THE THERMAL ENERGY OF THE EARTH
Temperature gradients, conductivity, and the rate at which
the earth loses energy—Kelvin’s attempt to estimate the age
of the earth—King’s treatment of the problem—Estimates by
Becker and Suzuki—The distribution of radium in rocks—The
thermal equilibrium of the earth—Concentration of radium
towards the surface—Constitution of the earth.
As the earth’s crust is penetrated by bore-holes, tunnels and mines, a
steady increase of temperature with depth is encountered. The rate of
increase varies greatly from place to place as the following records
show:
LOCALITY. DEPTH. TEMP. GRADIENT.
Anzin, France — 1° C. in 15·3 metres
Wigan 750 metres 1° C. “ 30·0 ”
Sperenberg 1700 ” 1° C. ” 36·5 ”
Mt. Cenis Tunnel 1600 ” 1° C. “ 43·0 ”
Minas Geraes, Brazil — 1° C. ” 86·0 ”
Calumet, Michigan 1430 ” 1° C. “ 12·20 ”
Such a variation would naturally be expected when the chemical
reactions of weathering, cementation and metamorphism are remembered,
and the unequal distribution of radioactive elements in the earth’s
crust. The solution of silicates by ground waters takes place with
liberation of heat, and it has been estimated that 120 calories are
released when one gram of rock is decomposed by weathering. The
processes of metamorphism take place at the expense of the earth’s
heat, and the net result of the complete cycle in which (a) an igneous
rock is eroded, (b) the resulting sediments buried and transformed
into schists and (c) the latter brought again to the surface by earth
movement and denudation, is a running down of energy involving a
permanent loss. Moreover, the earth loses heat from the interior, not
only by conduction, but also by convection. The circulation of ground
waters and the activities of vulcanism—which include the upward
movement of molten magmas, heated waters and gases—all result in the
transference of heat from the interior to the outer zones of the crust.
It is generally accepted, that when cooling by convection is left out
of account, the temperature gradient due to conduction alone is of the
order of 1° C. in 32 metres. If this estimate should be in error, it is
more likely to be too high than too low.
To calculate the rate at which heat escapes from the earth by
conduction from the interior, it is necessary to know the average
conductivity of rocks. This factor, in turn, varies greatly in
different materials, but in the case of the most predominant rocks the
conductivity is accurately known, and the value _k_ = 0·004 may
be accepted as very close to the true average value. The variation of
conductivity with increasing temperature and pressure scarcely affects
the problem. The former tends to diminish the conductivity, the latter
to augment it. As far as our present knowledge goes, these two effects
almost exactly balance each other, and the assumption that
conductivity remains fairly constant with depth is therefore justified.
If _r_ represent the earth’s radius, and _d_θ/_dr_ the
temperature gradient, θ being the temperature, then the quantity of
heat which passes from the surface per second is given by
_d_θ
4π_r_² × _k_ × ——— and can be readily calculated.
_dr_
According to the Laplacian hypothesis, the earth’s original store of
heat was derived from the nebula from which it separated. Kelvin made
the assumption that as the molten globe cooled down it was preserved
by convection currents at a temperature nearly uniform from centre to
surface. On attaining the point of solidification, it would gradually
become solid throughout, most probably starting from the centre, and
only when solidification was complete could the surface continue to
cool further. The problem which Kelvin set himself to solve was this:
Given a solid globe originally at a uniform temperature of 7000° F.
(3871° C.), and subsequently cooling down, to calculate the time which
would be required for the establishment of the present surface gradient.
Taking the most probable values for conductivity _k_, density
of rock ρ, specific heat of rock σ, and temperature gradient
_d_θ/_dr_, he applied his data to Fourier’s differential
equation for the linear conduction of heat—
_d_θ _k_ _d_²θ
——— = ——— · ———
_dt_ ρσ _dr_²
Solving this equation for the unknown factor _t_, the required
time period was found. The high initial temperature was chosen from
very meagre data, as representing a maximum figure for the
melting-point of rock. Kelvin was particularly anxious that his
treatment should provide an over- rather than an under-estimate of
time. The experiments of Dr. C. Barus have shown that diabase, a good
typical rock, becomes thoroughly liquid at 1200° C. If Kelvin had
used this temperature instead of his arbitrary 3871° C., his periods
of cooling would have been reduced to less than one-tenth of those
actually arrived at.
For enormous periods of time, the development of a temperature gradient
would be restricted to the earth’s outer zones. The interior, in
complete thermal isolation, would remain unaffected, its loss of heat
being quite insensible. The limited thickness of the outer shell, in
which cooling would make itself felt, is made clear by the following
figures:
160 miles in 100 million years.
