Chapter
9
Interiors
We now turn our attention to interiors, and therefore to practically all of the mass
that makes up the planets. Observing the interior structure of a planet is obviously not
an easy task, but we need to know as much as we can about interiors, in part because
such information affects nearly all other aspects of planetary science. As examples, the
interior chemistry of a planet provides information about how it formed and how it has
evolved, and the way in which a planet gets rid of its internal heat has a profound effect
on its surface geology or meteorology. In addition, the coupling of a planet’s rotation to
the motions of the conductive fluid in its interior gives rise to strong magnetic fields that
greatly extend the planet’s influence past its surface boundary. Also, the details of the
internal density distribution of a planet control the shape of its external gravitational field,
and this affects the orbital dynamics of its satellites and rings.
Since the interior of a planet is impossible to see directly, we must find indirect ways
by interpreting the external signatures caused by internal structure. The single most
important parameter that describes a planet’s interior is its bulk density. The bulk
density of a planet reveals whether it is mostly made of “rock” (silicates, metals) or “ice”
(water, ammonia, methane) or “gas” (hydrogen, helium). But to really understand a
planet’s interior, we must learn about the detailed distribution of mass inside it. The
planetary interior that is best understood is Earth’s because seismologists have observed
9–1
and modeled the pressure and shear waves that emanate from earthquake sites and pass
through Earth’s interior. There is even a modest amount of seismic data for the Moon.
For Venus, Mars, and the Moon we know the shapes of their external gravity fields from
orbiting spacecraft and we have high resolution images of the volcanic and tectonic features
on their surfaces. For the Moon, Mercury, and Galilean satellites we have estimates of
gravitational oblateness. For Pluto and the rest of the icy satellites of the outer planets,
we have estimates of their masses but no other detailed gravity observations. For all of
these bodies we have imaging of at least part of their surfaces, from which we can make
limited inferences about their internal structures and dynamics.
The interiors of the giant planets themselves are more closely related to stellar interiors
than to terrestrial-planet interiors. In fact, the Sun and the four giant planets constitute
five variations on a single theme of a large, hot, rotating sphere of hydrogen and helium gas.
Of these five bodies, our best information is for the Sun, because by using precise Doppler
techniques to observe the Sun’s surface vibrations, helioseismologists have been able to
infer basic properties of its interior structure. Efforts to apply helioseismology directly
to Jupiter have been hampered by a low signal-to-noise ratio, but the lessons learned from
the solar observations indirectly influence our thinking about the giant planets. The most
direct observational constraints on the internal structure of the giant planets come from
the shapes of their gravity fields, which we know partly as the result of flyby and orbital
spacecraft missions, but mostly from accurate ground-based observations of the orbital
precessions of satellites and narrow rings.
9.1 External Gravity Fields
Studying the shape of a planet’s external gravity field is one of the best ways to find
out about its internal density distribution. Because an object composed of spherically
symmetric shells has a gravity field that is identical to that of a point mass, and because
at a great distance any object has this property, we will need to rely on deviations from
spherical symmetry and will have to probe each planet’s external gravity field at close
range to learn about its inside makeup.
What we want is a way to perturb a planet under controlled conditions, so that we
can study how the planet reacts to the perturbation and thereby learn something about
its internal structure. As luck would have it, the centripetal acceleration caused by rapid
rotation provides a natural perturbation. Consider two rotating planets that have the
same average density, the same size and the same oblate shape. Let the density of one
of the planets be uniform throughout and the density of the other be differentiated into
a heavy core and a light outer shell. The one that is rotating faster is the one with the
denser core, as we will quantify below.
It is useful to describe a planet’s rotation rate in terms of a nondimensional parameter,
q, which is formed by comparing the magnitude of a planet’s centripetal acceleration to
the magnitude of its gravitational acceleration on the surface at the equator:
q≡
RΩ2
R 3 Ω2
,
=
GM/R2
GM
(9.1)
where R, Ω, and M are the planet’s equatorial radius, angular velocity, and mass, respectively. For the giant planets R is usually taken to be the 1 bar pressure level (1 bar
9–2
is Earth’s atmospheric pressure at sea level), and Ω is taken to be the rotation rate of
the magnetic field (historically called the System III rotation rate). Some large, rapidly
rotating stars have q values that actually approach unity, implying that those stars are
on the verge of flying apart. Saturn has q = 0.153 and is the most rotationally distorted
planet in our solar system, while Jupiter has q = 0.089, and the rest of the planets have
much smaller values, as listed in Table 9.1. In a later section we will derive an expression
involving q that describes the difference between the oblateness of a planet’s shape and
the oblateness of its gravity field.
9.1.1 Poisson’s Equation and the Gravitational Potential
To understand the gravity field of an entire planet, we will first build upon what we
know about two point masses. We will take one point mass to be our test mass, which
we can place at any position r in order to determine the local gravity field g(r). For the
second mass, we will take in turn each differential mass element dm at each location r0
inside the planet. If we put d ≡ r − r0 , then the gravity field g(r) associated with the mass
element dm is the acceleration felt by the test mass when placed at the location r:
g(r) = −
G dm
G dm r − r0
= − 2 dˆ.
0
2
0
|r − r | |r − r |
d
(9.2)
If we enclose dm with any surface S, then the component of g that is normal to each
surface element dS will be given by:
n̂ · g = −
G dm
n̂ · dˆ.
d2
(9.3)
where n̂ is the unit vector that is normal to dS. Notice that, as seen from dm, the surface
element dS subtends a differential solid angle dΩ (not to be confused with angular velocity
Ω) given by:
dS
dΩ = 2 n̂ · dˆ.
(9.4)
d
That means that the total surface integral of n̂ · g is simply:
Z
Z
Z
n̂ · dˆ d2
n̂ · g dS = −G dm
(9.5)
dΩ = −G dm dΩ = −4π G dm .
2
n̂ · dˆ
S
S d
S
By the nature of the cancellation of the orientation factor n̂ · dˆ and the distance factor
d2 in (9.5) we can see that any inverse-square vector field g will have the property that
the total flux n̂ · g through an arbitrary closed surface will be a constant. Now, take the
surface S to encompass the entire planet, and include all the mass elements dm = ρ dV .
Then (9.5) becomes:
Z
Z
n̂ · g dS = −4π G
S
ρ dV .
(9.6)
V
Recall Gauss’ divergence theorem for any vector field E:
Z
Z
∇ · E dV =
n̂ · E dS .
V
S
9–3
(9.7)
By combining (9.6) with the divergence theorem (9.7), we find:
Z
(∇ · g + 4π G ρ) dV = 0 .
(9.8)
V
Since (9.8) is true for any volume V that encloses the density distribution ρ, the integrand
of (9.8) must vanish, which implies:
∇ · g = −4π G ρ .
(9.9)
Finally, if we introduce the gravitational potential field Φ:
g = −∇Φ ,
(9.10)
then:
∇ · (−∇Φ) = −4π G ρ ,
⇒
∇2 Φ = 4πG ρ .
(9.11)
Equation (9.11) is called Poisson’s Equation. In Chapter 11 we will approximate the
gravitational potential of a planetary ring by using (9.11) with the density ρ taken to be
an infinitesimally thin disk of material modeled by a delta function.
9.1.2 Spherical Harmonics
In practice we are restricted to measuring the gravitational field from the exterior of
a planet, in which case ρ = 0 and Poisson’s equation (9.11) reduces to Laplace’s equation:
∇2 Φ = 0 .
(9.12)
The simplicity of (9.12) belies the fact that it applies to the gravitational potential outside
of any distribution of mass, no matter how complicated that distribution might be. It is
advantageous to pick a coordinate system whose geometry shares the same symmetries as
the density distribution that we are trying to study. The spherical coordinate system is the
most natural system for describing a planet’s external gravitational potential. We take the
origin to be the center of mass of the planet, and use r, θ, and λ for the radial coordinate,
co-latitude, and longitude, respectively. The co-latitude is defined such that θ = 0 refers
to the north pole. The coordinate origin is the planet’s center of mass. Laplace’s equation
(9.12) written in spherical coordinates takes the form:
µ
¶
µ
¶
1 ∂
∂2Φ
1
∂
∂Φ
1
2 ∂Φ
r
+
sin
θ
+
= 0.
(9.13)
r2 ∂r
∂r
r2 sin θ ∂θ
∂θ
r2 sin2 θ ∂λ2
The general solution to (9.13) is found by separation of variables, and may be written:
Φ(r, θ, λ) =
l
∞ X
h
X
l
αlm r + βlm r
l=0 m=−l
9–4
−(l+1)
i
Ylm (θ, λ) ,
(9.14)
where the αlm and βlm terms are the sought-after coefficients that describe the planet’s
density distribution. The Ylm (θ, λ) functions are called the spherical harmonics, and
represent the complete set of orthonormal functions on the surface of a sphere. The
integer l represents the number of nodal lines on one hemisphere and is a measure of the
fineness of structure. The integer m represents the distribution of lines between latitude
and longitude. When m=0 solutions have only a latitudinal dependence and are referred
to as zonal harmonics. When m = l solutions are only a function of longitude and are
called sectoral harmonics. Solutions in which 0 < m < l have both a latitudinal and
longitudinal dependence and are tesseral harmonics.
We will find that for most applications the full generality of (9.14) is not necessary.
In fact, since Φ satisfies the boundary condition that it does not go to infinity as r goes
to infinity, we can immediately eliminate the rl terms in (9.14) by setting all of the αlm
coefficients to zero. Notice that the remaining r−(l+1) terms have the property that the
higher l terms fall off with distance more rapidly than the lower l terms. This means
that the highest-order terms of a planet’s gravitational potential will hardly be detectable
outside of the planet.
The co-latitude appears only as cos θ in the explicit form of Ylm (θ, λ), and so it is
customary to define a new coordinate µ:
µ ≡ cos θ .
(9.15)
The Ylm are then given by:
s
Ylm =
2l + 1 (l − m)! m
P (µ) eimλ .
4π (l + m)! l
(9.16)
The dependence of Ylm on longitude is Fourier-like:
eimλ = cos(mλ) + i sin(mλ) .
(9.17)
The Plm (µ) in (9.16) satisfy the equation
·
¸
m2
d2 P
dP
(1 − µ ) 2 − 2µ
+ l(l + 1) −
P =0
dµ
dµ
1 − µ2
2
which reduces to Legendre’s equation for the case of m=0, which is discussed in more
detail later. The Plm (µ) are called the associated Legendre polynomials and are
generated recursively by Rodrigues’ Formula:
µ
Plm (µ)
=
(−1)m (1 − µ2 )m/2
2l l!
