Douce A.P. Thermodynamics of the Earth and Planets

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108 Energy sources in planetary bodies

The parameter h is one among various non-dimesional parameters that are used to describe

tides, called Love numbers (in honor of an early twentieth-century British physicist and

mathematician who studied tides). The tidal shear strain is then approximately given by:

ε ≈

δr

tidal

. (2.78)

The rotational component of tidal dissipation arises from displacement of the tidal bulge

on the secondary’s surface, but the magnitude of Φ

tidal

remains constant. Substituting

expressions for Φ

tidal

and g

(equations (2.75) and (1.7)) and noticing that we are only

interested in absolute values we get:

ε ≈

, (2.79)

where M

is the mass of the secondary, and all of the other symbols have already been

defined. We now substitute this equation for the shear strain in equation (2.74), and the

latter in (2.72), and get the following equation for the rate of dissipation of tidal energy by

rotation, per unit mass:

tidal,r

≈

tides

µh

2ρQ









, (2.80)

where ω

tides

is the characteristic rate with which the tidal bulge travels over the surface of

the secondary. For a body whose rotation is tidally locked to the primary, for instance the

Moon, ω

tides

=0 and there is no tidal heating from rotation. Otherwise ω

tides

=2π/τ, where

τ is the time elapsed between two consecutive maxima of the tidal bulge on the same point

on the secondary and on the same side relative to the primary (i.e. approximately 24 hours

and 50 minutes for the Earth’s tides).

So far we have considered a situation in which the tidal shear strain remains constant

in magnitude, but the locus of this strain moves in response to the secondary’s rotation. If

the orbit is eccentric we must also consider tidal dissipation as a result of time-dependent

variation in the magnitude of the tidal shear strain ε. This arises because the orbital radius

changes with time, and hence so does the tidal gravitational potential Φ

tidal

. The absolute

value of the change in Φ

tidal

with changes in the distance between secondary and primary

is given by:

dΦ

tidal

≈

3GM

, (2.81)

so that for a small but finite change in the orbital radius, R, we have:

Φ

tidal

≈

3GM

R = 3Φ

tidal

R

≈3EΦ

tidal

, (2.82)

where E ≈ R/R is the eccentricity of the orbit. The magnitude of the time-dependent

component of the tidal strain is then given by:

ε ≈

Φ

tidal

=3E

tidal

=3E

, (2.83)

109 2.7 Tidal dissipation of mechanical energy

which leads to the following rate of dissipation of tidal energy due to eccentricity, per unit

mass (you should work out the algebra to see where this comes from):

tidal,e

≈

9ω

orbital

µh

2ρQ









. (2.84)

The parameter ω

orbital

is the rate of orbital motion = 2π/τ, where τ is in this case the

orbital period of the secondary. This expression of course vanishes for a circular orbit

(E = 0). Substituting (2.80) and (2.84)in(2.73) we find the total rate of supply of internal

energy by tidal dissipation, per unit mass:

≈

µh

2ρQ











tides

+9E

orbital



. (2.85)

In order to obtain numerical estimates for tidal heating we need to substitute appropriate

values in this equation. All of the basic planetary properties (M

, M

, ρ,r) and orbital

parameters (R,E,ω

tides

and ω

orbital

) are well known for the present-day Solar System,

although for the early Solar System only educated guesses are possible for the orbital

parameters. Much of the uncertainty in the physics of tidal heating is thus contained in the

shear modulus, µ, the dissipation factor Q

and the Love number h. We can get estimates

of the possible ranges within which these parameters are likely to vary.

Mean parameter values for the bulk Earth are fairly well known (see Hubbard, 1984;

Kaula, 1964; Stacey, 1992; Turcotte & Schubert, 2002; Meyer & Wisdom, 2008). The bulk

shear modulus for the Earth is ≈ 7 ×10

bar (= 7 ×10

Pa), which can be taken to be

a characteristic value for solid rocky planets. Icy satellites, and rocky planets with greater

melt contents than the Earth (such as Io ?) are less rigid than the Earth and thus have lower

values of µ, of the order of 10

bar (10

Pa). A range in µ of 10

to 10

Pa may cover

most solid bodies in the Solar System (rocky and icy planets, satellites and minor bodies).