240 ” ” 240 ” ”
320 ” ” 600 ” ”
570 ” ” 1000 ” ”
The different periods which Kelvin favoured in his famous contributions
to this problem have already been mentioned, and are tabulated below.
1862. 96 million years (limits 20-400).
1876. 50-90 ” ”
1897. 20-40 ” ”
Clarence King, in 1893, applied a new criterion to the subject, taking
into consideration the effect of pressure in raising the melting-point
of rocks, and the necessity for an earth which should be stable under
the influence of tidal stresses. Barus had measured the melting-point
of diabase at various pressures, and a law of variation of
melting-point with depth was formulated on his experimental results.
If this law were to hold as far as the centre, diabase would there be
able to exist in the solid state at any temperature below 76,000° C.
Thus the hypothesis arose that solidification would begin at the centre
owing to the high pressure obtaining there. Under these conditions,
however, a temperature gradient was already developed. If the gradient
were to exceed the rate by which the fusion point of rocks is raised
by pressure, the former would catch up, and at a certain depth the
temperature would reach the fusion point and a zone of fluid rock would
be inevitable.
King accepted diabase as a representative rock, and rejected any
distribution of heat which would demand a liquid zone in that part of
the earth’s crust where diabase or similar rocks would be expected to
prevail. This procedure is justified by the consideration that were
such a zone to exist, the earth would be incapable of maintaining tidal
stability, and the crust would break down. King found that in the
admissible cases, the initial temperature of crustal solidification
would not exceed 2000° C., and that the period of cooling, which
would reduce the gradient to that of the present was limited by 24
million years. Higher initial temperature would involve fluidity, and
superior age necessitate a lower surface gradient. The following curves
represent the gradients of Kelvin’s earth of 100 million years, and
King’s earth of 20 million years, in relation to the diabase
fusion point curve. It will be seen that according to King’s argument,
Kelvin’s earth implies the impossible condition of a liquid zone from A
to B.
[Illustration: FIG. 16.
Temperature Gradients in relation to the Fusion point of Diabase.]
In 1910, Becker attempted to deal with the same problem without relying
on the temperature gradient, it being considered that owing to the
presence of radium in rocks the gradient could not be trusted. In its
place he assumes that the crustal strains associated with upheaval
and subsidence are completely relieved at the surface of easiest
fusion—and that according to the calculations of Hayford on isostatic
compensation, the present depth of that surface is 71 miles. At that
depth, therefore, the temperature curve and the diabase curve approach
most closely, so that the additional temperature required to produce
fusion and relief of strain there becomes a minimum. In the above
diagram, C would represent the point of easiest fusion. Becker
justifies his choice of diabase by showing that on the Laplacian law
of density, rocks of this type become predominant at depths greater
than 40 miles, the more acid rocks, lying above, being more refractory.
Tidal stability is provided for by rejecting any temperature curve
which crosses the diabase line in the zone of basic rocks. The most
probable earth, according to Becker, is one with an initial temperature
of 1300° C., which would attain a surface gradient of 1° C. in 42
metres in 60 million years. He concludes that only “a tenth of the heat
emitted by the earth can be ascribed to radioactivity plus all other
exothermic chemical transformations.”
A Japanese estimate of the time elapsed since the molten surface of the
earth began to solidify appeared in 1912. Suzuki makes the assumption
that a thin solid crust has gradually increased in thickness, so that
the latent heat of fusion liberated at the junction of solid and liquid
rock is equal to the heat lost at the surface. The present thickness
of the crust is assumed, on the authority of Milne and Arrhenius to
lie between 30 and 40 miles. Granting these postulates, the thermal
constants for basalt and granite lead to an age of 20 to 60 million
years according to the thickness of the crust and the material of which
it is composed.
It is surprising that Becker and Suzuki should have treated the
problem in this restricted way. The heat evolution attending atomic
disintegration was established in 1903, and in dealing with the earth
this phenomenon must be regarded as one of fundamental importance.
To ignore the significance of radio-thermal energy is to reduce the
problem to a mathematical exercise, interesting, no doubt, but with
little value in its geological application. Let us make a simple
calculation of the quantity of radium, which, if uniformly distributed
throughout the earth, would make good the loss of heat. If Q is the
heat generated per second by the radium in each cubic centimetre, then
we have
_d_θ
4π_r_² × _k_ × ———— = ⁴/₃ × π_r_³Q,
_dr_
whence, Q = 6 × 10⁻¹⁵ calories per second.
