3 Original and future orbit determinations

The orbital elements of each comet were recomputed on the basis of archive observations available at the Minor Planet Center (Cambridge, USA), with some additional data selected from the literature. For comets observed during the period of 1900-1950 the data collected in Warsaw in cooperation with Slovakian astronomers at the Astronomical Institute from Bratislava and Tatranska Lomnica were also taken into account. The observations used for the orbit determination in the present work are more complete than the data used in the MW Catalogue (see Col. 5 of Tables 1 and 2) except for two comets: 1898 V1 (Chase) and 1955 O1 (Honda). The observations for each comet were selected according to the objective criteria elaborated by Bielicki & Sitarski (1991).

 

 

Table 1: Reciprocals of semimajor axis for original, osculating and future orbits (in units of 10^-6 AU^-1) for pure gravitational motion and for motion with non-gravitational effects represent by radial, A₁, and transverse, A₂, and - in two cases - normal, A₃, components. The "original'' and "future'' values of reciprocal of the semimajor axis are given in barycentric coordinates. The tenth column gives the mean original $(1/a)_{\rm ori}$ obtained from a sample of 500 randomly selected orbits (with the mean rms residual given in the last column).

Comet	Standard and non-gravitational orbits determined from observations									Sample of 500 random orbits
design.	1/a			Interval of	No.	No.	rms	Model	Non-gravitational		Fitting
&	in units of			observations	of	of			parameters		to obs.
q	10-6 AU-1				obs.	res.			A1;  A2; ( A3)	$\left < (1/a)_{\rm ori}\right >$	with
	ori	osc	fut						(in units of		rms of
									10-8 AU day-2)
1975 V2	-56	- 7	0	19751113-19760126	70			MW97
0.219	-148	- 99	1125	19751113-19760209	85	164	1 $.\!\!^{\prime\prime}$64	M1975V2		$-148\,\pm\,23$	1 $.\!\!^{\prime\prime}$71
	+453	+503	1728	19751113-19760209	85	164	1 $.\!\!^{\prime\prime}$59	M1975V2n	+1.599 $\pm$ 4.372	$+408 \pm 822$	1 $.\!\!^{\prime\prime}$65
									+0.5699$\,\pm\,$.1642
1956 R1	0	-531	0	19561108-19580411	150			MW97	+1.7; +0.1
0.316	-104	-785	-636	19561108-19580411	249	458	1 $.\!\!^{\prime\prime}$78	M1956R1		$-104.5\pm 5.4$	1 $.\!\!^{\prime\prime}$82
	+118	-562	-387	19561108-19580411	249	458	1 $.\!\!^{\prime\prime}$55	M1956R1n	+1.5169$\,\pm\,$.1517	$+117.5\pm 19.7$	1 $.\!\!^{\prime\prime}$59
									+0.1170$\,\pm\,$.0237
1959 Y1	0	-288	+0	19600104-19600617	37			MW97	+5.9; -1.4
0.504	-135	-589	-586	19600104-19600617	37	74	3 $.\!\!^{\prime\prime}$14	M1959Y1		$-135\pm 26$	3 $.\!\!^{\prime\prime}$32
