Journal of Earth Science, Vol. 26, No. 1, p. 124–133, February 2015
Printed in China
DOI: 10.1007/s12583-015-0509-z
ISSN 1674-487X
Evaluation of the Heat, Entropy, and Rotational Changes
Produced by Gravitational Segregation during
Core Formation
Anne M. Hofmeister*, Robert E. Criss
Department of Earth and Planetary Sciences, Washington University, St. Louis MO 63130, USA
ABSTRACT: Core formation by gravitational segregation allegedly released sufficient interior heat
to melt the Earth. Analysis of the energetics, which compare gravitational potential energy (Ug) of a
fictitious, homogeneous reference state to Earth’s current layered configuration, needs updating to
correct errors and omissions, and to accommodate recent findings: (1) An erroneous positive sign
was used for Ug while maintaining the reference value of 0 at infinity, which results in an incorrect
sign for ΔUg, which is crucial in determining whether a process is endothermic or exothermic. (2)
The value of Ug for Earth’s initial state is uncertain. (3) Recent meteorite evidence indicates that
core formation began before the Earth was full-sized, which severely limits Ug. (4) Inhomogeneous
accretion additionally reduced Ug. (5) The potentially large effect of differential rotation between
the core and the mantle was not accounted for. (6) Entropy changes associated with creating order
were neglected. Accordingly, we revise values of Ug, evaluate uncertainties, and show that Ug was
converted substantially to configurational energy (TS). These considerations limit the large sources
of primordial heat to impacts and radioactivity. Although these processes may play a role in core
formation, their energies are independent of gravitational segregation, which produces order and
rotational energy, not internal heat. Instead, gravitational segregation promotes planetary cooling
mainly because it segregates lithophilic radioactive elements upward, increasing surface heat flux
while shortening the distance over which radiogenic heat diffuses outwards.
KEY WORDS: core formation, primordial heat, configurational energy, thermodynamics, rotational energy.
0 INTRODUCTION
The importance of ancient heat reservoirs to Earth’s thermal history depends on their magnitudes and the rates of heat
generation and loss to space. Core formation has been viewed
as augmenting accretionary (impact) heating and internal radioactive heat sources by release of gravitational potential
energy (Ug) during segregation (Anderson, 2007; Rubie et al.,
2007; Schubert et al., 2001). Previous calculations assumed
that Ug was positive (by changing the sign in the definition, see
below), leading to Ug being a huge, positive quantity, 2 333
to 2 780 J·g-1 (e.g., Stacey and Stacey, 1998; Flasar and Birch,
1973; Birch, 1965). The positive sign for Ug, if correct, indicates that core formation provided an independent, rapidly
released source of energy as heat. This amount of heat is
roughly double all radioactive heat generated over Earth’s
history. Such an energy burst would have influenced Earth’s
thermal history, with remnants possibly contributing to the
current heat budget (e.g., Galimov, 2005). However, Ug is
*Corresponding author: [email protected]
© China University of Geosciences and Springer-Verlag Berlin
Heidelberg 2015
Manuscript received February 26, 2014.
Manuscript accepted July 4, 2014.
negative, as discussed below, which makes its conversion to
heat questionable.
Subsequent meteoritic evidence also suggests that the
static analysis needs revisiting. For example, isotopic (Hf-W)
evidence indicates that Earth’s core formed early and rapidly,
within the first ~30 to ~125 Ma of its 4.5 Ga history and before
accretion was complete (reviewed by Kleine et al., 2009). The
static analyses of Ug predate this discovery, and so were
based on comparing Ug of the present Earth to that of a fictitious, full-sized homogeneous globe. Clearly, this latter imaginary configuration was never available to be converted to heat.
Additional restrictions on the magnitude of Ug would exist if
accretion were heterogeneous, with early, preferential deposition of iron in the central zone.
In addition, the existing static analyses (e.g., Flasar and
Birch, 1973) involve substantive errors that also need addressing.
(1) Positive values for Ug (e.g., Birch, 1965) were obtained by changing the sign in the defining formula from negative, as required by Newton’s Law of gravitation, to positive,
while keeping the reference value at infinity as zero. This sign
change leads to positive Ug. The correct way to obtain positive values of Ug is to add a constant reference value to Ug for
both the final and initial states. Had this approach been used,
the difference Ug would have been negative.
Hofmeister, A. M., Criss, R. E., 2015. Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation. Journal of Earth Science, 26(1): 124–133. doi:10.1007/s12583-015-0509-z
Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation
The correct, negative, sign of the energy difference is important in thermodynamic analysis, dictating that heat is lost to
the surroundings, per the definition of an “exothermic” transition. Use of a positive sign traces to early analyses of star formation from nebulae (e.g., Chandrasekhar, 1939; Eddington,
1916; Emben, 1907), wherein the positive quantity, “gravitational energy” was invoked to produce internal heat and explain starlight. However, stellar contraction provides far too
little energy to explain stellar emissions and it is now known
that nuclear fusion provides starlight. Consequently, more
recent analyses use the correct, negative sign for Ug to describe
nebula contraction (e.g., Lynden-Bell and Lynden-Bell, 1977).
For further discussion, see Hofmeister and Criss (2012a). Because the proper, negative, sign is now used in astronomy, the
analogous concepts in geophysics need to be reconsidered.
