On the radial velocity wave in the Galactic disk

TL;DR

The paper develops a compact analytic framework based on linear perturbation theory to predict the mean radial velocity response $\,\overline{v}_R\,$ in the Galactic disk to weak perturbations. The key result is a closed-form expression for $\overline{v}_R(\varphi,J_\varphi,t)$, featuring an effective frequency $\omega^{\mathrm{eff}}_{m,\pm}$ and a Dehnen drift term that induces phase shifts between dynamically hot and cold populations. Applying the theory to three perturbation archetypes shows that distant kicks and rigid bars struggle to reproduce the observed multi-component $J_\varphi$-$\overline{v}_R$ signal, while a transient spiral perturbation can. Fitting the spiral model to Gaia DR3 data via MCMC yields a set of parameters that reproduce the data well and are corroborated by test-particle simulations, though caveats about self-gravity, gas, and sample selection remain. Overall, the work demonstrates that linear perturbation theory can capture the essential physics of the $J_\varphi$-$\overline{v}_R$ wave and points to a transient spiral as the leading explanatory mechanism, while highlighting the need for more complete dynamical modeling.

Abstract

Stars in the Galactic disk have mean radial velocities $\overline{v}_R$ that oscillate as a function of angular momentum $J_\varphi$. This `$J_\varphi$-${\overline{v}}_R$ wave' signal also exhibits a systematic phase shift when stars are binned by their dynamical temperatures. However, the origin of the wave is unknown. Here we use linear perturbation theory to derive a simple analytic formula for the $J_\varphi$-$\overline{v}_R$ signal that depends on the equilibrium properties of the Galaxy and the history of recent perturbations to it. The formula naturally explains the phase shift, but also predicts that different classes of perturbation should drive $J_\varphi$-$\overline{v}_R$ signals with very different morphologies. Ignoring the self-gravity of disk fluctuations, it suggests that neither a distant tidal kick (e.g., from the Sgr dwarf) nor a rigidly-rotating Galactic bar can produce a qualitatively correct $J_\varphi$-$\overline{v}_R$ wave signal. However, short-lived spiral arms can, and by performing an MCMC fit we identify a spiral perturbation that drives a $J_\varphi$-${\overline{v}}_R$ signal in reasonable agreement with the data. We verify the analytic formula with test particle simulations, finding it to be highly accurate when applied to dynamically cold stellar populations. More work is needed to deal with hotter orbits, and to incorporate the fluctuations' self-gravity and the role of interstellar gas.

Figures (9)

• Figure 1: In panel (a), the black line shows the observed $J_\varphi$-$\overline{v}_R$ signal in the Galactic disk for stars in an azimuthal wedge centered on the Sun ($\vert \varphi - \varphi_\odot\vert < 0.2$ rad) and on dynamically cold orbits ($J_z < 3$ kpc km s$^{-1}$), following cao2024radial. The red line shows our best analytic solution assuming a transient spiral perturbation (Eq. \ref{['eqn:vR_spiral']}); the blue line shows the result of a test particle simulation with the same parameters. The gold dashed line is a ten-parameter fit to the data following Eq. \ref{['eqn:friske']}. See §\ref{['sec:fitting']} for details.
• Figure 2: Radial velocity signal for a dynamically very cold ($\epsilon=0.01$) population of $N=10^7$ test particles subjected to the impulsive quadrupolar perturbation \ref{['eqn:deltaphi_impulse']}. Colors show contours of the mean radial velocity $\overline{v}_R$ in real space $(X,Y)=(R\cos\varphi, R \sin \varphi)$ at three different times $t/T_8$, extracted from test particle simulations. The dashed lines in these panels show the loci of the curves $\varphi = (\Omega-\kappa/2) t$ respectively, where the frequencies are evaluated at the guiding radius $R_\mathrm{g}=R$.
• Figure 3: The $J_\varphi$-$\overline{v}_R$ signal (averaged over $\vert \varphi \vert < 0.2$) generated by the quadrupolar impulsive kick \ref{['eqn:deltaphi_impulse']}, measured at time $t=5.2 T_8$, for various values of $\eta$ and $\epsilon$. Colored lines show the results of $N=5\times 10^7$ test particle simulations, while black lines show the result of analytic theory (Eq. \ref{['eqn:vR_analytic_impulse']}).
• Figure 4: Similar to Fig. \ref{['fig:linearity_test']}, except using the rigidly-rotating bar potential \ref{['eqn:phi_bar']}, with $R_\mathrm{b}=2R_0$, $R_\mathrm{CR}=4R_0$, and $s=1$ (instantaneous switch on). We run the simulation to $t=2T_8$. The OLR location $\omega_{2,-}^\mathrm{eff} = 2\Omega_{\mathrm{p}}$ is shown with a vertical dot-dashed line in each panel.
• Figure 5: We plot the formula \ref{['eqn:vR_spiral']}, averaged over the azimuthal range $\vert \varphi \vert < 0.2$. The fiducial parameters for rows (a)-(f) are $m=2$, $\varphi_0=0$, $R_\mathrm{CR}=R_\mathrm{ref}=8R_0$, $\alpha = 15^\circ$, $\beta=0.35$, $\tau/T_8=0.3$ and $t/T_8=1.0$. In row (g) we perform the same calculation except for a shearing spiral with shear rate $\Gamma$ (see Eq. \ref{['eqn:shearing_wave']}). The $\Gamma=0$ case is equivalent to the fiducial case in the rows above. In each row we vary only a single parameter compared to the fiducial model, as indicated in the panel labels, and in each panel we plot two curves, one for $\epsilon=0.01$ (blue) and another for $\epsilon = 0.1$ (red). The fiducial panel is reproduced once in each row with a bold frame (but note each row has a unique vertical axis). Resonance locations for circular orbits are shown with black vertical lines.