Introduction

In this term paper, we will get acquainted with the Bessel equation and its application in the equations of mathematical physics. Bessel functions were first defined by the Swiss mathematician Daniel Bernoulli and named after Friedrich Bessel.

Friedrich Wilhelm Bessel

German mathematician and astronomer of the 19th century. Born July 22, 1784 in Minden. Independently studied mathematics and astronomy, in 1804 he calculated the orbit of Halley's comet. In 1806 he became an assistant to the prominent astronomer I. Schroeter in Lilienthal, and soon gained a reputation as a prominent astronomer-observer and calculator-mathematician. In this capacity, in 1810 he was invited to Königsberg University to organize an observatory, the director of which he remained until the end of his life. Believing that it is necessary to make corrections to the results of observations, taking into account the presence of the most insignificant factors, Bessel developed mathematical methods for correcting the results of observations. The first work in this direction was the correction of the positions of the stars in the catalog compiled in the 18th century. English astronomer J. Bradley. In the future, Bessel himself conducted observations of the stars; in 1821-1833 he determined the position of more than 75 thousand stars and compiled extensive catalogs that formed the basis of modern knowledge about the starry sky.

Bessel was one of the first to measure the parallaxes of stars and their distance. In 1838 he determined the distance to the binary star 61 Cygnus, which turned out to be one of the closest to the solar system. Observing for a number of years the bright stars Sirius and Procyon, Bessel discovered features in their trajectories that could only be explained by the presence of satellites. These assumptions were subsequently confirmed: in 1862, the satellite of Sirius was discovered, and in 1896, the satellite of Procyon. Known are Bessel's works in the field of geodesy (determining the length of the second pendulum, the invention of the basic instrument).

The Bessel equation arises while finding solutions to the Laplace equation and the Helmholtz equation in cylindrical and spherical coordinates. Therefore, Bessel functions are used in solving many problems of wave propagation, static potentials, etc., for example:

· electromagnetic waves in a cylindrical waveguide;

thermal conductivity in cylindrical objects;

Bessel functions are also used in solving other problems, for example, in signal processing.

Bessel equation

When solving many problems of mathematical physics, they come to a linear differential equation:

where is a constant. This equation is also encountered in many questions of physics, mechanics, astronomy, and so on. Equation (1) is called Bessel equation. Since equation (1) has a singular point x = 0, its particular solution should be sought in the form of a generalized power series:

Substituting series (2) into equation (1), we obtain

Equating to zero the coefficients at various powers of x, we will have:

From the first equality we find two values ​​for p: p 1 = and p 2 =-

If we take the first root p = , then from formulas (5) and (6) we get:

It follows that a 2k+1 =0 (k=2, 3, 4,…), and the coefficients with even indices are obviously determined by the formulas:

From which it is clear that the general expression for the coefficients has the following form:

As for the coefficient a 0 , which has been completely arbitrary so far, we choose it in this way:

where Г() is the gamma function, which is defined for all positive values ​​(as well as for all complex values ​​with a positive real part) as follows:

With this choice of a 0, the coefficient a 2k can be written as:

This expression can be simplified by using one of the main properties of the gamma function. To do this, we integrate the right side of equality (8) by parts; then we get the following basic formula:

Note that formula (10) makes it possible to determine the gamma function for negative values, as well as for all complex values.

Let k be some positive integer. Applying formula (10) several times, we obtain

Setting = 0 in this formula, we find, by virtue of the equality

another important property of the gamma function, expressed

G(k+1) = k! (12)

bessel equation function orthogonality

Using formula (11), expression (9) for the coefficient a 2k will take the following form:

Entering the found values ​​of the coefficients a 2k+1 and a 2k into series (2), we obtain a particular solution of equation (1). This solution is called the Bessel function of the 1st kind - order and is usually denoted by J V (x).

Thus,

Series (14) converges for any value of x, which is easy to verify by applying the d'Alembert test.

Using the second root p 2 =-, you can build the second particular solution of equation (1). It can be obtained, obviously, from the solution (14) by a simple replacement by -, since equation (1) contains only 2 and does not change when replaced by -:

If not equal to an integer, then particular solutions J V (x) and J-V (x). the Bessel equations (1) will be linearly independent, since the expansions on the right-hand sides of formulas (14) and (15) start from different powers of x. If there is a positive integer n, then in this case it is easy to detect a linear dependence of the solutions J n (x) and J -n (x). Indeed, with an integer for k = 0, 1, 2, ..., n- 1, the value - + k + 1 takes integer negative values ​​or zero. For these values ​​of k: Г(-+k+1)=, which follows from the formula:

Thus, the first n terms in expansion (15) vanish and we get

or, putting k= n + l, we get

It follows that for an integer n the functions J n (x) and J -n (x) are linearly dependent.

