The areas that set a planet's temperature

Start with the incoming power

Consider a spherical planet of radius $R_p$, a distance $a$ from a star. Let $F_\star$ be the stellar irradiance at the orbit, in watts per square metre measured on a surface perpendicular to the rays. Irradiance describes power arriving per area. The parallel bundle that hits the sphere has cross-section $\pi R_p^2$, even though it lands across a curved hemisphere.

The Bond albedo $A_B$ is the fraction of incident stellar power reflected back into space, summed over wavelengths and directions. The absorbed power is therefore

For a star described by effective temperature $T_\star$ and radius $R_\star$, its total emitted luminosity spreads over larger spheres as light travels outward. Equating the same luminosity through those spheres gives the geometric dilution factor:

Reading: Nicolas B. Cowan and Eric Agol, The Statistics of Albedo and Heat Recirculation on Hot Exoplanets · Nicolas B. Cowan and Eric Agol, A Model for Thermal Phase Variations of Circular and Eccentric Exoplanets

Let the whole sphere radiate

A blackbody surface at temperature $T$ emits power $\sigma T^4$ per unit area. Suppose the entire planet has one temperature and a prescribed internal heat flux $F_{\rm int}$. Its surface area is $4\pi R_p^2$. In equilibrium, emitted power equals absorbed starlight plus internal heat.

The radius cancels because both areas grow with its square. Moving the planet closer changes irradiance, however, and raising albedo removes some incoming energy before it can heat the layer. The fourth power makes temperature respond more slowly than power: sixteen times the absorbed flux produces twice the equilibrium temperature when internal heat is negligible.

A planet need not really be uniform for this temperature to be useful. It can be defined as the temperature that would radiate the same total thermal power from the same area. An arithmetic average of local temperatures generally differs because hot patches contribute disproportionately to the fourth power.

Reading: Nicolas B. Cowan and Eric Agol, The Statistics of Albedo and Heat Recirculation on Hot Exoplanets · Nicolas B. Cowan and Eric Agol, A Model for Thermal Phase Variations of Circular and Eccentric Exoplanets

Worked example with the demonstration's inputs

Use a stellar effective temperature of $4520\,\mathrm K$, an orbital distance $a/R_\star=4.88$ and $A_B=0.20$. For the internal source, set $F_{\rm int}=\sigma(150\,\mathrm K)^4$. These are fixed illustrative hot-Jupiter inputs. They are not a fit of the real MIRI observation.

Geometric dilution gives about $9.94\times10^5\,\mathrm{W\,m^{-2}}$ at the planet. Absorption removes one fifth, and distributing the remainder over the full sphere divides it by four. Taking the fourth root after adding internal heat gives $T_{\rm eq}\simeq1368\,\mathrm K$.

Now ask for the equilibrium of the patch directly under the star if it has no horizontal heat transport. Its incident rays arrive perpendicular to the surface, so there is no area factor of four. Ignoring the small internal contribution for this estimate gives about $1935\,\mathrm K$. A dark isolated patch would instead approach the specified internal-source temperature of $150\,\mathrm K$.

Those two patch temperatures are not a plausible description of every atmosphere. They are limiting cases that help locate an error: a no-transport model should heat the substellar region much more than the far side. A transport model should explain how some of that contrast changes.

Reading: Nicolas B. Cowan and Eric Agol, A Model for Thermal Phase Variations of Circular and Eccentric Exoplanets · Nicolas B. Cowan and Eric Agol, The Statistics of Albedo and Heat Recirculation on Hot Exoplanets

A representative nonuniform result

A synthetic steady layer with equatorial wind $2000\,\mathrm{m\,s^{-1}}$, synchronous body spin and areal heat capacity $10^7\,\mathrm{J\,m^{-2}K^{-1}}$ gives cell temperatures from about $735$ to $1920\,\mathrm K$. Its net nightside radiation is about $0.139$ of absorbed stellar power. The internal source has been subtracted in that nightside quantity.

These results come from the conservative longitude model derived in the next chapter, with 128 longitude cells and 24 equal-area latitude bands. The independent sparse reference script reproduces the same discrete steady solution. Increasing longitude resolution to 512 changes the nightside fraction to about $0.1377$. The model conserves global power much more accurately than this discretization difference, so a tiny conservation residual is not the same as a finely resolved atmosphere.

The result also illustrates why a single temperature is insufficient. The total emitted power satisfies the global equation, even though no observer ever sees the whole sphere at once. At different orbital phases, the observer sees different projected mixtures of those patch temperatures.

Reading: Nicolas B. Cowan and Eric Agol, A Model for Thermal Phase Variations of Circular and Eccentric Exoplanets · Nicolas B. Cowan and Eric Agol, The Statistics of Albedo and Heat Recirculation on Hot Exoplanets

From a physical temperature to a brightness temperature

A local blackbody spectrum is given by Planck radiance, a power per projected area, per solid angle and per wavelength. Integrating it over all wavelengths gives the Stefan–Boltzmann law. A real instrument instead measures a finite wavelength interval, with its own response.

A brightness temperature asks which blackbody temperature would reproduce that finite-band intensity. A mixture of hot and cool patches is usually not a single blackbody spectrum. Nor does one wavelength always escape from the same pressure as another. Thus the brightness temperature inferred from an atmospheric spectrum must be kept distinct from the one-layer temperature in a thermal transport calculation.

For a uniformly warm blackbody planet and a uniform stellar effective bolometric intensity, distance to the system cancels in their total observed flux ratio. The ratio is the squared radius ratio times the fourth power of the temperature ratio. A measured infrared ratio requires the corresponding band integral instead. Keeping the band in the calculation is essential before trying to use light to infer absorbed power.

Reading: Nicolas B. Cowan and Eric Agol, The Statistics of Albedo and Heat Recirculation on Hot Exoplanets · Taylor J. Bell et al., Nightside clouds and disequilibrium chemistry on the hot Jupiter WASP-43b