1 Introduction

A vast majority of the known short-period comets are subject to nongravitational forces (Marsden & Williams 1997). The method to determine nongravitational effects in the comet's orbital motion was proposed by Marsden et al. (1973), who assumed that the three components of a nongravitational force acting on the comet have the form:

$$F_i = A_ig(r) \>\>\> A_i = {\rm ~const ~~for~~} i=1,2,3\,$$ (1)

where F₁,F₂,F₃ represent the radial, transverse and normal component of the nongravitational force, respectively. The analytical function g(r) simulates the ice sublimation rate as a function of the heliocentric distance r. At least three consecutive apparitions are needed to determine the "constant'' nongravitational parameters A₁, A₂, A₃effective within investigated time interval. To examine whether these parameters are really constant or not, it is necessary to combine sets of three (at least) consecutive apparitions.

Figure 1: Temporal variation of the cometary distance from the Sun (thick solid curves) and from the Earth (thin solid curves) for six investigated comets. The distributions of observations with respect to perihelion passages and approaches of the comets to the Earth are shown on the bottom of each graph

Figure 2: Changes of the orbits of six comets caused by approaches to Jupiter; the dashed curves denote these parts of cometary orbit which are placed under the ecliptic plane

In most short-period comets long-term nongravitational effects are constant or slowly changing with time, however, in some comets these effects seem to be strongly variable (see Fig. 2 in Sekanina 1993a). Then, rapidly varying nongravitational perturbations are rather irregular (as in the spectacular case of Comet 21P/Giacobini-Zinner), but sometimes may show systematic trends in the motions (32P/Comas Sola and 37P/Forbes cases). It seems that some discontinuities are present in otherwise continuous nongravitational perturbations, thus comets exhibiting such a behaviour are called "erratic'' comets (Marsden & Sekanina 1971). From rich sample of known erratic short-period comets six are investigated in detail in this paper. Four of them, 31P/Schwassmann-Wachmann 2, 32P/Comas Solá, 37P/Forbes, and 43P/Wolf-Harrington, have been observed for almost seventy-year long time interval since their discoveries (see Table 1 and Fig. 1); the next two - 16P/Brooks 2 and 21P/Giacobini-Zinner - have been observed at 14 and 13 returns, respectively, for almost a century. All of them have experienced close approaches to Jupiter; in Fig. 2 we present two orbits for each comet to show changes of heliocentric orbits caused by approaches to Jupiter (more details are given in Sect. 5).

 

 

Table 1: Global range of orbital parameters (perihelion distance, q, eccentricity, e, and inclination, i) and close encounters with Jupiter for six erratic comets

Range of orbital parameters					close encounter
					with Jupiter
Year	P	q	e	i	date	dist.
[yr]	[yr]	[AU]		[$^\circ$]		[AU]
16P/Brooks 2
1886	37.5	5.453	0.513	7.134
					July 1886	.001
1911	7.10	1.963	0.469	6.070
					Feb. 1922	0.12
1994	6.89	1.843	0.491	5.541
21P/Giacobini-Zinner
1887	7.00	1.241	0.661	33.408
					Oct. 1898	0.19
1900	6.46	0.932	0.732	29.830
					Sep. 1969	0.57
1972	6.52	0.994	0.715	31.703
31P/Schwassmann-Wachmann 2
1920	9.29	3.565	0.194	0.741
					Mar. 1926	0.18
1994	6.39	2.070	0.399	3.753
					Mar. 1997	0.25
2002	8.72	3.408	0.195	4.550
32P/Comas Solá
1910	9.35	2.152	0.515	18.707
					May  1912	0.18
1927	8.52	1.772	0.575	13.766
1996	8.83	1.846	0.568	12.917
37P/Forbes
1929	6.38	1.528	0.556	4.645
1987	6.26	1.475	0.566	4.671
					Aug. 1990	0.34
1993	6.13	1.447	0.568	7.159
43P/Wolf-Harrington
1925	7.60	2.428	0.372	23.684
					June 1936	0.12
1939	6.20	1.448	0.571	18.322
					Jan. 1948	0.72
1952	6.50	1.599	0.541	18.493
1997	6.46	1.581	0.544	18.510