In order to find the general solution of equation (1), when equal to an integer n, it is necessary to find the second, linearly independent of J V (x), particular solution. To do this, we introduce a new function Y v (x), setting

It is obvious that this function is also a solution to equation (1), since it is a linear combination of partial solutions JV (x) and

J-V(x) of this equation. Then it is easy to see, on the basis of relation (16), that when n is equal to an integer, the right side of equality (17) takes an indefinite form. If we reveal this uncertainty according to L'Hospital's rule, then as a result of a series of calculations (which, due to their complexity, are not reproduced here), we obtain the following representation of the function Y n (x) for a positive integer n:

In a particular case, when n = 0, the function Y o (x) is represented as follows:

The function Y v (x) introduced here is called the Bessel function of the 2nd kind - order or the Weber function.

The Weber function Y v (x) is also a solution to the Bessel equation when is an integer.

The functions J V (x) and Y v (x) are obviously linearly independent, therefore, these functions for any - fractional or integer - form a fundamental system of solutions. This implies that the general solution of Eq. (1) can be represented as

where C 1 and C 2 are arbitrary constants.

German astronomer and surveyor, member of the Berlin Academy of Sciences (1812). R. in the city of Minden in a large family of a petty official. In his youth he was an amateur astronomer. Seriously engaged in self-education. In 1804 he independently calculated the orbit of Halley's comet, which earned him the praise of G. V. Olbers. In 1806 he became an assistant at the private observatory of I. I. Schroeter in Lilienthal. In 1810 he was invited to Königsberg to organize a new observatory, the director of which he worked until the last years of his life.

Bessel is one of the founders of astrometry. He developed a theory of instrument errors and consistently put into practice the idea of ​​the need to make appropriate corrections to the results of observations. When processing the results of observations, he widely used various mathematical methods, in particular, he used the results of probability theory and the least squares method. A. Pannekoek believes that Bessel set a new, higher standard both for designers of astronomical instruments and for the work of astronomers themselves. The methods of reduction of astronomical observations improved by Bessel are described by him in his work "Königsberg Tables" (1830).

Bessel's first major work was a revision of observations of the positions of stars in the famous Bradley catalog compiled in the 1840s and 50s. The results of this work were summarized in the work Fundamentals of Astronomy published in 1818, in which, in addition to a catalog of 3200 stars, the values ​​of the constants of refraction, precession and nutation obtained by Bessel are given with an accuracy much greater than in all previous definitions of these quantities. In the process of this work, fairly accurate refraction tables were compiled.

Bessel was one of the greatest observing astronomers. In 1821-1833, on the Reichenbach meridian circle he established, he observed more than 75,000 stars in the zone from +47 to -16 in declination. Over the course of a number of years, he observed the bright stars Sirius and Procyon on the meridian circle, and established in 1844 that the movement of these stars does not occur in a straight line, but along a wavy line. He suggested that each of these stars has an invisible companion, in other words, these are systems of two bodies revolving around a common center of gravity. This assumption was confirmed in 1862 by A. Clark, who managed to directly observe the satellite of Sirius, and in 1896 by J. Scheberle, who discovered the satellite of Procyon. In 1838, after analyzing his own observations of the binary star 61 Cygni on the Fraunhofer heliometer, Bessel measured its parallax, which turned out to be 0.37" (in 1840 he obtained a more accurate value of 0.35"). Almost simultaneously with Bessel, T. Henderson at the observatory at the Cape of Good Hope measured (1839) the parallax of α Centauri - 0.91", and V. Ya. Struve at the Derpt Observatory determined (1839) the parallax of α Lyra - 0.26". These works were the first successful measurements of parallaxes after centuries of astronomers' attempts to find distances to stars.

Childhood, adolescence and first discovery

In the room given to the student of the Kulenkamps, the only window faced the north side and was inconvenient for astronomical observations. This was the reason that Bessel installed his sextant in the house of his friend I. G. Gelle, where there was a room with large windows looking south, west and east. First, he tested his watch, and was surprised by the accuracy he received - he expected much larger errors from his device. But, as he himself noted, an even more valuable result of this was the ability to perform trigonometric calculations.

