SOME REMARKS ON HYPOTHESES CONCERNING AN INCREASE IN THE EARTH’S VOLUME AND ROTATIONAL SPEED

L. A. PUKHLYAKOV

(Presented by Professor A. G. Sivov)

Recently, in connection with the broad development of paleomagnetic studies, hypotheses concerning an increase in the Earth’s volume have gained great popularity; among them is the Jordan–Heezen hypothesis, which is based on the assumption, put forward by Dirac, that the gravitational constant decreases in proportion to the age of the Galaxy. Developing Dirac’s idea, Jordan in 1952 came to the conclusion that over the last 3.25 billion years the gravitational constant has decreased by a factor of two to three, as a result of which the circumference of the Earth has increased by 1100 miles [7]. Thus, since the beginning of the Cambrian—that is, over the last 510–570 million years—this quantity should have decreased by 11–21%.

Objecting to these authors, E. N. Lyustikh [6] notes that a decrease of the gravitational constant to such an extent should have been accompanied by some recession of all the planets from the Sun and by a decrease in the amount of solar radiation that could reach the Earth’s surface. This, in turn, should have led to a cooling of our planet’s climate. However, E. N. Lyustikh did not give a quantitative solution of this problem, and therefore it seems to some geotectonists that such cooling could not have been significant. In this connection there arises the need to consider the question from a quantitative point of view.

First of all, let us note that the greater the distance between the Sun and a planet, the greater the reserve of potential energy (relative to the Sun) possessed by that planet. To determine the amount of this energy, let us suppose that some planet has moved relative to the Sun by a distance \(ds\). Obviously, in doing so it must convert from potential energy into kinetic energy an amount of energy that can be expressed by the relation

\[ dW_p = 273.8 \cdot m \frac{R_{\odot}^{2}}{s^{2}}\, ds, \tag{1} \]

where \(273.8\) is the acceleration due to gravity at the surface of the Sun in \(\text{m}/\text{s}^2\); \(m\) is the mass of the given planet in kilograms; \(R_{\odot}\) is the adopted radius of the Sun, equal to \(695.3 \cdot 10^{6}\ \text{m}\), and \(s\) is the distance between the given planet and the Sun in meters. Integrating this expression over the limits from \(s_1\) to \(s_2\), we obtain

\[ W_p = 273.8 \cdot m \cdot R_\odot \left( \frac{R_\odot}{s_2} - \frac{R_\odot}{s_1} \right). \tag{2} \]

To obtain now the potential energy of a planet, it is sufficient to put into expression (2), as \(s_1\), the distance between it and the Sun at the present time, and as \(s_2\) the computed radius of the Sun. As a result, the first term in parentheses becomes unity, and the expression for potential energy itself takes the form

\[ W_p = 190{,}373{,}000{,}000\, m \left( 1 - \frac{R_\odot}{s_{\mathrm{pl}}} \right). \tag{3} \]

Every planet possesses not only potential but also kinetic energy. Its amount can be determined from the formula

\[ W_k = \frac{m v^2}{2}, \tag{4} \]

where \(v\) is the velocity of the planet’s motion relative to the Sun. This quantity is also a function of the distance between the planet and the Sun, for a planet can have a stable orbit only if its centrifugal acceleration, expressed by the relation

\[ a = \frac{v^2}{s}, \tag{5} \]

is equal to the centripetal acceleration

\[ g_\odot = 273.8 \frac{R_\odot^2}{s^2}, \tag{6} \]

whence

\[ \frac{v^2}{s} = 273.8 \frac{R_\odot^2}{s^2}, \tag{7} \]

or

\[ v^2 = 273.8 \frac{R_\odot^2}{s}. \tag{8} \]

Thus the kinetic energy of a planet relative to the Sun is expressed by the relation

\[ W_k = \frac{273.8}{2}\, m \frac{R_\odot^2}{s}. \tag{9} \]

Converting one of the solar radii contained in expression (9) into numerical form and carrying out the corresponding operations in it, we find that the quantity of interest to us can be expressed by the relation

\[ W_k = 190{,}373{,}000{,}000\, m \frac{R_\odot}{2s}. \tag{10} \]

Finally, adding expressions (3) and (10), we obtain the formula for the total energy of a planet relative to the Sun:

\[ W_\odot = 190{,}373{,}000{,}000\, m \left( 1 - \frac{R_\odot}{2s} \right). \tag{11} \]

In further calculations it will be more convenient for us to use not the planet’s total energy, but the energy per unit mass, or the specific energy

of the given planet relative to the Sun. Obviously, its amount will be expressed by the relation

\[ \Theta_{\odot}=190.373.000.000\left(1-\frac{R_{\odot}}{2s}\right). \tag{11-a} \]

To determine the specific energy of the Earth relative to the Sun, it is now sufficient to substitute into expression (11 a), as \(s\), \(149.5\cdot 10^9\) m and carry out the corresponding operations. This quantity turns out to be equal to 189,930,000,000 J/kg.

