[Paper Review] Local stability criterion for self-gravitating disks in modified gravity

Asiyeh Habibi, Mohammad Taghi Mirtorabi+1 more|arXiv (Cornell University)|May 25, 2014

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper derives a generalized local stability criterion for self-gravitating fluid and stellar disks in modified gravity theories that include a Yukawa-like correction to the Newtonian potential. By extending Toomre's criterion to account for the additional gravitational term in Modified Gravity (MOG), the authors show how the stability of differentially rotating disks is altered, with quantitative results dependent on MOG parameters such as α and μ.

ABSTRACT

We study local stability of self-gravitating fluid and stellar disk in the context of modified gravity theories which predict a Yukawa-like term in the gravitational potential of a point mass. We investigate the effect of such a Yukawa-like term on the dynamics of self-gravitating disks. More specifically, we investigate the consequences of the presence of this term for the local stability of the self-gravitating disks. In fact, we derive a generalized version of Toomre's local stability criterion for diferentially rotating disks. This criterion is complicated than the original one in the sense that it depends on the physical properties of the disk. In the case of MOdified Gravity theory (MOG), we use the current confirmed values for the free parameters of this theory, to write the generalized Toomre's criterion in a more familiar way comparable with the Toomre's criterion. This generalized Toomre's criterion may be used to study the global stability of stellar and fluid disks using computer simulations.

Motivation & Objective

• To address the instability of differentially rotating stellar and fluid disks in modified gravity theories that include a Yukawa-like potential correction.
• To resolve the longstanding problem of disk galaxy instability in the absence of dark matter halos by modifying the gravitational law.
• To generalize Toomre’s local stability criterion to include the effects of a Yukawa-like term in the gravitational potential.
• To provide a computationally usable form of the stability criterion for simulating global disk stability in MOG.
• To compare the modified criterion with the standard Toomre criterion using confirmed MOG parameter values.

Proposed method

• Derive linearized perturbation equations for fluid and stellar disks in the presence of a modified gravitational potential with a Yukawa-like term.
• Use the Poisson equation modified by two gravitational potentials: one Newtonian-like (φ) and one massive (ψ), governed by ∇²φ = 4πG₁ρ and (∇²−μ²)ψ = 4πG₂ρ.
• Apply a tightly wound spiral mode perturbation ansatz: Σ₁ ∝ e^{i(mϕ−ωt)} with radial dependence e^{ikR}, assuming m ≫ 1.
• Solve the modified Poisson equations for the perturbed potentials φ₁ and ψ₁ on the disk plane (z=0), using delta-function surface density.
• Obtain analytical expressions for the gravitational potential perturbation Φ₁ = φ₁ + ψ₁ in terms of surface density perturbation Σ₁, incorporating the effective coupling G* and modified wavenumber k*.
• Derive the generalized Toomre-like stability criterion by combining the perturbed Euler and continuity equations with the modified potential response.

Experimental results

Research questions

• RQ1 How does the inclusion of a Yukawa-like term in the gravitational potential affect the local stability of self-gravitating disks in modified gravity?
• RQ2 Can the modified gravity framework (specifically MOG) stabilize differentially rotating disks without requiring dark matter halos?
• RQ3 What is the form of the generalized Toomre stability criterion when the gravitational potential includes a massive exchange term?
• RQ4 How do the MOG parameters α and μ quantitatively alter the stability threshold compared to the standard Toomre criterion?
• RQ5 To what extent can the generalized criterion be used to predict global stability in disk galaxies via numerical simulations?

Key findings

• The generalized stability criterion for fluid disks in MOG includes a modified effective gravitational coupling G* = G(c₁ + c₂/√(1 + μ²/k²)) and a modified wavenumber k* = |k|√(1 + μ²/k²).
• For MOG, with confirmed parameters (α ≈ 0.05, μ ≈ 0.001 kpc⁻¹), the correction to the potential is significant at galactic scales, altering the stability threshold.
• The derived criterion reduces to the standard Toomre criterion in the limit where the Yukawa term vanishes (c₂ → 0, μ → 0).
• The stability condition becomes more stringent with increasing μ, indicating that massive gravity components suppress disk instabilities.
• The modified criterion depends explicitly on the disk’s physical properties such as surface density, rotation speed, and epicyclic frequency.
• The analytical form of the criterion allows for direct implementation in numerical simulations to study global stability of stellar and fluid disks in MOG.