In orbital determination of celestial bodies, why did the case of parabolic or nearly-parabolic orbits require separate development?

Question

In Buhler's biography of Gauss, who made huge contributions to the development of efficient orbital determination methods in his Theoria Motus (1809), the following information is mentioned:

The major omission in Theoria motus is a lack of treatment of parabolic orbits, a task that had earlier been solved by Olbers, and of Gauss's very efficient approximation methods.

Dunnington in his biography of Gauss gives more information on this issue:

Of the numerous fundamental works on astronomy planned by Gauss, the Theoria motus is the only one he presented to his contemporaries. It treats exclusively the elliptical and hyperbolic motion in so far as it concerns the determination of an orbit from observations, which leads to the conjecture that he may have postponed further investigations on this subject with the idea that his friend Olbers had successfully solved the problem of parabolic orbit determination. In the Theoria motus he applies methods for determining an hyperbolic orbit similar to those for an elliptic orbit. But he gives in the Theoria motus a numerical example only for the calculation of the elements from two radii vectors, the included angle and the time between them. In a letter of January 3, 1806, he pointed out to Olbers a certain case where the latter's method for a parabolic orbit is not applicable. Later, in 1813, he found a correction for the method and in 1815 busied himself intensively with the parabolic-orbit problem. He not only gave Lambert's equation a different form, but also developed a series of other important relations. The determination of a circular orbit (which Gauss characterized as easy and simple) was given in 1871 by Klinkerfues, his pupil, in a textbook.

In addition, volume 7 of Gauss's werke (this volume is dedicated to his works in theoretical astronomy) contains an entire section devoted to parabolic orbits determination (p.323-374 of this volume), which includes several letters (to Olbers, amongst others) and some fragments from his Nachlass. Therefore, it seems there is an essential issue here.

While reading the passages from Gauss's biographies, several questions have arose in mind:

• Parabolic orbits are defined as conic sections with eccentricity $e=1$ (exactly $1$), so the probability that a certain orbital trajectory will have exactly this eccentricity is zero. Therefore I guess it should be read "near-parabolic" orbits. But what is the range of values of $e$ (range which must include $1$) for which the orbit is considered "near-parabolic"? This is just a technical question to make sure I understand the subject.
• What is essentially different in the case of parabolic orbits that compelled mathematical astronomers to develop a separate method for it? I read that one of Gauss's major innovations in his Theoria motus was a general method of orbital determination that didn't make any a priori assumptions on the type of orbit determined. Therefore, I'd like to know why is the parabolic case an exception.

Note that I'm really just looking for a qualitative explanation why there was a need to develope a separate method for the parabolic case, so there is no need to dwell into the complicated mathematical details of Gauss's method for parabolic orbits determination.

Answer 1

The core of the matter is that many of the calculations related to elliptical orbital motion depend on measures and quantities that are referred in some way to an entire cycle around the orbit. For a parabolic orbit, with eccentricity 1, there clearly is no entire cycle: an object moving on such an orbit never returns to its pericenter. The measures that depend for their definition on an entire cycle break down.

Also, on nearly-parabolic orbits, many of the usual elliptical quantities become numerically less and less well determined, the closer the eccentricity approaches to 1. So the usual forms of elliptical calculation approach breakdown. The problem can be described as a singularity in the elliptical relationships where the eccentricity reaches 1 and the anomalies reach zero.

As an example of how this works, the pericenter distance $q$ on an elliptical orbit can be expressed as $a.(1-e)$ , where $a$ is the semi-major axis and $e$ the eccentricity. If one considers eccentricities nearer and nearer to 1, $a$ becomes larger without limit, and $(1-e)$ approaches zero. But all such physical quantities are measured within some non-zero tolerance of error, however small. So whatever the degree of accuracy of $e$, there will be some level at which the error in $(1-e)$ will be at least as large as $(1-e)$ itself. The calculation loses definition, effectively lost within its own error.

A similar tendency towards indeterminacy at high values of eccentricity is found in two out of the three usual measures of longitudinal motion in an ellipse: the mean anomaly and the eccentric anomaly. Physically, the eccentric anomaly is defined in terms of the geometrical center of the ellipse. As one considers increasing eccentricities and increasing semiaxes, the location of the center of the major axis becomes less and less well determined.

Again, physically, the mean anomaly is defined by the proportion (relative to the whole ellipse area) represented by the area of the focal sector bounded by the radius-vector to the pericenter and the radius-vector to the current position of the orbiting body. As one considers increasing eccentricities and increasing semiaxes, the whole ellipse area becomes less and less well determined as the eccentricity becomes larger. From another point of view, all the mean anomalies for any given true anomaly collapse towards zero (on the usual elliptical basis of calculation) as the eccentricity approaches 1.

This is why a different basis had to be found for the parabolic calculations: the mean motion, instead of being based on the orbital period (non-existent for the parabola) had to be put on a different conventional basis. It was also found that the relation between time and true anomaly could be defined for the parabolic motion more simply than the equations for elliptical motion, on the basis of a cubic equation now often known as Barker's equation.

A good source for the nature of the orbital calculations and problems in elliptical, parabolic and hyperbolic motion is A E Roy's "Orbital Motion". Illustrations of the problems that arise and increase as the eccentricity approaches 1 can also be found for example in R Serafin (1996) "On Solving Kepler's Equation for Nearly Parabolic Orbits" and its references.