Douce A.P. Thermodynamics of the Earth and Planets
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78 Energy sources in planetary bodies
we see that, with one notable exception, planetary binding energies are one to two orders
of magnitude greater than the observed energy outputs or, in other words, observed heat
fluxes are significantly less than those that could be expected from steady release of all of
the gravitational binding energy over the age of the Solar System. This discrepancy can be
the result of a combination of at least three factors: (i) early in their histories the planets
cooled at rates that were much faster than today’s, (ii) heat transfer inside the planets is
very inefficient, so that a large fraction of the dissipated binding energy is still stored inside
the planets, and (iii) only a fraction of their gravitational binding energies were retained as
internal energy, with the balance being radiated to space during planet formation.
The one notable exception to these conclusions is Io, for which measured heat flow is
two orders of magnitude higher than what can be accounted for from gravitational binding
energy. The conclusion in this case is that Io, which is the most volcanically active planetary
body in the Solar System (see, for example, Davies, 2001) must have a major heat source
that is not active to any significant extent in any of the other bodies included in Fig. 2.4.
Focusing now on the planets that appear to show an excess of gravitational binding
energy, we see that there are some subtle differences. The ratio of binding energy to heat
flow for Earth, Jupiter and Neptune is almost exactly the same (q
B
/q
M
≈ 40). Compared
to these three planets, Saturn puts out twice as much heat relative to its binding energy
(q
B
/q
M
≈18), whereas Uranus radiates one sixth less heat (q
B
/q
M
≈240). Saturn’s large
energy output is thought to reflect a process that is not significant at present in any other
planet (see Stevenson & Salpeter, 1977; Hubbard, 1980; see also Section 2.6).
Martian heat flow has not been measured yet. The value quoted in Table 2.1 and plotted in
Fig. 2.4 is an estimate based on the bulk chemical composition of Mars, which appears to be
enriched in K relative to the Earth. Lunar heat flow was measured in two of the Apollo land-
ing sites. If these values are representative of the entire Moon then, by comparison with other
terrestrial planets the Moon’s internal heat flux is greater than expected. This is rather sur-
prising, and may be a relic of a short-lived extreme heating event in the Moon’s early history.
2.4 Accretion
It is virtually certain that the chief process in the assemblage of the terrestrial planets was
accretion of smaller bodies (called planetesimals), which in turn were the products of accre-
tion of yet smaller bodies, and so on down to the smallest particles of dust and ice which
formed by condensation of gaseous elements in the solar nebula (see Weidenschilling, 1974,
1976, 1980, 2000; Wetherill, 1990, 1994).Although the details of the accretionary processes
that led from microscopic dust particles to Earth-size planets may not be fully understood
it is possible to place some constraints on its energetic aspects. The gravitational binding
energy of a planet is an upper bound to the amount of internal energy that can be derived from
dissipation of gravitational potential energy. The results of Worked Example 2.1 suggest,
however, that only a fraction of a planet’s gravitational binding energy may be converted to
internal energy. Some of the gravitational potential energy dissipated during accretion must
be radiated to space, however, and the radiated energy flux, given by the Stefan–Boltzmann
law (equation 2.1) is a very strong function of temperature, as it scales with the fourth-power
of T .
