Chapter

9 Interiors

We now turn our attention to interiors, and therefore to practically all of the massthat makes up the planets. Observing the interior structure of a planet is obviously notan easy task, but we need to know as much as we can about interiors, in part becausesuch information affects nearly all other aspects of planetary science. As examples, theinterior chemistry of a planet provides information about how it formed and how it hasevolved, and the way in which a planet gets rid of its internal heat has a profound effecton its surface geology or meteorology. In addition, the coupling of a planet’s rotation tothe motions of the conductive fluid in its interior gives rise to strong magnetic fields thatgreatly extend the planet’s influence past its surface boundary. Also, the details of theinternal 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 waysby interpreting the external signatures caused by internal structure. The single mostimportant parameter that describes a planet’s interior is its bulk density. The bulkdensity 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 aplanet’s interior, we must learn about the detailed distribution of mass inside it. Theplanetary interior that is best understood is Earth’s because seismologists have observed

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and modeled the pressure and shear waves that emanate from earthquake sites and passthrough 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 fromorbiting spacecraft and we have high resolution images of the volcanic and tectonic featureson their surfaces. For the Moon, Mercury, and Galilean satellites we have estimates ofgravitational 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 ofthese bodies we have imaging of at least part of their surfaces, from which we can makelimited inferences about their internal structures and dynamics.

The interiors of the giant planets themselves are more closely related to stellar interiorsthan to terrestrial-planet interiors. In fact, the Sun and the four giant planets constitutefive 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 Dopplertechniques to observe the Sun’s surface vibrations, helioseismologists have been able toinfer basic properties of its interior structure. Efforts to apply helioseismology directlyto Jupiter have been hampered by a low signal-to-noise ratio, but the lessons learned fromthe solar observations indirectly influence our thinking about the giant planets. The mostdirect observational constraints on the internal structure of the giant planets come fromthe shapes of their gravity fields, which we know partly as the result of flyby and orbitalspacecraft missions, but mostly from accurate ground-based observations of the orbitalprecessions 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 findout about its internal density distribution. Because an object composed of sphericallysymmetric shells has a gravity field that is identical to that of a point mass, and becauseat a great distance any object has this property, we will need to rely on deviations fromspherical symmetry and will have to probe each planet’s external gravity field at closerange to learn about its inside makeup.

What we want is a way to perturb a planet under controlled conditions, so that wecan study how the planet reacts to the perturbation and thereby learn something aboutits internal structure. As luck would have it, the centripetal acceleration caused by rapidrotation provides a natural perturbation. Consider two rotating planets that have thesame average density, the same size and the same oblate shape. Let the density of oneof the planets be uniform throughout and the density of the other be differentiated intoa heavy core and a light outer shell. The one that is rotating faster is the one with thedenser 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 tothe magnitude of its gravitational acceleration on the surface at the equator:

q ≡ RΩ2

GM/R2=R3Ω2

GM, (9.1)

where R, Ω, and M are the planet’s equatorial radius, angular velocity, and mass, re-spectively. 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 ofthe magnetic field (historically called the System III rotation rate). Some large, rapidlyrotating stars have q values that actually approach unity, implying that those stars areon the verge of flying apart. Saturn has q = 0.153 and is the most rotationally distortedplanet in our solar system, while Jupiter has q = 0.089, and the rest of the planets havemuch smaller values, as listed in Table 9.1. In a later section we will derive an expressioninvolving q that describes the difference between the oblateness of a planet’s shape andthe 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 weknow about two point masses. We will take one point mass to be our test mass, whichwe can place at any position r in order to determine the local gravity field g(r). For thesecond mass, we will take in turn each differential mass element dm at each location r′

inside the planet. If we put d ≡ r−r′, then the gravity field g(r) associated with the masselement dm is the acceleration felt by the test mass when placed at the location r:

g(r) = − Gdm

|r− r′|2r− r′

|r− r′| = −Gdmd2

d . (9.2)

If we enclose dm with any surface S, then the component of g that is normal to eachsurface element dS will be given by:

n · g = −Gdmd2

n · d . (9.3)

where n is the unit vector that is normal to dS. Notice that, as seen from dm, the surfaceelement dS subtends a differential solid angle dΩ (not to be confused with angular velocityΩ) given by:

dΩ =dS

d2n · d . (9.4)

That means that the total surface integral of n · g is simply:∫S

n · g dS = −Gdm∫S

n · dd2

d2

n · ddΩ = −Gdm

∫S

dΩ = −4πGdm . (9.5)

By the nature of the cancellation of the orientation factor n · d and the distance factord2 in (9.5) we can see that any inverse-square vector field g will have the property thatthe total flux n · g through an arbitrary closed surface will be a constant. Now, take thesurface S to encompass the entire planet, and include all the mass elements dm = ρ dV .Then (9.5) becomes: ∫

S

n · g dS = −4πG∫V

ρ dV . (9.6)

Recall Gauss’ divergence theorem for any vector field E:∫V

∇ ·E dV =∫S

n ·E dS . (9.7)

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By combining (9.6) with the divergence theorem (9.7), we find:∫V

(∇ · g + 4πGρ) dV = 0 . (9.8)

Since (9.8) is true for any volume V that encloses the density distribution ρ, the integrandof (9.8) must vanish, which implies:

∇ · g = −4πGρ . (9.9)

Finally, if we introduce the gravitational potential field Φ:

g = −∇Φ , (9.10)

then:∇ · (−∇Φ) = −4πGρ ,

⇒ ∇2Φ = 4πGρ . (9.11)

Equation (9.11) is called Poisson’s Equation. In Chapter 11 we will approximate thegravitational potential of a planetary ring by using (9.11) with the density ρ taken to bean 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 ofa planet, in which case ρ = 0 and Poisson’s equation (9.11) reduces to Laplace’s equation:

∇2Φ = 0 . (9.12)

The simplicity of (9.12) belies the fact that it applies to the gravitational potential outsideof any distribution of mass, no matter how complicated that distribution might be. It isadvantageous to pick a coordinate system whose geometry shares the same symmetries asthe density distribution that we are trying to study. The spherical coordinate system is themost natural system for describing a planet’s external gravitational potential. We take theorigin 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 refersto 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:

1r2

∂

∂r

(r2 ∂Φ∂r

)+

1r2 sin θ

∂

∂θ

(sin θ

∂Φ∂θ

)+

1r2 sin2 θ

∂2Φ∂λ2

= 0 . (9.13)

The general solution to (9.13) is found by separation of variables, and may be written:

Φ(r, θ, λ) =∞∑l=0

l∑m=−l

[αlm r

l + βlm r−(l+1)

]Ylm(θ, λ) , (9.14)

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where the αlm and βlm terms are the sought-after coefficients that describe the planet’sdensity distribution. The Ylm(θ, λ) functions are called the spherical harmonics, andrepresent the complete set of orthonormal functions on the surface of a sphere. Theinteger l represents the number of nodal lines on one hemisphere and is a measure of thefineness of structure. The integer m represents the distribution of lines between latitudeand longitude. When m=0 solutions have only a latitudinal dependence and are referredto as zonal harmonics. When m = l solutions are only a function of longitude and arecalled sectoral harmonics. Solutions in which 0 < m < l have both a latitudinal andlongitudinal 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 goesto infinity, we can immediately eliminate the rl terms in (9.14) by setting all of the αlmcoefficients to zero. Notice that the remaining r−(l+1) terms have the property that thehigher l terms fall off with distance more rapidly than the lower l terms. This meansthat the highest-order terms of a planet’s gravitational potential will hardly be detectableoutside of the planet.

The co-latitude appears only as cos θ in the explicit form of Ylm(θ, λ), and so it iscustomary to define a new coordinate µ:

µ ≡ cos θ . (9.15)

The Ylm are then given by:

Ylm =

√2l + 1

4π(l −m)!(l +m)!

Pml (µ) eimλ . (9.16)

The dependence of Ylm on longitude is Fourier-like:

eimλ = cos(mλ) + i sin(mλ) . (9.17)

The Pml (µ) in (9.16) satisfy the equation

(1− µ2)d2P

dµ2− 2µ

dP

dµ+[l(l + 1)− m2

1− µ2

]P = 0

which reduces to Legendre’s equation for the case of m=0, which is discussed in moredetail later. The Pml (µ) are called the associated Legendre polynomials and aregenerated recursively by Rodrigues’ Formula:

Pml (µ) =(

(−1)m(1− µ2)m/2

2l l!

