6 Comparing the individual models with other studies

6.1 Periodic comet Comas Solá

Comet 32P/Comas Solá was discovered in November 1926 and was observed extensively (from several months up to almost two years) during its all nine returns; last apparition was in 1995-97; next perihelion passage will occur in April 2005. No major orbital changes occurred during this time interval: an orbital period ranged between 8.5 and 9 yr and perihelion distance somewhat varied near 1.8 AU (see Table 2). This comet is classified as an erratic comet (Marsden et al. 1973; Sekanina 1993a) although the discontinuities in nongravitational perturbations of its orbital motions are less spectacular than for the Giacobini-Zinner. All orbital investigations of 32P/Comas Solá (Marsden et al. 1973; Forti 1983, 1989; KSS 1998b) have shown that the comet's deceleration reached maximum about 1940 and next fastly changed into acceleration. In other words, the tendency of A₂ to increase switched over to the tendency to decrease. This qualitative picture of the behaviour of A₂ may suggests that before the 1944 apparition something happened with cometary nucleus. Possibly an episodic outgassing event took place (Sekanina 1993a). This was confirmed by our numerical modelling of the real orbital motion of Comas Solá during 1926-1996. The best solution was obtained by setting the discontinuity of the time shift $\tau$ to appear at the moment of aphelion passage exactly in 1940 (KSS 1998b). This moment also coincides with the moment of transition from deceleration to acceleration visible in estimates of A₂ (see open circles in Fig. 3). In the previous paper on Comas Solá (KSS 1998b) we obtained two equivalent models representing oblate and prolate spheroidal nucleus. However, taking into account last astrometric observations carried out after perihelion passage in March 1996 (not included in KSS) it appears that corrected model (Table 3 and Fig. 4c) with prolate nucleus is better fitted to observations. Assumed discontinuity in $\tau$seems to cancel the peak of Marsden's A₂ (full circles in Fig. 3). The mean A₂ was calculated here by averaging the values of A₂ weighted by a function g(r'):

$$\overline A_2 = \int A_2(t)\, g(r') {\rm d}t /{\int g(r') {\rm d}t },$$ (3)

where the integrations are performed between given moments of two aphelion passages. These values of $\overline A_2$ are given as full circles and thick dashed horizontal lines in Fig. 3.

From CCD photometry taken at the heliocentric distance of 3.1 AU, Lowry et al. (1999) obtained an upper limit (because of coma contamination) of the nuclear radius of $3.2 \pm 0.4$ km. Thus, from our prolate forced precession model we obtain an upper limit for rotational period of 7.3 hr.

6.2 Periodic comet Forbes

Comet 37P/Forbes was discovered in August 1929 and is observed every 6.3 years except for its three returns in: 1935, 1955, and 1967. In Fig. 1, presenting the plots of comet's distances from the Sun and the Earth, one can see that there were unfavourable observational conditions during those three lost returns. In August 1990 the comet approached Jupiter to within 0.34 AU what caused some changes in orbital elements, especially in the angular elements (e.g. the inclination of the orbit plane increased from 4 $.\!\!^\circ$7 to 7 $.\!\!^\circ$2.

There was no problem linking satisfactorily all observations of the comet to one system of orbital elements and seven nongravitational parameters, although Comet Forbes may be treated as an erratic comet since the A₂ value changed its sign after 1960. This change of the secular deceleration into acceleration can be explained by the spin axis precession of the comet's nucleus as it is visible in Fig. 4. The nucleus of the comet appeared to be slightly prolate along the rotation axis, and the ratio of the polar radius to the equatorial one amounts to 1.06.

6.3 Periodic comet Brooks 2

The comet was discovered by Brooks in July 1889. It has been observed every seven years during consecutive returns to the Sun except for 1918 and 1967 when it was not rediscovered because of poor observational conditions (this is well documented in Fig. 1 where we plotted distances of the comet from the Sun and the Earth).

It was observed in 1889 that Comet Brooks 2 split into at least nine fragments. Then the principal comet was rediscovered in 1896 and has been observed during next 12 apparitions, the last time in 1994/95. The splitting was probably caused by an extremely close approach of the comet to Jupiter in 1886 to within 0.001 AU. It was carefully examined by Sekanina & Yeomans (1985) who also investigated the nongravitational motion of the comet. It seems that Comet Brooks 2 is the continuously splitting object (though now not in such a spectacular manner as in 1889) and therefore nongravitational forces acting on the comet are among the largest for short-period comets.

