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Combining constant mass flux in a collisional cascade with the rate in part (iv) givesSet the exponent to zero and multiply by three:or $21+b+p-\alpha (6+b-2p)=0$. Therefore the velocity-dependent collisional cascade hasFor $b=p=0$, this recovers the Dohnanyi collisional cascade slope $\alpha =7/2$. The denominator must be nonzero, and the resulting slope and destructive cutoff must satisfy the assumptions used for the collision-rate integral. A size-dependent vertical belt thickness would supply an additional volume factor and change this result.

With size-independent collision speed and catastrophic disruption threshold $Q_D^{\ast }\propto D^b$, the catastrophic planetesimal collision rate scales as $R\propto D^{3-\alpha +b(1-\alpha )/3}$. Multiplying by mass per logarithmic size bin gives processed mass proportional to $D^{7-2\alpha +b(1-\alpha )/3}$. Constant mass flux in a collisional cascade sets the exponent to zero, yielding the displayed differential-number index when $b\ne -6$. This is the zero-velocity-exponent case of a velocity-dependent collisional cascade.