George Howard Darwin (1845-1912)
and the Birth of the Moon
Robinson M. Yost
For thousands of years humans contemplated the heavens and looked for answers. How had the stars and planets come to be? Were the celestial bodies created by a god or gods, or had they always existed? Although the answers were diverse, it was not until the 17th and 18th centuries that naturalistic explanations emerged for the origins of our solar system. By the mid-19th century, those pondering cosmological questions brought various scientific ideas to bare on the answers. An example is the work of the English mathematical astronomer, George Howard Darwin. In particular, Darwin hypothesized on the genesis of the earth's moon. Many historians of astronomy have described his idea, but few have examined its development over time. What was Darwin's idea, where did it come from, and how did it change? Darwin's hypothesis, as we shall see, was inextricably linked with wider cosmological concerns.
George Howard, the second son of Charles and Emma Darwin, graduated from Trinity College, Cambridge in 1868 finishing Second Wrangler and second Smith's Prizeman. In 1873, Darwin abandoned a career in law for the life of a Cambridge academic. Elected a Fellow of the Royal Society in 1879, Darwin gained the Plumian Chair of astronomy four years later. Darwin won numerous academic honors including the Royal Society's Copley Medal in 1911 for his contributions to tidal research. He also served as the president of several scientific societies including the British Association for the Advancement of Science and Royal Astronomical Society.
Sympathetic to evolutionary ideas from early in life, Darwin was connected with his father's work: editing the 2nd edition of Descent of Man in 1874 and authoring several of his own articles on evolutionary-related topics. Evolution remained a theme as he shifted emphasis from living to inanimate matter. In Darwin's eyes, both biologist and physicist searched for modes of stability and instability in their subjects. However, since the conditions of the living world were much more complex, the biologist usually had to settle for qualitative conclusions. The physicist, on the other hand, was not satisfied until he obtained quantitative results. This preference for the quantitative remained a recurring theme as Darwin investigated the history of the earth, moon, and planets.
Several scientific traditions blended to form essential ingredients of Darwin's thought. One of them was mathematical physics. In 1742, Scottish mathematician Colin Maclaurin generalized Sir Isaac Newton's result on the shape of the earth. Maclaurin demonstrated that an ideal fluid rotating uniformly about a fixed axis formed increasingly flattened spheroids as the rotational velocity was increased. Almost a century later, Carl Jacobi found another possible figure of equilibrium with three unequal axes. As a Cambridge undergraduate, Darwin learned the mathematics of rotating fluids and other hydrodynamical ideas from Lucasian professor, George Gabriel Stokes.
Idealized mathematical solutions were often linked to the study of the earth's formation, interior and age as well as other problems in physical geology. Debate persisted in the 19th century over the processes by which a molten primeval earth had solidified, contracted, and cooled. Most geologists argued for a thin crust of 50 miles or less to explain surface activities such as volcanoes and earthquakes. However, in 1839, William Hopkins combined mathematical analysis with chemical and astronomical data to demonstrate that the earth must have a crust of 800 to 1000 miles thick. By the 1860s most "solidists", as they were called, followed the lead of physicist William Thomson. Using astronomical, thermodynamical and geophysical evidence, Thomson argued for a virtually solid, rigid earth around 100 million years old. His views profoundly influenced a generation of physicists including the young George Darwin. In fact, Darwin later regarded his own work on tidal friction and cosmology as the outcome of a conversation with Thomson in 1877.
Interdisciplinary "geophysical" problems were often intertwined with cosmology. In 1796 Simon-Pierre Laplace had proposed that the sun formed from the center of a hot rotating cloud of gas, while the outer gaseous layers cooled and shed discrete rings. These rings eventually condensed into smaller planetary nebulae. Analogously, the planetary clouds condensed and shed rings, becoming planets and their satellites. The "nebular hypothesis", as it was called, fit well with the astronomical observations of William Herschel. John Pringle Nichol, William Whewell, Herbert Spencer and others popularized modified Laplacian ideas and by the 1860s the nebular hypothesis was firmly-established in British thought. Laplace had given plausible explanations for the origins of the solar system, but his descriptions remained primarily qualitative. Darwin and others attempted to quantify the nebular hypothesis. In the process, they blended the mathematical, geophysical, and cosmological.
