 Cross-section of a power-law fragment population

ID: cross-section-of-a-power-law-fragment-population

For spherical fragments of common mass density

ρ

, with

n (D) = K D^{- α}

σ_{tot} = \frac{3 m}{2 ρ} \frac{4 - α}{α - 3} \frac{D _{m i n}^{3 - α} - D _{m a x}^{3 - α}}{D _{m a x}^{4 - α} - D _{m i n}^{4 - α}}, 3 < α < 4.

Indeed

m = (π ρ K /6) \int D^{3 - α} d D

and

σ_{tot} = (π K /4) \int D^{2 - α} d D

; eliminating

K

proves the formula. If both cutoff ratios raised to

α - 3

and

4 - α

are small, respectively, this reduces to

3 m (4 - α) D_{m i n}^{3 - α} D_{m a x}^{α - 4} / [2 ρ (α - 3)]

. Approaching either endpoint requires the corresponding logarithmic integral.

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