 

 

Table 2: Global characteristics of observations of six erratic comets. For Comet 16P/Brooks 2 observations till 1939 apparition were simulated; also the observations for 1926 and 1939/40 apparitions in the Comet 21P/Giacobini-Zinner case, were simulated; thus the mean residuals a priori for both of them are not given

Comet	No of						No	No	Mean
design.	apparitions	Observation interval					of	of	res.
							obs.	res.	a priori
16P	14	1889 Aug.	29	-	1995 Feb.	24	595	961	1 $.\!\!^{\prime\prime}$78
21P	13	1900 Dec.	24	-	1999 Apr.	30	1589	2529	2 $.\!\!^{\prime\prime}$04
31P	11	1929 Jan.	4	-	1995 July	4	485	921	1 $.\!\!^{\prime\prime}$31
32P	9	1926 Nov.	4	-	1997 June	25	582	939	1 $.\!\!^{\prime\prime}$41
37P	9	1929 Aug.	22	-	1999 Nov.	11	286	553	1 $.\!\!^{\prime\prime}$15
43P	9	1924 Dec.	22	-	1998 May	26	322	628	1 $.\!\!^{\prime\prime}$23

 

 

Table 3: Nongravitational parameters A₁, A₂, A₃ determined by linking three or four consecutive apparitions. The mean residual RMS and number of observations used to improve the orbits are given in the last two columns

