Galaxy rotation curve

A galaxy rotation curve gives circular speed as a function of radius. Its decomposition into stellar, gaseous, and dark-matter contributions is generally nonunique.

Angular-momentum estimate of a self-gravitating galactic disk ( $R_{d} / R_{h} = f_{j}^{2} / (κ f_{d}), v_{d} / v_{h} = κ f_{d} / f_{j}$ )

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Let a disk contain fraction

f_{d}

of its host mass and let its edge specific angular momentum be

f_{j} R_{h} v_{h}

. With halo relation

v_{h}^{2} = G M_{h} / R_{h}

and disk geometry coefficient

κ = v_{d}^{2} R_{d} / (G M_{d})

, combining centrifugal balance and the edge angular momentum gives the displayed estimates. The coefficient records the radial mass profile and thickness: enclosed mass does not determine a disc rotation curve. Treating

κ = 1

as a monopole estimate must not be mistaken for an exact thin-disk theorem.

Enclosed mass does not determine a disc rotation curve

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The spherical shell theorem yields

v_{c}^{2} = GM (< r) / r

only in spherical geometry. A razor-thin axisymmetric disc instead requires

v_{c}^{2} = R \partial_{R} Φ (R, 0)

, including forces from exterior annuli. The interior gravitational force of an exterior thin ring gives a concrete counterexample with identical enclosed mass but different circular speeds.

Interior gravitational force of an exterior thin ring ( $\partial_{R} ψ_{ring} = G m R / (2 a^{3}) + \dots$ )

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A thin ring of mass

m

and radius

a

has interior relative potential

G m [1 + R^{2} / (4 a^{2}) + \dots] / a

. Its leading radial acceleration is outward,

G m R / (2 a^{3})

, unlike the zero interior force of a spherical shell. Exterior disc annuli therefore affect a galaxy rotation curve.

Circular speed ( $v_{c}$ )

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In a spherical potential, the circular speed satisfies

v_{c}^{2} (r) = r \frac{d Φ}{d r} = \frac{GM ( < r )}{r} .

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It measures the gravitational field even when the system itself is not supported by circular motion.

Flat galaxy rotation curve

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A flat galaxy rotation curve has approximately constant circular speed

v_{c}

over a radial interval. Under spherical symmetry it implies

M (< r) = v_{c}^{2} r / G

and mass density

ρ (r) = v_{c}^{2} / (4 π G r^{2})

Spherical mass profile for a flat rotation curve ( $M (< r) = v_{0}^{2} r / G, ρ (r) = v_{0}^{2} / (4 π G r^{2})$ )

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In a spherically dominated Newtonian gravitational field, the circular orbit speed satisfies

v_{c}^{2} = GM (< r) / r

. A flat galaxy rotation curve therefore implies

M (< r) = v_{0}^{2} r / G

. Differentiating the enclosed mass gives

ρ = (4 π r^{2})^{- 1} d M / d r

, yielding the inverse-square density profile. This is a halo-region approximation, not a mass inversion for a flattened stellar disk. A flat curve extending to infinite radius would imply infinite mass, so a real halo profile must eventually change or truncate.

Exponential galactic disk

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A thin exponential galactic disk has surface density

Σ (R) = Σ_{0} e^{- R / R_{d}}

and total mass

2 π Σ_{0} R_{d}^{2}

Exponential-disk Keplerian asymptotic ( $v_{c}^{2} = GM / R [1 + 9 R_{d}^{2} / (2 R^{2}) + O (R^{- 4})]$ )

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For an exponential galactic disk, the first nonspherical term in the Legendre expansion of thin-disk gravity is positive and proportional to

R^{- 3}

v_{c}^{2}

. The missing enclosed mass and exterior-ring terms are exponentially small at large radius. Hence the galaxy rotation curve approaches the point-mass Keplerian curve from above.

Maximum disk

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The maximum disk hypothesis assigns the largest stellar mass-to-light ratio compatible with a measured rotation curve, conventionally making the disk supply about

85%

of the circular speed near

2.2 R_{d}

Disk--halo degeneracy

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The disk--halo degeneracy is the ability of different stellar mass-to-light ratios and dark-halo profiles to reproduce nearly the same total galaxy rotation curve.

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A galaxy rotation curve is a plot that shows how the orbital speeds of stars and gas in a galaxy vary with distance from the galaxy's center. Typically, the x-axis represents the distance from the galactic center (often measured in kiloparsecs or light-years), while the y-axis represents the orbital velocity (usually expressed in kilometers per second). In the context of galaxies, several key points can be highlighted: 1. **Expected vs.

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