	+186	-268	-265	19600104-19600617	37	74	1 $.\!\!^{\prime\prime}$47	M1959Y1n	+5.4803$\,\pm\,$.4558	+188$\,\pm\,$50	1 $.\!\!^{\prime\prime}$55
									-1.1497 $\pm$ .3218
1989 Q1	0	- 31	+0	19890824-19891224	180			MW97	+3.4; +0.6
0.642	-.15	-245	+195	19890824-19891224	231	454	2 $.\!\!^{\prime\prime}$03	M1989Q1		$-0.6\pm 15.1$	2 $.\!\!^{\prime\prime}$08
	+164	- 80	+359	19890824-19891224	231	454	1 $.\!\!^{\prime\prime}$78	M1989Q1n	+2.4160$\,\pm\,$0.7310	+161$\,\pm\,$52	1 $.\!\!^{\prime\prime}$83
									+0.6677$\,\pm\,$0.2728
1991 Y1	-94	-90	+1114	19911224-19920502	178			MW97
0.644	-98	-94	+1111	19911224-19920502	274	540	1 $.\!\!^{\prime\prime}$47	M1991Y1		$-98.1\pm 9.3$	1 $.\!\!^{\prime\prime}$50
	+ 6	+10	+1215	19911224-19920502	274	540	1 $.\!\!^{\prime\prime}$41	M1991Y1n	+1.051 $\,\pm\,$0.163	$+ 7 \pm 19$	1 $.\!\!^{\prime\prime}$44
1975 X1	-734	-1405	-1461	19751209-19760204	59			MW97
	-1071	-1743	-1799	19751209-19760204	82	157	2 $.\!\!^{\prime\prime}$48	M1975X1		$-1051 \pm 230$	2 $.\!\!^{\prime\prime}$57
	+1037	+367	+309	19751209-19760204	82	157	2 $.\!\!^{\prime\prime}$37	M1975X1n1	+8.3477$\,\pm\,$2.1851	+1048 $\pm$ 599	2 $.\!\!^{\prime\prime}$47
	+ 967	+297	+239	19751209-19760204	82	157	2 $.\!\!^{\prime\prime}$38	[M1975X1n2]	+8.2386$\,\pm\,$2.2067	+ 896 $\pm$ 647	2 $.\!\!^{\prime\prime}$47
									+0.7452$\pm$1.745
1955 O1	-727	-1071	-432	19550730-19551112	102			MW97
0.885	-635	-978	-339	19550802-19551113	60	105	1 $.\!\!^{\prime\prime}$15	M1955O1		$-635\pm 158$	1 $.\!\!^{\prime\prime}$21
	+2002	+1661	2295	19550802-19551113	60	105	0 $.\!\!^{\prime\prime}$91	M1955O1n	+6.2749$\,\pm\,$0.8157	+1986$\,\pm\,$381	0 $.\!\!^{\prime\prime}$96
									+1.6158$\,\pm\,$0.4874
1996 N1	-161	-621	+526	19960704-19961012	283			MW97
0.926	-160	-620	+527	19960704-19961103	316	614	0 $.\!\!^{\prime\prime}$81	M1996N1		$-160.2\pm 6.4$	0 $.\!\!^{\prime\prime}$83
	- 96	-556	+590	19960704-19961103	316	614	0 $.\!\!^{\prime\prime}$81	[M1996N1n]	+0.9394$\,\pm\,$0.8936	$-97\pm 60$	0 $.\!\!^{\prime\prime}$83
1998 P1	-128	+205	+1116	19980811-19990515	461	922	4 $.\!\!^{\prime\prime}$68	M1998P1		$-128\pm 10$	4 $.\!\!^{\prime\prime}$75
1.15	+797	+1127	+2036	19980811-19990515	461	922	1 $.\!\!^{\prime\prime}$18	M1998P1n	+32.611$\,\pm\,$0.417	$797\pm 13$	1 $.\!\!^{\prime\prime}$20
									+0.8765$\,\pm\,$.1368
									( $-1.194\pm .0549$)
1968 N1	-82	-573	+260	19680713-19681110	131			MW97
1.16	-153	-644	+189	19680713-19681110	147	291	1 $.\!\!^{\prime\prime}$19	M1968N1		$-147\pm 117$	1 $.\!\!^{\prime\prime}$22
	+1728	+1243	+2072	19680713-19681110	147	291	1 $.\!\!^{\prime\prime}$18	M1968N1n	+5.697 $\,\pm\,$3.001	+1759$\,\pm\,$913	1 $.\!\!^{\prime\prime}$22
									+15.406$\,\pm\,$7.061
