An Introduction to Modern Astrophysics · Problem 21.2

The Milky Way Galaxy — Carroll & Ostlie

Physics — Astrophysics · polytropic-jupiter-interior-model

Problem

Analytic functions can be derived for the pressure and density structure in the interior of Jupiter if an approximate relationship between pressure and density is assumed. A reasonable choice for a composition of pure molecular hydrogen is

P(r) = K\rho^2(r),

where

K

is a constant. This type of analytic model is known as a polytrope (see page 334ff for a discussion of polytropic stellar models). (a) By substituting the expression for the pressure into the hydrostatic equilibrium equation (Eq. 10.6) and differentiating, show that a second-order differential equation for the density can be obtained, namely

\frac{d^2\rho}{dr^2} + \frac{2}{r}\frac{d\rho}{dr} + \left(\frac{2\pi G}{K}\right)\rho = 0.

(b) Show that the equation is satisfied by

\rho(r) = \rho_c \left(\frac{\sin kr}{kr}\right),

where

\rho_c

is the density at the center of the planet and

k \equiv \left(\frac{2\pi G}{K}\right)^{1/2}.

R_J = 6.99 \times 10^7

m and assuming that the density goes to zero at the surface (i.e.,

kR_J = \pi

), determine the values of

k

and

K

. (d) Integrate Eq. (10.7) using the analytical solution for Jupiter's density as a function of radius to find an expression for the planet's interior mass,

M_r

, written in terms of

r

and

\rho_c

. Hint:

\int r(\sin kr)\,dr = \frac{1}{k^2}\sin kr - \frac{r}{k}\cos kr.

(e) Using the boundary condition that

M_r = M_J

at the surface, estimate the planet's central density. (The value obtained in this problem is lower than the result found from detailed numerical calculations, 1500 kg m

^{-3}

. One major reason for the difference is that Jupiter's composition is not purely molecular hydrogen.) (f) Make separate plots of the density and interior mass as functions of radius. (g) What is the central pressure of your model of Jupiter? (One detailed model gives a value of

8 \times 10^{12}

N m

^{-2}

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