(2) Contraction produces heat in certain situations (e.g.,
compression of gas in a piston-cylinder wherein pressure-volume
(P-V) work is externally applied). However, available calculations of core formation show that Earth’s contraction is negligibly small (e.g., Stacey and Stacey, 1998). (3) Computations of
rotational energy (Birch, 1965) neglected the attenuation of
Earth’s spin and did not consider that the core might spin independently and faster than the mantle, providing one possible
means to balance the change in Ug. Today, differential rotation is
small and its value is debated (see Dehant et al., 2003). Estimates
of the core’s super-rotation range from negligible (0.1 o/yr) to
small (1.1 o/yr). Finite values are compatible with the discrepancy between Earth’s spin axis and its magnetic axis. Due to
attenuation (e.g., viscous dissipation), a much larger difference
could have existed when the core first formed.
(4) Gravitational problems require conservation of Helmholtz free energy (F=E–TS, where E=internal energy, T=
temperature, and S=entropy) (Müller, 2003). But, the “uncompensated heat” (TS) of Clausius was not considered in core
formation. This omission is perplexing because analysis of
contraction of nebulae to form stars, upon which analyses of
core formation are based, considered changes in entropy (Emden, 1907).
Because the gravitational energy change depends only on
the initial and final density distributions, how the core forms is
of secondary importance (Rubie et al., 2007), and the possible
heating from gravitational segregation can be evaluated using
thermodynamics and classical mechanics. The purpose of the
present paper is to evaluate primordial heating associated with
configurational changes alone. We do not evaluate additional
and independent primordial heat sources, such as impacts.
The present paper begins by re-evaluating the energy difference between hypothetical initial states and the modern
layered Earth. We next explore production, retention, and dissipation of heat during core formation in the static picture by
applying thermodynamic principles. Part of our analysis is
qualitative because the initial state is fictitious. We show that if
the core spins up relative to the mantle, significant rotational
energy can be produced via conversion of Ug, and also that
entropy changes are large. Gravitational entropy is a confusing
and controversial topic in astrophysics (Wallace, 2010), and
thus we provide a detailed discussion of entropy changes due
to the gravitational process of core production. We estimate
125
heat produced by frictional work performed during boundary
layer settling and discuss formation of differential rotation via
gravitational settling. We finally show that redistribution of
material during core formation promoted cooling of the Earth.
In short, we will show that core formation was a spontaneous
process that could not have greatly heated the Earth, but instead facilitated the loss of heat to space. Such a transition is
consistent with thermodynamic principles, not at odds with
them.
1 POTENTIAL ENERGY CHANGES FROM A STATIC
VIEW OF CORE FORMATION
1.1 Previous Work
Calculation of Ug requires analysis of Earth’s layered
configuration and of its presumed initial state, constrained by
the mass, compressibility, and moment of inertia. Compared to
Earth’s present average radius of 6 370 km, Flasar and Birch
(1973) calculated a slightly smaller radius for its initial reference state (6 355 km). Stacey and Stacey (1998) found less
difference in their homogenization (6 364.62 km). In both papers, the small, estimated change in radius (<1%), provides
PV=+0.01 J·g-1 that is negligible compared to Ug.
Table 1 assigns the correct negative sign required by
Newton’s Law of gravitation, to previous numerical values.
Flasar and Birch (1973) overestimated |Ug| as 2 780 J·g-1.
Reduction of |Ug| to 2 333 J·g-1 by Stacey and Stacey (1998)
mostly reflects improved knowledge of Earth’s current state.
Use of equations of state to compute Ug is uncertain because
phase changes exist beyond those assumed (e.g., postperovskite: Murakami et al., 2004), and T of the fictitious homogenous Earth is unknown. Below we make various estimates to gauge uncertainties.
1.2 Change in Gravitational Potential Associated with a
Full-Size Proto-Earth
Because the volume change is negligible in the equation
of state calculations, we constrain Ug and estimate uncertainties by comparing Ug of the present state, represented by
PREM (Preliminary Earth Reference Model: Dziewonski and
Anderson, 1981), to several hypothetical density distributions
(Fig. 1; Table 1), while conserving mass and maintaining a
constant radius. Although not all of these distributions are realistic, the range of results permits evaluation of uncertainties.
Numerical results are cross-checked against two analytical
solutions. First, for a sphere of constant density (), the moment of inertia (I) and self-potential energy are
I0 
2
3 GM 2
Mr 2 ; U 0  
5
5 r
(1)
where M is mass, r is radius, G is the gravitational constant,
and the subscript 0 denotes the homogeneous state (Eddington,
1916). Alternatively, if density varies linearly from a at the
surface to c at the center, the average density is ave=
(c+3a)/4, and we find
I  I0
 c  5 a
13 c 2  65 c  a  90 a 2
; U g  U0
(2)
6  ave
168 ave 2
Anne M. Hofmeister and Robert E. Criss
126
Table 1 Gravitational potential results using a correct, negative sign
Model
Density
(103 kg·m-3)
Moment of inertia
(1037 kg·m2)
Gravitational potential
(1032 J)
Compositionally homogeneous
Constant density
Square root
5.51
9.70
-2.24
4.16–7.12†
9.25
-2.29
Birch (compressed)
4.09–7.12
-
-2.324#
Stacey & Stacey
3.98–7.17
9.02
-2.327#
Linear
4.16–9.57‡
8.91
-2.34
Mixture
2.74–8.67*
8.70
-2.37
Layered
Linearized PREM
2.65–14.10‡
8.02
-2.47
PREM
1.02–13.08§
8.01
-2.48
-
-
-2.490#
4.15, 12.48
8.02
-2.49
Flasar & Birch (layered)
Two-layer
Total Earth mass=5.974×1024 kg; average radius=6 370 km. *. Weighted average of silicate- and Fe-fractions,
where the density of the silicates with depth was obtained from PREM by extrapolating the trend in lower
mantle density with depth to the core, and where the Fe density was obtained extrapolating the trend in density with depth of the inner core to the surface. The low value at the surface results from the extrapolation. †.