And now let us see at what distance from the Sun the Earth should be if its attraction proves to be greater than the present one by 11%. Obviously, in this case in formula (11 a) one must substitute 211,314,000,000 as the coefficient. But however the gravitational constant may change, the specific energy of the Earth must remain unchanged; hence, to determine the distance between it and the Sun, in the expression obtained one must substitute, as \(\Theta_{\odot}\), 189,930,000,000 J/kg. As a result we obtain

\[ 189.930.000.000=211.314.000.000\left(1-\frac{R_{\odot}}{2s}\right), \tag{12} \]

whence we find that the quantity of interest to us must reach 4.941 solar radii, or 3.435 million km. Thus, if the Dirac—Jordan—Heisenberg hypothesis is correct, the distance between the Sun and the Earth at the beginning of the Cambrian must have been 43.52 times smaller than the present one, and the amount of solar rays that could reach the terrestrial surface must have exceeded the present amount by approximately 1900 times. It is, of course, impossible to agree with such an assumption, since the climate of our planet in the Cambrian was approximately the same as at the present time.

But perhaps the change in the gravitational constant was still smaller, for example, 0.1% over 510–570 million years. Such an assumption is clearly not capable of explaining the origin of the Atlantic Ocean. But it follows from it that the coefficient in formula (11 a) must be equal to 190,563,000,000, and the distance between the Earth and the Sun—143 solar radii, or 99.4 million km, which is 1.5 times less than the present one. And this means that the amount of solar energy received by the Earth in the Cambrian must have exceeded the present amount by 2.25 times. However, even this assumption cannot be accepted, for the climate of our planet at that time was approximately the same as now, and somewhat earlier, in the Precambrian, even glaciations occurred [13, p. 124].

Thus, an analysis of paleoclimatological data shows that the gravitational constant is indeed constant and therefore the hypothesis of an increase in the Earth’s volume, which is based on the assumption of its nonconstancy, may be considered untenable.

Another variant of the hypothesis of an increase in the Earth’s volume is the Kirillov—Neiman hypothesis, according to which cosmic rays, representing particles of matter, penetrate into its interior. It is understood that these rays are not retained in the Earth’s crust; as a result, according to I. V. Kirillov [4] and V. B. Neiman [7], only the internal geospheres of the Earth increase in volume, while its crust does not increase in volume and therefore cracks arise in it. As evidence for the hypothesis, the diminution of the animal world is cited—the disappearance of giant animals. Let us examine this question in more detail.

In fact, as was shown in his time by Ya. I. Perelman [8], the growth of animals is limited—they cannot reach arbitrary sizes. This follows from the fact that the mass of animals, with an increase

with an increase in the dimensions of their bodies and the preservation of proportionality among the individual members, increases in proportion to the volume, or to the third power of the length of their body:

\[ m=c_1\cdot V=c_2 l^3, \tag{13} \]

whereas the strength of individual members, for example the legs, is proportional to the area of their cross-section, that is, only to the second power of the body length:

\[ Q=c_3\cdot F=c_4 l^2 \tag{14} \]

Expressing from these two relations the quantity \(l\) (the body length), it is quite simple to obtain the dependence between the strength of an animal and its volume, and consequently also its mass:

\[ Q=c\cdot V^{\frac{2}{3}} \tag{15} \]

Proceeding from this dependence, one may conclude that, while increasing its mass and absolute strength, a large animal decreases its relative strength—becomes ever weaker in relation to its own weight. Finally, this weakness may reach such a degree that the animal will no longer be able to move on its own legs.

