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The midplane galaxy rotation curve fixes the radial derivative of the Newtonian gravitational potential, not every exterior boundary condition. Adding $a|z|$ leaves that radial derivative unchanged but adds a uniform sheet of surface density of a disk $a/(2\pi G)$. The pure scale-free Mestel disk excludes such extra sources.

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.

Dark matter is an inferred gravitating component whose mass is not accounted for by the observed luminous material. Galaxy rotation curves, motions in galaxy clusters and gravitational lensing probe its gravitational contribution through different measurements.^[ref]

For illustration, circular motion in a spherical mass distribution obeys $v_c^2=GM(<r)/r$. A roughly constant orbital speed over increasing radius therefore requires enclosed mass to keep increasing in this model. A realistic inference also models visible matter and the system's geometry; a rotation curve alone is not a direct identification of a new particle.

Dark-matter direct detection searches for interactions in terrestrial targets, while indirect detection of dark matter looks for secondary products. These tests address microscopic models rather than replacing the astronomical gravitational evidence.

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.

A galactic disk is a flattened stellar or gaseous component of a galaxy, commonly supported mainly by ordered rotation. Its radial mass profile determines its galaxy rotation curve; thickness, truncation and external forces also matter. A galactic disk need not be an exponential galactic disk, and its neutral-gas absorption radius need not equal a stellar scale length.

For an axisymmetric reflection-symmetric thin disk, the Laplace equation in cylindrical coordinates separates into regular radial Bessel functions and vertical decaying exponentials. The divergence theorem imposes ${\Phi }_z^{+}-{\Phi }_z^{-}=4\pi G\Sigma$. Hankel transform inversion gives $S(k)=-2\pi G{\int }_0^{\infty }\Sigma (R)J_0(kR)RdR$. Its radial derivative gives the galaxy rotation curve. Infinite scale-free models may require subtracting an additive divergent constant.

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.

Jacob Bekenstein developed tensor-vector-scalar gravity, a relativistic theory intended to realize modified Newtonian dynamics. The construction supplements the metric tensor with dynamical scalar fields and vector fields.

A nonrelativistic rule for stellar motion alone does not determine how light bends. His 2004 theory addressed both slow-particle dynamics and gravitational lensing, while examining the high-acceleration limit and Solar-System tests. This made the proposal a broader physical framework with multiple observational consequences, rather than merely a fit to a galaxy rotation curve.^[ref]

A Jacobi tidal radius estimates the radial saddle distance from a low-mass satellite on a circular orbit in a smooth spherical host. Let the orbital radius be $r_o$, the host potential be ${\Phi }_h$, andIn the satellite's rotating frame, linearizing the combined host and centrifugal acceleration along the host-satellite line gives $[{\Omega }^2-{\Phi }_h^{\prime \prime }(r_o)]x$. Balancing this outward acceleration against the satellite's inward gravity givesFor an extended satellite, $M_s$ is its enclosed mass at $r_t$, so the equation can be implicit.

Differentiate ${\Phi }_h^{\prime }=G{M}_h/r^2$ and write $\alpha =d\log M_h/d\log r$ at $r_o$. Then ${\Phi }_h^{\prime \prime }=(\alpha -2){\Omega }^2$ andA finite radial saddle in this leading approximation requires $3-\alpha >0$, as well as $r_t\ll r_o$. For a point-mass host, $\alpha =0$ gives the familiar denominator $3M_h$. A flat galaxy rotation curve has $\alpha =1$ and instead gives $2M_h$.

Inside an exactly constant-mass density core, $\alpha =3$ makes the linear radial stretching coefficient vanish. Dividing by zero is not a prediction of a finite tidal boundary there; higher-order structure or a different model is needed. The Jacobi tidal radius describes a rotating-frame equilibrium scale, not a spherical material wall or an exact stripping rule for eccentric, evolving orbits. The distinction from the inertial differential gravitational field is explained by the tidal-gravity coefficient versus rotating Jacobi coefficient.

For a thin axisymmetric astrophysical disk, angular integration of the Newtonian kernel eliminates odd Legendre polynomials. Define $I_n={\int }_0^R\Sigma (s)s^{n+1}ds$ and $J_n={\int }_R^{\infty }\Sigma (s)s^{-n}ds$. The even coefficient is ${\alpha }_n=\pi [n!/(2^n((n/2)!{)}^2){]}^2$. In differentiating the paired inner and outer contributions, their moving-boundary terms cancel. The resulting galaxy rotation curve includes an inward inner-ring contribution and an outward exterior-ring contribution. For smooth surface density, the force at the disk is interpreted with a symmetric Cauchy principal value or a vanishing-thickness regularization.

