The amount of energy generated by the falling together of the matter of the sun from universal diffusion to the dimensions which the sun has at present, is only about 13,000,000 times the amount now radiated per annum. In Thomson's paper Pouillet's estimate of the energy radiated per second is used, and this number is raised to 20,000,000. Taking the latter estimate, the whole potential energy exhausted by the condensation of the sun's mass to uniform density would suffice for only 20,000,000 years' supply. But the sun is undoubtedly of much greater density in the central parts than near the surface, and so the energy exhausted must be much greater than that stated above. This will raise the number of years provided for. On the other hand, a considerable amount of energy would be dissipated during the process of condensation, and this would reduce the period of radiation estimated. Thomson suggests that 50,000,000, or 100,000,000, years is a possible estimate.
It is not unlikely that the rate of radiation in past time, when the sun had not nearly condensed to its present size, was so much less than it is at present that the period suggested above may have to be considerably augmented. Another source of radiation, which seems to be regarded by some authorities as a probable, if not a certain, one, has been suggested in recent years—the presence of radio-active substances in the sun. So far as we know, Lord Kelvin did not admit that this source of radiation was worthy of consideration; but of course, granted its existence to an extent comparable with the energy derivable from condensation of the sun's mass, the "age of the sun's heat" would have to be very greatly extended. These are matters, however, on which further light may be thrown as research in radio-activity progresses. Lord Kelvin was engaged when seized with his last illness in discussing the changes of energy in a gaseous, or partially gaseous, globe, slowly cooling and shrinking in doing so; and a posthumous paper on the subject will shortly be published which may possibly contain further information on this question of solar physics.
But Thomson put forward a third argument in the paper on Geological Time, which has always been regarded as the most important. It is derived from the fact, established by abundant observations, that the temperature in the earth's crust increases from the surface inwards; and that therefore the earth must be continually losing heat by conduction from within. If the earth be supposed to have been of uniform temperature at some period of past time and in a molten state, and certain assumptions as to the conductive power and melting point of its material be made, the time of cooling until the gradient of temperature at the surface acquired its present value can be calculated. This was done by Thomson in a paper published in the Transactions, R.S.E., in 1862. We propose to give here a short sketch of his argument, which has excited much interest, and been the cause of some controversy.
In order to understand this argument, the reader must bear in mind some fundamental facts of the flow of heat in a solid. Let him imagine a slab of any uniform material, say sandstone or marble, the two parallel faces of which are continually maintained at two different temperatures, uniform over each face. For example, steam may be continually blown against one face, while ice-cold water is made to flow over the other. Heat will flow across the slab from the hotter face to the colder. It will be found that the rate of flow of heat per unit area of face, that is per square centimetre, or per square inch, is proportional to the difference of the temperatures in the slab at the two faces, and inversely proportional to the thickness of the slab. In other words, it is proportional to the fall of temperature from one face to the other taken per unit of the thickness, that is, to the "gradient of temperature" from one face to the other. Moreover, comparing the flow in one substance with the flow in another, we find it different in different substances for the same gradient of temperature. Thus we get finally a flow of heat across unit area of the slab which is equal to the gradient of temperature multiplied by a number which depends on the material: that number is called the "conductivity" of the substance.
Now, borings made in the earth show that the temperature increases inwards, and the same thing is shown by the higher temperatures found in deeper coal mines. By means of thermometers sunk to different depths, the rate of increase of temperature with depth has been determined. Similar observations show that the daily and annual variations of temperature caused by the succession of day and night, and summer and winter, penetrate to only a comparatively small depth below the surface—three or four feet in the former case, sixty or seventy in the latter. Leaving these variations out of account, since the average of their effects over a considerable interval of time must be nothing, we have in the earth a body at every point of the crust of which there is a gradient of increasing temperature inwards. The amount of this may be taken as one degree of Fahrenheit's scale for every 50 feet of descent. This gradient is not uniform, but diminishes at greater depths. Supposing the material of uniform quality as regards heat-conducting power, the mathematical theory of a cooling globe of solid material (or of a straight bar which does not lose heat from its sides) gives on certain suppositions the gradients at different depths. The surface gradient of 1° F. in 50 feet may be taken as holding for 5000 feet or 6000 feet or more.