Bessel's name became widely known in scientific circles, and more than once he received invitations to take this or that post in other observatories or educational institutions. He was invited to Düsseldorf, and to Greifswald, and to Leipzig, but Bessel does not dare to accept any of the offers. It was a pity to part with the aging Schroeter and, besides, he had no teaching experience.

In 1810 Bessel was invited to Königsberg. He was offered to build an observatory there, equip it with the necessary instruments at public expense, and then lead it. He was also provided with free housing at the future observatory and free fuel. Supported by Olbers. he accepts the invitation. March 27 Bessel leaves the observatory. That was the last time he and Schroeter saw each other. Napoleon, who did not trust his vassals in Germany, annexes part of the German territories, including Lilienthal, to France. During the hostilities, French troops entered Lilienthal and plundered the observatory, broke instruments, broke the chronometer. At that time, Schroeter was not there, and when he returned from Bremen, he was shocked. It was not possible to revive the observatory, and in 1840 the last traces of the observatory, which was one of the best in Europe, were demolished.

University and observatory

In Königsberg, Friedrich Wilhelm Bessel created an observatory and performed the most significant of his works, lectured at the university. It was with Königsberg that his social activities and family life were closely connected. Here he lived for 36 years, and here he was buried. Königsberg of the beginning of the 19th century is a provincial city of Prussia, with a Gothic cathedral of the 14th century, a university of venerable age, the Pillau sea harbor and an old knight's castle. Initially, the Königsberg University was supposed to become one of the ideological instruments in the colonial policy of the former conquering knights (Teutons). For the first two and a half centuries of the existence of the university, the life of the university passed among fruitless religious and dogmatic strife. If we add to this the harsh (by European standards) climate of East Prussia and the remoteness of the city from the main centers of German science, it becomes clear why by the beginning of the 19th century Königsberg University was scientifically one of the most backward academic institutions in the Prussian state. For example, only 11 students were studying at the philosophical and medical faculties of this university at that time, and in total there were 332 students at the university. There was no observatory there either, which seems even a little strange, given that Königsberg was a port city. After the signing of the Treaty of Tilsit, the Prussian king Friedrich Wilhelm III lived in Königsberg, which made him take a closer look at the needs of this city. Bessel's invitation was precisely one of the consequences of this. Fresh scientific forces are attracted to Königsberg University.

In Königsberg he was received very kindly. Despite the lack of pedagogical experience, the beginning of pedagogical work turned out to be quite successful: he read lectures very willingly and with a full audience. But there were still some problems - the leadership of the Faculty of Philosophy was negative about the fact that they teach a person without academic degrees. Bessel was made clear that he needed a diploma. As always in difficult times, Bessel turned to his friends Olbers and Gauss for help, writing to them about his problems. Thanks to them, Bessel received in absentia a doctoral degree from the Faculty of Philosophy of the University of Göttingen. These problems were due to tensions between the old conservative wing of the university and the new young forces. But it contributed to the rallying of young scientists. 2 editorial boards were created, natural science and humanitarian. The naturalists published the first volume of their works in 1812 under the title "Königsberg Archive of Natural Science and Mathematics". Bessel wrote 4 papers for this collection: 2 in mathematics and 2 in astronomy (on Saturn and "Some Results of Bradley's Observations"). Bessel's works were of exceptional importance for the development of astronomy. It is difficult to imagine a more complete combination of a brilliant theoretician with a brilliant practitioner; the methods of observation introduced by him and their processing served as unsurpassed models.