									( $-4.401\,\pm\, 2.029$)
1986 P1	0	-260		19860805-19890411	642			MW97	+1.8; +0.1
1.20	-.16	-335	+725	19860805-19890411	688	1359	1 $.\!\!^{\prime\prime}$66	M1986P1		$-.14\pm 1.02$	1 $.\!\!^{\prime\prime}$69
	+77	-258	+802	19860805-19890411	688	1359	1 $.\!\!^{\prime\prime}$33	M1986P1n1	+1.7984$\,\pm\,$0.0655	+76$\,\pm\,$3	1 $.\!\!^{\prime\prime}$35
	+75	-259	+801	19860805-19890411	688	1359	1 $.\!\!^{\prime\prime}$33	[M1986P1n2]	+1.7837$\,\pm\,$0.0672	+76$\,\pm\,$3	1 $.\!\!^{\prime\prime}$35
									+.03329$\,\pm\,$.03349
1971 E1	0	-628	+0	19710309-19710909	131			MW97n	+5.6; -2.
1.23	- 87	-766	-433	19710309-19710909	138	275	1 $.\!\!^{\prime\prime}$92	M1971E1		$-87 \pm 76$	1 $.\!\!^{\prime\prime}$97
	+372	-306	+ 27	19710309-19710909	138	275	1 $.\!\!^{\prime\prime}$30	M1971E1n1	+6.2140$\,\pm\,$0.3548	+375$\,\pm\,$ 58	1 $.\!\!^{\prime\prime}$34
	+447	-231	+101	19710309-19710909	138	275	1 $.\!\!^{\prime\prime}$30	[M1971E1n2]	+6.3224$\,\pm\,$0.3744	+447$\,\pm\,$101	1 $.\!\!^{\prime\prime}$34
									+0.5885$\,\pm\,$0.6348
1996 E1	-42	-654	+356	19960315-19961012	216			MW97
1.35	-41	-652	+357	19960315-19961012	249	494	1 $.\!\!^{\prime\prime}$06	M1996E1		$-40.9\pm 4.3$	1 $.\!\!^{\prime\prime}$09
	-22	+52	+377	19960315-19961012	249	494	0 $.\!\!^{\prime\prime}$88	M1996E1n1	+6.333$\,\pm\,$0.469	$-21.3\pm 9.0$	0 $.\!\!^{\prime\prime}$90
									+0.4137$\,\pm\,$0.1544
	+8	+82	+406	19960315-19961012	249	494	0 $.\!\!^{\prime\prime}$88	[M1996E1n2]	+7.169$\,\pm\,$1.093	+10$\,\pm\,$39	0 $.\!\!^{\prime\prime}$90
									$-0.0358\pm 0.5521$
									(+0.291$\,\pm\,$0.343)
1946 C1	-13	-678	+373	19460202-19470809	183			MW97
1.72	-4	-669	+382	19460129-19470809	498	807	2 $.\!\!^{\prime\prime}$92	M1946C1		$-3.5\pm 5.4$	2 $.\!\!^{\prime\prime}$97
	+36	-628	+423	19460129-19470809	498	807	2 $.\!\!^{\prime\prime}$84	M1946C1n	+2.242$\,\pm\,$0.853	+36.6$\,\pm\,$8.1	2 $.\!\!^{\prime\prime}$89
									+2.552$\,\pm\,$0.632
1993 A1	0	-918	+0	19930102-19940610	539			MW97n	+16.6; -2.2
1.94	-18	-1066	-539	19921126-19940817	723	1446	2 $.\!\!^{\prime\prime}$81	M1993A1		$-18.6\pm 3.8$	2 $.\!\!^{\prime\prime}$84
	+126	-922	-395	19921126-19940817	723	1446	1 $.\!\!^{\prime\prime}$20	M1993A1n	+15.961$\,\pm\,$0.200	+125.6$\,\pm\,$2.6	1 $.\!\!^{\prime\prime}$21
									$-2.2460\pm 0.2374$
1946 U1	-1	-393	+26	19461102-19481002	97			MW97
2.41	+3	-389	+107	19461101-19481002	143	260	2 $.\!\!^{\prime\prime}$09	M1946U1		+3.2$\,\pm\,$7.8	2 $.\!\!^{\prime\prime}$15
	+65	-327	+92	19461101-19481002	143	260	2 $.\!\!^{\prime\prime}$02	M1946U1n	+18.866$\,\pm\,$7.795	+65$\,\pm\,$16	2 $.\!\!^{\prime\prime}$08
									+34.829$\,\pm\,$9.600

