Jeans Anisotropic Models: Theory & Application

• Jeans Anisotropic Models are foundational dynamical galaxy models that generalize the classical Jeans approach by incorporating a b‐ansatz to account for anisotropy in velocity dispersion.
• They solve the steady-state Jeans equations in cylindrical or spherical coordinates to extract robust mass estimates and dark matter constraints, with the flattening parameter q and b parameter governing model behavior.
• The JAM method and related schemes offer computational efficiency and physical transparency, making them crucial for interpreting stellar kinematics in elliptical galaxies, disk galaxies, and globular clusters.

Jeans Anisotropic Models are a foundational class of dynamical galaxy models that generalize the classic Jeans approach to account for anisotropy in velocity dispersions within axisymmetric (and, under certain assumptions, spherical) stellar systems. The "Jeans Anisotropic Multi-Gaussian Expansion" (JAM) method, and the closely related $b$-ansatz framework, provide efficient, physically motivated schemes to model the internal kinematics and gravitational potential of galaxies using a parametrized anisotropy. These models are essential for extracting robust dynamical masses, constraining dark matter distributions, and interpreting integral-field or stellar kinematic data. They are widely used in the analysis of elliptical galaxies, disk galaxies, and globular clusters as well as in the interpretive modeling of resolved stellar populations and satellite systems.

1. Mathematical Formulation

The core of Jeans Anisotropic Models lies in solving the steady-state, axisymmetric Jeans equations under specified anisotropy assumptions. In cylindrical coordinates $(R, \varphi, z)$ and with the standard assumption $\langle v_R v_z \rangle = 0$, the equations read: $$\frac{\partial (\rho\,\sigma_R^2)}{\partial R} + \rho\,\frac{\sigma_R^2 - \overline{v_\varphi^2}}{R} = -\rho\,\frac{\partial\Phi}{\partial R}$$

$$\frac{\partial (\rho\,\sigma_z^2)}{\partial z} = -\rho\,\frac{\partial\Phi}{\partial z}$$

where $\rho(R,z)$ is the tracer density and $\Phi(R,z)$ is the total gravitational potential. The classical closure is the two-integral (isotropic in $R$ and $z$) case $\sigma_R^2 = \sigma_z^2$, but the generalized "b-ansatz" introduces

$\sigma_R^2(R,z) = b\,\sigma_z^2(R,z)$

with $b > 0$ a global or locally varying dimensionless anisotropy parameter (Deo et al., 2024, Deo et al., 4 Nov 2025).

The solution proceeds by first integrating the vertical equation: $$\sigma_z^2(R,z) = \frac{1}{\rho(R,z)} \int_z^\infty \rho(R,z')\,\frac{\partial\Phi}{\partial z'}\,dz'$$ Given $b$, the radial Jeans equation is then algebraic in $\overline{v_\varphi^2}(R,z)$: $$\overline{v_\varphi^2}(R,z) = b\,B(R,z) + R\,\frac{\partial\Phi}{\partial R}$$ where $B(R,z)$ is an auxiliary (model-independent) function of the tracer and potential (Deo et al., 4 Nov 2025). In the spherically-aligned case, JAM$_\mathrm{sph}$, the approach is analogous but expressed in spherical coordinates with two independent anisotropy parameters.

2. Closure Schemes and Physical Constraints

To ensure physical solutions (i.e., positive velocity moments and real streaming velocities), the allowed range of $b$ is tightly constrained by analytic conditions derived from the behavior of $B$, $D=\partial_R\sigma_z^2$, and $C=R\partial_R\Phi - \sigma_z^2$ in the $(R,z)$ plane:

• $\sigma_R^2 = b\,\sigma_z^2 \geq 0$;
• $\overline{v_\varphi^2} \geq 0$;
• $$\Delta = \overline{v_\varphi^2} - \sigma_R^2 \geq 0$$ if Satoh decomposition is to be applied.

The locus of zeroes and sign patterns of $B$, $C$, and $D$ partition the domain into regions where $b$ can be increased or decreased without violating these constraints. In practice, the maximum allowed $b$ for global constancy is set by $\min_z b_0(z)$ (to keep $\sigma_R^2 \geq 0$), $\min_z \beta_m(z)$ (to keep $\overline{v_\varphi^2} \geq 0$), and the requirement $b_1 \leq b \leq b_2$ (to keep $\Delta \geq 0$) (Deo et al., 4 Nov 2025).