III. The Fission Hypothesis
Lunar origins were not the original impetus behind Darwin's early work. In 1878 Darwin noted the great variety of observed planetary inclinations or obliquities, something inexplicable in terms of the nebular hypothesis. In his speculative attempt to explain how obliquities could change (and thereby precession), questions arose with respect to the stability of a rotating nebula and how (and when) it would shed rings. With these ideas in mind, Darwin set out to explain changes in precession by finding the mass-motion of viscous and imperfectly elastic spheroids produced by external disturbing bodies. Mathematical exigencies required simplifying the initial case to three bodies- the earth, sun and moon. The ultimate aim, however, was applying the end results to cosmological evolution.
Darwin's theoretical analysis described the influences of the sun and moon on the bodily tides of earth. He illustrated, for example, that there are a variety of positions of dynamical equilibrium (some stable, others unstable), depending on the obliquities and viscosities of the system. In every case, however, the moon slowly retarded the earth's rotation because tidal friction invariably acted as a braking mechanism. Tidal friction was earlier investigated by Laplace explaining the moon's secular acceleration, but it was Darwin who worked out its evolutionary consequences for the earth-moon system. Conservation of angular momentum and Newton's 3rd Law required that the earth exercise a counteracting force on the moon. In the distant future then, the moon would gradually be "pushed" away from the earth.
Darwin also traced the system into the past. A complete analytical solution was not possible, so he combined analytical with numerical methods. In the first solution, he fixed the degree of terrestrial viscosity and plugged in present values of year, day, month and obliquity to sets of differential equations. Darwin found that at a minimum of 54 million years in the past, the obliquity, day, and month were dramatically less than at present and the moon was considerably closer to the earth. Since a molten earth would cool and stiffen with time, Darwin's second attempt used variable viscosity. The new solution did not indicate elapsed time, but showed that the earth and moon were much closer in the very distant past. Darwin reported:
This periodic time of the moon of 5 hours 40 minutes corresponds to an interval of only 6000 miles between the moon's centre and the earth's surface... The conclusion, therefore, appears to me almost irresistible, that if the moon and earth were ever molten viscous bodies, then they once formed parts of a common mass.
Nevertheless, he warned:
A whole series of problems, some of them of great difficulty, still await solution; and not until they are solved will it be possible either decisively to accept or reject the modified form of the nebular hypothesis, to which my results obviously point.
Perhaps the moon and other satellites had not coalesced from a Laplacian ring. The fission hypothesis was born, yet the process of lunar separation remained mysterious. In 1880 Darwin wrote:
The planet... separates into two masses, the larger being the earth and the smaller the moon. I do not attempt to define the mode of separation, or to say whether the moon was initially more or less annular.
IV. The Pear-shaped Figure
Over the next 25 years, Darwin refined, modified and extended the fission hypothesis. For instance, in 1881 he reported that the satellites of Mars, Jupiter and Saturn could not be traced back to near contact. The case of the earth and moon "differed considerably", and was seemingly unique in the solar system. He concluded that modifications of "considerable importance" had been introduced to the nebular hypothesis.
After working on practical tidal problems during the mid 1880s, Darwin returned to the fission hypothesis. In 1887 his intent was:
to obtain a representation of the single form of equilibrium which must exist when two masses approach so near to one another as just to coalesce into a single mass.
Darwin found that as equal masses approached each other they became shaped like "flattened eggs." In one case, the smaller elongated ends actually overlapped, becoming a single mass of fluid. Darwin was not alone in proposing this "dumb-bell" shaped figure of equilibrium.
In his 1885 elaboration of William Thomson's discussion of rotating equilibrium figures, Jules Henri Poincaré concluded that a Jacobian ellipsoid could transform into a dumb-bell shape with unequal bulbs. The pyriform or pear-shaped figure came as a "revelation" to Darwin who had approached the same problem from the other end. When it came to applying Poincaré's work to the fission hypothesis, difficulties remained. The most pressing was determining if the pyriform was a stable figure of equilibrium. If the pyriform were never stable, it would undergo drastic deformations followed by a catastrophic disintegration. Therefore establishing stability was crucial to the fission hypothesis.
