Friedrich Wilhelm Bessel

How to use this course

Friedrich Wilhelm Bessel left school at fourteen and never attended a university. He became the astronomer who set the standard for what a measurement means: an observation is worth nothing until every source of error has been modelled, estimated and removed. With that discipline he turned James Bradley’s Greenwich observations into the first modern star catalogue, measured the distance of a star for the first time, inferred unseen stars from the wobble of visible ones, and determined the figure of the Earth used for a century of mapping. Along the way he systematized the functions that carry his name.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Angles are in arcseconds (″) unless stated; one radian is \(206\,265''\).

Timeline

Minden and Bremen, 1784–1805

Friedrich Wilhelm Bessel was born in Minden, in the Prussian province of Minden-Ravensberg, on 22 July 1784, the second son of a civil servant in a large family. He disliked the Latin-centred teaching of the Minden Gymnasium and left in January 1799, at fourteen. He was apprenticed as a clerk to the import-export house of Kulenkamp in Bremen.

The firm traded overseas, and Bessel taught himself geography, Spanish and English in his spare time, and then navigation. Navigation led to the problem of finding longitude at sea, which led to astronomy, and he taught himself the mathematics of orbits from books. He became a salaried accountant in the firm and studied in the evenings.

Halley’s comet

In 1804 he reduced the observations of the comet of 1607, made by Thomas Harriot and Nathaniel Torporley, and computed its orbit. The comet was Halley’s, on an earlier return. He took the work to Heinrich Wilhelm Olbers, a Bremen physician and the leading authority on comets. Olbers saw that the reduction was of professional quality and arranged its publication. He became Bessel’s patron and friend, and steered him toward astronomy as a career. The same year Bessel began a correspondence with Carl Friedrich Gauss that lasted almost forty years.

Lilienthal, 1806–1810

In 1806, on Olbers’s recommendation, Bessel left Kulenkamp to become assistant at the private observatory of Johann Hieronymus Schröter in Lilienthal, near Bremen, succeeding Karl Ludwig Harding. The salary was much lower than his commercial one. Schröter’s large reflectors were built for describing planets, not for measuring positions, and Bessel used his time there to master observing and, above all, reduction.

In 1807 he began the project that would define his career: the reduction of Bradley’s Greenwich observations of 1750–1762, published at Oxford in 1798 and 1805. Bradley had made the observations with the best instruments of his century, but nobody had extracted their full accuracy. That year he also met Gauss in Bremen. Gauss had just computed the orbit of Ceres, and the two men shared a conviction that observation and calculation belonged together.

Bessel’s work on Bradley’s data made his name. In 1809–10 King Frederick William III of Prussia, reorganizing the University of Königsberg after the defeat by Napoleon, appointed him director of a new observatory and professor of astronomy. He was twenty-five. Since he had no degree, the University of Göttingen awarded him an honorary doctorate on Gauss’s recommendation.

Building Königsberg, 1810–1818

Bessel arrived in Königsberg in May 1810. The observatory did not yet exist; it was completed in 1813, and in the meantime he taught. In 1812 he married Johanna Hagen, daughter of the Königsberg chemist and pharmacist Karl Gottfried Hagen. The physicist Franz Neumann later married her sister, so the two leading scientists of Königsberg became brothers-in-law. Bessel was elected to the Berlin Academy in 1812.

His first great prize, the Lalande Prize of the Paris Institute in 1811, was for refraction tables (Lesson 10). The great work of these years was the completion of the Bradley reduction. In 1818 he published Fundamenta Astronomiae pro anno 1755, positions of 3,222 stars for the epoch 1755. Every step was set out: Bradley’s instrumental errors, the refraction, precession, nutation and aberration constants, and the clock corrections. The catalogue is usually regarded as the beginning of modern astrometry, and it fixed reliable positions a lifetime earlier than any other source.

Reducing Bradley: from observed to mean place

For the general reader. The position of a star measured on a particular night is distorted by many small effects: the Earth’s motion (aberration), the slow swing of its axis (precession and nutation), the air (refraction) and the instrument. To compare observations made decades apart, every one must be corrected to a common standard. Bessel organized these corrections into a compact scheme of a few numbers per day that every observatory used for more than a century.

