ThatsMaths

On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without success to locate it. Without accurate knowledge of its orbit, the search seemed hopeless. How could its trajectory be determined from a few observations made from the Earth, which itself was moving around the Sun?

The mathematical problem is this: “given a set of observations, on what curve is a planet travelling?” We assume that the planet is following a Kepler orbit, an ellipse with the Sun at one focus, and we wish to determine the details of this ellipse so that predictions of future positions of the planet can be made.

Six Little Numbers

Six numbers are required to pin down a planet in a Kepler orbit. They are called the orbital elements. The orbital plane is fixed by giving the direction of the normal to it. This requires two numbers, like latitude and longitude on the celestial sphere. For Earth, the normal points towards Polaris. The ellipse in this plane is determined by giving its shape (eccentricity), size (major axis) and orientation (direction of perihelion). One more parameter is required to specify the position of the planet on the orbit.

The six orbital parameters cannot be observed directly but must be derived from other measurements. Two radius vectors r₁ and r₂ from the Sun to the planet, at two times, give six independent components which suffice to determine the orbital elements. But we cannot measure these vectors: with telescopic observations, we are able to determine only the longitude and latitude, or azimuth and elevation, of the planet relative to the moving Earth. We have no information whatsoever about distances.

So, we need three successive observations of azimuth and elevation to give six angles. A further complication is that the earth is moving between observations. However, we know about this and can easily convert the measurements to heliocentric angles. The Figure shows three “lines-of-sight” from the moving Earth to the planet.

Gauss Predicts the Orbit

In principle, three successive observations, giving six angles, are sufficient to compute the six orbital elements. The brilliant mathematician Carl Friedrich Gauss showed how this could be done (Gauss 1809). He used three observations like those shown in the Figure to calculate the orbit of Ceres.

But Gauss did not stop there. All measurements have errors; in order to minimise the effects of these, Gauss recomputed the orbit using several triplets of observations. He then combined these, using a method that he had devised earlier, to produce an improved prediction of the position of Ceres. His method is what we now call the method of least squares (*).

(*) Although it is often claimed, there seems to be some ambiguity as to whether Gauss actually applied the theory of least squares in computing the orbit of Ceres.

Many astronomers struggled to find the new dwarf planet, using the few observations published by Piazzi. Most were unsuccessful, but the predictions of Gauss were spectacularly accurate and, in December 1801 (on New Year’s Eve, some say) astronomers Franz Xaver von Zach and Heinrich Olbers were able to find Ceres using Gauss’s predicted orbit. Gauss, aged just 24, quickly earned the reputation throughout Europe as a genius.

The Titius-Bode Law

Ceres is one of a large group of minor planets or asteroids orbiting the Sun between Mars and Jupiter. In the background to the discovery of Ceres was the Titius-Bode Law, which implied that the distances of successive planets from the Sun increase in a regular fashion. The large gap between Mars and Jupiter indicated the possibility of a ‘missing’ planet, at a distance of 2.8 astronomical units from the Sun. In 1772, Johann Elert Bode suggested that an undiscovered planet could exist between the orbits of Mars and Jupiter. The gap between Mars and Jupiter had already been noted by Kepler in 1596. The mean distance of Ceres from the Sun is 2.77, in good agreement with the Titius-Bode Law.

Sources

Gauss, C. F., 1809: Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections (English translation by C.H. Davis), reprinted 1963, Dover, New York.

Le Corvec, Veronique, Jeffrey Donatelli and Jeffrey Hunt, 2007: How Gauss Determined the Orbit of Ceres. Math 221 Presentation. PDF.

Teets, Donald and Karen Whitehead, 1999: The Discovery of Ceres: How Gauss Became Famous. Math. Mag., 72, 2, 83-93