Mean Mass Density near the Sun from the Divergence Theorem and Pulsar Accelerations

TL;DR

This work introduces a non-parametric framework to map the Sun-centered mass distribution using pulsar line-of-sight accelerations via Gauss's law, enabling the mean density $\bar{\rho}(r)$ and, with a baryon model, the mean dark matter density $\bar{\rho}_{DM}(r)$ as a function of radius. The method eschews dynamical-equilibrium assumptions, instead leveraging interpolation and spherical-harmonic expansions to extract density asymmetries about the Galactic midplane. Current data yield results broadly compatible with kinematic models up to roughly 1–1.5 kpc but remain limited by sparsity and uncertainties; nevertheless, the framework provides a principled avenue to probe dark matter distribution and possible substructure near the Sun as more pulsar accelerations become available. The approach also outlines extensions to detect density asymmetries and dark subhalos, potentially offering new constraints on the local dark matter landscape and its deviations from axisymmetry or smoothness.

Abstract

We introduce a new, non-parametric method for estimating the mass enclosed within a sphere of arbitrary radius centered on the Sun. The method is based on the divergence theorem as applied to measurements of the line-of-sight accelerations of millisecond pulsars. We describe a procedure for inferring the mean mass density within a sphere of a given radius centered on the Sun and find results that are consistent with previous analyses. When combined with a model for the distribution of baryons, this provides the mean mass density of dark matter as a function of distance from the Sun, rather than a single value as is typically reported by kinematic studies. However, with the present pulsar data, the method cannot unambiguously measure a signal from the local distribution of dark matter at this time; such a measurement is expected to soon become possible as the amount of pulsar acceleration data grows and its precision improves. We derive an extension of the well-known shell theorem to a spherical-harmonics expansion of the density and potential, and use the result to obtain estimates for density asymmetries with respect to the Galactic midplane from the observed acceleration data. The predicted asymmetries do not follow the observed distribution of MW disk stars or gas; this can potentially be explained by a non-uniform distribution of dark matter in the Solar neighborhood.

Figures (11)

• Figure 1: Heliocentric distances of the pulsars in our sample. The black line gives a kernel density estimate of the blue histogram. Inside the dashed red line (1.1 kpc), there is sufficient density of pulsars to obtain a reasonable approximation of the overall acceleration field at a given point. Outside of this distance, the density of the data drops below 1 source per cubic kpc, and the sparseness of the data makes it difficult to effectively interpolate between datapoints.
• Figure 2: Performance of various interpolation algorithms in recovering the true $\bar{\rho}(z)$ profile from randomly shuffled data. Smaller values of $\log \mathcal{G}$ correspond to better recovery of the actual underlying accelerations. The distribution of each algorithm is a kernel density estimate over 100 Monte Carlo samples of the observed pulsar data. Top: Nearest neighbor interpolation performs the best out of these algorithms, so it is used for the remainder of this paper. Bottom: The inclusion of an extra point at the location of the Sun with $a_r = 0$ significantly improves the ability of the interpolation procedure to recover the underlying mean density profile.
• Figure 3: Examples of the line-of-sight acceleration field used in our calculation of the mean volume density. The accelerations shown here are for sphere with radius $r=0.75$ kpc centered on the Sun, calculated using the GalaMilkyWayPotential2022 model. We show the interpolated on-sky approximation of the acceleration field for different numbers of sources distributed randomly within a cube with side length 2 kpc centered on the Sun. Although the integrand only vaguely resembles the true acceleration field for 50 sources, we find that the integral is still accurate to within 10% on average. With 1000 sources, the observed acceleration distribution is similar to ground truth, and the integral is accurate to within 1%.
• Figure 4: Mean volume and surface density as a function of distance from the Sun, inferred from the pulsar data (blue, with $1\sigma$ uncertainty shaded region). A reference density model fit to observed MW data from the gala Python package is shown in red. The inferred volume and surface densities agree with the kinematic model until about 1.25 kpc from the Sun, when both densities become larger than the simple kinematic predictions.
• Figure 5: Power law fits to the GalaMWPotential2022 model and the observed pulsar data. The pulsar data is fit by a steeper slope, potentially implying a disk that is more vertically compact than in the kinematic model.