The start of the construction of the observatory was delayed: Bessel arrived in Königsberg with a project that seemed unsatisfactory to him, and a place for construction was not even chosen. Bessel chose between two places, he handed over the papers to the military department, where they also could not choose and sent them to Berlin, where they lay for another 6 weeks. As a result, the place was chosen - Butterberg Hill, in the highest western part of the old city rampart. Construction progressed slowly and unevenly. Difficulties primarily arose due to lack of funds, which is not surprising: the country was devastated by the recent war, the huge indemnity that Prussia paid to Napoleon devastated the state treasury, and the political situation was extremely difficult. In the summer of 1811 construction came to a complete halt. He had already begun to think about moving to another place, but still in the fall of 1811 he managed to get money, and he decided to stay in Königsberg. In the summer of 1812, Napoleon Bonaparte passed through Koenigsberg, wishing to explore the city. He was amazed that the Prussian king could think of such things (building an observatory) at such a time. By November 1813, the observatory was completed, and on November 12, Bessel made the first observations there. The observatory was modest in size. In plan, the building had the shape of a cross, slightly elongated from east to west. In this direction, the length of the building was 26 meters, the length of the “beam of the cross” in the north-south direction was 18.4 m. 5.8 m wide and a one-story western wing about 7x8 m in size. The main entrance was located in the center of the eastern wall and led to the first floor corridor. All rooms of the observatory were interconnected, and it was possible to get into any of them without going outside. The most important task of astronomy in the first decades of the 19th century was to determine the exact positions of the stars. Under the influence of this target setting, the instrumental base of most European observatories of that time was formed. The main instruments were “fixed” meridian instruments (passage, vertical circle, later - meridian circle), which served for absolute determination of the coordinates of the luminaries, as well as a “movable” refractor telescope with a micrometer for accurate differential measurements of small angles. With the help of a refractor, the positions of satellites of planets, binary stars were determined, comets and asteroids were observed. This trend was reflected both in the initial equipment of the Königsberg Observatory and in Bessel's subsequent acquisitions. Friedrich Bessel enthusiastically speaks of the happiness he feels in running such a magnificent institution that fully satisfies his desires. Bessel managed to collect many very good instruments for his observatory, and in subsequent years the instruments were updated and improved. Part of the equipment of the observatory was made according to the drawings of Bessel and served as a prototype and model for all other observatories of this era. A very large library for the observatory of that time was also created - for 2650 volumes, mainly on astronomy, mathematics and geography. The bulk of the books were written during Bessel's lifetime. The most important scientific periodicals are presented in full (Universal Geographical Ephemerides, Monthly Correspondences, Journal of Theoretical and Applied Mathematics, etc.). There were also books that were a bibliographic rarity, for example, "On the rotations of the celestial spheres" by N. Copernicus. The library was an exemplary book collection of this kind and evidence of the depth and versatility of the scientific interests of its collector.

Years of maturity and personality

Of Bessel's individual works, the most important was that he was one of the first astronomers to solve the age-old problem of the parallax of stars, of the scale of the universe. Following V. Ya. Struve, who in 1837 first determined the distance to the star Vega in the constellation Lyra, In 1838, using a heliometer (an astrometric instrument for measuring small (up to 1 °) angles on the celestial sphere), determined the parallax of the star 61 Cygnus, having measured thus distance to fixed stars. This star turned out to be one of the closest to the solar system. Bessel also developed the theory of solar eclipses, determined the masses of the planets and the elements of Saturn's satellites. Observing for a number of years the bright stars Sirius and Procyon, Bessel discovered in their motion such features that could only be explained by the fact that these stars have satellites. But these satellites are so faint in luminosity that they could not be seen with telescopes. Bessel's assumption was subsequently confirmed: in 1862, a satellite of the star Sirius was discovered, and in 1896, a satellite of Procyon. Bessel, studying the shape of the tail of Halley's comet, was the first to explain its direction by the action of repulsive forces emanating from the Sun (recall that the tails of comets are almost always directed in the opposite direction from the Sun).

Equally important are Bessel's work in geodesy. Here, just as in measuring astronomy, he developed tools and methods that are still used today, and left behind theoretical work. The triangulation of East Prussia (1832) made by him together with Bayer is considered a model of this kind of work. From 10 triangulations, Bessel calculated the dimensions of the earth's spheroid. Also known are such works by Bessel as determining the length of the second pendulum and the invention of the basic instrument. In mathematics, there are functions of his name, which are widely used in physics, technology and astronomy (for example, the so-called cylindrical functions of the 1st kind and the differential equation that they satisfy (Bessel equation), the inequality for the coefficients of the Fourier series (Bessel's inequality ), as well as one of the interpolation formulas). Bessel invented a basic device that greatly facilitated the measurement of line lengths on the ground.

Bessel enjoyed great prestige not only in Germany, but also far beyond its borders. Many astronomers from other countries aspired to Koenigsberg, wishing to get to know him and learn from him. Bessel gave public lectures almost annually for 12 years. These readings provided a systematic exposition of his views on many questions of astronomy. The Königsberg decades were a time of life and scientific maturity, when goals were clearly seen, when doubts about the correct choice of a life path were gone, and when hard everyday work rewarded Bessel with abundant fruits. The work of the scientist was the main content of his life. But Bessel was neither an armchair professor in science nor a pedant in everyday life. To a large extent, his achievements in the scientific field were facilitated by the atmosphere of mutual goodwill that Bessel's personality invariably formed among the people around him. He had close friends, numerous colleagues and correspondents, and was deeply revered by his students. He was happy in a family in which he always found rest and love. Bessel was very easy to deal with people, always friendly and attentive to the interlocutor. He appreciated these qualities in others as well. He organically could not stand hypocrisy and crookedness.

