1. Celestial mechanics from the Hipparchus to Le Verrier

Celestial mechanics has an ancient and distinguished history. It was continually refined over the centuries as the most precise of the physical sciences, until the development of modern physics as we know it today following the Galilean and Newtonian revolutions. Prime motivations for these endeavours were originally to keep track of time so that religious festivals could be celebrated on the correct dates and to maintain a track of time and latitude for the purposes of navigation, which will prove to be a significant aspect of this story.

1.2. Perturbations of the motions of the planets

A central theme of celestial mechanics concerns the perturbations to the orbits and axes of rotation of the planets due the gravitational attraction of other celestial bodies, such as planets, satellites and comets. The understanding of the major gravitational perturbations was of practical importance for the purposes of navigation and for maintaining an accurate track of time. Perturbations to the elliptical orbits of the planets needed more advanced mathematical tools than Newton's laws in their original form. It was known that there is no general closed-form solution to the problem of three bodies interacting under gravity, the intractable three-body problem – the orbits are generally chaotic (Murray & Dermott Reference Murray and Dermott2000). The first generally accessible account of the nature of the three-body problem was provided by Leonhard Euler (1707–1783) in his 1749 prize solution for the ‘theory of Jupiter and Saturn, explicating the inequalities that these planets appear to cause in each other's motions’.

In its most general form, the shape and orientation in space of a planet's elliptical orbit are described in terms of a planet's orbital elements – six of them are required. For the case of the Earth and the other planets, the orbits are taken to be ellipses assuming that the planets can be represented by point masses and in the absence of perturbations. The six elements are taken to be:

1. (1) the semi-major axis of the ellipse a;

2. (2) the eccentricity of the ellipse e;

3. (3) the inclination of the orbit to the plane of the ecliptic i;

4. (4) the longitude of the ascending node needed to fix a reference point on the Earth's orbit;

5. (5) the longitude of perihelion of the planet's orbit Ω;

6. (6) the time at which the planet passed perihelion T.

Superimposed upon these is the obliquity, or axial tilt ɛ, of the Earth's equatorial plane to the ecliptic plane, defined by the angle between these planes, with north defined by the right-hand rule. The longitude of the ascending node is the longitude at which the Earth's orbit crosses the equatorial plane, corresponding to the epoch of the vernal equinox, providing a reference point on the Earth's orbit about the Sun. The challenge to the celestial mechanicians was that all six orbital elements and the precession of the equinoxes of any planet are perturbed by the presence of all the other planets and bodies in the solar system. This proved to be a major mathematical challenge, but by the late 18th and 19th Centuries, major advances in analysis had been made with contributions from many of the greatest mathematicians, the majority belonging to the French and German schools of mathematical analysis. This can be appreciated from the French and German names of so many of the theorems and functions in analysis and calculus. From Germany, Gauss, Bessel, Jacobi, Dirichlet, Riemann, Kronecker, Weierstrass, Hilbert and Weyl; from France, d'Alembert, Lagrange, Laplace, Legendre, Fourier, Poisson, Cauchy, Liouville, Galois, Hermite, Poincaré, Hadamard and Lebesgue, among many others.

The deviations of the orbits of the planets from perfect ellipses are small and so perturbation theory is the obvious way of evaluating these changes. We will find that the ellipticity of the Earth's orbit e varies from about 0.02 to 0.078. Thus, to obtain accurate predictions about the small perturbations of the planetary orbits, it is convenient to expand the functions describing the orbits to high orders in the ellipticity, e, e ², e ³, e ⁴ …. Euler showed that the eccentricities of the planetary orbits are invariant to second-order, e ², in small quantities. The effects of the presence of the other planets were to be found to higher order in the eccentricity.

Building on the work of Euler, Joseph-Louis Lagrange (1736–1813), in his Traité de Mécanique Analytique (Lagrange Reference Lagrange1788), transformed mechanics into a branch of mathematical analysis. In the Preface to the Treatise, he wrote: ‘One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course’ (Lagrange Reference Lagrange1788, preface, page 1). In 1786, Pierre-Simon Laplace (1749–1827) had also proved that the eccentricities and inclinations of planetary orbits to each other always remain small, constant and self-correcting. These and many of his earlier results formed the basis of his great Traité de Mécanique Céleste (Laplace Reference Laplace1799–1825) published in five volumes, the first two in 1799. These contained an extended exposition of perturbation theory and its detailed application to the motion of the Moon and the planets. He demonstrated the need to extend the perturbations to high powers of the eccentricity e. He also resolved anomalies in the motions of Jupiter and Saturn as a result of a resonance between the 2:5 ratio of the periods of their planetary orbits.

