Anisotropy & the OM-vs-Michie formalism (B6)

Anisotropy & the OM–vs–Michie formalism (B6)¶

Velocity anisotropy is encoded in the Binney anisotropy $\beta(r) = 1 - \sigma_t^2/(2\sigma_r^2)$ — zero for isotropic orbits, rising toward 1 for radially-biased ones — and parametrized by an anisotropy radius $r_a$ . This demo asks two questions: how well can $r_a$ be measured, and does it matter which anisotropy formalism you assume?

Inputs and assumptions¶

The fit recovers one parameter, the anisotropy radius $r_a$ — but always under the assumed OM formalism. Part (b) makes the point that this assumed model form is itself the load-bearing input: fit the wrong formalism and $r_a$ is biased.

Table 1:Model inputs

Input	Meaning and role	Status (fiducial)
$r_a$	Osipkov–Merritt anisotropy radius — the science target; sets where orbits turn radial via $\beta_{\rm OM}=r^2/(r^2+r_a^2)$ .	recovered (1.5 pc; part (b) truth 6.0 pc fit under the wrong model)
anisotropy model form	The assumed fit model $\beta_{\rm OM}(r;r_a)=r^2/(r^2+r_a^2)$ — the load-bearing choice that part (b) deliberately violates.	known / fixed (model assumption)
$r_h$ / $W_0$ , $r_c$	Plummer half-mass radius (1 pc) for part (a); King $W_0=7$ , $r_c=1$ pc for the part-(b) Michie cluster — concentration/scale held at truth.	known / fixed
masses, $G$	Equal-mass tracers (1 $M_\odot$ ); $G$ (STELLAR) threaded into velocity sampling.	known / fixed
$N$ , bins, occupancy, box, MLE, $\beta$ -SE	$3\times10^4$ stars; 18 log radial bins (min 80/bin); $r_a\in(0.2,60)$ pc; 500 Adam steps; conservative per-bin $\beta$ error $(1+\|\beta\|)/\sqrt n$ .	numerical choices

Important

The anisotropy formalism is the load-bearing assumption — $r_a$ is not a single number read off $\beta(r)$ independent of the DF family. The fit always uses the OM closed form $\beta_{\rm OM}(r;r_a)=r^2/(r^2+r_a^2)$ , which assumes the DF depends on a single variable $Q=E-L^2/2r_a^2$ . A genuinely different formalism (Michie’s anisotropic-King DF, not a function of one $Q$ ) produces a different $\beta(r)$ shape. Fitting the OM form to a Michie sample both mis-estimates $r_a$ (8.9 vs 6.0 pc) and leaves a systematic shape residual (reduced $\chi^2$ inflated $12.9\times$ ). The model you assume shapes what you infer — the analogue, in model space, of B12’s known- $\epsilon$ requirement.

(a) Well-specified: recovering Osipkov–Merritt $r_a$ ¶

The Osipkov–Merritt (OM) ansatz Merritt, 1985 makes the DF a function of $Q = E - L^2/2r_a^2$ , giving the closed-form profile

\beta_{\rm OM}(r) = \frac{r^2}{r^2 + r_a^2}.

(1)

A Plummer+OM cluster ( $r_h=1$ , true $r_a=1.5$ pc) is sampled, $\beta(r)$ binned, and $r_a$ recovered by a $\chi^2$ fit of (1):

quantity	truth	recovered
$r_a$	1.5 pc	$1.55 \pm 0.035$ pc (pull +1.49)

The OM form fits an OM cluster well (reduced $\chi^2 = 0.43$ ; the conservative $\beta$ error $\sigma_\beta = (1+|\beta|)/\sqrt n$ deflates it), and the Fisher forecast gives $\sigma(r_a)\propto N^{-1/2}$ . Anisotropy this strong ( $r_a\sim r_h$ ) is detectable in a modest sample.