¶µ
d
dµ
¶l+m
(µ2 − 1)l .
(9.18)
The first factor on the right hand side normalizes the function such that Pl0 = +1. The
use of normalizations is common because numerical factors in the associated polynomials
increase rapidly with m. Normalizing the solution causes the coefficients in the harmonic
9–5
analysis relate more clearly to the physical process that they represent. The first five
Legendre polynomials, Pl ≡ Pl0 , are:
P0 = 1 ,
P1 = µ ,
1
P2 = (3µ2 − 1) ,
2
1
P3 = (5µ3 − 3µ) ,
2
1
P4 = (35µ4 − 30µ2 + 3) .
8
(9.19)
The form of P2 (µ) will play a role when we connect the rotation parameter q to the
oblateness of Φ.
Just as with any solution written as a series expansion, the functions that make up
the spherical harmonics are all well known, and it is the βlm coefficients that hold all of
the information about a planet’s density distribution. To connect the βlm coefficients to
the density distribution ρ(r), we express Φ(r) in its integral form:
Z
Φ(r) = −G
V
ρ(r0 )
dV 0 .
|r − r0 |
(9.20)
Outside a planet, where r > r0 , the expansion of |r − r0 |−1 in terms of spherical harmonics
is given by:
(
)
∞
l
l
X
X
1
r0
4π
∗
=
Ylm
(θ0 , λ0 )Ylm (θ, λ) ,
(9.21)
|r − r0 |
rl+1 (2l + 1)
l=0
m=−l
∗
is the complex-congugate of Ylm . Applying this series expansion to (9.20) yields
where Ylm
the spherical-harmonic series expansion for Φ(r, θ, λ):
Z
Φ(r) = Φ(r, θ, λ) = −G
×
∞
l
X
r0
r(l+1)
l=0
(
∞
r 0 =0
Z
π
θ 0 =0
Z
2π
ρ(r0 , θ0 , λ0 )
λ0 =0
)
l
X
4π
2
∗
0
0
Ylm (θ , λ )Ylm (θ, λ) r0 sin θ0 dr0 dθ0 dλ0 .
(2l + 1)
(9.22)
m=−l
Thus, the βlm coefficients in (9.14) are given by an integral involving the density:
βlm
4πG
=−
(2l + 1)
Z
∞
r 0 =0
Z
π
θ 0 =0
Z
2π
λ0 =0
∗
ρ(r0 , θ0 , λ0 ) Ylm
(θ0 , λ0 ) r0
l+2
sin θ0 dr0 dθ0 dλ0 .
(9.23)
The sensitivity of the βlm coefficients to the radial distribution of density comes from the
r0l+2 weighting factor, which causes information about the outer layers to be preferentially
contained in the higher l coefficients.
9–6
9.1.3 Zonal Harmonics
In practice, one usually requires only the leading terms of the complete spherical harmonic expansion (9.14) for Φ. Also, because planets display significant rotational symmetry
and north-south symmetry, not all of the (l, m) generality in (9.14) is necessary. In such
cases, it is customary to write the spherical harmonic expansion of Φ in an “unnormalized”
form, truncated to some number of leading terms, n:
)
(
n µ ¶l X
l
X
R
GM
Plm (θ) [Clm cos(mλ) + Slm sin(mλ)] , (9.24)
1−
Φ(r, θ, λ) ≈ −
r
r
m=0
l=2
where R and M are the planet’s equatorial radius and total mass, respectively. The
expression (9.24) represents the planetary gravitational potential or geoid. The form of
(9.24) emphasizes that Φ is dominated by the point-mass potential, −GM r−1 , and that
the higher-order terms provide small corrections to this leading-order term. Note that
the summation proceeds from l = 2 because a value of one gives an asymmetric potential
which can be avoided by choosing the planet’s center of mass as the coordinate origin.
Also note that the implicit complex notation of (9.14) is replaced in (9.24) by completely
real expressions. The Clm and Slm coefficients are dimensionless multipole moments of the
density distribution that are directly related to the βlm coefficients of (9.14).
The case of rotational symmetry corresponds to m = 0 and the coefficients of greatest
interest in (9.24) are the zonal harmonic coefficients Cl0 , which are traditionally given their
own symbol:
Jl ≡ Cl0 .
(9.25)
The most important of these parameters is J2 , the ellipticity coefficient, which is typically observed to have a magnitude of 1000 times that of higher degree and order terms in
a planet’s gravity field. Assuming that ρ does not depend on λ, then if we compare (9.24)
to (9.23) and (9.14) with m = 0, we will obtain the following relationship connecting the
zonal harmonic coefficients to the density:
Z R Z 1
2π
ρ(r, µ) Pl (µ) rl+2 dµ dr ,
(9.26)
Jl = −
M Rl r=0 µ=−1
where we have made use of the identity sin θ dθ = −dµ. The values of J2 and the rotational
parameter q are shown for the various planets in Table 9.1.
Notice from (9.19) that for odd l, the Pl (µ) are odd functions across the equator. Since
planets display strong north-south symmetry, the odd l terms in the series expansion of Φ
are usually negligible when compared with the even l terms. Therefore, the leading terms
of Φ for a planet with rotational symmetry and north-south symmetry may be expressed
simply as:
(
)
µ ¶2l
n
X
GM
R
Φ(r, µ) = −
1−
J2l
P2l (µ) ,
r
r
l=1
(
)
µ ¶2
µ ¶4
GM
1
1
R
R
(3µ2 − 1) − J4
(35µ4 − 30µ2 + 3) − . . . , (9.27)
1 − J2
=−
r
r
2
r
8
9–7
Table 9.1
Planetary Internal Structure Parameters
Planet
Bulk Density J2
q
Λ2 ≡ J2 /q
C/M R2
80
98
0.31
27
0.43
0.17
0.11
0.12
0.22
??
0.34
0.3335
0.391
0.366
0.26
0.25
0.23
0.23
(kg m−3 )
Mercury
Venus
Earth
Moon
Mars
Jupiter
Saturn
Uranus
Neptune
5420
5250
5515
3340
3940
1314
690
1190
1660
0.0000(8 ± 6) 0.000001
0.00000(6 ± 3) 0.000000061
0.0010826
0.0035
0.0002024
0.0000076
0.001959
0.0046
0.014733
0.089
0.01646
0.153
0.003352
0.027
0.004
0.018
with the J2l coefficients connected to ρ through (9.26).
9.1.4 Interpreting J2
The ellipticity coefficient J2 is the principal term required to describe the oblate shape
of a planetary or stellar body, because it provides information on the radial distribution
of mass in the body. Observations of satellite and narrow-ring precession rates, together
with flyby spacecraft trajectory data, yield the best measurements of the zonal harmonic
coefficients J2 and J4 , and sometimes even J6 , for the various planets. These coefficients
provide a description of the density distribution inside a planet that reveals the extent of
its mass differentiation and core formation. They also provide the primary means by which
planetary interior models are judged for accuracy. J2 is commonly expressed in terms of
the moments of inertia about the polar (C) and equatorial (A) axes as
J2 =
C −A
.
M R2
(9.28)
where the moment about the polar axis is
Z
¡ 2
¢
C=
x + y 2 dm
and corresponds to the integral over the mass distribution composed of small elements dm
Ä to the rotation axis. In spherical coordinates we
times the square of the distance from dm
may write
Z 2π Z π Z R
ρ(r0 )r04 cos3 θdr0 dθdλ
C=
0
0
0
and A can be similarly expressed. The derivation of (9.28) is given as a homework problem.
9–8
The J2 coefficient, as it appears in (9.26) and (9.27), describes what we have been
loosely refering to as the “oblateness” of Φ. We will now connect J2 to the rotational
parameter q and to the oblateness of the planet itself. The effective potential Φe at the
surface of a rotating planet includes the gravitational potential, Φ, and the centripetal
potential:
1
1
2
Φe = Φ − |Ω × r|2 = Φ − (Ω r sin θ) ,
(9.29)
2
2
where r sin θ is the cylindrical radius measured from the rotation axis to the planet’s
surface. Since we are using (9.27) to express Φ in terms of P2 (µ), it will be convenient
here to also express the centripetal potential in terms of P2 (µ):
P2 (µ) = P2 (cos θ) =
⇒
1
1
3
(3 cos2 θ − 1) = [3(1 − sin2 θ) − 1] = 1 − sin2 θ ,
2
2
2
2
⇒ sin2 θ = [1 − P2 (µ)] ,
3
1
1 2 2
− Ω r sin2 θ = − Ω2 r2 [1 − P2 (µ)] .
2
3
The effective potential at the surface of the planet is thus:
·
¸
R2
GM
1
Φe (r, µ) = −
1 − 2 J2 P2 (µ) − . . . − Ω2 r2 [1 − P2 (µ)] .
r
r
3
Grouping terms yields:
¶ µ
¶
µ
1 2 2
GM R2
GM
1 2 2
− Ω r +
J2 + Ω r P2 (µ) + . . .
Φe (r, µ) = −
r
3
r3
3
(9.30)
(9.31)
We will now assume that the planet is in hydrostatic equilibrium, such that the surface of the planet is an equipotential surface. This is a good approximation for giant
gaseous planets. The solid planets exhibit small (at most several percent) deviations from
hydrostatic behavior due to internal dynamics and interior structure. However these small
differences provide essential insight about the structures and geodynamical states of these
bodies. We now define the planetary flattening, f =(a-c)/a where a is the equatorial
radius and c is the polar radius. Equivalently we may write:
a ≡ R,
c ≡ (1 − f ) R .
For a planet in hydrostatic equilibrium, Φe (r, µ) satisfies the difference:
Φe ( R, 0 ) − Φe [ (1 − f ) R, ±1 ] = 0 .
(9.32)
If we now apply (9.30) to (9.31), and use the fact that:
1
P2 (equator) = − ,
2
the result is:
GM
0=−
R
·
1
1−
1−f
¸
P2 (pole) = 1 ,
1
− Ω2 R 2
3
9–9
·
¸
1 − (1 − f )
2
·
¸
·
¸
1
1 2 2
1
GM
−3
2
+ Ω R − − (1 − f ) .
J2 − − (1 − f )
+
R
2
3
2
(9.33)
For all the planets in the solar system, J2 , f , and q are each much less than unity. Therefore,
we can make the approximation:
(1 − f )n ≈ 1 − n f .
