Radially-Resolved 3D Velocity Dispersion Profiles

• The paper demonstrates a robust approach to constructing 3D velocity dispersion profiles, revealing a linear decline that constrains the mass distribution in galaxy halos.
• It employs precise distance estimation, heliocentric-to-Galactocentric transformation, phase-space clipping, and Monte Carlo error analysis to accurately derive the profile.
• The consistency between parametric and non-parametric techniques validates these profiles as essential diagnostics for understanding galaxy dynamics and formation history.

A radially-resolved three-dimensional (3D) velocity dispersion profile specifies how the statistical spread of all three orthogonal components of velocity varies as a function of distance from the center of an astrophysical system, such as a galaxy, stellar halo, or globular cluster. Such profiles are essential diagnostics of the underlying gravitational potential, and hence mass distribution, as well as of the dynamical status and formation history of the system. Recent advancements in measurement techniques and statistical methodologies have enabled the detailed construction and interpretation of radially-resolved 3D velocity dispersion profiles across a diverse range of galactic environments and physical scales.

1. Principles of Radially-Resolved 3D Velocity Dispersion Profiles

A 3D velocity dispersion profile, $\sigma_v(r)$, characterizes the root-mean-square spread in the velocities of tracer objects (stars, gas clouds, etc.) at a given Galactocentric radius ($R$ or $r$), decomposed into three orthogonal directions—typically radial, azimuthal, and vertical (or equivalent Cartesian or spheroidal axes). In the context of the Milky Way’s halo, the analysis is performed in the Galactocentric rest frame after applying appropriate coordinate transformations:

$$v_\mathrm{rf} = v_\mathrm{helio} + 220\,\sin l\,\cos b + (10\,\cos l\,\cos b + 5.2\,\sin l\,\cos b + 7.2\,\sin b)$$

where the terms are defined in Galactic coordinates and referenced to the Local Standard of Rest (0910.2242). The mapping of observed line-of-sight velocities to intrinsic 3D components typically assumes statistical isotropy unless proper motion measurements are available to decompose all directions.

The profile $\sigma_v(R)$ is usually constructed by binning tracers in narrow radial shells and estimating the velocity dispersion in each shell, yielding a function that captures both small-scale (local) and global dynamical structure.

2. Empirical Determination: Methodology and Statistical Framework

Radially-resolved 3D velocity dispersion profiles are derived from large kinematic surveys of tracer populations, such as the sample of 910 distant halo stars used for the Milky Way (0910.2242). Crucial elements of methodology include:

• Distance Estimation: Distances to individual stars are inferred by combining observed photometric colors, magnitudes, and metallicities with theoretical stellar evolution tracks. This enables an assignment of each star’s position in Galactocentric coordinates.
• Velocity Frame Transformation: Observed heliocentric velocities are transformed into the Galactocentric rest frame using established formulae to properly account for Solar motion and Galactic rotation.
• Phase Space Clipping: Outliers, particularly unbound hypervelocity stars, are excluded. In the parametric approach, this is accomplished using a Milky Way potential model to define a radius-dependent escape velocity:

  $$v_{\mathrm{esc}}(R) = -2.30 \times 10^{-4} R^3 + 0.0588 R^2 - 6.62 R + 500\;\mathrm{km\,s}^{-1}$$

  Stars exceeding $v_\mathrm{esc}(R)$ are clipped (0910.2242).

• Error Analysis: Distance (hence velocity) uncertainties are addressed with Monte Carlo simulations, and bootstrap resampling quantifies errors on the dispersion measurement.
• Radial Binning: Tracer samples are divided into radial bins (e.g., bins of width ~0.33$R$, each containing at least 70 stars for the Milky Way halo case), within which the velocity dispersion is computed.
• Parametric and Non-Parametric Techniques: Velocity dispersion can be derived via parametric modeling anchored in a physical potential (requiring equilibrium assumptions), or with non-parametric, phase-space-based approaches such as the caustic technique, which infers the local escape velocity directly from the observed “trumpet” shape in $(R, v)$ space (see explicit definition in Section 3).

3. Computational Approaches: Parametric and Non-Parametric Methods

Two distinct computational strategies are widely used to extract radially-resolved 3D velocity dispersion profiles:

A. Parametric Potential-Based Method

This approach uses an explicit Milky Way potential model to determine the escape velocity and thus define a membership criterion for the kinematic sample. After member selection, the velocity dispersion is estimated in radial bins, with model-dependent corrections for measurement errors and uncertainties. For the Milky Way halo (0910.2242), the resulting profile is well described by a linearly declining function:

$$\sigma_v(R) = (-0.30 \pm 0.10) R + (120 \pm 4.6)\ \mathrm{km\,s}^{-1}, \quad 15 < R < 75~\mathrm{kpc}$$

B. Non-Parametric Caustic Technique

Developed initially for galaxy clusters, the caustic technique leverages the observed distribution of galaxies or stars in projected position–velocity space without explicit equilibrium assumptions. The method identifies “caustics” via:

$f_q(r, v) = \kappa\ ,$

where $f_q(r, v)$ is the phase-space density and $\kappa$ is set through the relation

$$\langle v_\mathrm{esc}^2 \rangle_{\kappa,R} = \frac{\int_0^R {\cal A}_\kappa^2(r)\,\varphi(r)\,dr}{\int_0^R \varphi(r)\,dr} = 4\,\sigma^2,$$

with ${\cal A}_\kappa(r)$ the caustic amplitude and $\varphi(r)$ the projected density. Iterative rejection removes interlopers, converging to a robust member sample (838/910 for the halo star sample). The derived velocity dispersion profile exhibits a similar radial decline:

$$\sigma_v(R) = (-0.45 \pm 0.16) R + (110 \pm 6.0)\ \mathrm{km\,s}^{-1}, \quad 16 < R < 64~\mathrm{kpc}$$

Despite a systematic amplitude offset (~10%) between methods, both yield consistent insights regarding the global trend.

4. Observational Results and Synthesis

The key quantitative result for the Milky Way’s radially-resolved velocity dispersion profile is a mean decline of

$$-0.38 \pm 0.12~\mathrm{km\,s}^{-1}~\mathrm{kpc}^{-1}$$

over $15 < R < 75~\mathrm{kpc}$, as directly measured (0910.2242). The profile decreases from about $110~\mathrm{km\,s}^{-1}$ at $R \approx 15~\mathrm{kpc}$ to $\approx 85~\mathrm{km\,s}^{-1}$ at $R \approx 80~\mathrm{kpc}$.

A comparison with independent surveys utilizing different stellar tracers, such as Battaglia et al. (2005) and Xue et al. (2008), reveals close agreement, reducing the likelihood that the observed decline is due to sample selection or systematic error. This robust, monotonic decrease is a critical empirical constraint on the shape and total mass of the Galactic halo.

5. Dynamical and Physical Interpretation

The radially-resolved 3D velocity dispersion profile encodes the response of the tracer population (e.g., halo BHB stars) to the Galactic gravitational potential in the outer halo. Under the assumption that the stellar tracers respond dynamically to the mass distribution (i.e., obey the Jeans equations or analogous dynamical constraints), the declining velocity dispersion profile reflects a combination of mass density decline and the gravitational potential’s flattening at large radii.

Specifically, in the $\Lambda$CDM paradigm, the halo’s density profile can be parametrized by, e.g., an NFW profile, from which the potential and thus the expected velocity dispersion as a function of radius can be self-consistently derived. The observed linear decline matches predictions from CDM-motivated potentials and disfavors models in which halo mass continues to rise sharply at large $R$ (i.e., disfavors massive extended dark halos inconsistent with the observed kinematics).

Additionally, the profile’s absolute normalization and shape provide input to dynamical mass estimators, enabling total mass inferences when combined with models for the tracer population's orbital anisotropy and spatial distribution.

6. Methodological Implications and Systematic Considerations

The analysis underlines several essential methodological aspects:

• Interloper Removal: Both parametric (escape velocity) and non-parametric (caustic method) interloper clipping are critical to avoid artificial inflation of the velocity dispersion due to unbound or foreground/background contaminants.
• Error Propagation: Measurement uncertainties, notably in luminosity/distance estimates, can introduce non-Gaussian error distributions. Monte Carlo resampling is necessary to accurately estimate statistical confidence intervals.
• Consistency Checks: The similarity of results across distinct methodologies and in comparison with independent data sets fortifies the conclusion that the derived profile traces the genuine dynamical properties of the halo.
• Range of Validity: Profiles are well constrained over $15-80$ kpc; the regime at $R \gg 100$ kpc remains observationally challenging and is subject to greater uncertainty.

7. Implications for Mass Modeling and Galaxy Formation

Radially-resolved 3D velocity dispersion profiles are indispensable inputs for reconstructing the Galaxy’s mass distribution and assessing the nature of its dark matter halo. In the case of the Milky Way:

• The declining profile tightly constrains the mass–radius relation in the halo’s outer regions, improving the accuracy of total mass estimates.
• The robustness of the profile against tracer selection or interloper bias validates its use in analytical mass diagnostics.
• Agreement with CDM-motivated density profiles and lack of evidence for a sharp increase in dispersion at large radii support a picture of a virialized, massive Galactic halo with a relatively steep density fall-off.
• The empirical profile provides a reference against which theoretical models of halo assembly, baryonic–dark matter interplay, and dynamical heating mechanisms can be tested.

In summary, modern analyses leveraging large, well-characterized tracer catalogs, careful interloper rejection, and both parametric and non-parametric methodologies yield radially-resolved 3D velocity dispersion profiles of high fidelity. These profiles—exemplified by the linear decline observed in the Milky Way’s halo—are fundamental to galactic dynamics, mass modeling, and our broader understanding of galaxy formation within the $\Lambda$CDM framework (0910.2242).