Let us imagine a giant who is four times taller than an ordinary human being. The thickness of all the members of his body (including the legs) should, apparently, be 16 times greater than that of an ordinary human being. Thus, if the legs of an ordinary person are capable of supporting a load of 150 kg (meaning that the ordinary person’s own weight is 75 kg and the load he can carry is also 75 kg), then the giant’s legs will be capable of withstanding a load of no more than 2400 kg. Meanwhile, his own weight should be 64 times greater than the weight of an ordinary person, that is, it should reach 4800 kg. It follows that such a giant could not live under terrestrial conditions. But suppose that the giant considered has found himself on the Moon, where the force of gravity is 6 times less than on Earth. Obviously, his weight there should at once decrease to 800 kg. Now he will be able not only to walk by himself, but also to carry on himself a load weighing up to 1600 kg.

In order to establish the dependence between the acceleration due to gravity and the possibility of increasing the size of animals, let us suppose that in some locality it has decreased by a factor of \(\kappa\). This means that any animal, without losing mobility, will be able to increase its mass also by a factor of \(\kappa\). But an increase of mass by a factor of \(\kappa\), according to formula (15), must lead to an increase of its strength by a factor of \(\kappa^{\frac{2}{3}}\), and this, in turn, will allow the animal to increase its mass by another factor of \(\kappa^{\frac{2}{3}}\). Increasing its volume and mass by a factor of \(\kappa^{\frac{2}{3}}\), the animal, according to formula (15), must increase its strength by another factor of \(\kappa^{\left(\frac{2}{3}\cdot\frac{2}{3}\right)}\), and so on. As a result, the maximum possible increase in the mass of an animal when the acceleration due to gravity is reduced by a factor of \(\kappa\) will be expressed in the form of the following series

\[ M_{\mathrm{B}}=M_{\mathrm{D}}\cdot \kappa\cdot \kappa^{\frac{2}{3}}\cdot \kappa^{\left(\frac{2}{3}\cdot\frac{2}{3}\right)}\cdot \kappa^{\left(\frac{2}{3}\cdot\frac{2}{3}\cdot\frac{2}{3}\right)}, \tag{16} \]

where \(M_{\mathrm{D}}\) is the maximum possible mass of an animal for a given ratio of the various parts of its body and a given (not reduced) value of the acceleration due to gravity. For mammals under present-day conditions it may be taken as equal to 5 t (the mass of an African elephant).

Taking the logarithm of expression (16) and taking \(\lg k\) outside the brackets, we obtain

\[ \lg \frac{M_{\mathrm{B}}}{M_{\mathrm{D}}} = \lg k \left[ 1+\frac{2}{3}+\left(\frac{2}{3}\right)^2+\left(\frac{2}{3}\right)^3+\ldots+\left(\frac{2}{3}\right)^n+\ldots \right]. \tag{17} \]

Finally, carrying out the corresponding operations in the square brackets, we obtain

\[ \lg \frac{M_{\mathrm{B}}}{M_{\mathrm{D}}}=3\lg k, \tag{18} \]

whence we find that

\[ M_{\mathrm{B}}=M_{\mathrm{D}}\cdot k^3, \tag{19} \]

that is, with a decrease in the acceleration of gravity by a factor of \(k\), animals, without changing the proportions between the dimensions of the various parts of their body, can increase their mass and volume by a factor of \(k^3\). This is precisely where the explanation lies for the fact that the largest of modern animals, the whale, whose mass sometimes reaches 150 tons, lives in water, where a substantial part of the animal’s weight is lost owing to the buoyant force of the water. I. A. Efremov [3] also explains the growth of the largest reptiles of the Mesozoic by their semiaquatic way of life.

Thus, the increase in the Earth’s mass proposed by I. V. Kirillov [4] and V. B. Neiman [7] should in fact have led to a diminution of the animal world. It remains to establish whether the sizes of animals of past geological epochs correspond to the sizes that they should have had according to the hypothesis.

It is known that if two planets of different mass and size have the same density, then the acceleration of gravity on their surfaces will be proportional to their mean radii. Hence, if one proceeds from those dimensions of the Earth that V. B. Neiman gives for it [7, p. 72], then it must be assumed that the acceleration of gravity on its surface should have been, in the Neogene, 0.82; in the Paleogene, 0.67; in the Cretaceous, 0.62; in the Triassic and Jurassic, 0.53; in the Upper Paleozoic, 0.46; and in the Lower Paleozoic, 0.34 of the present-day acceleration of gravity. Hence the mass of land animals resembling an elephant in external structure, according to formula (19), should have reached 9 tons in the Neogene, 16.5 tons in the Paleogene, 20 tons in the Cretaceous, 33.5 tons in the Jurassic and Triassic, 51.5 tons in the Upper Paleozoic, and 128 tons in the Lower Paleozoic.