We want to find out what fraction of a planet’s gravitational binding energy is stored as
internal energy during accretion. Let us assume that a planetary body grows by accretion at
79 2.4 Accretion
a rate that is fast enough that no significant heat transfer occurs from the planetary interior to
its surface, so that heat is lost only by radiation from the material that makes up its surface at
any given instant. We will revisit this assumption in Chapter 3 and convince ourselves that it
is indeed a very good model for planetary accretion. We consider an infinitesimal increment
in the planet’s mass, dm, taking place in an infinitesimal time dt. The gravitational binding
energy dissipated by accretion of this mass raises its temperature from the initial temperature
that it had in the solar nebula, T
0
, to the instantaneous temperature at the current planetary
surface, T . Our assumption about negligible heat transfer rate from the planet’s interior
means that once this surface layer is buried by continued accretion its temperature remains
constant. We describe energy balance during accretion with the general formulation of the
First Law of Thermodynamics:
dE =dQ −dW . (2.11)
The change in internal energy, dE, corresponds to the difference between the internal energy
of the added mass at the instantaneous temperature of the planet’s surface and its internal
energy at the background temperature of the solar nebula. We consider accretion to be a
constant pressure process for which P = 0, as it occurs at the surface of a planet with no
atmosphere. We need an expression for (∂E/∂T )
P
. The method to obtain this expression
is explained in Appendix 2, where we show that:
∂E
∂T
P
=C
P
−PαV =C
P
, (2.12)
and the second identity results from setting P = 0. In terms of specific heat capacity
(equation (1.54)) and assuming that c
P
is constant between T and T
0
we rewrite (2.12) as:
dE =c
P
(
T −T
0
)
dm. (2.13)
The change in gravitational binding energy, dU
B
, is balanced by mechanical work performed
on the planet, −dW , i.e:
dU
B
+
(
−dW
)
=0 (2.14)
so that the work performed by the system (i.e. the planet) is, from equation (2.3):
dW =dU
B
=−
Gmdm
r
=−
4
3
πGr
2
ρdm, (2.15)
where r is the instantaneous radius of the growing planet at the time when its mass is m and its
surface temperature is T . A way of seeing why the signs must balance as in equation (2.14)
is to realize that the system always performs work against gravity. The work performed
by the system, dW , must then equal the change in gravitational potential energy. In order
for the gravitational potential energy to increase the system must perform (positive) work,
whereas negative work by the system corresponds to a decrease in gravitational potential
energy.
The planet loses thermal energy by radiation. The infalling mass would also radiate at
its nebular temperature T
0
, so that the heat loss that corresponds to dissipated gravitational
binding energy is the difference between the energy radiated at T and that radiated at
T
0
,F −F
0
(equation (2.1)). The net thermal energy lost over the entire surface area of the
planet and over the time increment dt is thus given by:
dQ =−4πr
2
(
F −F
0
)
dt =−4πr
2
/σ
T
4
−T
0
4
dt. (2.16)
80 Energy sources in planetary bodies
Substituting equations (2.13), (2.15) and (2.16)in(2.11) we obtain:
c
P
(
T −T
0
)
dm =−4πr
2
/σ
T
4
−T
0
4
dt +
4
3
πGr
2
ρdm. (2.17)
A slight rearrangement of this equation highlights its meaning: that the rate of dissipation
of gravitational binding energy equals the rate of increase in internal energy plus the rate
at which thermal energy is radiated from the planet’s surface:
4
3
πGr
2
ρ
dm
dt
=c
P
(
T −T
0
)
dm
dt
+4πr
2
σ
T
4
−T
0
4
. (2.18)
This is a fourth-degree equation in T , the temperature at the surface of a body when its
radius is r and the mass accretion rate is dm/dt. We can use equation (2.18) to construct
the temperature profile of a planet resulting from accretion, T = T(r), by specifying an
accretionary growth model that gives dm/dt as a function of r, or, equivalently, of accreted
mass, m. We do this in the following Worked Example.
Worked Example 2.2 Energy dissipation, radiation and storage during planetary accretion
As an example of the application of equation (2.18), we will use it to calculate possible initial
temperature profiles for the Earth, and to estimate what fraction of the Earth’s gravitational
binding energy might have been stored as internal energy during accretion. The problem
is solved numerically, by considering short time increments, calculating the mass, radius
and accretion rate at the end of each time increment, using those values to solve equation
(2.18), and iterating until the total mass of the planet has been accreted. A Maple procedure
that accomplishes this is described in Software Box 2.1.