A characteristic value of the dissipation factor for rocky planets is 100, whereas for icy

planets the value may be of order 10. The effective Q

for Earth is also 10, but this low

value is due to the fact that much of the Earth’s tidal dissipation takes place by turbulent

motion in shallow oceans and by friction between ocean water and land. The dissipation

factor for the solid Earth is of order 100. We will thus assume that Q

for solid bodies in

the Solar System is in the range 10–100. The dissipation factor for the giant planets, which

we do not consider in this section, is much greater, of the order of 10

–10

, reflecting their

low internal friction.

The Love number h is in fact not independent of a planet’s shear modulus. For Earth,

h ≈0.6. For rigid bodies smaller than Earth, h is given approximately by:

h ≈

10GM

38µr

. (2.86)

Derivation of this formula from first principles is beyond the scope of this book (see for

example Hubbard, 1984, Chapter 4), but we can give a justification of its plausibility a

posteriori. Substituting this expression into equation (2.77) and then into equation (2.78)

we get:

ε ≈

ρΦ

tidal

. (2.87)

110 Energy sources in planetary bodies

The product in the numerator, ρF

tidal

, has dimension of stress (force per unit area) and can

thus be interpreted as the stress arising from the tidal acceleration. This stress varies directly

with the magnitude of the tidal gravitational potential and, because gravity acts on mass,

with the mass contained in unit volume. Equation (2.87) therefore says that shear strain

is directly proportional to the tidal stress and inversely proportional to the planet’s shear

modulus, which agrees with equation (1.33).

Substituting equation (2.86) into (2.85) and simplifying we get:

≈

0.035G

ρM

µQ



tides

+9E

orbital



, (2.88)

in which the only uncertain value is the product µQ

. From our previous discussion of

likely values of µ and Q

we see that this product may be in the range 10

–10

Pa for

most solid bodies in the Solar System. Figure 2.19 shows possible ranges of tidal heat

generation for various satellites of Jupiter and Saturn, calculated with equation (2.88). The

vertical lines, extending over three orders of magnitude, correspond to the three order of

magnitude uncertainty in the product µQ

. Tidal heating values for the Earth and Moon,

in contrast, were calculated with equation (2.85) using the better constrained values of µ,

and h for these two bodies. The horizontal dashed lines show measured heat outputs

per unit mass for Io and Earth.

What can we conclude from this diagram? Focusing first on the Earth–Moon system,

we see that present day tidal heating in these bodies (arising from rotation of the Earth

–15

–14

–13

–12

–11

–10

–9

–8

Tidal heat dissipation per unit mass (WKg

–1

)

Io’s heat output

Terrestrial heat output

Europa

Ganymede

Titan

Enceladus

Mimas

Earth today

Moon today

Hadean

Earth?

Hadean

Moon?

Spinning

Hadean

Moon?

Tidal heating

Fig. 2.19

Possible ranges in tidal heating for various planetary bodies. See text for discussion.

111 2.7 Tidal dissipation of mechanical energy

and eccentricity of the lunar orbit, respectively) is negligible. The same may not have been

true in the early Solar System. The symbols labeled “Hadean Earth” and “Hadean Moon”

represent a hypothetical early stage in which the Moon’s rotation was already tidally locked

to the Earth, but in which its orbit was only 10 times the Earth’s radius (compared to the

present day’s ∼60) and the Earth’s day was only 10 hours long. Tidal dissipation in the

Earth resulted from the planet’s rotation, whereas that in the Moon was a result of orbital

eccentricity, assumed to have had its present day value of 0.055. The diagram suggests

that tidal dissipation may have been an important component of the Hadean internal heat

budgets for the two bodies. These calculations, however, assume present-day shear moduli,

dissipation factors and Love numbers, all of which may have been radically different if,

as seems likely, the Hadean Earth and Moon contained much greater proportions of melt

than today. Tidal heat generation in the Moon during tidal despinning may have been

horrendous, perhaps one order of magnitude greater than in present day Io (Schultz et al.,

