Accurate masses for dispersion-supported galaxies

Abstract: We derive an accurate mass estimator for dispersion-supported stellar systems and demonstrate its validity by analyzing resolved line-of-sight velocity data for globular clusters, dwarf galaxies, and elliptical galaxies. Specifically, by manipulating the spherical Jeans equation we show that the dynamical mass enclosed within the 3D deprojected half-light radius r_1/2 can be determined with only mild assumptions about the spatial variation of the stellar velocity dispersion anisotropy. We find M_1/2 = 3 \sigma_los^2 r_1/2 / G ~ 4 \sigma_los^2 R_eff / G, where \sigma_los^2 is the luminosity-weighted square of the line-of-sight velocity dispersion and R_eff is the 2D projected half-light radius. While deceptively familiar in form, this formula is not the virial theorem, which cannot be used to determine accurate masses unless the radial profile of the total mass is known a priori. We utilize this finding to show that all of the Milky Way dwarf spheroidal galaxies (MW dSphs) are consistent with having formed within a halo of mass approximately 3 x 10^9 M_sun in Lambda CDM cosmology. The faintest MW dSphs seem to have formed in dark matter halos that are at least as massive as those of the brightest MW dSphs, despite the almost five orders of magnitude spread in luminosity. We expand our analysis to the full range of observed dispersion-supported stellar systems and examine their I-band mass-to-light ratios (M/L). The M/L vs. M_1/2 relation for dispersion-supported galaxies follows a U-shape, with a broad minimum near M/L ~ 3 that spans dwarf elliptical galaxies to normal ellipticals, a steep rise to M/L ~ 3,200 for ultra-faint dSphs, and a more shallow rise to M/L ~ 800 for galaxy cluster spheroids.

• The paper presents a novel mass estimator that accurately calculates the mass within the half-light radius using the spherical Jeans equation.
• It validates the estimator across diverse dispersion-supported systems, from dwarf spheroidals to elliptical galaxies.
• The findings challenge traditional dark matter models by revealing uniform halo masses in Milky Way dSphs and constraining galaxy formation feedback processes.

A Mass Estimator for Dispersion-Supported Galaxies

The paper, "Accurate masses for dispersion-supported galaxies," explores a significant challenge in astrophysics: determining the mass distribution of dispersion-supported stellar systems. These systems include a variety of galactic structures, such as globular clusters and dwarf galaxies, which typically lack the rotational support to simplify dynamic analyses. The researchers propose an innovative mass estimator that accurately approximates the mass within the half-light radius, $r_{1/2}$, a critical radius for such analyses.

Fundamental Derivation and Analysis

By manipulating the spherical Jeans equation, the researchers derive a formula succinctly given by:

$$M_{1/2} = \frac{3}{G} \sigma_{\text{los}}^2 r_{1/2} \approx \frac{4}{G} \sigma_{\text{los}}^2 R_e,$$

where $\sigma_{\text{los}}$ is the luminosity-weighted line-of-sight velocity dispersion, $R_e$ is the 2D projected half-light radius, and $G$ is the gravitational constant. This equation is a milestone in that it provides a robust estimate of mass with minimal dependence on the stellar velocity anisotropy parameters $\beta$, thus overcoming one of the chief uncertainties in mass estimation for these systems. Unlike the classical virial theorem, this formula does not require prior knowledge of the mass distribution's radial profile, making it fairly generalizable to a broad class of stellar systems.

Model Validation and Findings

The paper thoroughly tests the formula using empirical data from a diverse set of dispersion-supported systems ranging from Local Group dwarf spheroidals (dSphs) to elliptical galaxies and even galaxies within clusters. Remarkably, the formula yields consistent mass estimates within the $r_{1/2}$ across several orders of magnitude in luminosity and mass scales, suggesting its broad applicability.

For the Milky Way’s dSphs, the analysis reveals an intriguing result: most of these galaxies can be considered to have formed in dark matter halos with a characteristic mass of approximately $3 \times 10^9 \, M_\odot$. This assertion challenges simplistic star formation models that might predict a correlation between galaxy luminosity and halo mass, as it is found that even the faintest dSphs inhabit similarly massive halos, highlighting a uniformity in the dark matter collapse processes at different scales and epochs.

The mass-to-light ratio, $\Upsilon_{1/2}$, appears to follow a U-shaped distribution as a function of galaxy mass, with a minimum spanning dwarf to normal ellipticals and steep rises noted in ultra-faint dSphs and the most massive galaxy clusters. This empirical result underscores a pivotal trend in galaxy formation, constraining models that seek to explain feedback processes within galaxies.

Implications and Future Prospective

This research offers a pivotal tool for astrophysicists, allowing for improved estimates of galaxy mass where now only kinematic analyses are feasible. This advance can better inform dark matter halo models and the processes governing baryonic matter within them, providing a critical window into understanding the mass distribution in galaxies without rotational support.

The implications extend to enhancing our comprehension of dark matter's interplay with stellar processes and providing a more detailed framework for testing cosmological simulations. Future research might explore extensions of this formula to account for deviations from spherical symmetry, more complex anisotropy profiles, or the inclusion of external fields or interactions in multi-galaxy systems. Such developments could provide deeper insights into the structural formation dynamics of the universe's most intriguing galaxies.

Overall, this work paves the way for new analytic tools in the field of galactic dynamics, offering precise methodological approaches that can be widely applied to improve our understanding of galaxy evolution and the dark matter halos in which they reside.