The steady-state 1/ -distribution and the problem of cometary fading

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Abstract
The problem of the 1/α-distribution including stellar perturbations and allowing for arbitrary fading per revolution is investigated. Inclusion of stellar perturbations means that the problem is strictly two-dimensional (comets can be scattered both in energy and angular momentum), but it is shown that this general problem can be reduced approximately to a tractable one-dimensional form. It is emphasized that conclusions about comet origins based on the 1/α-distribution (e.g. whether a steady-state primordial model or a transient recent-capture model applies) depend entirely on the assumptions made about fading. Assuming the steady-state model to apply we have solved the one-dimensional integral equation and compared the predicted 1/α-distribution with observations in order to constrain the fading function. The initial very sharp fall-off in the 1/α-distribution (⁠|$1/\alpha \lesssim10^{-4}$| AU−1) can be attributed to the injection spectrum of new comets from the Oort Cloud, but the subsequent decrease in the distribution requires strong fading for its explanation. This agrees with the results of previous authors, although the present work shows that strong fading is not restricted solely to ‘new’ comets. The required fading probability per revolution is described within a factor of order 2 by |$k(x)\simeq0.3[1+(x/4\times10^{-3})^2]^{-3/2}$|⁠, where x is 1/α in units AU−1. A physical model based on thermal effects associated with a long-period comet's perihelion passage is tentatively proposed which could account for this qualitative behaviour. It is emphasized that until a viable explanation for fading has been found, the validity of the primordial hypothesis for comet origins remains unresolved.

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