Jeans Equations in Oblate Axisymmetric Geometry

• Jeans equations in oblate axisymmetry are a framework that links observed stellar motions and number densities to the gravitational potential in flattened, rotating systems.
• The formulation in cylindrical and spherical coordinates involves treating anisotropic velocity dispersions and applying various closure schemes to close the system.
• Recent applications using survey data like Gaia and SDSS constrain dark matter halo shapes and masses, highlighting the impact of anisotropy and systematics in modeling.

The Jeans equations in oblate axisymmetric geometry constitute the foundational mathematical framework for connecting observed stellar kinematics and number densities to the gravitational potential in astrophysical systems exhibiting rotational flattening, such as galactic disks and halos. Originating from moments of the collisionless Boltzmann equation, these equations underlie a wide range of applications, from mapping mass distributions in the Milky Way to constraining the shape and density profile of dark matter halos on both galactic and cosmological scales. The oblate axisymmetric case introduces nontrivial geometric and dynamical complexity, requiring careful treatment of anisotropic velocity dispersions, the orientation of the velocity ellipsoid, and the specifics of the system’s flattening.

1. Mathematical Formulation in Cylindrical and Spherical Coordinates

The axisymmetric Jeans equations are derived under the assumptions of steady state and axial symmetry, typically formulated in cylindrical coordinates $(R, \phi, Z)$ or spherical polar coordinates $(r, \theta, \phi)$. In the classic cylindrical form, relevant for flattened (oblate) systems, the radial ($a_R$) and vertical ($a_Z$) components of the acceleration are given as:

$\begin{aligned} a_R =\ & \sigma_{RR}^2\,\frac{\partial\left(\ln\nu\right)}{\partial R} + \frac{\partial\sigma_{RR}^2}{\partial R} + \sigma_{RZ}^2\,\frac{\partial\left(\ln\nu\right)}{\partial Z} + \frac{\partial\sigma_{RZ}^2}{\partial Z} \ & + \frac{\sigma_{RR}^2}{R} - \frac{\sigma_{\phi\phi}^2}{R} - \frac{\bar v_\phi^2}{R} \[1.2em] a_Z =\ & \sigma_{RZ}^2\,\frac{\partial\left(\ln\nu\right)}{\partial R} + \frac{\partial\sigma_{RZ}^2}{\partial R} + \sigma_{ZZ}^2\,\frac{\partial\left(\ln\nu\right)}{\partial Z} + \frac{\partial\sigma_{ZZ}^2}{\partial Z} + \frac{\sigma_{RZ}^2}{R} \end{aligned}$

Here, $\nu=\nu(R,Z)$ is the stellar number density, $\bar v_\phi$ is the mean azimuthal velocity, and $\sigma^2_{ij}$ are the components of the velocity dispersion tensor.

In stellar dynamical modeling, the solution is non-trivial because, without further assumptions, the system is underdetermined: the Jeans equations link density and velocity moments but do not provide a closure relation for the anisotropy or velocity ellipsoid orientation. Several closure schemes are used, such as the spherically-aligned ellipsoid assumption (JAM$_{\text{sph}}$), the Satoh $k$-decomposition, or the phenomenological $b$-ansatz, each with distinctive dynamical implications (Cappellari, 2019, Deo et al., 10 Apr 2024, Caravita et al., 2021).

2. Acceleration Mapping and Evidence for Dark Matter

Applying the axisymmetric Jeans framework to large kinematic data sets, such as SDSS or Gaia, enables the construction of two-dimensional acceleration maps in the $(R, Z)$ plane. In practice, this involves:

• Binning the stellar sample in $R$ and $Z$,
• Estimating $\nu$, the full set of velocity dispersions, and $\bar v_\phi$ in each bin,
• Fitting local second-order polynomials to estimate spatial derivatives,
• Evaluating the governing equations to recover acceleration fields.

Morphological analysis of these maps for the Milky Way reveals that the measured accelerations at galactocentric distances of $\sim20$ kpc are a factor of $\sim3$ greater than can be explained by visible matter, unequivocally demonstrating the dynamical dominance of a dark matter component at these radii (Loebman et al., 2012).

A critical technique involves fitting analytic halo models—such as an isothermal-like distribution with iso-density surfaces parametrized as $1/\left[R^2 +(Z/q_{DM})^2\right]$—to the inferred acceleration or potential maps. In the referenced studies, this yields dark matter halo flattening parameters as low as $q_{DM}=0.47 \pm 0.14$ (oblate), providing strong evidence for a highly flattened dark halo within the inner 30 kpc (Loebman et al., 2012).

3. Anisotropy, Velocity Ellipsoid Alignment, and Closure Schemes

Oblate axisymmetric geometry compels dynamical models to account for anisotropic velocity dispersions and the possibility that the velocity ellipsoid is not aligned with either spherical or cylindrical coordinates. Several schemes have emerged:

• Spherically-aligned ellipsoid (JAM$_{\text{sph}}$): Assumes the principal axes of the velocity dispersion tensor align with spherical coordinates, enabling efficient numerical solutions and direct application to MGE-formulated galaxy models (Cappellari, 2019, Zhang et al., 14 May 2025).
• $b$-ansatz: Introduces $\sigma_R^2 = b(R,z)\sigma_Z^2$ as a closure condition, controlling the radial versus vertical anisotropy. Rigorous analytic criteria ensure physical solutions (e.g., positive $\overline{v_\phi^2}$) and facilitate exploration of parameter space before full Jeans equation integration (Deo et al., 10 Apr 2024).
• Satoh $k$-decomposition and its generalizations: Splits $\overline{v_\phi^2}$ between ordered streaming and random azimuthal components. The generalization accounts for cases where the classical Satoh approach would yield unphysical (imaginary) results, especially in models with multiple or negative density components (Caravita et al., 2021).