)(d

dµ

)l+m(µ2 − 1)l . (9.18)

The first factor on the right hand side normalizes the function such that P 0l = +1. The

use of normalizations is common because numerical factors in the associated polynomialsincrease rapidly with m. Normalizing the solution causes the coefficients in the harmonic

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analysis relate more clearly to the physical process that they represent. The first fiveLegendre polynomials, Pl ≡ P 0

l , are:

P0 = 1 ,P1 = µ ,

P2 =12

(3µ2 − 1) ,

P3 =12

(5µ3 − 3µ) ,

P4 =18

(35µ4 − 30µ2 + 3) . (9.19)

The form of P2(µ) will play a role when we connect the rotation parameter q to theoblateness of Φ.

Just as with any solution written as a series expansion, the functions that make upthe spherical harmonics are all well known, and it is the βlm coefficients that hold all ofthe information about a planet’s density distribution. To connect the βlm coefficients tothe density distribution ρ(r), we express Φ(r) in its integral form:

Φ(r) = −G∫V

ρ(r′)|r− r′| dV

′ . (9.20)

Outside a planet, where r > r′, the expansion of |r− r′|−1 in terms of spherical harmonicsis given by:

1|r− r′| =

∞∑l=0

r′l

rl+1

4π

(2l + 1)

l∑m=−l

Y ∗lm(θ′, λ′)Ylm(θ, λ)

, (9.21)

where Y ∗lm is the complex-congugate of Ylm. Applying this series expansion to (9.20) yieldsthe spherical-harmonic series expansion for Φ(r, θ, λ):

Φ(r) = Φ(r, θ, λ) = −G∫ ∞r′=0

∫ π

θ′=0

∫ 2π

λ′=0

ρ(r′, θ′, λ′)

×∞∑l=0

r′l

r(l+1)

4π

(2l + 1)

l∑m=−l

Y ∗lm(θ′, λ′)Ylm(θ, λ)

r′

2 sin θ′ dr′dθ′dλ′ . (9.22)

Thus, the βlm coefficients in (9.14) are given by an integral involving the density:

βlm = − 4πG(2l + 1)

∫ ∞r′=0

∫ π

θ′=0

∫ 2π

λ′=0

ρ(r′, θ′, λ′)Y ∗lm(θ′, λ′) r′l+2 sin θ′ dr′dθ′dλ′ . (9.23)

The sensitivity of the βlm coefficients to the radial distribution of density comes from ther′l+2 weighting factor, which causes information about the outer layers to be preferentiallycontained 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 har-monic expansion (9.14) for Φ. Also, because planets display significant rotational symmetryand north-south symmetry, not all of the (l,m) generality in (9.14) is necessary. In suchcases, it is customary to write the spherical harmonic expansion of Φ in an “unnormalized”form, truncated to some number of leading terms, n:

Φ(r, θ, λ) ≈ −GMr

1−

n∑l=2

(R

r

)l l∑m=0

Pml (θ) [Clm cos(mλ) + Slm sin(mλ)]

, (9.24)

where R and M are the planet’s equatorial radius and total mass, respectively. Theexpression (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 thatthe higher-order terms provide small corrections to this leading-order term. Note thatthe summation proceeds from l = 2 because a value of one gives an asymmetric potentialwhich 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 completelyreal expressions. The Clm and Slm coefficients are dimensionless multipole moments of thedensity 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 greatestinterest in (9.24) are the zonal harmonic coefficients Cl0, which are traditionally given theirown symbol:

Jl ≡ Cl0 . (9.25)

The most important of these parameters is J2, the ellipticity coefficient, which is typi-cally observed to have a magnitude of 1000 times that of higher degree and order terms ina 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 thezonal harmonic coefficients to the density:

Jl = − 2πMRl

∫ R

r=0

∫ 1

µ=−1

ρ(r, µ)Pl(µ) rl+2 dµ dr , (9.26)

where we have made use of the identity sin θ dθ = −dµ. The values of J2 and the rotationalparameter 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. Sinceplanets display strong north-south symmetry, the odd l terms in the series expansion of Φare usually negligible when compared with the even l terms. Therefore, the leading termsof Φ for a planet with rotational symmetry and north-south symmetry may be expressedsimply as:

Φ(r, µ) = −GMr

1−

n∑l=1

J2l

(R

r

)2l

P2l(µ)

,

= −GMr

1− J2

(R

r

)2 12

(3µ2 − 1)− J4

(R

r

)4 18

(35µ4 − 30µ2 + 3)− . . ., (9.27)

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Table 9.1

Planetary Internal Structure Parameters

Planet Bulk Density J2 q Λ2 ≡ J2/q C/MR2

(kg m−3)

Mercury 5420 0.0000(8± 6) 0.000001 80 ??Venus 5250 0.00000(6± 3) 0.000000061 98 0.34Earth 5515 0.0010826 0.0035 0.31 0.3335Moon 3340 0.0002024 0.0000076 27 0.391Mars 3940 0.001959 0.0046 0.43 0.366Jupiter 1314 0.014733 0.089 0.17 0.26Saturn 690 0.01646 0.153 0.11 0.25Uranus 1190 0.003352 0.027 0.12 0.23Neptune 1660 0.004 0.018 0.22 0.23

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 shapeof a planetary or stellar body, because it provides information on the radial distributionof mass in the body. Observations of satellite and narrow-ring precession rates, togetherwith flyby spacecraft trajectory data, yield the best measurements of the zonal harmoniccoefficients J2 and J4, and sometimes even J6, for the various planets. These coefficientsprovide a description of the density distribution inside a planet that reveals the extent ofits mass differentiation and core formation. They also provide the primary means by whichplanetary interior models are judged for accuracy. J2 is commonly expressed in terms ofthe moments of inertia about the polar (C) and equatorial (A) axes as

J2 =C −AMR2

. (9.28)

where the moment about the polar axis is

C =∫ (

x2 + y2)dm

and corresponds to the integral over the mass distribution composed of small elements dmtimes the square of the distance from Ädm to the rotation axis. In spherical coordinates wemay write

C =∫ 2π

0

∫ π

0

∫ R

0

ρ(r′)r′4cos3θdr′dθdλ

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 beenloosely refering to as the “oblateness” of Φ. We will now connect J2 to the rotationalparameter q and to the oblateness of the planet itself. The effective potential Φe at thesurface of a rotating planet includes the gravitational potential, Φ, and the centripetalpotential:

Φe = Φ− 12|Ω× r|2 = Φ− 1

2(Ω r sin θ)2

, (9.29)

where r sin θ is the cylindrical radius measured from the rotation axis to the planet’ssurface. Since we are using (9.27) to express Φ in terms of P2(µ), it will be convenienthere to also express the centripetal potential in terms of P2(µ):

P2(µ) = P2(cos θ) =12

(3 cos2 θ − 1) =12

[3(1− sin2 θ)− 1] = 1− 32

sin2 θ ,

⇒ sin2 θ =23

[1− P2(µ)] ,

⇒ −12

Ω2 r2 sin2 θ = −13

Ω2 r2 [1− P2(µ)] . (9.30)

The effective potential at the surface of the planet is thus:

Φe(r, µ) = −GMr

[1− R2

r2J2 P2(µ)− . . .

]− 1

3Ω2r2 [1− P2(µ)] .