= 2·16 × 10⁻¹¹ calories per hour.
Now 1 gram of radium in complete radioactive equilibrium emits 216
calories per hour and consequently all the heat would be supplied by
10⁻¹³ grams per cubic centimetre, or 1·8 × 10⁻¹⁴ grams per gram of
earth material.
RADIUM PER GRAM OF IGNEOUS ROCK IN BILLIONTHS (10⁻¹²) OF A GRAM.
+-----------------+------------+-------------+----------+-----------+
| OBSERVER. | ACID. |INTERMEDIATE.| BASIC. |ULTRABASIC.|
+-----------------+------+-----+-----+-------+----+-----+----+------+
|Strutt |11[14]| 2·59| 4 | 2·25 | 9 | 0·52| 4 | 0.46 |
|Farr and Florence| 3 | 1·83| 4 | 1·68 | 6 | 0·54| | |
|Buchner | 8 | 2·61| 15 | 1·64 | 4 | 0·73| | |
|Fletcher | 4 | 0·85| 20 | 0·85 | 5 | 0·71| | |
|Holmes | 8 | 2·80| | | 4 | 0·85| 10 | 0·51 |
+-----------------+------+----+------+-------+----+-----+----+------+
|Mean |34 | 2·63| 43 | 1·28 | 28 | 0·66| 14 | 0·50 |
|Joly |86 | 3·01| 48 | 2·57 | 31 | 1·28| | |
+------------------+-----+-----+-----+-------+----+-----+----+------+
[Footnote 14: The number of rocks examined in each case is given in the
first of each pair of columns.]
Turning to the rocks themselves, the actual amount of radium is found
to be a hundred times more than we want. Strutt was the first to
discover this embarrassing richness, and his results, with those of
later investigators, are summarised in the adjoining table. Joly’s
results are given apart from those of other observers, for they were
arrived at by the fusion method, and, moreover, separate rocks were
not examined. A composite mixture of typical rock specimens was made
up and a single analysis then sufficed to determine the average radium
content. It will be noticed that Joly’s results are consistently
higher than those found by the solution-method. Joly claims that his
own procedure is more reliable than that followed in the solution
method. Up to 1909, Joly had himself employed the latter method in
the examination of 126 igneous rocks. In striking disagreement with
the results of other workers, he found an average radium content of 7
× 10⁻¹² grams per gram of rock. An explanation of the discrepancy is
not yet forthcoming, but in the light of his most recent work, which
gives an average of 2·5 × 10⁻¹², he has now suggested that his earlier
results be set aside. Measurements of thorium in rocks are not yet
so plentiful as those of radium, and most of our present knowledge
of the distribution of this element is due to Joly and Fletcher. The
most probable averages of the data now available may be summarised as
follows:
------------------+-----------------------------------------
| AVERAGE PER GRAM OF ROCK.
TYPE OF ROCK. +---------------------+-------------------
| RADIUM. | THORIUM.
------------------+---------------------+-------------------
Igneous | 2·5 × 10⁻¹² grs. | 2·0 × 10⁻⁵ grs.
Sedimentary | 1·5 × 10⁻¹² ” | 1·0 × 10⁻⁵ ”
Metamorphic | 2·0 × 10⁻¹² ” | 1·5 × 10⁻⁵ ”
Deep-sea deposits | 5·0 × 10⁻¹² ” |
------------------+---------------------+-------------------
Accepting these figures for igneous rocks provisionally and combining
them with the respective heat emission of radium and thorium in
complete radioactive equilibrium, viz.,
Radium[15] per gram 6 × 10⁻² calories per second.
Thorium ” ” 7·5 × 10⁻⁹ ” ” ”
[Footnote 15: Equivalent to 3,000,000 grs. of uranium.]
it is clear that each gram of the earth’s crust is a source of heat
supplying on an average 15 × 10⁻¹⁴ calories per second on account of
its radium content, and 15 × 10⁻¹⁴ calories on account of its thorium
content. The total heat emission is therefore of the same order in each
case, and amounts altogether to 30 × 10⁻¹⁴ calories per second. The
whole mass of the earth is 6 × 10²⁷ grams, and if this were the source
of as much radio-thermal energy throughout, the supply of heat in 1000
million years would have been sufficient to raise its temperature to
about 40,000° C., and the present gradient should be many times greater
than it is. This conclusion cannot be reconciled with the evidence
afforded by the crustal rocks, both their structure and temperature
gradients being decisively against any such possibility.
