Strigari, Bullock & Kaplinghat (2007)

Abstract (paraphrased)¶

Strigari, Bullock & Kaplinghat show that adding stellar proper motions to the standard line-of-sight (LOS) velocity-dispersion data of a dwarf spheroidal (dSph) galaxy is a powerful probe of the dark-matter density profile. Allowing for a general (six-parameter) halo density profile and a constant stellar velocity anisotropy $\beta$ , they forecast (via a $6\times6$ Fisher matrix) that the log-slope of the dark-matter density at about twice the stellar core (King) radius, $r_\star \simeq 2 r_{\rm King}$ , can be measured to within $\pm0.2$ once the proper motions of $\sim$ 200 stars are combined with $\sim$ 1000 LOS velocities — a factor of $\sim$ 5 better than LOS data alone. The key physics is a manifest degeneracy between $\beta$ and the log-slope in the LOS dispersion that is broken by the tangential information carried in the two on-sky proper-motion components. For progenax, the load-bearing content is §2 “Mass Modeling”, where they write the three observable projected dispersions — one LOS and two proper-motion — as projections of the 3-D radial dispersion $\sigma_r(r)$ , generalising the line-of-sight Jeans projection of Binney & Mamon (1982) to the proper-motion channels.

The load-bearing equations (verified against the paper, §2, p. L2)¶

The 3-D velocity is decomposed (§2, p. L1) into line-of-sight ( $v_{\rm los} = v_r\cos\theta + v_\theta\sin\theta$ ) and the two in-sky components parallel and tangential to the projected radius vector $R$ : $v_R = v_r\sin\theta + v_\theta\cos\theta$ (on-sky radial) and $v_t = v_\phi$ (on-sky tangential). The dispersions are $\sigma_i^2 \equiv \langle v_i^2\rangle$ , with $\sigma_\phi^2 = \sigma_\theta^2$ assumed. Solving the Jeans equation for the 3-D radial dispersion $\sigma_r(r)$ and integrating along the line of sight gives the three resulting observable velocity dispersions (their Eqs. 1–3, p. L2):

Line-of-sight (Eq. 1, p. L2):

\sigma_{\rm los}^2(R) = \frac{2}{I_\star(R)}\int_R^\infty \left(1 - \beta\,\frac{R^2}{r^2}\right) \frac{\nu_\star\,\sigma_r^2\,r\,dr}{\sqrt{r^2 - R^2}}.

(1)

Proper-motion radial / on-sky radial (Eq. 2, p. L2):

\sigma_R^2(R) = \frac{2}{I_\star(R)}\int_R^\infty \left(1 - \beta + \beta\,\frac{R^2}{r^2}\right) \frac{\nu_\star\,\sigma_r^2\,r\,dr}{\sqrt{r^2 - R^2}}.

(2)

Proper-motion tangential (Eq. 3, p. L2):

\sigma_t^2(R) = \frac{2}{I_\star(R)}\int_R^\infty (1 - \beta)\, \frac{\nu_\star\,\sigma_r^2\,r\,dr}{\sqrt{r^2 - R^2}}.

(3)

Here $\beta(r) = 1 - \sigma_\theta^2/\sigma_r^2$ is the stellar velocity anisotropy, $I_\star(R)$ is the surface density of stars, and $\nu_\star(r)$ is the three-dimensional number (light) density (§2, p. L2). The paper states explicitly that “it is clear from inspection that each component depends on $\beta$ in a different fashion, and therefore they can be used together to constrain its value” — which is precisely the $\beta$ – $\gamma$ degeneracy break progenax’s OED exploits.

Scope — Strigari+2007 contains ALL THREE kernels explicitly (verified, whole Letter read)

Equations (1)–(3) (p. L2) of this Letter give, in closed form, the LOS kernel $(1-\beta R^2/r^2)$ and the two proper-motion kernels $(1-\beta+\beta R^2/r^2)$ (radial) and $(1-\beta)$ (tangential), all three sharing the identical projection measure $r/\sqrt{r^2-R^2}$ and the weight $\nu_\star\,\sigma_r^2$ , with the prefactor $2/I_\star(R)$ . This is exactly the form in project_dispersion. So unlike Binney & Mamon (1982) — which contains the LOS kernel only — this Letter is a primary source for the two proper-motion kernels. (The 4-page Letter does not derive Eqs. 1–3 from first principles; it presents them as the “three resulting observable velocity dispersions”, deferring the constant- $\beta$ Jeans solution to standard treatments. The earliest first-principles derivation of the proper-motion projection is Leonard & Merritt 1989, ApJ 339, 195; Strigari+2007 is the canonical dSph-astrometry statement of the same formulae and an entirely adequate primary citation for the kernels as written.)