Density values are fixed by surface density and Earth’s total mass using=a+b(6 370–r)1/2. ‡. Density values
are fixed by surface density and Earth’s total mass using=c+d(6 370–r). §. Dziewonski and Anderson (1981).
The initial value accounts for a very thin ocean. Initial density of the rock layers is 2.60 kg·m-3. #. We have
reversed the reported sign of the gravitational potential.
geneous Earth, relative to PREM gives Ug= -1 840±200 J·g-1.
Our result for |Ug| is 20%–33% smaller than previous estimates.
14
PREM
12
Two layers
Density (10 3 kg·m -3)
For the layered Earth, Ug from PREM is slightly larger
(more positive) than Ug obtained from a “two-layer” model
which assumes constant, average density for each of the core
and mantle, and is slightly smaller (more negative) than a
model which assumes that density increases linearly with depth
(Fig. 1; Table 1). These final Ug values are all similar, despite
the different assumed profiles (Fig. 1), suggesting that the homogeneous Earth can likewise be constrained from hypothetical density constructs (Table 1). Specifically, for the compositionally homogeneous Earth, a linear density increase from the
weighted average of silicate and metal should set a lower limit
as -2.34×1032 J. The model with constant density throughout
overestimates the upper limit because it grossly misrepresents
the central region. We model compression of a homogeneous
interior by extrapolating PREM values for the core outwards
and mantle inwards (Fig. 1). This mixture model, which gives
Ugi= -2.37±0.03×1032 J (Table 1), is our best representation of
the initial value for Ug.
Previous studies of Flasar and Birch (1973) and Stacey
and Stacey (1998), reported values for initial Ug that are larger
(more positive) than our estimates (Table 1). This difference
arises because densities calculated near the center of ~7.1×103
kg·m-3 in the previous models are smaller than in our constructs (cf., Eq. 2). Post-perovskite was not considered, which
would produce a higher central density. Temperatures are not
known, which also affect such equation of state estimates. Due
to the uncertainties associated with representing a fictitious
initial state by equation of state calculations, we estimated Ug
(above) by comparing density profiles.
Core formation from a fictitious, compositionally homo-
10
Linear
Mixture
Linearized
PREM
8
Birch
Square root
6
4
2
0
1 000
2 000
3 000 4 000
Radius (km)
5 000
6 000
Figure 1. Density profiles used to compute gravitational
potential energy in Table 1. Models for the layered Earth
are: grey solid curve. PREM; long dashes. linear model
similar to PREM; fine dots. two layered model; models for
the homogeneous Earth are: squares and fine line. density
listed by Birch (1965); solid line. square root model; short
dashes. linear model similar to Birch’s surface density;
dotted curve. mixture of silicates and Fe metal, see text.
Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation
1.3 Effect of Earth’s Size during Core Formation on
Gravitational Potential
Early core formation (e.g., Kleine et al., 2009) severely
limits the amount of accumulated gravitational potential, as is
obvious from recasting Eq. 1 in terms of the radius and density
of the homogenous proto-Earth (H)
U 0,early (r )  
16 2
2
π G H r 5
15
(3)
We estimate Ug,early(r) numerically by subtracting
Uo,early(r) with a linearized density profile from an Earth having
the same radius but with two homogeneous layers. Changes in
Ug,early are small until a proto-Earth radius exceeds ~3 000 km
(Fig. 2a), due to -Ug/M going approximately as r2.
1.4 Effect of Heterogeneity during Accretion on Gravitational Potential
Heterogeneous accretion is possible in view of meteorite
mineralogy. Proportions of silicate to metallic grains in enstatite chondrites differ substantially among specimens and contrast greatly with the proportions in C1 chondrites (e.g.,
Brearly and Jones, 1998), all of which are included in compositional models of the Earth (e.g., Lodders, 2000). Heterogeneous accretion could have greatly reduced the energy change
associated with core formation, if Fe were preferentially deposited early on. As an extreme case, if iron were the only material
deposited initially, forming a full-size core, Ug would be zero.
Figure 2b schematically illustrates the effect of iron being
preferentially deposited in the core region.
2 000
Late core
formation
- U g (J·g -1)
1 500
1 000
500
Early core
formation
(a)
0
0
1 000
2 000
3 000 4 000
Radius (km)
5 000
6 000
2 000
Homogeneous proto-Earth
(b)
- U g (J·g -1)
1 500
1 000
500
0
12.5
Heterogeneous
accretion
Completely
layered
proto-Earth
100
% Fe deposited in core region during accretion
Figure 2. Trends in the negative change in gravitational potential. (a) Dependence of -ΔUg on the radius of the homogeneous Earth when core formation began; (b) schematic of the
dependence of -ΔUg on the amount of Fe preferentially deposited in the region of the present-day core.
127
2 THERMODYNAMICS OF CORE FORMATION
A homogenous globe is a locally stable configuration.
Had this idealized configuration ever existed, externally derived energy would have been required to convert this metastable reference state to the gravitationally favored state of core
plus mantle. The final state has lower energy and lower entropy, so the difference in internal energy is negative and the
transition is exothermic (Nordstrom and Munoz, 1986). The
need for impetus has been recognized, as core formation was
suggested to occur subsequent to large-scale melting (e.g.,
Stevenson, 1990). However, the static model of Section 1 accounts only for the gravitational energy difference between the
current state and the presumed initial states, addressing neither
the mechanism of ordering (Rubie et al., 2007) nor the additional, externally derived energy needed to initiate the change.