It has been shown that barycentric elements of the orbit determined at a distance above 150-200 AU change insignificantly (Todorovic-Juchniewicz 1981). For this reason each comet is followed from its position at a given epoch (see Table 3) backwards (original orbit) and forwards (future orbit) until the comet reaches a distance of 250 AU from the Sun. The barycentric orbital parameters of incoming comets (before planetary perturbations) and outgoing comets (whose orbits have been perturbed) are called "original'' and "future'' quantities, respectively. In the numerical calculations the equations of motion are integrated in barycentric coordinates using recurrent power series method (Sitarski 1989) taking into account the perturbations by all nine planets. The respective values of original, osculating and future reciprocals of semimajor axis are given in Cols. 2-4 of Tables 1 and 2. The first line for each comet gives respective values of reciprocals of semimajor axis taken from the MW Catalogue.

The most hyperbolic original orbit in the sample was Comet Sato 1975 X1 ( $(1/a)_{\rm ori} = -1071$ in units of 10^-6 AU^-1used in this paper) with q=0.86, and Comet Honda 1955 O1 ( $(1/a)_{\rm ori} = -635$) with q=0.89 (see Table 1). In the next section we show that both these comets have strongly positive values of $(1/a)_{\rm ori}$ if the non-gravitational solutions are considered.

Our $(1/a)_{\rm ori}$ values differ substantially from those published by Marsden and Williams in their catalogue. These discrepancies in $(1/a)_{\rm ori}$ are caused by an enlargement of the observational data and by adoption of the objective data selection procedure (Bielicki & Sitarski 1991). The influence of data selection is seen for the first and third comet of Table 2 where the more restrictive selection procedure results in less negative values of $(1/a)_{\rm ori}$ for these comets. Generally, the present $(1/a)_{\rm ori}$ values are less negative than those given in the MW Catalogue (see Tables 1 and 2). Out of seventeen comets listed in Table 2, roughly half (nine comets) turned out to have positive $(1/a)_{\rm ori}$ values and none had a more negative value of $(1/a)_{\rm ori}$. Four comets with more negative values of $(1/a)_{\rm ori}$ are given in Table 1, but for all of them the non-gravitational effects change the original orbits from hyperbolical to elliptical (see the next section).

 

 

Table 2: Reciprocals of semimajor axis for incoming, osculating and future orbits (in units of 10^-6 AU^-1) for pure gravitational motion. The "original'' and "future'' values of reciprocal of the semimajor axis are given in barycentric coordinates. The eighth column gives the mean original $(1/a)_{\rm ori}$ obtained from a sample of 500 randomly selected orbits (with the mean rms residual given in the last column).