Software Box 2.1 Calculation of temperature profile and internal energy storage during
planetary accretion
The calculations discussed in Worked Example 2.2 are implemented in the Maple work-
sheet accretion.mw, which contains several procedures. The first procedure sets the
values of physical constants and model parameters, except for the accretion time. Plan-
etary mass and density, nebular temperature (T
0
), specific heat and emissivity must be
changed in this procedure, if desired. Procedure solveT uses Maple’s numerical solver
to find a positive real value of T from equation (2.18). It requires the values of r and
dm/dt . These are calculated by one of the following procedures, linear_growth,
exponential_growth or sine_growth, that implement equations (2.20) through
(2.22) and pass the values of m and dm/dt to solveT via the utility procedure
calc_out. The three procedures iterate to the specified accretion time using a time
step that can be adjusted by modifying the total number of steps desired. Output is
sent to a text file which can be read and manipulated by plotting or spreadsheet pro-
grams. The name and format of the output file can be modified inside each of the
procedures that describe the growth models. When execution of a model is finished
Maple returns total gravitational binding energy, energy stored as internal energy, and
the ratio between these two quantities – this information is not saved in the output files.
81 2.4 Accretion
The first task is to specify an accretionary growth model so as to obtain numerical
values for the mass accretion rate, dm/dt as a function of radius, r. We will consider three
distinct models: (i) a linear model in which accretion rate is constant, (ii) a model in which
accretion rate decreases exponentially with time (see Wetherill, 1990), and (iii) a model
in which accretion rate increases rapidly, reaches a maximum and then decreases rapidly
to zero (Hanks & Anderson, 1969). The accretion models are best described in terms of
accretion rates normalized to the total mass of the assembled planet, M. The three models
are described by the following equations, where µ = m/M:
dµ
dt
(i)
=
C
M
dµ
dt
(ii)
=
C
M
e
−kt
dµ
dt
(iii)
=
C
M
t
2
sin kt,
(2.19)
where the dimension of dµ/dt is [T ]
−1
, and the constants C and k are functions of the
accretion time t
a
. In every case we set µ = 0att = 0 and µ = m at t = t
a
. For the linear
model we easily find:
dµ
dt
(i)
=
1
t
a
(
µ
)
(i)
=
t
t
a
.
(2.20)
For the exponential model we set k = 1/t
a
, meaning that the accretion rate becomes
exponentially slow as the accretion time is reached. Integrating between 0 and t
a
we find:
dµ
dt
(ii)
=
1
t
a
e
−t/t
a
1 −e
−1
(
µ
)
(ii)
=
1 −e
−t/t
a
1 −e
−1
.
(2.21)
For the sinusoidal model we make dm/dt =0att =t
a
(see also Exercise 2.1) and we get:
dµ
dt
(iii)
=
1
t
a
π
3
π
2
−4
t
t
a
2
sin
π
t
t
a
(
µ
)
(iii)
=
1
π
2
−4
−π
2
t
t
a
2
cos
π
t
t
a
+2cos
π
t
t
a
+2π
t
t
a
sin
π
t
t
a
−2
.
(2.22)
Figure 2.5 shows dµ/dt and m/M as a function of time for the three accretion models,
for an accretion time t
a
= 10 million years. The function for planetary radius, r = r(t)
is obtained by substituting m = m(t) in equation (2.5), with the simplifying assumption
that the density of the planet remains constant during accretion – we will tackle density
inhomogeneities caused by contraction and differentiation in later sections.
82 Energy sources in planetary bodies
0
2*10
6
4*10
6
6*10
6
8*10
6
10
7
0
5*10
–8
10
–7
1.5*10
–7
2*10
–7
Time (yr)
Accretion rate per unit mass (yr
–1
)
C
Ce
– kt
Ct
2
sin (kt)
0
2*10
6
4*10
6
6*10
6
8*10
6
10
7
0
0.2
0.4
0.6
0.8
1
Time (yr)
Fraction of planetary mass accreted
C
Ce
– kt
Ct
2
sin (kt)
Fig. 2.5 Specific accretion rates (per unit mass) and accretionary histories for linear, exponential and sinusoidal accretion
models. Accretion time is 10 million years.
