Charles Messier

How to use this course

Charles Messier is remembered for a list he made to help himself. He wanted comets, and the sky kept offering him faint glows that looked like comets but did not move. He recorded them so that neither he nor anyone else would waste a night on them again. The list grew to 103 objects in 1781, and seven more were added in the twentieth century from his and Pierre Méchain’s notes. It contains the Crab Nebula, the Andromeda galaxy, the Orion Nebula, the Pleiades and the galaxy M87, whose central black hole was imaged in 2019.

Messier himself would have pointed to his comets. He observed more than forty, discovered about a dozen and a half, and recovered Halley’s comet on its first predicted return. He was not a theorist. His skill was patient, accurate observation, and the mathematics of this course is the mathematics that turned observations like his into orbits and physics.

Each technical lesson opens with an italic paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The companion modules on Caroline Herschel and William Herschel treat sweeping, sidereal time, precession and the surface brightness of nebulae in detail; this course refers to them rather than repeating them.

Timeline

Badonviller, 1730–1751

Charles Messier was born on 26 June 1730 in Badonviller, a small town on the western slopes of the Vosges. It then belonged to the principality of Salm, a pocket of the Holy Roman Empire inside Lorraine. He was the tenth of twelve children of Nicolas Messier, an usher of the princely court, and Françoise Grandblaise. Six of the children died young, and his father died in 1741, when Charles was eleven. He grew up with his elder brother Hyacinthe, who held an administrative post in the service of the princes of Salm.

Two events turned the boy to the sky. In 1744 a spectacular comet appeared, the great comet of 1743–44, whose fan of six tails rose before dawn in early March. On 25 July 1748 an annular eclipse of the Sun was visible from Badonviller. Messier had no formal training in mathematics or astronomy. What he had was good eyesight, patience and a neat hand, which proved to be his introduction to Paris.

In 1751 the principality of Salm was partitioned and the family’s position changed. Hyacinthe moved to Senones, and Charles, now 21, went to Paris to look for work.

Delisle’s clerk and Halley’s comet, 1751–1759

In Paris Messier found a place with Joseph-Nicolas Delisle, astronomer of the Navy and professor at the Collège de France. Delisle had spent more than twenty years in St Petersburg and had brought back a large collection of maps and observations. He needed a copyist, and Messier’s first work was to copy maps, including a map of Beijing and of the Great Wall of China. From 1754 Messier was paid as a clerk of the Navy’s depot of charts.

Delisle’s observatory was a small tower on the roof of the Hôtel de Cluny, the medieval town house of the abbots of Cluny near the Sorbonne. Messier learned to observe there. His first recorded observation is the transit of Mercury of 6 May 1753. He would live and observe at the Hôtel de Cluny for most of his life.

Waiting for the comet

Edmond Halley had predicted that the comet of 1682 would return around 1758. Alexis-Claude Clairaut, with Joseph-Jérôme Lalande and Nicole-Reine Lepaute, refined the prediction by computing the perturbations of Jupiter and Saturn. Delisle drew up a chart of the path the comet should follow across the sky, and from 1757 Messier searched along it. The chart was in error, and for months he searched in the wrong places. In August 1758, while following a different comet, he came upon a faint nebula near the star ζ Tauri that looked like a comet but did not move. He measured its position on 12 September. It would become the first entry in his catalogue, and is now known as the Crab Nebula.

On 21 January 1759 Messier found Halley’s comet. Johann Georg Palitzsch, a farmer and amateur astronomer near Dresden, had seen it on Christmas night 1758, so Messier’s sighting was not the first, but it was independent and well measured. Delisle did not allow him to announce it, and the observations were made public only on 1 April 1759, when the comet had passed perihelion and was moving away. The delay, which was not Messier’s choice, led some to doubt the observations. Lesson 06 computes the orbit he was searching for.

