[Paper Review] Detecting the growth of structures in Pure Stellar Disk Models

Diego Enríquez, I. Puerari+1 more|arXiv (Cornell University)|May 23, 2017

Stellar, planetary, and galactic studies1 references3 citations

TL;DR

This study uses 3D N-body simulations of pure stellar disks to investigate the growth and evolution of spiral and bar structures via 1D and 2D Fourier transforms. It reveals that galaxy morphology arises from the superposition of multiple coexisting spiral structures with distinct pitch angles, number of arms, and pattern speeds, while barred models show that approximately one-third of the disk mass is trapped in orbits near Lagrangian points $L_4$ and $L_5$.

ABSTRACT

We performed a series of 3D N-body simulations where the initial conditions were chosen to get two sets of models; unbarred and barred ones. In this work, we analyze the growth of spirals and bar structures using 1D, and 2D Fourier Transforms FT methods. Spectrograms and diagrams of the amplitude of the Fourier coefficients as a function of time, radius and pitch angle show that the general morphology, of our modeled galaxies, is due to the superposition of structures which have different values of pitch angle and number of arms. Also, we made in barred models a geometric classification of orbits from the bar reference frame showing that the barred potential and the Lagrangian points $L_4$ and $L_5$ catch approximately one-third of the total disk mass.

Motivation & Objective

• To investigate the formation and evolution of non-axisymmetric structures—specifically spirals and bars—in isolated, pure stellar disk systems.
• To determine the role of the Toomre $Q$ parameter and velocity dispersion in triggering spiral structure growth.
• To examine how multiple spiral modes with varying pitch angles, number of arms, and pattern speeds coexist and evolve over time.
• To analyze orbital dynamics in barred models, particularly classifying orbits relative to the bar and Lagrangian points.
• To assess the impact of numerical noise due to particle count on structural development, especially bar formation timescale.

Proposed method

• Conducted fully self-consistent 3D N-body simulations with 1.2 to 8 million particles using Kuijken-Dubinski initial conditions.
• Applied 1D Fourier Transform (FT1D) to analyze time-dependent amplitude, pitch angle, and mode number ($m$) of spiral structures at different radii.
• Used 2D Fourier Transform (FT2D) to map the amplitude of Fourier coefficients as a function of time, radius, and pitch angle, revealing coexisting structures.
• Defined and applied a geometric orbital classification in the bar’s rotating frame: compact, bar-aligned, and Lagrangian-point-trapped orbits.
• Tracked the evolution of pattern speeds and compared them to resonant curves ($\Omega \pm \kappa/m$) to assess confinement of structures.
• Varied key parameters: disk radial velocity dispersion ($\sigma_R$), scale height ($z_d$), particle count ($N$), and disk mass ($M_D$) to study their effects on structure development.

Experimental results

Research questions

• RQ1 How do spiral structures grow in pure stellar disks, and what role does the local $Q$ parameter play in their formation?
• RQ2 To what extent do multiple spiral structures with different pitch angles, number of arms, and pattern speeds coexist and interact in the same disk region?
• RQ3 What is the orbital dynamics of stars in barred stellar disks, particularly in relation to Lagrangian points $L_4$ and $L_5$?
• RQ4 How does numerical noise from limited particle counts affect the timing and development of bars and spirals?
• RQ5 Can the observed morphology of simulated galaxies be explained as a superposition of multiple, distinct, evolving spiral modes?

Key findings

• Spiral structures emerge in the intermediate disk region where the Toomre $Q$ parameter is minimal and grow outward more intensely than inward due to increasing $Q$ toward the center.
• The 2D Fourier transform reveals that the overall morphology of simulated galaxies results from the superposition of multiple spiral structures with different pitch angles, mode numbers ($m$), and pattern speeds.
• Multiple spiral structures with varying $m$ and $p$ (frequency) can coexist in the same radial region at the same time, indicating complex, dynamic interactions.
• Pattern speeds of amplified structures are confined between the $\Omega \pm \kappa/m$ resonant curves, confirming their dynamical stability and confinement.
• Approximately one-third of the total disk mass in barred models is found in orbits trapped near the Lagrangian points $L_4$ and $L_5$, with 15–16% in bar-like orbits and 10% in loop-like orbits.
• Orbital morphology evolves over time, with particles transitioning between compact, bar-aligned, and Lagrangian-point-trapped types, indicating dynamic redistribution of mass.