Urbain Le Verrier

How to use this course

Urbain Le Verrier is famous for one evening. On 23 September 1846 Johann Galle in Berlin pointed a telescope where Le Verrier told him to look and found an eighth planet. That evening rested on a decade of calculation, and it was followed by thirty years of more. Le Verrier rebuilt the theory of every major planet, found the anomaly in Mercury’s orbit that general relativity would later explain, and founded the French weather service. He was also among the most disliked administrators in French science.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow his career: the secular theory of 1839–1840 and the algorithm it contained, the expansion of the disturbing function, comets (1844), Neptune (1845–1846), Mercury (1859) and the planetary tables (1858–1877). Numerical values in the exercises use modern constants unless stated otherwise.

Timeline

Saint-Lô and Caen, 1811–1831

Urbain Jean Joseph Le Verrier was born in Saint-Lô, in the Manche department of Normandy, on 11 March 1811. His father, Louis-Baptiste Le Verrier, was a minor official in the administration of state property; his mother was Marie-Jeanne-Joséphine de Baudre. The family was of modest means. He had an older sister, Léontine, who would keep house for him in later years.

He spent eight years at the college of Saint-Lô and then, from 1827 to 1830, studied mathematics at the Collège Royal de Caen, where he finished at the top of his class. In 1830 he sat the entrance examination of the École polytechnique and failed.

A second attempt

He spent the next year in Paris at the Institution Mayer, a preparatory school directed by the mathematician Choquet, and at the Collège Louis-le-Grand. In 1831 he placed second in mathematics in the national Concours général and was admitted to the École polytechnique. The year in Paris also gave him a family: in 1837 he married Choquet’s daughter, Lucile.

Polytechnique, tobacco and chemistry, 1831–1837

The École polytechnique trained engineers for the state services. Le Verrier graduated in 1833, ranked eighth, and chose the state tobacco administration, a monopoly that needed engineers for its manufactories. The post did not hold him. In 1835 he left it to pursue research.

His first research was in chemistry. In the laboratory of Joseph Louis Gay-Lussac he studied combinations of phosphorus with hydrogen and with oxygen and published papers in the Annales de chimie et de physique in 1835 and 1837. To support himself he taught mathematics at the Collège Stanislas and gave private lessons.

In 1837 he applied for a post as répétiteur (teaching assistant) in chemistry at the École polytechnique and did not get it. A post in astronomy and geodesy, attached to the course of Félix Savary, was offered instead. He accepted, and from that accident of appointments came his career. Within two years he was presenting memoirs on celestial mechanics to the Académie des sciences.

Turning to the sky, 1837–1846

Le Verrier began where Pierre-Simon Laplace and Joseph-Louis Lagrange had left off. Their theory of secular variations showed that the eccentricities and inclinations of the planets oscillate within bounds. In memoirs presented to the Académie in September and October 1839 and published in 1840, Le Verrier recomputed the theory with better masses, extended it to Uranus, and derived explicit limits for each planet. He found that Jupiter, Saturn and Uranus are stable without restriction in that approximation. To solve the linear system he devised a method for the characteristic polynomial of a matrix that mathematicians still teach (Lesson 05).

When Savary died in 1841, Le Verrier was passed over for his post at the École in favour of Michel Chasles. He went on working. In 1843 he presented a new theory of Mercury, whose existing tables failed against observation. Next he turned to comets: from 1844 he studied the orbits of the comets discovered by Hervé Faye and Francesco de Vico, and on 11 November 1844 he presented a theory of Lexell’s comet of 1770, whose orbit had been transformed by Jupiter (Lesson 07).