This section amends the energy balance associated with the
previous static model to adhere to thermodynamic law.
2.1 Energy Conservation and Temperature Changes in the
Static Picture
In gravitational problems, conservation of Helmholtz free
energy
F=E–TS=Ug+R.E.–TS
(4)
fulfills the 1st law (Müller, 2003; also see Hofmeister and
Criss, 2012a), where R.E. is rotational energy=½I2 and  is
angular velocity. Helmholtz’s, not Gibb’s free energy (G), nor
internal energy (E) is used because: (1) F represents the
maximum amount of work that can be preformed, hence the
name “free energy.” In contrast, G excludes pressure-volume
(PV) work (Nordstrom and Munoz, 1986). (2) The strong dependence of Ug on radius and thus V is what drives core formation. Thus, V is one of the two independent variables. (3) The
remaining independent variable in core formation is T. Temperature is independent of changes in V upon exactly differentiating the Helmholtz free energy, whereas both P and S do
depend on V. The two dependent variables, P and S, are affected by T as well. (4) Because F governs situations where V
and T are the characteristic, independent variables (Pippard,
1974; Reif, 1965), it pertains to gravitational problems, and is
known as “the mechanical state function” (Anderson, 2007).
We emphasize that none of the four thermodynamic variables (V, P, T, S) are a priori constant; moreover, Maxwell’s
relations, and knowing F(V,T), provide all other thermodynamic quantities (Pippard, 1974; Reif, 1965). Although the
state functions (E, F, G, and enthalpy) can be parameterized in
other ways, this does not provide sufficient information to
solve the problem (e.g., Pippard, 1974). Problems associated
with previous use of internal energy, rather than F, are discussed further below.
Neglecting PV terms is theoretically justified: Pressure is
force per area, so PV terms go as G(M–m)m/3a~GMm/3a
where m is the mass of a very thin, outermost layer and a is the
surface radius of the planet. Because m is small compared to M,
PV is negligible compared to Ug~GM2/2a for the initial state
and also for the final state and thus (PV) << Ug.
We begin by comparing initial and final states in the reversible approximation. From energy conservation and Eq. 4,
Anne M. Hofmeister and Robert E. Criss
128
the transition is described as
UgEf+R.E.f –TEfSEf=UgEi+R.E.i–TEiSEi
(5)
where the subscript E indicates Earth, f denotes final and i
denotes initial states. Rearranging terms, adding the quantity
0=TEiSEf–TEiSEf, and regrouping terms gives
UgE= -R.E.+SEfTE+TEiSE.
(6)
Independent of all else, signs of the terms in Eq. 6 show
that TE rises insignificantly, as follows.
(1) Because UgE is negative and entropy is always positive, if no other changes occur, TE must be negative. Positive
TE is only possible if the other terms on the right side of Eq. 6
are negative and their magnitude is larger than |UgE|. Thus,
any negative change in gravitational potential must be balanced by other negative terms, either -R.E. (which is negative
given differences in I in Table 1) and/or TEiSE (which is negative because layering creates order, estimated below). Thus,
thermodynamic analysis does not require a temperature increase. Instead, Eq. 6 shows that core production foremost
involves conversion of potential energy to kinetic energy and
to production of order.
(2) If indeed SEfTE is positive and finite (as is possible in
the irreversible case), then TEiSE must be even more negative
to offset this term, because -R.E. is fixed by classical mechanics (Section 3). Therefore, neglecting SEfTE provides the
minimum change in entropy that permits the core to form (the
reversible approximation). A temperature increase of Earth’s
interior (SEfTE) is greatly restricted by the size of the other
terms in Eq. 6.
(3) The change in internal energy being negative means
that the conversion is exothermic. In the instantaneous (static)
picture, heat is released to space. The time-dependent (dynamic)
process of core formation which involves both heat production
and shedding is not addressed by our static model, wherein the
transition process is unspecified (cf., Rubie et al., 2007). Below, we briefly discuss the dynamic process of frictional heating during boundary layer settling.
2.2
Constraints Imposed by Entropy Changes and
Radiative Transfer to Space
Transition of Earth from a disordered (homogeneous)
state to an ordered (layered) state provides a negative sign for
SE. The Earth is not isolated but is tied to the surroundings by
radiative transfer. Specifically, Earth’s surface temperature
represents the energy balance of radiation received from the
Sun and re-radiated to space, which maintains a steady state.
Because received Solar irradiance is ~6 000 times Earth’s heat
flux from the interior, Earth’s surface temperature is essentially
constant and is independent of heat flux from the interior that
is lost to space. Thus, the 2nd law is fulfilled by Earth shedding heat to space
SO -SE
(7)
where subscript O indicates the universe other than Earth. Because a positive sign is required for SO, SE must be negative
for the reversible case (the equality). For the irreversible case,
the sign of SE remains negative for energy balance, but SO
has larger magnitude (see below).
Radiative transfer is the means to adhere to the 2nd Law.
Heat emitted from the Earth infinitesimally raises the temperature of the Universe, and thus SO, from the definition of heat
capacity. Release of heat would cool the Earth. We view such a
heat release as instantaneous in the static model, wherein the
transition process is unspecified (Rubie et al., 2007). In the
static model, the amount of heat released is -TSE, which
equals CPT and requires a temperature decrease. The size of
the temperature drop depends on the changes in entropy and in
rotational energy (Eq. 6), which are estimated below. In the
dynamic picture, this release of heat is gradual, but such a
process must exist to fulfill the 2nd Law.
