Quasi-Isothermal Distribution Function

• Quasi-isothermal distribution functions are modified isothermal models that incorporate turbulence, intermittency, and spatial variations to accurately describe astrophysical systems.
• They are applied in modeling galactic dynamics, stellar orbits, and molecular cloud structures, offering improved fits over classic lognormal or isothermal profiles.
• These functions enable unified analyses of vertical disc structure and long-range interactions, linking simulation data with theoretical frameworks in astrophysics.

A quasi-isothermal distribution function describes the equilibrium or steady-state statistical distribution of mass, energy, or stellar orbits under conditions that are approximately, but not exactly, isothermal. Such distributions play a pivotal role in stellar dynamics, molecular cloud structure, and galactic disc modeling, as they extend the classic isothermal assumption to systems exhibiting turbulence, intermittency, self-gravity, or non-trivial velocity dispersion profiles. Quasi-isothermal forms enable more accurate and physically meaningful description of observed density, velocity, or phase-space distributions across astrophysical contexts.

1. Theoretical Foundations of Quasi-Isothermal Distribution Functions

The isothermal assumption—constant temperature or velocity dispersion—yields tractable forms for density or phase-space distributions (e.g., the Maxwell-Boltzmann or $\mathrm{sech}^2$ profiles). In astrophysical environments, deviations arise due to turbulence, self-gravity, or the mechanical structure of large-scale stellar/gaseous systems. A quasi-isothermal distribution retains some isothermal properties, such as an underlying multiplicative or lognormal tendency in PDFs, but admits physically motivated departures through intermittency parameters, power-law tails, scale-dependent fluctuations, or spatially varying velocity dispersion.

Quasi-isothermal distributions can be derived from gas dynamics, via cascade models for turbulence, solutions of the Poisson and hydrostatic equilibrium equations (with non-isothermal modifications), or solutions of the steady-state Vlasov equation approached through either exact or approximate means (Hopkins, 2012, Gromov et al., 2015, Campa et al., 2016, Donkov et al., 2017, Sarkar et al., 2020).

2. Quasi-Isothermal Distributions in Turbulent Isothermal Gas

In isothermal turbulence, the core of the volumetric density PDF is often close to lognormal, reflecting the multiplicative aggregation of shocks and rarefaction events. However, high-resolution simulations reveal systematic deviations from lognormality, especially in the distribution wings. The proposed quasi-isothermal fitting function for the volume-weighted logarithmic density is: $$P_V(\ln\rho)\,d\ln\rho = I_1\Bigl(2\sqrt{\lambda\,u}\Bigr)\,e^{-(\lambda+u)}\,\sqrt{\frac{\lambda}{u}\,du}$$ with

$$u = \frac{\lambda}{1+T} - \frac{\ln\rho}{T}, \quad \lambda = \frac{S_{\ln\rho,V}}{2T^2}$$

where $I_1(x)$ is the modified Bessel function of the first kind, $S_{\ln\rho,V}$ is the volume-weighted variance in $\ln\rho$, and $T$ quantifies deviation from lognormality. In the limit $T \rightarrow 0$, the distribution recovers the lognormal.

The parameter $T$ is physically interpreted as the variance per intermittent event in the turbulent cascade, or the “rate parameter” in a continuous-time relaxation process governing the stochastic multiplicative evolution of density fluctuations. Simulations demonstrate that $T$ increases with the compressive Mach number, scaling approximately as $T \approx 0.05 \mathcal{M}_c$. This function dramatically outperforms pure lognormal and higher-moment lognormal-based models in matching simulation data, especially in the heavy-tailed regime, with root-mean-square log-density deviations as low as $0.05\,\mathrm{dex}$ compared to several tenths for classical models (Hopkins, 2012).

3. Quasi-Isothermal Models in Galactic Potentials and Stellar Dynamics

In galactic dynamics, the quasi-isothermal distribution function is a cornerstone for constructing analytic models of galactic gravitational potentials and systematic stellar orbits. The quasi-isothermal Stäckel’s model posits the galactic potential in the equatorial plane as: $$\Phi(R,0) = \Phi_0 \ln \left[1 + \frac{\beta}{\sqrt{1+\kappa^2 R^2}}\right]$$ with $w(R) = \sqrt{1+\kappa^2 R^2}$, $\Phi_0$ and $\kappa$ as amplitude and scale parameters, and $\beta$ (or $q = \beta/(\beta+1)$) governing the potential’s shape. As $\beta \rightarrow 0$ or $\infty$, the model approaches the Schuster–Plummer or Jaffe potential, respectively (Gromov et al., 2015).

The quasi-isothermal Stäckel potential’s key property is integrability, ensured by the existence of a third quadratic integral of motion $I_3$, responsible for the observed triaxiality in the stellar velocity ellipsoid. Best-fit parameters for the Milky Way based on HI rotation curves are $\Phi_0 \approx 258$ km$^2$/s$^2$, $\kappa \approx 0.32$ kpc$^{-1}$ (for $R_0 = 8$ kpc), and $q \to 1$. This model accurately traces the main trend of the Galactic rotation curve, with spatial density contours matching the dimensions of the bulge/bar.

The model’s vertical structure, constrained by a “vertical focus” parameter $z_0$ (set to $5.4$ kpc), allows full 3D density modeling:

• Central density and rapid radial/vertical fall-off.
• Equidensity contours paralleling Galactic bulge and disk structure.
• Analytical extensibility to multiple components (disk, bulge, halo).