The commission from Arago

In 1845 François Arago, director of the Paris Observatory and permanent secretary of the Académie, asked Le Verrier to examine the motion of Uranus. Alexis Bouvard’s tables of Uranus, published in 1821, had drifted from observation by an amount that could no longer be blamed on observers. Le Verrier presented his first memoir on 10 November 1845. It rebuilt the perturbations of Uranus by Jupiter and Saturn and showed that they could not explain the residuals. In January 1846, before the decisive memoirs, he was elected to the astronomy section of the Académie. Lesson 08 follows the calculation to its end.

Secular variations and the stability of the solar system

For the general reader. Planets tug on each other. Most tugs average out over a few orbits, but some do not: they slowly turn the long axis of each ellipse and make its shape breathe, over tens or hundreds of thousands of years. Laplace and Lagrange found that these slow changes are governed by a linear system. Le Verrier’s first great work was to solve it carefully for all the planets then known.

The secular system

Let planet \(j\) have mass \(m_j\) (in units of the solar mass), semi-major axis \(a_j\), mean motion \(n_j\), eccentricity \(e_j\) and longitude of perihelion \(\varpi_j\). Introduce the eccentricity vector

Average the mutual perturbations over the fast orbital angles and keep terms of first order in the masses and second order in eccentricities and inclinations. The result is a linear system with constant coefficients.

Each eccentricity is then a sum of uniformly rotating vectors, so it oscillates between bounds and never grows without limit. This is the classical statement of the stability of the solar system. Its weakness is plain: it is a statement about a truncated, linearized system.

Laplace’s invariant

The matrix \(A\) is not symmetric, but it becomes symmetric after weighting by \(\Lambda_j=m_j\sqrt{a_j}\), proportional to the circular angular momentum of planet \(j\): one checks \(\Lambda_jA_{jk}=\Lambda_kA_{kj}\). It follows that the real eigenvalue assumption holds and that

Laplace used this to argue for stability. The bound it gives is strong for a massive planet and weak for a light one. Jupiter and Saturn carry almost all of the weighted sum, so their eccentricities cannot change much. Mercury and Mars carry almost none, and the invariant alone does not stop their eccentricities from growing large. Le Verrier saw this and computed explicit bounds planet by planet. In 1989 Jacques Laskar showed by numerical integration of the averaged equations to higher order that the inner solar system is chaotic, with a Lyapunov time of about five million years.

The Le Verrier algorithm for characteristic polynomials

For the general reader. The secular frequencies of Lesson 04 are the roots of a polynomial built from a matrix. Writing out that polynomial for seven planets by expanding a determinant is a nightmare. In his 1840 memoir Le Verrier used a much better route: compute the traces of the powers of the matrix, which only needs multiplication, and recover the polynomial from them. A refined version of his method is now called the Faddeev–Le Verrier algorithm.

Traces and Newton’s identities

Let \(A\) be an \(n\times n\) matrix with eigenvalues \(\lambda_1,\dots,\lambda_n\) and characteristic polynomial \(p(\lambda)=\det(\lambda I-A)=\lambda^n+c_{n-1}\lambda^{n-1}+\cdots+c_0\). The power sums of the eigenvalues are traces: \(s_k=\sum_i\lambda_i^k=\operatorname{tr}(A^k)\). Isaac Newton’s identities connect the power sums with the coefficients.

So \(c_{n-1}=-s_1\), \(c_{n-2}=\tfrac12(s_1^2-s_2)\), and so on. This was Le Verrier’s method in 1840: compute \(A,A^2,\dots,A^n\), take traces, and solve the triangular system.

The Faddeev–Le Verrier recursion

In 1949 Dmitry Faddeev and Ivan Sominsky gave the method a compact matrix form that produces the adjugate as a by-product.

