7 Conclusions and final remarks

The main conclusions can be summarized as follow:

1. The forced precession model of the rotating cometary nucleus successfully describes a long-term motion of short-period comets, even "erratic'' comets strongly affected by nongravitational forces. In five of six erratic comets, however, we have detected certain changes in the geometry of the sublimation area on the nucleus surface. In our formalism we modelled such changes by varying the time shift parameter, $\tau$, which represents the moment of maximum activity with respect to perihelion time (see Table 4). Additionally, in the 21P/Giacobini-Zinner case we must assume that the global activity of comet (represented by a parameter A in our model) has changed between two consecutive perihelion passages in the years 1946-1959. It was the only way to link all the observed apparitions to a single consistent model. All these necessary assumptions involved additional model parameters which should be determined from positional data. Thus, the forced precession models for "erratic'' comets, except 37P/Forbes case, are more complicated than for "normal'' comets.

2. The asymmetric model of outgassing with variable time shifts (e.g. solutions with more than one $\tau$) caused an essential modification of temporary A₂ values. The best examples are here 21P/Giacobini-Zinner and 43P/Wolf-Harrington. It is clearly seen in Fig. 3 where we should compare the open and black dots. In such cases, however, the interpretation of the parameter A₂ is more complicated. Thus, values of nongravitational parameters determined as "constants'' by linking three consecutive apparitions should be treated as very uncertain. In general, forced precession models reveal that temporal variations of A₂ are considerably tempered for investigated "erratic'' comets.

3. Forced precession model gives some important information about shape and other physical properties of the cometary nucleus. These are oblateness of the nucleus and value of $P_{\rm rot}/R_{\rm a}$ ratio or $P_{\rm rot}/R_{\rm e}$ ratio, where $R_{\rm e}$ is the equivalent radius.

Among investigated comets are comets with equatorial radius, $R_{\rm a}$, greater than polar radius, $R_{\rm b}$, (models with positive parameter s in Table 4 meaning oblate spheroidal shape) as well as comets with the equatorial radius smaller than the polar one (prolate spheroidal shape of nucleus is represented by negative s). This is in a full agreement with observations which indicate the existence of elongated comets (for example 49P/Arend-Rigaux with probable axial ratio of $R_{\rm b}/R_{\rm a} = 1.59$ and 19P/Borrelly with $R_{\rm b}/R_{\rm a} = 2.55$).

The range of $P_{\rm rot}/R_{\rm a}$ ratios derived from forced precession models is also in a very good agreement with observations. If we draw lines representing these ratios on the rotational period vs. radius diagram, it turned out that all of them lie among the plotted comets with well-known sizes and rotational periods (see Fig. 5).

Acknowledgements

The authors would like to thank Professor Andrzej Wernik for his contributions to improve this paper. This work was supported by the Polish Committee for Scientific Research (the KBN grant 2.P03D.002.09).