Binney & Mamon (1982)

Abstract (paraphrased)¶

Given the projected surface brightness $\Sigma(R)$ and the line-of-sight velocity dispersion $\sigma_v(R)$ of a spherical galaxy as functions of projected radius, Binney & Mamon construct the unique constant mass-to-light-ratio model consistent with both, and recover as an output the radial dependence of the velocity-anisotropy profile $\beta(r)$ — without assuming isotropy. The method is an Abel-type inversion of the projected Jeans equation (their §2). Applied to M87, it returns a physically reasonable, radially anisotropic model with constant $M/L_V\approx7.6$ that needs no central black hole. The single relation this paper contributes to progenax is the line-of-sight projection integral (their Eq. 7), which maps an anisotropic 3-D model $(l, \sigma_r, \beta)$ onto the observed $\sigma_v(R)\,\Sigma(R)$ on the sky.

The load-bearing equations (verified against the paper, §2, p. 363)¶

Anisotropy parameter (Eq. 1, p. 363). With $\sigma_r,\sigma_\theta$ the radial and (single) tangential principal dispersions,

$\beta=0$ is isotropic; $\beta\to1$ is purely radial. (progenax supplies $\beta(r)$ from the Osipkov–Merritt law $\beta=r^2/(r^2+r_a^2)$ of Merritt (1985); BM82 itself is agnostic to the functional form of $\beta$.)

Surface-density / luminosity-density Abel pair (Eq. 2, p. 363). The luminosity density $l(r)$ is recovered from the surface brightness $\Sigma(R)$ by the standard Abel deprojection

Its forward companion — projecting a 3-D density $l(r)$ (or $\rho(r)$) onto the sky — is the inverse-Abel relation $\Sigma(R) = 2\int_R^{R_t} l(r)\,r\,(r^2-R^2)^{-1/2}\,dr$, which is what progenax evaluates for the projection normalisation $\Sigma$.

Line-of-sight projection integral (Eq. 7, p. 363) — the central result for progenax. “Elementary geometry” gives the observed line-of-sight dispersion as

The two integrals share the measure $r/(r^2-R^2)^{1/2}$ (the first carries weight $r$, the second weight $R^2/r$), so they combine into a single anisotropic kernel. Multiplying through by 2:

(4) is the form implemented in project_dispersion: the line-of-sight kernel $\bigl(1-\beta R^2/r^2\bigr)$, the weight $l\,\sigma_r^2$, the factor 2, and the $r/(r^2-R^2)^{1/2}$ projection measure are exactly Eq. (7) regrouped. (progenax uses the mass density $\rho$ as the projection weight under the paper’s own mass-follows-light assumption $\rho = A\,l$, Eq. 3; the constant $A$ cancels in $\sigma_v^2 = (\sigma_v^2\Sigma)/\Sigma$.)

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

project_dispersion (def at line 677) folds the $r/(r^2-R^2)^{1/2}$ pole away analytically via $r^2 = R^2 + u^2$ (so $r\,dr/(r^2-R^2)^{1/2} = du$) and integrates a smooth quadrature in $u$ (line 810, r = sqrt(R_i**2 + u**2)). With ratio = R²/r² (line 822) and w = rho * sigma_r2 (line 823):

The factor 2 (L829–832), the $l\,\sigma_r^2$ weight, and the substitution that removes $1/\sqrt{r^2-R^2}$ all match Eq. (7) cell-by-cell. 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 Eq. (7).

Verification verdict¶

MATCH (with an attribution caveat). The line-of-sight kernel (1 - beta*ratio) in S_los (L830) is algebraically identical to Binney & Mamon (1982) Eq. (7), regrouped as (4); the $\Sigma$ normalisation is the forward partner of their Eq. (2) Abel pair. No physics discrepancy in the $\sigma_{\rm los}$ / $\Sigma$ channels.

The only issue is provenance, not formula: the project_dispersion docstring (L678, L682, L825 “B&M82 kernels”) and the module header (L11–13, L21–22) credit all three projection kernels to Binney & Mamon (1982), but BM82 contains the line-of-sight kernel only. The proper-motion kernels $(1-\beta+\beta R^2/r^2)$ and $(1-\beta)$ are the standard later extension (Leonard & Merritt 1989, ApJ 339, 195; Strigari, Bullock & Kaplinghat 2007; Mamon & Łokas 2005). This is a docstring/citation fix for Phase 2 — no source-code logic change — flagged here for the closeout, not fixed in this note’s pass.

Use in progenax¶

• progenax.kinematics.dispersion.project_dispersion — line-of-sight projection of the anisotropic Jeans model ((4), BM82 Eq. 7) onto on-sky radii $R$, returning $\sigma_{\rm los}(R)$ and the projected surface density $\Sigma(R)$.

• Pairs with Merritt (1985) (the Osipkov–Merritt $\beta(r)$ supplied to the kernel) and the 3-D anisotropic Jeans solver that produces $\sigma_r^2(r)$.

• The isotropic-Plummer closed form $\sigma_{\rm los}^2(R) = (3\pi/64)\,GM/\sqrt{a^2+R^2}$ (Dejonghe 1987) is the tight absolute validation anchor for the $\sigma_{\rm los}$ channel in test_dispersion_physics.py.

Notes¶

• BM82’s broader algorithm (the Abel inversion that recovers $\sigma_r(r)$, $\beta(r)$ and $M/L$ from observed $\Sigma(R)$ and $\sigma_v(R)$, their Eqs. 8–20 and Appendices A–B) is the inverse problem; progenax uses only the forward projection, Eq. (7), as a differentiable forward model for optimal experimental design.

• The convention $\beta = (\sigma_r^2-\sigma_\theta^2)/\sigma_r^2$ (Eq. 1) uses a single tangential component $\sigma_\theta$; this is consistent with the Osipkov–Merritt $\beta(r)=r^2/(r^2+r_a^2)$ used in progenax (the single-vs-two-component factor of 2 is reconciled in Merritt (1985)).

• BM82 restricts to $\beta<1$ throughout (p. 372): the radial component never exceeds the total available — a physical bound the projection inherits.