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 ( 5 ) 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 ( 21 ) 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 .