Velocity dispersion ( $σ$ )

Velocity dispersion is the standard deviation of velocities about their mean. Its tensor form records anisotropy, while a line-of-sight measurement combines intrinsic kinematics with projection and observational weighting.

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Velocity ellipsoid ( $σ_{ij}^{2} = ⟨(v_{i} - \overset{v}{ˉ}_{i}) (v_{j} - \overset{v}{ˉ}_{j})⟩$ )

The velocity ellipsoid represents the positive velocity covariance tensor by its orthogonal principal axes and eigenvalue square roots. The uncentered second-moment tensor differs from covariance when a population streams. Vanishing mixed second moments align the second-moment ellipsoid with the chosen frame; for pure azimuthal streaming in a phase-mixed axisymmetric model, the off-diagonal covariance also vanishes.

Heliocentric projection of a spherical velocity dispersion

For a non-streaming spherical tracer population, the line of sight makes angle

χ

with the Galactocentric radial direction. The projected velocity dispersion is

σ_{los}^{2} = σ_{r}^{2} cos^{2} χ + σ_{t}^{2} sin^{2} χ = σ_{r}^{2} (1 - β sin^{2} χ),

where

σ_{t}^{2}

is one tangential variance and

β = 1 - σ_{t}^{2} / σ_{r}^{2}

is the velocity-anisotropy parameter. At a fixed radius, averaging with the observational angular weights gives

σ_{⊙} = σ_{r} 1 - β A

, where

A = ⟨ sin^{2} χ ⟩

.

Geometric correction for heliocentric velocity dispersion ( $A (r, R_{⊙})$ )

For tracers sampled uniformly on a Galactocentric shell of radius

r

, with the Sun at radius

R_{⊙}

, the correction in the heliocentric projection of a spherical velocity dispersion is

A = \frac{r ^{2} + R _{⊙}^{2}}{4 r ^{2}} - \frac{( r ^{2} - R _{⊙}^{2} ) ^{2}}{8 r ^{3} R _{⊙}} ln \frac{r + R _{⊙}}{r - R _{⊙}} .

The limiting value at

r = R_{⊙}

is

1/2

; it tends to

2/3

r / R_{⊙} \to 0

and to

2 R_{⊙}^{2} / (3 r^{2})

at large radius. Uniform heliocentric solid angle at fixed

r > R_{⊙}

instead gives

A = 2 R_{⊙}^{2} / (3 r^{2})

exactly. Specify the selection measure when applying the correction.

Velocity-anisotropy parameter ( $β$ )

For spherical radial dispersion

σ_{r}^{2}

and total two-component tangential dispersion

σ_{t}^{2}

, the velocity-anisotropy parameter is

β = 1 - \frac{σ _{t}^{2}}{2 σ _{r}^{2}} .

Positive values are radially biased, zero is isotropic, and negative values are tangentially biased.

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Velocity dispersion by

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Velocity dispersion is a measure of the range of velocities within a group of objects, such as stars in a galaxy or galaxies in a cluster. It quantifies how much the velocities of the objects deviate from the average velocity of the group. In a more technical sense, it is defined as the standard deviation of the velocities of the objects in the sample. In astrophysics, velocity dispersion is an important metric because it provides insights into the dynamics and mass distribution of celestial bodies.

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