Among Bessel's colleagues and friends there were many people who knew closely the Königsberg thinker Immanuel Kant during his lifetime. After the death of the philosopher, they organized the "Society of Kant's Friends", whose members were, among others, Bessel's father-in-law Professor K. G. Hagen, the philosopher Chr. J. Kraus, representatives of city authorities, etc. - only 20-30 people. Every year on April 22, Kant's birthday, the "Society" gathered for a gala dinner dedicated to this event. Over time, Bessel was also elected a member of the Society. He suggested reviving the traditional gatherings with a mock rite of election of the "bean king". A silver bean was baked into the cake served for dessert. The one who found a bean in his piece became the “bean king”, and his neighbors at the table to the left and right became “bean ministers”. The "King" was preparing a comic "bean" speech about the famous philosopher for the next meeting. This rite became a tradition of the "society of Kant's friends".

Friedrich Wilhelm Bessel(German Friedrich Wilhelm Bessel; July 22, 1784, Minden - March 17, 1846, Königsberg) - German mathematician and astronomer, student of Carl Friedrich Gauss.

Biography

Friedrich Wilhelm Bessel enrolled as an apprentice in one of the trading houses in Bremen, where he acquired knowledge of mathematics and became interested in astronomy. One astronomical work attracted the attention of Olbers, on whose recommendation he entered in 1806 to Schröter, in Lilienthal, at the observatory, where Bessel occupied the position of an observer for four years.

Without studying at the gymnasium and the university, he received a doctorate from the University of Göttingen. Albertina Professor (Königsberg University). He made a great contribution to the study of the scale of the Universe, including the study of parallax. Carried out calculations of the orbit of Halley's comet. Founder and director of the Königsberg Observatory. Determined the position of 75,000 stars and created extensive star catalogs. In 1838, he performed the first scientifically reliable measurements of the annual parallax for a star (61 Cygnus). The priority of discovering the annual parallax of stars is recognized by Bessel. In 1841, using the data of many measurements, he calculated the dimensions of the earth's ellipsoid, which were widely used in geodesy and cartography until the middle of the 20th century. In 1844, he predicted that Sirius and Procyon would have indistinguishable satellite stars.

Friedrich Wilhelm Bessel died on March 17, 1846 in the city of Königsberg (now Kaliningrad) and was buried in the cemetery in Königsberg. At the moment, the exact place of Bessel's burial is unknown. On the site where it was located, it is planned to build a multi-storey residential building.

Awards

• Lalande Prize (1811)
• Gold Medal of the Royal Astronomical Society (1829 and 1841)

Memory

• Bessel functions and Bessel's inequality are named after him.
• Bessel's name was borne by a school in Königsberg (German: Bessel-Ober-Realschule)
• Memorial marble slab in Kaliningrad (former Koenigsberg) on ​​a hill near the intersection of st. Bessel and st. General Galitsky.
• Monument in Bremen.
• Crater Bessel on the Moon.

German mathematician and astronomer. He made a great contribution to the study of the scale of the universe.

Biography

Born in 1874 in the small town of Minden in Germany in the family of a petty official. He did not receive an education - he did not even study at the gymnasium, but was diligently engaged in self-education. These studies were so successful that Bessel not only received a doctorate from the University of Göttingen and became a professor, but also made a significant contribution to science.

But he began his career as a clerk at the age of 15. The path to science was predetermined by his very character and mindset. Systematicity, thoroughness, innate mathematical talent turned the future businessman ... into a scientist: he was "too" seriously preparing for his trading career and, in addition to studying languages ​​​​(English, Spanish, French), geography and customs of peoples, considered it absolutely necessary to thoroughly study and master navigational astronomy. Soon he achieved his first successes: in 1803, by observing the occultation of the stars by the Moon, using crude home-made instruments, he was able to determine the longitude of Bremen.

Laplace's most difficult "Celestial Mechanics" and the higher mathematics necessary for its understanding, he could study only in the morning and night hours free from work. In 1804, he met the outstanding Bremen astronomer and physician G. V. M. Olbers, whom he introduced to the elements of the orbit of Halley's comet calculated by him according to the observations of T. Harriot and Lorporley in 1607. The work caused an enthusiastic review of Olbers, was published with his preface, in which he introduced Bessel to the scientific world, and marked the beginning of a great friendship between these two astronomers, which was supplemented by the friendship of Bessel and K. Gauss.