Comet	Standard orbits determined from observations								Sample of 500 random orbits
design.	1/a			Interval of	No.	No.	rms	Model		Fitting
&	in units of			observations	of	of				to obs.
q	10-6 AU-1				obs.	res			$\left < (1/a)_{\rm ori}\right >$	with
	ori	osc	fut							rms of
1899 E1	-109	-1093	-1253	18990305-18990811	124			MW97
0.33	-11	-961	-1154	18990305-18990710	240	457	4 $.\!\!^{\prime\prime}$29	M1898E1a	$-11\pm 19$	4 $.\!\!^{\prime\prime}$39
	-8	-958	-1154	18990305-18990710	240	449	3 $.\!\!^{\prime\prime}$93	M1898E1b	$-7\pm 19$	4 $.\!\!^{\prime\prime}$02
1952 W1	-125	-442	-283	19521210-19530718	64			MW97
0.778	+19	-297	-139	19521213-19530718	24	41	0 $.\!\!^{\prime\prime}$64	M1952W1	$+21\pm 37$	0 $.\!\!^{\prime\prime}$70
1987 A1	-121	-416	-164	19870108-19870524	33			MW97
0.92	-48	-342	-91	19870108-19870524	69	137	2 $.\!\!^{\prime\prime}$19	M1987A1a	$-45\pm 127$	2 $.\!\!^{\prime\prime}$28
	-43	-336	-91	19870108-19870524	69	136	2 $.\!\!^{\prime\prime}$12	M1987A1b	$-35\pm 125$	2 $.\!\!^{\prime\prime}$21
1892 Q1	-27	-452	-539	18920901-18930710	123			MW97
0.976	+84	-341	-428	18920902-18930622	15	29	2 $.\!\!^{\prime\prime}$38	M1892Q1	+80$\,\pm\,$57	2 $.\!\!^{\prime\prime}$62
1940 S1	-124	-1374	-1123	19401004-19410102	19			MW97
1.06	+4677	+3428	+3681	19401004-19410103	40	59	3 $.\!\!^{\prime\prime}$08	M1940S1	$+4658\pm 366$	3 $.\!\!^{\prime\prime}$28
1996 J1-B	-15	-714	+554	19960510-19970809	220			MW97
1.30	+20	-679	+569	19960510-19981217	529	1039	0 $.\!\!^{\prime\prime}$71	M1996J1	$+19.9 \pm 7.7$	0 $.\!\!^{\prime\prime}$72
1932 M1	-56	-365	-327	19320621-19321230	48			MW97
1.65	+33	-276	-237	19320621-19321201	78	153	2 $.\!\!^{\prime\prime}$93	M1932M1	$+34\pm 46$	3 $.\!\!^{\prime\prime}$05
1904 Y1	-75	-360	+170	19041218-19050502	48			MW97
1.88	+5	-280	+251	19041218-19050502	164	306	3 $.\!\!^{\prime\prime}$48	M1904Y1	$+12\pm 133$	3 $.\!\!^{\prime\prime}$58
1980 R1	-26	-815	-375	19800906-19810405	16			MW97
2.11	-4	-793	-352	19800906-19810405	18	35	2 $.\!\!^{\prime\prime}$16	M1980R1	$-2\pm 132$	2 $.\!\!^{\prime\prime}$35
1983 O2	-18	-87	+402	19830804-19840605	22			MW97
2.25	-18	-86	+402	19830804-19840605	39	68	1 $.\!\!^{\prime\prime}$58	M1983O1	$-18\pm 14$	1 $.\!\!^{\prime\prime}$68
1898 V1	-71	+216	+620	18981115-18990604	70			MW97
2.28	+0.09	+287	+691	18981115-18990604	39	72	2 $.\!\!^{\prime\prime}$37	M1898V1	$+2\pm 70$	2 $.\!\!^{\prime\prime}$51
1997 J2	-119	-360	-159	19970505-19970910	455			MW97
3.05	+47	-193	+6	19970505-19981004	1118	2236	0 $.\!\!^{\prime\prime}$81	M1997J2	$+47.3\pm 1.0$	0 $.\!\!^{\prime\prime}$82
1997 BA6	-484	-180	-112	19970111-19970504	102			MW97
3.44	+40	+345	+412	19970111-19980528	213	423	0 $.\!\!^{\prime\prime}$58	M1997BA6	$+40.7\pm 5.3$	0 $.\!\!^{\prime\prime}$60
1942 C2	-34	-774	-282	19420212-19430309	35			MW97
4.11	-26	-767	-280	19420212-19430311	48	88	1 $.\!\!^{\prime\prime}$65	M1942C2	$-24\pm 15$	1 $.\!\!^{\prime\prime}$74
1997 P2	-181	-6866	-2576	19970812-19970910	79			MW97
4.26	-27	-6654	-2342	19970812-19970930	95	185	0 $.\!\!^{\prime\prime}$65	M1997P2	$-27\pm 15$	0 $.\!\!^{\prime\prime}$67
1978 G2	-23	-340	-99	19780211-19800123	7			MW97
6.28	-23	-341	-100	19780211-19800123	7	14	0 $.\!\!^{\prime\prime}$82	M1978P2	$-23\pm 45$	0 $.\!\!^{\prime\prime}$94
1999 J2; 7.11	-15	-181	-123	19990512-19990909	329	657	0 $.\!\!^{\prime\prime}$61	M1999J2	$-15\pm 18$	0 $.\!\!^{\prime\prime}$62

The most positive value of $(1/a)_{\rm ori} = +4677$ belongs to Comet 1940 S1 Okabayashi-Honda (Table 2) whose orbit was calculated on the basis of 40 positional observations spanning a three months period. This orbit was poorly determined (the worst of whole sample), and significantly different from that given in the MW catalogue which was mainly caused by the three last observations of January 3rd, 1940. According to Kresak (1992) estimates of internal errors in 1/a for orbits of quality classes 1A-2B, this comet does not belong even to the lowest quality class 2B.

