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Astron. Astrophys. 363, 323-334 (2000)

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4. Nongravitational motion of comet 43P/Wolf-Harrington

The orbital motion of the comet has been investigated based on astrometric observations made during its nine observable apparitions from 1924 to 1998. The total number of the collected observations amounted to 415. From the last return of the comet to the Sun, 212 observations are available. For this dominated apparition the numbers of observations were decreased by taking into account so called normal places i.e. if more than two observations were made at the same day, they were replaced by one average comet position. For each apparition the observations were selected separately according to the mathematically objective criteria elaborated by Bielicki (Bielicki & Sitarski 1991). The mean residuals for each apparition have been calculated. The results from every apparitions have been combined and the "a priori" mean residual of [FORMULA], representing accuracy of the whole observational set, was obtained. Global characteristics of the observations are given in Table 1.


Table 1. Characteristics of the astrometric observations of 43P/Wolf-Harrington. Observations reduced by normal places are marked by [FORMULA].

4.1. Symmetrical models

The reasonably linkage of all astrometric observations of the comet 43P/Wolf-Harrington was impossible using the standard Marsden's model. This model was explicitly inconsistent with the real motion of the comet giving the mean residual [FORMULA].

To examine temporal variations of the nongravitational effects the constant nongravitational parameters [FORMULA] have been determined for each three successive observable apparitions. Values of parameters [FORMULA] for such observational intervals are listed in Table 2. The first interval of time has spanned six real apparitions of the comet because it was not observed during its three succesive returns to the Sun after 1924. A particular behaviour of the parametr [FORMULA] connected with the transverse component of the nongravitational force should be noticed. The nongravitational acceleration was slowly decreasing over a long period of time from 1924 to 1978 and then felt down reaching a small value close to zero. It is interesting to see that the acceleration did not switched into deceleration after the 1984 apparition but it started to increase. The parameter [FORMULA] is very poorly determined for almost all intervals of time.


Table 2. Nongravitational parameters [FORMULA] determined as constant values by linking of each three consecutive observable apparitions of the comet. The mean residuals are given in the last column.

An employment of the model (see Sect. 2) in which [FORMULA] are expressed in terms of angular parameters [FORMULA], I, [FORMULA] of the rotating cometary nucleus allowed to establish the perturbation in the nongravitational motion of the comet as a change in the nucleus orientation. When [FORMULA] increases from negative values to zero, the spin axis of retrograde rotating nucleus approaches the orbital plane. The values of the angular parameters obtained as results of the orbital linkages of four successsive observed apparitions, are presented in Table 3. The parameter [FORMULA] connected with [FORMULA] is rather poorly determined. Unstability in the comet's behaviour around 1984 did not allow to find the solution for the interval 1964-1984. The comet's observations from 1952 to 1991 have been linked previously (Szutowicz 1992) assuming linear variations of the angles I and [FORMULA]. Unfortunately, fitting of all apparitions of the comet to this model has failed.


Table 3. Angular parameters [FORMULA], I, [FORMULA] and A of the nucleus of the comet 43P/Wolf-Harrington determined within of sets of four consecutive apparitions. The last interval covers only three returns to the Sun.

4.2. Asymmetrical model

Asymmetrical model of the nongravitational acceleration understood as a perihelion shift of the function [FORMULA] (see Sect. 2) provided a much better orbital solution than the standard model. The orbital linkage of all comet's apparitions with the mean residual equal to [FORMULA] was obtained. However, it was necessary to introduce a linear variation of two parameters: the [FORMULA] describing the time shift of [FORMULA] with respect to the perihelion and the angle [FORMULA] connected with the comet's orientation. According to this orbital fit the function [FORMULA] reached peak 23 days before perihelion passage in 1925 and due to the linear evolution its maximum occured about 57 days after perihelion of the 1997 apparition. Linear variations of parameters remarkably limit the physical meaning of the model especially for the prediction of the future orbit.

To study a behaviour of the perihelion shift of [FORMULA] in shorter intervals of time, the asymmetric model was applied for each four successively observable apparitions of the comet. In Table 4, the values of [FORMULA] and [FORMULA] in the appropriate observational intervals are presented. The last interval includes only three apparitions, because of difficulty in linking of the observations from 1977 to 1998. One can see that a form of [FORMULA] variations is different than that for the symmetrical model. The maximum of the function [FORMULA] significantly peaks before perihelion passages for the comet's returns until the 1972. Then the acceleration decreases and the parameter [FORMULA] reaches positive values. The second change of [FORMULA] takes place after the 1991 apparition and the maximum of [FORMULA] is shifted again before perihelion. Nonlinear variations of [FORMULA] explain the poor accuracy of the orbital linkage of all comet's observations where the linear change of [FORMULA] was assumed. However, a general evolution of the shift of the maximum cometary activity, from negative to positive number of days in consecutive returns of the comet, has been confirmed.


Table 4. Constant nongravitational parameters [FORMULA], [FORMULA] and the parameter [FORMULA] describing the perihelion shift of the function [FORMULA] determined from linkages of each four consecutive apparitions of the comet. The last interval covers three apparitions.