Appearances	A1	A2	A3	Mean	No
	in units of 10-8 AU/day2			res	of
					obs.
16P/Brooks 2
1889/96/03	$4.11 \pm .92$	$-.3243 \pm .0021$	$-.271 \pm .229$	2 $.\!\!^{\prime\prime}$81	114
1896/03/11	$2.26 \pm .41$	$-.2870 \pm .0019$	$-.386 \pm .097$	1 $.\!\!^{\prime\prime}$47	108
1903/11/25	$0.978\pm .154$	$-.2410 \pm .0030$	$+.564 \pm .130$	1 $.\!\!^{\prime\prime}$44	118
1911/25/32	$1.01 \pm .15$	$-.2190 \pm .0066$	$+.523 \pm .103$	1 $.\!\!^{\prime\prime}$41	110
1932/39/46	$0.357 \pm .220$	$-.2517 \pm .0012$	$+.200 \pm .093$	1 $.\!\!^{\prime\prime}$89	81
1939/46/53	$1.34 \pm .27$	$-.2341 \pm .0011$	$+.121 \pm .080$	1 $.\!\!^{\prime\prime}$76	71
1946/53/60	$1.25 \pm .17$	$-.2055 \pm .0077$	$+.081 \pm .062$	1 $.\!\!^{\prime\prime}$73	63
1953/60/74	$1.35 \pm .14$	$-.2051 \pm .0020$	$+.169 \pm .061$	1 $.\!\!^{\prime\prime}$43	54
1960/74/80	$0.685 \pm .229$	$-.1614 \pm .0020$	$+.177 \pm .076$	1 $.\!\!^{\prime\prime}$67	40
1974/80/87	$0.584 \pm .235$	$-.1354 \pm .0019$	$+.253 \pm .073$	2 $.\!\!^{\prime\prime}$03	90
1980/87/94	$0.234 \pm .163$	$-.1574 \pm .0005$	$+.112 \pm .047$	1 $.\!\!^{\prime\prime}$61	181
21P/Giacobini-Zinner
1900/13/26	$0.632 \pm .136$	$+.0348 \pm .0001$	$-.037 \pm .095$	4 $.\!\!^{\prime\prime}$07	205
1913/26/33	$0.403 \pm .072$	$+.0351 \pm .0003$	$+.107 \pm .080$	3 $.\!\!^{\prime\prime}$85	197
1933/40/46	$0.179 \pm .055$	$+.0376 \pm .0023$	$-.069 \pm .055$	2 $.\!\!^{\prime\prime}$08	63
1940/46/59	$0.278 \pm .016$	$+.0393 \pm .0006$	$-.017 \pm .023$	1 $.\!\!^{\prime\prime}$71	85
1946/59/72	$0.244 \pm .011$	$+.1063 \pm .0001$	$-.169 \pm .015$	2 $.\!\!^{\prime\prime}$44	180
1959/72/79	$0.357 \pm .021$	$-.0182 \pm .0005$	$+.132 \pm .021$	2 $.\!\!^{\prime\prime}$25	144
1972/79/85	$0.398 \pm .015$	$-.0610 \pm .0013$	$-.077 \pm .016$	1 $.\!\!^{\prime\prime}$81	314
1979/85/92	$0.146 \pm .038$	$-.0600 \pm .0008$	$-.148 \pm .019$	1 $.\!\!^{\prime\prime}$30	228
1985/92/99	$0.218 \pm .010$	$-.0512 \pm .0018$	$-.022 \pm .012$	1 $.\!\!^{\prime\prime}$84	419
31P/Schwassmann-Wachmann 2
29/35/42/48	$0.863 \pm .225$	$-.1984 \pm .0033$	$-.082 \pm .074$	1 $.\!\!^{\prime\prime}$34	83
35/42/48/55	$1.264 \pm .059$	$-.1943 \pm .0013$	$-.052 \pm .049$	1 $.\!\!^{\prime\prime}$12	103
42/48/55/61	$0.805 \pm .082$	$-.1633 \pm .0012$	$+.076 \pm .046$	1 $.\!\!^{\prime\prime}$11	121
48/55/61/68	$1.347 \pm .052$	$-.1672 \pm .0014$	$+.007 \pm .053$	1 $.\!\!^{\prime\prime}$38	116
55/61/68/74	$1.811 \pm .063$	$-.2111 \pm .0014$	$-.088 \pm .050$	1 $.\!\!^{\prime\prime}$70	146
61/68/74/81	$2.003 \pm .068$	$-.1837 \pm .0015$	$-.026 \pm .062$	2 $.\!\!^{\prime\prime}$11	125
68/74/81/87	$1.576 \pm .092$	$-.2333 \pm .0025$	$+.040 \pm .088$	3 $.\!\!^{\prime\prime}$48	136
74/81/87/94	$1.306 \pm .031$	$-.2960 \pm .0009$	$+.005 \pm .029$	1 $.\!\!^{\prime\prime}$66	273
32P/Comas Solá
1926/35/44	$0.823 \pm .088$	$+.0033 \pm .0015$	$-.054 \pm .045$	2 $.\!\!^{\prime\prime}$02	172
1935/43/53	$0.688 \pm .055$	$+.0418 \pm .0011$	$-.143 \pm .035$	1 $.\!\!^{\prime\prime}$48	125
1943/52/62	$0.744 \pm .051$	$-.0159 \pm .0011$	$-.130 \pm .027$	1 $.\!\!^{\prime\prime}$33	116
1951/61/70	$0.691 \pm .046$	$-.0729 \pm .0010$	$-.211 \pm .028$	1 $.\!\!^{\prime\prime}$33	126
1960/69/79	$0.710 \pm .048$	$-.1253 \pm .0022$	$-.202 \pm .030$	1 $.\!\!^{\prime\prime}$52	114
1968/78/88	$0.605 \pm .104$	$-.1358 \pm .0010$	$+.037 \pm .049$	1 $.\!\!^{\prime\prime}$49	114
1977/87/96	$0.872 \pm .072$	$-.1125 \pm .0014$	$-.090 \pm .054$	1 $.\!\!^{\prime\prime}$41	162
37P/Forbes
1929/42/48	$0.416 \pm .186$	$+.0805 \pm .0028$	$+.044 \pm .040$	1 $.\!\!^{\prime\prime}$82	48
1942/48/61	$0.562 \pm .063$	$+.0488 \pm .0007$	$-.096 \pm .025$	1 $.\!\!^{\prime\prime}$56	46
1948/61/74	$0.694 \pm .036$	$-.0260 \pm .0001$	$-.072 \pm .018$	1 $.\!\!^{\prime\prime}$71	65
1961/74/80	$0.565 \pm .049$	$-.0784 \pm .0005$	$-.036 \pm .022$	1 $.\!\!^{\prime\prime}$69	68
1974/80/87	$0.829 \pm .060$	$-.0626 \pm .0017$	$-.046 \pm .022$	1 $.\!\!^{\prime\prime}$26	66
1980/87/93	$0.359 \pm .030$	$-.0391 \pm .0012$	$-.023 \pm .018$	1 $.\!\!^{\prime\prime}$14	111
1987/93/99	$0.409 \pm .027$	$-.0403 \pm .0013$	$-.012 \pm .018$	1 $.\!\!^{\prime\prime}$06	167
43P/Wolf-Harrington
1924/52/59	$0.109 \pm .077$	$-.0648 \pm .0040$	$-.022 \pm .041$	1 $.\!\!^{\prime\prime}$48	79
1951/58/65	$0.199 \pm .038$	$-.0513 \pm .0012$	$-.049 \pm .025$	1 $.\!\!^{\prime\prime}$18	91
1958/65/71	$0.343 \pm .034$	$-.0486 \pm .0011$	$-.031 \pm .023$	1 $.\!\!^{\prime\prime}$27	77
1965/71/78	$0.370 \pm .084$	$-.0471 \pm .0025$	$-.013 \pm .039$	1 $.\!\!^{\prime\prime}$60	59
1971/78/84	$0.333 \pm .056$	$-.0096 \pm .0028$	$-.000 \pm .037$	1 $.\!\!^{\prime\prime}$87	69
1978/84/91	$0.285 \pm .116$	$-.0178 \pm .0025$	$-.033 \pm .057$	1 $.\!\!^{\prime\prime}$86	71
1984/91/98	$0.406 \pm .021$	$-.0366 \pm .0008$	$-.009 \pm .015$	1 $.\!\!^{\prime\prime}$12	184