The ferret of comets, 1759–1771

After 1759 Messier devoted his nights to comets. Hunting them meant sweeping the evening and morning skies near the Sun, where comets brighten, again and again. In the 1760s he found a series of new comets, and his record became famous at court. Louis XV called him the furet des comètes, the ferret of comets. Today 13 comets are credited to him as discoverer, among them the comets of 1763 and 1764, and he observed many more. By Lalande’s count he observed 41 in his career.

Recognition came from abroad first. He was elected a Fellow of the Royal Society of London in 1764 and a foreign member of the Royal Swedish Academy of Sciences in 1769. He was later elected to academies in Germany, Russia and elsewhere. The Paris Academy of Sciences admitted him, after some difficulty, on 30 June 1770.

A catalogue of false alarms

The same sweeps that found comets turned up nebulae and clusters. In 1764 he found the Dumbbell Nebula in Vulpecula (M27) and observed the nebula in Andromeda (M31), described in the tenth century by the Persian astronomer Abd al-Rahman al-Sufi. He recorded each object with its position and a short description. On 16 February 1771 he presented a list of 45 to the Academy. It appeared in print in 1774, in the Mémoires for 1771. Of the 45, he had discovered 17 or 18 himself; the rest were known from earlier observers, including Edmond Halley, Jean-Philippe Loys de Chéseaux and Nicolas-Louis de Lacaille, whose positions he checked.

In 1770, at the age of forty, he married Marie-Françoise de Vermauchampt. Their son was born in 1772. His wife died, and the child died eleven days after his birth. A well-known anecdote says that Messier missed a comet, found by a rival, while he nursed his wife. In 1771 he succeeded Delisle as Astronomer of the Navy.

Small telescopes and faint light

For the general reader. Messier worked with instruments that a modern amateur would find poor. His telescopes had apertures of a few inches, and many had single lenses whose images were fringed with colour, or metal mirrors that reflected barely half the light. Yet he found comets and nebulae at the limit of what such instruments could show. This lesson explains what an aperture buys, why eighteenth-century telescopes were so long, and how faint a star such an instrument could reveal.

Messier’s instruments

He used a succession of instruments at the Hôtel de Cluny: refractors with focal lengths up to several metres, and Gregorian reflectors with speculum-metal mirrors up to about 20 cm across. His usual comet-seeking telescopes had apertures of about 10 cm. The reflectors, though larger, delivered less light than their size suggests, because speculum metal reflects only about 60 per cent of the light falling on it, and a Gregorian uses two mirrors.

The magnitude scale is logarithmic: a factor \(F\) in flux is \(2.5\log_{10}F\) magnitudes. The flux gain from the area is \((D/p)^2\), which gives the \(5\log_{10}(D/p)\) term. A star stays a point at any magnification, so, unlike a nebula (Lesson 10 of the Caroline Herschel module), its brightness grows with aperture.

Colour and the long telescope

A single lens of refractive index \(n\) has power \(P=1/f=(n-1)K\), where \(K\) depends only on the curvatures of its surfaces. Because \(n\) depends on wavelength, so does \(f\). Write \(V=(n_d-1)/(n_F-n_C)\) for the Abbe number of the glass, which measures how weakly it disperses. Then the spread in power between the blue F line and red C line is \(\Delta P=P/V\), so the focal lengths differ by about \(\Delta f\approx f/V\).

Sketch. (a) Red and blue rays focus \(\Delta f\approx f/V\) apart. Midway between the foci, a cone of rays of diameter \(D\) converging over a length \(f\) has diameter about \(D\,\Delta f/(2f)\), which subtends \(D/(2Vf)\) at the lens. (b) Powers of thin lenses in contact add, so the change in total power is \(P_1/V_1+P_2/V_2\), which must vanish. \(\square\)

Part (a) explains the absurdly long refractors of the seventeenth and eighteenth centuries: the blur in angle falls as \(1/f\). Part (b) is the principle of the achromatic doublet, a converging crown lens paired with a weaker diverging flint lens, patented by John Dollond in 1758. Achromats spread slowly, and Messier used both kinds of refractor.