(9.34)
If we apply (9.33) to (9.32), and divide by GM/R to bring out the rotation parameter q
as defined by (9.1), then we find the following relationship between J2 , f , and q:
J2 ≈
2
1
f − q.
3
3
(9.35)
Equation (9.35) provides a useful way to determine J2 for a hydrostatically balanced planet,
and hence an indication of the internal density distribution. The right-hand side of (9.35)
depends on the planet’s size, shape, mass, and rotation rate, all of which can be obtained by
ground-based observations. If J2 can be determined independently, deviations from (9.35)
provide a measure of the nonhydrostatic forces operating in the interior. In practice,
to analyze the small but significant departures of f from hydrostatic equilibrium for the
terrestrial planets it is necessary to retain terms to second order such that
µ
¶
µ
¶
f
1
3
2
2
− q 1− q− f .
(9.36)
J2 ≈ f 1 −
3
2
3
2
7
For Earth (9.36) yields f =3.3528x10−3 . The value of J2 is known to equal 1.082626x10−3 .
Note that in (9.28) J2 is not really a direct measure of the moment of inertia but
rather the difference between the polar and equatorial moments. To measure the moment
of inertia factor C/M R2 that describes the radial distribution of the interior mass another
parameter is needed. A spinning planet obeys the same laws of physics as a gyroscope.
Because of gravitational tugs from the Sun and moons, planetary spin axes wobble. For
Earth the period is 26,000 years so the direction of the North Pole as projected on the
celestial sphere changes. The rate of precession depends on a factor called the dynamic
elllipticity
C −A
H=
(9.37)
C
that is controlled by the polar and equatorial moments of inertia. We may substitute to
find
C
J2
=
.
(9.38)
H
M R2
In practice J2 is found from the rate at which a satellite orbit precesses around a planet,
and H is derived from measuring the rate of precession of the spin pole. As seen in Table
9.1 we know J2 for all of the terrestrial planets and giant planets. But we have measured
the spin axis precession rate only for the Earth and Mars. For solid planets the precession
rate can be found by measuring the orientation of the spin axis at two different times from
tracking surface landers or orbiting spacecraft. The precession rate for Mars was found
9–10
by comparing measurements found by the Viking landers to that found recently from the
Pathfinder lander.
9.2 Internal Heat
9.2.1 Sources of Planetary Heat
There are several sources of energy that contribute to the internal heat of a planet.
Early in planetary history, the accretional heat associated with impact bombardment
was a major source of heating. Also in early evolution, a significant amount of heat was
generated from the process of differentiation due to the release of gravitational potential
energy as planet’s heaviest components, like its nickel and iron, sink to the center and
form a dense core. The release of nuclear binding energy through the radioactive decay
of the uranium, thorium, and potassium found naturally in chondritic (solar) materials,
and hence in any silicate material derived from the protoplanetary nebula, is the most
important source of present-day internal heat for the terrestrial planets. Tidal dissipation is currently the dominant source of internal heat for Io, and is the most important
source of heat for most of the icy satellites of the outer solar system that display evidence
of recent tectonic and tectonic surface activity. In addition, tidal heating has played a role
in any planet that has at one time undergone substantial tidal despinning, like Mercury is
thought to have experienced. Also making a small contributions to the total heat budget
are certain types of phase changes in interior materials, and the ohmic dissipation of
internal electrical currents.
9.2.2 Mechanisms of Planetary Heat Loss
Given that planets are hot in their interiors, how do they get rid of their heat? If
there is not too much heat, a planet can cool off simply by conduction of its heat to the
surface, and then radiation into space. On the other hand, if there is more than a critical
amount of heat that needs to escape, then convection will ensue. Convection is a more
efficient means of transporting large amounts of heat than conduction because it involves
macroscopic motions of material under the simple principle that warm, buyoant material
deep in a planet will rise and cold, more dense material at depth will sink. Conduction,
on the other hand, is a diffusive process that operates by the transfer of kinetic energy
via molecular-scale collisions. Radiation is the most efficient heat loss mechanism because,
as we recall from Chapter 4, the heat flux (q) at a planetary surface scales as q = σT 4 .
However, because of the fourth power dependence on temperature, radiation is so efficient
that a hot molten surface will very rapidly (in a geologic sense) produce an insulating
crust that will cause the radiation process to cease. So in practice for the solid planets and
satellites radiation was very important during a short time during and post-accretion and
has been a minor contributor to planetary heat loss budgets in subsequent times.
For the terrestrial planets and icy satellites the loss of heat is the major factor that
controls the geologic expression of the surface. Planetary cooling causes volcanism and
tectonics. On the Earth the loss of heat by convection drives the global motions of the
Earth’s plates, producing earthquakes, mountain building, and rifting. The volcanic release
9–11
of volatiles trapped in the interior produces and modifies oceans and atmospheres. Given
the importance of the thermal state of planetary bodies for nearly all aspects of their
structure and evolution, it is clear why the study of heat budgets is a major focus of study
in compartive planetology.
9.2.3 Planetary Cooling Efficiency
An incomplete but instructive assessment of a planetary heat budget comes from
considering in the simplest possible way the global heat generation and heat flux. For
the terrestrial planets and the outer solar system satellites that have a significant rock
component, radioactive heating is the principal source of long term heating, and so the
heat generation scales with the planetary volume. Heat loss, by whatever mechanism,
scales with the planetary surface area. The ratio of the heat flux to the heat generation
is referred to as the cooling efficiency and has a form q/H ≈ 3/R, where R is the
planetary radius. We see that smaller planets produce less heat and cool faster, or more
efficiently, than larger planets, which produce more heat and cool more slowly. From this
simple approach we would expect that large planets that generate their heat primarily by
radioactive decay to be geologically active over more of their history than small planetary
bodies. Indeed we shall see that this is the case.
9.2.5 Conduction
The simplest approach to understanding conductive heat transport within a planet
comes from considering the flux of heat across a slab of material. Fourier’s Law takes
the form
dT
q = −k
(9.39)
dz
where q is the heat flux (not to be confused with the rotational parameter discussed
earlier), k is the thermal conductivity, a property of the material, and dT /dz is the
thermal gradient. This expression indicates that the flow of heat per unit area per unit
time is directly proportional to the temperature gradient at that point. The minus sign
indicates that heat flows in the direction (z) of decreasing temperature (T ). In this simple
case the gradient of temperature is linear with depth.
A relevant problem for the temperature state of small solid plaetary bodies is the radial
steady-state conduction of heat in a sphere. The solution, which derives from a straightforward energy balance for a symmetrically cooling body, is also relevant for spherical shells
and thus for conductively cooling lithospheres.
Consider a planet with surficial spherical shell of inner radius r and thickness δr. The
flux of heat out of the shell through the surface is
4π(r + δr)2 qr (r + δr)
and the flux into the base of the shell is
4πr2 qr (r)
9–12
where the subscript r on the flux indicates that we are considering radial heat flow only.
Since δr is small we expand the flux qr (r + δr) in a Taylor series
qr (r + δr) = qr (r) + δr
dqr
+ ...
dr
Neglecting powers of δr, the net flow of heat out of the shell is
µ
¶
£
¤
2
dqr
2
2
2
4π (r + δr) qr (r + δr) − r qr (r) = 4πr
qr +
δr.
r
dr
The rate of heat production per unit mass, H, within the shell is
4πr2 ρHδr
where ρ is density. Here 4πr2 δr is the volume of the shell which is approximated to first
order in r. By equating the rate of heat production in the shell to the net flux out of the
shell we arrive at the heat balance
dqr
2qr
+
= ρH.
dr
r
(9.40)
We now wish to relate the heat flux to the radial temperature gradient dT /dr. In spherical
geometry Fourier’s law has the form qr = −kdT /dr. Substituting into (9.40) we find
µ 2
¶
d T
2 dT
k
+
+ ρH = 0
dr2
r dr
or
k d
r2 dr
µ
r
2 dT
¶
+ ρH = 0
dr
The general expression for the temperature in a sphere or spherical shell comes from
integrating ?? twice
ρH 2 c1
T (r) = −
r +
+ c2
(9.41)
6k
r
where the constants c1 and c2 depend on the boundary conditions. For example we may
solve the for the temperature distribution in a spherical planet of radius R that has a
uniform rate of heat production. The boundary condition is that the outer surface of the
sphere has a temperature To . To have a finite temperature at the center of the planet c1
must vanish. To satisfy the surfical boundary condition we require
c2 = To +
ρHR2
6k
and so the temperature profile within the planet has the form
T (r) = To +
¢
ρH ¡ 2
R − r2 .
6k
9–13
(9.42)
From Fourier’s Law the the surface heat flux qo is given by
qo =
1
ρHR
3
(9.43)
This expression is based on conservation of energy and thus is valid for any specified mode
of internal heat transfer within the planet. The temperature distribution is shown in Figure
??.
9.2.5 Conduction and Convection
Earlier we noted that convection is a more efficient mecahnism of heat transfer than
convection. If this is the case why don’t planets (or anything else) always convect rather
than than conduct? In order to understand whether a planet’s interior is primarily losing its
heat through conduction or convection, we need a nondimensional number that measures
the relative contribution of each process.
Usually, an increase in temperature T will result in a decrease in density ρ. Water
near its freezing point is the most notable exception to this rule. Over small ranges of
temperature, we can approximate the dependence of density on temperature with a Taylor
series expansion:
ρ(T ) ≈ ρ0 [1 − α (T − T0 ) + . . .] ,
(9.44)
where ρ0 and T0 are a reference density and temperature, respectively, and α is called the
thermal expansion coefficient. Time-dependent heat conduction is a diffusive process,
and is described by a parabolic partial-differential equation:
D
T =
Dt
µ
¶
∂
+ v · ∇ T = κ ∇2 T .
∂t
(9.45)
The parameter κ is called the thermal diffusivity (= k/ρCp ). To get a feel for the
relative importance of conduction versus convection, consider the simple problem of a thin
slab of material that is confined between two horizontal plates, which are separated by a
vertical distance d. This simple convection problem was first investigated experimentally
in 1901 by H. Bénard, and analyzed theoretically in 1916 by Lord Rayleigh, and is now
refered to as Rayleigh-Bénard convection. The material is subject to a gravity field of
strength g, and is heated from below such that:
Tbot − Ttop ≡ ∆T > 0 .
(9.46)
Assume that the material has a kinematic viscosity, ν. Viscosity plays a role analogous
to the thermal diffusivity κ, but describes the rate of diffusion of momentum rather than
of heat. For small enough ∆T , the material’s viscosity will prevent convective motions,
and heat will be lost solely by conduction.