Proof sketch. Let \(B(\lambda)=\operatorname{adj}(\lambda I-A)=\sum_{k=1}^{n}M_k\lambda^{n-k}\), a matrix polynomial of degree \(n-1\). From \((\lambda I-A)B(\lambda)=p(\lambda)I\), comparing coefficients gives \(M_1=I\), \(M_k=AM_{k-1}+c_{n-k+1}I\) and \(AM_n+c_0I=0\). Jacobi’s formula gives \(p'(\lambda)=\operatorname{tr}B(\lambda)\). Comparing the coefficients of \(\lambda^{n-k-1}\) on both sides gives \((n-k)c_{n-k}=\operatorname{tr}M_{k+1}=\operatorname{tr}(AM_k)+n\,c_{n-k}\), which rearranges to \(c_{n-k}=-\frac1k\operatorname{tr}(AM_k)\). The last relation is the Cayley–Hamilton theorem in disguise.

The method costs about \(n^4\) operations, worse than Gaussian elimination, and it divides by \(k\), so it fails in small positive characteristic. Its virtue is that it uses only matrix products and traces, which makes it useful in parallel and symbolic computation.

The disturbing function and its expansion

For the general reader. To compute how one planet pulls another, astronomers write the extra gravitational energy as a long series of cosine terms, each tied to a combination of the planets’ positions. Le Verrier pushed this series further than anyone before him. By 1849 he had expanded it to the seventh power of the eccentricities and inclinations, a table of 469 terms that became the backbone of his planetary theories.

Definition

Take the Sun, a planet at heliocentric position \(\mathbf r\) and a perturbing planet of mass \(m'\) at \(\mathbf r'\). In heliocentric coordinates the planet’s acceleration is \(\ddot{\mathbf r}=-\mu\mathbf r/r^3+\nabla_{\mathbf r}\mathcal R\), with \(\mu=G(M_\odot+m)\) and

The first term is the direct part, the second the indirect part due to the acceleration of the Sun by \(m'\).

Laplace coefficients

For two circular, coplanar orbits with radii \(a<a'\), ratio \(\alpha=a/a'\) and angle \(\psi\) between the planets,

For eccentric, inclined orbits one expands in powers of \(e,e',\sin\tfrac i2,\sin\tfrac{i'}2\) about the circular case. The result is a sum of terms

where \(\lambda\) are mean longitudes, \(\varpi\) longitudes of perihelion and \(\Omega\) longitudes of the node.

A seventh-order expansion keeps every term whose total power in \(e,e',\sin\frac i2,\sin\frac{i'}2\) is at most seven. Le Verrier wrote each coefficient as a combination of Laplace coefficients and their derivatives in \(\alpha\). The table, with its 469 terms, was published in the first volume of the Annales de l’Observatoire de Paris and served astronomers for decades.

Small divisors

A term with argument \(\theta=j_1\lambda'+j_2\lambda+\dots\) produces a periodic perturbation whose amplitude is divided by \(\dot\theta\approx j_1n'+j_2n\), and for the mean longitude by \(\dot\theta^2\). When \(j_1n'+j_2n\) is small, a high-order term of tiny coefficient can produce a large effect. The classic case is the great inequality of Jupiter and Saturn, explained by Laplace in 1785: \(2n_J\approx5n_S\), and a third-order term produces perturbations in longitude of about \(20'\) for Jupiter and \(48'\) for Saturn.

Comets and Jupiter

For the general reader. A comet seen in one decade and another seen twenty years later might be the same body, if Jupiter changed its orbit in between. Le Verrier’s work of 1844–1847 on periodic comets was largely about this problem of identity: following an orbit backward through a close encounter with Jupiter and asking where it could have come from.

Lexell’s comet

The comet of 1770, discovered by Charles Messier, passed closer to the Earth than any other recorded comet. Anders Johan Lexell computed for it an orbit with a period of about five and a half years and showed that it had been moved into that orbit by Jupiter in 1767 and would pass close to Jupiter again in 1779. It was never seen again. Le Verrier’s theory of 1844 took up the question of where the 1779 encounter had sent it, and examined how far the uncertainty in the 1770 orbit spread once it was carried through the encounter. The same tools let him examine whether comets found by Faye and de Vico could be returns of earlier ones.