In 1795 the French Bureau des Longitudes in Paris, charged with the improvement of nautical navigation, standardisation of time-keeping, geodesy and astronomical observation, was founded with Lagrange and Laplace as the mathematicians, or geometers, among its founding members. Laplace subsequently became head of the Bureau and the Paris Observatory, which was to be responsible for producing formulae and predictions of the perturbations to the motion of the planets, which were to be used by Croll. The reason for the founding of the Bureau was entirely political, to counter the domination of the UK in mastery of the seas and its overseas territories.

William Rowan Hamilton (1805–1865), professor of astronomy at Trinity College, Dublin, Director of the Dunsink Observatory and Astronomer Royal for Ireland, presented his equations in the Philosophical Transactions of the Royal Society in 1834 and 1835 (Hamilton Reference Hamilton1834, Reference Hamilton1835). This developed into the theory of Hamiltonian mechanics and dynamics, in particular, into a pair for first-order dynamical equations replacing the second-order equations of the Euler–Lagrange approach:

$$\dot{q}_i = [ {H, \;\;q_i} ] \quad\;\dot{\!p}_i = [ {H, \;\;p_i} ] $$

H is the Hamiltonian and q _i and p _i the canonically conjugate spatial and momentum coordinates. The square brackets are Poisson brackets. Whereas the Euler–Lagrange equations require the integration of second-order differential equations, both Hamiltonian equations, being first order, are mathematically simpler. Note that these great papers of Hamilton's came out of his studies of the perturbations of the planetary orbits, part of his responsibilities as Astronomer Royal for Ireland.

In 1834, Carl Jacobi (1804–1851) continued the development of Hamiltonian mechanics, showing how to transform the Hamiltonian formalism from one system of coordinates to another. The resulting Hamilton–Jacobi equations were applied to celestial mechanics in 1834. Consequently, coordinate systems that were the most convenient for astronomical orbital problems could be chosen – for example, the use of parabolic coordinates to study perturbations of the planetary orbits. This was, however, distinctly higher mathematics and at the forefront of mathematical innovation.

2. Urbain Le Verrier

Urbain Le Verrier (1811–1877), a central figure in Croll's scientific work, worked at the Paris Observatory for most of his life. In 1845, François Arago (1786–1853), Director of the Paris Observatory, asked him to study the irregularities in the orbit of Uranus. In June 1846 he proved that the irregularities could not result from perturbations by the known planets and postulated that there must be an undiscovered planet beyond the orbit of Uranus. By August 1846, he predicted the orbit of the hypothetical planet.

On 18 September 1846, Le Verrier wrote to Johann Galle (1812–1910) at the Berlin Observatory asking him to search for the planet at its predicted position. Galle stated that it was: ‘… so novel a thing to undertake observations in reliance upon merely theoretical deductions; and that, while much labour was certain, success appeared very doubtful’ (Robertson & O'Connor Reference Robertson and O'Connor2021). Galle searched for the new planet on the same evening he received Le Verrier's letter. He found it that night within one degree of the predicted position. He reobserved it on the following night to be absolutely certain that it was a planet, writing to Le Verrier: ‘The Planet whose position you indicated really exists’ (Robertson & O'Connor Reference Robertson and O'Connor2021).

As one of Le Verrier's colleagues wrote: ‘… he discovered a star with the tip of his pen, without any instruments other than the strength of his calculations alone’ (Robertson & O'Connor Reference Robertson and O'Connor2021). Arago stated: ‘In the eyes of all impartial men, this discovery will remain one of the most magnificent triumphs of theoretical astronomy, one of the glories of the Académie and one of the most beautiful distinctions of our country’ (Robertson & O'Connor Reference Robertson and O'Connor2021). The discovery of the planet, which was named Neptune, was a magnificent achievement for the science of celestial mechanics, proving the correctness of the application of advanced mathematics to physical problems. This triumph of mathematical analysis helped make the public and governments aware of the importance of scientific research.