There are two important nondimesional numbers that can be formed from the parameters g, α, ∆T , κ, ν, and d. The first is the Rayleigh number, Ra, that measures the
tendency of a medium to convect. Ra measures the relative importance of buoyancy forces
9–14
(in the numerator) that drive convection and viscous resistance forces (in the denominator) that inhibit convection. The Rayleigh number can be expressed on various forms that
depend on the geometry of the medium and the nature of heating. However, a simple
representation of the Rayleigh number can be determined from a balance of forces on a
sphere in a uniform fluid medium. The sphere, or radius a will rise due to buoyancy forces
at a velocity V . Buoyancy occurs because the sphere has a density ρ − ∆ρ that is less than
the density of the fluid ρ. The buoyancy force scales with the volume of the sphere as
4 3
πa ∆ρg
3
Fb =
(9.47)
where g is the acceleration of gravity. As the sphere rises there will be a viscous resistance
to motion due drag over the spherical surface. This force can be written
µ
V
ν
a
Fr =
¶
4πa2
(9.48)
where µ is the dynamic viscosity of the medium, which is a measure of the resistance of
the medium to shear. It is a ratio of the stress to the strain rate in a viscous medium. By
equating the two forces we find the velocity at which the sphere will rise
V =
∆ρga2
.
3µ
(9.49)
If the sphere is less dense than its surroundings because it is warmer, we may express the
velocity in terms of a temperature difference. We write
∆ρ = −ρα∆T.
(9.50)
Substituting into (9.49) we find
V =−
ρα∆T ga2
.
3µ
(9.51)
It is apparent that the sphere will keep rising until it cools sufficiently such that the
buoyancy force is exceeded by viscous drag. The sphere will cool conductively at a time
that is proportional to the thermal diffusivity of the medium and the size of the sphere as
κt
=c
a2
(9.52)
where c is a constant. This is the characteristic cooling time of the sphere. Since
velocity is just distance over time we may now solve for the distance the sphere will rise
before it cools
ρα∆T ga2
d
(9.53)
V = =
t
3µ
9–15
For c=0.5, at which time the sphere will essentially be cool, the distance the sphere rises
may be written
µ
¶
1 ραg∆T a3
d=
a
(9.54)
6
µκ
The expression within the parentheses is dimensionless and is a ratio of buoyancy to drag.
It is a simple form of the Rayleigh number, Ra. Often Ra is expressed in terms of the
kinematic viscosity, which is related to the dynamic viscosity as ν = µ/ρ. Then
Ra =
αg∆T d3
.
κν
(9.55)
There is a critical value of Ra that marks the onset of convection, Rac , and this depends on
whether a medium’s bounding surfaces are made of highly conductive material, like copper,
or of highly insulating material, like plastic, or of material that has intermediate thermal
properties. The onset of convection also depends on whether the frictional boundary
conditions are taken to be “no slip” or “free slip.” For various geometries Rac falls in the
range 100-1000. The Rayleigh numbers for the terrestrial planets greatly exceed critical
values for either boundary condition and are thus expected to lose heat by convection.
The second important nondimensional number is the Prandtl number, P r:
Pr =
ν
,
κ
(9.56)
which compares the relative strengths of the two diffusion parameters. The Prandtl number is especially important in problems that combine rotation with convection, because
rotation can couple with the material’s viscosity and thermal diffusivity to a create a new
mode of instability called overstability. If the slab is rotating about a vertical axis with
angular velocity Ω, a third nondimensional parameter enters, usually taken to be the Taylor
number, T a:
d2
T a = 2Ω
(9.57)
ν
One effect of rotation is the Taylor-Proudman effect that tends to suppress vertical motions,
and consequently Rac generally increases as T a increases.
Linear analytical theory can predict Rac as well as the horizontal wavelength of the
convection at its onset. In Rayleigh’s original analysis, he assumed that the bounding
surfaces were perfect conductors. That way, the temperature on each boundary remains
constant even though there will be temperature fluctuations in the interior. With this
thermal boundary condition, convection begins as a series of overturing cells that are
approximately as long as they are tall. Interestingly, when the problem is set up using
insulating boundaries instead of conducting boundaries, convection first takes the form of
a single overturning cell that fills the experimental chamber. Actual geophysical systems
have thermal boundaries that are neither perfect conductors nor perfect insulators, and
so a mixture of both qualitative behaviors is possible. For large enough Ra, organized
convective cells give way to turbulent convection. The nature of turbulent convection
is especially important when considering the interior structure of the Sun and the giant
planets.
9–16
It may seem strange at first to think of the mantles of the terrestial planets as being
able to convect like fluids. After all, don’t we know from seismology that shear waves pass
through Earth’s mantle, and hence that Earth’s mantle is not fluid, but is made of solid
rock? The reason that there is no contradiction is that we must be careful to consider
the different timescales involved in the two processes. Seismic shear waves travel through
Earth’s interior in a matter of minutes. On the other hand, over geological timescales,
rocks are known to deform plastically when subject to a persistently applied strain rate.
If the characteristic time for this deformation is less than the age of the solar system, then
significant transport of heat by solid-state convection is possible.
Since much of a planet’s heat is generated by radioactive decay, it is appropriate to
consider a slab heated from within, rather than one heated from below. In that case, one
must specify the rate of internal heat generation per unit mass, H, and the mantle’s heat
capacity, Cp , and form a new Rayleigh number:
Ra =
g α d3 H d 2
.
κ ν κ Cp
(9.58)
The viscosity ν of Earth’s mantle is estimated from the rate of the glacial rebound of
the crust; remarkably, Earth’s crust is still rising back into its pre-ice age shape following
the most recent retreat of glaciers some 10,000 years ago. In 1935, N.A. Haskell used the
ages of elevated beaches in Scandinavia to calculate that the viscosity of Earth’s mantle
is about 1023 times greater than the viscosity of liquid water. With this value of ν and
various estimates of H and d, (9.49) yields Ra in the range 106 -109 for Earth’s mantle.
This is orders of magnitude larger than the critical value for this problem, Rac ≈ 3000,
which implies that Earth’s mantle is indeed convecting. Similar calculations indicate that
mantle convection is probably also taking place inside Venus and Mars, and to a lesser
extent inside Mercury and the Moon.
9.2.6 Banana Cell Convection
The nature of convection in a rapidly rotating gas-giant planet is only beginning to
be understood. One of the effects of rotation is to suppress motions along the direction of
the rotation axis, a consequence of the so-called Taylor-Proudman theorem that we will
discuss in a later lecture. Experiments on convection in a rapidly rotating sphere with
central gravity are difficult to perform. The best experiments to date were carried out
in the microgravity of Spacelab 3, as described in the paper “Laboratory experiments on
planetary and stellar convection performed on Spacelab 3,” by Hart et al. 1986. These
experiments, along with complimentary numerical experiments and theoretical work, indicate that convection occurs in the form of rolls aligned with the rotation axis and bent
by the sphericity of the planet. The convection cells have been dubbed “banana cells”
because of their shape.
If banana cell convection is in fact the correct description for deep convection in
the giant planets, then we can expect that such convection will have an influence on the
visible atmospheric dynamics. For example, it is not known why most Jovian vortices and
planetary-scale waves drift with velocities of only a few meters per second relative to the
9–17
planetary magnetic field reference frame (called the System III reference frame) whereas
the jet streams are typically moving at fifty to one hundred meters per second. Somehow,
the atmospheric features are sensitive to the rotation rate of the planet’s deep interior
where the magnetic field originates. One idea is that the banana cell convection produces
vertically propagating waves that act as a source of stationary forcing to the cloud-top
dynamics. In any event, a clear understanding of the surface weather patterns on the
giant planets will require a better understanding of the dynamics of their deep interiors, in
much the same way that surface tectonic features on the terrestrial planets are understood
to result from their specific interior dynamics and thermal histories.
9.3 Magnetic Dynamos
Whereas a planet’s external gravity field provides important density information about
the interior, a planet’s external magnetic field provides important velocity information
about the interior that would otherwise remain inaccessible. This information can be as
basic as providing a convenient reference frame from which to measure cloud-top wind
speeds on the giant planets, or as profound as indicating that Earth has significant fluid
motions in its liquid-iron core, and Uranus and Neptune have significant circulations in
their liquid-water mantles. In contrast, the conspicuous lack of strong magnetic fields for
either Venus or Mars provides important constraints on their internal dynamics.
To understand the mechanism of magnetic field generation we must first characterize
the general pattern of the field. Measurement of the magnetic field of the Earth, which is
by far the best-studied field in the solar system, shows field lines that emanate from the
south geomagnetic pole and enter at the north geomagnetic pole. The pattern is consistent
with a sphere that contains a powerful bar magnet at its center, though we shall discuss
that this is not a plausible explanation for any planetary magnetic field. Such a field is
referred to as a dipole field because it could be explained by a pair of magnetic poles
of equal strength a short distance apart. While there are important deviations from the
pattern expected for a dipole, to a first approximation a dipole aligned generally along the
polar axis is a good representation for the Earth’s and other planetary magnetic fields.
As for gravity, magnetism (Vm ) is a potential and so can be described by Laplace’s
equation
∇2 Vm = 0.
For planetary bodies, spherical coordinates are appropriate and so we may write a spherical
harmonic representation of the solution
Vm
∞ µ ¶l+1 X
l
a X R
=
Plm (cosθ) [glm cos(mλ) + hlm sin(mλ)] .
µo
r
m=0
(9.59)
l=1
that will effectively separate out the dipole and non-dipole components of the magnetic
field. Here the coefficients g and h have dimensions of the field (e.g. nanotesla or nT
=10−5 Gauss is usually used for Earth.) Note that the l=0 term vanishes, indicating that
there is no magnetic monopole. In the event that monopoles exist they do not significantly
contribute to planetary magnetism. The l=1 term is the dipole.
9–18
The dipole field can be conveniently represented at any position r from the dipole in
terms of Vm , where
1
cosθ
Vm =
m·r=
,
(9.60)
3
4πr
4πr2
where θ is the angle between the dipole axis and the radius vector r from the dipole to
the point of interest. The magnetic moment, m, represents the product of the pole
strength and separation. The magnetic field B(r) at any position r can be determined by
differentiating the magnetic potential
B(r) = −µo ∇V (r)
(9.61)
where µo (=4 π × 10−7 kg m Am−2 s−2 ) is the magnetic permeability of free space. Using
the same convention as we developed for gravity we resolve the radial and latitudinal
(zonal) components of the field
Br = −µo
µo 2m
∂Vm
=
cosθ
∂r
4π r3
(9.62)
µo m
µo ∂Vm
=
sinθ.
