Badminton Birdie-Like Aerodynamic Alignment of Drifting Dust Grains by Subsonic Gaseous Flows in Protoplanetary Disks

TL;DR

The paper addresses how protoplanetary disk grains can become toroidally aligned and effectively prolate under subsonic gas-dust drift. It develops a badminton birdie-like aerodynamic alignment mechanism using simple double-sphere and spheroid grain models to derive flow-induced restoring torques in Epstein drag, yielding damped oscillations that drive alignment within sub-orbital timescales. The authors apply the framework to a simple disk velocity field, showing that alignment depends on grain shape, center-of-mass offset, and Stokes number, with prolate and oblate grains able to align along the gas flow and produce azimuthal polarization patterns consistent with observations like HL Tau. The mechanism offers a testable link between grain geometry, aerodynamic flow, and disk polarization, providing a pathway to map subsonic dust-gas dynamics via polarized millimeter emission while highlighting caveats and directions for modeling complex disk structures and grain geometries.

Abstract

Recent (sub)millimeter polarization observations of protoplanetary disks reveal toroidally aligned, effectively prolate dust grains large enough (at least ~100 $μ$m) to efficiently scatter millimeter light. The alignment mechanism for these grains remains unclear. We explore the possibility that gas drag aligns grains through gas-dust relative motion when the grain's center of mass is offset from its geometric center, analogous to a badminton birdie's alignment in flight. A simple grain model of two non-identical spheres illustrates how a grain undergoes damped oscillations from flow-induced restoring torques which align its geometric center in the flow direction relative to its center of mass. Assuming specular reflection and subsonic flow, we derive an analytical equation of motion for spheroids where the center of mass can be shifted away from the spheroid's geometric center. We show that a prolate or an oblate grain can be aligned with the long axis parallel to the gas flow when the center of mass is shifted along that axis. Both scenarios can explain the required effectively prolate grains inferred from observations. Application to a simple disk model shows that the alignment timescales are shorter than or comparable to the orbital time. The grain alignment direction in a disk depends on the disk (sub-)structure and grain Stokes number (St) with azimuthal alignment for large St grains in sub-Keplerian smooth gas disks and for small St grains near the gas pressure extrema, such as rings and gaps.

Figures (12)

• Figure 1: Schematic of the double sphere model. The dust grain is composed of two spheres, labeled "black" (circle with a dark shade) and "white" (circle without a shade), with radii $\varsigma_{\bullet}$ and $\varsigma_{\circ}$, respectively. Point $O$ is the center of mass of the grain. The radius vectors from the center of mass to the center of each sphere are $\bm{r}_{\bullet}$ and $\bm{r}_{\circ}$, respectively. The distance between the centers of the spheres is $l$ denoted by a dashed line. $\bm{A}$ is the gas velocity relative to the center of mass of the grain, i.e., the aerodynamic flow from the perspective of the grain, and is along $\bm{\hat{e}}_{3}$. The spin of the grain is along $\bm{\hat{e}}_{2}$. $\theta$ is the angle of $\bm{r}_{\circ}$ from $\bm{\hat{e}}_{3}$ in the counter-clockwise direction in this figure. Since alignment occurs at $\theta=0$ when $m_{\bullet}/m_{\circ} > \varsigma_{\bullet}^{2} / \varsigma_{\circ}^{2}$, the black sphere has a larger mass and is smaller than the white sphere.
• Figure 2: A schematic of how the orientation of the grain relates to the potential well created by the existence of aerodynamic flow. Top panel: the orientation of an example double-sphere grain model that satisfies $m_{\bullet}/m_{\circ} > \varsigma_{\bullet}^{2} / \varsigma_{\circ}^{2}$ like in Fig. \ref{['fig:doublesphere_schem']} The heavier sphere is colored gray, while the less massive sphere is white. The green lines with arrows denote the direction of the gas flow. The circular arrow depicts the direction of the flow-induced torque, which always tries to align the grain and resists any angular displacement. As such, the flow-induced torque is a "restoring" torque. When aligned, the less massive sphere follows the direction of the flow, analogous to how the heavier head of the badminton birdie leads the less massive tail against a headwind. Bottom panel: the energy potential $U$ as a function of $\theta$. The depth of the potential well is $2P$. Alignment occurs at $\theta=0$ where $U$ is minimal.
• Figure 3: Dimensionless factors for the double-sphere model as a function of $\epsilon$ (ratio of radii between the two spheres) and $\kappa$ (ratio of material density). Panel a: the damping time dimensionless factor, $\Breve{t}_{d,d}$. Panel b: the oscillation time dimensionless factor, $\Breve{t}_{o,d}$. As $\epsilon\rightarrow1$ and $\kappa\rightarrow1$, $\Breve{t}_{o,d} \rightarrow \infty$ meaning there is no oscillation when the two spheres are equivalent.
• Figure 4: The phase portrait (trajectory map) of the double-sphere grain under the presence of gas flow $\bm{A}$. The horizontal axis is $\theta$, while the vertical axis is the angular velocity $\omega$ normalized by the oscillation frequency $\omega_{o}$. The black circles correspond to stable equilibrium points that serve as the attractors for alignment. The plus signs are the unstable equilibrium points. The blue curve is the separatrix that separates the bounded and unbounded regions. Regions inside the separatrix are bounded and oscillate around the attractors.
• Figure 5: Schematic of a prolate and an oblate where the center of mass is shifted along the axis of symmetry. We use ellipses to represent the cross-sections of the two spheroids in the $e_{1}e_{3}$-plane. The quantity $a$ is the length from the spheroid center along the axis perpendicular to the axis of symmetry, while $c$ is the length from the center along the axis of symmetry. For the prolate, the axis of symmetry is the long axis ($c > a$), while for the oblate, the axis of symmetry is the short axis ($c < a$). $\bm{\hat{e}}_{1}$, $\bm{\hat{e}}_{2}$, and $\bm{\hat{e}}_{3}$ are lab frame unit vectors where $\bm{\hat{e}}_{2}$ points out of the page. $\bm{\hat{b}}_{1}$, $\bm{\hat{b}}_{2}$, and $\bm{\hat{b}}_{3}$ are body frame unit vectors, where $\bm{\hat{b}}_{2}$ also points out of the page and $\bm{\hat{b}}_{3}$ is along the axis of symmetry. The offset of the center of the spheroid from the center of mass $O$ is $\bm{g}$. Since the center of mass is shifted along the axis of symmetry, $\bm{g} \parallel \bm{\hat{b}}_{3}$. The green arrow is the direction of the gas flow $\bm{A}$.