Uncover 3D Dark Matter Distribution of the Milky Way by an Empirical Triaxial Orbit-Superposition Model: Method Validation

Abstract

We introduce a novel dynamical model, named empirical triaxial orbit-superposition model, for the Milky Way halo. This model relies on minimal physical assumptions that the system is stationary, meaning the distribution function in 6D phase-space does not change when the stars orbiting in the correct gravitational potential. We validate our method by applying it to mock datasets that mimic the observations of the Milky Way halo from LAMOST + Gaia with stars' 3D position and 3D velocity observed. By removing the stellar disk and substructures, correcting the selection function, we obtain a sample of smooth halo stars considered as stationary and complete. We construct a gravitational potential including a highly flexible triaxial dark matter halo with adaptable parameters. Within each specified gravitational potential, we integrate orbits of these halo stars, and build a model by superposing the orbits together taking the weights of stars derived from the selection function correction. The goodness of the models are evaluated by comparing the density distributions as well as 3D velocity distributions numerically represented in the model to that in the data. The shape and radial density distribution of the underlying dark matter halo can be constrained well simultaneously. We apply it to three mock galaxies with different intrinsic shapes of their dark matter halos and achieved accurate recovery of the 3D dark matter density distributions for all.

Figures (13)

• Figure 1: The shapes of DM halos in Auriga 23, 5, 12, measured in the coordinate system where Z axis is perpendicular to the stellar disk, X and Y are the long and short axis of the DM halo in the disk plane. The DM halos of Auriga 23 and 5 are oblate aligned with the stellar disk ($p_{\rm DM}>q_{\rm DM}$) and vary little with radius. The DM shape of Auriga 12 varies from oblate aligned with the stellar disk ($p_{\rm DM} \sim 1$ and $p_{\rm DM}>q_{\rm DM}$) in the inner regions to be vertically aligned with the disk ($p_{\rm DM}<q_{\rm DM}$ and $q_{\rm DM} \sim 1$) at $r\gtrsim 20$ kpc.
• Figure 2: Left: MW observations from LAMOST + Gaia, and Right: mock data created for Auriga 23. Each dot represents one star/particle colored by weight obtained from selection function correction. There are about 15,000 k-giants in the final sample of smooth halo of the Milky Way. The mock data of Auriga 23 includes about 20,000 particles, which generally match the spatial distribution of the MW sample.
• Figure 3: Mock data for Auriga 23. The blue dots are disk stars that we excluded, i.e., stars with $\lambda_z>0.8$ or stars with $\lambda_z>0.5$ and $Z/Z_{\odot} > 0.5$. The black dots are taken as halo stars and kept in our sample. We have divided the data into $7 \times 6$ bins in $r_{\rm gc}$ versus $\theta$ with the intervals of $r_{\rm gc}=[5,8,12,16,22,30,40,50]$ kpc, $\theta$=[0,10,20,30,40,55,90$^\circ$], here we only show three columns to illustrate the data.
• Figure 4: Comparison of stellar density distribution from data and model for Auriga 23. The top panel from left to right are that constructed by observational data, and several models with different DM axis ratios of $q_{\rm DM}$ and $p_{\rm DM}$. The colors represent the number density as indicated by the colorbar, the dashed curves are iso-density contours. The second row are the uncertainty of the data, and model residuals. The data are well matched by the model with DM $q_{DM} = 0.64$ and $p_{DM} = 1$ which are well consistent with the ground truth. The stellar density distribution constructed by models are either too round or too flat if the shape of underlying DM changes. The stellar density distribution strongly constrains the 3D shape of the underlying DM halo.
• Figure 5: Comparison of velocity distributions from data and model for Auriga 23. The velocity distribution $v_r$, $v_\phi$, and $v_{\theta}$ are calculated in $7\times 6$ bins in $r$ versus $\theta$, but we only show three columns here as labeled. In each panel, the gray solid and dashed curves are the velocity distribution and uncertainty of observational data. The red solid curves are from the best-fitting model, blue and magenta dashed curves are models with different radial distribution of the underlying DM mass. The velocity distributions, especially $v_r$, strongly constrain the underlying DM radial density distribution. The best-fitting model (red solid) well reproduces the velocity distributions in all three components: $v_r$, $v_{\phi}$, and $v_{\theta}$.