Accurate mass estimates from the proper motions of dispersion-supported galaxies

TL;DR

This paper derives a new mass estimator for dispersion-supported galaxies using proper motion data, providing a practical tool to distinguish between dark matter models like CDM and SIDM.

Contribution

It introduces a radius-based mass estimator relying on proper motions and demonstrates its effectiveness with dwarf galaxy data and simulations.

Findings

Mass estimator accurate within 10-15% for simulated galaxies.

Can distinguish between SIDM and CDM models at small radii.

Consistent with cuspy NFW profiles in observed dwarf galaxies.

Abstract

Starting with the spherical Jeans equation, we show that there exists a radius where the mass enclosed depends only on the projected tangential velocity dispersion, assuming that the anisotropy profile slowly varies. This is well-approximated at the radius where the log-slope of the stellar tracer profile is $-2$: $r_{-2}$. The associated mass is $M(r_{-2}) = 2 G^{-1} \langle \sigma_{\mathcal{T}}^{2}\rangle^{*} r_{-2}$ and the circular velocity is $V^{2}({r_{-2}}) = 2\langle \sigma_{\mathcal{T}}^{2}\rangle^{*}$. For a Plummer profile $r_{-2} \simeq 4R_e/5$. Importantly, $r_{-2}$ is smaller than the characteristic radius for line-of-sight velocities derived by Wolf et al. 2010. Together, the two estimators can constrain the mass profiles of dispersion-supported galaxies. We illustrate its applicability using published proper motion measurements of dwarf galaxies Draco and Sculptor, and find that they are consistent with inhabiting cuspy NFW subhalos of the kind predicted in CDM but we cannot rule out a core. We test our combined mass estimators against previously-published, non-spherical cosmological dwarf galaxy simulations done in both CDM and SIDM. For CDM, the estimates for the dynamic rotation curves are found to be accurate to $10\%$ while SIDM are accurate to $15\%$. Unfortunately, this level of accuracy is not good enough to measure slopes at the level required to distinguish between cusps and cores of the type predicted in viable SIDM models without stronger priors. However, we find that this provides good enough accuracy to distinguish between the normalization differences predicted at small radii ($r \simeq r_{-2} < r_{\rm core}$) for interesting SIDM models. As the number of galaxies with internal proper motions increases, mass estimators of this kind will enable valuable constraints on SIDM and CDM models.

Tables

Table 1. Table 1: Observational measurements considered in this analysis.

Galaxy	$\sqrt{{⟨{\sigma }_{\mathrm{los}}^2⟩}^{\ast }}$	$\sqrt{{⟨{\sigma }_{ℛ}^2⟩}^{\ast }}$	$\sqrt{{⟨{\sigma }_{𝒯}^2⟩}^{\ast }}$	$R_e$	$r_{1/2}$	$M_{-3}$	$V_{-3}$	$M_{-2}$	$V_{-2}$
	$\left[\mathrm{km}{s}^{-1}\right]$	$\left[\mathrm{km}{s}^{-1}\right]$	$\left[\mathrm{km}{s}^{-1}\right]$	$\left[\mathrm{pc}\right]$	$\left[\mathrm{pc}\right]$	$\left[M_{\odot }\right]$	$\left[\mathrm{km}{s}^{-1}\right]$	[$M_{\odot }]$	$\left[\mathrm{km}{s}^{-1}\right]$
Draco	${}_{}{}^{(a,b)}10.1_{-0.5}^{+0.5}$	${}_{}{}^{\left(e\right)}11.0_{-1.5}^{+2.1}$	${}_{}{}^{\left(e\right)}9.9_{-3.1}^{+2.3}$	${}_{}{}^{(d,f)}214_{-2}^{+2}$	${279}_{-2}^{+2}$	${2.03}_{-0.20}^{+0.20}\times {10}^7$	${17.5}_{-0.9}^{+0.9}$	${7.80}_{-3.63}^{+4.89}\times {10}^6$	${14.0}_{-4.4}^{+3.3}$
Sculptor	${}_{}{}^{(a,b)}9.0_{-0.2}^{+0.2}$	${}_{}{}^{\left(c\right)}11.5_{-4.3}^{+4.3}$	${}_{}{}^{\left(c\right)}8.5_{-3.2}^{+3.2}$	${}_{}{}^{(d,f)}280_{-1}^{+1}$	${365}_{-1}^{+1}$	${2.11}_{-0.09}^{+0.09}\times {10}^7$	${15.6}_{-0.3}^{+0.3}$	${7.53}_{-5.67}^{+5.67}\times {10}^6$	${12.0}_{-4.5}^{+4.5}$

