Theory and numerical methods

Spherical Jeans equations

For a steady, spherical, nonrotating collisionless tracer, define \(\beta(r)=1-\sigma_\theta^2/\sigma_r^2\) with \(\sigma_\theta^2=\sigma_\phi^2\). The second-moment equation is

(1)\[\frac{d(\nu\sigma_r^2)}{dr}+\frac{2\beta}{r}\nu\sigma_r^2 =-\nu\frac{GM(r)}{r^2}.\]

The boundary condition is vanishing radial pressure at infinity. Projection gives

\[\Sigma(R)\sigma_{\rm los}^2(R)=2\int_R^\infty \left(1-\beta(r)\frac{R^2}{r^2}\right) \frac{\nu(r)\sigma_r^2(r)r\,dr}{\sqrt{r^2-R^2}}.\]

See References: Jeans equations for the dynamical foundations. Both backends offer numerical controls for refining the line-of-sight integrals. The route, node counts and infinity transformation are part of the numerical model and must be recorded. Agreement of two backends is a consistency check; an analytic solution or independent integral supplies stronger validation.

Axisymmetric Jeans equations

The Jeans equations follow from velocity moments of the collisionless Boltzmann equation. JeansPy uses cylindrical alignment and constant meridional anisotropy as formulated by Cappellari (2008), with spheroidal tracer and halo profiles used in dwarf-galaxy applications by Hayashi & Chiba (2012, 2015) and Hayashi et al. (2016). The earlier two-integral models and spheroidal force integrals are described by van der Marel et al. (1994); Binney & Tremaine (2008) provide the general dynamical background. See References: Jeans equations and axisymmetric dynamics.

The stellar and halo axes coincide, cross moments vanish, and \(\beta_z=1-\overline{v_z^2}/\overline{v_R^2}\) is constant. With \(P=\nu\overline{v_z^2}\),

\[P(R,z)=\int_{|z|}^\infty\nu(R,z')\Phi_z(R,z')\,dz',\qquad \overline{v_R^2}=\frac{P}{(1-\beta_z)\nu},\qquad \overline{v_\phi^2}=\frac{P+R\,\partial_RP}{(1-\beta_z)\nu}+R\Phi_R.\]

Neither finite positive moments at observed positions nor successful MCMC prove that a positive distribution function exists everywhere. Rotation, tilted velocity ellipsoids, spatially varying cylindrical anisotropy, Satoh decomposition, triaxiality, stellar self-gravity, PSF convolution and spatial-bin averaging are outside the current axisymmetric model.

Spheroidal force and finite cutoff

The homoeoidal force has D(t)²=1+(Q²-1)t², m(t)²=t²*(R²+z²/D(t)²), and

\[ \Phi_R=4\pi GQR\int_0^{t_{\max}}\frac{\rho(m)t^2}{D(t)}\,dt, \qquad \Phi_z=4\pi GQz\int_0^{t_{\max}}\frac{\rho(m)t^2}{D(t)^3}\,dt. \]

Without a cutoff, t_max=1. Outside a finite halo, m(t_max)=r_t. The analytic radial derivative includes the moving endpoint contribution integrand(t_max)*dt_max/dR; omitting it would give incorrect azimuthal moments outside the cutoff. The pressure boundary condition is P -> 0 at infinity, and stellar tracers continue outside the finite halo.

The axisymmetric geometry guide defines the parameter domains and observer coordinates. See numerical accuracy for quadrature settings and refinement.

Likelihood and interpretation

The standard unbinned likelihood conditions on positions and independent Gaussian velocity errors. It approximates the velocity distribution by a Gaussian with the predicted second moment:

\[\log L=\sum_i\log\mathcal N\!\left(v_i\mid v_{\rm sys}, \sqrt{\sigma_{\rm los}^2(\boldsymbol x_i)+\epsilon_i^2}\right).\]

This does not imply that Jeans equations uniquely determine the full velocity distribution. Membership cuts, binaries, contaminants, equilibrium and the choice of tracer are scientific assumptions that require sensitivity checks. Priors are defined in the explicitly named sampling coordinates. A uniform prior in log10_rs_pc is log-uniform in physical radius; no extra Jacobian is added to redefine it as a uniform physical-radius prior.