Hypervirial model

The hypervirial family has relative potential

Ψ (r) = \frac{GM}{( r ^{p} + a ^{p} ) ^{1/ p}}, 0 < p \leq 2,

ρ (r) = \frac{( p + 1 ) M a ^{p}}{4 π} \frac{r ^{p - 2}}{( r ^{p} + a ^{p} ) ^{2 + 1/ p}} .

Its name comes from satisfying the scalar virial relation pointwise.

Hypervirial distribution function

L^{p - 2} E^{(3 p + 1) /2} .

β = 1 - p /2

⟨ v^{2} ⟩ = Ψ/2 = - Φ/2

. The local kinetic- and potential-energy densities

K = \frac{1}{2} ρ ⟨ v^{2} ⟩, W = \frac{1}{2} ρ Φ

therefore obey

2 K + W = 0

at every radius, a stronger statement than the integrated virial theorem.

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