The Jacobi integral

Model the Sun and Jupiter as masses \(1-\mu\) and \(\mu\) on circular orbits of unit radius about their barycentre, with unit angular velocity, and treat the comet as massless. In the rotating frame the comet’s energy is not conserved, but a combination is.

Far from Jupiter, the comet moves on a nearly Keplerian heliocentric orbit with elements \(a,e,i\), and \(C_J\) can be written in terms of them. Neglecting terms of order \(\mu\) gives the criterion that Félix Tisserand, a later director of the Paris Observatory, published in 1889.

Derivation sketch. Use units with \(a_J=1\) and neglect \(\mu\). The rotating velocity is \(\mathbf v_{\text{rot}}=\mathbf v-\boldsymbol\omega\times\mathbf r\) with \(\boldsymbol\omega=\hat{\mathbf z}\), so \(v_{\text{rot}}^2=v^2-2\,\hat{\mathbf z}\cdot(\mathbf r\times\mathbf v)+x^2+y^2\). Hence \(C_J\approx 2/r-v^2+2h_z\). The vis-viva law gives \(2/r-v^2=1/a\), and \(h_z=\sqrt{a(1-e^2)}\cos i\). So \(C_J\approx T_J\). Comets with \(2<T_J<3\) form the Jupiter family.

The inverse problem: Uranus and Neptune

For the general reader. Usually astronomers know the planets and compute how they disturb each other. Le Verrier reversed the question. Uranus was drifting off its predicted path; what unseen body, where and how massive, would make it drift that way? His answer pointed to a spot in the sky, and a planet was there.

The residuals of Uranus

William Herschel discovered Uranus in 1781, and earlier sightings recorded as stars, back to John Flamsteed in 1690, gave a long arc. Bouvard’s tables of 1821 could not fit both the old and new observations, and by the 1840s Uranus was more than a minute of arc from its tabulated place. Le Verrier’s memoir of 10 November 1845 recomputed the action of Jupiter and Saturn and ruled them out. On 1 June 1846 he gave the position of a perturbing planet, and on 31 August its mass and orbit.

Formulation

Let \(\delta\lambda_{\text{obs}}(t_i)\) be the residuals of Uranus’s longitude after known perturbations are removed. Two kinds of unknown explain them: corrections \(\Delta E_k\) to the six elements of Uranus, and the parameters of the new planet (mass \(m'\), semi-major axis \(a'\), eccentricity \(e'\), perihelion \(\varpi'\), mean longitude at epoch \(\varepsilon'\)). To first order in \(m'\),

The problem is linear in \(\Delta E_k\) and \(m'\) but nonlinear in the orbit of the perturber. Le Verrier took the new orbit in the plane of the ecliptic, started from a distance about twice that of Uranus, as the Titius–Bode pattern of planetary distances suggested, and studied how the fit changed as the distance varied within limits. For each trial orbit he solved for the remaining unknowns and compared the fits with the observational errors.

Why it worked

The largest effects on Uranus come near conjunction, when the two planets are closest. The timing and sign of the residuals fix when the conjunction happened, and hence where the perturber was then. Its slow motion since that date is set roughly by Kepler’s third law, and it is insensitive to small errors in distance. Mass and distance trade off against each other: a farther, heavier planet pulls much like a nearer, lighter one. Le Verrier’s distance and mass were both too large, but the direction was right.

Le Verrier wrote to Johann Galle at the Berlin Observatory on 18 September 1846. The letter arrived on 23 September, and that night Galle, helped by the student Heinrich d’Arrest and a new star chart, found an object not on the chart. Its heliocentric longitude was \(327^\circ24'\), about \(52'\) from the prediction. The true orbit is nearly circular with \(a\approx30.1\) AU, and the mass of the Neptune system is about \(1/19\,400\) of the Sun’s.