Grouping terms yields:

Φe(r, µ) =(−GM

r− 1

3Ω2r2

)+(GMR2

r3J2 +

13

Ω2r2

)P2(µ) + . . . (9.31)

We will now assume that the planet is in hydrostatic equilibrium, such that the sur-face of the planet is an equipotential surface. This is a good approximation for giantgaseous planets. The solid planets exhibit small (at most several percent) deviations fromhydrostatic behavior due to internal dynamics and interior structure. However these smalldifferences provide essential insight about the structures and geodynamical states of thesebodies. We now define the planetary flattening, f=(a-c)/a where a is the equatorialradius 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:

P2(equator) = −12, P2(pole) = 1 ,

the result is:

0 = −GMR

[1− 1

1− f

]− 1

3Ω2R2

[1− (1− f)2

]9–9

+GM

RJ2

[−1

2− (1− f)−3

]+

13

Ω2R2

[−1

2− (1− f)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 qas defined by (9.1), then we find the following relationship between J2, f , and q:

J2 ≈23f − 1

3q . (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 byground-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 theterrestrial planets it is necessary to retain terms to second order such that

J2 ≈23f

(1− f

2

)− 1

3q

(1− 3

2q − 2

7f

). (9.36)

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 momentof inertia factor C/MR2 that describes the radial distribution of the interior mass anotherparameter 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. ForEarth the period is 26,000 years so the direction of the North Pole as projected on thecelestial sphere changes. The rate of precession depends on a factor called the dynamicelllipticity

H =C −AC

(9.37)

that is controlled by the polar and equatorial moments of inertia. We may substitute tofind

J2

H=

C

MR2. (9.38)

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 Table9.1 we know J2 for all of the terrestrial planets and giant planets. But we have measuredthe spin axis precession rate only for the Earth and Mars. For solid planets the precessionrate can be found by measuring the orientation of the spin axis at two different times fromtracking surface landers or orbiting spacecraft. The precession rate for Mars was found

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by comparing measurements found by the Viking landers to that found recently from thePathfinder 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 bombardmentwas a major source of heating. Also in early evolution, a significant amount of heat wasgenerated from the process of differentiation due to the release of gravitational potentialenergy as planet’s heaviest components, like its nickel and iron, sink to the center andform a dense core. The release of nuclear binding energy through the radioactive decayof the uranium, thorium, and potassium found naturally in chondritic (solar) materials,and hence in any silicate material derived from the protoplanetary nebula, is the mostimportant source of present-day internal heat for the terrestrial planets. Tidal dissipa-tion is currently the dominant source of internal heat for Io, and is the most importantsource of heat for most of the icy satellites of the outer solar system that display evidenceof recent tectonic and tectonic surface activity. In addition, tidal heating has played a rolein any planet that has at one time undergone substantial tidal despinning, like Mercury isthought to have experienced. Also making a small contributions to the total heat budgetare certain types of phase changes in interior materials, and the ohmic dissipation ofinternal 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? Ifthere is not too much heat, a planet can cool off simply by conduction of its heat to thesurface, and then radiation into space. On the other hand, if there is more than a criticalamount of heat that needs to escape, then convection will ensue. Convection is a moreefficient means of transporting large amounts of heat than conduction because it involvesmacroscopic motions of material under the simple principle that warm, buyoant materialdeep 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 energyvia 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 efficientthat a hot molten surface will very rapidly (in a geologic sense) produce an insulatingcrust that will cause the radiation process to cease. So in practice for the solid planets andsatellites radiation was very important during a short time during and post-accretion andhas 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 thatcontrols the geologic expression of the surface. Planetary cooling causes volcanism andtectonics. On the Earth the loss of heat by convection drives the global motions of theEarth’s plates, producing earthquakes, mountain building, and rifting. The volcanic release

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of volatiles trapped in the interior produces and modifies oceans and atmospheres. Giventhe importance of the thermal state of planetary bodies for nearly all aspects of theirstructure and evolution, it is clear why the study of heat budgets is a major focus of studyin compartive planetology.

9.2.3 Planetary Cooling Efficiency

An incomplete but instructive assessment of a planetary heat budget comes fromconsidering in the simplest possible way the global heat generation and heat flux. Forthe terrestrial planets and the outer solar system satellites that have a significant rockcomponent, radioactive heating is the principal source of long term heating, and so theheat 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 generationis referred to as the cooling efficiency and has a form q/H ≈ 3/R, where R is theplanetary radius. We see that smaller planets produce less heat and cool faster, or moreefficiently, than larger planets, which produce more heat and cool more slowly. From thissimple approach we would expect that large planets that generate their heat primarily byradioactive decay to be geologically active over more of their history than small planetarybodies. Indeed we shall see that this is the case.

9.2.5 Conduction

The simplest approach to understanding conductive heat transport within a planetcomes from considering the flux of heat across a slab of material. Fourier’s Law takesthe form

q = −kdTdz

(9.39)

where q is the heat flux (not to be confused with the rotational parameter discussedearlier), k is the thermal conductivity, a property of the material, and dT/dz is thethermal gradient. This expression indicates that the flow of heat per unit area per unittime is directly proportional to the temperature gradient at that point. The minus signindicates that heat flows in the direction (z) of decreasing temperature (T ). In this simplecase the gradient of temperature is linear with depth.

A relevant problem for the temperature state of small solid plaetary bodies is the radialsteady-state conduction of heat in a sphere. The solution, which derives from a straightfor-ward energy balance for a symmetrically cooling body, is also relevant for spherical shellsand thus for conductively cooling lithospheres.

Consider a planet with surficial spherical shell of inner radius r and thickness δr. Theflux of heat out of the shell through the surface is

4π(r + δr)2qr(r + δr)

and the flux into the base of the shell is

4πr2qr(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) + δrdqrdr

+ . . .

Neglecting powers of δr, the net flow of heat out of the shell is

4π[(r + δr)2qr(r + δr)− r2qr(r)

]= 4πr2

(2rqr +

dqrdr

)δr.

The rate of heat production per unit mass, H, within the shell is

4πr2ρHδr

where ρ is density. Here 4πr2δr is the volume of the shell which is approximated to firstorder in r. By equating the rate of heat production in the shell to the net flux out of theshell we arrive at the heat balance

dqrdr

+2qrr

= ρH. (9.40)

We now wish to relate the heat flux to the radial temperature gradient dT/dr. In sphericalgeometry Fourier’s law has the form qr = −kdT/dr. Substituting into (9.40) we find

k

(d2T

dr2+

2r

dT

dr

)+ ρH = 0

ork

r2

d

dr

(r2 dT

dr

)+ ρH = 0

The general expression for the temperature in a sphere or spherical shell comes fromintegrating ?? twice

T (r) = −ρH6k

r2 +c1r

+ c2 (9.41)

where the constants c1 and c2 depend on the boundary conditions. For example we maysolve the for the temperature distribution in a spherical planet of radius R that has auniform rate of heat production. The boundary condition is that the outer surface of thesphere has a temperature To. To have a finite temperature at the center of the planet c1must 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

6k(R2 − r2

). (9.42)

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From Fourier’s Law the the surface heat flux qo is given by

qo =13ρHR (9.43)

This expression is based on conservation of energy and thus is valid for any specified modeof 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 thanconvection. If this is the case why don’t planets (or anything else) always convect ratherthan than conduct? In order to understand whether a planet’s interior is primarily losing itsheat through conduction or convection, we need a nondimensional number that measuresthe relative contribution of each process.

Usually, an increase in temperature T will result in a decrease in density ρ. Waternear its freezing point is the most notable exception to this rule. Over small ranges oftemperature, we can approximate the dependence of density on temperature with a Taylorseries 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 thethermal expansion coefficient. Time-dependent heat conduction is a diffusive process,and is described by a parabolic partial-differential equation:

D

DtT =

(∂

∂t+ v · ∇

)T = κ∇2T . (9.45)

The parameter κ is called the thermal diffusivity (= k/ρCp). To get a feel for therelative importance of conduction versus convection, consider the simple problem of a thinslab of material that is confined between two horizontal plates, which are separated by avertical distance d. This simple convection problem was first investigated experimentallyin 1901 by H. Benard, and analyzed theoretically in 1916 by Lord Rayleigh, and is nowrefered to as Rayleigh-Benard convection. The material is subject to a gravity field ofstrength 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 analogousto the thermal diffusivity κ, but describes the rate of diffusion of momentum rather thanof 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 param-eters g, α, ∆T , κ, ν, and d. The first is the Rayleigh number, Ra, that measures thetendency of a medium to convect. Ra measures the relative importance of buoyancy forces

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(in the numerator) that drive convection and viscous resistance forces (in the denomina-tor) that inhibit convection. The Rayleigh number can be expressed on various forms thatdepend on the geometry of the medium and the nature of heating. However, a simplerepresentation of the Rayleigh number can be determined from a balance of forces on asphere in a uniform fluid medium. The sphere, or radius a will rise due to buoyancy forcesat a velocity V . Buoyancy occurs because the sphere has a density ρ−∆ρ that is less thanthe density of the fluid ρ. The buoyancy force scales with the volume of the sphere as

