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The vertical Jeans equation determines $K_z={\partial }_z\Phi$ from a tracer's density and velocity dispersion, with a correction for the tilted velocity ellipsoid. Integrating the axisymmetric Poisson equation for Newtonian gravity givesfor $z>0$ and reflection symmetry. Subtracting independently estimated stellar and gas surface densities constrains any additional gravitating component; a dark matter halo also contributes, so an excess is not automatically a dark disk.

First consider a genuinely one-dimensional, time-independent collisionless stellar system with Hamiltonian $E_z=v_z^2/2+\psi (z)$. Choose $\psi (0)=0$. For bound vertical oscillations, a phase-mixed population has equal occupations of the upward and downward branches of each orbit. By Jeans theorem, its stationary distribution is a function of the conserved vertical energy, $f_z(z,v_z)=F_z(E_z)$. Equivalently, the one-dimensional Collisionless Boltzmann equation $v_z{\partial }_z{f}_z-{\psi }^{\prime }(z){\partial }_{v_z}{f}_z=0$ makes $f_z$ constant along the energy curves. Branch-dependent incoming and outgoing populations or time-dependent waves would not satisfy this assumption.

Integrate both signs of $v_z$ and change variable to $E=v_z^2/2+\psi$:The density is thus an Abel transform of the vertical-energy distribution. To invert it without simply quoting an inversion rule, apply a second integral and exchange its order:The inner integral is $\pi$, as follows by setting $\psi =E+(t-E){\sin }^2\theta$. Writing $J(E)={\int }_0^{\infty }\nu (E+s)s^{-1/2}ds$ before differentiating avoids an artificial infinite endpoint term. Under sufficient differentiability and decay,The vertical-energy Abel inversion of a stellar distribution is thereforeThis proves the normalization, including the factor from the two velocity signs. The additive origin of $\psi$ merely changes the origin of energy.

For this expression to define a physical phase-space distribution function, $\nu$ must be expressible as a single-valued function of $\psi$, the integrals and differentiated integral must converge, and the resulting $F_z$ must be nonnegative. Reflection symmetry and a potential increasing with $|z|$ provide the usual single-valued relation on either side of the disk. A sufficiently smooth decreasing $\nu (\psi )$ is a sufficient condition for nonnegativity; it is not asserted to be necessary for every positive energy distribution. If there is a finite escape energy, replace the infinite limits by that energy and impose the appropriate cutoff and boundary conditions. The displayed infinite-limit formula presumes either a confining potential or a distribution extended by zero above its allowed energy, with no omitted boundary population.

A three-dimensional galactic disk does not automatically meet these hypotheses. Exact separation $\Phi (R,z)={\Phi }_{\parallel }(R)+\psi (z)$ makes $E_z$ an integral of motion. If the full galactic distribution function factorizes into an in-plane factor and a vertical factor $F_z(E_z)$, integrating the in-plane velocities gives the required one-dimensional distribution after its normalization is absorbed into $\nu$. More generally the vertical marginal must depend only on $E_z$ in the sample being analysed. Near the solar radius, approximate separation can be useful when the sampled heights and radial excursions are small relative to the scales over which the vertical force changes with $R$. If only a slowly varying vertical action is conserved, replacing it by one fixed $E_z$ requires an additional controlled local approximation.

The population must be sufficiently stationary and phase mixed, with negligible mean vertical flow, and the data must measure a well-defined tracer population with a known selection function. Radial coupling and the tilted velocity ellipsoid must be negligible or modelled. In the full vertical Jeans equation, that coupling is $R^{-1}{\partial }_R(R\nu \overline{v_R{v}_z})$; it cannot generally be discarded merely because a mixed moment vanishes at the midplane. The tracer number density $\nu$ is not the mass density producing the potential.

To infer the local mass, measure the tracer density as a function of height and its vertical velocity distribution near the midplane. With $\psi (0)=0$, the latter directly samples $F_z(v_z^2/2)$. Insert that measured distribution into the forward integral for $\nu (\psi )$, and match it to the observed $\nu (z)$ to infer $\psi (z)$. Equivalently, for trial potentials compute $d\nu /d\psi =(d\nu /dz)/(d\psi /dz)$ away from the plane, apply the inversion, and require agreement with the measured velocities. Density data alone do not determine both an unknown potential and an unknown energy distribution.

Once the vertical force is known, use the full axisymmetric Poisson equation for Newtonian gravity, rather than identifying every vertical derivative with density in isolation:At the midplane, $v_c^2(R)=R{\partial }_R\Phi (R,0)$, soIn terms of the circular-orbit Oort constants, $A=\frac{1}{2}(v_c/R-d{v}_c/dR)$ and $B=-\frac{1}{2}(v_c/R+d{v}_c/dR)$, the radial term equals $2(B^2-A^2)$. It vanishes for a locally flat rotation curve, not for an arbitrary rotating disk. An integrated version gives the dynamical surface density from vertical stellar motions:Here the actual vertical acceleration is $-{\psi }^{\prime }(z)$. The sign of the radial correction follows directly from the Poisson equation for Newtonian gravity.

For an illustrative isothermal tracer with constant vertical velocity dispersion $s$, takeThe velocity integral gives $\nu (z)={\nu }_0{e}^{-\psi (z)/s^2}$, hence $\psi (z)=-s^2\ln [\nu (z)/{\nu }_0]$. For a locally flat rotation curve this yieldsThis example shows explicitly how a measured tracer scale and velocity scale constrain the gravitating density; using the tracer's own density as the mass source would instead impose an unjustified self-gravitating model.

