Work
While the observatory was still in construction Bessel elaborated the Fundamenta Astronomiae based on Bradley's observations. As a preliminary result he produced tables of atmospheric refraction that won him the Lalande Prize from the French Academy of Sciences in 1811. The Königsberg Observatory began operation in 1813.
Starting in 1819, Bessel determined the position of over 50,000 stars with a meridian circle from Reichenbach, assisted by some of his qualified students. The most prominent of them was Friedrich Wilhelm Argelander, his successors were Otto August Rosenberger and August Ludwig Busch.
Bessel determined the first reliable value for the distance between a star and the Solar System with a heliometer from Fraunhofer using the method of stellar parallax. In 1838 he published a parallax of 0.314 arcseconds for 61 Cygni, which indicated that the star is 10.3 ly away. Compared with the current measurement of 11.4 ly, Bessel's figure had an error of 9.6%. Thanks to these results astronomers had not only enlarged the vision of the universe well beyond the cosmic magnitude, but after the discovery in 1728 by James Bradley of the aberration of light a second empirical evidence of the Earth's relative movement was produced. A short time later Friedrich Georg Wilhelm Struve and Thomas Henderson reported the parallaxes of Vega and Alpha Centauri.
Precise measurements with a new meridian circle from Adolf Repsold allowed Bessel to notice deviations in the motions of Sirius and Procyon, which must be caused by the gravitational attraction of unseen companions.
His announcement of Sirius's "dark companion" in 1844 was the first correct claim of a previously unobserved companion by positional measurement, and eventually led to the discovery of Sirius B by Alvan Graham Clark in 1862, the first discovery of a white dwarf. John Martin Schaeberle discovered Procyon B in 1896.
Bessel was the first scientist who realized the effect later called personal equation, that several simultaneously observing persons determine slightly different values, especially recording the transit time of stars.
In 1824, Bessel developed a new method for calculating the circumstances of eclipses using the so-called Besselian elements. His method simplified the calculation to such an extent, without sacrificing accuracy, that it is still in use.
On Bessel's proposal the Prussian Academy of Sciences started the edition of the Berliner Akademische Sternkarten (Berlin Academic Star Charts) in 1825 as an international project with Johann Franz Encke as executive editor. One unpublished new chart enabled Johann Gottfried Galle to find Neptune near the position calculated by Le Verrier in September 1846 at Berlin Observatory.
In the 18th century a series of cylinder functions were introduced to solve string vibration problems by Daniel Bernoulli, Leonhard Euler and others. These functions were used by Joseph-Louis Lagrange and Bessel to solve the Kepler's equation. Bessel later systematically studied the mathematical properties of these functions, now known as Bessel functions. These functions are critical for the solution of certain differential equations, these functions are used throughout both classical and quantum physics. In his studies of this functions, Bessel published for the first time the Fourier series, before Joseph Fourier who had worked on it but his work was published decades later.
A correction term in the formula for the sample variance estimator is named in his honour. This is the use of the factor n − 1 in the denominator of the formula, rather than just n. This occurs when the sample mean rather than the population mean is used to centre the data and since the sample mean is a linear combination of the data the residual to the sample mean overcounts the number of degrees of freedom by the number of constraint equations — in this case one.
Like numerous astronomers of his time Bessel dealt on the field of geodesy, too,
first theoretically, when he published a method for solving the main geodetic problem.
In 1830 he got the royal order for the survey of East Prussia with the purpose to connect the yet existing Prussian and Russian triangulation networks. This work was carried out in cooperation with Johann Jacob Baeyer, then major of the Prussian army; the final report was published in 1838.
He also obtained an estimate of increased accuracy for the Earth's ellipsoid, nowadays called the Bessel ellipsoid, based on several arc measurements.