Measuring a comet’s place

For the general reader. A comet’s orbit is computed from its positions on the sky over several nights. Messier did not measure those positions from scratch. He compared the comet with a nearby star whose position was already catalogued, measuring the small differences in right ascension and declination. He then had to correct for the bending of starlight by the atmosphere, which lifts objects near the horizon, where comets close to the Sun are usually found.

Differential positions

With a fixed telescope and cross-wires, an observer times the passage of the comet and of a comparison star across a wire perpendicular to the diurnal motion, and measures their separation in declination with a micrometer. If a clock keeping mean solar time shows an interval \(\Delta t\) seconds between the two transits, the difference in right ascension is \(\Delta\alpha=1.0027379\,\Delta t\) seconds of sidereal time, and the angular separation along the parallel of declination is

The advantage of differential measurement is that most errors of the instrument and the clock cancel, because they affect star and comet alike. Its weakness is that the result is only as good as the star’s catalogue position, which for Messier often came from Lacaille or Flamsteed.

Proof. At each horizontal interface Snell’s law gives \(n_k\sin z_k=n_{k+1}\sin z_{k+1}\), so \(n\sin z\) is the same in every layer. Above the atmosphere \(n=1\) and the angle is \(z_t\); at the ground it is \(n_0\sin z\). Writing \(z_t=z+R\) with \(R\) small, \(\sin z+R\cos z\approx n_0\sin z\), so \(R\approx(n_0-1)\tan z\). \(\square\)

The result is independent of the structure of the atmosphere, which is why a single constant serves for most of the sky. It fails near the horizon, where the Earth’s curvature matters and \(\tan z\) diverges; the true refraction at the horizon is about \(35'\), and it varies with temperature and pressure. Refraction raises objects without changing their azimuth, so it shifts both right ascension and declination.

Halley’s comet on its ellipse

For the general reader. Halley’s comet spends almost all of its 76-year orbit far beyond the planets, too faint to see. It is visible only for a few months near perihelion, its closest approach to the Sun. In 1758 observers knew roughly when it would come back but not exactly where to look. This lesson computes where the comet was on the night Messier found it, using the equation Kepler devised to locate a body on an ellipse at a given time.

Proof. Put the ellipse’s centre at the origin with the Sun at \((ae,0)\), and parametrize by \(x=a\cos E\), \(y=b\sin E\) with \(b=a\sqrt{1-e^2}\). The area swept from perihelion, seen from the Sun, is the area of the elliptic sector from the centre, \(\tfrac12abE\), minus the triangle with vertices at the centre, the Sun and the body, \(\tfrac12\,ae\,b\sin E\). By Kepler’s second law this area is the fraction \((t-T)/P\) of the total \(\pi ab\):

The distance is \(r^2=(a\cos E-ae)^2+b^2\sin^2E=a^2(1-e\cos E)^2\). The half-angle formula follows from \(r\cos\nu=a(\cos E-e)\) and \(r\sin\nu=b\sin E\). \(\square\)

Solving the equation

Kepler’s equation has no closed-form solution, but Newton’s method converges rapidly: starting from \(E_0=M\) (or \(E_0=\pi\) for large \(e\) near aphelion), iterate

Because \(f(E)=E-e\sin E-M\) is increasing with \(f'(E)=1-e\cos E\ge1-e>0\), the root is unique. For \(e\) close to 1 and small \(M\), the equation is nearly cubic: \(M\approx(1-e)E+eE^3/6\), which is the elliptic counterpart of Barker’s equation for parabolas (Lesson 05 of the Caroline Herschel module).