(9.63)
r ∂θ
4π r3
Note that by symmetry the Bλ component of the field vanishes.
Earlier we noted that the dipole field may arise from a bar magnet, but early on such
a possibility was ruled out as an explanation for the Earth’s magnetic field. For one thing
the Earth’s field exhibits secular variations that include: a decrease in the magnetic
moment of 0.05% longitude yr−1 , a westward drift of the dipole of 0.05◦ yr−1 and of the
non-dipole component of the field of 0.02◦ longitude yr−1 , a rotation of the dipole toward
the geographic axis of 0.02◦ yr−1 , and the growth and decay of various features of the nondipole field of order 10 nT yr−1 . In addition, the field exhibits geomagnetic reversals.
All of these mechanisms are consistent with variable magnetization and any mechanism
for magnetic field generation must allow for time variations. As seismology began to show
evidence for a fluid outer core within the Earth, theoretical work has converged to indicate
that a self-sustaining dynamo due to motions in the fluid core was the most plausible
mechanism of magnetic field generation. There are various models of dynamo action,
but this general mechanism is now widely believed to apply to all planets that exhibit
significant magnetic fields.
To be more specific, the differential motion of electrically conductive fluid in a planet’s
interior couples to the planet’s rotation to give rise to a self-sustaining magnetic field, a
process called the magnetic dynamo process, which is described by principles of magnetohydrodynamics. After decades of research, satisfactory models of this phenomenon are
only beginning to be achieved. We do know that Earth’s magnetic field is not simply the
result of permanent magnetism in its rocks, because the paleomagnetic record shows
that Earth’s magnetic field has reversed its polarity many times. There is no obvious
pattern to Earth’s polarity reversals, but the average period is on the order of 106 yrs. At
midocean ridges, the irregular pattern of magnetic polarity is organized into stripes that
parallel the ridges with mirror-image symmetry — this was the key fact that convinced
Bθ = −
9–19
the geophysical community in the mid 1960’s of the reality of plate tectonics. The sun also
exhibits magnetic polarity reversals. But unlike Earth, the Sun’s reversals occur regularly
in a 22-year cycle, a process that is not understood either.
The mechanism by which a geomagnetic field is believed to be generated has its basis
on the frozen flux principle, illustrated in Figure ??. The essential element of this
principle is that if a combination of the conductivity, magnitude and velocity of fluid
motion is sufficient, then a magnetic field is transported and deformed with the fluid. A
characteristic of this mechanism is that the kinetic energy of fluid motion is converted into
magnetic field energy.
There are two kinds of motions. The first is differential rotation within the core that
is a consequence of the tendency of a convecting fluid to conserve angular momentum as
it is transported radially in response to buoyancy forces. Thus the inner part of the core
rotates with a higher angular velocity than the outer part and it draws out the lines of a
poloidal field to produce an additional toroidal field. This mechanism is known as the
omega (ω) effect. Radial convective motion is required to regenerate the initial poloidal
field from the toroidal field by a second process, knwn as the alpha (α) effect. The
significant aspect of the motion that drives the alpha effect is that it is helical or spiral
in nature. Such motion arises as upwelling material is deflected by planetary rotation (i.e.
the Coriolis effect). Note that the two different kinds of motion re-enforce each other,
leading the dynamo to be self-sustaining.
A simple mechanical conceptualization for the self-sustaining dynamo is given by two
interconnected disk dynamos, shown in Figure ??. Each disk rotates in an axial field produced by a coil carrying a current driven by the other disk. Rotation in the field generates
an electromotive force between the perimeter of the disk and the axis, providing the current
that drives the other disk. The system is self-maintaining when current generation exceeds
the ohmic dissipation via an appropriate combination of velocity, size and loop conductance. Note that the model is symmetrical with respect to the polarities of the currents
and fields, and that the system works equally well if the polarities are reversed. Instabilities in the system, including spontaneous current reversals, have been demonstrated in
mechanical systems and computer models of more complex systems. This effect may be an
analog for the record of magnetic field reversals whose record is preserved in the Earth’s
geologic record.
If the action of the dynamo is to be robust the generation of the magnetic field must
be rapid enough to overcome the diffusion of the field out of the conductor. There are two
relevant time constants, the first of which is for the time τv for regeneration of the field
and is written
L
(9.64)
τv = .
v
Here L is the length scale over which field lines can be distorted and v is the relative
velocity of motion of the conductive fluid. The second time constant is τΩ , which describes
the decay of the field by ohmic dissipation out of the conductor
τΩ =
B
(−dB/dt)
9–20
(9.65)
where (−dB/dt) is the rate of free decay of a field that is not being maintained, such as
for a static conductive fluid. The necessary condition for dynamo action is
τv < τΩ .
(9.66)
Equation (9.61) is an unambiguous quantity only for a specific class of fields that maintain
their forms as their amplitudes decay.
The expression used in the mathematical formulation of dynamo theory is the magnetic
induction equation
∂B
= ηm ∇2 B + ∇ × (v × B).
(9.67)
∂t
where ηm is the magnetic diffusivity of the conductive medium. In (9.67) the first term on
the right hand side represents the diffusion or ohmic dissipation of the field and the second
describes the interaction with velocity that allows regeneration. Equation (9.67) is a more
physical description of the competition between generation and dissipation than is (9.66).
In the magnetic dynamo mechanism differential rotation of the conductive fluid causes
a winding up of the magnetic field lines, while organized motions in the fluid, perhaps
capitalizing on the same nonlinear fluid dynamics that organizes and maintains centuriesold storms like Jupiter’s Great Red Spot, tend to align with and amplify the dipole field
in a self-sustaining manner. While the details are not sufficiently worked out to make
useful quantitative predictions, we can estimate the order of magnitude of a magnetic field
that would be produced by such a dynamo process. The essential balance is between the
Coriolis force, actually part of the acceleration in a rotating coordinate system, and the
magnetic force:
1
2Ω × v ≈
(∇ × B) × B ,
(9.68)
4πρ
where Ω is the planet’s rotational velocity. We can estimate the order of magnitude of the
magnetic field from (9.68) to be
Ωv ∼
1 B
B
ρ R
⇒
B 2 ∼ ρRΩv ,
(9.69)
where the planet’s radius R estimates the effect of the gradient operator in (9.68), and v is
the typical differential velocity in the fluid. If we assume that v scales with the rotational
velocity ΩR from one planet to the next, and that the magnetic moment M scales with
BR3 , then
M ∝ ρ1/2 Ω R4 .
(9.70)
Table 9.2 provides a summary of magnetic field information for the planets and Figure ??
plots log(M ) versus log(ρ1/2 Ω R4 ). There is generally good agreement between (9.70) and
the observations, although Mars and Venus have conspicuously low magnetic moments.
Table 9.2 shows that among the terrestrial planets and the Moon, Earth’s magnetic
field stands out as having by far the greatest intensity. In fact, Earth’s field is more
comparable in magnitude to the giant gaseous planets than to its rocky counterparts. The
other terrestrial planet fields are much weaker and, except for possibly Mercury, do not
9–21
Table 9.2
Magnetic Fields of Planets and Satellites
Planet
Mercury
Venus
Earth
Moon
Mars
Jupiter
Saturn
Uranus
Neptune
Io
Europa
Ganymede
Equatorial Surface
Field
Bo (nT )
300
< 30
30300
< 25
<5
428000
21800
22800
13300
1800
240
100
Magnetic Moment
m/mEarth
m (A − m2 )
4.18 × 1019
< 6 × 1019
7.856 × 1022
< 1.3 × 1018
< 2.0 × 1018
1.46 × 1027
4.3 × 1025
3.7 × 1024
2.0 × 1024
6 × 10−4
< 8 × 10−4
1
< 1.7 × 10−5
< 2.5 × 10−5
1.87 × 104
550
47
25
Adapted from Stacey (1992).
require active dynamos. These fields are most likely due to remnant magnetization of
their crusts by formerly active fields. The strong field for the Earth is consistent with
the existence of its liquid outer iron-nickel core, in motion due to convective cooling and
rotation. Venus lacks a magnetic field but this in itself does not require that the core has
solidified. It is possible that that planet’s slow rotation may preclude dynamo action. There
has been a weakly-constrained estimate of the tidal Love number k2 for Venus based on
analysis of tracking data from the Magellan spacecraft that suggests that Venus has a liquid
core, but more detailed work will be required to verify this. The Mars Global Surveyor
spacecraft recently placed an upper limit on Mars’ main field of 5 nT . That spacecraft
also includes an electron reflectometer that can measure crustal remnant magnetization
and has discovered multiple magnetic anomalies of small spatial scale (≈ 100 km) but
large intensity (≈ 1.6 × 1016 A-m−2 ). Correlation of the magnitude of this remanent
field with the relative ages of surface units indicates that Mars likely had a strong main
field (and thus vigorous convection in a fluid core) early (≈ 4 Ga ago) in its history. For
Mercury, experiments are currently being planned that would allow an orbiting spacecraft
to ascertain the core state and critically evaluate the hypothesis that that planet has an
active dynamo. The Moon lacks a present-day main field but remnant magnetization was
detected by Apollo orbiting spacecraft. This is interesting in light of the fact that the
Moon is known to contain very little iron. It is debated whether what little iron the Moon
has is enough to have allowed a limited core dynamo early in its evolution, but alternative
mechanisms for magnetization of the surface must also be considered. Possibilities include
the interaction of the lunar surface with the solar wind, or magnetization by large impacts.
9–22
On the basis of a limited number of laboratory experiments it has been suggested that it
is possible to generate magnetization during the impact process, but the results have been
debated.
Observations of the giant planets indicate that the strength and orientation of a large
dipole best reproduces the observed external magnetic fields of those planets as well. Voyager discovered that both Uranus and Neptune have equivalent magnetic dipoles with
extraordinarily large tilts and central offsets. Athough there does not yet exist a satisfactory theory to account for the magnetic fields of Uranus and Neptune, or of any planet
with the possible exception of Earth for that matter, it is supposed that the large tilts
indicate that the magnetic fields of Uranus and Neptune are produced by a dynamo that
exists not in their cores, as is the case for Earth, but in their mantles.
The Galileo spacecraft recently detected magnetic fields around three of the Galilean
satellites. Ganymede exhibits what is believed to be a dipole field generated in an iron
core. This was the first inkling that Ganymede’s interior has a significant iron content.
