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.