Angular domains in cometary capture and anisotropy

Abstract

In the earlier work (1994) of the authors a theoretical formulation was developed for the capture process under the influence of external perturbations. The parameter hyperspace was scanned for physically acceptable values of σ, V 0, U′, and n;α= 2δ being the most probable speed of comets, V 0 the Sun's speed, U′ the perturbative component of velocity and n the number density of comets per pc 3; so that the computed capture rate and anisotropy conform to their observed values. The present paper is an extension of the above said earlier work in the sense that it gives an insight into the cause leading to anisotropy in capture. Different capture conditions obtained here for capture from the antapexial and apexial directions reveal that a larger 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. The observed anisotropy, i.e. the ratio of cometary aphelia in the solar antapex direction to those in solar apex direction is reproduced by Valtonen and Innanen ( Astrophys. J. 255, 307, 1982). In their theoretical formulation of the capture process they used a Maxwellian distribution for cometary speeds and only two directions from which comets approach the solar system, i.e. apexial and antapexial directions. 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 relative velocities of comets with respect to the Sun. Moreover, the present formulation includes the integrated effect of all the directions of approach of comets towards capture from apexial and antapexial directions. This has been achieved by deriving an expression for a parameter (ϵ) after integration over the whole of the geometric as well as velocity space. The total captivity parameter ϵ depends upon the effective relative speed of comets with respect to the Sun. ϵ is split into partial captivity parameters ϵ 1 and ϵ 2 for apexial and antapexial directions, respectively, according to Sun's different relative speeds for the two directions. Here ϵ 1 and ϵ 2 include the integrated contribution from all the directions towards apexial and antapexial captures. It is found that effective relative speed for comets in the former group is always greater than that for comets in the later group. Accordingly, different capture conditions in velocity space have been obtained for the two directions. Such domains in velocity space where these capture conditions are violated (fulfilled) are called here prohibited (allowed) angular domains. It is seen that the capture conditions render a larger allowed domain for capture of antapexial comets than that for capture of apexial comets. Hence the observed anisotropy can be visualized as a consequence of existence of these angular domains in cometary capture. In short, the observed anisotropy is in favour of antapexial capture since the allowed angular domain for antapexial capture is always greater than that for apexial capture.