2.3 Relationships of Internal Energy to Heat and Work
Previous analyses allegedly conserved internal energy, essentially by equating the erroneous positive change in Ug to positive heat (dQ) received by the Earth, then invoking an internal
temperature increase by integrating heat capacity. From elementary thermodynamics (Reif, 1965), dE=TdS–PdV=dQ+dW. Expansion was calculated to provide negligible energy, +0.01 J·g-1,
so negligible work was preformed, and dE=TdS=dQ=dU, which
is negative. From the approach of conserving internal energy
only, and using Newton’s Law, requires a decrease in internal
energy, which is as mandated for any spontaneous process
(Nordstom and Munoz, 1986). In marked contrast to previous
analysis of this process, core formation must either cool or order
the planet. Involvement of ordering is obvious in the layering,
was included in analysis of gravitational contraction of nebulae
(Emden, 1907), and is estimated in Section 3.3. Cooling accompanies ordering, to meet the 2nd Law.
3 ESTIMATION OF ROTATIONAL ENERGY, ENTROPY, AND PRE-EXISTING TEMPERATURES
We now evaluate the two negative terms on the right side
of Eq. 6 (-R.E.+TEiSE) which are capable of offsetting the
negative change in Ug during gravitational segregation.
The difficulty of calculating S for macroscopic bodies is a
fundamental problem in modern thermodynamics (Lieb and
Yngvason, 2003), as exemplified by heated debates regarding
black holes (Wald, 2001). Estimates of S between the homogeneous and layered states of the Earth are therefore very
rough. We use two approaches detailed below, after discussing
spin, which previously was underestimated due to attenuation
being neglected. More importantly, large changes in R.E. are
possible if the early core rotated faster than the mantle.
3.1 Rotational Energy Associated with Synchronous Rotation of the Core and Mantle
Angular momentum (L) was conserved (Iii = Iff) during
the transition from the homogeneous to layered states. For
synchronous rotation of the core and mantle, we derive
ΔR.E. 
 I 
 I 
1
I f f2 1  f   R.E.f 1  f 
2
 Ii 
 Ii 
(8)
Because Earth’s polar axis is tilted to the special axis of the
rotating Solar System, spin components exist in all three direc-
Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation
R.E.z(t) =1/2 If (2/p)2= R.E.z0 e-t/
(9)
where p is period, R.E.0 is spin energy at formation and  is a
decay constant, gives 1/=4.6×10-10 yr-1. At the time of core
formation, the “day” was 8.0±0.5 hours, and R.E. was ~330
J·g-1. Our estimate lies near the median of literature values
which range from 2.5 to 16.8 h (Lathe, 2006).
From Table 1, the factor in brackets in Eq. 8 is 0.08 to
0.11, which makes R.E. small. An upper limit is provided by
multiplying Eq. 8 by 3 to account for the three axes existing,
which provides 80 J·g-1, which is 10 times previous estimates,
but still only 4% of |Ug|. Thus, for synchronous rotation, configurational energy is the key term in core formation (Eq. 6).
The size of the proto-Earth does not alter this conclusion because R.E. and Ug both go as M5/3.
3.2 Effect of Differential Rotation of the Core and Mantle
on Rotational Energy
To roughly gauge the effect of the core spinning up independently of the mantle, we again conserve angular momentum
during core formation
L = Li = Icorecore + Imantlemantle
(10)
A possible mechanism is discussed in Section 4.3. A mantle of constant density and mass Mm has a moment of inertia of
I mantle 
2
a 5  b5
Mm 3 3
5
a b
(11)
where a is the surface radius and b is the radius of the coremantle boundary. For a core of constant density and mass Mc
we use I=2Mcb2/5. The change in rotation energy is then
R.E.=½Icorecore2+½Imantlemantle2–½Iii2
(12)
For the fictitious, initially homogeneous Earth, we use an
initial period of 8 hours and I of about 9×1037 kg·m2 (Table 1).
After core formation, I became 8×1037 kg·m2, and if the core
and mantle were rotating synchronously, conservation of angular momentum (I i=I f) gives the synchronous value upon
core formation as 6.9 h, which is within the range of estimates
(e.g., Lathe, 2006). Figure 3 shows that the mantle spinning
slightly more slowly is connected with the core spinning at a
substantially more rapid rate. Figure 3 (heavy arrow) also
shows that for R.E. to match Ug would require a ~24 h spin
period for the mantle upon core formation. This degree of differential spin is not possible because the spin rate provided for
the mantle is the same as is today’s spin, and thus does not
provide for subsequent attenuation. Even for this excessive
differential rotation, the period for the core remains within an
16
1.09
14
1.25
Core
12
1.45
10
1.75
8
2.18
6
2.91
T (h)
×10 -4 (rad/sec)
tions, although the polar component with its high spin rate dominates. Although it is difficult to evaluate spin without a fixed
frame of reference, the moment of inertia does not change much
(Table 1), which severely limits R.E., as follows.
The spin rate of interest is not today’s rate of spin, but
rather the spin at the time of core formation. Tides dissipate
energy, and have slowed Earth’s rotation down. The current
attenuation rate is 0.002 s per century. Using this rate, the current spin, and assuming exponential decay
129
Synchronous
4
4.37
6.9
8.7
Mantle
2
24
0
0
500
1 000
R.E. (J·g -1)
1 500
Figure 3. Relationship of the change in rotational energy
during core formation with the spin of the decoupled core
and mantle for the full-sized Earth. Grey curve. angular
velocity of the core; black curve, angular velocity of the
mantle; right y-axis provides the corresponding periods;
dotted arrow points to synchronous conditions; heavy arrow points to the energy at which ΔR.E. equals -ΔUg.
order of magnitude of the initial spin rate. From Fig. 3, we
conclude that differential rotation, if this was large in the early
Earth, could have offset a substantial amount of the energy
released by core formation, but not the majority.