Fb =43πa3∆ρg (9.47)

where g is the acceleration of gravity. As the sphere rises there will be a viscous resistanceto motion due drag over the spherical surface. This force can be written

Fr =(νV

a

)4πa2 (9.48)

where µ is the dynamic viscosity of the medium, which is a measure of the resistance ofthe medium to shear. It is a ratio of the stress to the strain rate in a viscous medium. Byequating 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 thevelocity in terms of a temperature difference. We write

∆ρ = −ρα∆T. (9.50)

Substituting into (9.49) we find

V = −ρα∆Tga2

3µ. (9.51)

It is apparent that the sphere will keep rising until it cools sufficiently such that thebuoyancy force is exceeded by viscous drag. The sphere will cool conductively at a timethat is proportional to the thermal diffusivity of the medium and the size of the sphere as

κt

a2= c (9.52)

where c is a constant. This is the characteristic cooling time of the sphere. Sincevelocity is just distance over time we may now solve for the distance the sphere will risebefore it cools

V =d

t=ρα∆Tga2

3µ(9.53)

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For c=0.5, at which time the sphere will essentially be cool, the distance the sphere risesmay be written

d =16

(ραg∆Ta3

µκ

)a (9.54)

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 thekinematic viscosity, which is related to the dynamic viscosity as ν = µ/ρ. Then

Ra =αg∆Td3

κν. (9.55)

There is a critical value of Ra that marks the onset of convection, Rac, and this depends onwhether 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 thermalproperties. The onset of convection also depends on whether the frictional boundaryconditions are taken to be “no slip” or “free slip.” For various geometries Rac falls in therange 100-1000. The Rayleigh numbers for the terrestrial planets greatly exceed criticalvalues for either boundary condition and are thus expected to lose heat by convection.

The second important nondimensional number is the Prandtl number, Pr:

Pr =ν

κ, (9.56)

which compares the relative strengths of the two diffusion parameters. The Prandtl num-ber is especially important in problems that combine rotation with convection, becauserotation can couple with the material’s viscosity and thermal diffusivity to a create a newmode of instability called overstability. If the slab is rotating about a vertical axis withangular velocity Ω, a third nondimensional parameter enters, usually taken to be the Taylornumber, Ta:

Ta = 2Ωd2

ν(9.57)

One effect of rotation is the Taylor-Proudman effect that tends to suppress vertical motions,and consequently Rac generally increases as Ta increases.

Linear analytical theory can predict Rac as well as the horizontal wavelength of theconvection at its onset. In Rayleigh’s original analysis, he assumed that the boundingsurfaces were perfect conductors. That way, the temperature on each boundary remainsconstant even though there will be temperature fluctuations in the interior. With thisthermal boundary condition, convection begins as a series of overturing cells that areapproximately as long as they are tall. Interestingly, when the problem is set up usinginsulating boundaries instead of conducting boundaries, convection first takes the form ofa single overturning cell that fills the experimental chamber. Actual geophysical systemshave thermal boundaries that are neither perfect conductors nor perfect insulators, andso a mixture of both qualitative behaviors is possible. For large enough Ra, organizedconvective cells give way to turbulent convection. The nature of turbulent convectionis especially important when considering the interior structure of the Sun and the giantplanets.

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It may seem strange at first to think of the mantles of the terrestial planets as beingable to convect like fluids. After all, don’t we know from seismology that shear waves passthrough Earth’s mantle, and hence that Earth’s mantle is not fluid, but is made of solidrock? The reason that there is no contradiction is that we must be careful to considerthe different timescales involved in the two processes. Seismic shear waves travel throughEarth’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, thensignificant transport of heat by solid-state convection is possible.

Since much of a planet’s heat is generated by radioactive decay, it is appropriate toconsider a slab heated from within, rather than one heated from below. In that case, onemust specify the rate of internal heat generation per unit mass, H, and the mantle’s heatcapacity, Cp, and form a new Rayleigh number:

Ra =g α d3

κ ν

H d2

κCp. (9.58)

The viscosity ν of Earth’s mantle is estimated from the rate of the glacial rebound ofthe crust; remarkably, Earth’s crust is still rising back into its pre-ice age shape followingthe most recent retreat of glaciers some 10,000 years ago. In 1935, N.A. Haskell used theages of elevated beaches in Scandinavia to calculate that the viscosity of Earth’s mantleis about 1023 times greater than the viscosity of liquid water. With this value of ν andvarious 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 thatmantle convection is probably also taking place inside Venus and Mars, and to a lesserextent 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 tobe understood. One of the effects of rotation is to suppress motions along the direction ofthe rotation axis, a consequence of the so-called Taylor-Proudman theorem that we willdiscuss in a later lecture. Experiments on convection in a rapidly rotating sphere withcentral gravity are difficult to perform. The best experiments to date were carried outin the microgravity of Spacelab 3, as described in the paper “Laboratory experiments onplanetary and stellar convection performed on Spacelab 3,” by Hart et al. 1986. Theseexperiments, along with complimentary numerical experiments and theoretical work, in-dicate that convection occurs in the form of rolls aligned with the rotation axis and bentby 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 inthe giant planets, then we can expect that such convection will have an influence on thevisible atmospheric dynamics. For example, it is not known why most Jovian vortices andplanetary-scale waves drift with velocities of only a few meters per second relative to the

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planetary magnetic field reference frame (called the System III reference frame) whereasthe 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 interiorwhere the magnetic field originates. One idea is that the banana cell convection producesvertically propagating waves that act as a source of stationary forcing to the cloud-topdynamics. In any event, a clear understanding of the surface weather patterns on thegiant planets will require a better understanding of the dynamics of their deep interiors, inmuch the same way that surface tectonic features on the terrestrial planets are understoodto result from their specific interior dynamics and thermal histories.

9.3 Magnetic Dynamos

Whereas a planet’s external gravity field provides important density information aboutthe interior, a planet’s external magnetic field provides important velocity informationabout the interior that would otherwise remain inaccessible. This information can be asbasic as providing a convenient reference frame from which to measure cloud-top windspeeds on the giant planets, or as profound as indicating that Earth has significant fluidmotions in its liquid-iron core, and Uranus and Neptune have significant circulations intheir liquid-water mantles. In contrast, the conspicuous lack of strong magnetic fields foreither Venus or Mars provides important constraints on their internal dynamics.

To understand the mechanism of magnetic field generation we must first characterizethe general pattern of the field. Measurement of the magnetic field of the Earth, which isby far the best-studied field in the solar system, shows field lines that emanate from thesouth geomagnetic pole and enter at the north geomagnetic pole. The pattern is consistentwith a sphere that contains a powerful bar magnet at its center, though we shall discussthat this is not a plausible explanation for any planetary magnetic field. Such a field isreferred to as a dipole field because it could be explained by a pair of magnetic polesof equal strength a short distance apart. While there are important deviations from thepattern expected for a dipole, to a first approximation a dipole aligned generally along thepolar 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’sequation

∇2Vm = 0.

For planetary bodies, spherical coordinates are appropriate and so we may write a sphericalharmonic representation of the solution

Vm =a

µo

∞∑l=1

(R

r

)l+1 l∑m=0

Pml (cosθ) [glm cos(mλ) + hlm sin(mλ)] . (9.59)

that will effectively separate out the dipole and non-dipole components of the magneticfield. 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 thatthere is no magnetic monopole. In the event that monopoles exist they do not significantlycontribute to planetary magnetism. The l=1 term is the dipole.

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The dipole field can be conveniently represented at any position r from the dipole interms of Vm, where

Vm =1

4πr3m · r =

cosθ

4πr2, (9.60)

where θ is the angle between the dipole axis and the radius vector r from the dipole tothe point of interest. The magnetic moment, m, represents the product of the polestrength and separation. The magnetic field B(r) at any position r can be determined bydifferentiating 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. Usingthe same convention as we developed for gravity we resolve the radial and latitudinal(zonal) components of the field

Br = −µo∂Vm∂r

=µo4π

2mr3cosθ (9.62)

Bθ = −µor

∂Vm∂θ

=µo4π

m

r3sinθ. (9.63)

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 thingthe Earth’s field exhibits secular variations that include: a decrease in the magneticmoment of 0.05% longitude yr−1, a westward drift of the dipole of 0.05 yr−1 and of thenon-dipole component of the field of 0.02 longitude yr−1, a rotation of the dipole towardthe geographic axis of 0.02 yr−1, and the growth and decay of various features of the non-dipole 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 mechanismfor magnetic field generation must allow for time variations. As seismology began to showevidence for a fluid outer core within the Earth, theoretical work has converged to indicatethat a self-sustaining dynamo due to motions in the fluid core was the most plausiblemechanism 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 exhibitsignificant magnetic fields.

