CHAPTER IX. The Thermal Energy of the Earth

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 dθ/dr the temperature gradient, θ being the temperature, then the quantity of heat which passes from the surface per second is given by

dθ 4πr² × 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 dθ/dr, he applied his data to Fourier’s differential equation for the linear conduction of heat—

dθ k d²θ ——— = ——— · ——— dt ρσ dr²

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.

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

dθ 4πr² × k × ———— = ⁴/₃ × πr³Q, 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| 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| | | +------------------+-----+-----+-----+-------+----+-----+----+------+

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 per gram 6 × 10⁻² calories per second. Thorium ” ” 7·5 × 10⁻⁹ ” ” ”

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:

hrρ r θᵣ = ————(D - ———) κ 2

At the base r becomes equal to D and the temperature θᴰ is therefore given by

hρD² θᴰ = ————— 2κ

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.