(b) Misspecified: an OM fit to a Michie cluster¶

Anisotropy comes in different formalisms. The Michie (1963) anisotropic-King DF Michie, 1963

f \;\propto\; \exp\!\left(-\frac{J^2}{2 r_a^2 \sigma^2}\right) \left[\exp\!\left(-\frac{E}{\sigma^2}\right) - 1\right]

(2)

is not a function of a single $Q$ , so its $\beta(r)$ shape differs from (1). Sampling a Michie cluster ( $W_0=7$ , true $r_a=6$ pc) and fitting it with the OM form:

quantity	value
OM-fit $r_a$	$8.90 \pm 0.27$ pc (vs true Michie $r_a = 6.0$ )
OM-fit reduced $\chi^2$	5.58 — a 12.9× inflation over the well-specified fit

So assuming the wrong anisotropy formalism does two things: it mis-estimates the anisotropy radius, and it leaves a systematic $\beta(r)$ residual (the $12.9\times$ $\chi^2$ inflation) that detects the mismatch. The anisotropy model you assume shapes what you infer — anisotropy is not a single number read off $\beta(r)$ independent of the DF family.

Figure¶

Anisotropy & the OM–vs–Michie formalism (scripts/demo_anisotropy.py, ALL
PASS). (a) A Plummer+OM cluster: \beta(r) (points, conservative errors) with
the recovered OM fit — a good match (reduced \chi^2=0.43). (b) A Michie
cluster fit with the OM form: the OM curve cannot follow the Michie \beta(r)
shape (reduced \chi^2=5.58, 12.9\times worse), and the best-fit r_a=8.9
mis-locates the true 6.0. (c) Forecast: \sigma(r_a)\propto N^{-1/2} for the
well-specified OM case. — Figure 1:**Anisotropy & the OM–vs–Michie formalism** (`scripts/demo_anisotropy.py`, ALL PASS). **(a)** A Plummer+OM cluster: $\beta(r)$ (points, conservative errors) with the recovered OM fit — a good match (reduced $\chi^2=0.43$ ). **(b)** A Michie cluster fit with the OM form: the OM curve cannot follow the Michie $\beta(r)$ shape (reduced $\chi^2=5.58$ , $12.9\times$ worse), and the best-fit $r_a=8.9$ mis-locates the true 6.0. **(c)** Forecast: $\sigma(r_a)\propto N^{-1/2}$ for the well-specified OM case.

Caveats¶

Warning

No “King+OM” model. King’s anisotropic form is Michie, so there is no fixed-density OM-vs-Michie pair. The misspecification here is an OM fit to a Michie sample on the $\beta(r)$ channel — not a self-consistent fixed-density refit. It demonstrates the formalism-dependence of the inferred $r_a$ , which is the point.
$\beta(r)$ channel only. The demo fits anisotropy alone; it does not co-fit concentration or the density (the $r_a$ –concentration degeneracy needs the $\sigma(r)$ channel, already exercised in B3).
Conservative $\beta$ errors. $\sigma_\beta=(1+|\beta|)/\sqrt n$ deflates the reduced $\chi^2$ ; the misspecification result is the ratio ( $12.9\times$ ), robust to the error scale.
The forecast $N\sim150$ is the asymptotic CRLB for this strong anisotropy ( $r_a\sim r_h$ ); the small- $N$ bound is optimistic, and weaker anisotropy needs far more stars.
Perfect 3-D phase space, equal-mass tracers, clean census. $\beta$ is built from full 3-D radial/tangential velocity moments — no line-of-sight projection, no proper-motion or RV errors; all stars are equal-mass ( $1\,M_\odot$ ) tracers with no mass-dependent anisotropy or segregation; and there is no foreground/membership contamination (only the per-bin occupancy cut applies).

How to run¶

env -u VIRTUAL_ENV uv run --no-sync python scripts/demo_anisotropy.py

References¶

Osipkov–Merritt anisotropy is Merritt (1985); the Michie anisotropic-King DF is Michie (1963). The OM and Michie velocity DFs are documented on the anisotropy and rotation page; B3’s two-component OM recovery is at halo + core.

References¶

Merritt, D. (1985). Spherical stellar systems with spheroidal velocity distributions. The Astronomical Journal, 90, 1027–1037. 10.1086/113810
Michie, R. W. (1963). On the distribution of high energy stars in spherical stellar systems. Monthly Notices of the Royal Astronomical Society, 125, 127–139. 10.1093/mnras/125.2.127