To be more specific, the differential motion of electrically conductive fluid in a planet’sinterior couples to the planet’s rotation to give rise to a self-sustaining magnetic field, aprocess called the magnetic dynamo process, which is described by principles of magne-tohydrodynamics. After decades of research, satisfactory models of this phenomenon areonly beginning to be achieved. We do know that Earth’s magnetic field is not simply theresult of permanent magnetism in its rocks, because the paleomagnetic record showsthat Earth’s magnetic field has reversed its polarity many times. There is no obviouspattern to Earth’s polarity reversals, but the average period is on the order of 106 yrs. Atmidocean ridges, the irregular pattern of magnetic polarity is organized into stripes thatparallel the ridges with mirror-image symmetry — this was the key fact that convinced

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the geophysical community in the mid 1960’s of the reality of plate tectonics. The sun alsoexhibits magnetic polarity reversals. But unlike Earth, the Sun’s reversals occur regularlyin 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 basison the frozen flux principle, illustrated in Figure ??. The essential element of thisprinciple is that if a combination of the conductivity, magnitude and velocity of fluidmotion is sufficient, then a magnetic field is transported and deformed with the fluid. Acharacteristic of this mechanism is that the kinetic energy of fluid motion is converted intomagnetic field energy.

There are two kinds of motions. The first is differential rotation within the core thatis a consequence of the tendency of a convecting fluid to conserve angular momentum asit is transported radially in response to buoyancy forces. Thus the inner part of the corerotates with a higher angular velocity than the outer part and it draws out the lines of apoloidal field to produce an additional toroidal field. This mechanism is known as theomega (ω) effect. Radial convective motion is required to regenerate the initial poloidalfield from the toroidal field by a second process, knwn as the alpha (α) effect. Thesignificant aspect of the motion that drives the alpha effect is that it is helical or spiralin 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 twointerconnected disk dynamos, shown in Figure ??. Each disk rotates in an axial field pro-duced by a coil carrying a current driven by the other disk. Rotation in the field generatesan electromotive force between the perimeter of the disk and the axis, providing the currentthat drives the other disk. The system is self-maintaining when current generation exceedsthe ohmic dissipation via an appropriate combination of velocity, size and loop conduc-tance. Note that the model is symmetrical with respect to the polarities of the currentsand fields, and that the system works equally well if the polarities are reversed. Instabil-ities in the system, including spontaneous current reversals, have been demonstrated inmechanical systems and computer models of more complex systems. This effect may be ananalog for the record of magnetic field reversals whose record is preserved in the Earth’sgeologic record.

If the action of the dynamo is to be robust the generation of the magnetic field mustbe rapid enough to overcome the diffusion of the field out of the conductor. There are tworelevant time constants, the first of which is for the time τv for regeneration of the fieldand is written

τv =L

v. (9.64)

Here L is the length scale over which field lines can be distorted and v is the relativevelocity of motion of the conductive fluid. The second time constant is τΩ, which describesthe decay of the field by ohmic dissipation out of the conductor

τΩ =B

(−dB/dt) (9.65)

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where (−dB/dt) is the rate of free decay of a field that is not being maintained, such asfor 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 maintaintheir forms as their amplitudes decay.

The expression used in the mathematical formulation of dynamo theory is the magneticinduction equation

∂B

∂t= ηm∇2B +∇× (v ×B). (9.67)

where ηm is the magnetic diffusivity of the conductive medium. In (9.67) the first term onthe right hand side represents the diffusion or ohmic dissipation of the field and the seconddescribes the interaction with velocity that allows regeneration. Equation (9.67) is a morephysical description of the competition between generation and dissipation than is (9.66).

In the magnetic dynamo mechanism differential rotation of the conductive fluid causesa winding up of the magnetic field lines, while organized motions in the fluid, perhapscapitalizing on the same nonlinear fluid dynamics that organizes and maintains centuries-old storms like Jupiter’s Great Red Spot, tend to align with and amplify the dipole fieldin a self-sustaining manner. While the details are not sufficiently worked out to makeuseful quantitative predictions, we can estimate the order of magnitude of a magnetic fieldthat would be produced by such a dynamo process. The essential balance is between theCoriolis force, actually part of the acceleration in a rotating coordinate system, and themagnetic force:

2Ω× v ≈ 14πρ

(∇×B)×B , (9.68)

where Ω is the planet’s rotational velocity. We can estimate the order of magnitude of themagnetic field from (9.68) to be

Ωv ∼ 1ρ

B

RB ⇒ B2 ∼ ρRΩv , (9.69)

where the planet’s radius R estimates the effect of the gradient operator in (9.68), and v isthe typical differential velocity in the fluid. If we assume that v scales with the rotationalvelocity ΩR from one planet to the next, and that the magnetic moment M scales withBR3, then

M ∝ ρ1/2 ΩR4 . (9.70)

Table 9.2 provides a summary of magnetic field information for the planets and Figure ??plots log(M) versus log(ρ1/2 ΩR4). There is generally good agreement between (9.70) andthe observations, although Mars and Venus have conspicuously low magnetic moments.

Table 9.2 shows that among the terrestrial planets and the Moon, Earth’s magneticfield stands out as having by far the greatest intensity. In fact, Earth’s field is morecomparable in magnitude to the giant gaseous planets than to its rocky counterparts. Theother terrestrial planet fields are much weaker and, except for possibly Mercury, do not

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Table 9.2

Magnetic Fields of Planets and Satellites

Planet Equatorial Surface Magnetic Moment m/mEarth

FieldBo(nT ) m (A−m2)

Mercury 300 4.18× 1019 6× 10−4

Venus < 30 < 6× 1019 < 8× 10−4

Earth 30300 7.856× 1022 1Moon < 25 < 1.3× 1018 < 1.7× 10−5

Mars < 5 < 2.0× 1018 < 2.5× 10−5

Jupiter 428000 1.46× 1027 1.87× 104

Saturn 21800 4.3× 1025 550Uranus 22800 3.7× 1024 47Neptune 13300 2.0× 1024 25Io 1800Europa 240Ganymede 100

Adapted from Stacey (1992).

require active dynamos. These fields are most likely due to remnant magnetization oftheir crusts by formerly active fields. The strong field for the Earth is consistent withthe existence of its liquid outer iron-nickel core, in motion due to convective cooling androtation. Venus lacks a magnetic field but this in itself does not require that the core hassolidified. It is possible that that planet’s slow rotation may preclude dynamo action. Therehas been a weakly-constrained estimate of the tidal Love number k2 for Venus based onanalysis of tracking data from the Magellan spacecraft that suggests that Venus has a liquidcore, but more detailed work will be required to verify this. The Mars Global Surveyorspacecraft recently placed an upper limit on Mars’ main field of 5 nT . That spacecraftalso includes an electron reflectometer that can measure crustal remnant magnetizationand has discovered multiple magnetic anomalies of small spatial scale (≈ 100 km) butlarge intensity (≈ 1.6 × 1016 A-m−2). Correlation of the magnitude of this remanentfield with the relative ages of surface units indicates that Mars likely had a strong mainfield (and thus vigorous convection in a fluid core) early (≈ 4 Ga ago) in its history. ForMercury, experiments are currently being planned that would allow an orbiting spacecraftto ascertain the core state and critically evaluate the hypothesis that that planet has anactive dynamo. The Moon lacks a present-day main field but remnant magnetization wasdetected by Apollo orbiting spacecraft. This is interesting in light of the fact that theMoon is known to contain very little iron. It is debated whether what little iron the Moonhas is enough to have allowed a limited core dynamo early in its evolution, but alternativemechanisms for magnetization of the surface must also be considered. Possibilities includethe interaction of the lunar surface with the solar wind, or magnetization by large impacts.