On March 19, 1806, Bessel began his scientific activity in Lilienthal by checking all the measuring instruments and instruments of the observatory and revising the methods of mathematical processing of observational results, although at the same time he continued to study comets and received a prize for calculating the orbit of the 1807 comet.

He worked at the University of Königsberg, where an observatory was built under his leadership, of which he remained the director until the end of his life. He created the German school of precise observations in astronomy. In Russia, the Pulkovo Observatory followed in many ways in his footsteps.

Bessel entered the history of science as one of the great mathematicians. Bessel published about 400 scientific papers and left a large correspondence with scientists, where he also presented his ideas and results. He also gave popular lectures on physics and astronomy. The merits of the scientist were highly appreciated by being elected members of many academies, including Berlin (1812), and foreign honorary members of the St. Petersburg Academy of Sciences (1814), as well as many scientific societies.

Achievements in astronomy

Already at the age of 20, he calculated the orbit of Halley's comet.

Bessel - one of the founders of astrometry

Astrometry- a branch of astronomy, the main task of which is the study of the geometric, kinematic and dynamic properties of celestial bodies. Bessel believed that it was necessary to introduce corrections into the results of observations, taking into account any, the most insignificant factors that reduce the accuracy of astrometric measurements. He developed mathematical methods for correcting the results of observations. Bessel's first major work was the revision of the results of observations of the positions of stars in a catalog compiled in the 18th century by the English astronomer D. Bradley. He determined the position of 75,000 stars and created catalogs that became the basis of modern knowledge of the starry sky.

Parallax measurement

Parallax- change in the visible position of the object relative to the distant background depending on the position of the observer, we have repeatedly talked about this on our website.

Bessel was one of the first astronomers to measure the parallaxes and thus the distances to the stars: in 1838 he measured the distance to the star 61 Cygni. This star turned out to be one of the closest to the solar system. In 1841, using the data of many measurements, he calculated the dimensions of the earth's ellipsoid, which were widely used in geodesy and cartography until the middle of the 20th century. In Europe, it is used in Germany, Austria, Switzerland, the Czech Republic and the countries of the former Yugoslavia, as well as in Indonesia, Japan, Eritrea and Namibia.

Discovery of the satellites of Sirius and Procyon

In 1844, Friedrich Bessel discovered that Sirius, the brightest star in the sky, periodically, albeit very weakly, deviates from a rectilinear trajectory of motion in the celestial sphere. Bessel came to the conclusion that Sirius must have an invisible "dark" companion, and the period of revolution of both stars around a common center of mass must be about 50 years. The message was met with skepticism, since the dark satellite remained unobservable, and its mass had to be quite large - comparable to the mass of Sirius.

In January 1862, Alvan Graham Clark at the University of Chicago observatory discovered a dim star in the immediate vicinity of Sirius. It was the dark satellite of Sirius, Sirius B, predicted by Bessel.

Although the main star constellation Canis Minor- yellowish Procyon - inferior to Sirius and in size, and in temperature, and in luminosity, there is something in common between these stars. Both of them lead small constellations in which none of the stars can compete with them in brightness. Both stars have white dwarfs as companions, the discovery histories of which are very similar.

Observing the bright stars Sirius and Procyon for some time, Bessel discovered features in their motion that could only be explained by the fact that these stars have satellites. But these satellites are so weak in luminosity that they could not be seen in the telescopes of that time. Bessel's assumptions were confirmed: in 1862, the satellite of Sirius was discovered, and in 1896, the satellite of Procyon. The closest other star to Sirius is Procyon.

But what do we know about Procyon and his satellite?

Procyon, a yellowish star 0.5M, has a luminosity only 5.8 times greater than the luminosity sun. It is somewhat larger than the Sun and slightly hotter - its surface temperature is close to 7000 K. Like Sirius, Procyon is one of the stars neighboring us: the distance to it is 3.5 pc. This star is in itself unremarkable, and if it were not for its proximity to the Earth (and therefore significant apparent brightness), we would not pay any attention to it. Another thing is Procyon's satellite. Consider this asterisk 11th star. magnitude, located at an average distance of 4 from Procyon, is a completely impossible task for an ordinary amateur astronomer. This small star emits almost 10 times less light than the satellite of Sirius, and is an even denser white dwarf than Puppy. But the similarity of two strange communities of completely different stars (Sirius and Procyon with their dwarf satellites) is undeniable.