Figure 2.6 shows accretion temperature profiles for the Earth for an accretion time of
10 million years and nebular temperature T
0
= 100 K, and assuming that the emissivity
of the accreting planet was 1. This means that the figure shows minimum tempera-
tures, as any value of ε<1 will cause a larger fraction of the binding energy to be
stored as internal energy (equation (2.18)). Accretion models predict that the center of
the planet was initially much colder than its outer layers. This is to be expected from
the fact that gravitational binding energy varies as m
2
/r or, equivalently, as the fifth
power of the radius (equation (2.8)), so that little heating takes place during the initial
stages of growth. Thermal convection would not take place with these temperature profiles
(Chapter 3), even if the planet were fully molten (which at these temperatures an iron–
silicate planet would not). Linear and exponentially decaying growth predict a very steep
inverted temperature gradient out to a radius of ∼1000 km, and relatively constant tempera-
tures from that depth to the planet’s surface. The differences between both models are minor.
83 2.4 Accretion
0 1000 2000 3000 4000 5000 6000
100
200
300
400
500
600
Radius (km)
Temperature ( K)
Ce
–kt
C
Ct
2
sin (kt)
Thermal profiles for quasi-static accretion of the Earth,
t
a
=10
7
yr
Fig. 2.6 Initial temperature profiles for quasi-static accretion of the Earth, assuming an accretion time of 10 million years and
nebular temperature =100 K. Other parameter values are: ρ = 5500 kg m
−3
, c
P
=900 J kg
−1
K
−1
and / = 1.
0
1000 2000 3000 4000 5000 6000
10
–3
10
–2
10
–1
1
Radius (km)
Fraction of binding energy stored as internal energy
Ct
2
sin (kt)
Ce
– k t
C
Quasi-static accretion of the Earth, t
a
=10
7
yr
Fig. 2.7 Fraction of gravitational binding energy that is converted to internal energy during accretion of the Earth. Model
parameters as in Figure 2.6.
The sinusoidal growth model results in a more gradual temperature increase outwards from
the planet’s center, and a marked temperature peak near the planet’s surface, which reflects
the late peak in mass accretion rate (compare Fig. 2.5, see also Hanks & Anderson, 1969).
If release of gravitational binding energy increases strongly with radius (equation
(2.8)), then why do accretionary temperatures flatten out and eventually decrease close
to the planet’s surface? The answer lies in the fact that thermal radiation varies as the
fourth power of temperature, so that as temperature increases a rapidly growing fraction
of the gravitational energy that is dissipated is radiated rather than converted to internal
energy. This can be seen in Fig. 2.7, which shows the fraction of binding energy
84 Energy sources in planetary bodies
that is converted to internal energy as a function of planetary radius. The balance is
radiated.
Integrating equation (2.13) along the accretionary history we obtain the total amount of
internal energy stored during accretion, and the ratio of this quantity to the total gravitational
binding energy (Software Box 2.1). This ratio is shown in Fig. 2.8 as a function of accretion
time, which for terrestrial planets is thought to be in the range 10
5
–10
7
years (seeWetherill &
Inaba, 2000). For accretion times in this range only a very small fraction of the binding
energy, of the order of 10
−2
, is stored as internal energy. The rest is radiated to space. It is
thus not necessary to postulate exorbitant planetary heat flows in the early solar system to
explain the discrepancies noted in Fig. 2.4.
Internal energy varies by a factor of ∼3 over the two orders of magnitude range in
accretion times (Fig. 2.8). As the total planetary mass is the same in all cases, this must
0.1 0.2 1 2 1
0
0.01
0.02
0.03
Total accretion time ( 10
6
yr)
Fraction of total binding energy stored as internal energy
Ce
– kt
C
Ct
2
sin (kt)
Accretion of the Earth
Fig. 2.8 Total fraction of binding energy converted to internal energy for the three Earth accretion models in Figure 2.7, but for
variable accretion times, 10
5
–10
7
years.