Code cross-reference (`src/progenax/kinematics/dispersion.py`)¶

project_dispersion (def at line 677) folds the $r/\sqrt{r^2-R^2}$ pole away analytically via $r^2 = R^2 + u^2$ (so $r\,dr/\sqrt{r^2-R^2} = du$ ) and integrates a smooth quadrature in $u$ (line 810, r = sqrt(R_i**2 + u**2)). With ratio = R²/r² (line 822), w = rho * sigma_r2 (line 823), and beta = r²/(r²+r_a²) (Merritt (1985), line 820):

Quantity	Code (line)	Kernel in code	Paper source
$\sigma_{\rm los}^2$	`S_los`, L830: `(1 - betaratio)w`	$1 - \beta\,R^2/r^2$	Eq. (1) (1) (= Binney & Mamon (1982) Eq. 7) ✓
$\sigma_{\rm pm,R}^2$	`S_pmr`, L831: `(1 - beta + betaratio)w`	$1 - \beta + \beta\,R^2/r^2$	Eq. (2) (2) ✓
$\sigma_{\rm pm,T}^2$	`S_pmt`, L832: `(1 - beta)*w`	$1 - \beta$	Eq. (3) (3) ✓
$\Sigma(R)$	`Sigma`, L829: `2trapezoid(rhojac, grid)`	$2\int \rho\, r/\sqrt{r^2-R^2}\,dr$	$I_\star(R)$ inverse-Abel (BM82 Eq. 2)

Cell-by-cell: the factor 2 (L829–832), the $\nu_\star\,\sigma_r^2$ weight (here rho * sigma_r2, $\rho = A\,\nu_\star$ under mass-follows-light, $A$ cancelling in the dispersion ratios), the $\beta(r)$ multiplier, and the substitution that removes $1/\sqrt{r^2-R^2}$ all match Eqs. (1)–(3) exactly. The isotropic limit ( $r_a=$ None $\Rightarrow\beta=0$ , L818) collapses all three kernels to 1, so $\sigma_{\rm los}=\sigma_{\rm pm,R}=\sigma_{\rm pm,T}$ — the correct $\beta=0$ behaviour of Eqs. (1)–(3) (anisotropy lives entirely in the kernel ratios).

Verification verdict¶

Verdict: MATCH — Strigari+2007 is the primary source for both PM kernels

$\sigma_{\rm pm,R}$ ⇒ MATCH. (1 - beta + beta*ratio) (L831) is algebraically identical to Strigari+2007 Eq. (2) kernel $(1 - \beta + \beta R^2/r^2)$ (p. L2). ✓
$\sigma_{\rm pm,T}$ ⇒ MATCH. (1 - beta) (L832) is identical to Strigari+2007 Eq. (3) kernel $(1 - \beta)$ (p. L2). ✓
$\sigma_{\rm los}$ ⇒ MATCH (also given here as Eq. 1; the dedicated LOS primary remains Binney & Mamon (1982) Eq. 7).
All three share the projection measure $r/\sqrt{r^2-R^2}$ , the weight $\nu_\star\,\sigma_r^2$ , and the factor 2 — exactly the project_dispersion implementation. No physics discrepancy.

Phase 2 (Anna-gated) provenance fix. The project_dispersion docstring (L11–13, L21–22, L678, L682, L825 “B&M82 kernels”) currently credits all three kernels to Binney & Mamon (1982), but BM82 contains the LOS kernel only. Recommended attribution after this verification: $\sigma_{\rm los}\to$ Binney & Mamon (1982) Eq. 7; $\sigma_{\rm pm,R}\to$ Strigari+2007 Eq. 2; $\sigma_{\rm pm,T}\to$ Strigari+2007 Eq. 3 (earliest derivation: Leonard & Merritt 1989). No src/ edits were made in this task.

Use in progenax¶

progenax.kinematics.dispersion.project_dispersion — proper-motion projection of the anisotropic Jeans model: $\sigma_{\rm pm,R}(R)$ from (2) (Strigari+2007 Eq. 2) and $\sigma_{\rm pm,T}(R)$ from (3) (Strigari+2007 Eq. 3) onto on-sky radii $R$ .
Pairs with Binney & Mamon (1982) (the LOS channel, Eq. 7) and Merritt (1985) (the Osipkov–Merritt $\beta(r)=r^2/(r^2+r_a^2)$ supplied to all three kernels).
The two proper-motion channels are what make the dispersion forward model a genuinely three-method observable for the OED Fisher, breaking the $\beta$ –log-slope degeneracy exactly as Strigari+2007 demonstrate (their §4 and Fig. 2).

Notes¶

Anisotropy convention. Strigari+2007 use $\beta(r) = 1 - \sigma_\theta^2/\sigma_r^2$ with a single tangential component $\sigma_\theta$ (and $\sigma_\phi^2=\sigma_\theta^2$ ) — the same convention as Binney & Mamon (1982) Eq. 1 and consistent with the Osipkov–Merritt law used in progenax. The Letter’s Fisher forecast assumes a constant $\beta$ ; progenax instead supplies the radius-dependent OM $\beta(r)$ , but the projection kernels (1)–(3) hold for an arbitrary $\beta(r)$ (the kernels are evaluated inside the radial integral).
The Letter’s headline science (the $6\times6$ Fisher matrix forecasting $\sigma(\gamma_\star)$ , Table 1; the $\beta$ – $\gamma$ contours, Fig. 2) is the motivation for progenax’s OED dispersion arc — the same “add proper motions to break the anisotropy degeneracy” argument — but progenax implements only the forward projection (Eqs. 1–3), not the paper’s specific six-parameter dark-halo parameterisation (their Eq. 4).
Equations above verified against the published PDF.

References¶

Strigari, L. E., Bullock, J. S., & Kaplinghat, M. (2007). Determining the Nature of Dark Matter with Astrometry. The Astrophysical Journal Letters, 657, L1–L4. 10.1086/512976