In 1847, Le Verrier began the development of the huge project ‘of embracing in a single work the whole of the planetary system.’ The aim was to construct the theoretical apparatus and tables for the motions of the planets and the determination of their masses taking into account all their mutual gravitational interactions. This work, which he did not complete until a month before his death in 1877, occupies more than 4000 pages of the Annales de l'Observatoire de Paris.

Le Verrier began by establishing the expansion of the perturbing forces in their most general form. The expansions extended to the seventh power of the eccentricities, e ⁷, and inclinations and included 469 terms dependent on 154 special functions. This work was the basis for his later investigations. He then turned to the first four planets, from Mercury to Mars, the theory and tables of which he completed in 1861. Le Verrier's early work on this project, as it was published, was used by Croll (Reference Croll1864, Reference Croll1875).

In 1854, Le Verrier became Director of the Paris Observatory which, under Arago's leadership, had not performed to the standards that Le Verrier expected. On taking over as director, he immediately insisted on strict discipline, his drive for efficiency making him very unpopular. Staff could only work on the projects assigned to them and all publications went out under the name of the Director. The new regime was strongly opposed by the staff, almost all of whom resigned. As stated by a contemporary: ‘I do not know whether M Le Verrier is actually the most detestable man in France, but I am quite certain that he is the most detested’ (Robertson & O'Connor Reference Robertson and O'Connor2021). These unhappy circumstances are reflected in the title of James Lequeux's biography: Le Verrier: Magnificent and Detestable Astronomer (Lequeux Reference Lequeux2015).

Le Verrier hoped that his work would lead to further planetary discoveries analogous to that of Neptune. He already knew by 1843 that the observed motion of Mercury did not agree with theory. In 1859, he showed that the perihelion of Mercury advances at a rate of about 43 arcsec per century, once the perturbing effects of the known planets were taken into account. He suggested that this might be due to the presence of a planet, Vulcan, within the orbit of Mercury. Despite searches, no such planet was observed. No satisfactory explanation was found until 1915, when Albert Einstein showed that, according to general relativity, the perihelion of Mercury should advance at a rate of about 43 arcsec per century because of the distortion of space-time by the Sun. Le Verrier's analysis turned out to be one of the key observations establishing the correctness of Einstein's general theory of relativity.

3. Croll on Le Verrier

By the time Croll began his studies of the influence of perturbations of the Earth's motion because of the presence of the other planets, the first results of Le Verrier's programme had become available and these were the basis for Croll's research. His motivation was to use the inferred past history of the Earth's motion as a means of providing secure dates for the geological history of the Earth through the dating of the ice ages. In his pioneering works of 1864 and 1867, he described the importance of both the changing ellipticity of the Earth's orbit and the precession of the equinoxes as contributors to climate change and the origin of the ice ages (Croll Reference Croll1864, Reference Croll1867a, Reference Crollb). His aims are summarised in the introduction to Chapter 19 of his influential book of 1875, ‘Climate and Time in their Geological Relations: a Theory of Secular Changes of the Earth's Climate’ (Croll Reference Croll1875). Many more details are provided by the other essays in this volume. The introductory paragraph reads as follows:

Croll used Le Verrier's expression for the variation of the eccentricity of the Earth's orbit about the Sun due to the gravitational perturbations of the planets and includes it as a footnote in his treatise (Croll Reference Croll1875, p. 312). He used the expressions contained in Le Verrier's Memoir ‘Connaisance des Temps’ of 1843 (Le Verrier Reference Le Verrier1843) (Fig. 1).

Figure 1 Le Verrier's expressions for the time-evolution of the eccentricity of the Earth's orbit about the Sun.

Croll used the expression in Fig. 1 to work out the values of the eccentricity of the Earth's orbit every 50,000 years from −3 M years before the present era to 1 M years into the future. The period from −700,000 years to the present is compared with present estimates in Fig. 2. The agreement is remarkably good, attesting to the quality of Le Verrier's and Croll's analyses.

Figure 2 Comparison of Croll's estimate of the variation of the ellipticity of the Earth's orbit about the Sun according to the expression in Fig. 1 (top) with present estimates (bottom) for the last 700,000 years. To make the comparison, Croll's graph has been flipped about the vertical axis. The vertical lines show the present epoch and 700,000 years in the past. (Source: Incredio, courtesy of creative commons: https://commons.wikimedia.org/w/index.php?curid=6930545.)