Equations

$$-\frac{d\Phi (r)}{dr}=\frac{1}{n_{\star }(r)}\frac{d}{dr}\left(n_{\star }{\sigma }_r^2(r)\right)+\frac{2\beta {\sigma }_r^2(r)}{r},$$

$$\beta (r)$$

$${\sigma }_{\mathrm{tot}}^2(r)={\sigma }_r^2+{\sigma }_{\theta }^2+{\sigma }_{\varphi }^2=(3-2\beta ){\sigma }_r^2(r).$$

$$M(r|\beta )=\frac{r{\sigma }_r^2(r)}{G}\left({\gamma }_{\star }+{\gamma }_{\sigma }-2\beta \right),$$

$${\gamma }_{\star }:=-\frac{d\log n_{\star }}{d\log r};$$

$${\gamma }_{\beta }$$

$$v_{\mathrm{los}}$$

$$v_{\mathcal{R}}$$

$${\Sigma }_{\star }{\sigma }_{\mathrm{los}}^2(R)$$

$${\Sigma }_{\star }{\sigma }_{\mathcal{R}}^2(R)$$

$${\Sigma }_{\star }{\sigma }_{\mathcal{T}}^2(R)$$

Full text

††pubyear: 2020††pagerange: Accurate mass estimates from the proper motions of dispersion-supported galaxies–LABEL:lastpage

1 Introduction

The $\Lambda$CDM cosmogony, while successful in describing the large scale structure of our universe, still suffers from potential discrepancies in modeling the properties on small scales, primarily for dark matter halos that are expected to host the observed dwarf galaxies. For example, Milky-Way satellites have significantly lower dark matter densities in the inner regions compared to the corresponding subhalos in cosmological $N$-body simulations — this is known as the Too Big To Fail problem (boylan2011too). A potentially related issue concerns the inner dark matter density profiles inferred from the rotation curves of small disk galaxies, many of which are observed to be cored/flat, while simulated $\Lambda$CDM halos are cusped/rising — this is the Cusp-core problem (flores1994; moore1994; de2010core). Feedback from star formation can potentially explain this discrepancy in larger dwarf galaxies (governato2010bulgeless; pontzen2012supernova). However, if dark matter cores exist within galaxies that have had too little star formation ($M_{\star}\lesssim 10^{6}\ M_{\odot}$) to affect the dark matter density slopes (di2013dependence; chan2015impact; tollet2016nihao), then this could be an indication that the dark matter is something other than CDM (see bullock2017small, and references there in).

Though particularly important, the question of whether or not the smallest galaxies have cusps or cores is notoriously difficult to answer owing to the fact that they are dispersion supported. While it is possible to quantify the detailed mass profiles of spheroidal galaxies through the use of kinematic measurements of individual stars in 3D (e.g. wilkinson2002dark; strigari2007astrometry), until recently we have been limited to data sets that include only 1D velocities along the line-of-sight. This introduces a degeneracy between the inferred mass profile slope and the underlying velocity dispersion anisotropy parameter $\beta$, which quantifies the intrinsic difference between the radial and tangential velocity dispersions.

One robust measurement that is possible with line-of-sight velocities is the integrated mass within a single characteristic radius for each galaxy. This idea was first emphasized by walker2009universal, who used spherical Jeans modelling to show that the integrated mass within an effective radius was independent of assumed $\beta$ for a wide variety of assumptions for many galaxies. wolf2010accurate extended this idea, also using Jeans modeling, to show that there exists, analytically, an idealized radius within which the mass inferred from line-of-sight velocities is formally insensitive to $\beta$. Under mild assumptions, this radius is where the log-slope of the stellar tracer profile is equal to $-3$. Both the Walker and Wolf mass estimators do remarkably well when compared to ab initio cosmological simulations of (non-spherical) dwarf galaxies in (campbell2017knowing; gonzalez2017dwarf). They are also used extensively to interpret observed line-of-sight velocity dispersion measurements (see simon2019, and references there in).