In England, John Couch Adams had worked on the same problem since 1843 and communicated results to James Challis and George Biddell Airy in 1845. After Le Verrier’s June prediction, Airy asked Challis to search with the Cambridge telescope; Challis recorded the planet in August without recognizing it. Adams acknowledged Le Verrier’s priority before the Royal Astronomical Society in November 1846. Adams’s papers, missing for decades and recovered in 1999, suggest that his predictions were less definite than the English account had claimed. Some critics, among them Benjamin Peirce at Harvard, argued that the agreement was partly fortunate, since the true orbit differed so much from the predicted one. John Herschel replied that the elements had served their purpose: they located the body.

Mercury’s perihelion and Vulcan

For the general reader. Mercury’s elliptical orbit slowly turns. Almost all of the turning is explained by the pull of the other planets and by the slow wobble of the Earth’s axis. In 1859 Le Verrier found a small remainder that nothing known could explain. He proposed an unseen planet near the Sun. There was none. The remainder was the first observed effect of general relativity.

The budget

In his report of 12 September 1859, Le Verrier analysed transits and meridian observations of Mercury. In modern figures, the perihelion advances by about \(5600''\) per century relative to the moving equinox. About \(5026''\) of this is the precession of the equinoxes, a property of the reference frame, and the perturbations of the planets account for about \(531''\). Le Verrier, with the planetary masses of his day, found an unexplained remainder of about \(38''\) per century. In 1882 Simon Newcomb revised it to about \(43''\).

Le Verrier proposed a planet, or a ring of small bodies, between Mercury and the Sun. Late in 1859 Edmond Lescarbault, a doctor and amateur astronomer at Orgères, reported seeing a dark round body cross the Sun earlier that year. Le Verrier visited him, was convinced, and the planet was named Vulcan. Searches during eclipses and transits over the following decades found nothing reliable.

Relativity

In November 1915 Albert Einstein computed the perihelion advance in general relativity and found \(43''\) per century without any new matter. Karl Schwarzschild’s exact solution followed in 1916.

Derivation sketch. With \(u=1/r\), angular momentum per unit mass \(h\) and \(\phi\) the orbital angle, the geodesic equation becomes

Without the last term the solution is the Kepler ellipse \(u_0=\frac{GM}{h^2}(1+e\cos\phi)\). Substituting \(u_0\) into the small term produces a resonant forcing \(\propto\cos\phi\), whose secular response is \(u\approx\frac{GM}{h^2}\big(1+e\cos((1-\delta)\phi)\big)\) with \(\delta=3G^2M^2/(c^2h^2)\). Perihelion returns after \(2\pi/(1-\delta)\approx2\pi(1+\delta)\), an advance of \(2\pi\delta=6\pi GM/(c^2a(1-e^2))\), using \(h^2=GMa(1-e^2)\).

Planetary tables and the scale of the solar system

For the general reader. Le Verrier’s life work was a set of tables giving the positions of the Sun and planets for any date. Building them meant fitting a century of observations to theory, and the fit revealed the masses of the planets and even the distance from the Earth to the Sun. He concluded that the accepted distance was about 3% too large, and a laboratory measurement of the speed of light soon agreed.

The plan

In 1847 Le Verrier planned a single, consistent theory of all the planets, with common masses and constants. He compiled observations back to 1750 and finished the compilation in 1852. Theories and tables followed: the Sun (1858), Mercury (1859), Venus and Mars (1861), then Jupiter, Saturn, Uranus and Neptune between 1870 and 1877. For Jupiter and Saturn, where the great inequality makes the series converge slowly, he combined analytic and interpolated values, and his student Jean-Baptiste Gaillot completed parts of these theories. The tables formed the basis of the French ephemeris, the Connaissance des Temps, until about 1912.

Solar parallax from planetary theory

The solar parallax \(\pi_\odot=R_\oplus/a\) (equatorial radius over the astronomical unit) fixes the scale of the solar system. Until the 1870s the standard value was Johann Franz Encke’s, from the transits of Venus of 1761 and 1769. Planetary theory gives two independent routes.

