The Milky Way's circular velocity curve measured using element abundance gradients

TL;DR

This study introduces a novel, data-driven method to empirically measure the Milky Way's circular velocity curve by exploiting correlations between element abundances and stellar orbits in the low-$\alpha$ disk. Centered on orbital torus imaging in the radial plane, the approach uses mean [Mg/Fe] gradients within narrow $L_z$ bins to locate the guiding-center radii and infer $v_c$ via $v_c = L_z/R_g$, without assuming a parametric Galactic potential. The authors report a Solar-radius circular velocity of $v_{c,\odot}=235.3^{+2.8}_{-3.7}$ km s$^{-1}$, and derive epicyclic and azimuthal frequencies, as well as Oort constants, with results broadly consistent with prior work, while also identifying features likely shaped by disequilibrium (bars/spirals). Validation on smooth and live galaxy simulations demonstrates the method's robustness and highlights how local dynamical perturbations can imprint distinct signatures in the inferred curves. Overall, the work provides a fully empirical, abundance-kinematic pathway to map the Milky Way's mass distribution and orbital structure, leveraging current and forthcoming spectroscopic surveys.

Abstract

Spectroscopic surveys now supply precise stellar label measurements such as element abundances for large samples of stars throughout the Milky Way. These element abundances are known to correlate with orbital actions or other dynamical invariants. We present a new data-driven method for empirically measuring the circular velocity curve of the Galaxy that uses element abundance gradients in the plane of radial kinematics. We use stellar surface abundances from the $\textit{APOGEE}$ survey combined with kinematic data from the $\textit{Gaia}$ mission. Our results confirm the ordered structure of the Milky Way disk in terms of average [Fe/H] and [Mg/Fe] abundance ratios, and suggest that $\langle$[Fe/H]$\rangle$ traces the radial position of stars in the disk, while $\langle$[Mg/Fe]$\rangle$ traces the orbital excursions around this radius. Our method uses the radial orbit structure in the Galaxy to enable an empirical measurement of the circular velocity curve, epicyclic and azimuthal frequencies, and kinematic gradients across the Milky Way disk. From these measurements, we infer a value of the circular velocity curve at the Solar radius of $v_{c,\odot} = 235.3^{+2.8}_{-3.7}$ km s$^{-1}$ using the most constraining abundance ratio, [Mg/Fe]. We also measure the radial and azimuthal frequencies for a circular orbit at the solar radius, $κ_{0,R_\odot}=36.9^{+0.8}_{-1.0}$ km s$^{-1}$ kpc$^{-1}$ and $Ω_{0,R_\odot}=28.5_{-0.1}^{+0.4}$ km s$^{-1}$ kpc$^{-1}$, respectively. These values lead to an estimate of the Oort constants of $A = 16.5^{+0.1}_{-0.1}$ km s$^{-1}$ kpc$^{-1}$ and $B=-11.9^{+0.1}_{-0.3}$ km s$^{-1}$ kpc$^{-1}$. We measure the radial acceleration at the Solar radius to be $(\frac{\partial Φ}{\partial R})_{\odot} = a_{R_\odot}=7.0^{+0.2}_{-0.1}$ pc Myr$^{-2}$.

Figures (14)