Io also shows evidence of an internal dynamo, with thermal energy continuously supplied
to the interior due to tidal forcing discussed earlier. A main field signature has also been
detected for Europa and the mechanism is under investigation. Of special interest is the
possibility that it may reflect convection in a liquid water ocean.
9.4 Terrestrial Planets
The various bodies in the solar system can be divided into four categories when considering internal structure: i) comets, ii) small bodies, iii) terrestrial planets, and iv) gas
giants. Comets are primitive, icy bodies that have their own distinctive composition and
history, and we will discuss them in detail in Chapter 2. The small-body category contains
objects with radii smaller than about 200 km, and includes dust, ring particles, small satellites, and most asteroids. These objects have had enough time over the age of the solar
system to lose the internal heat they started with as a byproduct of accretion, and are not
large enough to generate substantial internal heat from radioactive decay. Therefore, small
bodies are not likely to have convecting interiors or dynamically active magnetic fields at
the present time. We have already touched on the compositions of meteorites and asteroids
in Chapters 2 and 4, and we will discuss ring particles in Chapter 11. The terrestrial-planet
category includes Mercury, Venus, Earth, and Mars, the large icy satellites of the giant
planets, like Ganymede and Callisto, and the Pluto-Charon system. The fourth category
is the gas-giant category, which counts Jupiter, Saturn, Uranus, Neptune, and the Sun
as its five members. We will concentrate on the interiors of the terrestrial planets in this
section, and will discuss what is known about the gas giants in the next section.
9.4.1 Mercury
Mercury is the least-well observed terrestrial planet in the inner solar system. Most of
the data we have that bear on Mercury’s interior are photographs of its surface obtained in
1974-75 by three successive flybys of the Mariner 10 spacecraft. We do not have detailed
observations of Mercury’s heat flow, gravity field, or seismic activity. However, using
the Mariner 10 photographs, we can make indirect inferences based on observed surface
tectonic features that are the signature of changes in the size and shape of the planet.
9–23
It is likely that Mercury started out with a much faster rotation rate than it currently
has, and that it has undergone tidal despinning. This is believed to have occurred because
Mercury’s 58.65 day rotation rate is low, and in fact is in a 3:2 commensurability with
its 87.97-day orbital period, implying tidal evolution. A large change in its rotational
parameter q could have caused a large change in its figure, depending on whether its
interior has been mostly liquid or mostly solid during its lifetime.
On the surface, Mercury and Earth’s Moon are remarkably similar in appearance.
However, their internal chemical compositions are much different, because Mercury has a
mean density of 5440 kg m−3 , which is the second highest planetary density after Earth’s
5520 kg m−3 . Considering that Mercury is only one-third the size of Earth, its high density
implies a chemical composition of 60-70% metal and 30-40% silicate by weight. Mercury
contains about twice as much iron in relation to its total weight than any other planet,
which implies a core that extends out to 75% of its total radius — larger than the volume
of Earth’s Moon. Almost nothing can be said about the detailed structure of Mercury’s
thin outer silicate layer, although it is likely to be differentiated into a crust and a mantle.
Mariner 10 discovered that Mercury has a magnetic field. It forms a magnetosphere
that is about 7.5 times smaller than Earth’s when normalized to planetary radius, which
makes it a relatively weak magnetic field, but nevertheless right on the prediction of the
scaling relation (9.52). During periods when the solar wind is particularly strong, Mercury’s field probably gets depressed all the way to the surface. The source of the field
is not firmly established. If Mercury currently has a molten core, then its magnetic field
may be produced by an active dynamo. Another possibility is that Mercury no longer has
a liquid core and an active dynamo, but instead the magnetic field of a past dynamo is
trapped in the outer iron-bearing rocks. This second scenario is not as likely, because the
compressional tectonic features seen on Mercury’s surface are insufficient to support the
idea that a once-molten core has cooled and shrunk until it is now nearly solid.
If Mercury retained the amount of uranium and thorium expected to have been present
in the solar nebula at the time of its formation, then it should have experienced enough
radioactive heating to have differentiated, no matter what its initial temperature was.
We argued above that Mercury’s high mean density implies a large core. The release of
gravitational potential energy during the formation of the core probably raised the internal
temperature by 700◦ C and melted the mantle. Given the melting of the mantle, the volume
of the planet would have increased significantly, causing extensional fracturing in the crust
and volcanism, of which there is some evidence in the intercrater plains. Subsequent
cooling of the mantle is probably the explanation for Mercury’s compressional features.
9.4.2 Venus
Venus rotates in the retrograde direction once in 243 days, and its orbital period is
224.7 days. The equatorial bulge caused by Venus’ slow rotation is only a fraction of a
meter, which is much too small to provide any information about the distribution of mass
in the deep interior. However,the gravity field contains some information about the upper
mantle, because it is found to be correlated to surface topography. In contrast, Earth’s
gravity field is not well correlated with the surface topography. There is no gravity anomaly
at a point over a mountain compared to a point over an area far from the mountain if the
9–24
vertical integral of density is the same at both points. Such an equivalence is also the
condition necessary for a mountain to actually be floating, that is, in isostatic balance
with its surroundings. Thus, the mountains on Earth are in large measure supported by
buoyancy forces, but the mountains on Venus are not. It may be that mountains on Venus
are supported by the rigidity of the mantle. Or, it may be that Venus is even more dynamic
than Earth and forms mountains faster than they can relax back into the crust. Vigorous
mantle convection and dynamic support of topography has been proposed to explain the
lack of isostatic balance on Venus, which alleviates the need to assume that the upper
layers of Venus are significantly more rigid than those of Earth.
Since we have little information about the thermal structure in the interior of Venus, it
is difficult to calculate important parameters like the thickness of the lithosphere. The (former) Soviets successfully landed four Venera spacecraft on the surface of Venus. Gamma
ray analysis of rock samples suggests that the surface rocks on Venus have about the same
abundances of uranium and thorium as on Earth, implying similar internal heat sources
due to radioactive decay. If we assume that the temperature gradient in the interior of
Venus is similar to that determined for Earth, the lithosphere on Venus should actually
be about half the thickness of Earth’s lithosphere. This calculation tends to favor the
dynamic support model of the non-isostatic balance seen on Venus, because a thinner
lithosphere implies less mechanical rigidity. However topography and gravity information
from the Magellan mission has led to the surprising recognition that the thickness of Venus’
present-day mechanical lithosphere is comparable to that of Earth. One possible explanation is that the essentially complete lack of water in Venus’ near-surface results in much
higher strength of surface rocks. Such behavior has been demonstrated in the laboratory
for rocks of common crustal and upper mantle composition but at much higher strain rates
than are rheologically relevant.
The primary way that Earth gets rid of its internal heat is through the mechanism
of mantle convection and plate tectonics. It is thought that a lithospheric plate on Earth
subducts because it cools down enough to become denser than the underlying mantle, and
sinks. It is possible that the thin lithosphere on Venus can not cool down sufficiently to
take advantage of such a gravitational instability, in which case Earth-style plate tectonics
would not be occurring on Venus.
9.4.3 Earth
Earth’s surface is young compared to the other surfaces in the solar system. This is
largely due to the fast recycling of crustal material by plate tectonics, and to the rapid rate
of erosion caused by the abundance of liquid water. Only 29% of Earth’s surface stands
above liquid water (65% of that land is presently located in the northern hemisphere).
Most planetary surfaces are dominated by the circular structures associated with impact
craters and volcanism. In contrast,plate tectonics has caused the largest topographic highs
and lows on Earth to be linear. Examples include the linear mountain ranges that form
as one plate collides into another, the oceanic trenches that mark the location where plate
subduction occurs, and the mid-ocean ridges that delineate the seam where two plates are
separating as new oceanic crust rises up from the mantle. Thus, we see that the large-scale,
surface features on Earth are controlled by the planet’s interior dynamics.
9–25
Seismology has allowed us to map the interior of Earth to a level of detail that far
exceeds our knowledge of the interior of any other planet. Besides seismic data, we have
other clues including measurements of Earth’s gravity field and moments of inertia, and
chemical analyses of inclusions of material of deep origin found in volcanic rocks. Earth’s
crust accounts for only about 0.5% of the total mass. The so-called Mohorovicic discontinuity, called the Moho for short, defines the base of the crust. Oceanic crust is about
5-7 km thick, whereas continental crust is 35-40 km thick. The Moho marks an abrupt
change in velocity for both shear (S) and pressure (P) seismic waves, where the aluminum
rich rocks of the crust give way to the magnesium and iron rich (ultramafic) rocks that
makes up the mantle. Typical densities in the upper mantle are 3.2-3.6 g cm−3 , and in the
lower mantle are 5-6 g cm−3 . The mantle accounts for about 67% of the total mass. The
mantle’s lower boundary is called the Gutenberg discontinuity, which occurs at a depth of
about 2900 km. At this interface, P-waves experience a sharp drop in velocity and S-waves
are not transmitted at all, signalling the presence of the liquid outer core. Density rises to
9-10.5 g cm−3 in the outer core, and increases to a central density of 12-14 g cm−3 . The
core accounts for about 32% of the total mass.
Small samples of Earth’s upper mantle are actually accessible for laboratory studies,
because they have been carried up to the surface elevator-style inside of igneous rocks.
These ultramafic (rich in Mg and Fe) inclusions can be proven to have originated in
the mantle because they contain high-pressure minerals, like diamonds, and because they
contain certain mineral pairs that can only coexist when formed at high pressure. These
inclusions allow for relatively accurate estimates of the chemical composition of the upper
mantle to a depth of about 200 km. The mineralogical structure of the deep mantle is much
less certain, because the temperatures are poorly constrained, and the preferred phases of
the minerals at depth have not yet been determined in the laboratory.
The core can be subdivided into an outer core and an inner core. The outer core is
known to be liquid because it does not transmit shear waves. However, at the so-called
Lehmann discontinuity at 5200 km, the core material again transmits shear waves, but at
low velocities, indicating that the inner core is partly molten or near the melting point. The
core is less dense than pure iron-nickel liquid, most likely because it contains significant
amounts (9-12%) of sulfur, or possibly oxygen.
9.4.4 The Moon
magma ocean, no iron, max core size of 300 km. All that good stuff and more.