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On the basis of a limited number of laboratory experiments it has been suggested that itis possible to generate magnetization during the impact process, but the results have beendebated.

Observations of the giant planets indicate that the strength and orientation of a largedipole best reproduces the observed external magnetic fields of those planets as well. Voy-ager discovered that both Uranus and Neptune have equivalent magnetic dipoles withextraordinarily large tilts and central offsets. Athough there does not yet exist a satisfac-tory theory to account for the magnetic fields of Uranus and Neptune, or of any planetwith the possible exception of Earth for that matter, it is supposed that the large tiltsindicate that the magnetic fields of Uranus and Neptune are produced by a dynamo thatexists 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 Galileansatellites. Ganymede exhibits what is believed to be a dipole field generated in an ironcore. 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 suppliedto the interior due to tidal forcing discussed earlier. A main field signature has also beendetected for Europa and the mechanism is under investigation. Of special interest is thepossibility 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 con-sidering internal structure: i) comets, ii) small bodies, iii) terrestrial planets, and iv) gasgiants. Comets are primitive, icy bodies that have their own distinctive composition andhistory, and we will discuss them in detail in Chapter 2. The small-body category containsobjects with radii smaller than about 200 km, and includes dust, ring particles, small satel-lites, and most asteroids. These objects have had enough time over the age of the solarsystem to lose the internal heat they started with as a byproduct of accretion, and are notlarge enough to generate substantial internal heat from radioactive decay. Therefore, smallbodies are not likely to have convecting interiors or dynamically active magnetic fields atthe present time. We have already touched on the compositions of meteorites and asteroidsin Chapters 2 and 4, and we will discuss ring particles in Chapter 11. The terrestrial-planetcategory includes Mercury, Venus, Earth, and Mars, the large icy satellites of the giantplanets, like Ganymede and Callisto, and the Pluto-Charon system. The fourth categoryis the gas-giant category, which counts Jupiter, Saturn, Uranus, Neptune, and the Sunas its five members. We will concentrate on the interiors of the terrestrial planets in thissection, 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 ofthe data we have that bear on Mercury’s interior are photographs of its surface obtained in1974-75 by three successive flybys of the Mariner 10 spacecraft. We do not have detailedobservations of Mercury’s heat flow, gravity field, or seismic activity. However, usingthe Mariner 10 photographs, we can make indirect inferences based on observed surfacetectonic features that are the signature of changes in the size and shape of the planet.

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It is likely that Mercury started out with a much faster rotation rate than it currentlyhas, and that it has undergone tidal despinning. This is believed to have occurred becauseMercury’s 58.65 day rotation rate is low, and in fact is in a 3:2 commensurability withits 87.97-day orbital period, implying tidal evolution. A large change in its rotationalparameter q could have caused a large change in its figure, depending on whether itsinterior 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 amean density of 5440 kg m−3, which is the second highest planetary density after Earth’s5520 kg m−3. Considering that Mercury is only one-third the size of Earth, its high densityimplies a chemical composition of 60-70% metal and 30-40% silicate by weight. Mercurycontains 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 volumeof Earth’s Moon. Almost nothing can be said about the detailed structure of Mercury’sthin 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 magnetospherethat is about 7.5 times smaller than Earth’s when normalized to planetary radius, whichmakes it a relatively weak magnetic field, but nevertheless right on the prediction of thescaling relation (9.52). During periods when the solar wind is particularly strong, Mer-cury’s field probably gets depressed all the way to the surface. The source of the fieldis not firmly established. If Mercury currently has a molten core, then its magnetic fieldmay be produced by an active dynamo. Another possibility is that Mercury no longer hasa liquid core and an active dynamo, but instead the magnetic field of a past dynamo istrapped in the outer iron-bearing rocks. This second scenario is not as likely, because thecompressional tectonic features seen on Mercury’s surface are insufficient to support theidea 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 presentin the solar nebula at the time of its formation, then it should have experienced enoughradioactive 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 ofgravitational potential energy during the formation of the core probably raised the internaltemperature by 700C and melted the mantle. Given the melting of the mantle, the volumeof the planet would have increased significantly, causing extensional fracturing in the crustand volcanism, of which there is some evidence in the intercrater plains. Subsequentcooling 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 is224.7 days. The equatorial bulge caused by Venus’ slow rotation is only a fraction of ameter, which is much too small to provide any information about the distribution of massin the deep interior. However,the gravity field contains some information about the uppermantle, because it is found to be correlated to surface topography. In contrast, Earth’sgravity field is not well correlated with the surface topography. There is no gravity anomalyat a point over a mountain compared to a point over an area far from the mountain if the

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vertical integral of density is the same at both points. Such an equivalence is also thecondition necessary for a mountain to actually be floating, that is, in isostatic balancewith its surroundings. Thus, the mountains on Earth are in large measure supported bybuoyancy forces, but the mountains on Venus are not. It may be that mountains on Venusare supported by the rigidity of the mantle. Or, it may be that Venus is even more dynamicthan Earth and forms mountains faster than they can relax back into the crust. Vigorousmantle convection and dynamic support of topography has been proposed to explain thelack of isostatic balance on Venus, which alleviates the need to assume that the upperlayers of Venus are significantly more rigid than those of Earth.

Since we have little information about the thermal structure in the interior of Venus, itis difficult to calculate important parameters like the thickness of the lithosphere. The (for-mer) Soviets successfully landed four Venera spacecraft on the surface of Venus. Gammaray analysis of rock samples suggests that the surface rocks on Venus have about the sameabundances of uranium and thorium as on Earth, implying similar internal heat sourcesdue to radioactive decay. If we assume that the temperature gradient in the interior ofVenus is similar to that determined for Earth, the lithosphere on Venus should actuallybe about half the thickness of Earth’s lithosphere. This calculation tends to favor thedynamic support model of the non-isostatic balance seen on Venus, because a thinnerlithosphere implies less mechanical rigidity. However topography and gravity informationfrom 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 explana-tion is that the essentially complete lack of water in Venus’ near-surface results in muchhigher strength of surface rocks. Such behavior has been demonstrated in the laboratoryfor rocks of common crustal and upper mantle composition but at much higher strain ratesthan are rheologically relevant.

The primary way that Earth gets rid of its internal heat is through the mechanismof mantle convection and plate tectonics. It is thought that a lithospheric plate on Earthsubducts because it cools down enough to become denser than the underlying mantle, andsinks. It is possible that the thin lithosphere on Venus can not cool down sufficiently totake advantage of such a gravitational instability, in which case Earth-style plate tectonicswould not be occurring on Venus.

9.4.3 Earth

Earth’s surface is young compared to the other surfaces in the solar system. This islargely due to the fast recycling of crustal material by plate tectonics, and to the rapid rateof erosion caused by the abundance of liquid water. Only 29% of Earth’s surface standsabove liquid water (65% of that land is presently located in the northern hemisphere).Most planetary surfaces are dominated by the circular structures associated with impactcraters and volcanism. In contrast,plate tectonics has caused the largest topographic highsand lows on Earth to be linear. Examples include the linear mountain ranges that formas one plate collides into another, the oceanic trenches that mark the location where platesubduction occurs, and the mid-ocean ridges that delineate the seam where two plates areseparating 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.