In reality, however, the largest animal of the Oligocene—the indricothere (baluchithere), whose height from the soles of the forefeet to the withers exceeded 5 m, and whose length was 8 m [2, pp. 412–414]—should by mass have reached 20 tons. One of the ancestors of the modern elephant, Deinotherium, which lived in Miocene time, reached approximately the same dimensions. The Jurassic brontosaur, reaching 20 m in length and 5 m in height, should have weighed about 50 tons [14, pp. 120–139].

Thus, in all the cases considered, the actual sizes of the largest animals turned out to be greater than follows from the Kirillov–Neiman hypothesis. However, this is only half the trouble, for the Jurassic brontosaur may in fact have been a semiaquatic animal. Matters are worse when the question concerns the sizes of animals of the Upper and especially the Lower Paleozoic. Instead of being larger than Jurassic and later animals, they turn out to be smaller. Thus, the data on the sizes of animals of past geological epochs do not so much confirm the Kirillov–Neiman hypothesis as contradict it.

There is a fact that none of the hypotheses of an increase in the Earth’s volume can explain—the narrowing of the Atlantic Ocean as it goes on asegofi.

approaches the North Pole. In fact, its width between Africa and South America is approximately six thousand kilometers, whereas between Greenland and Scandinavia it is only about one and a half thousand. Meanwhile, if it had formed as a result of an increase in the Earth’s volume, its width would have been approximately the same along its entire length.

The situation is entirely different with the hypothesis of an increase in the Earth’s rotational speed, which was proposed in 1877 by E. V. Bykhanov in the book Astronomical Prejudices [1] and developed in detail by the author [10, 11, 12]. In particular, this hypothesis explains the narrowing of the Atlantic Ocean toward the North Pole by the fact that, as the Earth’s rotational speed increased, the most intensive displacement of the continents should have occurred in the equatorial region.

The same is true of giant animals. Namely, for the width of the Atlantic Ocean under the equator to reach 6 thousand km, it is necessary that the equatorial radius of the Earth lengthen by approximately 950 km. According to calculations carried out using Clairaut’s formula, such a lengthening can be achieved if the rotational speed is increased by a factor of 8.65 relative to the present one. In this case the acceleration of gravity at the equator should decrease to \(6.18 \, m/sec^2\) [10], which is 1.7 times less than the present value. Hence, according to formula (19), the maximum possible mass of animals in the epoch when the increase in the Earth’s rotational speed was replaced by the reverse process should have reached 20 t. The Eocene [11, 12] may be considered the epoch of the Earth’s maximum rotational speed. The Oligocene indricothere weighed about 20 t (see above). If now, in accordance with I. A. Efremov [3], Jurassic brontosaurs are considered semiaquatic animals, then it may be said that the agreement is satisfactory.

Comparing the atmospheres of different planets with one another, one may note that they prove to be denser the greater the mass of the planets. Thus, the Moon has no atmosphere at all; Mercury’s atmosphere amounts to no more than \(3 \, g/cm^2\); the atmospheric pressure on Mars reaches 5–20 mm of mercury; the atmosphere of Venus is tens of times denser than the Earth’s atmosphere. Finally, the atmospheres of the large planets—Uranus, Neptune, Saturn, and Jupiter—are so extensive that they significantly reduce their mean densities, which turn out to be close to the density of water. The reason for this phenomenon lies in the fact that the escape of gases from the surfaces of planets occurs the more intensely, the lower the escape velocity relative to the given planet [9, pp. 282–283]. And this regularity, theoretically substantiated and confirmed by factual data for most planets, proves to be violated in the case of our planet. Namely, its mass is greater than the mass of Venus, while the density of its atmosphere, on the contrary, is lower.

This paradox turns out to be quite explicable if one agrees that 50–70 million years ago the Earth’s rotational speed exceeded the present one by a factor of 8.65, and its equatorial radius thereby reached 7300 km. In fact, the linear velocity of motion of particles located on its equator, in such a case, should have reached approximately \(4.47 \, km/sec\); hence, in order to attain the escape velocity (\(11.2 \, km/sec\)), gas particles would have had to acquire an additional velocity of only about \(6.73 \, km/sec\). For comparison, let us note that the escape velocity relative to Venus is approximately \(10 \, km/sec\), and relative to Mars approximately \(5 \, km/sec\).