9.4.5 Mars
Mars rotates once in 24.6 hours, which is rapid enough to allow us to apply the methods
of the previous lecture to infer the response coefficient Λ2 ≡ J2 /q = 0.43. This value is
closer to the 0.5 value of a uniformly dense sphere than is Earth’s value of 0.31, implying
that Mars has a dense core, but is somewhat less centrally condensed than Earth. There
is not much other geophysical data. Mars lacks an appreciable magnetic field. The Viking
landers did not have sensitive seismometers, and although they did perform useful chemical
analyses on rock samples, the emphasis of the Viking mission was biological instead of
geological. Although Viking’s search for the chemical signature of living organisms proved
9–26
negative, the Martian soil was found to contain a surprisingly high abundance of Fe2 O3 ,
about 18% by mass (thereby demonstrating that Mars has a rusty color because it is
covered with rust). Models of the interior predict that the mantle is also rich in iron.
But due to a lack of internal structure data there is an 800-km uncertainty in the size of
the Martian core. Future seismic observations or pole positions from Doppler or optical
tracking of surface landers will be required to clarify our understanding of the planet’s
internal structure.
Mars has the largest volcanoes in the solar system. The volcanoes are of the shield
variety, implying that they were formed from low-viscosity (probably basaltic) lava that
could flow freely for large distances. Calculations based on geological evidence suggest
that the Martian lava is as much as ten times less viscous than the thinnest lava observed
on Earth, which is consistent with the properties of exceptionally iron-rich rock. One
reason that the volcanoes on Mars are larger than those on Earth is that on Earth plates
move across relatively stationary mantle upwellings, causing chains of volcanoes like the
Hawaiian Islands to form, instead of allowing a single volcanoe to grow in place. There is
no evidence of plate tectonics on Mars. In addition, the colder Martian interior results in
a thicker mechanical lithosphere that can support such large surface loads.
One quarter of the Martian surface is dominated by the Tharsis bulge, a topographic
high that stands about 7 km above the planet’s average radius, and contains most of
the major shield volcanoes. The Tharsis region is surrounded by radial fractures that
are the signature of tensional (stretching) forces. Both the Tharsis bulge and its largest
shield volcanoes can be associated with positive gravity anomalies, which means that the
topography on Mars is only partially supported by isostatic balance, at least at shallow
depths. The complex tectonic signature associated with Tharsis provides a rich data set
that can be combined with gravity and surface topography to study the origin of this
distinctive feature. Unfortunately at present there are no models that are consistent with
all observations. It is almost certain that models for the formation of Tharsis may require
multiple temporal stages that may include processes such as flexural loading, isostatic uplift
and mantle convection. There may also be a requirement for decoupling of the upper crust
and upper mantle by a weak crustal layer in the center of the province, which may have a
weaker crust. Data from upcoming Mars missions are likely to provide observations that
will greatly aid in reconciling the range of possible models.
9.4.6 Titan
The most important observation that describes a planetary body’s interior is its bulk
density. For Titan, this is the only observation we have that bears on its interior. Titan’s
bulk density is 1880 kg m−3 , which neatly falls midway between the densities of the Galilean
satellites Callisto and Ganymede, and is quite typical of a large, icy satellite. Based on
its bulk density, Titan is expected to be composed of roughly equal amounts of “rock”
(silicates and iron) and “ice” (water ice, in this case).
We have seen how the giant-planet atmospheres are essentially a continuation of their
interiors, whereas terrestrial-planet atmospheres are certainly not. Titan may fall in an
intermediate class, because it is likely that its atmosphere and interior are quite similar in
chemical composition. The rock component will tend to settle into a dense core, and to
9–27
provide heat through radioactive decay. It is not clear whether there is sufficient heat to
cause convection in an icy mantle similar to the solid-state convection in terrestrial-planet
mantles.
Much of the volatile inventory in Titan’s interior may be in the form of clathrates.
A clathrate is a crystal lattice that contains voids that hold guest molecules. Water ice
can hold many different molecules in a clathrate, like CH4 , N2 , CO, and noble gases.
The formation of CH4 -rich clathrate in an early, hot Titan could conceivably enrich the
atmosphere in N2 , because N2 would not be incorporated into clathrate until most of the
CH4 was removed.
9.4.7 Icy Satellites
9.4.8 Pluto-Charon
9.5 Giant Planets
The solar system’s four giant planets are in many ways more closely related to the
Sun than to the terrestrial planets. Each has a large collection of satellites that resembles
a miniature solar system. Jupiter, Saturn, and Neptune each have their own intrinsic
luminosities, and radiate about twice as much energy as they receive from the Sun. On
the other hand, Uranus radiates ≤ 6% more energy than it receives from the Sun, which
results in the curiosity that the effective temperatures of Uranus and Neptune are both
about 59K. Jupiter has more mass than all the other planets put together, 71% of the
total planetary mass. If Jupiter had been just eighty times more massive, then its central
pressure and temperature would have been high enough to ignite nuclear fusion. More
than half the local stars are paired together in binary systems, so the fact that Jupiter is
almost a star is not too surprising.
9.5.1 Polytropes
How the internal pressure and density of a planet vary with radial distance, from
the planet’s center to its surface, is of fundamental importance to our understanding of
the planet’s internal structure. Consider a nonrotating, spherical body in hydrostatic
equilibrium, such that:
dP
GM (r)
= −ρ g(r) = −ρ
,
(9.71)
dr
r2
where M (r) is the mass contained within radius r, which can be found by re-arranging
(9.71) as
r2 dP
.
(9.72)
M (r) = −
ρG dr
The mass in a spherical shell dM is related to the density ρ by the product of volume and
density as:
dM = 4πρ r2 dr .
(9.73)
9–28
By combining (9.72) and (9.73) we may write an expression for the change in mass as a
function of radius:
µ
¶
dM
d
r2 dP
=
−
= 4π ρ r2 ,
dr
dr
ρG dr
µ
¶
d r2 dP
⇒
= −4π G ρ r2 .
(9.74)
dr ρ dr
Provided we know an equation of state connecting the pressure to the density, P = P (ρ),
(9.74) is a second-order differential equation that can be solved to obtain ρ(r).
In general, an equation of state will involve more than one variable, for instance the
ideal gas law P = ρRT is in the form P = f (ρ, T ). If P and ρ are directly related by
an equation of state P = P (ρ) the system is said to be barotropic. For some barotropic
problems, a power-law relationship between P and ρ holds. This called a polytrope and
historically is written in the form:
P = k ρ1+1/n ,
(9.75)
where k and n are called the polytropic constant and polytropic index, respectively. Equation (9.75) implies that ρ ∝ P n/(n+1) , and so n = 0 refers to the case of constant density.
Other examples of polytropes include a non-relativistic degenerate gas, like in a white
dwarf star, for which n = 3/2 (P = k ρ5/3 ), and a relativistic degenerate gas, like in a
neutron star, for which n = 3 (P = k ρ4/3 ). Jupiter is well modeled by the equation of
state for metallic hydrogen, which follows an n = 1 polytrope:
P = k ρ2 ;
k = 2.72 × 1012 dyne cm4 g−2
(metallic hydrogen) .
Assuming a polytropic equation of state, (9.74) becomes:
¶
µ
¶
µ
1 d
2 −1+1/n dρ
r ρ
= −4π G ρ r2 ,
k 1+
n dr
dr
(9.76)
(9.77)
called the Lane-Emden equation. Explicit solutions for (9.77) are known for n = 0, 1, 5;
other values of n are found by numerical integration. The dependence of radius on mass
for a given polytropic index can be found from the dependence of central pressure on R
and M , assuming hydrostatic balance:
Pcen
M GM
∝ 3 2
R
R R
⇒
Pcen ∝
M2
,
R4
combined with the polytropic assumption:
µ
Pcen ∝
M
R3
¶1+ n1
,
yields:
R ∝ M (n−1)/(n−3) .
9–29
(9.78)
Notice that for 1 < n < 3, R actually decreases as M increases. The case n = 1, which
happens to be the one relevant to Jupiter, has the special property that the radius is
independent of the mass. The explicit solution to (9.77) when n = 1 is:
³q
sin
ρ(r) = ρcen
q
2πG
k r
´
,
(9.79)
2πG
k r
where ρcen is the central density. It is a homework problem to verify (9.79), and to find R
for a metallic hydrogen planet. Jupiter’s radius is within 12% of this radius, which provides
support for the idea that Jupiter is composed primarily of hydrogen. Also, a calculation
of the response coefficient Λ2 ≡ J2 /q for an n = 1 polytrope yields:
µ
Λ2 =
5
1
−
2
π
3
¶
= 0.173 ,
(9.80)
which, as shown in Table 9.1, compares well with the value Λ2 = 0.166 observed for Jupiter.
We expect the interiors of Jupiter, Saturn and Neptune to be vigorously convecting in
order to transport internal heat to their surfaces, where it is eventually radiated to space.
Since only a small superadiabaticity is required for effective convection, an unstable atmosphere tends to develop a lapse rate that is only marginally larger than the adiabatic lapse
rate. The vertical temperature profiles of the giant planets are observed to be adiabatic
at depth, to within measurement error.
In an adiabatic planet, small temperature fluctuations are efficiently removed by selfregulating convection. This is taken as the explanation for the remarkable lack of temperature variation seen on the surfaces of the giant planets. Even though the solar insolation
varies by a sizeable factor from equator to pole, the observed equator-to-pole temperature
differences are only a few degrees, in sharp contrast to the many tens of degrees seen on
Earth. Apparently, the adiabatic interiors of the giant planets act as a short circuit by
arranging for more internal heat to come out at the poles than at the equator, in just
the right proportion to balance the uneven distribution of incoming solar energy. The
importance of a sizeable heat source located below the atmosphere is one of the primary
distinguishing characteristics of giant planet atmospheric dynamics.
9.5.6 The Sun
Our understanding of the internal structure of the Sun has improved most significantly in about the last ten years due to very accurate measurements of the frequencies of
its acoustic modes of oscillations (p-modes) determined from helioseismology. In practice,
these oscillations are detected by Doppler shifts in images of the photosphere. Approximately five thousand normal modes of oscillations of the Sun have been identified thus far
and their frequencies have been measured with an accuracy of about 0.01%. The observed
periods of these modes lie between 3 and 15 minutes, and their lifetimes are typically a few
days. In addition to the p-modes, there are also standing internal gravity waves known
as g-modes with periods that exceed 40 minutes. The g-modes penetrate more deeply
9–30
than the p-modes, but none have been definitively observed. Coherent p-modes have been
used to determine the sound speed in the outer approximately 60% of the solar interior
(by radius) where these modes are trapped. Solar p-modes, like the energy states of the
hydrogen atom, are uniquely identified by specifying the number of nodes in the radial
direction, the spherical harmonic degree l, and the order m. For a non-rotating spherically
symmetric Sun the normal mode oscillation frequencies are degenerate with respect to m.