Route (ii) amplifies errors gently, since a 3% error in the Earth’s mass gives a 1% error in the parallax. Le Verrier’s analyses pointed to a parallax larger than Encke’s, hence a smaller astronomical unit. In 1862 Léon Foucault, physicist at the Paris Observatory, measured the speed of light in the laboratory and found it smaller than the accepted value, consistent with Le Verrier’s smaller astronomical unit.

The Observatory, the weather and the Empire, 1846–1870

Neptune made Le Verrier famous overnight. The Royal Society awarded him the Copley Medal in 1846 and elected him a foreign member in 1847; King Louis-Philippe made him an officer of the Légion d’honneur; a chair of celestial mechanics was created for him at the Paris Faculty of Sciences. Arago proposed to name the planet after him. The name Neptune prevailed.

Politics

Le Verrier was a conservative. In the revolution of 1848 he joined the National Guard on the side of order, and on 13 May 1849 he was elected deputy for the Manche. He supported Louis-Napoléon’s coup of December 1851 and was named a senator of the new Empire in 1852, the year he also became inspector general of higher education. In 1850 he chaired a commission on the École polytechnique whose report pushed teaching toward applications and cut the great course in analysis and mechanics. Joseph Liouville and Michel Chasles resigned their posts at the École in protest. From 1858 to 1870 he presided over the general council of the Manche.

Director of the Paris Observatory

Arago died in 1853, and on 30 January 1854 Le Verrier became director of the Paris Observatory. He reversed Arago’s priorities, turning the Observatory toward precise positional astronomy in support of his tables. He ran it with strict hierarchy and constant pressure, paid observers by results, and quarrelled with staff. Many resigned. He also hired Léon Foucault as the Observatory’s physicist, and from 1865 he made the observatory at Marseille a branch of Paris.

The weather service

On 14 November 1854, during the Crimean War, a storm struck the French and British fleets off Sevastopol and sank dozens of ships. The minister of war asked Le Verrier to study it. Collecting reports from across Europe, he showed that the storm had crossed the continent over several days and could have been tracked by telegraph. With his deputy Emmanuel Liais he built a network of telegraph-linked stations. A daily bulletin from French and foreign stations appeared from 1857, storm warnings for France and continental Europe from 1863, and from 1872 an international bulletin. After his death these services became the Bureau central météorologique, the ancestor of Météo-France.

Dismissal, return, death and legacy, 1870–1877

Complaints from the staff accumulated through the 1860s. Early in 1870 the imperial government removed Le Verrier as director, despite his loyalty to the regime, and appointed his rival Charles-Eugène Delaunay, the author of the lunar theory. After his dismissal he also resigned from the general council of the Manche and from the Bureau des longitudes. In August 1872 Delaunay drowned in a boating accident near Cherbourg. In 1873 Le Verrier was reinstated, this time under the supervision of a council that limited his authority.

He spent his last years finishing the theories of the outer planets despite a painful liver disease. The Royal Astronomical Society awarded him its Gold Medal a second time in 1876, for the theories of Jupiter, Saturn, Uranus and Neptune; he had received the first in 1868 for the inner planets. In 1876 he was too ill to attend, and the medal was presented in his absence by the society’s president, John Couch Adams. He completed the tables of Uranus and Neptune in 1877 and died at the Observatory on 23 September 1877, 31 years to the day after Galle’s discovery. He is buried in the Montparnasse cemetery under a stone celestial globe.