Lexell’s comet and Jupiter

For the general reader. On 14 June 1770 Messier found a faint comet in Sagittarius. On 1 July it passed the Earth at about six times the Moon’s distance, the closest approach of any comet in recorded history, and crossed the sky so fast that its motion could be seen in an hour. Anders Johan Lexell, working in St Petersburg, showed that it moved on a 5.6-year ellipse and argued that Jupiter had thrown it onto that orbit in 1767. In 1779 Jupiter threw it off again, and it was never seen after.

Parallax

A body at distance \(d\) is seen from points on the Earth’s surface in directions that differ by up to the horizontal parallax \(\pi_h=R_\oplus/d\). For the Moon this is about a degree; for the Sun about \(8.8''\). A comet at 0.015 AU has a parallax of almost ten arcminutes, so observations from different places, and at different hours from the same place, give visibly different positions. Lexell had to correct for this before an orbit would fit.

The Tisserand parameter

Consider a comet of negligible mass moving under the Sun and a planet on a circular orbit of radius \(a_J\). In a frame rotating with the planet at angular speed \(n_J\), the Jacobi integral

is exactly conserved, where \(r_1,r_2\) are the distances to the Sun and planet and \(v_{\text{rot}}\) the speed in the rotating frame.

Sketch. Far from the planet, drop the \(GM_J/r_2\) term. The inertial velocity is \(\mathbf v=\mathbf v_{\text{rot}}+n_J\hat z\times\mathbf r\), so \(v_{\text{rot}}^2=v^2-2n_J(\hat z\cdot\mathbf r\times\mathbf v)+n_J^2(x^2+y^2)\). Substituting, \(C_J\approx 2GM_\odot/r-v^2+2n_Jh_z\), with \(h_z=\sqrt{GM_\odot a(1-e^2)}\cos i\). The vis-viva equation gives \(2GM_\odot/r-v^2=GM_\odot/a\). With \(n_J^2a_J^3=GM_\odot\), dividing by \(GM_\odot/a_J\) gives \(T_J\). \(\square\)

Tisserand’s parameter is how astronomers decide whether a newly found comet might be an old one in disguise: if two orbits have very different \(T_J\), no encounter with Jupiter can connect them.

How bright is a comet?

For the general reader. A planet shines by reflecting sunlight, so its brightness depends on its distances from the Sun and from us. A comet does more: as it nears the Sun its ices evaporate and the gas and dust spread into a coma thousands of times larger than the nucleus. Its brightness therefore rises steeply as it approaches the Sun. This is why comet hunters, Messier among them, searched the twilight skies, low in the west after sunset and in the east before dawn.

Derivation. The flux received by the body is proportional to \(r^{-2}\), and the reflected flux reaching us to \(r^{-2}\Delta^{-2}\). In magnitudes this is \(2.5\log_{10}(r^2\Delta^2)=5\log_{10}r+5\log_{10}\Delta\). For a comet the amount of reflecting dust and fluorescing gas itself grows as the comet nears the Sun, roughly as a power of \(r\); writing the total as \(r^{-n}\) gives the second form, with \(n=2\) for an inert body. \(\square\)

The second statement follows from the triangle Sun–Earth–comet: the elongation is largest when the line of sight is tangent to the sphere of radius \(r\) about the Sun. Mercury, at \(r\approx0.39\), never strays more than about \(23^\circ\) to \(28^\circ\) from the Sun. A comet at perihelion inside Mercury’s orbit is similarly confined to the twilight sky.

What the Messier objects are

For the general reader. To Messier, all his objects were the same kind of thing: faint patches that were not comets. We now know that his list samples almost every kind of object in the universe that a small telescope can show. It contains clusters of young stars in our Galaxy, ancient globular clusters orbiting it, clouds of glowing gas where stars form, shells thrown off by dying stars, the debris of a supernova, and about forty galaxies millions of light years away.