3.3 Estimates for the Entropy of Core Segregation
Calculating S for macroscopic bodies is difficult and a fundamental problem in modern thermodynamics (Lieb and Yngvason, 2003). As a first order approximation, we evaluate entropy by approximating Earth as two families of non-interacting
particles (e.g., ideal gases), conceptually representing silicate and
metal fractions. The entropy change is given by
S=-nR(x1lnx1+x2lnx2)
(13)
where n is the number of moles, R=8.31 Jmol-1K-1 is the gas
constant, and xi is the mole fraction of each of the components.
Note that the term “entropy of mixing” is a misnomer because
the entropy increase is not due to “mixing” per se but to the
greater volume available to each component (Keesing, 1986).
For this reason, size of particles is irrelevant. Hence, although
derivations for Eq. 13 generally consider ideal gases, this result
holds for mixing of polymers (Flory, 1941; Huggins, 1941), as
well as to homogenizing fat globules of milk (Swendson,
2008). Hence, Eq. 13 should pertain to core formation, regardless of particle size. Equation 13, however, must be considered
a rough estimate of entropy.
To allow for the effect of pressure, we consider an alternative formulation, also valid for macroscopic particles. In
Joule’s experiment, free expansion of an ideal gas is isothermal
(e.g., Pippard, 1974). Instead, entropy increases during this
process, in accord with
S=nRln(Vf/Vi)
(14)
This is ideal mixing, but to first order, Eq. 14 also de-
Anne M. Hofmeister and Robert E. Criss
130
scribes non-ideal mixing processes, because the largest contribution to S is configurational. The volume in Eq. 14 refers to
the volume available to the molecules, rather than the sum of
the volumes occupied by each molecule.
The core’s contribution is then
Score=(McoreR/ core)ln(Vcore/VEarth)
(15)
where  is the average molecular weight and M is the mass.
The result is Score= -0.276 Jg-1K-1, and analogously for the
mantle, Smantle= -0.045 Jg-1K-1. The change in entropy accompanying core formation mostly arises from the restricted volume experienced for iron in the core.
Due to the logarithmic nature of Eqs. 13–15, configurational energy changes are relatively small unless T is high (Fig.
4). How much mixing energy (TS) is associated with core
formation depends on pre-existing interior temperature (Fig. 4).
The present Earth has a mantle temperature near 2 000 K and T
of iron melting at high pressures indicates core temperature
exceeds 3 000 K (e.g., Boehler, 2001). To match Ug (Eq. 6,
with R.E. and heating being negligible) requires T~5 700 K,
which is reasonable since this corresponds to temperatures
considered for the present inner core (Boehler, 2001) and
higher temperatures for the proto-Earth are possible due to ~5fold larger heat emissions from K, U and Th 4.5×109 years ago
(e.g., Hofmeister and Criss, 2013; Van Schmus, 1985).
ω core- ω mantle (deg/day)
7 000 6 000 5 000 4 000 3 000 2 000 1 000
1 840
Energy (J·g -1)
1 600
0
ΔR . E .
if differential
(upper axis)
T of present - day core
400
-TΔS
volume estimate
(lower axis)
0
0
1 000
4 IMPLICATIONS OF MECHANISMS OF SETTLING
ON PRODUCTION OF HEAT AND DIFFERENTIAL
ROTATION
This section considers two possible mechanisms for gravitational settling. First, we set limits on heat evolved by considering actual work that could be preformed during boundary
layer settling. Next, we discuss inherent limitations of friction
producing heat. Finally, we discuss how gravitational settling
in the upwelling and downwelling currents in convection cells
provide a means of transferring angular momentum from the
mantle to the core, thereby creating differential rotation.
4.1 Possible Work Performed in Boundary Layer Settling
The early Earth with abundant radioactive isotopes should
have been hot and convecting. Gravitational sorting could occur within the bottom boundary layer of the convecting protomantle or magma ocean (e.g., Li and Agee, 1996). Percolation
deep in the Earth at high pressures has also been inferred (Shi
et al., 2013). Sorting in the upper boundary layer near the surface is ineffective because sinking metal particles would be
caught up in flow inside the convection cells and concurrent
accretion would disrupt the sorting process near the surface.
For each particle, the work performed is
W=mgBLh
1 200
800
TS is required for energy balance. Given the approximations
used and the rough nature of the calculations, TS is larger
than R.E., but these terms have the same order of magnitude,
and both offset Ug. In addition, Eq. 7 indicates that the temperature decreases in the static model to meet the 2nd law. This
energy loss also offsets Ug, per Eq. 6. The temperature can
increase but only if offset by R.E. and TS.
2 000 3 000 4 000
Temperature (K)
5 000
6 000
Figure 4. Energy changes in the full-sized proto-Earth, which
has the maximum ΔUg of 1 840 J·g-1. Solid line and lower
x-axis= -TΔS from the ideal gas analogy (Eq. 15), which accounts for compression of Fe. Dotted line and upper x-axis=
-ΔR.E. for the full-size Earth, assuming differential rotation.
For temperatures of the present-day core (grey box),
entropy-energy accounts for ΔUg.