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Seismology has allowed us to map the interior of Earth to a level of detail that farexceeds our knowledge of the interior of any other planet. Besides seismic data, we haveother clues including measurements of Earth’s gravity field and moments of inertia, andchemical analyses of inclusions of material of deep origin found in volcanic rocks. Earth’scrust accounts for only about 0.5% of the total mass. The so-called Mohorovicic discon-tinuity, called the Moho for short, defines the base of the crust. Oceanic crust is about5-7 km thick, whereas continental crust is 35-40 km thick. The Moho marks an abruptchange in velocity for both shear (S) and pressure (P) seismic waves, where the aluminumrich rocks of the crust give way to the magnesium and iron rich (ultramafic) rocks thatmakes up the mantle. Typical densities in the upper mantle are 3.2-3.6 g cm−3, and in thelower mantle are 5-6 g cm−3. The mantle accounts for about 67% of the total mass. Themantle’s lower boundary is called the Gutenberg discontinuity, which occurs at a depth ofabout 2900 km. At this interface, P-waves experience a sharp drop in velocity and S-wavesare not transmitted at all, signalling the presence of the liquid outer core. Density rises to9-10.5 g cm−3 in the outer core, and increases to a central density of 12-14 g cm−3. Thecore 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 inthe mantle because they contain high-pressure minerals, like diamonds, and because theycontain certain mineral pairs that can only coexist when formed at high pressure. Theseinclusions allow for relatively accurate estimates of the chemical composition of the uppermantle to a depth of about 200 km. The mineralogical structure of the deep mantle is muchless certain, because the temperatures are poorly constrained, and the preferred phases ofthe 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 isknown to be liquid because it does not transmit shear waves. However, at the so-calledLehmann discontinuity at 5200 km, the core material again transmits shear waves, but atlow velocities, indicating that the inner core is partly molten or near the melting point. Thecore is less dense than pure iron-nickel liquid, most likely because it contains significantamounts (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 methodsof the previous lecture to infer the response coefficient Λ2 ≡ J2/q = 0.43. This value iscloser to the 0.5 value of a uniformly dense sphere than is Earth’s value of 0.31, implyingthat Mars has a dense core, but is somewhat less centrally condensed than Earth. Thereis not much other geophysical data. Mars lacks an appreciable magnetic field. The Vikinglanders did not have sensitive seismometers, and although they did perform useful chemicalanalyses on rock samples, the emphasis of the Viking mission was biological instead ofgeological. Although Viking’s search for the chemical signature of living organisms proved

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negative, the Martian soil was found to contain a surprisingly high abundance of Fe2O3,about 18% by mass (thereby demonstrating that Mars has a rusty color because it iscovered 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 ofthe Martian core. Future seismic observations or pole positions from Doppler or opticaltracking of surface landers will be required to clarify our understanding of the planet’sinternal structure.

Mars has the largest volcanoes in the solar system. The volcanoes are of the shieldvariety, implying that they were formed from low-viscosity (probably basaltic) lava thatcould flow freely for large distances. Calculations based on geological evidence suggestthat the Martian lava is as much as ten times less viscous than the thinnest lava observedon Earth, which is consistent with the properties of exceptionally iron-rich rock. Onereason that the volcanoes on Mars are larger than those on Earth is that on Earth platesmove across relatively stationary mantle upwellings, causing chains of volcanoes like theHawaiian Islands to form, instead of allowing a single volcanoe to grow in place. There isno evidence of plate tectonics on Mars. In addition, the colder Martian interior results ina thicker mechanical lithosphere that can support such large surface loads.

One quarter of the Martian surface is dominated by the Tharsis bulge, a topographichigh that stands about 7 km above the planet’s average radius, and contains most ofthe major shield volcanoes. The Tharsis region is surrounded by radial fractures thatare the signature of tensional (stretching) forces. Both the Tharsis bulge and its largestshield volcanoes can be associated with positive gravity anomalies, which means that thetopography on Mars is only partially supported by isostatic balance, at least at shallowdepths. The complex tectonic signature associated with Tharsis provides a rich data setthat can be combined with gravity and surface topography to study the origin of thisdistinctive feature. Unfortunately at present there are no models that are consistent withall observations. It is almost certain that models for the formation of Tharsis may requiremultiple temporal stages that may include processes such as flexural loading, isostatic upliftand mantle convection. There may also be a requirement for decoupling of the upper crustand upper mantle by a weak crustal layer in the center of the province, which may have aweaker crust. Data from upcoming Mars missions are likely to provide observations thatwill 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 bulkdensity. For Titan, this is the only observation we have that bears on its interior. Titan’sbulk density is 1880 kg m−3, which neatly falls midway between the densities of the Galileansatellites Callisto and Ganymede, and is quite typical of a large, icy satellite. Based onits 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 theirinteriors, whereas terrestrial-planet atmospheres are certainly not. Titan may fall in anintermediate class, because it is likely that its atmosphere and interior are quite similar inchemical composition. The rock component will tend to settle into a dense core, and to

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provide heat through radioactive decay. It is not clear whether there is sufficient heat tocause convection in an icy mantle similar to the solid-state convection in terrestrial-planetmantles.

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 icecan 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 theatmosphere in N2, because N2 would not be incorporated into clathrate until most of theCH4 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 theSun than to the terrestrial planets. Each has a large collection of satellites that resemblesa miniature solar system. Jupiter, Saturn, and Neptune each have their own intrinsicluminosities, and radiate about twice as much energy as they receive from the Sun. Onthe other hand, Uranus radiates ≤ 6% more energy than it receives from the Sun, whichresults in the curiosity that the effective temperatures of Uranus and Neptune are bothabout 59K. Jupiter has more mass than all the other planets put together, 71% of thetotal planetary mass. If Jupiter had been just eighty times more massive, then its centralpressure and temperature would have been high enough to ignite nuclear fusion. Morethan half the local stars are paired together in binary systems, so the fact that Jupiter isalmost a star is not too surprising.

9.5.1 Polytropes

How the internal pressure and density of a planet vary with radial distance, fromthe planet’s center to its surface, is of fundamental importance to our understanding ofthe planet’s internal structure. Consider a nonrotating, spherical body in hydrostaticequilibrium, such that:

dP

dr= −ρ g(r) = −ρ GM(r)

r2, (9.71)

where M(r) is the mass contained within radius r, which can be found by re-arranging(9.71) as

M(r) = − r2

ρG

dP

dr. (9.72)

The mass in a spherical shell dM is related to the density ρ by the product of volume anddensity as:

dM = 4πρ r2 dr . (9.73)

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By combining (9.72) and (9.73) we may write an expression for the change in mass as afunction of radius:

dM

dr=

d

dr

(− r2

ρG

dP

dr

)= 4π ρ r2 ,

⇒ d

dr

(r2

ρ

dP

dr

)= −4πGρ r2 . (9.74)

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 theideal gas law P = ρRT is in the form P = f(ρ, T ). If P and ρ are directly related byan equation of state P = P (ρ) the system is said to be barotropic. For some barotropicproblems, a power-law relationship between P and ρ holds. This called a polytrope andhistorically 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. Equa-tion (9.75) implies that ρ ∝ Pn/(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 whitedwarf star, for which n = 3/2 (P = k ρ5/3), and a relativistic degenerate gas, like in aneutron star, for which n = 3 (P = k ρ4/3). Jupiter is well modeled by the equation ofstate for metallic hydrogen, which follows an n = 1 polytrope:

P = k ρ2 ; k = 2.72× 1012 dyne cm4 g−2 (metallic hydrogen) . (9.76)

Assuming a polytropic equation of state, (9.74) becomes:

k

(1 +

1n

)d

dr

(r2 ρ−1+1/n dρ

dr

)= −4πGρ r2 , (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 massfor a given polytropic index can be found from the dependence of central pressure on Rand M , assuming hydrostatic balance:

Pcen

R∝ M

R3

GM

R2⇒ Pcen ∝

M2

R4,

combined with the polytropic assumption:

Pcen ∝(M

R3

)1+ 1n

,

yields:R ∝M (n−1)/(n−3) . (9.78)

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Notice that for 1 < n < 3, R actually decreases as M increases. The case n = 1, whichhappens to be the one relevant to Jupiter, has the special property that the radius isindependent of the mass. The explicit solution to (9.77) when n = 1 is:

ρ(r) = ρcen

sin(√

2πGk r)

√2πGk r

, (9.79)

where ρcen is the central density. It is a homework problem to verify (9.79), and to find Rfor a metallic hydrogen planet. Jupiter’s radius is within 12% of this radius, which providessupport for the idea that Jupiter is composed primarily of hydrogen. Also, a calculationof the response coefficient Λ2 ≡ J2/q for an n = 1 polytrope yields:

Λ2 =(

5π2− 1

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 atmo-sphere tends to develop a lapse rate that is only marginally larger than the adiabatic lapserate. The vertical temperature profiles of the giant planets are observed to be adiabaticat depth, to within measurement error.

