Merritt (1985) - progenax — differentiable initial conditions for N-body simulations

Abstract (paraphrased)¶

A method for deriving families of anisotropic distribution functions (DFs) consistent with any spherically symmetric density profile. Each family is labelled by a single free parameter $r_a$ , the anisotropy radius, and the radial-to-tangential velocity-dispersion ratio is $\sigma_r^2/\sigma_t^2 = 1 \pm r^2/r_a^2$ (Eq. 15). The models are isotropic in the centre and become radially ( $+$ ) or tangentially ( $-$ ) anisotropic outward. The radially anisotropic (“Type I”) branch is what is now universally called the Osipkov–Merritt model (Osipkov 1979; Merritt 1985). The construction reduces to a single Abel/Eddington inversion of an augmented density, so analytic solutions follow whenever the isotropic Eddington integral is analytic.

The method (verified against the paper, §II)¶

For an isotropic system the configuration density and phase-space density are related by

\rho(r) = 4\pi \int_{U(r)}^{0} dE\,\sqrt{2[E-U(r)]}\,f(E), \qquad f(E) = \frac{\sqrt 2}{4\pi^2}\frac{d}{dE}\int_{E}^{0}\frac{d\rho}{dU}\frac{dU}{\sqrt{U-E}}

(1)

(Eqs. 1–2; the second is Eddington 1916). Merritt’s key step: assume the DF depends on energy and angular momentum only through the single variable

Q_+ \equiv E + \frac{J^2}{2 r_a^2} \qquad\text{(Eq. 4a), with}\qquad f = f(Q_+).

(2)

Then the augmented density

\rho_1(r) \equiv \left(1 + \frac{r^2}{r_a^2}\right)\rho(r) \qquad\text{(Eq. 9)}

(3)

plays exactly the role $\rho$ plays in the isotropic problem, so the anisotropic DF is recovered by the same Eddington inversion applied to $\rho_1$ :

f_{\mathrm I}(Q_+) = \frac{\sqrt 2}{4\pi^2}\frac{d}{dQ_+} \int_{Q_+}^{0}\frac{d\rho_1}{dU}\frac{dU}{\sqrt{U-Q_+}} \qquad\text{(Eq. 11).}

(4)

This is the heart of architecture (C) in progenax: a single differentiable augmented-density Eddington core, with the isotropic case recovered as $r_a \to \infty$ (then $\rho_1 \to \rho$ and (4) $\to$ (1)).

The anisotropy law (verified, Eqs. 15 & 17)¶

The velocity anisotropy of every Type I solution is independent of the density profile:

\frac{\sigma_r^2}{\sigma_t^2} = 1 + \frac{r^2}{r_a^2} \quad\Longleftrightarrow\quad \beta(r) \equiv 1 - \frac{\sigma_t^2}{\sigma_r^2} = \frac{r^2}{r^2 + r_a^2}.

(5)

$\beta(0)=0$ (isotropic centre), $\beta(r_a)=\tfrac12$ , $\beta\to 1$ (radial) as $r\to\infty$ ; radial motions already dominate by a factor $\sim2$ at $r=r_a$ .

The analytic anisotropic Plummer model (§III — progenax’s validation anchor)¶

For the Plummer ( $n=5$ polytrope) density $\rho \propto (1+r^2/r_0^2)^{-5/2}$ (Eq. 39) with potential $U(r) = -6\sigma_0^2(1+r^2/r_0^2)^{-1/2}$ (Eq. 40), the inversion is analytic. The isotropic DF is

f(E) = \frac{\sqrt 2}{378\,\pi^3 G r_0^2 \sigma_0}\left(\frac{-E}{\sigma_0^2}\right)^{7/2}, \qquad -6 \le E/\sigma_0^2 \le 0

(6)

(Eq. 42; Eddington 1916) — i.e. $f(E)\propto(-E)^{7/2}$ , confirming progenax.kinematics.PlummerVelocityDF. The Osipkov–Merritt (Type I) Plummer DF is

f_{\mathrm I}(Q_+) = \frac{\sqrt 2}{378\,\pi^3 G r_0^2 \sigma_0} \left(\frac{-Q_+}{\sigma_0^2}\right)^{7/2} \left[\,1 - \frac{r_0^2}{r_a^2} + \frac{63}{4}\frac{r_0^2}{r_a^2}\left(\frac{-Q_+}{\sigma_0^2}\right)^{-2}\right] \qquad\text{(Eq. 45).}

(7)

The matched radial dispersion is $\sigma_r(r) = \sigma_0(1+r^2/r_0^2)^{-1/4} [1 + \tfrac12 (r^2+r_0^2)/(r^2+r_a^2)]^{1/2}$ , $\sigma_t = \sigma_r/\sqrt{1+r^2/r_a^2}$ (Eq. 47). (7) is the closed form against which progenax’s numerical augmented-density inversion (and its $\beta(r)$ profile) are validated.

Scope notes¶

progenax uses only the radially anisotropic Type I branch ( $\beta\ge0$ ). Merritt’s tangentially anisotropic Type II/IIa/IIb solutions ( $Q_-\equiv E - J^2/2r_a^2$ , Eqs. 19–38) carry a velocity-dispersion discontinuity at $r=r_a$ and are out of scope.
Merritt notes that Eddington’s (1914) generalized isothermal sphere $f(E,J^2)\propto\exp[-(E+\beta J^2)]$ obeys the same anisotropy law — the lowered version of this is the Michie (1963) anisotropic King model, the route for an anisotropic KingVelocityDF (augmented-density inversion of the King density gives the same $\beta(r)$ ).
Linear superpositions of different- $r_a$ solutions (Eq. 50) give a weighted-average $\beta(r)$ and a smooth way to avoid the Type II discontinuity — a possible future extension.

Use in progenax¶

progenax.kinematics — Osipkov–Merritt radial anisotropy ( $\beta(r)=r^2/(r^2+r_a^2)$ ). The shared augmented-density Eddington core ((3), (4)) is the basis of the true $f(Q_+)$ sampler that replaces the earlier heuristic velocity reshuffle.
Plummer (1911) · King (1966) · Elson, Fall & Freeman (1987) — the spatial densities whose augmented forms are inverted.
(7) + the $r_a\ge0.75\,r_0$ bound — the validation anchor for the anisotropic Plummer DF and its realized $\beta(r)$ profile.

Notes¶

The Osipkov–Merritt model is the standard one-parameter route to radial anisotropy in spherical ICs precisely because it reduces anisotropy to one Abel inversion of an augmented density, keeping the construction (and, in progenax, its gradients) as cheap and differentiable as the isotropic Eddington case. The analytic Plummer solution and its explicit $f\ge0$ bound make it an unusually clean validation target.