 Solution

diverges. The model therefore requires an outer truncation for a finite galaxy, and does not describe a central constant-density core. Its gravitational potential cannot be normalized to zero at infinity.

E = v^{2} /2 + Φ

L = x \times v

. Since the gravitational potential is stationary and spherical,

\frac{d E}{d t} = v \cdot (\dot{v} + \nablaΦ) = 0, \frac{d L}{d t} = x \times (- Φ^{'} \frac{x}{r}) = 0.

Thus both the specific energy and the entire angular-momentum vector are conserved.

\frac{d F}{d t} = \partial_{t} F + v \cdot \nabla_{x} F - \nablaΦ \cdot \nabla_{v} F = 0.

In a steady system

\partial_{t} F = 0

, so the value of the galactic distribution function is constant along every trajectory: it is itself an integral of motion. Conversely, if

I_{1}, \dots, I_{m}

are time-independent integrals of motion, then

F = h (I_{1}, \dots, I_{m})

satisfies

d F / d t = \sum h_{, a} \dot{I}_{a} = 0

. Locally, a complete set of invariant orbit labels expresses any steady galactic distribution function this way. This is the weak form of Jeans theorem; a stronger assertion about a small set of isolating integrals needs additional orbital regularity or mixing assumptions. In particular, the theorem does not say that every steady galaxy must have a galactic distribution function depending on only

E

and

L

v_{T} = (v_{θ}^{2} + v_{ϕ}^{2})^{1/2}

and

L = r v_{T}

. Introduce a polar angle

ψ

in the tangential velocity plane. Then

d^{3} v = d v_{r} v_{T} d v_{T} d ψ

, so the apparent

1/ L

r > 0

Let

t = e^{- Φ/ v_{0}^{2}} = a / (a + r)

c_{1} = 1

c_{2} = 2

, and

α_{k} = k / (2 v_{0}^{2})

. Each exponential component contributes

ρ_{k} = \frac{c _{k} t ^{k}}{8 π ^{3} G a r} \int_{- \infty}^{\infty} e^{- α_{k} v_{r}^{2}} d v_{r} \int_{0}^{\infty} e^{- α_{k} v_{T}^{2}} d v_{T} \int_{0}^{2 π} d ψ = \frac{v _{0}^{2}}{4 π G a r} \frac{c _{k}}{k} t^{k} .

Here the two Gaussian integrals and the angular integral have product

π^{2} / α_{k}

. Consequently,

ρ = \frac{v _{0}^{2}}{4 π G a r} (t + t^{2}) = \frac{v _{0}^{2}}{4 π G} \frac{2 a + r}{r ( a + r ) ^{2}} .

The galactic distribution function is nonnegative, gives the required density, and depends on conserved

E, L

; by Jeans theorem it is a self-consistent steady solution.

Differentiating the Gaussian integral with respect to

α_{k}

shows that each component has

⟨ v_{r}^{2} ⟩_{k} = 1/ (2 α_{k}) = v_{0}^{2} / k

, and the same value for

⟨ v_{T}^{2} ⟩_{k}

. Angular averaging gives

⟨ v_{θ}^{2} ⟩_{k} = ⟨ v_{ϕ}^{2} ⟩_{k} = v_{0}^{2} / (2 k)

. Weighting these by

ρ_{1}, ρ_{2}

σ_{r}^{2} = v_{0}^{2} \frac{t + t ^{2} /2}{t + t ^{2}} = \frac{v _{0}^{2}}{2} \frac{3 a + 2 r}{2 a + r}, σ_{θ}^{2} = σ_{ϕ}^{2} = \frac{v _{0}^{2}}{4} \frac{3 a + 2 r}{2 a + r} .

The radial numerator includes

2 r

, as in the original PDF. All mean velocities vanish by parity. The same parity, or the zero angular average of

sin ψ cos ψ

, gives $⟨ v_{r} v_{θ} ⟩ = ⟨ v_{r} v_{ϕ} ⟩ = ⟨ v_{θ} v_{ϕ} ⟩ = 0$ .

β = 1 - \frac{σ _{θ}^{2} + σ _{ϕ}^{2}}{2 σ _{r}^{2}} = \frac{1}{2} .

The constant-anisotropy distribution function therefore favors radial motion at every radius: the radial variance is twice either individual tangential variance. It still has a substantial population with nonzero angular momentum, rather than exclusively radial orbits. The

L^{- 1}

factor enhances low-angular-momentum stars. The radial velocity dispersion approaches

3 v_{0}^{2} /4

at the centre and

v_{0}^{2}

far out; each individual tangential variance is half as large. There is no mean rotation, despite the well-defined galaxy rotation curve for hypothetical circular orbits.

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