Croll drew particular attention to epochs when the ellipticity of the Earth's orbit had extended maxima at epochs −2.5 M and −1.85 M years in the past and about 1 M years in the future. These were identified as periods when the ice ages occurred, and will occur in the future. Quoting Fleming (Reference Fleming2006, p. 37),

This model is presented in the frontispiece to Croll's book (Croll Reference Croll1875) (Fig. 3). Le Verrier had shown that the maximum eccentricity of the Earth's orbit was 0.07775 and this was the value used by Croll on the coloured frontispiece and also as the upper bound of the eccentricity in Fig. 2. During the period considered by Croll, an eccentricity of 0.0722 was inferred at the epoch −2.4 M years before the present epoch, not much less than the maximum value. At maximum eccentricity, there is a 17% change in the distance from the Sun from perihelion to aphelion, corresponding to a 36% change in insolation.

Figure 3 The frontispiece of Croll's book, Climate and Time in their Geological Relations: a Theory of Secular Changes of the Earth's Climate (1875). The list of figures in Croll's book states ‘Earth's orbit when eccentricity is at its superior limit’. These values refer to the maximum ellipticity predicted by Le Verrier (see Croll's book). The dynamical parameters have been superimposed on the diagram for clarity.

This is only the beginning of a complex story. Croll fully appreciated that this is only one part of a whole series of climatic interactions between variations in the heat input to the atmosphere and the resulting climatic phenomena. Feedback mechanisms include:

• • radiative effects of the ice fields;

• • enhanced formation of cloud and fog;

• • changes in sea level and the mixing and redirection of warm and cold ocean currents.

These would enhance the climatic changes initiated by small changes in the Earth's orbit and inclination. As Croll wrote:

These were controversial claims at the time, and it was not until the revival of interest in these topics by Milutin Milanković in 1930 that the theory began to be taken seriously again. The reasons for the eclipse of Croll's theory are summarised by Fleming (Reference Fleming2006, p. 50): ‘However, because of uncertainties in the timing of ice ages and in the stratigraphic record, and because Croll's theory predicted glaciation in only one hemisphere, the theory was largely disregarded for at least three decades.’ The papers in this volume provide many more details of the convoluted history of the reception of Croll's pioneering research, which influenced the thinking of a number of the most distinguished scientists of the age involved in these studies, including Charles Lyell, John Herschel, Charles Darwin and the brothers Archibald and James Geikie.

I conclude this section with a tribute to my colleague Nick Shackleton whose pioneering ice core analyses of 1976 with J. D. Hays and John Imbrie showed convincingly the impact of variations in the Earth's orbital eccentricity, its obliquity and the precession of the perihelion upon climate change in their paper, ‘Variations in the Earth's orbit: pacemaker of the ice ages’ (Hays et al. Reference Hays, Imbrie and Shackleton1976). Using Fourier techniques, they showed that the three periodicities with which the Earth's orbit change (100,000 years, 40,000 years and 21,000 years) are all present in the temperature, isotopic and fossil records. Figure 4 illustrates the various perturbations of the Earth's dynamics and their relative amplitudes, as well as recent measurements of indicators of climate change such as carbon dioxide concentration and global temperature measures.

Figure 4 Comparison of the variations of orbital parameters with various measures of the global temperature for the epochs −800 to +800 Kyears. From the top, the seven graphs show: ε the obliquity of the Earth's orbit; e the eccentricity of the Earth's orbit; Ω the precession of the perihelion; the combined impact of the variations of the ellipticity and the precession of the perihelion; the mean daily insolation; a measure of the atmospheric mean carbon dioxide variations in the Earth's atmosphere; and the mean global temperature variations. (Source: Incredio, courtesy of creative commons: https://commons.wikimedia.org/w/index.php?curid=6930545.)

