n > 0

and fixed equation-of-state constant

K

, let

ξ_{1}

be the first zero of the regular Lane-Emden equation solution. A finite-radius model requires such a zero; for the usual nonnegative indices this holds for

n < 5

. The surface radius and mass follow from

r = α ξ

and

ρ = ρ_{c} θ^{n}

R = α ξ_{1}, M = 4 π α^{3} ρ_{c} \int_{0}^{ξ_{1}} ξ^{2} θ^{n} d ξ = 4 π α^{3} ρ_{c} [- ξ_{1}^{2} θ^{'} (ξ_{1})] .

(1)

The last equality integrates the Lane-Emden equation; define the positive Lane-Emden surface mass constant

ω_{n} = - ξ_{1}^{2} θ^{'} (ξ_{1})

. Substituting

α = C_{1} ρ_{c}^{(1 - n) / (2 n)}

gives

R = C_{1} ξ_{1} ρ_{c}^{(1 - n) / (2 n)}, M = 4 π C_{1}^{3} ω_{n} ρ_{c}^{(3 - n) / (2 n)} .

(2)

The central-density exponents cancel when the required powers are taken. Thus the polytropic mass-radius relation is

M^{n - 1} R^{3 - n} = C_{2} = (4 π C_{1}^{3} ω_{n})^{n - 1} (C_{1} ξ_{1})^{3 - n} .

(3)

It is independent of

ρ_{c}

, with

n, K

and composition held fixed. For

n = 1

the radius is fixed; for

n = 3

the mass is fixed. These limiting powers need no division by a vanishing exponent.

For

n = 0

, the literal pressure-density power and the supplied expression for

α

are singular. Interpret this case separately as an incompressible planetary interior with constant density. Then

M = 4 π ρ R^{3} /3

, or $R \propto M^{1/3}$ at fixed density. A weakly compressed rocky body, or a rough uniform-density approximation to Earth, is an example; realistic terrestrial planets are stratified and compressible.

For

n = 3/2

P \propto ρ^{5/3}

. This describes a cold nonrelativistic degenerate electron gas of fixed composition, or a fully convective monatomic ideal gas at fixed entropy. Examples are a nonrelativistic white dwarf, a sufficiently cooled partly degenerate brown dwarf as an approximation, and an approximately fully convective low-mass star for the ideal-gas version. The fixed-

K

scaling is $R \propto M^{- 1/3}$ . It must not be applied to an entire main-sequence stellar sequence with different entropies; the entropy dependence of a polytropic mass-radius relation explains the distinction.

b

 0  0

Solution

 0  0

The mass-radius curve of solar-composition substellar objects reflects the transition from weak compression to pressure ionization and electron degeneracy pressure, followed by sustained hydrogen burning. A schematic joining representative object classes is:

Figure 1.
Schematic mass-radius sequence from ice giants through gas giants and brown dwarfs to low-mass stars
.

The illustration is not an age-specific numerical evolutionary model. In particular, ice giants contain much more heavy material than a solar-composition giant, so a single uniform-composition equation of state does not describe the entire joined curve.

At the low-mass, weak-compression end, a fixed-density or fixed-composition approximation gives $R \propto M^{1/3}$ . Ice giants such as Neptune and Uranus have substantial water/rock-rich interiors and modest H/He envelopes; changing envelope fraction changes the radius markedly. Their heat comes from retained formation energy, contraction and radiogenic heating of heavy material. Fluid interiors generally convect, while composition stratification can impede mixing; outer radiative layers release the heat.
Ordinary gas giants reach radii of order $R_{J}$ over a broad range around Jovian masses: an effective $n ≃ 1$ polytrope explains the approximate $R \propto M^{0}$ segment. Increased mass compresses material enough to offset the added volume. Cooling and Kelvin-Helmholtz contraction, with additional differentiation energy such as helium settling in Saturn, supply the intrinsic luminosity. Their deep envelopes are usually convective, with radiative photospheres.
More massive brown dwarfs become increasingly supported by electron degeneracy pressure. The cold nonrelativistic $n = 3/2$ limit gives $R \propto M^{- 1/3}$ , but finite entropy and Coulomb effects flatten actual giant/brown-dwarf curves and their radii depend on age. They cool and contract; temporary deuterium fusion occurs above a composition-dependent deuterium-burning mass near $13 M_{J}$ . This threshold does not cause a sharp structural kink or permanent stellar luminosity. Interiors are largely convective and surface emission is radiative.
Near the hydrogen-burning minimum mass, roughly $0.075$ – $0.08 M_{⊙}$ or $75$ – $85 M_{J}$ for near-solar composition, sustained fusion prevents indefinite cooling into a degenerate object. The low-mass main-sequence star branch turns upward, with approximately $R \propto M$ over the illustrative interval. Its entropy is not constant across masses, so this branch is compatible with an approximately $n = 3/2$ internal profile. Hydrogen fusion through the proton–proton chain provides energy; the lowest-mass main-sequence stars are fully convective, capped by radiative atmospheres.

Planet/brown-dwarf naming conventions and deuterium burning do not define a universal discontinuity in the mass-radius relation. Composition, age and irradiation move the curves; the hydrogen-burning transition changes the long-term energy source more fundamentally.

c

 0  0

Solution

 0  0

For an incompressible planetary interior of density

ρ

, the enclosed mass and local gravity are

M (r) = 4 π ρ r^{3} /3

and

g (r) = 4 π Gρ r /3

. Hydrostatic equilibrium therefore gives

\frac{d P}{d r} = - \frac{GM ( r ) ρ}{r ^{2}} = - \frac{4 π G ρ ^{2}}{3} r .

(1)

Integrating inward from negligible surface pressure at

r = R_{p}

yields the uniform-density planetary pressure profile:

P (r) = \frac{2 π G ρ ^{2}}{3} (R_{p}^{2} - r^{2}) = P_{c} (1 - \frac{r ^{2}}{R _{p}^{2}}), P_{c} = \frac{2 π G ρ ^{2} R _{p}^{2}}{3} .

(2)

The surface gravity is

g_{s} = GM / R_{p}^{2} = 4 π Gρ R_{p} /3

. Eliminate

ρ R_{p}

to obtain

P_{c} = \frac{3 g _{s}^{2}}{8 π G} .

(3)

The units of

g_{s}^{2} / G

are pressure. For a nonzero imposed surface pressure, add

P_{s}

throughout and interpret the boxed central value as

P_{c} - P_{s}

. The constant-density approximation is crucial; real centrally concentrated rocky planets need a different profile and generally a larger central pressure at the same mass and radius.

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Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 59 / 3

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