[Paper Review] On the stability criteria for equatorial circular orbits in Galactic Dynamics: I.Newtonian Thin Disks

Ronaldo S. S. Vieira, Javier Ramos-Caro|arXiv (Cornell University)|Jun 27, 2012

Quantum chaos and dynamical systems4 references3 citations

TL;DR

This paper re-evaluates vertical stability criteria for equatorial circular orbits in Newtonian thin disk models, showing that the necessary and sufficient condition for vertical stability is simply that the surface mass density Σ > 0, due to the discontinuity in the gravitational field. Radial stability remains governed by the standard epicyclic criterion κ² > 0, and the authors construct stable finite thin disk models for nine galaxies using the Morgan & Morgan family of potentials.

ABSTRACT

We make a revision of the stability criteria for equatorial circular orbits, obtained from the epicyclic approximation, which is widely used in Newtonian models for axisymmetric galaxies. We find that, for the case of thin disk models, the criteria of vertical stability must be reformulated, due to the discontinuity in the gravitational field. We show that, for a model characterized by a surface mass density $Σ$, the necessary and sufficient condition to have vertically stable orbits is that $Σ>0$. On the other hand, the criteria for radial stability is the same as in thick diks, i.e. that the epicyclic frequency is positive.

Motivation & Objective

• To re-express the stability criteria for equatorial circular orbits in thin disk models under the epicyclic approximation, accounting for the discontinuity in the gravitational field.
• To resolve inconsistencies in standard stability criteria when applied to razor-thin disks, particularly regarding vertical oscillations.
• To construct finite, stable thin disk models for nine galaxies using the Newtonian version of the Morgan & Morgan family of potentials.
• To identify the existence of integrable disk-crossing orbits in thin disk systems, characterized by a third integral of motion involving ZΣ¹ᐟ³.
• To clarify the role of the surface mass density Σ and epicyclic frequency κ² in determining both radial and vertical stability in axisymmetric galactic models.

Proposed method

• Applies the epicyclic approximation to analyze small perturbations in the meridional plane (R, z) around circular orbits in thin disk models.
• Uses the effective potential Φ_eff(R, z) to derive stability conditions based on second derivatives: ∂²Φ_eff/∂R² > 0 for radial stability and ∂Φ_eff/∂|z| > 0 for vertical stability.
• Analyzes the discontinuity in the gravitational field at z = 0 due to the thin disk, leading to a reformulation of the vertical stability condition.
• Derives the condition Σ(R₀) > 0 as necessary and sufficient for vertical stability, based on energy and momentum conservation during disk crossings.
• Constructs finite thin disk models by superposing members of the Morgan & Morgan family of potentials, ensuring Σ > 0 and κ² > 0 for stability.
• Identifies a third integral of motion of the form ZΣ¹ᐟ³, where Z is the z-amplitude, to describe integrable disk-crossing orbits in stable configurations.

Experimental results

Research questions

• RQ1 What is the correct criterion for vertical stability in Newtonian thin disk models, given the discontinuity in the gravitational field?
• RQ2 How does the standard epicyclic approximation need to be modified when applied to razor-thin disks?
• RQ3 Can finite thin disk models be constructed that satisfy both radial and vertical stability criteria?
• RQ4 What is the role of the surface mass density Σ in determining vertical stability for thin disks?
• RQ5 Do stable thin disk models admit a family of integrable disk-crossing orbits, and if so, what is the form of the third integral of motion?

Key findings

• The necessary and sufficient condition for vertical stability of equatorial circular orbits in thin disk models is Σ(R₀) > 0, which arises due to the discontinuity in the gravitational field.
• The radial stability criterion remains unchanged from thick disk models: the epicyclic frequency squared must be positive, i.e. κ²(R₀) > 0.
• For small enough initial vertical velocity |v₀z|, the vertically perturbed orbit remains bounded and oscillates periodically around the equatorial plane, with amplitude bounded by (v₀z)² / (2α_R₀).
• The effective potential Φ_eff(R, z) has a strict local minimum at (R₀, 0) when both Σ(R₀) > 0 and κ²(R₀) > 0, ensuring Liapunov stability under small perturbations.
• Any galactic model with a thin disk component admits a wide class of integrable disk-crossing orbits, approximately described by a third integral of motion of the form ZΣ¹ᐟ³.
• Finite thin disk models for nine galaxies were successfully constructed as superpositions of Morgan & Morgan potentials, satisfying both Σ > 0 and κ² > 0, thus qualifying as stable configurations in the first approximation.