Effects of Resolution and Local Stability on Galactic Disks: 2. Halo Resolution and Softening on Bar Formation

This study uses N-body simulations to demonstrate that while dark matter halo resolution has a limited impact on bar formation in unstable disks, large gravitational softening suppresses bar growth by flattening the central density profile and hindering angular momentum transfer, while simultaneously exacerbating buckling instability through inhibited vertical heating.

The Gist

The Big Picture: Simulating the Universe's Dance Floor

Imagine you are trying to film a massive, chaotic dance party inside a giant, invisible ballroom (the Dark Matter Halo) where thousands of people (stars) are dancing. You want to see if a specific formation—a long, straight line of dancers (a Galactic Bar)—naturally forms out of the chaos.

To do this, scientists use supercomputers to run a "movie" of this dance. But here's the catch: computers can't track every single atom or person perfectly. They have to make shortcuts. This paper asks: Do these shortcuts ruin the movie? Specifically, does the way we count the invisible "ghost" dancers (Dark Matter) or how we smooth out their movements change whether the bar forms or not?

The Two Main "Shortcuts" (The Villains)

The paper focuses on two specific ways the computer simulation might cheat:

1. Low Resolution (The "Pixelated" Ghosts):

   • The Analogy: Imagine the invisible ballroom is filled with ghosts. In a high-quality simulation, there are millions of tiny, lightweight ghosts. In a low-quality one, there are only a few giant, heavy ghosts.
   • The Problem: If you have giant ghosts, they bump into the real dancers too hard. It's like a few bowling balls rolling through a crowd of people. They knock everyone around, heating up the dance floor and making it harder for the dancers to organize into a neat line.
   • The Finding: Surprisingly, having fewer, heavier ghosts didn't stop the bar from forming completely, but it did make the dance floor a bit messier and the bar slightly weaker.

2. Gravitational Softening (The "Blurry Lens"):

   • The Analogy: This is the big one. Imagine the computer has a rule: "If two ghosts get too close, we can't calculate the exact force between them. Instead, we just pretend they are a little fuzzy cloud." This "fuzziness" is called softening.
   • The Problem: In the center of the galaxy, things get very crowded. If your "blurry lens" is too big (large softening), the computer can't see the tight, intense interactions happening right in the middle. It's like trying to see a needle in a haystack while wearing thick winter goggles.
   • The Finding: This was the deal-breaker. If the "blur" was too big, the bar never formed. The center of the galaxy stayed round and smooth because the computer couldn't resolve the intense tug-of-war needed to stretch the stars into a bar.

The Key Discovery: The "Handshake" in the Center

The paper reveals a crucial mechanism for how bars form, which they call Angular Momentum Exchange.

• The Analogy: Think of the stars in the bar as a group of ice skaters spinning in a circle. To make the line longer and stronger, they need to push against the invisible ghosts (Dark Matter) in the center. It's like a handshake or a push-off. The stars push their spin onto the ghosts, and the ghosts push back, helping the stars stretch out into a bar.
• The Result: When the "blurry lens" (softening) was too big, this handshake couldn't happen. The computer couldn't calculate the push in the very center. Without that push, the stars couldn't stretch out, and the bar remained a tiny, weak oval or didn't form at all.

The Twist: Even if the galaxy was naturally unstable (like a wobbly tower ready to fall), a big "blur" could still stop the bar from growing. It's like having a wobbly tower, but if you glue the base to the floor with a thick, soft foam (the softening), the tower can't topple over into the shape you expect.

The "Buckling" Effect: When the Bar Breaks Its Back

Once a bar forms, it sometimes gets too strong and starts to wobble up and down, like a rug being kicked. This is called Buckling Instability. It turns a flat bar into a 3D "peanut" shape.

• The Analogy: Imagine a long, thin stick. If you push the ends together, it eventually snaps and bends upward.
• The Finding: The "blurry lens" made this worse. Because the computer couldn't see the center heating up properly (due to the blur), the center stayed too "cold" and thin. This made the bar much more likely to snap and buckle violently.
• The Consequence: In simulations with high softening, the bars didn't just fail to form; when they did, they broke apart or became peanut-shaped much faster than they should have.