1976; Turcotte et al., 1977; Peale & Cassen, 1978; see also Fig. 2.19). The possibility exists

that the Moon, and perhaps all satellites of the major planets (and Mercury?) may have had

extreme heating episodes of very short duration as their fast initial spins were braked by

the tidal pull of their planets (akin to the red-hot glowing brakes in Formula 1 race cars!).

The apparently anomalous lunar heat flow relative to its binding energy (Fig. 2.4) may be

a relic of this event. Direct observational evidence for this extreme heating event could

come if we managed to catch a glimpse of a young extrasolar planetary system at the right

evolutionary stage, which may be difficult, given the very short tidal despinning times,

∼10

–10

years.

The Jovian satellites Io, Europa and Ganymede are locked in a 1:2:4 orbital resonance. Io’s

observed heat output is at the higher end of possible tidal dissipation for the satellite. This

means that Io’s anomalously large heat flow (Fig. 2.4) may be entirely of tidal origin (Peale

et al., 1979; Segatz et al., 1988), but it does not prove that this is the case. Depending on the

still poorly known details of Io’s interior, tidal heating could be two orders of magnitude less

than observed heat output, in which case other explanations would be needed to account for

the missing energy flow. The case for Europa, which has a very young and tectonically active

crust composed of water ice, is similar. Tidal heating of Europa could provide energy at a

rate comparable to the observed terrestrial heat output, which would certainly be sufficient

to fuel Europa’s tectonism and cryovolcanism (Ruiz, 2005). Whether this would also be

possible at the lower end of the calculated range is, however, not clear. Ganymede shows

some recent tectonic activity, but the satellite is clearly nowhere near as active as Europa. Is

the energy supply at the higher end of Ganymede’s range sufficient to explain its relatively

young surface?

The moons of Saturn are a deeper enigma than the Jovian satellites. The hydrocarbon

content of Titan’s atmosphere requires an active supply of methane from the satellite’s

interior, because the fast rate at which hydrocarbons are destroyed by solar ultraviolet

radiation would otherwise render the atmosphere free of hydrocarbons over a time scale of

years. The Cassini–Huyghens mission has shown that cryovolcanoes may account, at

least in part, for the methane supply mechanism and has also revealed that Titan has a young,

tectonically active surface (see Lopes et al., 2010). Under the right circumstances, Titan’s

supply of tidal heat could be comparable to Europa’s and perhaps close to the terrestrial heat

output. As for Io, however, this does not prove that tidal heating is the chief energy source,

given the large uncertainties in the physics of Titan’s interior. A bigger problem arises from

the contrast between the smaller moons, Mimas and Enceladus. The latter is known to have

considerable internal activity, as shown by the presence of active water plumes near the

112 Energy sources in planetary bodies

satellite’s south pole (Hansen et al., 2008). Mimas, in contrast, appears to be geologically

dead. Possible tidal heating of both satellites is comparable, with Mimas actually having a

greater potential for tidal heating than Enceladus, because of its greater orbital eccentricity

and smaller distance to Saturn (Fig. 2.17). Explanations that have been advanced for the

paradox include (see Ross & Schubert, 1988; Nimmo et al., 2007; Meyer & Wisdom, 2007,

2008; Schubert et al., 2007): non-steady state thermal histories (i.e. Enceladus’s present-day

activity may be driven by heat stored during an earlier epoch of greater tidal dissipation

resulting from different orbital parameters), tidal dissipation localized in narrow high-strain

domains (i.e. faults), and sources of heat other than tidal.

