5 Results and limits for the nuclei sizes and rotational periods

The nongravitational equations of the comet motion have been integrated numerically by the recurrent power series method, taking into account all the planetary perturbations (Sitarski 1979, 1984). Satisfactory solution for each of the six comets were found.

Numerical values of the nongravitational parameters as well as values of the osculating orbital elements are given in Table 4. Solutions with the positive values of $f_{\rm p}$ and oblateness s for comets 31P/Schwassmann-Wachmann 2 and 21P/Giacobini-Zinner (Table 4b) represent the cometary model with the oblate spheroidal nucleus; the negative values of $f_{\rm p}$ and s obtained for comets 43P/Wolf-Harrington, 37P/Forbes 16P/Brooks 2 and 32P/Comas Solá (Table 4a) represent the prolate spheroid (the nucleus rotates around its longer axis). According to our solution the nucleus of 21P/Giacobini-Zinner with oblateness along the spin-axis of about 0.29 (equatorial to polar radius of 1.41) is the most oblate among all six investigated comets and nucleus of 32P/Comas Solá with the ratio of equatorial to polar radius of 0.78 being the most prolate one.

Figure 4: Temporal variation of angle I, $\phi$ and components F1, F2, F3 of the nongravitational force due to the spin axis precession of cometary nucleus; dashed horizontal line on the upper panels divide models with prograde rotation (consistent with the sense of the cometary orbit around the Sun) when I<90$^\circ$ from models with retrograde rotation when I>90$^\circ$; dashed horizontal line on the middle panels indicate changes of configuration of poles in the perihelia

Figure 4: continued

The motion of the nucleus rotation axis represented by angles I and $\phi$ and the qualitative variations of the nongravitational force components F₁, F₂, F₃ acting on the comet during its successive returns to the Sun are presented in Figs. 4a-f. According to our models six comets precess in different manner and at different rates; among them the precession of 21P/Giacobini-Zinner is the most rapid one. One can see that changes of F₁, F₂ and F₃ for all comets studied are significantly different. For comet Wolf-Harrington we notice a spectacular increase of nongravitational force after reducing the perihelion distance from 2.42 AU to 1.45 AU due to a close approach to Jupiter in 1936. In the Schwassmann-Wachmann 2 case two large changes of nongravitational force are visible; both due to a close encounter with Jupiter (March 1926 within 0.179 AU and March 1997 within 0.246 AU): the first - before comet discovery, the second - after the last detection of the comet.

The present precession models impose some constraints on physical properties of the nuclei of the comets. Using solutions shown in Tables 4a-b it is possible to calculate values of Sekanina's (1984) torque factor $f_{\rm tor}=f_{\rm p}/s$. Then we are able to calculate the value of $P_{\rm rot}/R_{\rm a}$ ratio from the formula: $P_{\rm rot}/R_{\rm a}=4\pi \cdot f_{\rm p}/(5s)$, where the equatorial radius, $R_{\rm a}$, is related to the oblateness and the polar radius, $R_{\rm b}$: $R_{\rm b}/R_{\rm a}=1-s$. Our models give:

$$P_{\rm rot}/R_{\rm a} = \left\{ \begin{array}{rl} 2.42 \pm 0... ... \\ 31.6~ \pm 1.0~ & \mbox{ Forbes} \\ \end{array} \right.$$

where $P_{\rm rot}$ is in hours and $R_{\rm a}$ in kilometers.

Figure 5: Logarithm of the nucleus radius (e.g. effective radius) is plotted versus the logarithm of the rotational period for a sample of well-observed cometary nuclei. The plotted nuclei are H - Halley, S-T - Swift-Tuttle, W-H - Wilson-Harrington, E - Encke, A-R - Arend-Rigaux, T2 - Tempel 2, N1 - Neujmin 1, S-W1 - Schwassmann-Wachmann 1, S-W2 - Schwassmann-Wachmann 2, L - Levy, IAA - IRAS-Araki-Alcock, B - Borrelly, H-B - Hale-Bopp (C/1995 O1), Hya - Hyakutake (C/1996 B2); open circles denote the upper limits of the nuclear radius. Linear relationships for presented erratic comets are plotted by solid straight lines (where G-Z - Giacobini-Zinner). Analogous ralationships obtained for comets 30P/Reinmuth 1, 45P/Honda-Mrkos-Pajdusáková, 46P/Wirtanen and 51P/Harrington are shown by dashed lines

These relationships lie within the region occupied by comets with well-determined sizes and rotation periods (see Fig. 5). In Fig. 5 are also shown relationships between effective radii and rotation periods resulting from other investigations undertaken by Sitarski and collaborators for some short-period comets.