4.3. Forced precession model

The analysis of the nongravitational parameters determined as constant values within sets of three or four consecutive apparitions has shown long-term variations of the nongravitational perturbations in the orbital motion of Comet Wolf-Harrington. One of the posssible explanation of these temporal variations could be forced precession of a rotating nonspherical cometary nucleus. Therefore the orbital version of the forced precession model (see Sect. 2) was applied. The model has been preliminary employed to link all of the comet apparitions (Krolikowska et al. 1998a), where the last return was represented by observations made only till December 1997. Values of six model parameters has been found for a prolate spheroid nucleus of the comet. The comparison of the mean residual of that orbital solution which reached [FORMULA] with the `a priori' residual of [FORMULA] indicated that the forced precession model needs some additional parameters to give better approximation to the real motion of the comet. As it follows from Table 2 the nongravitational perturbations exibit some irregular behaviour represented by unexpected changes not only in [FORMULA] but in [FORMULA], too. Time shift of the function [FORMULA] described by [FORMULA] can be determined together with the basic parameters of the forced precession model. To model discontinuities in the nongravitational motion of the comet changes of values of [FORMULA] close to the selected moments of aphelion passage has been assumed. After some numerical calculations two moments of discontinuities in 1975 and in 1988 has been established. The model parameters: A, [FORMULA], [FORMULA], [FORMULA], [FORMULA], s, [FORMULA], [FORMULA], [FORMULA] as well as orbital elements were determined together in a process of the iterative improvement of the orbit. All parameters are listed in Table 5 as Model C. The orbital solution was fitted to all observations of the comet with the mean residual of [FORMULA]. According to the model, the nucleus is a slightly prolated spheroid with the ratio of the longer axis to shorter one being equal to [FORMULA]. Precessional variations of the equatorial obliquity, I, and the solar longitude at the perihelion, [FORMULA], during 72 years of the comet's motion are shown in Fig. 1. Time dependences of the acceleration's orbital components [FORMULA], [FORMULA], [FORMULA] during the same interval of time are presented in Fig. 2. They are compared with appropriate evolutions of [FORMULA] obtained from the orbital solution for the spotty nucleus (Model A).

[FIGURE] Fig. 1. Temporal variation of angles I and [FORMULA] for Comet 43P/Wolf-Harrington due to the spin-axis forced precession of the comet's nucleus.

[FIGURE] Fig. 2. Orbital components [FORMULA] of the nongravitational acceleration as functions of time for 43P/Wolf-Harrington obtained from dicrete outgassing model (Model A) and forced precession model (Model C)


Table 5. Parameters describing the comet nucleus and the orbit obtained from fitting the forced precession model to all positional observations of the comet 43P/Wolf-Harrington

Based on discontinuities in the nongravitational perturbations the comet 43P/Wolf-Harrington may be classified as an "erratic" comet. Its orbital motion was compared with that of five comets which exhibit strongly variable nongravitational effects (Krolikowska et al. 1999, 2000). The satisfactory precessional models were found for all of these comets.

4.4. Spotty nucleus model

Time dependent shifts of a maximum activity seem to play essential role in the nongravitational motion of the comet Wolf-Harrington. Attemps to model the nongravitational acceleration as a result of the outgassing from one emission source which is active in the same degree over the whole interval motion of the comet 43P/Wolf-Harrington, have not been successful even taking into account a linear precession of the spin axis. Extensive numerical experiments with using a model of the spotty-nucleus described in Sect. 3, allowed to ascertain that processes of activation and/or deactivation of emission sources on the nucleus surface took place at least three times. From numerical fitting of the model parameters to positional observations, the lag angle, [FORMULA], the orientation of the nucleus in space described by the angles I and [FORMULA], the localization of three active regions given by cometocentric latitudes, [FORMULA], and appropriate values of the [FORMULA], [FORMULA] [FORMULA] parameters have been derived. Two orbital solutions called Model A and Model B were found on the assumption of slightly different lifetimes of active regions. In Table 6 both solutions are represented by the model parameters and orbital elements. Time of activation and deactivation of the regions are given too. The models have been satisfactory fitted to real motion of the comet, with the same mean residual of [FORMULA] as in the case of the precessional model. From presented solutions follows that the appearance and disappearance of regions located on the southern hemisphere of the comet nucleus are resposible for the variation of the nongravitational behaviour of the comet.


Table 6. Physical parameters of the nucleus and orbital elements obtained from linking all apparitions of the comet 43P/Wolf-Harrington by using of discrete outgassing model. Two possible orbital solutions called Model A and Model B are presented. The active regions are denoted by I, II, III.

In both cases the first region (I) is situated on the northern hemisphere of the nucleus at the latitude of about [FORMULA]. It was found to be the largest and persistent active, whereas on the southern hemisphere the variations of activity have been discovered. The second region (II), which was localized near to the cometary equator, became active about 150 days after perihelion of the 1965. According to scenario related to the Model A the total active area increased once again before 1978 apparition due to activation of the third region (III) on the latitude of [FORMULA]. This region decayed about 160 days after perihelion passage of the 1991. While from the Model B follows that the second region vanished after perihelion in 1978 and the third one was localized nearer to the equator than the analogue region of Model A.

Taking into account that the activation of new regions could disturb the nucleus spin axis, the orbital programme was run to search for the possible change of the angles I and [FORMULA]. Solution with the discontinuity in the equatorial obliquity around 1978 was found. However, the mean residual did not change in spite of introducing new parameters, therefore the new solution was given up.

In Fig. 2 variations of three orbital components of the nongravitational acceleration during the whole investigated period of the comet's motion resulting from the Model A are shown. One easily recognizes that in the case of the forced precession model the component [FORMULA] is always negative or close to zero, whereas for the spotty - nucleus model [FORMULA] is strongly variable near perihelions and reaches either positive and negative values.

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Online publication: December 5, 2000