We can draw the conclusion that tidal dissipation of mechanical (kinetic) energy can

be an important, perhaps even dominant, source of planetary internal energy. However,

in the absence of accurate knowledge of a body’s physical parameters, in particular its

rheological properties, it is very hard to be certain that tidal heating is responsible for most

of its internal activity. Mercury and Triton could be cases in which more definitive answers

appear possible (see end-of-chapter problems).

2.8 Dissipation of electrical energy

A variable magnetic field induces an electric field, and the latter will drive an electric

current across a conductive material. Dissipation of electric currents called ohmic heating,

see Section 1.8.3) results from friction between moving charges, and takes place in any

material with finite electrical conductivity. Because the Solar System is awash with magnetic

fields, originating both in the Sun and in the various planets with active internal dynamos,

it is in principle possible that ohmic heating may make non-negligible contributions to

planetary internal energies. In particular, the process has been invoked in order to explain

fast melting and differentiation of asteroids in the early Solar System, but there is no

agreement on whether or not ohmic heating was ever a significant planetary heat source.

This is so because of large uncertainties in the values of the parameters that control ohmic

heating.

There are two ways in which electric currents can be generated in a planetary body by

electromagnetic induction. A time-variable magnetic field that permeates a planet gener-

ates electric currents that circulate entirely inside the planet. These are called eddy currents.

Ohmic heating derived from eddy currents is thought to be negligible for all reasonable

ranges of parameters in the Solar System (Colburn, 1980). The other way in which electro-

magnetic induction can occur is more promising and arises from relative motion between

a conductive planet and a steady magnetic field. The currents induced in this case are not

confined to the planet, and circulate through the interplanetary medium in order for the

electric circuit to close (Fig. 2.20).

The chief contributor to interplanetary magnetic fields is the solar wind, which is a

stream of charged particles (protons and electrons, with some heavier ions as well) that

move radially away from the Sun at hypersonic speeds (10

–10

km s

−1

). The origin of the

solar wind is expansion of the solar corona. The solar wind is a low-density plasma which

is able to conduct currents induced in planetary bodies (Fig. 2.20). An important property

of the solar wind for our purposes is that it carries with it the interplanetary magnetic field,

which is also rooted in the Sun. As the solar wind streams past a planet the relative velocity

between planet and magnetic field is essentially equal to the velocity of the solar wind,

113 2.8 Dissipation of electrical energy

solar wind

Fig. 2.20

Electromagnetic induction in a planetary body. The interplanetary magnetic ﬁeld, with intensity B, is directed towards

the reader, perpendicular to the page. The solar wind carries the interplanetary magnetic ﬁeld and moves from left to

right, so that the electrical conductor (planet) moves towards the left of the page with velocity v relative to the

magnetic ﬁeld. The induced current J moves upwards across the planet, and the circuit closes via the interplanetary

plasma.

given that orbital velocities are typically one order of magnitude smaller. Electromagnetic

induction arises because of this relative velocity, but only if the interplanetary magnetic

field is able to penetrate the planetary body. This imposes an important limitation on the

type of body that can be a candidate for ohmic heating. It must not have an intrinsic magnetic

field, for if it does then the solar wind and the interplanetary magnetic field are deflected

around the planet. If the planetary body has an atmosphere then there is an additional

limitation on ohmic heating, which is that ionization of the outermost atmospheric layers

by solar ultraviolet radiation generates an electrically conductive ionosphere. Ionospheric

conductivity is typically much greater than that of solid planetary materials, so that most of

the induced current is in this case carried by the ionosphere and ohmic heating is shorted

out. Given these two limitations – no intrinsic magnetic field and no atmosphere – we

can expect that only small airless planetary bodies, such as asteroids, are candidates for

electromagnetic induction heating.

