Gravitational Potential of the Galactic Disk

Updated 8 January 2026

Gravitational potential of the galactic disk is the cumulative influence of stellar, gas, and dark matter components that dictates the spatial structure and dynamical behavior of disk galaxies.
Modeling involves decomposing mass into stellar, gas, and dark halo contributions while coupling Poisson and equilibrium equations for both analytic and numerical solutions.
Observational techniques, including photometry, kinematic profiling, and phase-space spiral methods, provide critical constraints on disk mass distribution and vertical equilibrium.

The gravitational potential of the galactic disk determines the spatial structure, stability, and dynamical evolution of disk galaxies, including the vertical distribution of stars and interstellar gas, the response of orbiting objects, and the interpretation of kinematic and photometric data. Modeling this potential requires simultaneously accounting for multiple mass components (stellar disk, gas disk, and dark-matter halo), coupling of Poisson and equilibrium equations in three dimensions, and fitting to extensive observational constraints from photometry and stellar/gas dynamics. The detailed structure of the disk potential underpins precise inferences of matter content and distribution, star formation regulation, and the dynamical response of different stellar populations.

1. Disk–Halo–Gas Mass Decomposition and Model Formulation

The total gravitational potential of a disk galaxy is derived from its mass distribution, which is customarily decomposed as

$\rho_\text{tot}(R, z) = \rho_\star(R, z) + \rho_g(R, z) + \rho_\text{DM}(r),$

where $\rho_\star$ is the stellar component, $\rho_g$ is the gas (atomic + molecular), and $\rho_\text{DM}$ is the dark matter halo component (usually with $r=\sqrt{R^2+z^2}$ ).

Stellar Disk

The standard model adopts an exponential distribution in $R$ ,

$\Sigma_\star(R) = \Sigma_{\star,0}\,\exp\left(-R/R_d\right),$

and a vertical exponential or $\operatorname{sech}^2$ profile,

$\rho_\star(R,z) = \frac{\Sigma_\star(R)}{2H_\star}\,\exp\left(-|z|/H_\star\right),$

with radial scale length $R_d$ and vertical scale height $H_\star$ determined from star-count or luminosity-profile analyses. The total stellar mass and thickness of the disk follow from these parameters.

Gas Disk

Gas is traced from HI and CO surveys, giving

$\Sigma_g(R) = \Sigma_\text{HI}(R) + \Sigma_\text{H%%%%8%%%%}(R)$

and is often treated as a razor-thin layer, $\rho_g(R,z) \simeq \Sigma_g(R)\,\delta(z)$ , though some models incorporate finite thicknesses for atomic and molecular layers.

Dark Matter Halo

A commonly adopted profile is the Navarro-Frenk-White (NFW) form: $\rho_\text{DM}(r) = \frac{\rho_0}{(r/R_s)\,(1 + r/R_s)^2},$ with scale radius $R_s$ and density normalization $\rho_0$ . Alternative models include pseudo-isothermal spheres and cored profiles. The cumulative halo mass $M_\text{DM}(<r)$ then yields the potential either by direct integration or the Poisson equation.

2. Analytical and Numerical Potential Solutions

Axisymmetric and Separable Potentials

In axisymmetric form, the galactic potential is a function of $R$ and $z$ , $\Phi(R, z)$ , with total

$\Phi_\text{tot}(R, z) = \Phi_\star(R, z) + \Phi_g(R, z) + \Phi_\text{DM}(r).$

Component potentials can be written using Hankel transforms for exponential disks, analytic kernel integrals for razor-thin (e.g., Binney & Tremaine Eq. 2.265), or Miyamoto–Nagai-type forms for thickened disks: $\Phi_\text{disk}(R, z) = -\frac{G M_d}{\sqrt{R^2 + \left(a_d + \sqrt{z^2 + b_d^2}\right)^2}}.$ Here $M_d$ is disk mass, $a_d$ is radial scale, $b_d$ is vertical scale. Complex mass models utilize sums of higher-order Miyamoto–Nagai terms to fit multiple observed disk subcomponents, including thin/thick stars, HI, and H $_2$ gas, codified in models such as those in Barros et al. (Barros et al., 2016).