We are entering a new era of astrometry, such that the internal proper motions in distant dwarf spheroidal galaxies are now becoming possible to measure with the advent of GAIA (brown2016gaia; prusti2016gaia; brown2018gaia; helmi2018gaia). Additionally, LSST may provide similar advances (lsst2009). Measurements of stellar velocities along the plane-of-the-sky promise an important new window into the mass and density structure of dwarf galaxies. The results of massari20173d and massari2019stellar provide an exciting first look at what we expect to measure in the coming years by providing plane of the sky velocity dispersion measurements for Sculptor and Draco, respectively.

The article is outlined as follows: In Section 2, we briefly introduce the spherical Jeans equation and the coordinate system used as the basis of our analysis. Section 3 derives the mass estimators by combining the Jeans equation and proper motions measured from the plane of the sky, which includes the key assumptions considered therein. Section LABEL:sec:observation.results demonstrates the use of the combined mass estimators to provide an implied mass-density slope for currently available proper motions of Draco and Sculptor. Section LABEL:sec:mock.observations assesses our estimators with mock observations constructed from high-resolution simulations. In Section LABEL:sec:discussion, we discuss possible biases that might arise due to Jeans modelling, and finally, Section LABEL:sec:conclusion summarizes our results and we provide concluding remarks.

2 Prelimnaries

In what follows, lower case $r$ represents the (physical) three-dimensional radius and the upper case $R$ represents the (physical) two-dimensional projected radius.

2.1 The Spherical Jeans Equation

For a spherically symmetric steady-state system, the first moment of the collisionless Boltzmann equation for a stellar phase-space distribution takes the form of the spherical Jeans equation (binney2011galactic):

[TABLE]

which relates the total gravitational potential, $\Phi(r)$, of the stellar system to its two tracers: the intrinsic radial velocity dispersion, $\sigma_{r}^{2}:=\langle v_{r}^{2}\rangle-\langle v_{r}\rangle^{2}$, and the three-dimensional stellar number density, $n_{\star}(r)$. The quantity,

[TABLE]

is a measure of the velocity dispersion anisotropy of the tracer population, where $\sigma_{\theta}$ and $\sigma_{\phi}$ are the intrinsic velocity dispersion tangential to radius $r$. We will assume that $\sigma_{\theta}=\sigma_{\phi}$. Radially biased systems tend to have $\beta\rightarrow 1$ while $\beta\rightarrow-\infty$ constitutes tangentially biased measurements. In addition, the total intrinsic velocity dispersion follows

[TABLE]

The total mass profile of the dynamical system is an implied quantity of Eq. (1), such that,

[TABLE]

where $G$ is Newtons gravitational constant and the logarithmic slopes are defined as

[TABLE]

2.2 Coordinate System of Measurements

We use the coordinate system discussed in strigari2007astrometry such that the three-dimensional components of stars’ velocity in a spherically, steady-state systems are comprised of the components radial, $v_{r}$, and transverse, $v_{\theta}$ and $v_{\phi}$. The projected proper motions are composed of these three-dimensional velocities, that is, along the measured line-of-sight,

[TABLE]

where $\vec{r}\cdot\vec{z}=\cos\theta$ and $\vec{z}$ is the line-of-sight direction, and along the plane of the sky, the components parallel and transverse to the projected radius $R$ are

[TABLE]

respectively. Here, the variances of the velocity dispersions are given by $\sigma_{i}^{2}:=\langle v_{i}^{2}\rangle$ and $\sigma_{\phi}=\sigma_{\theta}$ is assumed. The derived mapping of the observable proper motions to the deprojected, three-dimensional tracer profiles are

[TABLE]

The combination of the proper motions also satisfy $\sigma_{\rm tot}^{2}=\sigma_{\mathcal{R}}^{2}+\sigma_{\mathcal{T}}^{2}+\sigma_{\rm los}^{2}$. For an observed galaxy, $\Sigma_{\star}(R)$ is the projected stellar density, which is related to the three-dimensional $n_{\star}(r)$ via an Abel inversion, Eq (LABEL:eq:abel.inversion). For the relevance of the proceeding text, we will focus on the additional constraint imposed by

3 Mass Estimators from Proper Motions

In this section, quantities enclosed in brackets with an asterisk as $\langle\cdots\rangle^{*}$ indicates a measurement to be luminosity-weighted, $r_{1/2}$ is the three-dimensional, deprojected half-light radius, and $R_{e}$ is the two-dimensional effective radius.