Why Do Some Galaxies Lack a Spiral Bar?

Spiral galaxies often display a central “bar”, but not all of them have it. Astronomers have found that the bar’s presence depends on two key factors in computer models: the softening parameter (ε) and the disc mass fraction (md). The ε parameter controls how softened gravitational forces are between simulated particles; values that are too small cause “numerical heating”, meaning particles accelerate excessively at the disc’s centre. This creates chaotic orbits and prevents bar formation. On the other hand, increasing md usually delays bar appearance, but only if ε is moderate (≥10 pc). If ε is very small, even a more massive disc can’t spawn a bar because numerical heating dominates. Experiments with millions of particles (N≈10⁷) show that with ε ≤5 pc the bar is almost always suppressed, regardless of md. Another useful clue is the disc’s vertical acceleration profile: when this profile spikes, it signals numerical heating interfering. In short, the galactic bar results from a delicate balance between disc mass and the choice of ε in simulations, and understanding this helps explain the variety of shapes seen in real galaxies.

N‑body simulations are the primary tool for studying galactic dynamics. In these simulations, each particle represents a portion of matter (stars, gas, or dark matter) and interacts gravitationally with the others. The softening parameter ε is introduced to avoid numerical divergences when two particles get too close: the gravitational force is “smoothed” over a scale ε, making the potential more regular. However, if ε is too small relative to the simulation’s spatial resolution, short‑range interactions become overly intense, generating artificial accelerations (numerical heating). This phenomenon is especially evident in high‑resolution models (N≈10⁷), where particles have very low masses and are thus more sensitive to variations in ε. The result is a population of near‑circular orbits at the disc centre, which do not support the growth of a bar‑like structure.

The disc mass fraction md influences dynamical stability: a more massive disc increases self‑gravity, favoring bar instability. The theoretical dependence is often expressed as bar strength ∝ md⁻¹, but simulations show that this dependence is modulated by ε. When ε is moderate (≥10 pc), increasing md reduces the drift of the vertical acceleration profile without significantly altering bar growth. Conversely, for ε ≤ 5 pc, numerical heating dominates, suppressing the bar even in very massive discs.

An effective diagnostic is the disc’s vertical acceleration profile a_z(z). In the absence of numerical heating, a_z(z) remains stable and follows the mass distribution predicted by the equilibrium model. A sudden spike or exponential growth of a_z signals that particles are receiving non‑physical energy kicks, typical of numerical heating. Monitoring this profile allows real‑time identification of overly small ε, enabling corrective actions such as increasing ε or reducing the time step to keep the simulation physically reliable.

Finally, choosing ε must balance two needs: minimizing numerical heating while preserving the simulation’s ability to resolve scales of interest, such as bar width (often ~1‑2 kpc). Advanced techniques, like adaptive softening (ε varying with local density), are emerging to optimize this trade‑off, allowing more precise study of bar formation and evolution within a broader cosmological context.