Vertical and Radial Forces

The vertical force per unit mass is

$K_z(R, z) = -\frac{\partial \Phi_\text{tot}}{\partial z},$

governing vertical oscillations and hydrostatic support in the disk.

Analytic expressions for $K_z$ and $F_R$ from Miyamoto–Nagai disks facilitate closed-form predictions and data fitting: $F_z(R, z) = -\frac{G M_d\, (a_d + \sqrt{z^2 + b_d^2})\, z}{\sqrt{z^2 + b_d^2} \, \left[R^2 + (a_d + \sqrt{z^2 + b_d^2})^2\right]^{3/2}}$ with similar expressions for $F_R(R, z)$ (Bajkova et al., 2018, Kenyon et al., 2018).

Numerical computation is required for multi-component systems, non-axisymmetric potentials (e.g., spiral/barred disks), and for evaluating the integrals in the Hankel/Bessel transforms and gas self-gravity.

3. Vertical Hydrostatic Equilibrium, Gas Scale Heights, and Disk Weight

Vertical structure of the disk (and ISM) arises from hydrostatic balance in the vertical potential, with isothermal velocity dispersion $\sigma_z(R)$ : $\frac{d}{dz}\left[\,\rho_g\,\sigma_z^2\,\right] = -\rho_g\,K_z(R, z).$ A one-zone approximation gives

$H_g(R) \simeq \frac{\sigma_z^2}{K_z(R, 0)},$

i.e., the scale height of the gas layer is set by local vertical force balance. More accurate solutions include effects of gas self-gravity, finite stellar disk thickness, and the dark halo. For a three-component disk + halo, the equilibrium leads to a cubic equation for $H_g$ , as in (Vijayakumar et al., 27 Jun 2025).

The ISM "weight" (momentum flux per unit area confining the gas) is

$W(R) = \int_{-\infty}^{+\infty} \rho_g(R, z)\,K_z(R, z)\,dz,$

splitting as contributions from gas self-gravity, stellar gravity, and halo. Closed-form approximations are available: $W_\text{gas} = \frac{\pi}{2} G \Sigma_g^2, \quad W_\star = \pi G \Sigma_g \Sigma_\star \frac{H_g}{H_g + H_\star}, \quad W_\text{DM} = \zeta \Sigma_g \Omega_\text{DM}^2 H_g,$ with $\zeta \sim 1/3$ and $\Omega_\text{DM} \equiv V_\text{DM}(R)/R$ .

Scale height $H_\star$ and the stellar mass-to-light ratio (M/L) determine the midplane stellar density and thus influence $K_z(R, 0)$ —higher $H_\star$ reduces vertical restoring force and inflates $H_g$ by $30$–$40$\% if $H_\star$ varies by a factor of $3$.

4. Constraints from Observations and Dynamical Tracers

Vertical Kinematics and Forces

Detailed constraints on $K_z(z)$ and $\Phi(z)$ are obtained from vertical density and velocity-dispersion profiles (e.g., SEGUE K-dwarfs, red clump stars), utilizing the axisymmetric vertical Jeans equation: $\frac{1}{\nu(z)} \frac{d}{dz}\left[\,\nu(z)\,\sigma_z^2(z)\,\right] = -\frac{d\Phi}{dz},$ with fits to dynamical tracers providing estimates of disk surface density, scale heights, and local dark matter density (e.g., $z_h \lesssim 300\,\mathrm{pc}$ , $\Sigma_{\mathrm{tot}}(|z|<1\,\mathrm{kpc}) \simeq 66-67\,M_\odot\,\mathrm{pc}^{-2}$ , $\rho_\mathrm{DM} \sim 0.006 - 0.009\,M_\odot\,\mathrm{pc}^{-3}$ (Zhang et al., 2012, Sanchez-Salcedo et al., 2015, Widmark et al., 2021)).