What Does This Mean for Real Science?

The authors are basically telling other scientists: "Be careful with your settings!"

Many modern simulations of the universe use "blurry lenses" (large softening lengths) to save computer power. This paper warns that if you make the lens too blurry:

1. You might miss the formation of bars entirely (the "Missing Bar" problem).
2. You might make bars that break too easily.
3. You might get the wrong shape for the center of the galaxy.

The Recommendation: To get a realistic movie of a galaxy, you need to use a high-resolution camera (many small particles) and a sharp lens (small softening), especially for the very center of the galaxy. If you want to see how galaxies evolve over billions of years, you can't afford to have your "goggles" too thick.

Summary in One Sentence

If you simulate a galaxy with a "blurry" view of the center, you might accidentally stop the stars from forming a bar, or make that bar break apart, because the computer misses the crucial "push" needed to stretch the galaxy out.

Technical Summary

1. Problem Statement

Numerical artifacts in $N$-body simulations, specifically those arising from dark matter (DM) halo resolution (particle mass ratios) and gravitational softening lengths, have long been known to bias galaxy evolution outcomes. While previous studies established that low resolution causes spurious heating and artificial gas expansion, the specific impact of these numerical parameters on the formation and evolution of galactic bars and buckling instabilities remains a critical concern, particularly for modern cosmological simulations (e.g., NewHorizon, IllustrisTNG).

The paper addresses two main questions:

1. How does the ratio of DM particle mass to stellar particle mass ($m_{DM}/m_{\star}$) affect bar formation epochs and growth?
2. How does the gravitational softening length ($\epsilon_{DM}$) of the DM halo influence the central angular momentum exchange required for bar growth and the subsequent buckling instability?

2. Methodology

The authors conducted a controlled set of isolated disk-halo $N$-body simulations using the AREPO code.

• Initial Conditions:
  • Stellar Disk: Identical parameters across all models ($M_d = 5 \times 10^{10} M_{\odot}$, $R_d = 3$ kpc, $z_d = 0.3$ kpc, $N_{\star} = 5 \times 10^6$ particles). The softening length for stars was fixed at $\epsilon_{\star} = 0.03$ kpc.
  • Dark Matter Halo: Hernquist profiles with a fixed total mass ($M_{DM} = 1.14 \times 10^{12} M_{\odot}$).
  • Stability Variation: Two halo concentration parameters ($c=14$ and $c=16$) were used to create disks with different intrinsic stabilities (Toomre $Q$ values of 0.875 and 0.830, respectively). Lower $c$ (c14) results in a less stable, more bar-prone disk.
• Variable Parameters:
  • DM Resolution: Varied the number of DM particles ($N_{DM}$) to achieve mass ratios of $m_{DM}/m_{\star} = 10$ (high resolution) and $100$ (low resolution).
  • Softening Length ($\epsilon_{DM}$): Tested values of $0.03$kpc (standard),$0.60$kpc, and$0.96$ kpc. The larger values mimic common practices in cosmological simulations to mask noise.
• Simulation Duration: 4 Gyr with 400 snapshots.
• Analysis: Fourier decomposition ($F_m$) to track bar strength ($m=2$ mode), angular momentum transfer ($L_z$), and kinematic diagnostics ($\sigma_z$, $\sigma_R$, $\sigma_z/\sigma_R$) to identify buckling instabilities.

