Three observational signatures constrain exoplanet atmospheric dynamics:

A displaced maximum of the exoplanet thermal phase curve. If the thermal maximum precedes exoplanet secondary eclipse, the hottest region is displaced east of the substellar longitude in the usual synchronously rotating geometry. The observed offset for HD 189733 b is consistent with eastward heat advection and atmospheric superrotation; the early infrared phase-curve measurement established this effect.
Warm nightside emission and a reduced day-night contrast. The amplitude of an exoplanet thermal phase curve constrains day-night heat redistribution. For HD 189733 b, the same $8 μ m$ observations found hemisphere brightness temperatures roughly $970$ and $1210 K$ , supporting substantial transport to the nightside instead of purely local reradiation.
Atmospheric line shifts after orbital motion is removed. A residual Doppler shift measures winds projected along the line of sight. A carbon monoxide signal blueshifted by roughly $2 km s^{- 1}$ in HD 209458 b was interpreted as day-to-night winds in high-resolution transit spectroscopy. Rotational broadening and different limb velocities can supply additional dynamical information.

Opacity, exoplanet cloud distributions, and pressure-dependent brightness temperature affect the phase-curve interpretation. These measurements constrain circulation through atmospheric models; a single offset or line shift does not uniquely determine a three-dimensional wind field.

Solution

 0  0

The geometric albedo

A_{g}

compares the planet's reflected brightness at full phase with that of a perfectly reflecting Lambertian surface in the form of a flat disc of the same projected area, illuminated by the same incident flux. The Bond albedo

A_{B}

is the fraction of total incident stellar power reflected in all directions and over all wavelengths. They need not be equal: the planetary phase integral and the incident spectral weighting connect them.

For a blackbody star, the incident flux at orbital distance

a

S = \frac{L _{⋆}}{4 π a ^{2}} = σ_{SB} T_{⋆}^{4} (\frac{R _{⋆}}{a})^{2} .

(1)

Assume negligible intrinsic luminosity, thermal emissivity one, and efficient day-night heat redistribution giving uniform emission over the whole planet. The absorbed and emitted powers are

π R_{p}^{2} S (1 - A_{B}) = 4 π R_{p}^{2} σ_{SB} T_{e}^{4} .

(2)

Using the Stefan–Boltzmann law gives the planetary equilibrium temperature

T_{e} = T_{⋆} \frac{R _{⋆}}{2 a} (1 - A_{B})^{1/4} .

(3)

For a representative emitting area encoded by a stellar flux redistribution factor

f

, the equivalent expression is

T_{e} = T_{⋆} R_{⋆} / a [f (1 - A_{B})]^{1/4},

(4)

with

f = 1/4

for global uniform emission and

1/2

for uniform dayside-only emission. The Bond albedo, rather than the geometric albedo, belongs in the energy budget. Internal heat, wavelength-dependent emissivity and imperfect redistribution require corresponding modifications.

c

 0  0

Solution

 0  0

Use the single-layer greenhouse model with a blackbody surface and one isothermal atmospheric layer. The layer is transparent to incoming stellar radiation, absorbs a fraction

α

of the surface's thermal radiation, and has the same thermal emissivity

α

by Kirchhoff's law of thermal radiation. Here

0 \leq α \leq 1

, and

T_{e}

is the planetary equilibrium temperature defined by the absorbed global stellar flux

σ_{SB} T_{e}^{4}

. Neglect convection, latent heat transport and intrinsic heat.

The layer absorbs

α σ_{SB} T_{s}^{4}

and emits

α σ_{SB} T_{a}^{4}

upward and downward. For

α > 0

, its energy balance gives

α σ_{SB} T_{s}^{4} = 2 α σ_{SB} T_{a}^{4}, T_{a}^{4} = T_{s}^{4} /2.

(1)

The surface receives the absorbed stellar flux plus downward atmospheric emission, so

σ_{SB} T_{s}^{4} = σ_{SB} T_{e}^{4} + α σ_{SB} T_{a}^{4} .

(2)

Eliminating

T_{a}

yields

T_{s} = (\frac{2}{2 - α})^{1/4} T_{e} .

