The measured mass and radius illustrate the exoplanet interior-composition degeneracy. Three plausible structures are:

a predominantly rocky iron-silicate interior carrying a small hydrogen-helium envelope whose low density inflates the radius;
a water-rich differentiated planet with rock and metal below high-pressure ice, supercritical water, and a steam atmosphere;
an iron-poor silicate-rich interior with a high-mean-molecular-mass secondary atmosphere, possibly including water and carbon dioxide.

An exoplanet transmission spectrum can distinguish a large-scale-height hydrogen atmosphere from compact steam or carbon-dioxide atmospheres. Molecular abundances, clouds, atmospheric mean molecular mass, and atmospheric escape constrain the envelope mass and evolution. An exoplanet emission spectrum constrains temperature, albedo, and internal cooling. A detected hydrogen-rich atmosphere favors the first scenario, a water-dominated high-molecular-weight spectrum supports the second, and a thin secondary atmosphere supports the third, although clouds can preserve degeneracies.

b

Solution

Potential surface habitability requires a stable liquid-water interface, a temperature and pressure outside runaway-greenhouse and high-pressure-ice regimes, and an

H_{2}

inventory large enough for collision-induced greenhouse warming but not so deep that pressure and temperature sterilize the ocean. Long-term atmospheric retention, tolerable stellar ultraviolet and particle flux, nutrient cycling, and chemical disequilibrium compatible with metabolism are also needed. A location in the circumstellar habitable zone alone is insufficient.

For

M_{p} = 5 M_{\oplus}

and

R_{p} = 2 R_{\oplus}

g = \frac{5}{2 ^{2}} g_{\oplus} ≃ 12.3 m s^{- 2} .

Taking

T = 300 K

μ = 2.3

H = \frac{k _{B} T}{μ m _{H} g} ≃ 88 km .

If a strong feature spans

N_{H} = 5

R_{s} = 0.5 R_{⊙}

Δ D ≃ \frac{2 R _{p} N _{H} H}{R _{s}^{2}} ≃ 9 \times 1 0^{- 5} ≃ 90 ppm .

Clouds, hazes, refraction, and smaller molecular abundance reduce this estimate.

\frac{d P}{d r} = - \frac{G m ρ}{r ^{2}}, \frac{d m}{d r} = 4 π r^{2} ρ .

For

P = K ρ^{1 + 1/ n}

ρ = ρ_{c} θ^{n}, r = a ξ, a^{2} = \frac{( n + 1 ) K}{4 π G} ρ_{c}^{1/ n - 1} .

Eliminating

m

\frac{1}{ξ ^{2}} \frac{d}{d ξ} (ξ^{2} \frac{d θ}{d ξ}) = - θ^{n} .

ξ_{1}

is its first zero, then

R = a ξ_{1}, M = 4 π a^{3} ρ_{c} [- ξ_{1}^{2} θ^{'} (ξ_{1})] .

Eliminating

ρ_{c}

at fixed

K

gives

M^{n - 1} R^{3 - n} = C,

so one convenient choice is

α = n - 1, β = 3 - n

. Equivalently, one may take

β = 1

and

α = (n - 1) / (3 - n)

The index

n = 0

describes an incompressible constant-density body, approximating a weakly compressed small rocky planet. The index

n = 3/2

describes a fully convective monatomic ideal gas or nonrelativistic electron degeneracy, as in a low-mass star, brown dwarf, or nonrelativistic white dwarf.