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

## 2. Modelling of the nongravitational acceleration

The nongravitational acceleration observed in the motion of periodic comets is due to the momentum transferred to the nucleus by the anisotropic outgassing. When a comet approaches the Sun the heating of a nucleus surface by solar radiation causes sublimation of its icy layers. Nongravitational forces cannot be accurately predicted by theoretical analysis because of difficulty in modelling such factors as a nonspherical shape of nuclei, surface topography, distribution of icy areas exposed for vaporization, or inhomogeneity of cometary material.

In the standard model of nongravitational acceleration it is assumed that the whole surface of a rapidly rotating and smooth spherical cometary nucleus undergoes sublimation of the water snow. Marsden et al. (1973) introduced a method of determination of the nongravitational effects in the orbital investigations. According to this method the nongravitational terms are directly included into the equations of a comet's motion:

where is the radius vector, k is Gaussian gravitational constant, R is the planetary disturbing function, and vector is defined by . The orbital components of the nongravitational acceleration in the radial, transverse and normal direction are , and , respectively. In the Marsden's formalism, they are given by: , where the constants can be determined along with orbital elements from the least square fit to positional observations of a comet. The positive value of transverse component indicates an additional deceleration in a comet's motion. Likewise, a nongravitational acceleration is associated with negative value of the transverse component. The value of the nongravitational force is changing with heliocentric distance of a comet, r, according to symmetric with respect to the perihelion function which simulates the ice water sublimation rate. An existence of nonradial terms of the nongravitational force is connected with rotation of the comet nucleus.

For a cometary nucleus rotating around a fixed axis, Sitarski (1990) developed a procedure to treat the nongravitational parameters as functions of time:

where the argument of the Sun is: . The solar longitude at the perihelion is measured along the orbit from its ascending node on the equator in the sense of increasing true anomaly of the comet. The angle I is the obliquity of the orbit plane to the nucleus equator and is the lag angle of the outgassing maximum behind the subsolar meridian. The formulae for direction cosines have been introduced by Sekanina (1981). The parameters: A, , I and are determined from the observational equations of a comet in an iterative process of the orbit improvement.

Both, the standard model of the nongravitational acceleration and the model with the angular parameters, were used to investigate the motion of 43P/Wolf-Harrington in intervals of time covering either three or four of its returns to the Sun (Sect. 4.1).

A long-term variation of the nongravitational effects detected in the motion of some short-period comets can be interpreted in terms of spin axis precession. Quantitative model of the forced precession of the nuclear spin axis has been derived by Whipple & Sekanina (1979) to explain observable temporal variations in the nongravitational motion of the comet 2P/Encke. Then the model was developed and applied to comets with a strange behaviour of (Sekanina 1984, 1985a, 1985b, Sekanina & Yeomans 1985). According to the forced precession model, nonlinear variations with time of the direction of the spin axis are caused by changes in the reactive force acting on the nonspherical nucleus. The precession rate is a function of: the lag angle, , the nucleus orientation defined by angles I and , the modulus of the reactive force, the nucleus oblateness, s, and the precession factor, , which depends on a rotational period and a nucleus size. The forced precession model was adopted for orbital computation i.e. the precession formulae were transformed in such a way to include them directly into equations of the comet's orbital motion (Krolikowska et al. 1998b). In the orbital version of the model instead of the water production rate derived from the visual magnitudes estimations the function was used. Values of the model parameters: A, , I, , s, could be determined simultaneously with six orbital elements from the observational equations by the least squares method. The forced precession model was successfully employed to modelling the long-term motion of a certain number of short period comets (Sitarski 1995, 1996, Krolikowska & Sitarski 1996, Krolikowska et al. 1998a).

Sect. 4.3 presents the successfull linkage of all apparitions of the comet 43P/Wolf-Harrington based on the forced precession model in which irregular time shifts of the function with respect to perihelion were taken into account.

Yeomans & Chodas (1989) modified the symmetrical model of the nongravitational acceleration assuming that the water vaporization curve may reach its peak a certain number of days before or after perihelion. They varied the time shift of the function with respect to the perihelion passage to find the best fit to the astrometric observations of eight short-period comets. In the asymmetric outgassing model the function , at any time t, is replaced by , where . Thus, the maximum of the gas production rate is shifted by days with respect to the perihelion time. Furthermore, it is worth while to notice that the value of this shift can suffer a change over a long interval of the comet's motion. The method of determination of and its constant with time change, , in a process of the orbit improvement was developed by Sitarski (1994). In this approach the displacement is a linear function of time: , where is the osculation epoch of the initial orbit. In many cases the orbital motion of short period comets can be better approximated by the asymmetrical model of the nongravitational acceleration than by the symmetrical one. The Comet 43P/Wolf-Harrington is here a good example (see Sects. 4.2, 4.3).

The perihelion asymmetry of the gas production curve or the cometary light curve can be explained by outgassing from discrete sources on the nucleus surface. For such spotty nucleus, the maximum sublimation rate will take place when the subsolar point is closest to the active region what may not occur at the perihelion passage. Effects of discrete outgassing on the shape of the gas production curve and on nongravitational parameters were discussed in details by Sekanina (1993a, 1993c). In the Sekanina's model, an active region on the surface of a rotating nucleus experiences diurnal and seasonal variations. The seasonal effects cause the perihelion asymmetry of the gas production curve. Sekanina has shown that for a comet with isolated centres of activity, the sign of is not related to the direction of nucleus rotation and the nonradial components of the nongravitational acceleration do not vanish even when the lag angle is assumed to be zero. The erratic discontinuities in the nongravitational perturbations of comets or long-term changes in has been interpreted by Sekanina (1993b) as initiation of new active areas or deactivation of existing ones on the nucleus surface.

The first attempt to introduce the spotty nucleus model directly into the orbital computation was made for the comet 6P/d'Arrest by Szutowicz & Rickman (1993). However, in this approach the observed form of the comet light curve instead of the theoretical gas production curve was used. Actually, the Sekanina's model of discrete sources of a gas emission has been modified and adopted for orbital calculations. Details of the model are presented in Sect. 3. This model with some additional assumptions concerning a lifetime of the active regions, gave satisfactory orbital linkage of all apparitions of the comet 43P/Wolf-Harrington (Sect. 4.4). The spotty nucleus model has been successfully used to link all of observations of comet 46P/Wirtanen assuming temporal variations of the active fraction of the nucleus surface (Szutowicz 1999b, 1999c). The model allowed also to explain a dramatic jump of the nongravitational effects in the motion of the comet 71P/Clark by means of a redistribution of the surface active regions and the change of the nucleus orientation (Szutowicz 1999a).

© European Southern Observatory (ESO) 2000

Online publication: December 5, 2000