3.4 Energy Conservation
Our entropy calculations are rough, but show that preexisting high temperatures are required for core segregation for
energy balance. Figure 4 also shows that with a significant
change in R.E. due to differential rotation, a smaller change in
(16)
where m is the mass contrast between equivalent volumes of
silicate and oxide particles vs. metal, and gBL is the gravitational acceleration in the boundary layer. The minimum
amount of work is done when each grain falls double its own
diameter, h~4b where b is grain radius, to segregate a core (Fig.
5a). Because energy would be released when the heavier particles sank, the net work is negative (Table 2). Work is required
to lift a heavy object, not to drop it.
To provide estimates on the high side, we use g=6 m·s-2 at
the middle of the present core, and densities from ambient
conditions of forsterite and Fe. We use an average b of 1 mm.
The number of particles in the core is N=(rcore/b)3. Because
m=4/3b3, the total work is Wtot=4/3gavehrcore3 ~2×1022
J for h~4b. If all work is dissipated as heat, as little as -3×10-6
J·g-1 can be released in boundary layer settling. A more realistic gravitational acceleration, appropriate to a rough estimate of
1 000 km as the radius where core formation onsets, gives
about half this amount of work. Particle size is unimportant,
although it sets a lower limit for the boundary layer thickness.
Within uncertainties, work done if the particles fall about
half Earth’s present radius (Table 2) equals -Ug from the
static model. This equivalence validates our settling model.
Boundary layer sizes in the present-day Earth are 100–250
km (the lithosphere and layer D” above the core), compatible
Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation
Table 2 Approximate work* from boundary-layer sorting
½ Q friction
Mantle
h
Layer thickness
Boundary
Layer
g T
½ Q friction
(a)
Core
(b)
After core formed
Initial
Today
Polar spin
Axial
symmetry
(c)
Initial
131
During core formation
Figure 5. Schematics of processes in core formation. (a)
Gravitational sorting during convection. In the bottom
boundary layer of the convecting proto-mantle (within
horizontal lines), relatively dense metal grains (dark dots)
sink while light silicate and oxide grains (open circles) rise.
For a pair crossing (largest circles), frictional heating is
divided evenly in the center of mass, so half the heat goes
up and half down (stippled arrows). Rising lighter grains,
including those with radionuclides (starbursts), are swept
up by the convection current of the proto-mantle (shaded
arrows), whereas the sinking metal grains become caught
up in the core convection cell. Segregation requires that a
grain settle through only the boundary layer. (b) Redistribution of radioactive elements during core formation. Stipple represents radioactive elements which are concentrated
upwards upon core formation and further concentrated
with differentiation of the crust from the mantle. The
thickness of the continental crust (heavy stipple) is exaggerated. (c) Gravitational sorting within the convecting
protomantle and formation of a differentially rotating core
and mantle. Left, axially symmetric convection patterns in
the homogeneous proto-Earth. Open arrows=polar, plumelike upwellings. Black arrows=equatorial downflow radiating towards the center of the plane. Right, settling during
the partially formed core. Light stipple=silicates plus some
iron. Heavy stipple=boundary layers where settling occurs.
White oval=silicates outpacing iron in the upwelling currents, because these are axial no angular momentum is
removed. Grey oval=Fe particles outpacing the silicates in
the equatorial plane of downwelling, these carry angular
momentum downwards.
Work (J·g-1)
1 mm
-10-6
1m
-10-3
1 km
100 km
2 000 km
-1
-100
-3 000
*The sign is negative because energy is released.
with work of -100 to -250 J·g-1, i.e., 5%–14% of Ug in the static
picture. Again, the sign is negative because energy is released.
Under plausible conditions, energy released is below -100 J·g-1,
which is comparable to the uncertainty of Ug. Even if all such
heat were retained, the possible temperature increase is small.
4.2 Frictional Heating
Importantly, boundary layer settling presumes application
of an external force, gravity, in an open system. In this depiction,
the boundary layer is where heat evolves, whereas the core region and overlying developing mantle are the surroundings
which receive heat. This leaky system is “open” permitting exchange of heat, entropy, matter, and temperature.
Importantly, friction heats both the sinking and rising particles: each component receives ½ the evolved heat (Sherwood
and Bernard, 1984). Therefore, only ~½ of the heat possibly
evolved in core formation is cast into the core region. The light
component moving upward transports heat closer to the surface
thereby accelerating cooling of the Earth. Redistribution of the
particles further alters the distribution of radioactive elements
(Fig. 5b). Both aspects promote cooling. In addition, the sinking
iron particles are limited in the distance that they can sink. As
discussed by Hofmeister and Criss (2013), a hot iron particle
cannot “sink” into a region where the iron particles are colder
and thus denser. In boundary layer sorting, then, the frictional
heat is evolved at the top of the core, which maximizes the rate
of cooling.
The effectiveness of frictional heating at high temperatures
is questionable. Melts are lubricants making sliding less resistant.
Rolling of grains also reduces friction. The calculated work (Table 2) is an upper limit on heating. From Eq. 6, these positive
values must be offset by an increase in the term TEiSE. which
means that more order is produced.
4.3 Settling in Convection Cell Margins as a Possible Origin
of Differential Rotation
In addition to the basal boundary layers, gravitation settling
can also occur in proto-mantle downwellings and upwellings
(Fig. 5c). Due to axial symmetry, convection in the hot environment of the proto-Earth would occur as a simple pattern. Upwellings would be associated with the polar axes due to reduced
gravitational acceleration pressure at the surface (due to rotational flattening and the consequent high thermal gradient) as
well as to the known existence of plume-like structures in convecting systems and to the adherence to symmetry. Downwellings would then occur radially inward in the equatorial plane.