• Figure 1: Our parent sample of APOGEE-Gaia red giant branch stars in the low-$\alpha$ disk used in this work. The left panel shows a 2D histogram of the positions of the stars in our sample projected onto the Galactic plane (i.e., in Galactocentric [$x$, $y$] coordinates). The approximate position of the Sun is marked with the $\odot$ symbol. The middle panel shows the selection used to define the low-$\alpha$ sequence using a cut in element abundance space, i.e., the [Mg/Mn] vs. [Al/Fe] plane. This chemical cut was performed in order to restrict our sample to stars that are part of the low-$\alpha$ disk, and thus likely part of the Galaxy’s thin disk. The right panel shows the spectroscopic stellar parameters, effective temperature $T_{\mathrm{eff}}$ and surface gravity $\log~g$. In the middle and right panels, the full APOGEE–Gaia sample is shown as the faint background histogram.
• Figure 2: Our low-$\alpha$ sample projected in present day $R$--$z$ (side-on view). Each panel shows the sample binned, where each pixel represents the mean value of [Fe/H] (left) and [Mg/Fe] (right). $\langle$[Fe/H]$\rangle$ shows a gradient that is primarily dependent on Galactrocentric radius, $R$. Conversely, $\langle$[Mg/Fe]$\rangle$ shows a gradient that primarily changes with vertical position, $z$ (although there is some small dependence on $R$). This figure illustrates that the Milky Way low-$\alpha$ disk is well structured both chemically and kinematically for these element abundance ratios.
• Figure 3: Low-$\alpha$ sample in the $v_\phi$--$R$ plane. The left panel shows a 2D density histogram, the middle panel shows the data pixelated by average [Fe/H], and the right panel shows the data pixelated by average [Mg/Fe]. The left panel illustrates the kinematic structure in the Milky Way's disk; the jagged features (or ridges) are a result of features in the in-plane dynamics of stars, whereas the vertical lines around $R\approx7$-$8$ kpc are caused by the spatial selection function of the APOGEE survey. The middle panel shows how $\langle$[Fe/H]$\rangle$ traces angular momentum, $L_z = R\,v_\phi$, leading to stripes of constant $\langle$[Fe/H]$\rangle$ (e.g., dotted white line). The right panel shows how $\langle$[Mg/Fe]$\rangle$ traces the out-of-plane motions of stars in the low-$\alpha$ disk. Here, the ridges of constant $\langle$[Mg/Fe]$\rangle$ are qualitatively equal to the ones in density. The approximate location of the Sun is marked by the $\odot$ symbol.
• Figure 4: The parent low-$\alpha$ disk sample used in this work illustrated in the plane of radial kinematics ($R$--$v_R$), binned as a function of angular momentum, $L_z = R \, v_\phi$. In each panel, the data is pixelated, where each bin shows the mean value of [Mg/Fe] for all stars in each pixel. In the first panel (top left), there is a stripe of stars with $R\sim8$ kpc across all $v_R$, which appears due to the spatial selection function of the survey and the fact that more stars on higher non-circular orbits at their apocenter will be observed in bins closer to the inner Galaxy. In all bins the distribution of mono-abundance contours (i.e., the gradient with mean [Mg/Fe]) appears to trace a guitar-pic or arrow-head pattern, as expected for the epicyclic motion of stars on one side of the Milky Way disk Hunt2024. The shape of this guitar-pic structure however changes across different $L_z$ bins. At lower $L_z$, the distribution of the data is broader in $v_R$ and narrower in $R$; at larger $L_z$, the data show a broader distribution in $R$ and a narrower range in $v_R$. This change in shape is caused by the larger enclosed mass in the inner regions of the Galaxy. At larger radii, where the enclosed mass is lower, stars with the same or even smaller radial velocities reach larger radial excursions. Interestingly, we don't find this guitar-pic pattern for mono-abundance contours when examining the gradients with average [Fe/H] (see Figure \ref{['fig_app_lz_feh']}).
• Figure 5: This figure demonstrates that, for stars in a narrow slice of $L_z$, contours of constant element abundance ratio nearly delineate orbits in $\{R, v_R\}$ space, and trace the epicyclic excursions for stars with near-circular orbits in the Milky Way's low-$\alpha$ disk. It also shows how it is possible to empirically measure the circular velocity curve by finding the inflexion point in the element abundance trend. Left: Cartoon of the epicyclic motion of a star (red star symbol) in the Milky Way's disk on a near-circular orbit. Here, the solid gray line delineates the guiding-center radius, $R_g$, of a star orbiting around the Galactic center, and the dashed black line delineates the epicyclic motion of that star around its guiding-center radius. The difference between the star's observed position and its guiding-center radius, $\Delta R = R - R_g$, defines the magnitude of the radial excursion of that near-circular orbit. Center left: This panel shows seven orbits with different values of the radial action $J_R$ computed in a simple Milky Way mass model. For all of these orbits, the value of the other actions are set such that the orbits have zero vertical action, $J_z=0$, and a constant azimuthal action set to the solar value, $J_\phi = J_{\phi, \odot}$. Orbits with lower orbital action trace smaller trajectories (darker lines), are more elliptical, and reach smaller maximum excursions from the guiding-center radius, $R_g$. We note that this is our only usage of a parameterized Galactic mass model in this work, and this is only for demonstration purposes. Center right: We paint element abundances onto the simulated stars with a linear dependence on radial action $J_R$ with a slope similar to that of [Mg/Fe] (see the right panel). The filled contours then show curves of constant mean element abundance ratio for a simulated population of orbits in the radial phase space, going from smaller average [Mg/Fe] in the center to higher values on the outskirts (similar to what is seen for Milky Way data in Figure \ref{['fig_abun_lz_bins']}). Right: Milky Way APOGEE-Gaia data for low-$\alpha$ disk stars in a narrow slice of $|v_R|<10$ km s$^{-1}$ around the Sun ($1900 < L_z< 2050$ kpc km s$^{-1}$), and with $|z|<0.1$ kpc, $|v_z|<10$ km s$^{-1}$. Here, we show a 2D density distribution of the data and its dependence of $\langle$[Mg/Fe]$\rangle$ with $R$, overlaying the running median as a dashed red line. As can be seen from the figure, for stars in a small bin of $L_z$, the relationship between average chemical abundance with $R$ in a thin slice of $v_R$ around $v_R\approx0$ leads to an inflexion point in the mean abundance distribution that aligns with the guiding-center radius for stars in that bin (the vertical shaded line). Since $v_c = L_z/R_g$, this approach provides a way of empirically measuring the Milky Way's circular velocity curve using element abundance gradients.