature and luminosity. At corresponding radii let $r^{\prime }=\xi r$, $m_r^{\prime }=m{m}_r$, $T^{\prime }=tT$, $P^{\prime }=pP$ and $L_r^{\prime }=l{L}_r$. The printed $R^{\prime }=\xi r$ is understood as this local radial scaling, with surface radius $R^{\prime }=\xi R$.

Let $q={\rho }^{\prime }/\rho$ and keep composition, opacity coefficient and nuclear coefficient fixed first. Mass conservation gives $q=m/{\xi }^3$, and hydrostatic equilibrium gives $p=mq/\xi =m^2/{\xi }^4$. The ideal gas law then gives $t=p/q=m/\xi$. Scaling the stellar energy-generation rate equation and the stellar radiative temperature gradient gives respectivelyThe second follows from $dT/dr\propto -\kappa \rho L_r/(r^2{T}^3)$, not from energy production. Equating the two luminosity scalings givesThis assumes the denominator is nonzero and a consistent homologous family exists. If the mean molecular weight differs by $u={\mu }^{\prime }/\mu$, then $t=um/\xi$ andRatios of the opacity and energy-generation coefficients multiply the right-hand side. Thus fixed composition is a real restriction, not an automatic property of every stellar sequence.

Use the Stefan–Boltzmann law $L=4\pi R^2\sigma T_{\mathrm{eff}}^4$ to locate the family on a Hertzsprung-Russell diagram. It implies $T_{\mathrm{eff}}\propto M^{(Y-2X)/4}$ andFor proton–proton chain burning take the usual local approximation $\eta =4$, with Kramers' opacity law $\lambda =1$, $\nu =-7/2$. ThenFor the CNO cycle with electron-scattering opacity, $\lambda =\nu =0$, soA conventional local choice $\eta =16$ gives $R\propto M^{15/19}$ and slope $76/9\simeq 8.44$. Choosing $\eta =17$ instead gives slope $60/7\simeq 8.57$. The source specifies no numerical nuclear exponents, and the effective exponent changes with temperature; the general expression is the unambiguous answer.

The Hertzsprung-Russell diagram places hotter stars to the left. Its branches rise toward higher luminosity and mass, and the CNO cycle/electron-scattering opacity branch has the steeper logarithmic slope. Their illustrative joining point and normalization are arbitrary because the proportional opacity and reaction laws do not specify absolute stellar scales. These are the fully radiative ideal-gas homology predictions, not exact observed main sequence relations: convection and increasing radiation support limit those assumptions in real stars.

The chosen opacity law makes $\kappa L_r/m_r=\alpha L/M$ spatially constant. This closes the Eddington standard model relation between its radiation-pressure and total-pressure gradients. The four roman-numbered proofs use the whole model, not only this opacity assumption.

Both gas and radiation pressures are nonnegative, so $0\le \beta \le 1$. Rearranging the constant-fraction result givesThis is the Eddington luminosity for the effective opacity constant of this model. Positive gas support gives a strict inequality, with equality only in the radiation-only limiting case. Exceeding the limit would require a negative gas-pressure fraction, which is incompatible with the assumed equation of state.

Divide the radiation-pressure gradient by the total-pressure gradient. Radiative diffusion in a star and hydrostatic equilibrium giveThis last quantity is independent of radius under the specified opacity assumption. Integration gives $P_r=fP+C$. With the standard idealized zero-pressure outer boundary, where both components vanish, $C=0$. The stellar gas-pressure fraction is thereforeconstant throughout the model. This is the Eddington standard model idealization. A finite photospheric pressure offset would need its boundary treatment; the differential relation alone does not set the integration constant to zero.

Interpret the quoted band amplitude as an absolute transit-depth increment $\Delta D=0.0003$, or $300$ parts per million. For a thin atmospheric annulus of extra height $\Delta z$, the atmospheric spectral-feature amplitude isLet the water band span $N_H$ atmospheric scale heights, so $\Delta z=N_{H}H$. For a nearly isothermal atmosphere with constant gravity and mean molecular weight $\mu$, $H=k_{B}T/(\mu m_{u}g)$. The temperature from a transmission-band amplitude is thereforeHere $T_0,H_0,{\mu }_0,g_0$ refer to the chosen Jupiter reference layer. Assuming $g=g_J\simeq 25\mathrm{m}{\mathrm{s}}^{-2}$, an H/He exoplanet atmosphere with $\mu \simeq 2.3$, and a clear strong band with $N_H\simeq 5$, gives $H\simeq 210\mathrm{km}$ and $T\simeq 1.4\times 10^3\mathrm{K}$. Equivalently, a Jovian reference with $T_0\simeq 165\mathrm{K}$ has $H_0\simeq 24\mathrm{km}$, giving $H/H_0\simeq 8.7$.

The assumptions are hydrostatic, approximately isothermal limb gas, constant gravity, negligible cloud truncation, H/He composition, and an effective opacity contrast yielding about five scale heights. In an ideal clear limb, $N_H\simeq \log ({\kappa }_{\mathrm{band}}/{\kappa }_{\mathrm{cont}})$ follows from the slant optical depth of an isothermal atmosphere. The data fix $N_{H}H$, not $H$ separately: the temperature is conditional on the band opacity contrast and composition. For $N_H=3$–$7$ with the same composition and gravity, the estimate is about $1000$–$2400\mathrm{K}$.