3. Key Results

A. Impact of DM Resolution ($m_{DM}/m_{\star}$)

• Bar Formation Epoch: Increasing DM resolution (lowering $m_{DM}/m_{\star}$ from 100 to 10) slightly delays bar formation.
  • In stable disks ($c=16$), the delay is $\sim0.9$ Gyr.
  • In unstable disks ($c=14$), the delay is minimal ($\sim0.1$ Gyr).
• Mechanism: Low resolution ($m_{DM}/m_{\star}=100$) introduces spurious heating via massive DM particles passing through the disk. This global heating stabilizes the disk slightly, delaying the onset of instability.
• Evolutionary Path: Once formed, bars in low-resolution models ($m_{DM}/m_{\star}=100$) show similar growth to high-resolution models but may suffer from gradual weakening later in evolution due to accumulated numerical noise.
• Conclusion: For highly unstable disks driven by strong self-instabilities, a mass ratio of $m_{DM}/m_{\star} \le 100$ is acceptable, but for stable disks, higher resolution ($m_{DM}/m_{\star} \le 10$) is necessary to accurately capture formation timing.

B. Impact of Gravitational Softening ($\epsilon_{DM}$)

The softening length has a dominant and detrimental effect on bar formation compared to resolution alone.

• Suppression of Bar Formation: Large softening lengths ($\epsilon_{DM} \ge 0.60$ kpc) flatten the central DM density profile (creating a "core" instead of a "cusp").
  • This flattening inhibits the transfer of angular momentum from the bar to the DM halo in the central region ($R < \epsilon_{DM}$).
  • In stable models ($c=16$) with $\epsilon_{DM} = 0.96$ kpc, no significant bar forms; only a weak inner oval appears.
  • In unstable models ($c=14$) with $\epsilon_{DM} = 0.96$ kpc, a bar forms due to high initial instability, but its growth is capped ($F_2 \approx 0.3$) because the central dynamical friction is unresolved.
• Mechanism: Bar growth relies on the seed bar exchanging angular momentum with the DM halo. If the softening length is too large, this exchange is artificially suppressed, preventing the bar from strengthening and lengthening.

C. Impact on Buckling Instability

• Triggering Stronger Buckling: Large softening lengths ($\epsilon_{DM} \ge 0.30$ kpc) lead to stronger buckling instabilities.
• Mechanism:
  1. Standard simulations show gradual vertical heating ($\sigma_z$) in the central region during bar growth, which stabilizes the bar against buckling.
  2. Large softening prevents this gradual central heating, leaving the central disk "cold" vertically.
  3. This creates a large anisotropy in velocity dispersion ($\sigma_z/\sigma_R$), triggering a sudden, violent buckling event that significantly weakens the bar strength.
• Observation: Models with $\epsilon_{DM} = 0.60$ kpc exhibited a sharp drop in bar strength ($F_2$) and a sudden spike in vertical velocity fluctuations, unlike the gradual evolution seen in models with $\epsilon_{DM} = 0.03$ kpc.

4. Significance and Recommendations

• The "Missing Bar" Problem: The study suggests that the lack of strong bars in some cosmological simulations (e.g., NewHorizon) may not be solely due to gas physics or merger history, but significantly due to excessive DM softening lengths and low DM resolution. These parameters suppress the central angular momentum exchange required for bar growth.
• Numerical Artifacts: The paper highlights that large softening lengths create a "double penalty": they flatten the central density (reducing stability) and suppress the dynamical friction needed for bar growth, while simultaneously inducing artificial buckling.
• Practical Recommendations:
  • Mass Ratio: For Milky Way-mass galaxies, use $m_{DM}/m_{\star} \le 10$.
  • Stellar Particles: Use $N_{\star} \ge 5 \times 10^6$ ($m_{\star} \le 10^4 M_{\odot}$).
  • Softening: Adopt $\epsilon_{DM} < 0.30$ kpc, particularly in the central regions. The authors suggest that standard cosmological softening values ($\sim 0.3 - 0.6$ kpc) are too large to resolve the critical central interactions driving bar formation and stability.
  • Adaptive Softening: Future simulations should consider adaptive softening lengths scaled to the galaxy's effective radius to properly resolve central DM-star interactions.

Conclusion

The paper demonstrates that while DM resolution affects the timing of bar formation, gravitational softening is the critical factor determining whether a bar can form and grow to maturity. Excessive softening prevents the necessary central angular momentum exchange and triggers premature, destructive buckling instabilities. Accurate modeling of galactic bars requires resolving the central DM halo with softening lengths significantly smaller than the bar's scale length.