[Paper Review] Fluid stationary models for the extra-planar gas

M. Barnabe', R. Sancisi+2 more|arXiv (Cornell University)|Jan 1, 2006

Galaxies: Formation, Evolution, Phenomena55 citations

TL;DR

This paper proposes baroclinic fluid stationary models to explain the observed negative vertical gradient in rotational velocity of extra-planar gas in spiral galaxies like NGC 891. By relaxing the barotropic assumption (where pressure depends only on density), the model allows pressure to vary independently of density, enabling equilibrium configurations with decreasing rotation speed at higher altitudes. The method successfully reproduces the observed kinematics of NGC 891's gaseous halo, suggesting that pressure gradients—not just gravity—can sustain such flows, and potentially replacing the need for drag forces in ballistic models.

ABSTRACT

We show that fluid stationary models are able to reproduce the observed, negative vertical gradient of the rotation velocity of the extra-planar gas in spiral galaxies. We have constructed models based on the simple condition that the pressure of the medium does not depend on density alone (baroclinic instead of barotropic solutions: isodensity and isothermal surfaces do not coincide). As an illustration, we have successfully applied our method to reproduce the observed velocity gradient of the lagging gaseous halo of NGC 891. The fluid stationary models discussed here can describe a hot homogeneous medium as well as a "gas" made of discrete, cold HI clouds with an isotropic velocity dispersion distribution. Although the method presented here generates a density and velocity field consistent with observational constraints, the stability of these configurations remains an open question.

Motivation & Objective

• To explain the observed negative vertical gradient in rotational velocity of extra-planar gas in spiral galaxies, which standard fluid models fail to reproduce.
• To develop fluid stationary models that can account for the kinematics of gaseous halos without relying solely on ballistic dynamics.
• To investigate whether baroclinic solutions—where pressure does not depend solely on density—can reproduce the observed velocity decline with height.
• To assess the physical viability of such models, including their potential to replace or supplement ballistic or viscous drag mechanisms.
• To explore the applicability of these models to both hot, homogeneous media and cold H i cloud systems with isotropic velocity dispersion.

Proposed method

• The authors use axisymmetric, stationary hydrodynamic equations with a non-barotropic (baroclinic) equation of state, allowing pressure to vary independently of density.
• They solve the force balance equations in cylindrical coordinates, incorporating the total gravitational potential (stars, dark matter) and centrifugal force.
• The model assumes non-self-gravitating gas and uses a prescribed density distribution to derive pressure and rotational velocity fields via integration of the hydrostatic and radial force equations.
• For NGC 891, the method is applied numerically using a bilogarithmic grid and finite-difference integration to compute pressure, velocity, and cooling rates.
• A simplified analytical toy model is derived using homeoidal expansions of power-law density profiles to demonstrate the existence of velocity gradients in baroclinic configurations.
• The numerical code computes luminosity, cooling time, and edge-on surface brightness by integrating emissivity over grid cells using bilinear interpolation of cooling rates.

Experimental results

Research questions

• RQ1 Can baroclinic fluid models reproduce the observed negative vertical gradient in rotational velocity of extra-planar gas in spiral galaxies?
• RQ2 Is it possible to construct stationary, hydrostatic equilibrium solutions where isodensity and isothermal surfaces do not coincide, enabling velocity decline with height?
• RQ3 Can such baroclinic models explain the kinematics of NGC 891’s gaseous halo without invoking viscous drag or ballistic assumptions?
• RQ4 What is the physical interpretation of the pressure field in these models, and can they describe both hot plasma and cold H i clouds?
• RQ5 Do these models offer a viable alternative or complement to ballistic fountain models in explaining halo gas dynamics?

Key findings

• The baroclinic fluid model successfully reproduces the observed negative vertical gradient in rotational velocity for the extra-planar gas in NGC 891, matching kinematic observations.
• The model demonstrates that a pressure gradient independent of density can sustain a velocity decline with height, resolving a key limitation of standard barotropic models.
• The analytical toy model shows that even in a simplified configuration, rotational velocity decreases with increasing z, confirming the feasibility of the mechanism.
• The model suggests that pressure gradients can naturally provide the 'drag' effect previously invoked to explain discrepancies in ballistic models, without requiring explicit viscous or magnetic interactions.
• The method is applicable to both hot, homogeneous media and a collection of cold H i clouds with isotropic velocity dispersion, described by the stationary Jeans equations.
• The computed total luminosity and surface brightness profiles are consistent with observational constraints, supporting the model's physical plausibility.