4. Quasi-Isothermal Distributions in Molecular Clouds: Fractal-Turbulent Self-Gravity

In star-forming molecular clouds, the quasi-isothermal concept adapts to the coupling of fractal turbulence, isothermal microphysics, and self-gravity (Donkov et al., 2017). The density PDF is determined by ensemble-averaged energy conservation and scale hierarchies: $$\frac{d}{ds} \left[\frac{1}{2} \langle v^2 \rangle + \langle \phi \rangle + s\right] = 0,\quad s = \ln\left(\frac{\rho}{\rho_c}\right)$$ with $L(s)$ and $Q(s)$ (the dimensionless scale and its cubic root) linking geometric scale to density PDF $p(s) = -3 Q^2 Q'$.

Solving the corresponding nonlinear integral equation for $Q(s)$ yields two regimes:

• For cloud envelopes (accretion/turbulence balance): $Q(s) \propto \exp(-s/2)$ leads to $p(s) \propto \exp(-3s/2)$, a power law with slope $-1.5$.
• For high-density cores (free-fall collapse): $Q(s) \propto \exp(-2s/3)$ gives $p(s) \propto \exp(-2s)$, a power law with slope $-2$.

These analytical predictions explain dual-slope power-law tails seen in simulations and observations, corresponding to equilibrium and collapse, respectively. The formalism unifies fractal geometry, energetic equilibrium, and isothermal thermodynamics in setting the quasi-isothermal PDF structure.

5. Non-Isothermal Modifications to Quasi-Isothermal Disc Models

Classic vertical disc structure assumes an isothermal (constant) vertical velocity dispersion: $\rho(z) = \rho_0 \,\mathrm{sech}^2(z/z_0)$ arising from joint hydrostatic equilibrium and Poisson equations. Observationally, however, $\sigma_z(z)$ rises with height; parameterizing as $\sigma_z(z) = \sigma_{z,0} + C z$, the modified equilibrium equation

$$\frac{d^2\rho}{dz^2} = \frac{-4\pi G \rho^2 [\sigma_{z,0} + Cz]^2 + (1/\rho)\left(\frac{d\rho}{dz}\right)^2 - (2C/[\sigma_{z,0}+Cz])\frac{d\rho}{dz} - [2C^2\rho/(\sigma_{z,0}+Cz)^2]}{[\sigma_{z,0} + C z]^2}$$

alters the density profile.

Key consequences (Sarkar et al., 2020):

• Mid-plane density is $\sim$30–40% lower (at the solar radius) in the non-isothermal model for stars-alone discs.
• The scale height (half-width at half-maximum) increases by $\sim$37% at $R=8.5$ kpc for typical gradients.
• The profile departs from $\mathrm{sech}^2(z/z_0)$, developing extended wings that fit well with a sum of two $\mathrm{sech}^2$ functions; this can mimic a thick + thin disc in external galaxies, even for a single quasi-isothermal system.
• In composite models including gas and halo gravity, these effects are reduced but persist.

This demonstrates that observed double-component disc profiles can be explained, at least in part, by non-isothermal modifications of a singular quasi-isothermal distribution rather than requiring distinct populations.

6. Quasi-Isothermal and Quasi-Stationary States in Systems with Long-Range Interactions

The “quasi-isothermal” terminology also appears in the kinetic theory of long-range interacting systems. In such systems, as described by the Vlasov equation, the collisionless mean-field evolution quickly brings the distribution function to a quasi-stationary state (QSS), especially after “violent relaxation.” Quasilinear theory for the Vlasov equation provides the evolution: $$\partial_t f_0(p,t) = \partial_p [D(p,t) \partial_p f_0(p,t)]$$ with the diffusion coefficient $D(p,t)$ depending on the instantaneous growth rate $\omega_I$: $$D(p,t) = \frac{2 \chi(t)\omega_I(t)}{p^2 + \omega_I^2(t)}$$ where $f_0(p,t)$ is the angle-averaged velocity distribution. For weakly unstable initial states, this description accurately captures the slow diffusion toward a QSS characterized by plateaued velocity distributions and a kinetic temperature/energy-magnetization transition.

For strongly unstable regimes, numerical simulations show the QSS is better described by polytropic (Tsallis) distributions rather than Boltzmann or Lynden-Bell equilibria: $$f(\epsilon) \propto [\mu - (q-1)\beta\epsilon]^{1/(q-1)}$$ with $n = 2$ (Gaussian initial condition) or $n = 1$ (semi-elliptical case). These polytropic quasi-isothermal forms can account for negative specific heat regions in caloric curves, a distinct property of QSSs in long-range systems (Campa et al., 2016).

7. Implications and Applications

Quasi-isothermal distribution functions offer a unified conceptual and analytic framework for:

• Turbulent star-forming regions, where intermittency and cascade dynamics require non-lognormal PDFs.
• Galactic mass models, enabling accurate analytic fits to rotation curves and 3D density structures, with flexibility to encompass classical models as limiting cases.
• The interpretation of observed disc structure in galaxies, revealing the role of non-isothermal dispersions in mimicking multi-component discs.
• Long-range dynamical systems, where QSSs emerge from collisionless evolution and polytropic quasi-isothermal functions naturally fit extended states that deviate from thermodynamic equilibrium.

These formulations systematically connect observed astrophysical distributions with underlying physics—turbulent intermittency, self-gravity, hierarchical scaling, and non-collisional or non-isothermal energy input—leading to rigorous improvements over “pure” isothermal or lognormal models and clarifying apparent discrepancies in structural fits, moment analyses, and system stability.