In 1900 Darwin, as the president of the Royal Astronomical Society, presented Poincaré with the Society's gold medal and summarized the 1885 paper. Poincaré had said that mechanical systems in equilibrium could be arranged in families according to some measurable quantity (e.g., Maclaurin spheroids arranged by their angular velocities). Other systems could form families with completely different configurations (e.g., the Jacobian ellipsoids), but sometimes a member of one family closely resembled that of another. This was called a bifurcation point because the form belonged to both families. In passing through a bifurcation there would be an "exchange of stabilities" between the two families. One branch becoming unstable and the other taking on stability.
Darwin applied this "exchange of stabilities" to the lunar separation process. Hence, a particular Jacobian ellipsoid on the stable branch of bifurcation would acquire an unsymmetrical furrow becoming pear shaped. The furrow would go on deepening and at last divide the shape into two separate bodies. This process depended on the initial stability of the pear shape. In 1901, Darwin addressed the stability question utilizing a method developed in an earlier paper. He concluded: "The pear-shaped bodies are almost certainly stable, but a rigorous and conclusive proof is wanting..." Poincaré reached similar conclusions using different methods. Further approximations strengthened Darwin's conviction in pyriform stability. In 1902 he remarked:
I cannot restrain myself from joining [Poincaré] in seeing in this almost life-like process a counterpart to at least one form of the birth of double stars, planets, and satellites.
V. Criticisms and Conclusions
While Darwin addressed pyriform stability, others challenged the validity of the nebular hypothesis. In 1900 American astronomer F. R. Moulton and geologist T. C. Chamberlin launched an attack against Laplacian cosmology. They did this by persuasively presenting numerous criticisms, some dating back to the mid-19th century. Moulton wrote:
[The nebular hypothesis] has been accepted with a few slight modifications almost without question by the highest authorities, as Helmholtz, Kelvin, Newcomb, and Darwin, and it has been made the basis for the most sweeping conclusions. It has held such a dominant sway over the thoughts of investigators... that doubtless in many instances facts have been warped and perverted, and questionable methods of reasoning employed...
By 1905 Chamberlin and Moulton had abandoned the nebular hypothesis in favor of their own planetesimal hypothesis. This idea (and the tidal theories of English physicists James H. Jeans and Harold Jeffreys) depended on a massive star passing near the sun and pulling out material which accreted forming the planets and satellites. These new cosmologies were plausible alternatives to the nebular hypothesis because they avoided some of its pitfalls. Although aware of the criticisms, Darwin continued to support the nebular hypothesis. In 1905 he wrote, "enough has been said to show that the Nebular Hypothesis cannot be considered as a connected intelligible whole." Nonetheless, he said, "we still have to rely on such theories as that of Laplace for the explanation of the main outlines of the solar system."
That same year Russian mathematician Aleksandr Lyapunov proved that the pyriform figure was always unstable. Darwin remarked:
Liapounoff now stated that he is able to prove the pear-shaped figure to be unstable from the beginning. For the present at least I still think it is stable, and this belief receives powerful support from Mr. Jeans' researches.
A former pupil of Darwin's, Jeans had investigated the stability of an infinite rotating liquid cylinder and found that a pyriform cross section of the cylinder was stable. In 1911 Darwin remained convinced writing: "According to my calculations this series of figures is stable..."
After Darwin's death in 1912, doubts continued to grow. In 1916 Jeans noted:
I was able to discover a quite simple error which invalidated Darwin's discussion of the problem. ...I confirmed the conclusion already reached by Liapounoff, that the pear-shaped series was initially unstable...
Harold Jeffreys supported the fission hypothesis as late as 1924, but in 1930 his new researches were "sufficient to the invalidate the theory." The fission hypothesis, linked to an increasingly questionable nebular hypothesis and fraught with its own problems, was no longer widely accepted by the 1930s.
As we have seen, G. H. Darwin's ideas were a complex mixture of mathematics, geophysics and cosmology. Initially, he had worked on a theoretical problem typical for a Cambridge Wrangler. Eventually, Darwin sought to quantify aspects of a problem that had remained primarily qualitative, the nebular hypothesis. This led to his theory of evolution by tidal friction and the fission hypothesis. In doing the complex mathematical investigations, Darwin had concrete cosmological applications in mind. Challenges to his ideas were difficult to accept. In fact, Darwin's confidence was often great enough for him to reject ideas conflicting with his own. In this respect, he was merely one in a generation of astronomers (and other scientists) confronting emerging ideas as the twentieth century began.