This gradient of diminution of temperature outwards leads inevitably to the conclusion that heat must be constantly flowing from the interior of the earth towards the surface. This is as certain as that heat flows along a poker, one end of which is in the fire, from the heated end to the other. The heat which arrives at the surface of the earth is radiated to the atmosphere or carried off by convection currents; there is no doubt that it is lost from the earth. Thus the earth must be cooling at a rate which can be calculated on certain assumptions, and it is possible on these assumptions to calculate backwards, and determine the interval of time which must have elapsed since the earth was just beginning to cool from a molten condition, when of course life cannot have existed on its surface, and those geological changes which have effected so much can hardly have began.
Considering a globe of uniform material, and of great radius, which was initially at one temperature, and at a certain instant had its surface suddenly brought to, let us say, the temperature of melting ice, at which the surface was kept ever after, we can find, by Fourier's mathematical theory of the flow of heat, the gradient of temperature at any subsequent time for a point on the surface, or at any specified distance within it. For a point on the surface this gradient is simply proportional to the initial uniform temperature, and inversely proportional to the square root of the product of the "diffusivity" of the material (the ratio of the conductivity to the specific heat) by the interval of time which has elapsed since the cooling was started. Taking a foot as the unit of length, and a year as the unit of time, we find the diffusivity of the surface strata to be 400. If we take the initial temperature as 7000 degrees F.—which is high enough for melting rock—and take the interval of time which has elapsed as 100,000,000 years, we obtain at the surface a gradient approximately equal to that which now exists. A greater interval of time would give a lower gradient, a smaller interval would give a higher gradient than that which exists at present. A lower initial temperature would require a smaller interval of time, a higher initial temperature a longer interval for the present gradient.
With the initial temperature of 7,000 degrees F., an interval of 4,000,000 years would give a surface gradient of 1° F. in 10 ft. Thus, on the assumption made, the surface gradient of temperature has diminished from 1⁄10 to 1⁄50 in about 96,000,000 years. After 10,000 years from the beginning of the cooling the gradient of temperature would be 2° F. per foot. But, as Thomson showed, such a large gradient would not lead to any sensible augmentation of the surface temperature, for "the radiation from earth and atmosphere into space would almost certainly be so rapid" as to prevent this. Hence he inferred that conducted heat, even at that early period, could not sensibly affect the general climate.
Two objections (apart from the assumptions already indicated) will readily occur to any one considering this theory, and these Thomson answered by anticipation. The first is, that no natural action could possibly bring the surface of a uniformly heated globe instantaneously to a temperature 7000° lower, and keep it so ever after. In reply to this Thomson urged "that a large mass of melted rock, exposed freely to our earth and sky, will, after it once becomes crusted over, present in a few hours, or a few days, or at most a few weeks, a surface so cool that it can be walked over with impunity. Hence, after 10,000 years, or indeed, I may say, after a single year, its condition will be sensibly the same as if the actual lowering of temperature experienced by the surface had been produced in an instant, and maintained constant ever after." The other objection was, that the earth was probably never a uniformly heated solid 7000° F. above the present surface temperature as assumed for the purpose of calculation. This Thomson answers by giving reasons for believing that "the earth, although once all melted, or melted all round its surface, did, in all probability, really become a solid at its melting temperature all through, or all through the outer layer which has been melted; and not until the solidification was thus complete, or nearly so, did the surface begin to cool."
Thomson was inclined to believe that a temperature of 7000° F. was probably too high, and results of experiments on the melting of basalt and other rocks led him to prefer a much reduced temperature. This, as has already been pointed out, would give a smaller value for the age of the earth. In a letter on the subject published in Nature (vol. 51, 1895) he states that he "is not led to differ much" from an estimate of 24,000,000 years founded by Mr. Clarence King (American Journal of Science, January 1893) on experiments on the physical properties of rocks at high temperatures.