Within this pattern, light silicate particles would rise at a
Anne M. Hofmeister and Robert E. Criss
132
faster rate than heavy Fe particles in the polar upwellings (Fig.
5c). These ascending silicate grains would carry no angular momentum upwards due to their axial position in a thin, plume-like
axial structure. Thus, the relatively fast-rising, light and hot silicates when added to the mantle decrease the angular momentum
of the mantle. In the equatorial downwellings, Fe particles would
sink faster than the silicates. These carry angular momentum
downwards, but do not increase L of the core. It is the loss of
silicates from the core region upwards which creates differential
rotation.
5 DISCUSSION
5.1 Timing is Everything
Early formation of the core, during accretion, strongly limits the available gravitational potential (Fig. 2). Meteorite evidence favors early core formation (e.g., Kleine et al., 2009) and
thus low |Ug|. Similarly, preferential internal deposition of iron
is also limiting.
5.2 Gravity Controlled Ordering
Equation 6 was arrived at from purely theoretical considerations. Estimates of changes in gravitational potential, configurational energy, and differential rotation support this approximate
equality. Estimates from boundary layer settling confirm that
heat evolved deep in the Earth is small (Table 2). More importantly, the process transports half of the frictional heat generated
plus radionuclides towards the surface, (Fig. 5b, discussed below), thus hastening cooling.
Equation 6 and our estimates of S have some important implications. Equation 13 shows that entropy changes are larger for
purer cores. Essentially, higher purity means a more restricted
core volume (Eqs. 14–15).
The purity of a planetary core depends on the temperature
of the object, its mass and size. These variables are interrelated.
Small planets will have more light elements in their cores.
Unless the object is hot and convecting, the metastable state of
the homogeneous Earth cannot be overturned and a core cannot
form. Smaller planets being colder means that differentiation is
limited, as is formation of cores. Core size in rocky moons and
planets is correlated with g and is consistent with Stokes settling
in boundary layer sorting (Hofmeister and Criss, 2012b).
5.3 Freezing of the Inner Core
Differentiation of Earth’s homogeneous core into inner
(solid) and outer (liquid) portions is also accompanied by simultaneous decrease in gravitational potential energy as well as
release of latent heat upon freezing. In this spontaneous process,
like core formation, E has two components, namely Ug and
the latent heat of crystallization. Because both of these quantities
are negative, heat must be shed to the surroundings. The Fe inner
core is an ordered phase and would store considerable uncompensated heat due to its high T, offsetting the decrease in Ug. Just
as water in a bird bath freezes as temperature of the surroundings
falls, the inner core forms as the Earth cools. That is, core freezing is a passive response to the gradual wind down of Earth’s
radioactivity.
5.4 Implications for Earth’s Thermal Evolution
Core formation promotes planetary cooling in two important ways. First, radioactive isotopes in a homogenous Earth
would be distributed uniformly through the interior, whereas
the ordered configuration strongly partitions these species into
the outer, silicate-rich layers (Figs. 5a, 5b). The upwards
movement of radioactive isotopes is evident in today’s continental crust containing almost all the radioactive isotope inventory expected if the bulk silicate Earth has a chondritic composition (Henderson, 1982). Because the evolved radiogenic heat
is transported over a shorter distance subsequent to core formation, cooling is more rapid. Using conductive cooling for illustrative purposes, dimensional analysis relates the cooling time
() of a body to its size (L) and thermal diffusivity (D) (e.g.,
Hofmeister, 2010)
~ L2/D
(17)
Because the mantle is about ½ the size of the whole Earth, the
length scale after core formation, decreases by a factor of 2, and
thus the cooling time decreases (very roughly) by a factor of 4.
Heat produced by friction (Fig. 5a) during settling is
equally shared by ascending and descending particles (Sherwood and Bernard, 1984). Only half this heat can be stored in
the core region: the other half is carried upwards. Placement
near the surface permits more rapid cooling, per Eq. 17. The
heat evolved is small for any reasonable boundary layer size
(Table 2), and does not offset the huge cooling effects discussed above.
Aspects of Earth’s internal reorganization (including core
formation, core freezing, production of continental crust, convection, and volcanism) are means of dissipating energy, while creating order, and promoting cooling. Heating is realized by the
very different processes of impacts and decay of radionuclides.
However, the self-gravitational potential of the Earth, like the
other rocky planets, is accounted for by its orbital rotational
energy (Hofmeister and Criss, 2012a). Impact heating is therefore limited to late stage collisions, after the Earth was almost
fully assembled and after the core had begun to form.
6 CONCLUSIONS
We have shown that one possible primordial source of heat
(gravitational segregation during core formation) is negligible
from the perspective of thermodynamics when the proper (negative) sign for Ug is used. Rather, the energy in the static picture
was lost to space or stored in the ordered configuration and possibly also in differential rotation; although the latter is small
today (e.g., Dehant et al., 2003) it could have once been significant and subsequently attenuated. Importantly, because core
formation occurred during accretion (e.g., Kleine et al., 2009),
the hypothetical homogenous Earth never existed, and the huge
amount of energy calculated by earlier workers by comparing
today’s layered Earth to this state was never available. No remnant of this heat can augment today’s heat budget.
ACKNOWLEDGMENTS
This study was supported by EAR-0757841. We thank
Genevieve Criss and Warren Hamilton for discussion and
comments.
Evaluation of the Heat, Entropy, and Rotational Changes Produced by Gravitational Segregation during Core Formation
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