85 2.4 Accretion
0 1000 2000 3000 4000 5000 6000
0
500
1000
1500
Radius (km)
Temperature ( K)
Ce
– kt
0.1 m.yr.
Ce
– kt
1 m.yr.
Ce
– kt
10 m.yr.
Ct
2
sin (kt)
0.1 m.yr.
Ct
2
sin (kt)
1 m.yr.
Ct
2
sin (kt)
10 m.yr.
Quasi-static accretion of the Earth
Fig. 2.9 Effect of accretion time (10
5
–10
7
years) on initial temperature profile of the Earth, for the exponential and sinusoidal
models in Figure 2.6.
reflect a difference of comparable magnitude in planetary temperatures (equation (2.13)).
Figure 2.9 shows that even for very fast accretion (10
5
years), however, maximum temper-
atures attained in the absence of other energy sources (∼1500 K) may not be sufficient to
melt an undifferentiated metal–silicate planet. Recall that these calculations assume ε =1.
You can explore other parameter combinations in the end-of-chapter exercises.
A conclusion that we can draw from this example is that accretion may not in general be
an efficient means of converting gravitational binding energy to planetary internal energy,
unless proto-planetary material has an emissivity significantly lower than 1, but there is an
important exception to this statement, which we discuss in the following section.
2.4.1 The effect of large impacts
The picture of accretion as a quasi-static process, in which mass is added to the growing
planet in infinitesimal increments, ignores the possibility that catastrophic planet-sized
impacts may take place late in the assembly of planetary bodies (see Wetherill, 1985).
Such impacts can dissipate enormous amounts of kinetic energy virtually instantaneously.
Consider two celestial bodies that collide inelastically. Kinetic energy is dissipated and
becomes thermal energy, which is stored as a combination of sensible heat (increase in
temperature) and latent heat (melting). Assume that two bodies of masses m and M, and
moving with relative speed v, collide head on. In this one-dimensional geometry v is simply
the magnitude of the velocity vector
v
a
(Section 1.3.2). The kinetic energy dissipated is
given by equation (1.15):
U
k
=−
1
2
mv
2
M
M +m
. (2.23)
86 Energy sources in planetary bodies
Because this process transforms kinetic energy to internal energy, we must add a kinetic
energy term to the First Law of Thermodynamics (equation (1.56)):
dE =dQ −PdV −dU
k
, (2.24)
where dU
k
is the change in the system’s kinetic energy. The impact is an adiabatic process,
so dQ = 0. If we also assume that the energetic consequences of volume changes are
negligible compared to the energy dissipated during a collision, we see that the change in
internal energy equals the kinetic energy that is dissipated:
dE =−dU
k
. (2.25)
The increase in internal energy per unit total mass of the system, E
m
, is given by:
E
m
=
1
2
mv
2
M
M +m
1
M +m
=
v
2
2
Mm
(
M +m
)
2
. (2.26)
Let the ratio between M and m be k, such that m =kM. Equation (2.26) can then be written
as follows:
E
m
=
v
2
2
k
(
k +1
)
2
. (2.27)
This equation shows that, for a given collision speed, the increase in internal energy per
unit mass, and therefore the average temperature increase, depends only on the ratio of
the masses of the colliding bodies, and not on the absolute values of M and m. E
m
vanishes as k →0, so that the average temperature increase is negligible when one of the
colliding bodies is much smaller than the other one. Of course, this is a global result that
ignores the local details: the K–T impact may have caused a negligible increase in the total
internal energy content of the Earth, but the local increase in temperature was certainly not
negligible. But this is also the point: accretion of a small body onto a much larger one does
not lead to a significant increase in the latter’s thermal budget. We then ask whether there
is a value of k for which E
m
(and T ) is maximum. On symmetry grounds, we can argue
that this must happen when the two bodies have the same mass, i.e. for k = 1. In order to
check this we look for the extrema of the function:
