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Bessel's Astronomical Contributions Bessel’s Astronomical Contributions By Dean Pandelaras Friedrich Wilhelm Bessel was one of the most influential astronomers of the early nineteenth century. Like many of the great mathematicians at the time and those that preceded him, Bessel’s interest in mathematics was motivated by his interest in studying the physical world. Through his clear passion and dedication towards understanding the physical world, Bessel contributed immensely to the fields of astronomy and mathematics. Bessel was born on July 22nd, 1784, the son of a poor civil servant in Westphalia (present day Germany).5 As a young child, he attended the Gymnasium in Minden for four years, appearing to be not very talented and struggling with Latin.5 At the age of 14, he began an apprenticeship with an import-expert firm where he developed his accounting skills.4 Interested in travel, Bessel studied geography, Spanish, English, and navigation.5 The principles of navigation led Bessel to study astronomy and mathematics in order determine longitude, after considering the problem of finding the position of a ship at sea.4 In 1804, Bessel wrote a paper in which he calculated the orbit of Halley’s Comet from observations made by Harriot in 1607.5 He sent his results to Olbers, a renowned comet expert at the time, who recognized the quality of Bessel’s work and had his paper published.4 Olbers recommended that Bessel relinquish his affluent career at the import-export firm to pursue astronomy professionally, which he did in 1806.5 While Bessel’s life prior to his academic career cannot attest to the extent of his achievements as a scientist, it does provide insight into his character and aptitude. Despite leaving his formal education at the age of 14 with a lackluster record, Bessel pursued his interests in astronomy and mathematics in his free time. Notice, his interests in astronomy and mathematics were derived initially from the practical problem of sea navigation. This shows that even at a young age, Bessel’s interest in the physical world motivated much of his exploration. By the age of twenty years old, he was astoundingly able to gain the respect of comet expert Olbers. With this in mind, one could conclude that his poor performance during formal education was likely a result of disinterest, rather than ability. Furthermore, his choice to forgo the financial safety of working for the import-export firm, for the willful poverty that was professional astronomy prior to significant accomplishment, was a decision that not all men would be willing to undertake. His overwhelming desire to pursue knowledge must have superseded most other interests and values in his life. Overall, Bessel’s passion, dedication, and aptitude towards understanding the physical world and in particular astronomy was evident even in adolescence. Much of Bessel’s achievements as a scientist relate to the observation, calculation, and documentation of star positions and motions. In 1807, he began correcting the measurement errors (related to refraction) in James Bradley’s observations of the mean positions of 3,222 stars.4 This work propelled Bessel to the international stage, where he was awarded a doctorate by the University of Göttingen on the recommendation of Gauss as well as the honor of the Lalande Prize from the Institut de France.5 Shortly after, he was also elected to the Berlin Academy.5 Following this, Bessel became a professor of astronomy at the Königsberg Observatory where he undertook the task of determining the positions and proper motions of over 50,000 stars.5 This eventually led to one of Bessel’s most profound achievements: the accurate demonstration of the parallax of 61 Cygni in 1838.4 The profundity of these accomplishments cannot be overlooked. Bessel had dedicated most of his adult life contributing to the most accurate and largest astronomical database of the time. By rigorously accounting for optical, mechanical, and meteorological sources of error, Bessel revolutionized the field of positional astronomy with more accurate and precise observations. However, his fairly accurate demonstration of the parallax of 61 Cygni is by far Bessel’s most historically significant contribution to our understanding of the universe. It provided the first solid evidence to support Copernicus’s claim that the Earth revolved around the Sun.2 Simply put, parallax is the displacement of an observation caused by the change of viewpoint.2 An exercise to help understand the phenomenon of parallax is to observe an object close to eye level. One would first observe the position of such object by covering one eye. Then, by observing the same object with just the other eye, one should notice a shift in position caused by the change of perspective. This shift is called parallax. It can also be observed that the parallax phenomenon increases the closer the object is to the viewer. Therefore, it was theorized that parallax should be observable if the Earth revolved around the Sun. Understanding this, Bessel chose 61 Cygni, a relatively dim star, but one he deduced to be nearby via its proper motion.5 Bessel ultimately calculated a value of 0.314”, which is very close to the correct value of parallax that is known today, 0.292” (measured in seconds of arc).5 Finally, Bessel’s most well-known accomplishment and his greatest contribution to the field of mathematics is his generalization of what are known as Bessel functions. A Bessel function is defined as a solution to the differential equation, called the Bessel Equation.3 1 1 Bessel Equation Bessel Function (First Kind, p = n) ∞ 푑2푦 푑푦 (−1)푘 푥 2푘+푛 푠푠푠푠푠푠푥2 + 푥 + (푥2 − 푝2)푦 = 0 퐽 (푥) = ∑ ( ) 푑푥2 푑푥 푛 푘! (푘 + 푛)! 2 푘=0 Qualitatively, graphs of Bessel functions (first kind) of order n resemble a damped function, oscillating over the x-axis. It can also be thought to be similar to a sine or cosine function decreasing in amplitude as x increases. Bessel first introduced these functions in his study of Kepler’s problem of determining the motion of three bodies moving under mutual gravitation.4 However, particular cases of Bessel functions had been developed earlier by Daniel Bernoulli while studying the oscillations of a hanging chain, and Euler while studying the vibrations of a membrane.3 Later, Bessel further developed Bessel functions in order to study planetary perturbations, the disturbance in orbit of planets caused by additional bodies.5 Following the development of Bessel functions, several of their applications became apparent in describing physical phenomena, including the flow of heat in a cylinder, the diffraction of light, fluid mechanics, and the deformations of elastic bodies.3 Overall, Bessel contributed immensely to the fields on astronomy and mathematics. Motivated by his study of the physical world, Bessel was first and foremost an astronomer. However, the practical need for a rigorous language to describe the motion of celestial bodies and analyze observational data required Bessel to be an equally strong mathematician. With the shoulders of Newton, Euler, and several others to stand on, Bessel was able to accomplish two uniquely difficult feats in astronomy and mathematics: the first accurate observation of parallax of a star and the generalization of Bessel functions. Through his lifelong dedication and passion for understanding the physical world, Bessel revolutionized the fields of positional astronomy and applied mathematics. Works Cited 1Asmar Nakhlé H.. Partial Differential Equations with Fourier Series and Boundary Value Problems. 2nd ed., Pearson Education, Inc., 2000. 2"1838: Friedrich Bessel Measures Distance to a Star." Carnegie Science, Carnegie Science, cosmology.carnegiescience.edu/timeline/1838. 3"Bessel function." Encyclopaedia Britannica, Encyclopaedia Britannica, Inc., 22 Oct. 2007, www.britannica.com/science/Bessel-function. 4"Friedrich Wilhelm Bessel." Encyclopaedia Britannica, Encyclopaedia Britannica, Inc., 18 July 2019, www.britannica.com/biography/Friedrich-Wilhelm-Bessel. 5O'Connor, J. J., and E. F. Robertson. "Friedrich Wilhelm Bessel." MacTutor History of Mathematics archive, MacTutor, www-history.mcs.st-andrews.ac.uk/Biographies/Bessel.html. .
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