Earth and Planetary Physics ? Problems ? 2 Planetary interiors

Jacobs University Bremen
Joachim Vogt
Spring 2009
2 Planetary interiors
Earth and Planetary Physics
— Problems —
2.1 Heat production of a granite layer (Q)
What is the thickness of a granite layer (density ρ = 2.6 · 10 3 kg m −3 and heat production rate
1.07 · 10 −9 J kg −1 s −1 ) that yields the observed heat flow Q = 75 · 10 −3 W m −2 from the Earth’s
interior through the surface ?
2.2 Central pressure for uniform density and constant gravity (Q)
To obtain a first crude estimate for the pressure Pc at the center of a spherical body, assume not
only the density ρ to be uniform, but also the gravity g to be constant throughout the body (and
equal to its surface value). Use the hydrostatic equilibrium condition to compute Pc.
2.3 Central pressure for uniform density and linear gravity profile (E)
Derive the formula given in class for the pressure Pc at the center of a spherical uniform body
in hydrostatic equilibrium, and verify the estimates for the minimum radii of spherical bodies.
(a) Show that the radial gravity profile g = g(r) is linear in the radial coordinate r, i.e.,
g(r) = gs
where gs = −GM/R 2 denotes the surface value of the gravity.
Hint: Note that in general ∇ · g = −4πGρ where ρ is the mass density, and G is the
gravitational constant.
(b) The resulting expression for g(r) will be used to integrate the hydrostatic equilibrium
condition, and to obtain the radial pressure profile P(r). Express the uniform mass density
ρ in terms of the total mass M of the planet.
1
r
R

(c) To estimate the minimum radii of (quasi-)spherical objects, insert numerical values of
the material strengths for Pc. For rocky bodies, use ∼ 2 · 10 8 Pa and an (uncompressed)
density ρ = 3.5 · 10 3 kg/m 3 . For iron bodies, the material strength is ∼ 4 · 10 8 Pa and
ρ = 7.9 · 10 3 kg/m 3 .
2.4 Earth’s moment of inertia and core density (E)
Use the values RE and ME of the spherical Earth model and the core radius Rc = 3480 km (infered
from seismological recordings) to determine the core density ρc and the mantle density
ρm of a simplified two-layer model of the Earth’s interior. The model comprises only the mantle
and the core. Suppose that both regions are characterized by homogeneous mass density
functions ρm and ρc.
Obviously, you need additional information to solve the problem.
(a) Use a typical surface rock density value as a proxy for the mantle density (ρm = 3 g/cm 3 )
and then determine ρc.
(b) Compute the moment of inertia for the spherical two-layer Earth model and identiy this
theoretical value with the measured value of the Earth’s (main) moment of inertia C given
in class. Now determine, both, the mantle density ρm and the core density ρc from the two
equations for the total mass and the total angular momentum.
Hint: The equations become simpler if normalized variables are used. For normalization of the
densities you may use the mean density of the whole Earth:
ρ E =
mass of the Earth
volume of the Earth
and the core radius can be normalized by RE. This yields
ˆρm = ρm/ρ E ,
ˆρc = ρc/ρ E ,
ˆRc = Rc/RE .
3ME
=
4πR3 ,
E
2.5 Temperature, pressure, and density in the Earth’s mantle [H3]
In this problem you are supposed to determine approximately the profiles of temperature T,
pressure P, and mass density ρ in the Earth’s mantle.
(a) In order to find the temperature profile T = T(r), assume that the gradient is adiabatic:
dT
α g
= − T .
dr
cp
adiab
2

Use constant values for the parameters in this equation: gravity g = 9.81 m s −2 = go
(surface value), specific heat (at constant pressure) cp = 10 3 J kg −1 K −1 , and thermal expansion
coefficient α = 3 · 10 −5 K −1 . The temperature boundary condition at the surface
is To = 300 K.
(b) Compute P = P(r) and ρ = ρ(r) using the hydrostatic equilibrium condition and a fixed
value of the adiabatic compressibility: β = 4.3 · 10−12 Pa −1 . Note that
β = 1
∂ρ
.
ρ ∂P
adiab
The boundary values are ρo = 3.3 · 10 3 kg m −3 and Po = 0 Pa.
(c) Plot ρ, P and T as functions of radial distance r.
2.6 Interior of gas giants [H3]
This is problem 6.28 in the textbook Planetary Sciences of Imke de Pater and Jack J. Lissauer.
Approximate Jupiter and Saturn by pure hydrogen spheres, with an equation of state P = Kρ 2
and K = 2.7 · 10 12 cm 5 g −1 s −2 .
(a) Determine the moment of inertia for the planets.
(b) Assume the planets each have a core of density 10 g cm −3 , and the moment of inertia
I/MR 2 = 0.253 for Jupiter and 0.227 for Saturn. Determine the mass of the cores.
Submission due date for homework problems: 9 March 2009
3