Legacy

• Celestial mechanics: the discovery of Neptune, the first planet found by prediction; complete, consistent theories of all eight major planets; the seventh-order disturbing function.
• Physics: the anomaly of Mercury’s perihelion, explained in 1915 by general relativity and still one of its classic tests.
• Mathematics: the trace method for characteristic polynomials, now the Faddeev–Le Verrier algorithm.
• Meteorology: a telegraphic network of weather stations and the first storm warnings in France and continental Europe.
• Method: inferring an unseen mass from perturbations. The same reasoning underlies the detection of exoplanets from timing variations and the hypothesis, proposed in 2016, of a distant ninth planet shaping the orbits of Kuiper belt objects.
• Honours: Copley Medal (1846), RAS Gold Medals (1868, 1876), one of the 72 names on the Eiffel Tower, a statue by Henri Chapu at the Paris Observatory, a lunar crater, a ring of Neptune and the asteroid (1997) Leverrier.

Connections

People linked to Urbain Le Verrier across this collection. A name without a link has no page here yet. See Urbain Le Verrier in the network of all 337 lives →

Teachers and advisors

• François Arago 1845-1853
• Joseph Louis Gay-Lussac 1835-1837

Students

• Jean-Baptiste Gaillot

Influenced by

• Joseph-Louis Lagrange
• Pierre-Simon Laplace

Influenced

• Albert Einstein
• Félix Tisserand

Colleagues and collaborators

• George Biddell Airy 1846
• John Herschel
• Joseph Liouville
• Léon Foucault 1855-1868
• Emmanuel Liais
• Félix Savary 1837-1841
• Heinrich d'Arrest 1846
• James Challis 1846
• Johann Gottfried Galle 1846

Rivals

• Michel Chasles
• Charles-Eugène Delaunay 1854-1872
• John Couch Adams 1845-1877

Also mentioned in this life

• Arthur Cayley
• Jean le Rond d’Alembert
• John Flamsteed
• Joseph Fourier
• William Rowan Hamilton
• Carl Gustav Jacob Jacobi
• Johannes Kepler
• Aleksandr Lyapunov
• Charles Messier
• Isaac Newton
• William Herschel

Fields

• celestial mechanics 16 lives
• astronomy 50 lives
• meteorology 6 lives

On CoursesHub

• Classical Mechanics
• Astrophysics for Beginners

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Urbain Jean Joseph Le Verrier, MacTutor History of Mathematics.
2. J. Lequeux, Le Verrier, Dictionary of Scientific Biography (MacTutor copy).
3. Urbain Le Verrier, Wikipedia; Urbain Le Verrier, Wikipédia (for his political and meteorological career).
4. J. Lequeux, Le Verrier: Magnificent and Detestable Astronomer, Springer, 2013.
5. Discovery of Neptune, Wikipedia; W. Sheehan, N. Kollerstrom and C. B. Waff, The case of the pilfered planet, Scientific American, December 2004.

Science

1. U. J. J. Le Verrier, Sur les variations séculaires des éléments des orbites pour les sept planètes principales, Journal de mathématiques pures et appliquées 5 (1840).
2. U. J. J. Le Verrier, Recherches sur les mouvements d’Uranus, Comptes rendus 22 (1 June 1846); Sur la planète qui produit les anomalies observées dans le mouvement d’Uranus, Comptes rendus 23 (31 August 1846).
3. U. J. J. Le Verrier, Théorie du mouvement de Mercure, Annales de l’Observatoire de Paris 5 (1859).
4. J. Lequeux, Des premiers travaux de Le Verrier à la découverte de Neptune, arXiv:1710.03688.
5. “That star is not on the map”: the German side of the discovery of Neptune, arXiv:2108.06305.
6. Faddeev–LeVerrier algorithm, Wikipedia; S. H. Hou, A simple proof of the Leverrier–Faddeev characteristic polynomial algorithm, SIAM Review 40 (1998).
7. C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999 (disturbing function, secular theory, Tisserand’s criterion).
8. N. T. Roseveare, Mercury’s Perihelion from Le Verrier to Einstein, Oxford, 1982.
9. J. Laskar, A numerical experiment on the chaotic behaviour of the solar system, Nature 338 (1989).