2 Original orbit calculation

If we want to know what the cometary orbit was before it entered the planetary system it is necessary to calculate its past motion among planets, taking into account all types of perturbations that act on a comet in the solar neighbourhood. As the dynamical model we used heliocentric, post-Newtonian equations of motion, similar to those used by Yau et al. (1994) in their investigation of the past motion of comet 109P/Swift-Tuttle. As the source of planetary positions we used the JPL DE406 Planetary Ephemeris (Standish 1998), using the software (in C) which is made available as free-ware by the author at the URL: ftp://ftp.astro.amu.edu.pl/pub/jpleph. We integrated numerically the equations of motion of each comet using Everhart's RA15 integrator (Everhart 1985) but some test calculations were also performed using the D2RKD7 routine based on the method of Dormand & Prince (1978), which was made available to the author by Ken Fox (see Fox 1984). We also performed some tests using the RA27 routine by Everhart (1985), in quadruple precision on the Alpha-based Linux workstation. Those tests were extremely time consuming but they allowed us to separate dynamical sensitivity to initial conditions from both truncation and round-off numerical errors in some difficult cases.

 

 

Table 2: Original 1/a distribution of the growing population of long-period comets, 1990 and 1999 data are restricted to comets with the perihelion distance greater than 1.5 AU an qualified as class 1 orbits.

1/a orig.	Oort	Oort	This paper
[10-6AU-1]	1950	1990	(1999)
<10	14	42	60
10-20	2	1	5
20-30		3	6
30-40	2	1	4
40-50		3	6
50-60		5	7
60-70	1	1	3
70-80		1	2
80-90		2	2
90-100	0	1	1
>100		6	20
Total:	19	66	116

The backward integration of the motion of each comet was stopped when a comet crossed the PPL or when it reached its aphelion at a smaller distance. It should be stressed here that none of the comets which came from a distance greater than the PPL made any close approaches (distance less than 0.1 AU) to the Solar System planets, so their original elements are obtained without initial errors amplified by such approaches. Additionally, for each comet we performed a set of numerical integration runs with starting data slightly modified (on the level of one tenth of the least significant digit published in Marsden's catalogue). We wanted to find those cases, in which abnormally high sensitivity to the starting data can be observed, disqualifying them from further investigation.

As a result of this part of the work, we have obtained a set of original orbital elements for 327 comets, calculated at the epoch they crossed the PPL. Among them there are 44 comets that cannot reach this distance - for them, we include the orbital elements at their first (going backward) aphelion. The ASCII file containing these data is available to readers at the URL: http://main.amu.edu.pl/~dybol/DH/org_elem.dat. The accuracy of these data is comparable to the starting elements, taken from the Marsden catalogue (Marsden & Williams 1997), typically six or seven significant digits.