We have estimated the uncertainties of $(1/a)_{\rm ori}$ values. This was made by means of the following numerical simulations. The orbital elements (hereafter standard, or nominal elements) were determined by an iterative, least squares process from equations based on the positional observations. This "best solution'' orbit is one of the potential orbits allowed by observations of the comet. Since we do not know the true cometary orbit, we will construct a set of randomly selected orbits which all fit well with the observations used for the orbit determination. The orbit selection procedure was adopted from Sitarski (1998) where the entire method was described in details. According to this procedure, we randomly selected a set of 500 orbits belonging to the same celestial body, e.g., a series of orbits in which the comet "could'' move. A random number generator with Gaussian distribution was applied to select random values of the right sides of the elimination equations described in Sect. 3 of Sitarski's paper (1998). Then new values of orbital elements were calculated from the elimination equations. Each set of six orbital elements describing the chosen orbit differs from the nominal values, but they always fit the observations with an accuracy defined by residuals calculated according to Sitarski's method. The mean rms residuals of randomly selected orbits are given in the last column of Tables 1 and 2. Next, for each random orbit we calculated the original orbit. Thus, for each comet we obtained the distribution of original reciprocals of semimajor axes (Figs. 1 and 2). The normal distribution was then fitted to the histogram of original $1/a_{\rm ori}$.

Examples of the Gaussian function fitting to the obtained $(1/a)_{\rm ori}$ distributions are given in Fig. 1 for Comet 1997 P2 (Spacewatch), and in Fig. 2 for three other comets. In this way we calculated the mean value of $1/a_{\rm ori}$ and its standard deviation $\sigma$. The goodness of fit was tested by a $\chi ^2$ method and fitting by normal distributions was very good in all cases. The respective values of $\left< 1/a_{\rm ori}\right>$ with their $\sigma$'s are given in Tables 1 and 2. One finds that the planetary perturbations also contribute to the calculated uncertainties. However, for all the comets from the sample the uncertainties in original and future orbits are similar to uncertainties in randomly selected orbits and reflect the internal errors of orbital elements derived from astrometric data. This is illustrated in Fig. 1 for Comet 1997 P2 (Spacewatch) which underwent the strongest energy perturbation (among the group of analyzed objects) during its passage through the planetary system. Before its discovery this comet passed close to Jupiter (at a distance of about 0.65 AU) and in the middle of 2000 it passed 2.7 AU from Saturn losing $\delta (1/a) = (1/a)_{\rm fut} - (1/a)_{\rm ori} = -2315$ (see Table 2), mostly due to these two close encounters. Thus, this comet suffers planetary perturbations of almost a factor of 4 greater than the typical (mean) value of energy perturbations of long-period comets (about $\delta (1/a) =670$; Yabushita 1979). For all 500 randomly selected orbits we obtained, the calculated uncertainties of $(1/a)_{\rm fut}$ and $\delta (1/a)$ are equal to 36 and 21, respectively. This means that the observational uncertainties usually do not affect our estimates of planetary perturbations and do not contribute substantially to the uncertainties of calculated $(1/a)_{\rm ori}$and $(1/a)_{\rm fut}$. Therefore, uncertainties of $(1/a)_{\rm ori}$ given in Tables 1 and 2 are closely connected with the orbital quality classes 1A-2B.

Figure 1: Distributions of reciprocals of semimajor axes for Comet 1997 P2 (Spacewatch). The left panel shows distribution of heliocentric $(1/a)_{\rm osc}$ for 500 dynamical clones of the osculating orbit (from the normal distribution given by black dots) which were then integrated backwards and forwards to the distance of 250 AU from the Sun. In this way the distributions of $(1/a)_{\rm ori}$ and $(1/a)_{\rm fut}$ for barycentric orbits were derived and these are shown in the middle and in the right panels, respectively. The Gaussian functions fitted to the $(1/a)_{\rm ori}$ and $(1/a)_{\rm fut}$ distributions are represented by open circles.

Figure 2: Distribution of original reciprocals of semimajor axes $(1/a)_{\rm ori}$ (in units of 10-6 AU-1) derived from integrations of 500 dynamical clones of the osculating orbit of three comets. In each panel, the left histogram corresponds to the pure gravitational solution and the right one - to the non-gravitational solution. The Gaussian functions fitted to the $(1/a)_{\rm ori}$ distributions are shown with squares; the mean values of Gaussian distributions $\left < (1/a)_{\rm ori}\right >$and their uncertainties are given in the tenth column of Table 1.