(3)

The outgoing top-of-atmosphere flux is

(1 - α) σ_{SB} T_{s}^{4} + α σ_{SB} T_{a}^{4} = σ_{SB} T_{e}^{4}

, confirming overall energy conservation. At

α = 0

T_{s} = T_{e}

and the decoupled layer's temperature is not determined; at

α = 1

T_{s} = 2^{1/4} T_{e}

. This warming is the restriction of thermal escape by the absorbing and emitting layer.

d

 0  0

Solution

 0  0

A mass-radius curve of solar-composition substellar objects must specify age, irradiation and composition. For a mature, weakly irradiated hydrogen–helium sequence, the qualitative mass-radius relation has three main trends:

At low giant-planet masses, weak compression allows a rough $R \propto M^{1/3}$ scaling at nearly fixed mean mass density. Radius rises toward a broad maximum around one to a few Jupiter masses.
Through massive gas giants and brown dwarfs, radii remain near Jupiter's and eventually decline as compression and electron degeneracy pressure become important. A nonrelativistic degenerate $n = 3/2$ stellar polytrope gives the approximate polytropic mass-radius relation $R \propto M^{- 1/3}$ ; finite thermal pressure and real equations of state flatten and modify this idealized scaling.
Above the hydrogen-burning minimum mass, sustained hydrogen burning supplies the energy that supports a rising low-mass main-sequence branch. For near-solar composition, the threshold is around $80$ Jupiter masses (about $0.075$ solar masses), with composition-dependent variation. Red dwarfs then have radii increasing with mass, approximately linearly over part of the low-mass sequence.

The conventional deuterium-burning mass near

13

Jupiter masses marks a commonly used giant-planet/brown dwarf classification scale, not a sharp discontinuity in radius or a universal formation boundary. Young brown dwarfs are larger and cool by contraction; irradiation, a heavy-element core and hot-Jupiter radius inflation shift planets away from a single solar-composition curve.

Figure 1.
Schematic mass-radius trends for solar-composition bodies
. **Details**.

The drawing is a qualitative mature sequence, not an interpolation of a specified evolutionary model grid. Three relevant observations, using representative historical measurements, are:

Inflated radii: HD 209458 b has a radius around $1.35 R_{J}$ at a mass around $0.7 M_{J}$ , illustrating hot-Jupiter radius inflation relative to a simple mature unirradiated sequence. Early transit photometry measured its unusually large radius.
Compact metal-enriched planets: HD 149026 b was measured near $0.36 M_{J}$ and $0.73 R_{J}$ . Its small radius for its mass and irradiation implies substantial heavy-element enrichment in interior models, showing the importance of composition in the planetary mass-radius relation; see the discovery analysis.
Jupiter-like sizes at substellar masses: CoRoT-3b was measured near $22 M_{J}$ but only $1.0 R_{J}$ , supporting the broad near-constant-radius region extending into brown dwarf masses. Its measured mass and radius also illustrate why mass alone does not establish a formation history.

Gas-giant radius growth gives way to a nearly Jupiter-sized compressed branch, followed by rising radii in hydrogen-burning stars.

e

 0  0

Solution

 0  0

Three major research directions, appropriate to the scientific questions posed in 2018, are:

Characterizing small-planet atmospheres and habitability. Measure an exoplanet transmission spectrum and an exoplanet emission spectrum to establish whether rocky planets retain atmospheres, determine their composition and climate, and test exoplanet habitability. Interpreting an exoplanet biosignature requires a planetary and stellar context: photochemical false positives, exoplanet cloud obscuration and atmospheric escape can all affect apparently promising molecules.
Connecting atmospheres and interiors to planet formation. Combine planetary mass-radius relations with ages, host-star abundances, atmospheric metallicity of a giant planet and atmospheric carbon-to-oxygen ratio to test growth and migration histories. Exoplanet interior-composition degeneracy, chemical processing and atmospheric condensate rainout prevent a molecular abundance or bulk radius from being a unique record of birth conditions.
Understanding atmospheric circulation and thermal evolution across populations. Combine exoplanet thermal phase curves, Doppler spectroscopy and pressure-dependent spectra to test day-night heat redistribution, exoplanet cloud formation and disequilibrium chemistry in an exoplanet atmosphere. Relate these measurements to hot-Jupiter radius inflation, internal cooling and atmospheric escape, using comparative observations to distinguish mechanisms instead of fitting each planet in isolation.

These are research goals rather than claims that the relevant mechanisms or habitable atmospheres have already been established.

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Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 315 / 4

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