Various interpretations have been proposed to explain the long-term variations in nongravitational perturbations of comet's orbital motion. In this paper the nongravitational motion of these erratic comets during their whole observational intervals assuming that sublimation of the rotating and precessing icy cometary nucleus is a source of nongravitational effects and their temporal variations. We will show that it is possible to link all apparitions of each comet on the basis of forced precession model with physically reasonable parameters. Thus, we conclude that the forced precession model of the rotating nonspherical cometary nucleus adequately explains variations of the nongravitational effects observed in investigated "erratic'' comets and is suitable to make predictions of the future returns. Preliminary analysis including five of six comets discussed here, was published elsewhere (Królikowska et al. 1999). These investigations are a continuation of studies of nongravitational effects undertaken by Sitarski and collaborators (e.g. Sitarski 1992, 1994b; Królikowska & Sitarski 1996; Królikowska et al. 1998a, 1998b; Królikowska & Szutowicz 1999; Szutowicz 1999, 2000).

Figure 3: Temporal variations in the nongravitational parameter A2 for six short-period comets. The open circles represent values of A2 determined as constant values within sets of at least three consecutive apparitions (see Table 3); these circles are referred to the middle of the time intervals covered by calculations (which are shown as thin solid horizontal lines). The solid circles are the mean (e.g. $\overline A_2$ from Eq. (3)) averaged over three consecutive revolutions around the Sun) values of A2 (e.g. resulting from ours forced precession models discussed in the Sects. 3-5). Solid circles are drawn on the middle of thick dashed horizontal lines which represent the time interval taking into account for A2 averaging. Upright arrows indicate the moments of discontinuities of $\tau$ (and also A in 21P/case) introduced to the forced precession models (see Sect. 4 and Table 4)