Let $z$ increase outward and define the inward optical depth by $d\tau /dz=-\bar{\kappa }\rho$. With constant outward internal radiative flux $F=\sigma T_{\mathrm{int}}^4$, radiative diffusion givesThe constant $q$ is a boundary condition; diffusion alone does not determine it. For an unirradiated grey atmosphere with the Eddington closure approximation and Eddington surface boundary condition, $q=2/3$, soThe internal effective temperature of a planet is defined by its intrinsic cooling flux, not by its incident stellar heating. Since $dT/d\tau >0$ and $d\tau /dz<0$, temperature decreases outward. Towards the thin upper layers, $T\to 2^{-1/4}{T}_{\mathrm{int}}\simeq 0.84T_{\mathrm{int}}$; if the density and optical-depth gradient vanish there, $dT/dz\to 0$. This is the upper nearly isothermal atmosphere. The diffusion approximation itself fails at small optical depth: the grey boundary closure supplies the approximate continuation. Small irradiation perturbs this intrinsic-flux solution.

Young, self-luminous gas giants at wide orbital separations can have intrinsic cooling dominate their photospheric budget. They are favorable for exoplanet direct imaging, especially in the infrared, where their own thermal radiation is easier to separate from the host light. A strongly irradiated hot Jupiter instead has a large stable outer radiative region set mainly by stellar heating; it can be nearly isothermal over a broad pressure range or develop an atmospheric thermal inversion if stellar light is absorbed sufficiently high. Inversion is not inevitable for every strongly irradiated planet. Deep convection begins below its radiative-convective boundary.

The intrinsic curve follows the grey formula. The irradiated curves illustrate possible shapes only; they are not solutions for a specified opacity model. Smaller optical depth corresponds to greater altitude.

In local thermodynamic equilibrium, thermal intensity samples the Planck function near an optical depth of order unity. The Eddington-Barbier relation makes this explicit: $I_{\lambda }(0,\mu )\simeq B_{\lambda }[T({\tau }_{\lambda }=\mu )]$. A molecular band has greater opacity than its adjacent continuum and therefore samples a higher layer. A band in emission relative to the continuum implies that this higher layer is hotter: the line-forming region has an atmospheric thermal inversion under the assumed LTE, thermal interpretation.

The continuum is thermal radiation from an optically thick, deeper photosphere, with comparatively smooth opacity. In an H/He hot Jupiter, collision-induced absorption by H2-H2 and H2-He collisions supplies an important continuum; weak overlapping molecular lines and opaque exoplanet clouds can contribute too. It is not a separate blackbody emitter floating above the gas. A strongly isothermal layer would erase LTE molecular contrast rather than generate emission peaks.

In the emitting inversion, $dT/dz>0$, whereas the dry adiabatic lapse rate has $dT/dz=-g/C_p<0$. Thereforewhich lies on the stable side of the Schwarzschild criterion. An upward-displaced parcel cools and becomes denser than the ambient hot upper gas. The region can thus carry and redistribute thermal energy by radiative transfer, not by unstable thermal convection. Winds may transport energy horizontally; stability rules out the specified buoyant vertical convection, not every possible motion.

An atmospheric thermal inversion is an altitude interval with $dT/dz>0$. Absorption of incoming stellar radiation above the usual thermal-emitting layers can heat the upper gas faster than it cools, producing an inversion. In a semi-grey irradiated atmosphere, a large shortwave-to-infrared opacity ratio favors such high-altitude energy deposition; local infrared emitters and the intrinsic flux also matter.

In the Solar system, Earth and all four giant planets have well-known stratospheric inversions. The ozone layer absorbs ultraviolet sunlight on Earth; methane and photochemical hydrocarbons absorb solar radiation in the giant planets, with aerosols contributing. The giant planets are Jupiter, Saturn, Uranus and Neptune. This refers to their stratospheric temperature rise, not to the gradient at every atmospheric level.

For hot Jupiters, influential factors include the stellar flux and spectrum; the abundances of high-altitude absorbers such as titanium monoxide and vanadium monoxide; atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio; thermal dissociation, atmospheric photochemistry and condensation; a atmospheric cold trap or atmospheric condensate rainout that removes absorbers; replenishment by vertical mixing; exoplanet clouds and atmospheric hazes; and heat redistribution by circulation. The ratio of visible heating to infrared cooling, rather than a single chemical species in isolation, determines whether an inversion persists.

The hot-Jupiter radius inflation problem is that some strongly irradiated giant planets have radii much larger than standard age-, mass- and composition-dependent cooling models predict. Greater internal entropy generally means a larger radius at fixed mass. The two broad classes of explanation are retaining existing heat by delaying cooling and depositing additional energy into the deep planet.

For delayed cooling of an inflated giant planet:

For heating of an inflated giant planet:

Surface or upper-atmosphere heating that is promptly reradiated need not raise deep entropy. The depth and long-term power budget distinguish an effective inflation mechanism from one that merely changes a photospheric temperature.