An exact solution of the electromagnetic induction equations and the resulting ohmic

heating is mathematically quite complex and beyond the scope of this book. A simpli-

fied treatment, however, allows us to identify which are the crucial controlling parameters

and to analyze the possible magnitude of this heat source. The intensity of the electric

field, /, induced by motion with velocity v relative to a magnetic field of intensity B is

given by:

/ = vB sin α, (2.89)

where α is the angle between the velocity vector and the magnetic field vector (this is a

slightly disguised version of the Lorentz force equation, see Section 1.8.3). The induced

114 Energy sources in planetary bodies

electric field is maximum if motion is perpendicular to the magnetic field, and van-

ishes if motion is parallel to the magnetic field – we will assume for simplicity that the

former is always the case (α = π/2, see Fig. 2.20). An electric field acting across a mate-

rial with conductivity σ causes an electric current to flow, with the current density, J ,

given by:

J = σ/ = σ vB sin α. (2.90)

The current density equals the intensity of the electric current (I , measured in Amperes

in the SI system) per unit of cross-sectional area perpendicular to the current flow. We

recall from introductory physics that the power dissipated by ohmic heating, W , is given

by I

R, where R is the total resistance of the material through which the current flows.

The resistance is in turn given by R = z/aσ, where a is the cross-sectional area of the

current-carrying conductor (so that J = I /a) and z is the length of conductor over which

the current flows. If I is the total current that flows through the planet then ohmic dissipa-

tion of this current must equal the rate of increase of the planet’s internal energy, dE/dt,

so that:

R = a

aσ

. (2.91)

The product az is the volume of material heated by ohmic dissipation, therefore the rate of

increase of internal energy per unit mass, de/dt, is given by:

ρσ

σ v

sin

, (2.92)

where ρ is, as usual, the density of the material. The SI unit for the intensity of the mag-

netic field is called the Tesla (the intensity of the Earth’s magnetic field at the planet’s

equator is approximately 3 × 10

−5

Tesla), and conductivity in SI units is measured in

mho m

−1

. Using these units and the standard kg–m–s for density and velocity we obtain

de/dt inJs

−1

The magnitudes of v , the velocity of the solar wind, and B, the strength of the inter-

planetary magnetic field, are well known in the present-day solar system. Typical values

at the radius of the Earth’s orbit are v = 500 km s

−1

and B = 5 ×10

−9

Tesla. Electrical

conductivity of rocks is widely variable and is an exponential function of temperature.

For example, the conductivity of carbonaceous chondrites may be given approximately by

σ = 10

−9

exp(0.014T)– see Herbert and Sonett (1979). From 100 K to 1000 K, σ for car-

bonaceous chondrites varies by almost 6 orders of magnitude, from ∼4×10

−9

mho m

−1

∼10

−3

mho m

−1

. Assuming a density of 3000 kg m

−3

, ohmic heating of a cold chondritic

asteroid at the Earth’s orbit could account for ∼10

−17

−1

, which is 6 orders of mag-

nitude less than the present day terrestrial heat output, and 2–3 orders of magnitude less

than present day tidal heating in the Earth–Moon system (Fig. 2.19). Heating rate would

increase to ∼10

−12

−1

if the temperature of the asteroid were raised to 1000 K.

The conclusion appears to be that, starting with cold material, ohmic dissipation by itself

is unlikely to cause a temperature increase of that magnitude.

115 2.8 Dissipation of electrical energy

Worked Example 2.5

It was noticed by Gradie and Tedesco (1982) that the distribution of asteroid types suggests

that there was a gradient in maximum temperatures across the asteroid belt. Asteroids that

appear to be composed of strongly metamorphosed or melted materials are concentrated in

the inner part of the belt (at heliocentric distances smaller than ∼2.7 A.U.), whereas those

that appear to be composed of undifferentiated early Solar System material, including a

significant proportion of ices, dominate the outer part of the belt (distances greater than

∼3.4 A.U.). The concentration of asteroids that have undergone relatively low-temperature

hydrothermal alteration peaks in between these two distances. Meteorite ages show that

thermal processing of asteroidal bodies was a very early event in the history of the Solar

System, and was probably completed within the first 10

years. Electromagnetic induction

heating is in principle consistent with these observations because (a) the interplanetary mag-

netic field, and hence ohmic heating, varies inversely with heliocentric distance and (b) it

is known that, during the transition from gravitational collapse to thermonuclear burning,

Sun-like stars go through a brief stage of increased solar wind activity known as the T-

Tauri phase. Can early thermal processing of asteroids be a consequence of electromagnetic

induction heating during the T-Tauri phase?