In an adiabatic planet, small temperature fluctuations are efficiently removed by self-regulating convection. This is taken as the explanation for the remarkable lack of temper-ature variation seen on the surfaces of the giant planets. Even though the solar insolationvaries by a sizeable factor from equator to pole, the observed equator-to-pole temperaturedifferences are only a few degrees, in sharp contrast to the many tens of degrees seen onEarth. Apparently, the adiabatic interiors of the giant planets act as a short circuit byarranging for more internal heat to come out at the poles than at the equator, in justthe right proportion to balance the uneven distribution of incoming solar energy. Theimportance of a sizeable heat source located below the atmosphere is one of the primarydistinguishing characteristics of giant planet atmospheric dynamics.

9.5.6 The Sun

Our understanding of the internal structure of the Sun has improved most signifi-cantly in about the last ten years due to very accurate measurements of the frequencies ofits acoustic modes of oscillations (p-modes) determined from helioseismology. In practice,these oscillations are detected by Doppler shifts in images of the photosphere. Approxi-mately five thousand normal modes of oscillations of the Sun have been identified thus farand their frequencies have been measured with an accuracy of about 0.01%. The observedperiods of these modes lie between 3 and 15 minutes, and their lifetimes are typically a fewdays. In addition to the p-modes, there are also standing internal gravity waves knownas g-modes with periods that exceed 40 minutes. The g-modes penetrate more deeply

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than the p-modes, but none have been definitively observed. Coherent p-modes have beenused 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 thehydrogen atom, are uniquely identified by specifying the number of nodes in the radialdirection, the spherical harmonic degree l, and the order m. For a non-rotating sphericallysymmetric 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 result-ing frequency splitting provides information about the rotation rate in the solar interior.Various workers have used this technique to estimate the solar quadrupole moment andto map the rotation rate in the outer 60% of the Sun. The rotation rate of the solar coreremains 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 solarsystem evolution depend on accurate knowlege of the radial density distribution of thesolar interior. For example, it is known from helioseismology that the J2 of the Sun is toosmall, 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 toboth present-day solar dynamics as well as stellar evolution. Another central question insolar dynamics is the how convection interacts with the mean rotation to yield differentialrotation. This question has implications of this interaction for the generation of the solardynamo. With regard to solar evolution, a rapidly rotating deep interior may be a relic ofthe 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 giant-planet, a range that goes from T ≈ 50–150 K and P ≈ 1 bar at the cloud-top levelto T ≈ 10, 000–30, 000 K and P ≈ 50 megabars at the center, we may expect that theprimary constituent, hydrogen, is present in different phases at different levels. There isno firm agreement as to the phase diagram for hydrogen over such a large range, primarilybecause of the difficulty in obtaining high-pressure data in the laboratory. At pressuresabove about 3 megabars, normal molecular hydrogen is suspected to have its electrons andprotons squeezed so close together that it becomes a metal. The temperatures at thesedepths are thought to be sufficient for metallic hydrogen to be in the liquid state. Sincemetallic hydrogen is an alkaki metal, it is reasonable to assume that its electrical propertieswill be similar to other alkali metals like lithium. It is not known whether the transitionfrom molecular to metallic hydrogen occurs gradually or abruptly, and this uncertaintytranslates into an uncertainty for the interior models of Jupiter and Saturn. Recentlymetallic hydrogen was experimentally detected by H.H. Mao and colleagues. Their resultsrepresented an important first order verification of models of the internal structures of thegiant 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 hydrogentransition in Jupiter should occur at about 70–80% of the total radius. This implies thatthe bulk of Jupiter is in the metallic hydrogen phase. Saturn’s metallic hydrogen core is

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calculated to have a radius that is about half of its total radius. Both Uranus and Neptuneare expected to have extensive mantles composed of liquid water, which is a relativelyconductive fluid. Thus, the primary ingredients for a magnetic dynamo exist on all fourgiant 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 differentiationin the planets. An important source of internal energy available to Jupiter and Saturn isthe separation of helium and hydrogen. The formation of a “helium core” can in principledouble the cooling lifetime for these planets, because of the large amount of gravitationalpotential energy stored in an undifferentiated hydrogen and helium mixture.

A simple scenario for the differentiation of helium is as follows. We begin with aplanet composed of hydrogen and helium that is well mixed. The planet cools down untilit reaches the temperature for the liquid phase transition in helium, at which point dropletsof helium condense. Since these droplets are denser than their surroundings, they fall ashelium rain into the deeper layers. The rain continues moving inward until it reaches alevel where the temperature is high enough to cause it to go back into solution with thehydrogen. This phenomenon of rain being reabsorbed is actually observed on Earth overdeserts where rain falls from cool, high altitudes but reevaporates before it can hit the hotground.

In contrast to the formation of an iron core on a terrestrial planet, helium differenti-ation on a giant planet is a stable, self-regulating process. Since differentiation liberatesgravitational potential energy and thereby heats the fluid, it will cause some of the heliumto go back into solution. This negative-feedback mechanism acts as a thermostat andtends to maintain the planet’s temperature at a relatively constant level, until the supplyof upper-level helium is depleted.

On Jupiter, the surface helium abundance is close to the solar composition level, andthe predicted cooling rate for Jupiter does not seem to require an additional source ofheat from the helium rain mechanism. Thus, for Jupiter, helium differentiation appearsto not yet have begun. There is some evidence that helium is depleted in Saturn’s outerlayers relative to Jupiter and the Sun, and the filtering effects of a conductive, differentiallyrotating helium core on Saturn have been invoked to help explain the unusual symmetryof the external magnetic field.

9.5.8 Uranus and Neptune

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Problems

9-1. HelioseismicityThe 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 thestudy 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 observedon the Sun’s surface? What is their typical frequency? Where are the best observationsbeing performed? What are the preliminary results on the internal velocity field and massstructure of the Sun? What are the prospects for future work, and for extending thetechnique to the giant planets?

9-2. Rotational oblatenessa) Compare the relationship between f , q, and J2 in (9.35) to the actual values, J2a,

for Earth, Mars, the Moon, Jupiter and Saturn, by calculating (J2 − J2a)/J2a. Whichplanets are farthest from hydrostatic equilibrium?

b) A planet or a star becomes severely distorted if the rotational parameter q ap-proaches 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 itsdensity ρ.

c) Estimate how short the length of day must be on Earth and Saturn before theseplanets would fly apart.

9-3. Internal density structuresConsider 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 polarmoments of inertia, Ix and Iz, respectively, which put constraints on interior models.Assume ρ is independent of λ, and show that:

J2 =Iz − IxMR2

, (9.81)

where M and R are the planetary mass and equatorial radius, respectively. [Hint: Startwith cartesian-coordinate integrals for Ix and Iz, and compare integrals without solvingthem.]

9-5. Conductive heat flowa) 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 heatgeneration H, and a surface temperature To. The lithosphere extends from Rc < r < R.

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For r < Rc heat loss is by solid state convection which maintains the radial thermalgradient 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 atR = Rc.

9-6. Heat loss in a conducting planetAs 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 theminimum 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 productionH=6.2x10−12 W kg−1 is the long-lived chondritic heat production and k=3.3 W m−1K−1

is the thermal conductivity. Assume a temperature difference (∆T=T − To) of 1100C isrequired for melting to occur.

b) Now suppose that the heat production is four times higher than the long-livedchondritic rate. What is the minimum body size?

c) Discuss the results.

9-7. Convective heat lossDervive 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 temperaturethat increases linarly with depth as γ=dT/dz. [Hint: Calculate how fast the sphere mustrise to remain more buoyant than its surroundings.]

9-8. Jupiter and the n = 1 polytropeThe density of a spherical, hydrostatically balanced planet that follows a polytropic

equation of state:p = k ρ1+1/n , (9.82)

satisfies the Lane-Emden equation:

k

(n+ 1n

)d

dr

(ρ−1+1/n r2 dρ

dr

)= −4π r2Gρ , (9.83)

with boundary conditions ρ(0) = ρcen and ρ(R) = 0.a) Verify that for n = 1, the solution to (9.83) is:

ρ(r) = ρcen

sin(√

2πGk r

)√

2πGk r

. (9.84)

Plot the function sinc(x) ≡ sin(x)/x for the relevant interval x ∈ 0, π.

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b) Apply ρ(R) = 0 to (9.84) to find R in terms of k. The radius does not depend onthe 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 andSaturn.

d) By integrating (9.84) over the spherical volume of the planet, find the total massM , and show that:

ρcen =π2

3ρavg , (9.86)

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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