Proof. Let \(G=\sum_i\mathbf p_i\cdot\mathbf r_i\). Then \(\dot G=\sum m_iv_i^2+\sum\mathbf F_i\cdot\mathbf r_i=2K+W\), where the second sum equals \(W\) because the gravitational potential is homogeneous of degree \(-1\) in the positions (Euler’s theorem on homogeneous functions). For a bound system \(G\) stays bounded, so the time average of \(\dot G\) over a long interval tends to zero. \(\square\)

For a cluster of mass \(M\), one-dimensional velocity dispersion \(\sigma\) and characteristic radius \(R\), \(K=\tfrac32M\sigma^2\) and \(W=-\alpha GM^2/R\) with \(\alpha\) of order 1, so \(M\approx3\sigma^2R/(\alpha G)\). The precise factor depends on the cluster’s density profile and on which radius is used, so such masses are good to a factor of about two.

M1: from nebula to pulsar

For the general reader. The first object in Messier’s catalogue, found in 1758 while he waited for Halley’s comet, has become one of the most studied objects in the sky. John Bevis had seen it in 1731. In the 1840s Lord Rosse drew it with crab-like arms and gave it its name. In the twentieth century astronomers found it was expanding, traced the expansion back to a star recorded by Chinese astronomers in 1054, and in 1968 found at its centre a neutron star spinning thirty times a second. It is the remnant of a supernova.

Expansion age

A feature moving outward at speed \(v\) perpendicular to the line of sight, at distance \(d\), has proper motion \(\mu=v/(4.74\,d)\) arcseconds per year, with \(v\) in km/s and \(d\) in parsecs. If the nebula has expanded uniformly from a point, a feature at angular distance \(\theta\) from the centre is \(\theta/\mu\) years old. In 1921 John Charles Duncan compared photographs taken years apart and found the nebula expanding; in the same year Knut Lundmark noted that it lay near the position of the guest star of 1054. Careful modern measurements trace the expansion back to several decades after 1054, which implies that the nebula has been accelerated, pushed from inside by the pulsar wind.

The pulsar

The Crab pulsar has period \(P\approx33.7\) ms, increasing at \(\dot P\approx4.2\times10^{-13}\) s per second.

Proof. With \(\Omega=2\pi/P\), \(\dot\Omega=-2\pi\dot P/P^2\), so \(-I\Omega\dot\Omega=I(2\pi/P)(2\pi\dot P/P^2)=4\pi^2I\dot P/P^3\). For the age: \(\dot\Omega=-k\Omega^3\) means \(P\dot P=2\pi\cdot\)const, so \(d(P^2)/dt=2P\dot P\) is constant. Integrating from \(P_0\) to \(P\), \(t=(P^2-P_0^2)/(2P\dot P)\approx P/(2\dot P)\). \(\square\)

The dipole formula for spin-down also gives the surface magnetic field at the equator, \(B\approx3.2\times10^{19}\sqrt{P\dot P}\) gauss, with \(P\) in seconds, for \(I=10^{45}\) g cm\(^2\) and radius 10 km.

Academy, catalogue, Revolution, 1771–1817

As Astronomer of the Navy from 1771 Messier had a salary, a royal pension and the observatory on the Hôtel de Cluny. He kept observing comets, eclipses, occultations and transits, and contributed papers regularly to the Academy’s Mémoires. In 1775 Lalande gave his name to a small constellation he drew on a celestial globe, Custos Messium, the guardian of the harvests, a pun on Messier’s name. It did not survive into modern star atlases.

From about 1774 he worked with Pierre Méchain, a younger astronomer and later director of the Paris Observatory, who found many new nebulae and passed them to Messier. The catalogue grew to 68 objects in the Connaissance des Temps published in 1780, and to 103 in the edition for 1784, completed in 1781. The last entries, many of them Méchain’s, were added hastily and some were not checked. Around 1781 Messier fell about eight metres into an ice cellar and was badly injured. He recovered only slowly, and the catalogue stopped growing.

