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Published March 10, 2012 | Published
Journal Article Open

Self-consistent Size and Velocity Distributions of Collisional Cascades

Abstract

The standard theoretical treatment of collisional cascades derives a steady-state size distribution assuming a single constant velocity dispersion for all bodies regardless of size. Here we relax this assumption and solve self-consistently for the bodies' steady-state size and size-dependent velocity distributions. Specifically, we account for viscous stirring, dynamical friction, and collisional damping of the bodies' random velocities in addition to the mass conservation requirement typically applied to find the size distribution in a steady-state cascade. The resulting size distributions are significantly steeper than those derived without velocity evolution. For example, accounting self-consistently for the velocities can change the standard q = 3.5 power-law index of the Dohnanyi differential size spectrum to an index as large as q = 4. Similarly, for bodies held together by their own gravity, the corresponding power-law index range 2.88 < q < 3.14 of Pan & Sari can steepen to values as large as q = 3.26. Our velocity results allow quantitative predictions of the bodies' scale heights as a function of size. Together with our predictions, observations of the scale heights for different-sized bodies for the Kuiper belt, the asteroid belt, and extrasolar debris disks may constrain the mass and number of large bodies stirring the cascade as well as the colliding bodies' internal strengths.

Additional Information

© 2012 The American Astronomical Society. Received 2011 November 1; accepted 2012 January 3; published 2012 February 22. It is a pleasure to thank Re'em Sari for helpful discussions. For H.S. support for this work was provided by NASA through Hubble Fellowship Grant HST-HF-51281.01-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contact NAS 5-26555. M.P. was partially supported by NSF Grant AST-0909210.

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