The Star-Gazer

The year 1801 was one of the best of Gauss’ life. It is the year that defined the beginning of his remarkable career as he published his monumental textbook Disquisitiones Arithmeticae and rediscovered the planet Ceres, all at just age 24.

On January 1st 1801, Giuseppe Piazzi discovered Ceres, a dwarf planet. However, he could only keep it in his sights as it moved nine degrees across the sky. Then, on February 12th, it disappeared behind the Sun. According to Piazzi’s calculations, it would take a few months to reappear. But it never did. With such little information collected while visible, Piazzi failed to relocate the planet (Tennenbaum & Director, 2019).

A portrait of Giuseppe Piazzi ([Piazzi], 2019).

By October, Gauss had heard about the problem. He decided to try his hand at it and so stepped into a new field - astronomy. Three months later he had his answer and made it public. On December 31st, Franz Xaver von Zach and Heinrich Olbers had both used Gauss’ calculations and spotted the planet again. Gauss had been less than half a degree off. This feat from a non-astronomer seemed fantastical, especially considering Gauss did not let anyone in on his methods until several years later (Biographical, 1991).

Gauss’ brilliance was his recognition that the planet followed an elliptical orbit. Piazzi had failed because he was relying on bad precedent. In 1781, the planet Uranus had faced a similar problem. After its discovery by William Herschel, astronomers looked for a model of its orbit and landed on a circular one. Coincidentally, what we know now to be an elliptical orbit had such a small eccentricity that the circular approximation was satisfactory (Tennenbaum & Director, 2019). When faced with an ellipse with a larger eccentricity (as in the case of Ceres), a circle would not suffice. Even worse, there was much less data from Ceres than Uranus because it was visible for such a short time. Mathematicians such as Euler, Lambert, Lagrange, and Laplace had worked with ellipses before, but none had found a method for deriving an elliptical equation with such little data available. Laplace had gone as far as to state the problem unsolvable (Weiss, 1999).

The orbits of planets in our solar system including Uranus and Ceres ([Dwarf Planet], 2018).

To create such an elliptical orbit, Gauss’ first challenge was to determine the distance between the Earth and Ceres. Over the 42 days Piazzi kept track of the planet, he made 19 separate observations about its position in the sky. Each observation consisted of a timestamp and two angles that determined the direction of the planet in relation to Earth. Gauss’ initial work used three of these - data coming from the day of discovery (January 1st), the last day seen (February 11th), and a day in the middle (January 21st). He chose the middle day as his median with which to calculate the distance from Earth (Tennenbaum & Director, 2019).

In another diversion from the Piazzi, Gauss made use of Kepler’s hypotheses. First, that an orbit is independent of a planet itself; thus, no details are needed about mass. Second, assuming an orbit does not come near another large body, the Sun will be a focal point of the orbit. With these in mind, the orbit is dependent on 5 variables; the relative scale of the orbit, the eccentricity of the orbit, the shortest distance from the orbit to the Sun, the relative tilt of the main axis of the orbit, and the plane of the orbit relative to that of the Earth. Hence, Gauss assumed nothing besides the three observations he had taken from Piazzi. Such a set up led to an equation of degree eight. The execution to find a solution would be complex; it required over 100 hours of manual calculations and approximations.

First, a basic approximation of its solution was found using the three chosen observations. Then, he used the method of least squares, which he had developed at age 18 but failed to publish. He slowly increased precision by adding in the rest of the observations until his solution fit them all. From this, he created an ellipse using the Sun as one focus and keeping his three original data points lying on the ellipse. Then, as Kepler’s Second Law told him how to calculate the length of an arc and he had data telling him how long it took to get from one point to another, the current location could be found (Weiss, 1999).

Gauss’ drawing of the orbits of several planets, including Ceres (Weiss, 1999).

Overwhelmed by the success and recognition this feat won him, Gauss continued work in astronomy. Eight years after the fact, Gauss revealed his methods. In 1809, he had collected and refined his work to meet his satisfaction for publication. His book became Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections. He simplified so much of the orbital calculations that existed at the time that his work is still in use today, requiring only a few modifications to be applied to computers. He introduced the Gaussian gravitational constant and successfully proved the method of least squares for normally distributed errors; a theory known as the Gauss-Markov theorem, although there are disputes about Gauss’ priority.

Since Ceres, over 1,500 celestial bodies have been discovered (though Ceres remains the largest). By improving his methods, Gauss calculated orbits for many throughout his life with increasing speed. In 1807, Olbers discovered the planet Vesta and Gauss calculated its orbit in only 10 hours - much faster than the 3 months it took for Ceres. His methods also worked for the parabolic orbits of comets and these came even faster. Gauss could find them in a single hour, while Euler’s previous methods took three days (Weiss, 1999).

The Göttingen Observatory where Gauss was director ([The Old Observatory], 2014).

This success changed the course of Gauss’ life. As noted in previous posts, Gauss had been receiving a stipend from the Duke of Brunswick, allowing him to leave home and focus on mathematics full-time. However, growing up financially unstable, he always doubted the security the stipend provided. As democratic revolutions were happening around the globe (a cause Gauss did not believe in), he saw the risk they posed to his well-being. After Gauss’ success with Ceres, the Duke increased his stipend, but it was not enough to assuage his insecurities. The obvious path was to become a professor - a path he resisted as he was not one that enjoyed working with students. However, now that he was being celebrated by astronomers, he had another option. The life of a professional astronomer included little teaching and more time for research. He had also seen firsthand the public recognition astronomy could win him, something that was comparatively lacking in mathematics. Gauss had a great ambition for fame and dreamed of rising to the level of renown in the sciences that only comes once in a generation. Thus, he cleared the path for himself and in 1807 was appointed Professor of Astronomy and Director of the observatory at Göttingen. This was certainly a stable position and he held it for the rest of his life (Biographical, 1991).

With two extraordinary accomplishments in a single year, Gauss was made famous in the academic world. While his father had downgraded him as the “star-gazer,” it was a name that ended up becoming true (Eisenhart, 2008, pg. 1).

References

Biographical dictionary of mathematicians : Reference biographies from the dictionary of scientific biography (Vol. 2). (1991). New York, NY: Charles Scribner’s Sons.

Dwarf planet orbit [Digital image]. (2018, January). Retrieved from https://superstarfloraluk.com/9487485-Make-Make-Dwarf-Planet-Orbit.html

Eisenhart, C. (2008). Gauss, Carl Friedrich. Retrieved from https://www.encyclopedia.com/people/science-and-technology/mathematics-biographies/carl-friedrich-gauss

Giuseppe Piazzi [Digital image]. (2019, January 1). Retrieved from http://scihi.org/giuseppe-piazzi-ceres/

Tennenbaum, J., & Director, B. (2019). How Gauss determined the orbit of Ceres. Schiller 
Institute.

The old observatory of Carl Friedrich Gauss [Digital image]. (2014, August 13). Retrieved from https://www.flickr.com/photos/unigoettingen/15175546641

Weiss, L. (1999). Gauss and Ceres. Retrieved from http://sites.math.rutgers.edu/~cherlin/History/Papers1999/weiss.html