Besselian day numbers

Precession and nutation depend on the date and on the star only through simple trigonometric factors. So does annual aberration. Bessel therefore separated each correction into a part depending only on the date and a part depending only on the star. He tabulated the date parts in the Tabulae Regiomontanae (1830).

Derivation of the aberration terms. The Earth’s unit velocity on a circular orbit is \((\sin\lambda_\odot,-\cos\lambda_\odot,0)\) in ecliptic coordinates; rotating about the \(x\)-axis by \(\varepsilon\) gives the equatorial vector \(\hat v=(\sin\lambda_\odot,-\cos\lambda_\odot\cos\varepsilon,-\cos\lambda_\odot\sin\varepsilon)\). The displacement \(\kappa\,\hat v_\perp\) has components \(\kappa\,\hat v\cdot\mathbf e_\alpha\) and \(\kappa\,\hat v\cdot\mathbf e_\delta\), with \(\mathbf e_\alpha=(-\sin\alpha,\cos\alpha,0)\) and \(\mathbf e_\delta=(-\sin\delta\cos\alpha,-\sin\delta\sin\alpha,\cos\delta)\). Then

Dividing the first line by \(\cos\delta\) gives \(Cc+Dd\). The precession and nutation terms follow in the same way from the formulas of the Bradley module.

Proper motion

With Bradley’s positions for 1755 reduced to a common system, comparison with later catalogues gave reliable proper motions for thousands of stars. Giuseppe Piazzi had already noticed in 1792, by comparison with Bradley, that 61 Cygni moves exceptionally fast. Bessel’s system made such comparisons routine, and the proper motions in turn selected the stars most likely to be near (Lesson 07).

Errors of observation

For the general reader. No two measurements agree exactly. Bessel insisted on asking why, and on separating two kinds of error. Random errors scatter and can be beaten down by averaging. Systematic errors push every measurement the same way and must be found and corrected. He discovered one systematic error in the observers themselves: different astronomers, watching the same star cross the same wire, record different times.

Least squares

Adrien-Marie Legendre published the method of least squares in 1805, and Gauss justified it by probability in 1809. Bessel was among its most systematic users. Given observations \(y_i=\mathbf x_i^{\mathsf T}\boldsymbol\theta+\epsilon_i\) with independent errors of variance \(\sigma^2\), the estimate minimizing \(\sum\epsilon_i^2\) solves the normal equations \(X^{\mathsf T}X\hat{\boldsymbol\theta}=X^{\mathsf T}\mathbf y\).

Proof. \(\sum(y_i-\bar y)^2=\sum(y_i-\mu)^2-n(\bar y-\mu)^2\). The first term has expectation \(n\sigma^2\), the second \(n\cdot\sigma^2/n=\sigma^2\). For the general case, the residual vector is \((I-H)\boldsymbol\epsilon\) with \(H=X(X^{\mathsf T}X)^{-1}X^{\mathsf T}\) a projection of rank \(p\), so \(\mathbb E\|(I-H)\boldsymbol\epsilon\|^2=\sigma^2\operatorname{tr}(I-H)=(n-p)\sigma^2\).

The factor \(n-1\) is called Bessel’s correction. Gauss had the result in 1823, and the name is a convention, but it reflects Bessel’s constant attention to the honest estimate of errors. He reported results with their probable error, the half-width of the interval containing half the errors; for a normal distribution it is \(0.6745\,\sigma\). In Fundamenta Astronomiae he compared the distribution of residuals in Bradley’s observations with Gauss’s law of errors and found close agreement, and in 1838 he argued that the normal law arises when an error is the sum of many small independent causes.

The personal equation

In 1796 Nevil Maskelyne, the Astronomer Royal, dismissed his assistant David Kinnebrook, whose transit times were persistently about \(0.8\) s later than his own. Bessel read of the episode and in the early 1820s compared his own transit times with those of other skilled observers. He found consistent differences of a substantial fraction of a second, stable for each pair of observers. This personal equation was a systematic error of the human observer, and correcting for it became standard practice. It was also one of the starting points of experimental psychology’s study of reaction times.