Each closure impacts the predicted rotation curve, spatial anisotropy, and the inferred mass distribution, especially when adopting multi-component models (e.g., disk+halo+bulge) or in regions where the flattening is pronounced.

4. Application to Galactic Dark Matter Halo Shape and Mass

Combining high-precision kinematic data with the oblate axisymmetric Jeans formalism allows for robust constraints on the shape and mass of the Galactic halo. Modern studies employ:

• Multi-Gaussian Expansion (MGE) for baryonic and dark matter mass models,
• Combination of multiple stellar tracers (e.g., K giants, BHB stars) with known density profiles,
• Anisotropy parameters (often radially varying) and different flattening prescriptions for the halo.

Recent results indicate that the velocity ellipsoid in the Galactic halo is nearly spherical in the $(R, |z|)$ plane, justifying spherical-alignment closure in the Jeans analysis. The most consistent dark halo models allow the flattening parameter $q_h$ to vary with galactocentric distance: $q_h$ decreases (more oblate) for $r_{gc}<20$ kpc and increases (rounder halo) for $r_{gc}>20$ kpc. Quantitative estimates include $$M_{\rm tot}(<60~{\rm kpc}) = 0.533^{+0.061}_{-0.054} \times 10^{12}\,M_\odot$$, $r_{200} = 188\pm15$ kpc, and $$M_{200} = 0.820^{+0.210}_{-0.186} \times 10^{12}\,M_\odot$$ (Zhang et al., 14 May 2025).

Consistency between different tracers and models is achieved primarily when radially varying halo flattening is permitted, suggesting an evolution in the dark halo shape with radius.

5. Limitations, Systematic Effects, and Methodological Advances

While the axisymmetric Jeans equations enable powerful dynamical inferences, several sources of systematic uncertainty are intrinsic to their application in oblate geometry:

• Assumption of equilibrium and axisymmetry: Real galaxies (including the Milky Way) show disequilibria due to, e.g., satellite impacts, bars, spiral arms, and warps. Simulations indicate that departures from equilibrium and axisymmetry can bias inferred rotation curves by 10–15% at large radii (Koop et al., 29 May 2024).
• Tracer density modeling: Mistakes in the assumed spatial distribution of the tracer population (e.g., adopting an untruncated exponential when the disk is actually truncated or flared) can artificially induce a fake decline in the inferred rotation curve, complicating the interpretation of dark matter halo profile and mass.
• Sensitivity to the shape of the velocity ellipsoid: The choice between different closure schemes (JAM$_{\text{sph}}$, JAM$_{\text{cyl}}$, $b$-ansatz, Satoh $k$-decomposition) can affect the inferred dynamical quantities, especially in the presence of significant anisotropy or multiple kinematic components.

Recent methodological advances focus on:

• Efficient numerical schemes for solving the Jeans equations under complex anisotropy with spherically or cylindrically aligned velocity ellipsoids (Cappellari, 2019),
• Analytical frameworks enabling prior determination of physically allowed parameter space before resource-intensive Jeans equation integration (Deo et al., 10 Apr 2024),
• Scaling and modular decomposition strategies for rapid model inference across large multi-component parameter grids (Caravita et al., 2021).

6. Extensions: Instabilities, Nonlinear Dynamics, and External Tides

The Jeans equations in oblate axisymmetric geometry also underpin analyses of dynamical stability and the fragmentation of self-gravitating systems. Kinetic theory extensions reveal:

• In oblate geometry, the critical Jeans length and mass are axis-dependent, with preferred instability and collapse directions set by geometric and anisotropy parameters (Kremer, 2015, Stupka, 2016).
• External tidal fields further introduce direction-dependent modifications to the Jeans instability. Under anisotropic tides, the fragmentation mass and length scales become axis-dependent; strong tides can confine collapse to fewer than three dimensions, favoring filamentary structure formation (Li, 5 Mar 2024).
• The interplay between magnetic fields, rotation, and geometry yields anisotropic collapse (e.g., preferential filamentary collapse in magnetized, rotating, oblate nebulae), with the dispersion relation coupling the classic Jeans mode to magneto-rotational instability modes (Montani et al., 2017).

7. Future Prospects and Observational Implications

The accelerating growth in high-precision astrometric, photometric, and spectroscopic surveys (e.g., LSST, Gaia) will supply the necessary data to map acceleration fields, velocity moments, and matter distributions well beyond current observational limits (Loebman et al., 2012). These developments will:

• Enable direct dynamical mapping of dark matter distributions and halo shapes to distances $>10\times$ larger than previously feasible,
• Reduce uncertainties in the baryon-to-dark matter fraction,
• Refine constraints on the 3D structure and assembly histories of galaxies,
• Permit the joint application of Jeans modeling and complementary dynamical tools (e.g., orbital spectral analysis) for converging potential estimates.

The core challenge remains the accurate translation of observed number densities and multiple-component velocity moments, through appropriately closed Jeans equations, into robust constraints on the gravitational potential and mass structure in oblate axisymmetric systems. The ongoing development and public deployment of efficient, flexible Jeans solvers—incorporating anisotropy, multi-component modeling, and robust assessment of systematics—are central to future progress in the quantitative dynamical inference of galactic structure and evolution.