Solar rotation causes this degeneracy to be removed and the measurement of the resulting frequency splitting provides information about the rotation rate in the solar interior.
Various workers have used this technique to estimate the solar quadrupole moment and
to map the rotation rate in the outer 60% of the Sun. The rotation rate of the solar core
remains undetermined.
The current best estimates of J2 for the whole sun range from 1.5 to 1.7 × 10−7
with error estimates of almost a factor 2. However, certain outstanding questions in solar
system evolution depend on accurate knowlege of the radial density distribution of the
solar interior. For example, it is known from helioseismology that the J2 of the Sun is too
small, by almost a factor of 103 , to be responsible for the perihelion advance of Mercury.
Another key question is whether the Sun has a rapidly rotating core, which is relevant to
both present-day solar dynamics as well as stellar evolution. Another central question in
solar dynamics is the how convection interacts with the mean rotation to yield differential
rotation. This question has implications of this interaction for the generation of the solar
dynamo. With regard to solar evolution, a rapidly rotating deep interior may be a relic of
the rapid spin our Sun was born with, as very young stars are observed to rotate rapidly.
9.5.7 Jupiter and Saturn
Because of the large range of temperatures and pressures encountered in a giantplanet, a range that goes from T ≈ 50–150 K and P ≈ 1 bar at the cloud-top level
to T ≈ 10, 000–30, 000 K and P ≈ 50 megabars at the center, we may expect that the
primary constituent, hydrogen, is present in different phases at different levels. There is
no firm agreement as to the phase diagram for hydrogen over such a large range, primarily
because of the difficulty in obtaining high-pressure data in the laboratory. At pressures
above about 3 megabars, normal molecular hydrogen is suspected to have its electrons and
protons squeezed so close together that it becomes a metal. The temperatures at these
depths are thought to be sufficient for metallic hydrogen to be in the liquid state. Since
metallic hydrogen is an alkaki metal, it is reasonable to assume that its electrical properties
will be similar to other alkali metals like lithium. It is not known whether the transition
from molecular to metallic hydrogen occurs gradually or abruptly, and this uncertainty
translates into an uncertainty for the interior models of Jupiter and Saturn. Recently
metallic hydrogen was experimentally detected by H.H. Mao and colleagues. Their results
represented an important first order verification of models of the internal structures of the
giant planets.
Assuming that the Jupiter pressure–density relationship follows an n = 1 polytrope,
as we have already indicated is likely, the radius for the molecular-to-metallic hydrogen
transition in Jupiter should occur at about 70–80% of the total radius. This implies that
the bulk of Jupiter is in the metallic hydrogen phase. Saturn’s metallic hydrogen core is
9–31
calculated to have a radius that is about half of its total radius. Both Uranus and Neptune
are expected to have extensive mantles composed of liquid water, which is a relatively
conductive fluid. Thus, the primary ingredients for a magnetic dynamo exist on all four
giant planets, and in fact, each giant planet has a strong magnetic field.
Formation of an iron core is not the only important example of mass differentiation
in the planets. An important source of internal energy available to Jupiter and Saturn is
the separation of helium and hydrogen. The formation of a “helium core” can in principle
double the cooling lifetime for these planets, because of the large amount of gravitational
potential energy stored in an undifferentiated hydrogen and helium mixture.
A simple scenario for the differentiation of helium is as follows. We begin with a
planet composed of hydrogen and helium that is well mixed. The planet cools down until
it reaches the temperature for the liquid phase transition in helium, at which point droplets
of helium condense. Since these droplets are denser than their surroundings, they fall as
helium rain into the deeper layers. The rain continues moving inward until it reaches a
level where the temperature is high enough to cause it to go back into solution with the
hydrogen. This phenomenon of rain being reabsorbed is actually observed on Earth over
deserts where rain falls from cool, high altitudes but reevaporates before it can hit the hot
ground.
In contrast to the formation of an iron core on a terrestrial planet, helium differentiation on a giant planet is a stable, self-regulating process. Since differentiation liberates
gravitational potential energy and thereby heats the fluid, it will cause some of the helium
to go back into solution. This negative-feedback mechanism acts as a thermostat and
tends to maintain the planet’s temperature at a relatively constant level, until the supply
of upper-level helium is depleted.
On Jupiter, the surface helium abundance is close to the solar composition level, and
the predicted cooling rate for Jupiter does not seem to require an additional source of
heat from the helium rain mechanism. Thus, for Jupiter, helium differentiation appears
to not yet have begun. There is some evidence that helium is depleted in Saturn’s outer
layers relative to Jupiter and the Sun, and the filtering effects of a conductive, differentially
rotating helium core on Saturn have been invoked to help explain the unusual symmetry
of the external magnetic field.
9.5.8 Uranus and Neptune
9–32
Problems
9-1. Helioseismicity
The best observations of the internal structure of the Sun come from helioseismicity.
These observations are vitally important, not only to stellar dynamics, but also to the
study of the interiors of the giant planets. Find out what is involved with helioseismicity,
and summarize your findings in a 1-2 page report. What kind of surface waves are observed
on the Sun’s surface? What is their typical frequency? Where are the best observations
being performed? What are the preliminary results on the internal velocity field and mass
structure of the Sun? What are the prospects for future work, and for extending the
technique to the giant planets?
9-2. Rotational oblateness
a) Compare the relationship between f , q, and J2 in (9.35) to the actual values, J2 a ,
for Earth, Mars, the Moon, Jupiter and Saturn, by calculating (J2 − J2 a )/J2 a . Which
planets are farthest from hydrostatic equilibrium?
b) A planet or a star becomes severely distorted if the rotational parameter q approaches unity. This q ≤ 1 limit puts a constraint on the angular velocity Ω of a planet.
Estimate the maximum Ω you would expect of a rapidly rotating planet in terms of its
density ρ.
c) Estimate how short the length of day must be on Earth and Saturn before these
planets would fly apart.
9-3. Internal density structures
Consider Planet X in which gravity is independent of depth. How would density vary
with radius? Express the result in terms of the planetary radius and the mean density.
9-4. Interpreting J2
The gravitational-oblateness parameter J2 can be related to the equatorial and polar
moments of inertia, Ix and Iz , respectively, which put constraints on interior models.
Assume ρ is independent of λ, and show that:
J2 =
Iz − Ix
,
M R2
(9.81)
where M and R are the planetary mass and equatorial radius, respectively. [Hint: Start
with cartesian-coordinate integrals for Ix and Iz , and compare integrals without solving
them.]
9-5. Conductive heat flow
a) Using a steady state heat balance, derive (9.43).
b) Derive an expression for the temperature at the center of a planet with a conductive
lithosphere. Assume that the planet has a radius R, uniform density ρ, internal heat
generation H, and a surface temperature To . The lithosphere extends from Rc < r < R.
9–33
For r < Rc heat loss is by solid state convection which maintains the radial thermal
gradient dT /dr at a constant adiabatic value -Γ. To solve for a physically reasonable T (r)
it is necessary that both the temperature and the radial heat flux qr are continuous at
R = Rc .
9-6. Heat loss in a conducting planet
As discussed in Chapter 2, eucrites are a class of meteorite composed of basalt and
likely formed in a parent body that had undergone partial melting. Consider how the
minimum size of the parent body can be constrained from simple heat flow arguments.
Assume a spherical, homogeneous silicate parent body with a density ρ of 3300 kg m−3
characterized by chondritic abundances of long-lived heat-producing (U, Th, K) elements.
a) Calculate the smallest radius for which melting will occur at the center of the body
if the sole mechanism of heat loss is steady state conduction. Assume the heat production
H=6.2x10−12 W kg−1 is the long-lived chondritic heat production and k=3.3 W m−1 K−1
is the thermal conductivity. Assume a temperature difference (∆T =T − To ) of 1100◦ C is
required for melting to occur.
b) Now suppose that the heat production is four times higher than the long-lived
chondritic rate. What is the minimum body size?
c) Discuss the results.
9-7. Convective heat loss
Dervive a modified form of the Rayleigh Number given in (9.55) assuming that the
temperature of the medium in which the sphere rises is characterized by a temperature
that increases linarly with depth as γ=dT /dz. [Hint: Calculate how fast the sphere must
rise to remain more buoyant than its surroundings.]
9-8. Jupiter and the n = 1 polytrope
The density of a spherical, hydrostatically balanced planet that follows a polytropic
equation of state:
(9.82)
p = k ρ1+1/n ,
satisfies the Lane-Emden equation:
¶
µ
¶
µ
d
n+1
−1+1/n 2 dρ
ρ
= −4π r2 G ρ ,
r
k
n
dr
dr
with boundary conditions ρ(0) = ρcen and ρ(R) = 0.
a) Verify that for n = 1, the solution to (9.83) is:
³q
´
2πG
sin
k r
q
.
ρ(r) = ρcen
2πG
r
k
Plot the function sinc(x) ≡ sin(x)/x for the relevant interval x ∈ {0, π}.
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(9.83)
(9.84)
b) Apply ρ(R) = 0 to (9.84) to find R in terms of k. The radius does not depend on
the total mass for an n = 1 polytrope, so what happens when more mass is added?
c) Metallic hydrogen satisfies an n = 1 polytrope with:
k = 2.72 × 1012 dyne cm4 g−2 .
(9.85)
Use this k to calculate the radius R, and compare with the equatorial radii of Jupiter and
Saturn.
d) By integrating (9.84) over the spherical volume of the planet, find the total mass
M , and show that:
π2
ρavg ,
(9.86)
ρcen =
3
where ρavg is the planet’s bulk density. [Hint: Use integration by parts.] Use (9.82), (9.85),
and (9.86) to estimate Jupiter’s central density and pressure.
References
Carr, M.H., 1984, The Geology of the Terrestrial Planets, NASA SP-469.
Cox, A.N., W.C. Livingston, & M.S. Matthews, 1991, Solar Interior and Atmosphere,
University of Arizona Press.
Gehrels, T. & M.S. Mattews, 1984, Saturn, University of Arizona Press.
Gill, A.E., 1982, Atmosphere-Ocean Dynamics, Academic Press.
Holton, J.R., 1992, An Introduction to Dynamic Meteorology, 3rd ed., Academic Press.
Hubbard, W.B., 1984, Planetary Interiors, Van Nostrand Reinhold.
Turcotte, D.L. & G. Schubert, 1982, Geodynamics, Wiley & Sons.
Whitham, G.B., 1974, Linear and Nonlinear Waves, Wiley.
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