Kepler’s equation and Bessel functions

For the general reader. To predict where a planet will be, one must solve Kepler’s equation, which links time to the angle along the orbit. There is no formula in elementary functions, but the solution can be written as a sum of sine waves whose amplitudes are values of special functions. Bessel studied these functions systematically, first for Kepler’s problem and then for planetary perturbations. They turned out to describe vibrating drums, waves in pipes and light through a circular lens as well.

Kepler’s equation

For an elliptic orbit of eccentricity \(e<1\), the mean anomaly \(M=n(t-t_0)\) increases uniformly with time and the eccentric anomaly \(E\) satisfies

The difference \(E-M=e\sin E\) is an odd, \(2\pi\)-periodic function of \(M\).

Proof. Write \(E-M=\sum b_n\sin nM\) with \(b_n=\frac2\pi\int_0^\pi(E-M)\sin nM\,dM\). Integrate by parts; the boundary terms vanish because \(E-M=0\) at \(M=0\) and \(M=\pi\):

since \(\int_0^\pi\cos nM\,dM=0\). Now change variable to \(E\), which also runs from \(0\) to \(\pi\), and substitute \(M=E-e\sin E\): \(b_n=\frac{2}{n\pi}\int_0^\pi\cos(nE-ne\sin E)\,dE=\frac2n J_n(ne)\).

Properties

The integral form is the one Bessel used in his 1824 memoir on the part of planetary perturbations caused by the motion of the Sun, where he derived the main properties. Particular cases had appeared earlier, in Daniel Bernoulli’s hanging chain (1732), in Euler’s vibrating membrane, in Lagrange’s 1770 series for Kepler’s equation, and in Fourier’s cooling cylinder. Bessel’s treatment made them a single family, and Peter Andreas Hansen introduced the generating function in 1843.

The parallax of 61 Cygni

For the general reader. For three centuries astronomers had tried to measure the distance of a star by watching it shift as the Earth goes round the Sun. The shift is smaller than a third of an arcsecond even for the nearest stars. Bessel succeeded in 1838 by choosing a star likely to be near, measuring only its distance from faint neighbouring stars with a special split-lens telescope, and modelling every error. The answer, about ten light years, gave the first true scale of the stellar universe.

Choosing the star

Brightness is a poor guide to distance, since stars differ greatly in luminosity. Bessel chose 61 Cygni because its proper motion, more than \(5''\) per year, was the largest he knew. Tangential velocity is proportional to proper motion times distance, and stars move at broadly similar speeds, so large proper motion suggests nearness. The star is also a wide double, which helped confirm that both components share the motion.

The heliometer

A heliometer is a refractor whose objective lens is cut along a diameter; the two halves slide along the cut, producing two images of the field whose separation is read from a precise scale. Bringing the image of one star onto the other image of another measures their separation with great accuracy. The Königsberg heliometer was the last instrument designed by Joseph von Fraunhofer and was installed in 1829, after his death. Measuring only separations of nearby stars cancels refraction, aberration, nutation and precession to first order, because they affect both stars almost equally.

Derivation. The observer sits at \(-\mathbf R_\odot\), where \(\mathbf R_\odot=(X,Y,Z)\) points from Earth to Sun, so the apparent direction is that of \(D\hat s+\mathbf R_\odot\), and to first order the displacement is \(\varpi(\mathbf R_\odot-(\mathbf R_\odot\cdot\hat s)\hat s)\). Taking components along \(\mathbf e_\alpha=(-\sin\alpha,\cos\alpha,0)\) and \(\mathbf e_\delta\) and noting that the observer’s displacement is opposite to the Sun’s vector gives the formulas.

Bessel’s measurement

Over more than a year in 1837 and 1838 Bessel measured the separation of 61 Cygni from two faint comparison stars, one about \(462''\) and the other about \(706''\) away, in more than a hundred sets. He modelled each separation as a constant plus a linear term (the relative proper motion) plus the projection of the parallactic displacement onto the line joining the stars, and fitted the unknowns by least squares. His result, published in the Astronomische Nachrichten in 1838, was \(\varpi=0.314''\pm0.020''\). A remeasurement with his assistant Schlüter gave \(0.348''\). The modern value is \(0.286''\). Shortly afterwards Thomas Henderson announced a parallax for α Centauri from observations at the Cape of Good Hope, and Friedrich Georg Wilhelm Struve, who had given a tentative value for Vega in 1837, published an improved one in 1840.