The inferred density includes luminous and faint stars, gas and dark matter. Comparing it with an independently estimated baryonic census tests for extra gravitating mass. An excess need not be a dark disk: a dark matter halo also contributes to the local force. Conversely, agreement with the baryonic density locally does not exclude a halo at other radii or heights. Vertical disequilibrium, radial mixing, errors in selection or tracer populations, and neglected velocity tilt can mimic a mass discrepancy, so the one-dimensional equilibrium assumptions must be tested when interpreting such an inference.

The thin-sheet field used above requires the gas to sample heights beyond most of the stellar mass, while the stellar galactic disk still dominates vertical gravity and is approximately locally plane-parallel. In inner regions a cold gas layer can lie inside the thicker stellar layer, where reflection symmetry gives $K_z\simeq {\nu }_z^{2}z$ rather than a constant field. The vertical Jeans equation, or gas hydrostatic equilibrium, then givesHere $H_g$ is the Gaussian width and ${\nu }_z$ is the vertical epicyclic frequency; in a locally flat galaxy rotation curve, ${\nu }_z^2\simeq 4\pi G{\rho }_{\mathrm{total}}(R,0)$. A stellar layer of approximately fixed thickness then gives $H_g\propto {\Sigma }_{\ast }^{-1/2}$, rather than the inverse surface density law.

At large radii, the dark matter halo or the gas's own gravity can exceed the dwindling stellar restoring force, so replacing the total force by $2\pi G{\Sigma }_{\ast }$ fails. A nearly spherical dark matter halo instead gives ${\nu }_z\simeq v_c/R$ near its plane, and $H_g\simeq {\sigma }_{g}R/v_c$ for a flat galaxy rotation curve, much slower than exponential galactic disk flaring. Warps, radial gradients across a thick gas layer, non-equilibrium disturbances, and height-dependent velocity dispersion also invalidate the locally plane-parallel steady isothermal approximation. The inverse stellar surface-density law applies only while its force and geometric assumptions apply.

Assume a steady, locally plane-symmetric galactic disk, with negligible radial tilt term in the vertical Jeans equation. Write $\rho (z)$ for the stellar mass density, and $K_z={\partial }_z\Phi$ for the upward gradient of the Newtonian gravitational potential; the actual vertical gravitational acceleration is $-K_z$. Constant vertical velocity dispersion givesIntegrating the plane-parallel Poisson equation for Newtonian gravity from the symmetry plane gives $K_z(z)=4\pi G{\int }_0^z\rho (z^{\prime })d{z}^{\prime }$. Outside most of the mass, this tends to $2\pi G\Sigma$ for $z>0$, where the surface density includes both sides of the galactic disk. Treating this field as constant and integrating the vertical Jeans equation yieldsThis exponential is the thin-sheet approximation or the outer tail, not the exact profile at every height. To see the distinction without an additional assumption, set $\rho ={\rho }_0{e}^{-u}$ with $u=(\Phi -\Phi (0))/{\sigma }^2$. The Poisson equation for Newtonian gravity becomes $u^{\prime \prime }=(4\pi G{\rho }_0/{\sigma }^2)e^{-u}$, with $u(0)=u^{\prime }(0)=0$, and is solved bySince $\Sigma =2{\rho }_0{z}_0$, the exact self-gravitating isothermal slab has $z_0={\sigma }^2/(\pi G\Sigma )$. At large $|z|/z_0$, ${\mathrm{sech}}^2(z/z_0)\sim 4e^{-2|z|/z_0}$, recovering precisely the boxed scale height; near the plane its slope vanishes, whereas an exponential has a cusp.

Measure the number-density profile $n(z)$ and vertical velocity dispersion ${\sigma }_z(z)$ of a tracer stellar population. The vertical Jeans equation estimates the restoring field fromThe second term accounts for the tilt of the velocity ellipsoid; an exponential tracer with constant velocity dispersion and negligible tilt gives $K_z={\sigma }_z^2/h$ above the plane. The tracer need not supply all the gravitating mass. To infer the dynamical surface density from vertical stellar motions, integrate the axisymmetric Poisson equation for Newtonian gravity:For a nearly flat galaxy rotation curve with a small radial correction, this reduces to $K_z/(2\pi G)$. Estimate stellar surface density independently from counts, the initial mass function and faint objects, and add the gas surface density and stellar remnants. The residuallimits the allowed unseen material within the same height interval. A fitted dark matter halo contributes too: measurements at several heights help distinguish its extended density from a thin dark component in the galactic disk. Under a specified decomposition, the disc dark-matter fraction is the inferred dark-disc surface density divided by the total disc surface density, with uncertainty or an upper limit if the residual is consistent with zero. Selection effects, disequilibrium, the tilt term and uncertain baryonic counts must enter the error budget; an excess alone does not establish that all of it is in the galactic disk.

A plane-symmetric, self-gravitating stellar population with constant vertical velocity dispersion satisfies the vertical Jeans equation and the Poisson equation for Newtonian gravity. With ${\Phi }^{\prime }(0)=0$, their solution isIts surface density is $\Sigma =2{\rho }_0{z}_0$, so $z_0={\sigma }_z^2/(\pi G\Sigma )$. Its outer density tail is exponential with scale height $z_0/2$; its central profile is not exponential.