The Shape of the Vertical Action Distribution Locates the Scatterers that Heat the Galactic Disc

Technical Summary: The Shape of the Vertical Action Distribution Locates the Scatterers that Heat the Galactic Disc

Problem Statement
The vertical velocity dispersion of stars in the Milky Way's disc increases with age, a phenomenon known as disc heating. While the existence of this heating is well-established, the identity and spatial distribution of the "scatterers" responsible remain unresolved. Two primary classes of models exist:

1. Volume-filling scatterers: Perturbations such as spiral structure, the Galactic bar, or dark matter subhaloes that act on a star throughout its entire orbit.
2. Midplane-confined scatterers: Perturbations, such as Giant Molecular Clouds (GMCs), that are confined to a thin layer around the Galactic midplane and only interact with stars during their vertical crossings.

Previous attempts to distinguish between these mechanisms using the age–velocity dispersion relation or the shape of the velocity ellipsoid have been inconclusive due to degeneracies between the scatterer abundance, mass, and the heating history. The paper argues that the vertical action ($J_z$), an adiabatic invariant, offers a superior diagnostic because it labels the orbit rather than the instantaneous velocity and is insensitive to the growth of the disc potential over time.

Methodology
The authors derive the theoretical distribution of vertical action for a coeval population of stars heated by scatterers with an arbitrary vertical height distribution.

• Theoretical Derivation:

  • The heating process is modeled as a diffusion in vertical action space using the Fokker–Planck equation.
  • The authors derive the diffusivity $D(J_z)$ as a function of the scatterer's height profile $q(z)$. They show that the diffusivity scales as a power law $D \propto J_z^b$, where the exponent $b$ is determined solely by the geometry of the scatterers.
  • Volume-filling case: If scatterers fill the volume, the diffusivity grows linearly with action ($b=1$), resulting in an exponential distribution $p(J_z) \propto \exp(-J_z/J_0)$.
  • Midplane-layer case: If scatterers are confined to a thin layer, the interaction time decreases as the star's crossing speed increases. This reduces the growth of diffusivity to $D \propto J_z^{1/2}$ ($b=1/2$), resulting in a "stretched exponential" distribution $p(J_z) \propto \exp[-(J_z/J_0)^{3/2}]$.
  • The general solution for a coeval population is $p(J_z) \propto \exp[-(J_z/J_0)^{2-b}]$, where the shape (tail exponent) depends only on geometry ($b$), while the scale ($J_0$) encodes the heating amplitude and history.

• Data and Fitting:

  • The model is fitted to a sample of 7,589 low-$\alpha$ red clump stars from the APOGEE DR14 survey, cross-matched with Gaia DR2 astrometry.
  • The sample is restricted to stars with effective radii $5 \le R_{\text{eff}} \le 10$ kpc and ages $2 \le \tau \le 8$ Gyr.
  • A critical component of the methodology is the rigorous treatment of the selection function, which preferentially misses high-action stars near the survey's magnitude limit. The authors apply a per-star selection function derived in Ting & Rix (2019) directly to the model likelihood.
  • The fitting procedure marginalizes over measurement errors in age, action, and radius using Gaussian quadrature. Crucially, the scale parameter $J_0$ is left free at each age (parameterized by four knots and a radial slope). This allows the measurement of the shape exponent $b$ without assuming a specific heating history or birth dispersion, effectively decoupling the geometry from the heating rate.

Key Results

1. Measured Exponent: The fit yields a diffusivity slope of $b = 0.51^{+0.06}_{-0.07}$.

   • This value is consistent with the prediction for a midplane-confined layer ($b=0.5$) but strongly inconsistent with the volume-filling prediction ($b=1$).
   • Synthetic tests confirm that the method can recover the correct exponent unbiasedly and separates the two geometries at a significance of $5.6\sigma$.
   • The result holds across different radial bins and age cuts, and is robust against variations in assumed action errors.

2. Effective Scatterer Mass:

   • Using the measured free scales ($J_0$) and the derived exponent ($b \approx 0.5$), the authors calculate the heating amplitude.
   • For the 2–4 Gyr age bin, comparing the heating amplitude to the present-day molecular surface density yields an effective scatterer mass of $2.7 \times 10^6 M_\odot$.
   • This value falls within the range ($1.4\text{--}3.4 \times 10^6 M_\odot$) derived from independent cloud catalogues and mass functions (e.g., Miville-Deschênes et al. 2017).
   • For older stars, the apparent effective mass increases if compared to the current gas density. However, when corrected for the declining star-formation history (and thus declining gas density) over the last several gigayears, the effective mass for all age bins converges to the range $1.9\text{--}2.8 \times 10^6 M_\odot$.

3. Heating History:

   • The free scales grow with age as $J_0 \propto \tau^{1.12}$. This is faster than the $\tau^{0.67}$ expected for a constant heating rate with a cold birth, indicating that the disc heated more strongly in the past.

Significance and Claims
The paper claims to resolve the long-standing debate regarding the location of Galactic disc scatterers by utilizing the shape of the vertical action distribution rather than just its mean or velocity dispersion.

• Localization of Scatterers: The measured exponent $b \approx 0.5$ provides direct evidence that the primary mechanism for vertical heating is localized to the Galactic midplane. This rules out volume-filling mechanisms (such as dark matter subhaloes, bending waves, or direct spiral perturbations) as the dominant source of diffusive heating. The authors estimate that any volume-filling component contributes less than 7% to the heating.
• Identification of Scatterers: The combination of the midplane geometry and the derived effective mass ($\sim 10^6 M_\odot$) identifies Giant Molecular Clouds (GMCs) as the scatterers. The mass is consistent with the upper end of the GMC mass function, and the geometry matches the thin layer of molecular gas.
• Resolution of Previous Discrepancies: The paper explains why previous studies (e.g., Binney & Lacey 1988) inferred $b=1$ (volume-filling). Those studies analyzed mixed-age populations or velocity dispersions without resolving the shape at fixed age. The authors demonstrate that mixing populations with different heating scales artificially flattens the tail of the distribution, mimicking a volume-filling geometry ($b \approx 1$) even if the true underlying geometry is a thin layer ($b=0.5$).
• Mechanism vs. Energy Source: The authors clarify that while the scattering occurs at the midplane (via clouds), the energy may originate from in-plane motions (e.g., driven by spiral structure) that are redirected vertically during cloud crossings. The exponent measures the location of the redirection, not the ultimate source of the energy.

In conclusion, the paper establishes that the Milky Way's disc is heated near the plane by an evolving population of objects with the mass of Giant Molecular Clouds, and that the shape of the vertical action distribution is a robust, model-independent diagnostic for the spatial distribution of scattering agents.