There are three cases which may be considered. The earth may be in
thermal equilibrium, gaining as much heat as it loses and cooling only
as the slow decay of the radio-elements permits; or it may be growing
hotter, or, which is very unlikely, it may be cooling more rapidly than
it would do if in radio-thermal equilibrium. The first case is the one
now regarded with most favour. If the earth has cooled at all, and
there seems to be no sound reason why we should altogether abandon that
venerable conception, it must at some time have attained a condition
of equilibrium. With the slow march of atomic disintegration its own
rate of cooling would then keep time. The temperature gradient would be
maintained solely by radioactivity for an immeasurably long period.
The superabundance of radium which seemed to be implied by Strutt’s
original work is certainly, as he then suggested, restricted to the
surface rocks. The interior of the earth must be relatively free from
radium and thorium. It is easy to calculate the thickness of the outer
zone of the earth’s crust, which would suffice to supply the stream
of heat passing to the surface. The temperature θᵣ at any distance
_r_ from the surface is given by the following equation, where
_k_ is conductivity, _h_ the heat production of radium and
thorium per second, in each gram of rock, ρ the density of the rock,
and D the total depth of the radioactive layer:
_hr_ρ _r_
θᵣ = ————(D - ———)
_κ_ 2
At the base _r_ becomes equal to D and the temperature θᴰ is therefore
given by
_h_ρD²
θᴰ = —————
2_κ_
Using the figures given above, the thickness of the radioactive layer
would be restricted to about ten miles and the basal temperature would
reach only 250° C. This result cannot be held to express the facts, for
there can be no doubt that the radium and thorium content decreases
with depth for the same reason that the type of rock varies with
depth. A glance at the table on p. 130 will show that there is a rough
proportionality between the acidity or percentage of silica of a rock
and its radium content. The more basic rocks are much poorer in radium,
and, as would be expected, in thorium also. Now we have good reason to
suppose that the more deep-seated rocks of the earth’s crust are of
basic and ultra-basic composition, and that below the 30-mile crustal
zone they are exclusively ultra-basic, perhaps similar in composition
to the material of stony meteorites, with which they agree almost
exactly in density (3·4). This information, which it might be thought
would be for ever withheld from us, is derived from the study of
earthquake waves. The latter in passing through the earth’s interior
carry with them a record of the type of material they have penetrated.
Within the stony zone, which extends downwards for several hundred
miles, and separated from it somewhat sharply, lies the heavy core of
the earth (density about 7·8), probably of metallic composition, like
the iron meteorites. If we may judge from the latter, this nucleus is
entirely free from radium, and that there is safety in this analogy is
indicated by the very low radium content of such native iron as has
found its way in basaltic magmas up to the surface.
We have already seen (p. 30) that a first differentiation of the
original heterogeneous material from which the earth was built would
result in the formation of a metallic core surrounded by a stony zone.
The further differentiation of the latter, whereby the crust with its
abundant variety of acid and basic rocks was developed, is of too
complex a nature to be considered here. It is very probable that as the
more siliceous constituents separated and became concentrated towards
the surface, they carried with them their store of radio-elements.
In this way, basing our ideas on evidence quite independent of the
temperature gradient, we are led to the remarkable conclusion that the
radium and thorium of the earth are to be found almost exclusively in
the earth’s crust. The most probable depth of the radioactive layer
may therefore be placed at 30 miles and the basal temperature in this
case would be about 750° C., which would be more in accordance with the
requirements of volcanic phenomena. Moreover, it must not be forgotten
that the heat lost by the upward movement and convection currents
of rock magmas, heated waters and gases, has also to be accounted
for. The radio-thermal equivalent must be substantially increased to
include this phase of the subject. The basal temperature of 750° C.
is only a minimum, and the higher temperatures demanded by geology
are not therefore inconsistent with the facts. However, until more
data are accumulated, it would be rash to attempt to deduce the exact
distribution of the radio-elements in the crust, but already we may
assert with confidence that the crustal average is somewhat lower than
that of the surface rocks in which granitic types form so large a
proportion. What the average actually may be cannot yet be decided.
Kelvin’s problem must now be reversed. It is impossible to deduce the
earth’s age from its thermal condition. We can only say that the age
must be very much greater than Kelvin calculated. The new problem which
presents itself is to determine the thermal history of the earth,
accepting its antiquity as a known or partially known factor. The
science of radioactivity is a welcome addition to the tools which the
geologist employs in his difficult task of elucidating the earth’s
history, and it is peculiarly valuable in helping him just where he has
hitherto had most cause for despair.