Values of the parameter A₂, as determined from observations of three consecutive apparitions, show irregular variations in time. Sekanina & Yeomans (1985), analysing also the light curves of the comet, tried to fit such parameters of the rotating comet's nucleus to predict the observed variations of A₂. They found a nearly spherical, slowly precessing, nucleus (they assumed the oblateness of 0.01).

We obtained our solution for the forced precessing nucleus by linking all the apparitions of Comet Brooks 2. The varying activity of the comet was simulated by four different values of $\tau$ for different intervals describing an asymmetry of the function g(r) with respect to perihelion. Values of all the remaining parameters have been obtained by the least squares solution of 961 observational equations. In our model the nucleus of the comet appeared to be slightly prolate along the spin axis, the ratio of the polar radius to the equatorial one amounts to 1.06. It is interesting that our determined value of the lag angle $\eta=21$ $.\!\!^\circ$35 is very close to that assumed by Sekanina & Yeomans (1985); also their value of the torque factor is comparable with ours: $f_{\rm tor}=f_{\rm p}/s=0.14~ 10^8$. However, mean values of A₂ calculated for every perihelion time from our model of the nongravitational motion have rather regular run with time and do not contain such jumps as in case of A₂determined by linking of three consecutive apparitions, presented in Table 3 and in Fig. 3. We also should notice that the plot of the nucleus radius versus rotational period for Comet Brooks 2 lies very well among the plots for other periodic comets in Fig. 5. Radius of the comet's nucleus was estimated by Sekanina & Yeomans (1985) as 0.4 km, which gives the rotation period of 2.3 hours, what seems to be unacceptably short. On the other hand, if we take following Sekanina & Yeomans (1985), the rotation period equal to 5.69 hours, we obtain the equatorial radius of the comet's nucleus about 1 km.

6.4 Periodic comet Wolf-Harrington

The comet 43P/Wolf-Harrington was discovered twice. First time it was observed by M. Wolf in December 1924 and only after twenty seven years in October 1951 it was rediscovered by R. G. Harrington. The identity of both comets was supported by Wisniewski (1964). Between 1924 and 1951 the comet three times returned to the Sun and approached Jupiter twice: in June 1936 to within 0.123 AU and in January 1948 to within 0.716 AU. These approaches caused considerable changes in the comet's orbit. After 1936 the perihelion distance was reduced from 2.43 to 1.45 AU and the orbital period dropped from 7.6 to 6.2 years. The second encounter to Jupiter caused the opposite changes but much smaller then the first one. Since 1952 the comet has been observed at every return to perihelion every 6.5 years. The perihelion distance of the comet will decrease into 1.36 AU and its orbital period will reach 6.1 years in the 2010 apparition due to approach Jupiter in June 2007. The orbit evolution is shown in Fig. 1.

The orbital motion of the comet is subjected to variable nongravitational perturbations (Szutowicz 1987, 1992). The Marsden's nongravitational acceleration was slightly decreasing until 1978 and then rapidly changed reaching value close to zero (see Table 3 and open circles in Fig. 3). The general tendency of this variation was no longer kept and the acceleration started to increase after the next apparition in 1984.

Preliminary linkage of all comet's apparitions by the forced precession model proved that the model should be enriched by additional parameters (Królikowska et al. 1998a, 1998b). The forced precession model was reasonably fitted to all observations when two changes of the time shift of the function g(r') with respect to the perihelion were assumed. Thus, our orbital solution is represented by six basic precessional parameters and three values of time shifts (see Table 4). It is interesting that evolution of the perihelion shift of g(r') from negative to positive and again negative number of days, has been confirmed by a model of other type. All observations of the comet have been also successfully linked by the spotty nucleus model where discontinuities in the nongravitational perturbations are connected with a surface redistribution of the active areas (Szutowicz 2000). The moments of discontinuities in $\tau$ roughly coincide with time of activation and deactivation of discrete sources of a gas emission established on the basis of the spotty nucleus model. It seems that the irregular perihelion shifts of the maximum activity play an essential role in the nongravitational perturbations of this comet. It should be noticed that the asymmetric model of outgassing for the precessing nucleus of the comet Wolf-Harrington essentially changed the meaning of transverse component of the nongravitational force. As it follows from Fig. 3 the parameter A₂ is not so strongly variable with time as in the case of symmetric model.

There are not too many nuclear radius estimates of the comet 43P/Wolf-Harrington. Lowry et al. (1999) derived the nuclear radius of 3.3 km from photometric observations of the comet at a heliocentric distance of 4.87 AU. They also estimated the lower limit of the nuclear radius of 0.42 km based on the measured OH production rates given by A'Hearn et al. (1995). Based on this last value our forced precession model gives a lower limit of rotational period of 11.4 hrs.

