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.

An empirical scaling between a disk galaxy's total baryonic mass and its characteristic flat rotation speed. The ideal isolated deep-MOND limit predicts $v_f^4=G{M}_b{a}_0$. Observational tests must specify the speed estimator, distances, gas masses and stellar mass-to-light ratios; the empirical relation alone does not determine a unique theory of gravity.

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 $\kappa =v_d^2{R}_d/(G{M}_d)$, 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 $\kappa =1$ as a monopole estimate must not be mistaken for an exact thin-disk theorem.

The spherical shell theorem yields $v_c^2=GM(<r)/r$ only in spherical geometry. A razor-thin axisymmetric disc instead requires $v_c^2=R{\partial }_R\Phi (R,0)$, 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.

A thin ring of mass $m$ and radius $a$ has interior relative potential $Gm[1+R^2/(4a^2)+\cdots ]/a$. Its leading radial acceleration is outward, $GmR/(2a^3)$, unlike the zero interior force of a spherical shell. Exterior disc annuli therefore affect a galaxy rotation curve.

In a spherical potential, the circular speed satisfiesIt measures the gravitational field even when the system itself is not supported by circular motion.

For a test body on a circular Kepler orbit about a point mass $M$, the centripetal acceleration $v_c^2/r$ equals $GM/r^2$, so $v_c=\sqrt{GM/r}$. For a spherical extended source use the enclosed mass $M(<r)$ instead. The formula is not valid for an arbitrary nonspherical gravitational potential.

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 $\rho (r)=v_c^2/(4\pi G{r}^2)$.

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 $\rho =(4\pi r^2{)}^{-1}dM/dr$, 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.

A spherical halo with flat circular speed $v_c$ has $M(r)=v_c^{2}r/G$ and $\rho =v_c^2/(4\pi G{r}^2)$. If it is sharply truncated at $r_{\ast }$, the work per unit mass from $r<r_{\ast }$ to the boundary is $v_c^2\log (r_{\ast }/r)$ and from the boundary to infinity is $v_c^2$. Adding them gives the displayed escape velocity. A bound star's observed radial speed is only a lower bound on its total speed and on the escape speed, so one maximum radial measurement generally gives a lower bound on $r_{\ast }$ rather than an exact halo edge.

Subtracting a finite stellar mass from the distant spherical mass required for a flat galaxy rotation curve gives the displayed asymptotic offset. Extending it to $r=0$ would give negative enclosed mass, so it cannot define a physical inner halo. Keeping only its positive leading term instead gives a singular isothermal halo with density proportional to $r^{-2}$ and a nonzero limiting central circular speed. These are different extrapolations, and neither is forced by an outer rotation curve.

A thin exponential galactic disk has surface density $\Sigma (R)={\Sigma }_0{e}^{-R/R_d}$ and total mass $2\pi {\Sigma }_0{R}_d^2$.

For an exponential galactic disk, the first nonspherical term in the Legendre expansion of thin-disk gravity is positive and proportional to $R^{-3}$ in $v_c^2$. 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.

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 $2.2R_d$.

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.