Phase-Space Spiral Methods

Non-equilibrium features such as phase-space spirals in the $(z, v_z)$ -plane encode information on $\Phi(z)$ independent of steady-state assumptions, as shown in the Widmark et al. series (Widmark et al., 2021, Widmark et al., 2021, Widmark et al., 2021, Widmark et al., 2022). These techniques use the “winding” of the spiral to determine vertical oscillation periods $P(E_z)$ and thus reconstruct the vertical potential out to $z \sim 1$ kpc, with spatial resolution and systematics control superior or complementary to classic Jeans approaches.

5. Non-Axisymmetries and Spiral–Bar Perturbations

Realistic galactic potentials include non-axisymmetric features (spiral arms, bars) that contribute localized perturbations. Axisymmetric backgrounds are commonly modeled as logarithmic or exponentially flattened disks plus halos, to which is added a spiral potential of the form

$\Phi_1(R, \varphi) = A(R) \cos\left(m\varphi + m \ln\frac{R}{h_{R,1}}/\tan p\right),$

with pitch angle $p$ , number of arms $m$ , and amplitude $A(R)$ related to the underlying spiral mass-density contrast (typically $|\Sigma_1/\Sigma_0|\simeq0.1$ at the solar circle) (Eilers et al., 2020, Junqueira et al., 2012).

The impact of spiral/bar perturbations is strongest near resonances (ILR, corotation, 4:1), affecting orbital stability, disk response, and features such as arm bifurcations and bar formation.

6. Empirical Constraints, Inferred Parameters, and Model Degeneracies

Inversions from Kinematic Data

Combining stellar density and dispersion profiles, velocity-ellipsoid tilts (e.g., $\alpha_0\approx0.68$ measuring flattening of the potential (Sun et al., 2023)), and detailed fits to measured rotation curves yields constraints on:

Disk mass and scale length: $M_d \sim 6\times10^{10}\,M_\odot$ , $R_d = 2.2$ –$4.5$ kpc
Disk vertical scale: $H_\star \sim 0.3$ –$0.4$ kpc
Gas and stellar vertical force at $z=1.1$ kpc: $|K_{z=1.1}|/2\pi G \simeq 77\,M_\odot\,\mathrm{pc}^{-2}$
Local dark matter density: $\rho_\mathrm{DM} \geq 0.006\,M_\odot\,\mathrm{pc}^{-3}$
Surface density to $z=1$ kpc: $\sim67\,M_\odot\,\mathrm{pc}^{-2}$

Systematics and Model Choices

Model degeneracies arise between disk and halo contributions, disk flare and warp, and in the assumed scale lengths and population selection. Time-varying disturbances (breathing modes, phase-space spirals) can lead to departures from equilibrium-based estimates, biasing local mass-density inferences if not accounted for (Widmark et al., 2020).

7. Applications and Theoretical Implications

The full 3D gravitational potential of the galactic disk framework is foundational for:

Deriving ISM vertical scale heights and weights, regulating pressure–SFR relations (Vijayakumar et al., 27 Jun 2025).
Constraining the local dark-matter distribution and disentangling baryonic/halo contributions (Sanchez-Salcedo et al., 2015, Widmark et al., 2021).
Predicting orbit families, disk stability, migration and the formation and persistence of spiral structure.
Modeling dynamical responses to satellite impacts, bar/spiral evolution, and outer-disk flaring.
Providing input potentials for N-body and hydrodynamical simulations with realistic disk–halo coupling and non-axisymmetries (Shin et al., 2017).

Empirical models matched to Milky Way data are readily extendable to external disks, enabling comparative studies of disk dynamical equilibrium, flaring, and non-axisymmetric response in varied mass and structural regimes.

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