4. The ages of the Sun and Earth

In the second half of the 19th Century, astronomical and geological estimates of the age of the Earth came into conflict (Longair Reference Longair2006). Geological estimates of the age of the Earth were made by James Hutton (1726–1797) in 1785 and Charles Lyell (1797–1875) in 1830 based on geological evidence. Hutton claimed an age of many millions of years based on erosion and sedimentation rates, while Lyell, a good friend of Croll's, suggested at least 500 million years from sedimentation rates in the oceans. But Croll was cautious. As he wrote in his comprehensive book of 1875,

The origins of the theory of stellar structure and evolution can be traced to the understanding of the first law of thermodynamics in the early 1850s by Rudolph Clausius (1822–1888) and William Thomson, later Lord Kelvin (1824–1907), ‘energy is conserved when heat is taken into account’. Applying the law to the stars, Hermann von Helmholtz (1821–1894) and Thomson proposed that the contraction of the Sun itself would provide an enormous reservoir of potential gravitational energy. The Sun was conceived of as a liquid sphere which gradually contracted and cooled. Energy transport through the Sun was assumed to be by convection.

Kelvin and Helmholtz could then estimate the age of the Sun since its present luminosity is known and its gravitational potential energy can be estimated. This Kelvin–Helmholtz time-scale t _KH for cooling of the Sun can be estimated by dividing its gravitational potential energy by its present luminosity L,

$$t_{{\rm KH}}\approx \displaystyle{E \over L}\approx \displaystyle{{{\it G}{\it M}^2} \over {{\it LR}}}$$

where M is its mass and R its radius. For the Sun, this time-scale is about 10⁷ years. Thomson used heat flow arguments to estimate the age of the Earth. He knew the temperature gradient in the outer layers of the Earth and estimated how long it would take the Earth to cool by thermal conduction through its surface. He found an age of 20–40 million years – not so different from the Kelvin–Helmholtz time-scale for the Sun.

The astrophysical estimates were considerably shorter than those favoured by the geologists who suggested that the age of the Earth was greater than 100 million years, over 500 million years according to Lyell, but these estimates were subject to some uncertainty, as noted by Croll. Kelvin's changing estimates of age of the Sun and the Earth are set out in some detail by Smith & Wise (Reference Smith and Wise1989) in their biography of William Thomson, Energy and Empire: A Biographical Study, eventually settling down to values of about 20 to 40 million years.

Everything changed with the discovery of radioactivity in 1895 by Henri Becquerel (1852–1908). Here was a continuing source of energy for the interior of the Earth and the Sun and so the cooling estimates could no longer be maintained. Even more remarkably, in 1904, Ernest Rutherford (1871–1937) used the relative abundances of long-lived radioactive and stable isotopes of heavy elements such as uranium to estimate the age of the Earth. His estimate was at least 700 million years, much to the relief of the geologists. Rutherford announced his first results in 1904 in his Friday Evening Discourse at the Royal Institution. In his reminiscences, he remembers:

The first person to investigate the internal structure of the Sun as a gaseous, rather than a liquid, body was J. Homer Lane (1819–1880). During the American Civil War, he made the first calculations to determine whether or not the Sun's surface properties were consistent with it being considered to be a sphere of a perfect gas. In 1869, he reversed the calculations and, assuming the material of the Sun to be a perfect gas, attempted to reproduce its surface properties and to determine the variation of its density, temperature and pressure with radius. This programme was not particularly successful. Nonetheless, he was the first person to adopt the correct equations of hydrostatic equilibrium and mass conservation for the stars.

In the late 1870s, similar calculations were carried out by Augustus Ritter (1826–1908) at Aachen, who identified the initial phase of evolution of the star as the contraction of a perfect gas sphere, which then cooled according to the Kelvin prescription. The culmination of these early physical models for stars was the treatise by Robert Emden Gaskugeln, published in 1907. Emden (1826–1908) went further than Lane and Ritter by introducing polytropic solutions, which allowed a much wider range of stellar models to be investigated. The equation of state of the gas was assumed to be of power-law form p = κρ^γ. Emden showed that the solutions result in stars with a boundary at a finite radius. These theories did not gain the recognition they deserved because astronomers argued that the physical conditions inside the stars and the process of energy generation were unknown.

The solution of the problems of understanding the energy source of the Sun had to await the possibility of nuclear energy generation following Einstein's discovery of the mass-energy identity E = mc ² and accurate estimates of the mass deficit in the conversion of hydrogen to helium, proposed by Eddington (1882–1944) in 1920. This process ensured that the lifetime of the Sun would be about 10¹⁰ years. Likewise, internal heat sources in the Earth removed the second of Kelvin's arguments. In a letter to me in 1993, William McCrea (1904–1999) wrote: