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What else are asteroids good for?

 

The January 1, 1801 discovery of the dwarf planet Ceres set the stage for one of the great dramas in the history of astronomy.  An Italian monk, Giuseppe Piazzi (1746-1846), discovered a faint, nomadic object through his telescope in Palermo, correctly believing it to lie in the orbital region between Mars and Jupiter where Kepler registered a gap in his harmonic scheme.

Piazzi watched the object for 41 days but then fell ill, and shortly thereafter the wandering star star strayed into the halo of the Sun and was lost to observation.

The newly-discovered planet had been lost, and astronomers had a mere 41 days of observation covering a tiny arc of the night from which to attempt to compute an orbit and find the planet again.

Yet limited scope of the measurements doomed attempts to compute the orbit in the context of contemporary mathematical methods. The dean of the French astrophysical establishment, Pierre-Simon Laplace (1749-1827), declared that it simply could not be done.

In Germany, the 24 year old German mathematician Carl Friedrich Gauss had considered that this type of problem – to determine a planet's orbit from a limited handful of observations – "commended itself to mathematicians by its difficulty and elegance."

Gauss discovered a method for computing the planet's orbit using only three of the original observations and successfully predicted where Ceres might be found.

 

The prediction catapulted him to worldwide acclaim, due, in the words of biographer W. K. Bühler, "to the popular appeal which astronomy has always enjoyed," and launched one of the most fruitful careers in the history of science.

Nevertheless, the original method by which Gauss determined the orbit of the nomadic star is a subject of considerable debate that will probably not be resolved with existing sources.  Gauss never published the method by which he computed the orbit and in his later work, the Theoria Motus, he declared that he had practically erased all traces of it in his drive to build a mathematically elegant structure for celestial mechanics.

Gauss' biographer Dunnington notes that the scientist's "earliest notes on Ceres...lack clearness."   Gauss' colleagues asked for a candid review of the original solution, and in response he produced an essay in 1802, not published until years later, Summarische Übersicht der Bestimmung der Bahnen der beiden neuen Hauptplaneten angewandten Methoden (Gauss Werke, v. 6, pp. 148-165). The method presented still had undergone certain refinements.  For instance, we know from other sources that Gauss used a method of "successive approximations" to compute the original orbital elements, but in the Summarische Übersicht the orbit sails out from a system of linear equations. The paper itself is regarded as the birth of modern linear algebra. 

This leaves us to speculate about Gauss' original train of thought. 

The outlines of the problem are straightforward in principle.  Gauss needed to know at least two spatial positions for the planet in order to obtain an orbit. Yet this requirement defined a formidable task since Gauss only only knew a handful of Earth-centered angles along a tiny arc of the night sky. 

Two spatial positions are sufficient to determine a unique ellipse whose focus lies at the center of the Sun.  Based upon Kepler's investigations we know that a planet's orbit is an ellipse lying in a plane.

We seek to demonstrate, in principle, how this kind of problem would be solved. 

As related in the Summarische Übersicht... Gauss' solution to computing the orbit with three observations rested on Kepler's hypothesis that the properties of the orbit (as expressed by Kepler's "Three Laws") solely determine the motion of a celestial object.  No information is needed about mass, velocity, or other details relating to the object itself.

We will first consider an application of the Area-Time principle ("Second Law"): the idea that the area swept out on the orbit between the observations must be proportional to the time between the observations.  By all accounts this relation was central to Gauss' original method.

Gauss knew the ratio of the areas of two orbital sectors between his three observations, because he knew the time between the observations.

To make headway, Gauss would have had to translate a relation expressed in terms of areas into one expressed in terms of distances, since what he was looking for was a way to get at least two Sun-centered spatial positions for the planet, in order to determine a unique ellipse.

Gauss would probably have focused on determining the middle of the three heliocentric distances, since with this in hand it is possible to determine the other positions, and hence the orbit, by relation.

Below we offer a conjecture of one way Gauss might have determined the central spatial position based on the idea that he would have initially concieved of and sought to relate the variables of the problem through a system of parallel displacements.

Using parallel displacements we can relate a set of rectangular areas to the central distance measure.  The triangles generated by division of the parallelograms give us area variables that we can closely (though not exactly) relate to the sectors of the orbit. 

The points "a" and "b" on the construction divide the two Sun-centered distance measures on which they lie in the same ratio as the shaded areas in the graphic above divide the whole area of the construction.

 

In the process of describing the central observation in terms of the other two observations, we can thus find a way to relate a proportion known in terms of areas to one corresponding to distances.  The areas of the blue and purple triangles are almost, though not precisely, equal to the areas of the sectors swept out on the orbit.

Projecting the hypothetical construction down onto the plane of the orbit of the Earth, we find that:

Since the distances from the Earth to the Sun are known, by the properties of similar triangles (watch the animation above) we can decompose the original two displacements into two more and define a plane on which the planet must lie.  The planet itself would be found at the intersection of the observed angle to Ceres from the second observation with the plane (see the animation).

Problems of accuracy would arise, due to the discrepancy between the areas of the orbital sectors and the triangles in the model (see diagram at right) as we would be relating the area on the orbit between the first and third observations to the maroon-colored triangle in the geometric model. 

If Gauss could have found a way to correct for the largest source of error in this comparison, namely the difference between the maroon triangle (at right) and the orbital sector between the 1st and 3rd positions (which we find he did in his Theoria Motus), he could have dampened the error and presumably brought the result of his approximation into an acceptable range of certainty.

 

His expression for this differential correction would be written in terms of the parameters of the orbit, and these were unknown. Thus Gauss would have used a method of successive approximations to find values for the key parameters that were self-consistent.

To implement the calculation he would have needed to translate his model into algebraic variables on a system or system(s) of coordinates.

Gauss' method of "least squares," celebrated as the key to this problem, would have speeded up an otherwise extremely tedious trial-and-error type of calculation to determine orbital elements.

In view of the contradictory nature of his original notes it is difficult to say what Gauss might have done for certain.  Nevertheless, these kinds of speculations can enrich our understanding of Gauss' later published works, after he had streamlined the various relations he had discovered into systems of formulas.

 

Basic Orbital Mechanics

Determining the orbit on the basis of Gauss' three observations alone would have been difficult in practice because the proximity of the positions would magnify small errors in the initial measurements, skewing the final calculation of the ellipse.

Using a device well known to astronomers, Gauss could substitute Kepler's Area-Time principle into his Third Law, to determine the ratio of area swept out to time elapsed, a constant for any given orbit, and define this constant in terms of the parameters of the ellipse.

This ratio of area swept out to time-elapsed enables the astronomer to determine a dimension of the ellipse known as the "half parameter" (see the diagram above).

In the case of the problem in question, knowing the magnitude of this parameter would provide another dimension from which to help determine the orbit.

Here Gauss would again have used his method of successive approximations, since he did not specifically know the areas of the orbital sectors in question although he knew the time between the observations.

We can conjecture that he might estimate the area of the orbital sector between observations 1 and 3 using the postulated parallel displacement model (which would correspond to one unique value for "half parameter"), then using this value for H (and two of his Sun-Ceres distance measures) he could determine a unique elliptical orbit.

With the orbit in hand he could compute the area on the orbit between observations 1 and 3 that he used to compute the "half parameter" in the first place and see if the values matched.

More on the Ceres Orbit

D. Teats and K. Whitehead undertake a comprehensive and lucid review of Gauss' solution as presented in the Summarische Übersicht in a 1999 article in Mathematics Magazine (Vol. 72, No. 2 (Apr., 1999): pp. 83-93), the journal of the Mathematical Association of America.   The 58 page pamphlet by F. Klein, M. Brendel and L. Schlesinger, Materialien für eine wissenschaftliche Biographie von Gauss (B.G. Teubner, Leipzig, 1920) hints at elements of the original method. 

Biographies of Gauss also provide useful background on his determination of the orbit of Ceres.  We recommend: G. W. Dunnington, Carl Freidrich Gauss: Titan of Science (New York, NY: Exposition Press, 1955); W. K. Bühler, Gauss, A Biographical Study, (New York, NY: Springer-Verlag, 1981); and T. Hall, Carl Friedrich Gauss: A Biography (Cambridge, MA: MIT Press, 1970).

Background resources on the web framing the historical context of Gauss' achievement include Ivars Peterson's Math Trek article, Gauss's Orbits, and this well-written term paper from a student at Rutgers. 

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