The effect of flattening $q$ (intrinsic axis ratio) controls how large $b$ can be: flatter models tolerate larger $b>1$, i.e., higher radial anisotropy. The inclusion of a moderate dark halo increases the critical $b$ values by only $\sim10\%$ (Deo et al., 4 Nov 2025).

3. Streaming Decomposition: Satoh Ansatz

The mean streaming velocity $\overline{v_\varphi}(R,z)$ is not uniquely fixed by the Jeans equations and requires a separate ansatz. The classical Satoh decomposition applies

$$\overline{v_\varphi}(R,z) = k(R,z)\,\sqrt{\overline{v_\varphi^2}(R,z) - \sigma_R^2(R,z)} \quad,\quad 0 \leq k \leq 1$$

such that

$$\sigma_\varphi^2 = \sigma_R^2 + (1 - k^2)\left[\overline{v_\varphi^2} - \sigma_R^2\right]$$

or, in the generalized Caravita et al. framework, using a $k$-decomposition more robust when $\Delta<0$ locally (Caravita et al., 2021). This parameter $k$ can be constant, spatially varying, or chosen to model specific features such as counter-rotation.

4. Impact of Anisotropy and Flattening

The physically allowed $b$-range and resulting kinematic maps are strongly set by the flattening parameter $q$:

• Flatter oblate systems ($q\ll 1$) can support much larger $b>1$ than nearly spherical or round systems.
• In ellipsoidal models, the region in $(R,z)$ where $\overline{v_\varphi}$ could become negative for too high $b$ shrinks to the $z=0$ plane as $q\to 1$.
• The presence of a dark matter halo (e.g., spherical or SIS) shifts the allowed values but does not usually dominate the physical constraints compared to flattening.

Empirical fits to fast rotator early-type galaxies (ATLAS$^\mathrm{3D}$, MaNGA) find $b \gtrsim 1$ and $\sigma_\varphi^2/\sigma_R^2\approx 1$ inside one effective radius, with a general upper bound

$b \lesssim \frac{1}{0.3 + 0.7\,q}$

giving $b \approx 1$ for nearly spherical systems, and larger $b$ allowed only by invoking higher flattening ($q \ll 1$) (Deo et al., 4 Nov 2025).

5. Analytic Solution in Ellipsoidal and Two-Component Systems

When the tracer density is stratified on similar ellipsoids, i.e., $\rho(R,z) = \rho_0(m)^{-\gamma}$ with $m^2 = R^2 + z^2/q^2$ and the potential is ellipsoidal or spherical, all necessary derived fields $(\sigma_z^2, D, B, C)$ are obtainable via one-dimensional integrals over a latent variable $u$, permitting rapid mapping of the kinematic structure (Deo et al., 4 Nov 2025). For more general potentials (e.g., Sersic or $\gamma$-models plus halo), these can be computed numerically once and used to bracket the behavior of all plausible models.

6. Practical Modeling Guidelines

Selecting $b$ for a dynamical model involves:

• Evaluating $\min_z b_0(z)$ for $\sigma_R^2\ge 0$ and $\min_z\beta_m(z)$ for $\overline{v_\varphi^2}\ge 0$, as well as ensuring $b_1 \leq b \leq b_2$ for $\Delta \geq 0$, possibly via curves as a function of $q$ (see Figures in (Deo et al., 4 Nov 2025)).
• If stellar kinematics demand $b>1$ (radial anisotropy), using sufficiently flattened models; round systems force $b \to 1$ or $b < 1$.
• For observed galaxies, the empirical envelope $b \lesssim 1/(0.3+0.7\,q)$ provides a practical upper bound in the region of interest.

Summary: the $b$-ansatz, when applied to axisymmetric (and ellipsoidal) systems, yields a bracketed one-parameter family of physically valid models whose entire kinematic structure can be constructed before any detailed numerical solution of the Jeans equations. The method's limitations are manifest only near boundaries of parameter space (e.g., for extreme anisotropy or nearly spherical systems), and the choice and effect of $b$ are dominated by the intrinsic flattening. This gives the JAM family both computational speed and physical transparency in the dynamical modeling of galaxies (Deo et al., 4 Nov 2025).