Invisible companions

For the general reader. Bessel’s catalogues were good enough to show that the bright stars Sirius and Procyon do not move in straight lines: they weave slightly from side to side with a period of decades. He concluded that each orbits an unseen companion, a dark star in his words. Sirius’s companion was seen in 1862 and Procyon’s in 1896. Both are white dwarfs, small and faint but as massive as stars like the Sun.

Derivation. In the centre-of-mass frame \(m_A\mathbf r_A+m_B\mathbf r_B=0\), so each star traces a scaled copy of the relative orbit, scaled by \(m_B/(m_A+m_B)\) for A. Kepler’s third law for the relative orbit is \(a^3/P^2=G(m_A+m_B)/4\pi^2\), which in units of AU, years and solar masses is \(a^3/P^2=m_A+m_B\).

Bessel had only the visible star’s positions, so he could find \(P\) and the wobble \(a_A\) but not \(a\) or the masses separately. He published the argument in 1844, noting that a star need not be luminous to have mass. Christian August Friedrich Peters computed the orbit of Sirius about the centre of mass in 1851, and Alvan Graham Clark saw the companion while testing a new lens in 1862. The method is the ancestor of the astrometric detection of exoplanets.

Arc measurement and the Bessel ellipsoid

For the general reader. The Earth is slightly flattened. To make maps, surveyors need its exact size and shape. Bessel measured an arc of meridian in East Prussia and then combined it by least squares with the other arc measurements made around the world. His ellipsoid of 1841 served as the reference for the maps of Germany, Austria-Hungary, Japan and other countries for more than a century.

Derivation. Parametrize the meridian ellipse by \(x=a\cos\varphi/W\), \(z=a(1-e^2)\sin\varphi/W\) with \(W=\sqrt{1-e^2\sin^2\varphi}\); one checks that the normal at this point makes angle \(\varphi\) with the equator. Then \(M=ds/d\varphi=\sqrt{x'^2+z'^2}\). A direct computation gives \(x'=-a(1-e^2)\sin\varphi/W^3\) and \(z'=a(1-e^2)\cos\varphi/W^3\), hence \(M=a(1-e^2)/W^3\).

The East Prussian arc

In 1830 a royal order asked Bessel to connect the Prussian and Russian triangulations. With Johann Jacob Baeyer he measured a chain of triangles in 1831–32 from a carefully measured baseline, determined latitudes and azimuths astronomically at the ends, and published Gradmessung in Ostpreussen in 1838. The analysis introduced methods of adjustment that became standard in geodesy. In 1841 he combined it with the other arc measurements then available, by least squares, to obtain

Refraction and the seconds pendulum

For the general reader. Two of Bessel’s most lasting contributions were corrections that everyone else needed. His refraction tables told observers how much the air lifts each star, depending on altitude, temperature and pressure. His pendulum experiments fixed the strength of gravity at Königsberg and Berlin, and they exposed a subtle error: a swinging pendulum drags air along with it, which makes it swing more slowly than a pendulum in vacuum.

Refraction

The leading term follows from Snell’s law in a plane-parallel atmosphere, \(n_0\sin z=\sin(z+R)\), which gives \(R\approx(n_0-1)\tan z\); the curvature of the layers produces the \(\tan^3z\) term. The scaling with \(P/T\) comes from the Gladstone–Dale relation \(n-1\propto\rho\) and the ideal gas law. Bessel’s tables, first in Fundamenta Astronomiae and then in Tabulae Regiomontanae, were the standard for decades.

The seconds pendulum

In 1826 Bessel measured the length of the seconds pendulum at Königsberg. He used one ball on a fine wire at two lengths differing by a precisely measured amount, so that the uncertain position of the centre of oscillation cancelled. He also showed that the air affects the period in two ways: buoyancy reduces the effective weight, and the air carried along increases the effective inertia. The second effect had been missed by earlier experimenters.