Accretionary heating of small planetary bodies is negligible (Section 2.4), so we will

assume that the initial temperature of theasteroids was 100 K. We first calculate the mean rate

of supply of heat necessary to melt an asteroid, assuming a uniform melting temperature of

1500 K and values for the specific heat capacities and enthalpy of fusion of c

=10

Jkg

−1

and h

=5 ×10

Jkg

−1

, respectively. Melting the asteroid in 10 million years then

requires a mean heat supply rate of ∼6 ×10

−9

−1

. This is somewhat higher than

Io’s present day heat output (Fig. 2.19). It is also a minimum estimate because we have

ignored heat loss to space, but on the other hand initial temperature in the solar nebula may

have been higher than 100 K, so that the two sources of uncertainty may partly cancel each

other out.

Let us assume a density of 3000 kg m

−3

and a conductivity of 10

−8

mho m

−1

, consistent

with a low initial temperature. If ohmic dissipation was the only source of heat, then it

requires that the product vB be ∼40 Tesla m s

−1

(equation (2.92)). Observation of present-

day T-Tauri stage stars, and models, suggest that the velocity of the T-Tauri wind was not

significantly different from present-day solar wind velocity, 10

–10

km s

−1

(although the

plasma density was of course much higher). The required magnetic field was then of the

order of 10

−4

Tesla at the orbital radius of the asteroid belt. This value is tantalizingly similar

to what little is known about the possible intensity of the interplanetary magnetic field in

the early Solar System. Magnetic paleointensities measured in a wide range of meteorites

are consistent with inducing fields of the order of 10

−4

–10

−6

Tesla (Stacey, 1976; Acton

et al., 2007). Direct observations suggest magnetic fields of ∼0.1 Tesla in the cores of

protostellar disks, and comparable values are obtained from proto-star modeling (Donati

et al., 2005; Machida et al., 2007). Because the intensity of the magnetic field falls with the

square of the distance, the ratio between field intensity at the asteroid belt (∼3 A.U.) and at

the proto-solar surface (∼0.01 A.U.) would be ∼10

−5

, which is generally consistent with

paleointensities measured in meteorites (see also Herbert, 1989). Electromagnetic induction

could have contributed to early heating of asteroids, and might even have been capable of

causing large-scale melting, if magnetic field intensities in the early Solar System were

at the upper end of the likely ranges and/or if planetesimal conductivities were greater

116 Energy sources in planetary bodies

than the 10

−8

mho m

−1

assumed in this calculation. This highlights the problem with the

hypothesis of planetesimal melting by ohmic heating: it is in principle possible, but the

controlling parameters can vary over such wide ranges that it is very difficult to test (see

Grimm & McSween, 1989). And there is another nagging problem, which is that the ratio

between field intensity at 3.4 A.U. and 2.7 A.U. is ∼0.63, i.e. not too different from 1.

Explaining the large difference in the degree of thermal processing of asteroids across the

asteroid belt would require an uncomfortably fine parameter tuning.

2.9 Radioactive heating

Decay of radioactive isotopes converts nuclear binding energy to kinetic energy, chiefly

of subatomic particles, that is dissipated as the particles collide with atoms, slow down

and eventually come to a stop. The energy released by decay of one nucleus of radioactive

parent, called the decay energy, 

, is given by the difference between the total rest mass

of the decay products (daughter nucleus plus subatomic particles) and the rest mass of the

parent nucleus (this difference is also called the Q value of the decay). This is the mass

equivalent of the amount of nuclear binding energy liberated by the decay. The decay rate

is given by:

=−λN , (2.93)

where N is the number of atoms of parent isotope and λ is the decay constant, which is equal

to ln 2 divided by the half life of the isotope. The rate at which decay energy is liberated is

then given by:



=−λN 

, (2.94)

so that the rate of heat production per kg of isotope, η

, is:

=6.02 ×10



, (2.95)

where w

is the proper mass of the nucleus of decay constant λ

, and I have dropped

the negative sign because this is heat absorbed by the system (equation (1.55)). In many

cases decay from the parent nucleus to the stable daughter nucleus occurs via a number of

intermediate steps, called a radioactive decay series. The decay series consists of isotopes

with half lives much shorter than that of the initial isotope, so that their concentrations at

equilibrium are negligible compared to the concentration of the initial parent isotope. The

decay energies of the intermediate decay products, however, are not negligible. The energies

liberated by all of the members of the decay series add up so that the general expression for

the total rate of radioactive heat production per kg of the initial parent isotope is:

=6.02 ×10





d,k

, (2.96)

117 2.9 Radioactive heating

Table 2.2 Important heat-generating radioisotopes

Half life Isotopic abundance Isotopic abundance

Isotope (J s

−1

)(s

−1

) (yrs) (present) (4.56 Ga)

238

U 9.46 ×10

−5

4.19 ×10

−18

4.47 ×10

0.992 75 —

235

U 5.69 ×10

−4

3.12 ×10

−17

7.04 ×10

0.007 20 —

232

Th 2.64 ×10

−5

1.56 ×10

−18

1.41 ×10

1—

K 2.92 ×10

−5

1.72 ×10

−17

1.28 ×10

1.17 ×10

−4

—

Al 4.55 ×10

−1

3.06 ×10

−14

7.17 ×10

0 5.8 ×10

−5

Fe 7.19 ×10

−2

1.46 ×10

−14

1.50 ×10

07×10

−7

Mn 6.38 ×10

−3

5.87 ×10

−15

3.74 ×10

09×10

−6

where λ

and w

are the decay constant and proper mass of the initial (longer-lived) parent

isotope in the series, 

d,k

is the decay energy of the kth isotope in the decay series, and

we assume that the decay series is in secular equilibrium (Chapter 12). Table 2.2 lists the

values of η

and λ

(as well as the corresponding half lives) for a number of isotopes that are

important planetary heat sources. The isotopes

235

238

232

Th and

K, with half lives

of the order of 10

–10

years, have been important heat producers throughout the entire

history of the Solar System, and are still important today. In contrast,

Al,

Fe and

Mn,

with half lives of approximately 0.72, 1.5 and 3.75 million years, respectively, have been

extinct for most of the age of the Solar System but could have had dramatic thermal effects

during the first 5 million years or so. There are two reasons for this. First, owing in part

to their short half lives, the rates of heat production per unit mass of

Al,

Fe and

are several orders of magnitude greater than those of the long-lived isotopes (Table 2.2).

Second, all three are isotopes of major elements.

If the concentration of radioisotope i in a rock is C

, then the heat generated by this

isotope per kg of rock is C

. The concentration of a radioactive isotope in a closed system

decays with time according to equation (2.93), and so does the rate of heat production. Let

the concentration of isotope i in a rock at the time of formation of the Solar System be C

Assuming that the rock remains closed to chemical exchange with its environment, the rate

of radioactive heat production by isotope i per unit mass of rock at time t after the formation

of the Solar System is given by (Chapter 12):

−λ

. (2.97)

The initial concentrations of

235

238

232

Th and

K can be determined from their

present-day concentrations, C

, as follows (Chapter 12):

(2.98)

where t

is the age of the Solar System. The present day concentration of active isotopes,

in turn, is the product of the corresponding present day isotopic abundance (also given in

Table 2.2) and the elemental concentration in the rock. The initial concentrations of the

extinct isotopes can obviously not be determined from equation (2.98). The initial isotopic

abundances of

Al,

Fe and

Mn at the time of formation of the Solar System (shown in

Table 2.2) have been estimated on the basis of minute anomalies in the isotopic abundances

of their stable daughter isotopes (

Mg,

Ni and

Cr, respectively) measured in meteorites