ANGULAR DOMAINS IN COMETARY CAPTURE AND ANISOTROPY

In the earlier work (1994) of the authors a theoretical formulation wa s developed for the capture process under the influence of external pe rturbations. The parameter hyperspace was scanned for physically accep table values of sigma, V-o, U', and n; alpha = root 2 sigma being the most probable speed of comets, V-o the Sun's speed, U' the perturbativ e component of velocity and n the number density of comets per pc(3); so that the computed capture rate and anisotropy conform to their obse rved values. The present paper is an extension of the above said earli er work in the sense that it gives an insight into the causes leading to anisotropy in capture. Different capture conditions obtained here f or capture from the antapexial and apexial directions reveal that a la rger velocity domain is allowed for capture of comets from the former direction. This disparity in allowed velocity domains for the capture from the two directions is found to explain the observed anisotropy. T he observed anisotropy, i.e. the ratio of cometary aphelia in the sola r antapex direction to those in solar apex direction is reproduced by Valtonen and Innanen (Astrophys. J. 255, 307, 1982). In their theoreti cal formulation of the capture speeds and only two directions from whi ch comets approach the solar system, i.e. apexial and antapexial direc tions. The authors of the present paper have also explained anisotropy (1994) on the basis of the Maxwellian distribution of cometary speeds , but by incorporating the effect of external perturbations on relativ e velocities of comets with respect to the Sun. Moreover, the present formulation includes the integrated effect of all the directions of an tapexial directions. This has been achieved by deriving an expression for a parameter (epsilon) after integration over the whole of the geom etric as well as velocity space. The total captivity parameter epsilon depends upon the effective relative speed of comets with respect to t he Sun. epsilon is split into partial captivity parameters epsilon(1) and epsilon(2) for apexial and antapexial directions, respectively, ac cording to Sun's different relative speeds for the two directions. Her e epsilon(1) and epsilon(2) include the integrated contribution from a ll the directions towards apexial and antapexial captures. It is found that effective relative speed for comets in the former group is alway s greater than that for comets in the later group. Accordingly, differ ent capture conditions in velocity space have been obtained for the tw o directions. Such domains in velocity space where these capture condi tions are violated (fulfilled) are called here prohibited (allowed) an gular domains. It is seen that capture conditions render a larger allo wed domain for capture of antapexial comets than that for capture of a pexial comets. Hence the observed anisotropy can be visualized as a co nsequence of existence of these angular domains in cometary capture. I n short, the observed anisotropy is in favour of antapexial capture si nce the allowed angular domain for antapexial capture is always greate r than that for apexial capture. Copyright (C) 1996 Elsevier Science L td.