Königsberg, 1818–1846

With a Reichenbach meridian circle installed in 1819, Bessel began in 1821 a survey of the sky in zones, assisted above all by Friedrich Wilhelm Argelander. By the mid-1830s the zones contained 75,011 observations in 536 zones between declinations \(-15^\circ\) and \(+45^\circ\). In 1825 he proposed the Berlin Academy star charts, completed under Johann Franz Encke’s editorship; one of them helped Johann Gottfried Galle find Neptune in 1846. Bessel had himself argued from the irregularities of Uranus that a more distant planet existed.

He also made Königsberg a centre of teaching. Although he had never studied at a university, he helped Carl Gustav Jacob Jacobi and Franz Neumann reform instruction in mathematics and physics, and Neumann carried Bessel’s methods of measurement and error analysis into the mathematical-physics seminar he directed with Jacobi. His doctoral students included Argelander (1822), Heinrich Scherk and Carl August von Steinheil. His correspondence with Gauss, begun in 1804, continued until 1843, though it was strained by a quarrel in 1837 about Gauss’s reluctance to publish.

Bessel and Johanna had two sons and three daughters. Their elder son, an architect, died in 1840 at 26, and the younger died in infancy. Their daughter Marie married the physicist Georg Adolf Erman. In 1842 Bessel travelled to England for the British Association meeting in Manchester, with Jacobi and Erman, and met John Herschel and other British scientists. His health was failing, and after a long illness he died at the observatory on 17 March 1846.

Legacy

The Royal Astronomical Society gave Bessel its Gold Medal for the second time in 1841, for the parallax of 61 Cygni. His deeper legacy was a method. Bessel showed that the precision of a result depends less on the instrument than on the completeness of the model of its errors, and he made that the working standard of astronomy and geodesy.

• Astrometry: the reduction of Bradley’s observations, the Besselian day numbers, the zone catalogues and the systematic treatment of instrumental and personal errors.
• Distances: the first stellar parallax and the first true scale of the stellar system.
• Dynamics: the inference of unseen masses from motions, applied to Sirius and Procyon and, much later, to exoplanets and dark matter.
• Geodesy: the Bessel ellipsoid, used for national mapping in central Europe and Japan into the twentieth century.
• Mathematics: the functions \(J_n\), which appear in every problem with cylindrical symmetry, and Bessel’s inequality for orthonormal expansions.
• Honours: Berlin Academy (1812), Lalande Prize (1811, 1816), Royal Society of Edinburgh (1823), Fellow of the Royal Society (1825), RAS Gold Medal (1829, 1841), Order Pour le Mérite (1842). A lunar crater and the asteroid 1552 Bessel bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Friedrich Wilhelm Bessel, MacTutor History of Mathematics.
2. Friedrich Bessel, Wikipedia.
3. Biographical Encyclopedia of Astronomers, Bessel, Friedrich Wilhelm.
4. Mathematics Genealogy Project, Friedrich Wilhelm Bessel.
5. K. Lawrynowicz, Friedrich Wilhelm Bessel 1784–1846, Birkhäuser, 1995.

Science

1. F. W. Bessel, Fundamenta Astronomiae pro anno MDCCLV, Königsberg, 1818.
2. F. W. Bessel, Untersuchung des Theils der planetarischen Störungen, welcher aus der Bewegung der Sonne entsteht, Abh. Akad. Wiss. Berlin (1824).
3. F. W. Bessel, Tabulae Regiomontanae reductionum observationum astronomicarum ab anno 1750 usque ad annum 1850 computatae, Königsberg, 1830.
4. F. W. Bessel and J. J. Baeyer, Gradmessung in Ostpreussen und ihre Verbindung mit Preussischen und Russischen Dreiecksketten, Berlin, 1838.
5. F. W. Bessel, Bestimmung der Entfernung des 61sten Sterns des Schwans, Astronomische Nachrichten 16 (1838).
6. The first stellar parallaxes revisited, arXiv:2009.11913.
7. 61 Cygni and Bessel function, Wikipedia.
8. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge, 1944, chapter 1 (history).
9. W. M. Smart, Textbook on Spherical Astronomy, 6th ed., Cambridge, 1977, chapters on refraction, star reductions and parallax.