The Oort Constants

Why the Oort constants matter

Before 1927 it was not settled that the Milky Way rotates at all, let alone how. The Oort constants were the first quantitative proof that our Galaxy spins differentially — that stars closer to the Galactic Centre complete their orbits faster than stars farther out — and they remain the cleanest local handle on that motion.

• They proved the Galaxy rotates. Oort's A > 0, B < 0 result confirmed Bertil Lindblad's proposal that the stellar system is a rotating, flattened disk rather than a static swarm.
• They anchor the local rotation curve. A − B gives the angular speed Ω₀ and A + B its radial slope, so together they fix both the value and the local gradient of the Milky Way's rotation curve at the Sun.
• They calibrate the Galactic Centre distance. Combined with the Sun's proper motion relative to Sgr A*, the constants help tie down R₀ ≈ 8.2 kpc and Θ₀.
• They feed dynamics. The epicyclic frequency κ (used for spiral density waves, Lindblad resonances and disk stability) is built directly from the Oort constants: κ² = −4B(A − B).
• They test for non-axisymmetry. Deviations from the pure A/B fit reveal the fingerprints of the Galactic bar, spiral arms and stellar moving groups.

How differential rotation writes itself onto the sky

Imagine standing on a merry-go-round where inner horses turn faster than outer ones. If you look at horses just inside your radius, they slide ahead of you; horses just outside fall behind. The Milky Way's disk behaves the same way, and that relative sliding is exactly what the Oort constants quantify. Here is the reasoning Oort followed:

1. Set up the geometry. Place the Sun at Galactocentric radius R₀ moving on a circular orbit at speed Θ₀. A nearby star at distance d and Galactic longitude ℓ sits at a slightly different radius R and moves at speed Θ(R).
2. Project the velocity difference. Take the star's velocity relative to the Sun and resolve it into the line-of-sight (radial) and tangential (proper-motion) components, keeping only first-order terms in the small ratio d/R₀.
3. Collect the sinusoids. The radial component turns out to vary as sin(2ℓ) and the tangential component as cos(2ℓ) plus a constant offset. The amplitude of the double-sine term is the Oort constant A; the constant offset in the proper motions is B.
4. Fit real stars. Bin nearby stars by longitude, measure their mean radial velocities and proper motions, and least-squares fit the sinusoids. The fitted amplitude and offset are A and B directly.

The double-sine pattern is the signature. Toward Galactic longitudes ℓ = 45°, 135°, 225° and 315° the differential streaming is maximal; toward the Galactic Centre and anticentre (ℓ = 0°, 180°) and toward pure rotation and anti-rotation (ℓ = 90°, 270°) the radial signal vanishes. Seeing that four-lobed sky pattern is what convinced astronomers the disk shears.

The governing equations

The two Oort formulae relate the observed heliocentric motions of nearby disk stars to A and B. To first order in distance:

v_r = A · d · sin(2ℓ)

μ = A · cos(2ℓ) + B  (equivalently v_t = d · [A·cos(2ℓ) + B])

where:

• v_r — heliocentric radial velocity of the star (km/s), corrected to the local standard of rest.
• v_t — heliocentric tangential velocity (km/s); the proper motion μ = v_t/d.
• d — distance from the Sun to the star (kpc).
• ℓ — Galactic longitude of the star (degrees), measured in the disk from the Galactic Centre direction.
• A, B — the Oort constants (km/s/kpc). A is the shear amplitude, B the vorticity offset.

The constants connect to the circular-velocity curve Θ(R) evaluated at the Sun (R = R₀):

A = ½ ( Θ₀/R₀ − dΘ/dR )   and   B = −½ ( Θ₀/R₀ + dΘ/dR )

Two combinations do the heavy lifting. The difference gives the local angular rotation rate:

Ω₀ = Θ₀/R₀ = A − B

and the sum gives the negative slope of the rotation curve:

A + B = − dΘ/dR |_R₀

Here Θ₀ is the circular speed at the Sun (km/s), R₀ the Sun's Galactocentric distance (kpc), and dΘ/dR the radial gradient of the rotation speed (km/s/kpc). A flat rotation curve has dΘ/dR = 0, forcing A + B = 0 and A = −B. Rigid-body rotation has Θ ∝ R, so A = 0 and the disk carries pure vorticity B. A Keplerian point-mass field (Θ ∝ R^−1/2) is shear-dominated. The real Milky Way sits close to the flat-curve limit, with A ≈ −B to within a few km/s/kpc.

Worked example: from A and B to the Milky Way's spin

Take the Gaia-era values A = 15.3 km/s/kpc and B = −11.9 km/s/kpc and turn them into physical quantities.

