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 Incoming links: Hypervirial model

Hypervirial distribution function Created 2026-09-28 Updated 2026-10-06

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.

Put

Ψ = - Φ = GM (r^{p} + a^{p})^{- 1/ p}

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

E = - E = Ψ - v^{2} /2

and

n = (3 p + 1) /2

. Direct velocity integration gives

ρ = A r^{p - 2} \int_{0}^{2Ψ} v^{p} (Ψ - v^{2} /2)^{n} d v \int_{0}^{2 π} d φ \int_{0}^{π} sin^{p - 1} α d α = A 2^{(p + 1) /2} π^{3/2} \frac{Γ ( p /2 ) Γ ( 3 ( p + 1 ) /2 )}{Γ ( 2 p + 2 )} r^{p - 2} Ψ^{2 p + 1} .

Matching this expression to the density fixes

A = \frac{( p + 1 ) a ^{p} Γ ( 2 p + 2 )}{2 ^{(p + 5) /2} π ^{5/2} G ^{2 p + 1} M ^{2 p} Γ ( p /2 ) Γ ( 3 ( p + 1 ) /2 )} .

This coefficient is positive for

0 < p \leq 2

, so the distribution function self-consistently generates the stated density and potential.

Plummer model Created 2026-10-06 Updated 2026-10-14

The spherical Plummer model has

ψ = GM / a^{2} + r^{2}

and

ρ = 3 M a^{2} / [4 π (a^{2} + r^{2})^{5/2}]

. It is the

p = 2

hypervirial model. In unit scales its isotropic distribution is

242 E^{7/2} / (7 π^{3})