y =
k
(
k +1
)
2
. (2.28)
The first derivative of this function vanishes at k =1, and the second derivative is negative
at k =1, showing that this extremum is a maximum. The increase in internal energy per unit
mass is indeed maximum when the masses of the colliding bodies are equal. The maximum
value for a given collision speed is obtained by setting k = 1 in equation (2.27):
E
m,max
=
v
2
8
. (2.29)
We must specify the relative speed of the collision. In general, the maximum speed for
an inelastic collision is of the order of the escape velocity at the surface of a body with
a mass equal to the combined masses of the impactors, M +m. The escape velocity of a
planet is the minimum velocity that a free-falling object must have in order to be able to
move away from the planet indefinitely. By “free falling” we mean that no other forces
87 2.4 Accretion
besides the gravitational attraction of the attracting planet act on the moving body. The
escape velocity is therefore the speed at which the kinetic energy of the body equals its
gravitational potential energy (Exercise 2.4). If two bodies collide with a relative speed that
is significantly greater than the escape velocity of their combined masses then gravity is
not capable of holding the mass together, and the impactors will tend to shatter rather than
merge. I will give a more quantitative demonstration of this in a moment. First, note that
the escape velocity at the surface of a body of mass M and radius r is given (Exercise 2.4)
by V
e
=
√
(2GM)/r, which we can write in terms of planetary radius and density as:
V
e
=
8
3
πGρr
2
1/2
. (2.30)
Substituting in (2.29) we get the following expression for the increase in internal energy per
unit mass, for an impact at the escape velocity of the combined masses of the impactors:
E
m,V
e
=
1
3
πGρr
2
. (2.31)
From equation (2.10) we write the binding energy per unit mass in terms of density and
radius:
u
B
=−
4
5
πGρr
2
. (2.32)
We see that (2.31) and (2.32) are of the same order of magnitude. If the energy dissipated
were significantly greater than the binding energy then there would not be enough gravi-
tational attraction to hold the mass together during the collision. But (2.31) is the energy
dissipated by an impact at the escape velocity of the combined masses, showing that the
escape velocity is the approximate limiting velocity for an inelastic collision.
Because E
m,Ve
corresponds to a collision between two bodies of equal mass (and,
we will assume, equal density) we can write (2.31) in terms of the radius of each of the
impactors, r
i
, as follows:
E
m,V
e
=
2
2/3
3
πGρr
2
i
. (2.33)
This function is plotted in Fig. 2.10 for bodies of two compositions: silicate–metal mixtures
with ρ =4500 kg m
−3
, corresponding to chondritic impactors (undifferentiated early Solar
System material), and ice bodies with ρ = 1000 kg m
−3
. We can compare these energy
values to the specific heat capacities and enthalpies of melting of planetary materials. The
composition of chondritic meteorites is roughly one third metal and two thirds silicate.
Specific heat capacities per unit mass of silicate minerals are of the order of 10
3
JK
−1
kg
−1
and of iron about 5 ×10
2
JK
−1
kg
−1
, so that a typical value for chondritic material may be
0.85 kJ K
−1
kg
−1
. The value for H
2
O at typical nebular temperatures of 200 K is ∼1.5 kJ
K
−1
kg
−1
. The vertical scales on the left of the graph show characteristic temperature
increases for both types of materials, based on these heat capacities and the energy dissi-
pation per unit mass given in the vertical axis. We see that for bodies up to 100–200 km
in radius the kinetic energy dissipated by collisions is insignificant on a global scale, even
though local effects can be devastating. Aglobal temperature increase of the order of 100 K,
which may be important in ice-rich bodies at nebular temperatures as it may heat them to
their melting point, requires a collision between 1000-km sized bodies.