Asymmetric Drift

Asymmetric drift (v_a) is the systematic kinematic lag between the mean azimuthal rotational velocity (⟨v_φ⟩) of a given stellar population in a galactic disk and the circular orbital velocity (v_c) of a hypothetically cold, zero-dispersion test particle orbiting at the same galactocentric radius R. In rotationally supported disk galaxies such as the Milky Way, young O- and B-type stars and cold molecular clouds move on nearly circular orbits in the Galactic midplane with small random velocity dispersions (σ_R ≈ 10 km s⁻¹), rotating at nearly the true circular velocity of the Local Standard of Rest (v_c ≈ 220–233 km s⁻¹). Older stellar populations that have experienced cumulative vertical disk heating and in-plane radial scattering possess much larger radial velocity dispersions (σ_R ≈ 35–65 km s⁻¹) and exhibit a progressively larger backward lag (v_a = v_c − ⟨v_φ⟩ ≈ 10–50 km s⁻¹) relative to the Local Standard of Rest, creating a skewed, asymmetric distribution of azimuthal velocities in the solar neighborhood.

Although asymmetric drift behaves macroscopically like a pressure-support term in fluid dynamics, individual stars in a galactic disk are collisionless and experience no physical drag force. Instead, asymmetric drift arises from the combination of two radial gradients across an exponentially declining galactic disk:

1. Exponentially Declining Stellar Number Density: The surface number density of disk stars falls exponentially with galactocentric radius R as ν(R) = ν₀ exp(−R / h_R), where the radial scale length is h_R ≈ 2.6 kpc in the Milky Way thin disk. Consequently, within any radial annulus near the Sun's radius R₀ ≈ 8.18 kpc, there are substantially more stars whose guiding-center radii (R_g) lie in the denser inner galaxy (R_g < R₀) and are currently near the outer turning point (apocenter) of their eccentric epicyclic orbits than there are stars from the sparser outer galaxy (R_g > R₀) currently near their inner turning point (pericenter).
2. Conservation of Angular Momentum Along Epicycles: Each star conserves its vertical angular momentum L_z = R v_φ as it oscillates radially between pericenter and apocenter. A star visiting R₀ from an inner guiding center (R_g < R₀, near apocenter) possesses lower angular momentum than a circular orbit at R₀, so its instantaneous azimuthal speed at R₀ is slower than v_c(R₀). Conversely, a star visiting R₀ from an outer guiding center (R_g > R₀, near pericenter) moves faster than v_c(R₀).
3. Outwardly Declining Velocity Dispersion: In addition to higher stellar density, the inner disk has a higher radial velocity dispersion σ_R(R) ∝ exp(−R / h_σ), giving inner-disk stars larger radial epicyclic excursions ΔR ≈ 2^1/2 σ_R / κ. Because inward-born apocentric visitors outnumber outward-born pericentric visitors in any local solar-neighborhood volume, averaging v_φ over all stars currently passing through R₀ yields a net mean azimuthal velocity ⟨v_φ⟩ that lags behind v_c(R₀).

The quantitative formula for asymmetric drift is derived directly from the steady-state, axisymmetric radial Jeans equation (obtained by taking the first velocity moment of the collisionless Boltzmann equation in cylindrical coordinates (R, φ, z)):

(1 / ν) ∂(ν σ_R²) / ∂R + ∂⟨v_R v_z⟩ / ∂z + (σ_R² − σ_φ² − ⟨v_φ⟩²) / R = −∂Φ / ∂R

By definition, the circular velocity v_c(R) at the Galactic midplane (z = 0) balances the radial gravitational potential gradient: v_c² / R = ∂Φ / ∂R. Substituting the definition of asymmetric drift, v_a ≡ v_c − ⟨v_φ⟩, we have:

v_c² − ⟨v_φ⟩² = (v_c − ⟨v_φ⟩)(v_c + ⟨v_φ⟩) = 2 v_c v_a − v_a² ≈ 2 v_c v_a

In the thin-disk regime where v_a ≪ v_c (so the quadratic term v_a² is negligible compared to 2 v_c v_a), rearranging the radial Jeans equation yields Strömberg's asymmetric drift equation:

v_a = v_c − ⟨v_φ⟩ ≈ (σ_R² / 2v_c) [ (σ_φ² / σ_R²) − 1 − ∂ ln(ν σ_R²) / ∂ ln R − (R / σ_R²) ∂⟨v_R v_z⟩ / ∂z ]

For a stellar population with an exponential radial density profile ν(R) ∝ exp(−R / h_R) and an exponential radial dispersion profile σ_R²(R) ∝ exp(−2R / h_σ), the logarithmic derivative is ∂ ln(ν σ_R²) / ∂ ln R = −(R / h_R + 2R / h_σ). Furthermore, the ratio σ_φ² / σ_R² is fixed by the Oort constants via −B / (A − B) ≈ 0.44, while the vertical tilt term ∂⟨v_R v_z⟩ / ∂z takes a value between 0 (if the velocity ellipsoid stays aligned with the cylindrical R–z axes) and (σ_R² − σ_z²) / R (if it points radially toward the Galactic Center). Thus the bracketed factor k_AD is nearly constant across all main-sequence disk subpopulations, reducing the relation to a linear scaling with σ_R²:

v_a = k_AD σ_R² / (2 v_c) ≈ σ_R² / (D_AD)

where the empirical proportionality constant in the solar neighborhood is D_AD ≈ 80 ± 5 km s⁻¹.