6.5 Periodic comet Schwassmann-Wachmann 2

This comet is a particularly appropriate candidate for a detailed investigation of nongravitational forces because of two observational facts: 1.) it has been observed at all eleven apparitions since its discovery in 1929;   2.) the astrometric material covered the substantial arcs of revolution periods at almost each return (in the last three apparitions even more than 2.5 yr out of the orbital period of 6.5 yr). This comet was discovered after its close encounter with Jupiter (within 0.179 AU) when the perihelion distance reduced from 3.56 AU to 2.09 AU, the eccentricity increased from 0.19 to 0.39 and orbital period shortened from 9.3 yrs to 6.4 yrs (see Table 2). In March 1997 - after the last comet detection - the second close approach to Jupiter occurred (within 0.246 AU) which moved comet back to the larger perihelion distance of 3.41 AU; next perihelion passage will occur in 2002. During almost 70 years of observations only moderate orbital changes were detected: the perihelion distance varied between 2.09 AU (1929) - 2.16 AU (1961) - 2.07 AU (1994). Nevertheless, this comet, in spite of rather large perihelion distance, exhibits strong variations of A₂ and additionally belongs to a group of comets with largest negative (and absolute) values of A₂ (see Fig. 2 in Sekanina 1993a).

Sekanina (1993b) investigated this comet assuming outgassing from a few isolated active areas. His analysis was based on observed light curves and eight discrete values of nongravitational parameter A₂ derived from orbital calculations. The apparent light curves of consecutive apparitions were compiled in four groups: 1929-35, 1942-48, 1955-68, 1974-87. He found a possible solution with four active sources: one persistent and three short-lived appearing in 1929, 1955 and 1961. Forced precession model presented in Table 4 requires two discontinuities in time shift of maximum of activity, however occurring not before 1968 perihelion passage. Thus, seven former appearances carried out between 1929-68 could be linked to a single set of seven parameters A, $\eta, I_0$, $\phi _0, f_{\rm p}$, s and $\tau$. Before 1970 and after 1978 aphelion passages we derived small positive time shifts about 3.3 days and 1.2 days, respectively, in significant contrast to a large time shift of about 62 days during 1974 perihelion passage (e.g. in time interval between 1970-1978). These discontinuities can be interpreted as some sudden changes in the geometry of activity related to the activation of a new discrete source and disappearance/decline of some old one; however global cometary activity (described by parameter A in forced precession model) can remain unaltered.

Unfortunately, it is not possible to verify such large temporal changes with observed light curves because only on five apparitions: 1929, 1942, 1955, 1968 and 1981 (Fig. 4 in Sekanina 1993b) the comet was observed photometrically for several months around the perihelion. In the remaining returns the comet was detected for many months before and many months after perihelion passages. Taking into account more complete light curves around perihelion passage at 1929, 1942, 1955 it seems that maximum occurred slightly before these passages. However, we do not have any information about the position of maximum light curves in 1935, 1948 and 1961 perihelion passages, which contributed to our slightly positive value of time shift before 1970.

Orbital calculations based on Marsden's A₁, A₂, A₃ show local maximum of A₂ around 1974 perihelion passage. This coincides with two model discontinuities of $\tau$ at 1970 and 1978 aphelion passages which are essential in linking all eleven returns of 31P/Schwassmann-Wachmann 2 (hereafter SW2). As in the Comas Solá case, assumed discontinuities of $\tau$ cancel this local peak of Marsden's A₂ (compare open circles with full ones in Fig. 3).