• Angular rotation rate: Ω₀ = A − B = 15.3 − (−11.9) = 27.2 km/s/kpc. In SI-friendly units that is about 27.2 km/s per kiloparsec ≈ 8.8 × 10⁻¹⁶ rad/s.
• Circular speed: with R₀ = 8.2 kpc, Θ₀ = Ω₀ · R₀ = 27.2 × 8.2 ≈ 223 km/s — matching the ~220–235 km/s obtained by other methods.
• Rotation-curve slope: dΘ/dR = −(A + B) = −(15.3 − 11.9) = −3.4 km/s/kpc. The curve is falling very gently near the Sun — essentially flat.
• Galactic year: the Sun's orbital period is P = 2πR₀/Θ₀ ≈ 2π × 8.2 kpc / 223 km/s ≈ 226 million years — one "galactic year."
• Epicyclic frequency: κ = √(−4B(A − B)) = √(−4 × (−11.9) × 27.2) ≈ 36 km/s/kpc, the rate at which a slightly perturbed star oscillates radially about its guiding centre.

Oort constants across a century of measurement

Source / epoch	A (km/s/kpc)	B (km/s/kpc)	Ω₀ = A − B
Oort 1927 (original)	~19	~−24	~43*
IAU 1985 standard	14.4 ± 1.2	−12.0 ± 2.8	26.4
Hipparcos (Feast & Whitelock 1997)	14.8 ± 0.8	−12.4 ± 0.6	27.2
Gaia DR1 / TGAS (Bovy 2017)	15.3 ± 0.4	−11.9 ± 0.4	27.2

*Oort's 1927 absolute scale differed because the distance scale of the day was less secure; his key result was the sign and existence of A and B, not the precise magnitude.

Shear versus vorticity, made concrete

The two constants isolate two different ways a velocity field can be non-uniform. Drop a small circular blob of gas into the disk and watch it deform over one orbit:

• A (shear) stretches the blob into an ellipse because its inner edge orbits faster than its outer edge. Shear is why gas clouds, star-forming regions and tidal streams get sheared into trailing arcs, and why the Milky Way's spiral arms are transient and wound up rather than rigid spokes.
• B (vorticity) spins the blob about its own centre. The local vorticity of the disk is 2B ≈ −24 km/s/kpc; its negative sign encodes that the disk rotates clockwise as seen from the North Galactic Pole. Vorticity, through the epicyclic frequency κ, controls the stability of the disk against clumping (the Toomre criterion) and the locations of Lindblad resonances.

Common misconceptions

• "The Oort constants describe the whole Galaxy's rotation." No — they are strictly local first-order quantities valid within a few hundred parsecs of the Sun. Extrapolating them to the outer disk is invalid.
• "A and B are universal constants like G or c." They are local values at R₀; every radius in the Galaxy has its own A and B.
• "A positive A means the disk speeds up outward." It is the opposite: A > 0 with the sign convention here reflects that inner material orbits faster, producing trailing shear.
• "You need the distances to the stars to measure A and B." The proper-motion relation μ = A·cos(2ℓ) + B is distance-independent, so A and B can be recovered from proper motions alone; distances are only needed to convert to physical km/s/kpc units and to isolate v_r.
• "The rotation curve being flat means A + B = 0 exactly." A + B ≈ 3 km/s/kpc is small but not zero — the local curve declines very gently, and that residual carries real information.
• "They confirm the Sun feels the Galactic Centre's pull directly." They measure the velocity field, not the force; the enclosed-mass interpretation comes only after assuming near-circular orbits.

A note on history and the local standard of rest

Jan Hendrik Oort, then a young astronomer at Leiden, published the constants in 1927 in the Bulletin of the Astronomical Institutes of the Netherlands, building on Bertil Lindblad's 1925–1927 model of the Galaxy as a set of nested rotating subsystems. Oort's decisive move was to correct stellar motions to the local standard of rest (LSR) — the fictitious point co-located with the Sun that moves on a perfectly circular orbit at Θ₀. Because the Sun itself has a peculiar velocity of about 18 km/s toward the solar apex (roughly U ≈ 11, V ≈ 12, W ≈ 7 km/s in the modern Schönrich–Binney–Dehnen calibration), that solar motion must be subtracted before the clean A·sin(2ℓ) and B pattern emerges. Get the LSR wrong and the constants are biased. A century later, Gaia's billion-star astrometry has turned Oort's four-lobed sky pattern from a marginal statistical detection into a precise map — one that now also resolves the small non-axisymmetric C and K terms betraying the bar and spiral arms Oort could not see.