In 1924 and 1925, Swedish-American astronomer Gustaf Strömberg discovered empirically that when nearby stars are binned by spectral type or velocity dispersion, their mean heliocentric azimuthal velocity ⟨V⟩_⊙ varies linearly with the square of their radial velocity dispersion σ_R². Because the observed heliocentric azimuthal velocity of any stellar group i is the sum of the solar azimuthal peculiar velocity V_⊙ (relative to the Local Standard of Rest) and the group's asymmetric drift v_a,i:

−⟨V_i⟩ = V_⊙ + v_a,i = V_⊙ + (1 / D_AD) σ_R,i²

plotting −⟨V_i⟩ against σ_R,i² and extrapolating linearly to a hypothetical zero-dispersion population (σ_R² → 0) determines the Sun's azimuthal peculiar velocity V_⊙ above the Local Standard of Rest.

Using 14,000 stars from the Hipparcos catalog, Walter Dehnen and James Binney (1998) applied Strömberg's linear extrapolation to derive the classic solar motion vector (U_⊙, V_⊙, W_⊙) = (10.00 ± 0.36, 5.25 ± 0.62, 7.17 ± 0.38) km s⁻¹ (where U points radially inward toward the Galactic Center, V points in the direction of Galactic rotation, and W points vertically toward the North Galactic Pole). In 2010, Ralph Schönrich, James Binney, and Walter Dehnen demonstrated that accounting for the radial metallicity gradient of the Milky Way disk breaks a systematic degeneracy in Strömberg's relation: because lower-metallicity stars from the outer disk have cooler dispersions than inner-disk stars of the same age, binning stars by color without metallicity biases the slope at small σ_R². Incorporating chemical abundances and Gaia DR3 astrometry yields the modern benchmark solar peculiar motion of (U_⊙, V_⊙, W_⊙) = (11.1 ± 0.7, 12.24 ± 0.47, 7.25 ± 0.37) km s⁻¹, alongside non-axisymmetric perturbations visible in vertex deviation and phase mixing.

In external spiral and lenticular galaxies observed via integral-field spectroscopy (such as MaNGA, CALIFA, and SAMI) or H I 21-cm synthesis imaging, asymmetric drift plays a major role in mass modeling and dark matter halo inference. While cold ionized gas (Hα) or neutral hydrogen (H I) traces the circular velocity v_c(R) directly in high-mass spirals, stellar absorption-line kinematics measure the slower streaming velocity ⟨v_φ⟩ = (v_c² − k_AD σ_R²)^1/2. Similarly, in low-mass dwarf irregular galaxies where the turbulent gas velocity dispersion (σ_gas ≈ 8–12 km s⁻¹) is comparable to the peak rotation speed (v_max ≈ 15–30 km s⁻¹), applying an asymmetric drift correction (v_c² = v_rot² + σ_D²) is necessary to recover the true gravitational potential and avoid underestimating the inner dark matter density slope.

1. Strömberg, G. (1924). "The asymmetry in stellar motions and the existence of a velocity-restriction in space." The Astrophysical Journal, 59, 228–251. https://doi.org/10.1086/142813
2. Dehnen, W., & Binney, J. J. (1998). "Local stellar kinematics from Hipparcos data." Monthly Notices of the Royal Astronomical Society, 298(2), 387–394. https://doi.org/10.1046/j.1365-8711.1998.01600.x
3. Schönrich, R., Binney, J., & Dehnen, W. (2010). "Local kinematics and the local standard of rest." Monthly Notices of the Royal Astronomical Society, 403(4), 1829–1833. https://doi.org/10.1111/j.1365-2966.2010.16253.x
4. Golubov, O., Just, A., Bienaymé, O., Bland-Hawthorn, J., Gibson, B. K., Grebel, E. K., Munari, U., Navarro, J. F., Parker, Q., Reid, W., Seabroke, G., Siebert, A., Siviero, A., Steinmetz, M., Watson, F., Williams, M., Wyse, R., & Zwitter, T. (2013). "The asymmetric drift, the local standard of rest, and implications from RAVE data." Astronomy & Astrophysics, 557, A92. https://doi.org/10.1051/0004-6361/201321228