From the near aphelion CCD photometry of SW2 Luu & Jewitt (1992) found a best-fit lightcurve period of $5.58 \pm 0.03$ hr which they interpreted as the rotation period of the nucleus. Since the geometric albedo of the nucleus of SW2 is unknown, Luu & Jewitt assumed a typical albedo for other known nuclei of 0.04, which corresponds to an upper limit (because of the presence of coma in their data) of effective radius of 3.1 km. Next, from about 0.5 magnitude variation of the lightcurve they conclude that the axis ratio $R_{\rm a}/R_{\rm b}$ (equatorial to polar radius) must be greater than 1.6 (rather extreme asphericity). For such nucleus parameters Luu & Jewitt derived the critical density (resulting from assumption that nucleus must be stable against centripetal disruption) of about 460 kg m^-3. They conclude that this value is greater than other cometary estimates of mean density (see Fig. 6 therein). In fact substituting their values of $P_{\rm rot}$ and axis ratio into Eq. (9) in their paper we obtain even greater value of 530 kg m^-3. (It seems that 460 kg m^-3 corresponds to axis ratio of about 1.4.) Our model is in some contradiction with axis ratio postulated by Luu & Jewitt. Our best forced precession model gives slightly oblate nucleus with a ratio $R_{\rm a}/R_{\rm b}$ of $1.10\pm 0.03$ what leads to a smaller critical density of about 380 kg m^-3 for the rotation ratio of 5.58 hr. This still rather high density is mainly due to a short rotation ratio. Substituting this value of rotation period into $P_{\rm rot}/R_{\rm a}$ relation, our model of SW2 requires very small effective radius of 0.40 km (equatorial radius of 0.41 km). However, A'Hearn et al. (1995) from two near-aphelion observations of SW2 in 1981 estimated a water production rate of $\log Q {\rm [H_2O]} = 27.85$, which corresponds to an active area of 7.9 km². This gives a lower limit on the nucleus radius of 0.8 km with the assumption that a whole surface is active. Since SW2 recently undergone large reduction of perihelion distance thus it should be significantly more active than others short-period comets. Therefore, divergence from the forced precession model based on rich positional material could be interpreted as resulting from underestimation of coma contamination in photometric data of SW2.

6.6 Periodic comet Giacobini-Zinner

The comet was discovered in December 1900 by Giacobini, and rediscovered in October 1913 by Zinner. Its return to the perihelion in 1907 was lost for observers because of poor observational conditions, what is very well visible in Fig. 1. Because of the unfavourable position of the comet with respect to the Earth and Sun, the comet was not observed in 1920 and 1953 too; in October 1939 only two observations have been made, and five observations in September 1965 (made at one observatory) did not fit, unfortunately, to any of our complete numerical models of the comet's motion and have had to be rejected. It is interesting that Comet Giacobini-Zinner could be a candidate for the Earth-comet collision since the ascending node of the comet's orbit lies almost in the Earth's orbit, and in December 1926 the minimum distance between orbits of both celestial bodies amounted to 0.0009 AU only. In 1985 the International Cometary Explorer reached the comet beyond 7800 km from the nucleus. However, during the comet fly-by only physical observations of cometary tails were made.

Nongravitational effects in the motion of Comet Giacobini-Zinner show an irregular behaviour in time if we observe values of the parameter A₂ as determined by linking of three consecutive apparitions of the comet (see Table 3): in the period 1900-1999 A₂ changed its sign after 1960. It was also shown that the time-shift $\tau$ plays an essential rôle when investigating the nongravitational motion of the comet (Yeomans & Chodas 1989; Sitarski 1994a), and that $\tau$ changed its sign after 1960 too.

Sekanina (1985) carefully examined the nongravitational motion of Comet Giacobini-Zinner trying to explain its "erratic'' character by the precessional motion of spin axis of the comet's nucleus. However, Sekanina has had to assume the unrealistically large oblateness of nucleus equal to 0.88. In our forced precession model for the comet's motion we obtained the reasonable oblateness s=0.29, and got a small lag angle $\eta=6\hbox{$.\!\!^\circ$ }5$ in some agreement with Sekanina who assumed $0^\circ$ for the $\eta$value. An irregular nongravitational behaviour of the comet results in the complicated numerical model of the comet's motion: we obtained two values of the nongravitational parameter A: A⁽¹⁾ before and A⁽²⁾ after 1956, and four different values of $\tau$ with different signs (we got the positive values for $\tau _1$ and $\tau _2$ and negative ones for $\tau _3$ and $\tau _4$). This model successfully links all observations of 1900-1999. We notice that the mean values of A₂ calculated from our model show quite regular run with time as it is seen in Fig. 3. On the other hand the wobbling of the nucleus spin axis is more rapid than that in Sekanina's results (see I(t) in Fig. 4a), and variations of $\phi(t)$ and F₃ express an extremely speedy spin axis precession. The ratio of the nucleus radius to the rotational period for Comet Giacobini-Zinner, resulting from our solution, has a very good position in Fig. 5.

Assuming rotational period of about $9.5 \pm 0.2$ hr detected by Leibowitz & Brosch (1986), our forced precession model gives equatorial radius of $1.99 \pm 0.08$ km and polar radius of $1.41 \pm 0.07$ km. Belton (1991), however, suggests that periodicity found by Leibowitz & Brosch (1986) refers to second harmonic giving the probable period of about 19 hr. This period provides twice as large nucleus with effective radius of 3.7 km (4.0 km and 2.8 km for equatorial and polar radius, respectively).