Revolution and Empire

The Revolution swept away the institutions that paid him. His pension and salary ended, the Academy was abolished in 1793, and he was left poor. He continued to observe. When the Republic rebuilt its scientific institutions, he was appointed to the Bureau des Longitudes in 1796 and became a member of the Institut. In 1806 Napoleon gave him the cross of the Legion of Honour. Two years later Messier published, at his own expense, a memoir on the great comet of 1769, connecting it with the year of the Emperor’s birth. The flattery shows how much he still needed patronage in his late seventies.

His eyesight and health failed in his last years. In 1815 a stroke left him partly paralysed. He died at his home in Paris on 12 April 1817, aged 86, and was buried in the Père Lachaise cemetery. Jean-Baptiste Delambre wrote an account of his life for the Institut.

The catalogue after Messier

Soon after 1781 William Herschel obtained a copy of Messier’s final catalogue. Its hundred-odd objects showed him that the nebulae were a field worth surveying, and with Caroline Herschel he swept the sky from Slough for twenty years and catalogued about 2,500. Their work, extended by John Herschel to the southern sky, became the General Catalogue of 1864 and the New General Catalogue of J. L. E. Dreyer in 1888, which gives every Messier object an NGC number. Messier’s list nevertheless kept its place, because it gathers the brightest and most striking objects visible from northern latitudes.

The list also grew. In the twentieth century historians found, in Messier’s and Méchain’s notes, observations of objects that belonged in it. Camille Flammarion added M104 in 1921; Helen Sawyer Hogg added M105 to M107 in 1947; Owen Gingerich added M108 and M109 in 1960; and Kenneth Glyn Jones added M110, a companion of the Andromeda galaxy, in the 1960s. The standard list has 110 entries. In early spring the Sun lies in a part of the sky with no Messier objects, and amateur astronomers attempt a Messier marathon, observing all of them between dusk and dawn.

Legacy

• Comets: about 13 discoveries now credited to him, a careful recovery of Halley’s comet in 1759, and the discovery of Lexell’s comet, the closest comet ever recorded.
• The catalogue: 110 objects, the first systematic list of non-stellar objects, now a standard observing list and a sampler of astrophysics, from star formation to supermassive black holes.
• Honours: Fellow of the Royal Society (1764), member of the Paris Academy (1770), of the Bureau des Longitudes (1796) and of the Legion of Honour (1806). The lunar crater Messier and the asteroid 7359 Messier are named after him.

Sources and further reading

Biography

1. Charles Messier, Wikipedia (with the list of comets credited to him).
2. Observatoire de Paris, Bibnum, Charles Messier (1730–1817).
3. Charles Messier and Charles Messier, French and German Wikipedia.
4. J. J. O’Connor and E. F. Robertson, Charles Messier, MacTutor History of Mathematics.
5. J.-B. J. Delambre, Histoire de l’astronomie au dix-huitième siècle, Paris, 1827 (contains his account of Messier).
6. K. G. Jones, Messier’s Nebulae and Star Clusters, 2nd ed., Cambridge University Press, 1991.
7. Encyclopaedia Britannica, 11th ed. (1911), article “Messier, Charles”.

Astronomy

1. C. Messier, Catalogue des nébuleuses et des amas d’étoiles, Mémoires de l’Académie royale des sciences for 1771 (published 1774); final version in Connaissance des Temps for 1784.
2. Messier object, Wikipedia (editions, additions, classification).
3. Lexell’s Comet and Crab Nebula, Wikipedia.
4. Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope results. I. The shadow of the supermassive black hole, arXiv:1906.11238.
5. J. M. A. Danby, Fundamentals of Celestial Mechanics, 2nd ed., Willmann-Bell, 1988 (Kepler’s equation, Tisserand’s criterion).
6. C. D. Murray and S. F. Dermott, Solar System Dynamics, Cambridge University Press, 1999 (Jacobi integral and Tisserand parameter).
7. D. R. Lorimer and M. Kramer, Handbook of Pulsar Astronomy, Cambridge University Press, 2005 (spin-down and characteristic age).