M. Schwarzbach [13, ...] arrives at the conclusion that in the Mesozoic and Paleozoic the Earth’s atmosphere was denser than at the present time,

p. 137] and other scientists, proceeding from the sizes of the wings and the mass of the inhabitants of the air environment of that time.

Considering the development of the Earth’s atmosphere, J. P. Kuiper [5] notes that of the original amount of neon (atomic weight 20), apparently, one hundred-billionth remains; of the heavy gases—krypton (atomic weight 84) and xenon (atomic weight 131)—one ten-millionth. The question arises as to how the Earth could have lost almost its entire gas envelope of the protoplanet. In J. P. Kuiper’s opinion [5, p. 25], “for a non-rotating Earth” the solution of this problem is impossible.

An analogous conclusion follows from E. Ferguson’s data on the problem of helium. Namely, proceeding from the reserves of uranium and thorium and the rate of their decay, he finds that the quantity of helium contained in the Earth’s atmosphere could have formed in only a few million years. The quantity that could have formed over the entire history of the Earth must exceed the amount present by approximately 1000 times. From this E. Ferguson concludes that “the Earth’s atmosphere was in some way devastated (emptied) of helium several million years ago” [15].

Thus, the latest data on the study of the composition of the atmosphere not only do not contradict the hypothesis of an increase in the Earth’s rotational speed, but, on the contrary, confirm it. However, a new question arises: what could have caused an 8.65-fold increase in the Earth’s rotational speed? In earlier works on this question [11, 12] it was assumed that in the past the Earth had a second satellite, Perun, which at first slowly approached and then joined it. Its approach began in the Devonian, and the joining occurred at the end of the Upper Cretaceous. The territory covered by its fragments subsided and formed a gigantic depression of the Earth’s surface—the Pacific Ocean. The level of the world ocean immediately after this event dropped sharply, as a result of which submarine canyons began to form everywhere, and all the continents except Australia turned out to be connected with one another. Subsequently, however, the sialic rocks buried beneath the fragments of Perun, which contained much bound water, metamorphosed and transformed this water into a free state. Thanks to this, the level of the world ocean rose again and eventually reached its present position.

LITERATURE

1. Astronomical Prejudices and Materials for the Compilation of a New Theory of the Formation of the Planetary System (author E. V. Bykhanov). Livny, 1877.

2. L. Sh. Davitashvili. A Short Course in Paleontology. Gosteoltekhizdat, 1958.

3. I. A. Efremov. Problems in the study of dinosaurs. Priroda, No. 6, 1953.

4. I. V. Kirillov. The hypothesis of the development of the Earth, its continents and oceanic basins. Bulletin of the Moscow Society of Naturalists, Geological Section, vol. XXXIII, issue 2, 1958.

5. J. P. Kuiper. Origin, Age, and Possible Ultimate Fate of the Earth. Collection Planet Earth (translated from English). IL, 1963.

6. E. N. Lyustikh. Some remarks on the use of physics in geotectonic constructions. Izvestiya of the USSR Academy of Sciences, Geological Series, No. 1, 1962.

7. B. B. Neiman. The Expanding Earth. Geografgiz, 1962.

8. Ya. I. Perelman. Entertaining Mechanics. Gostekhizdat, 1951.

9. P. I. Popov, K. L. Baev, B. A. Vorontsov-Velyaminov, R. V. Kunitskii. Astronomy. Uchpedgiz, 1940.

10. L. A. Pukhlyakov. Basic Provisions of the Hypothesis of Earth Deformations. Asino, 1957.

11. L. A. Pukhlyakov. On the question of the origin of mountains and oceans (the hypothesis of an increase in the Earth’s rotational speed). Proceedings of the Tomsk Polytechnic Institute, vol. 120, 1961 (1962).

12. L. A. Pukhlyakov. On the question of the origin of the Pacific Ocean. Proceedings of the Tomsk Polytechnic Institute, vol. 127, issue 2, 1965.

13. M. Schwarzbach. Climates of the Past (translated from German). IL, 1955.

14. A. A. Yakovlev. Life of the Earth. Gostoptekhizdat, 1949.

15. What happened to the Earth’s helium? New Scientist, vol. 24, No. 420, 1964.