A Mestel disk is a thin axisymmetric astrophysical disk with surface density inversely proportional to radius. Its galaxy rotation curve is exactly flat, $v_c=v_M$, and $v_c^2=GM(R)/R$ despite its nonspherical geometry. In the Legendre expansion of thin-disk gravity, all positive-degree interior and exterior terms cancel. Its total mass is infinite, so only potential differences, rather than a potential zero at infinity, are defined.

A low-acceleration modification associated with Mordehai Milgrom. For an isolated spherical system, $g\mu (g/a_0)=g_N$, with $\mu (x)\to 1$ at high $x$ and $\mu (x)\to x$ at low $x$. An isolated point-like baryonic distribution then gives $v_c^4=G{M}_b{a}_0$ and a flat asymptotic galaxy rotation curve. The algebraic spherical relation is not a complete relativistic field theory and should not be assumed to hold vectorially for arbitrary mass distributions.^[ref]

Mordehai Milgrom proposed modified Newtonian dynamics as an alternative explanation of the discrepancy between visible mass and inferred gravitational effects in galaxies. The modification becomes important below a characteristic acceleration scale $a_0$.

In the low-acceleration spherical limit, $a^2/a_0=GM/r^2$. For a circular orbit, substituting $a=v^2/r$ gives $v^4=GM{a}_0$, independent of orbital radius. The same limiting rule therefore links a flat outer galaxy rotation curve with a relation between baryonic mass and orbital speed. These are consequences of the proposed dynamics, not a direct detection of the underlying mechanism.^[ref]

For a spherical mass distribution, the shell theorem says the circular gravitational acceleration is $GM(<R)/R^2$. Balancing it against the centripetal acceleration $V_c^2/R$ givesAt $R=16\mathrm{kpc}=4r_0$, the galaxy rotation curve gives $V_c=200(1-e^{-4})=196.337\mathrm{km}{\mathrm{s}}^{-1}$. With $R=4.93708\times 10^{20}\mathrm{m}$,This is the total dynamically inferred enclosed mass, including any dark matter; the galaxy rotation curve alone does not separate its components.

At large radius $V_c\to V_0$, so the spherical enclosed mass is $M(<r)\simeq V_0^{2}r/G$. A spherical mass density satisfies $dM/dr=4\pi r^2\rho (r)$, and henceEquivalently, a local power law $\rho \propto r^{-p}$ with enclosed mass dominated by that extended profile has $M\propto r^{3-p}$ and $V_c^2\propto r^{2-p}$. A flat galaxy rotation curve requires $p=2$.

Differentiating the exact mass profile verifies the limit and the finite central density:As $x\to 0$, the bracket is $3x^2+O(x^3)$, so $\rho (0)=3V_0^2/(4\pi G{r}_0^2)$. As $x\to \infty$, it tends to one, recovering the $r^{-2}$ tail. Extending the flat galaxy rotation curve to infinity makes total mass diverge linearly; a real finite halo must eventually depart from this idealization.

For approximately circular motion in a spherically dominated Newtonian gravitational field, $v_c^2(r)=GM(<r)/r$. If most gravitating mass followed the concentrated luminous galaxies, then beyond their main light distribution the enclosed mass would nearly saturate, giving a declining Keplerian galaxy rotation curve, $v_c\propto r^{-1/2}$.

Instead, extended galaxy rotation curves commonly remain roughly flat. The spherical mass profile for a flat rotation curve then requiresThe inferred mass continues growing where the luminous contribution is small, so the mass-to-light ratio increases outward. Under the assumed gravitational dynamics this is evidence for an extended dark matter halo. Measurements of neutral-gas motion can probe radii beyond the bright stellar disk. A realistic disk contribution must be computed using its flattened geometry; the spherical formula describes a halo-dominated region and does not turn every disk rotation curve into a spherical mass profile.

Observed galaxy rotation curves commonly remain approximately flat across much of the measured outer luminous disk. A pure NFW profile is not an exactly flat curve: it rises in the inner cusp and ultimately declines. Nevertheless, its broad maximum can look nearly flat over a finite radial range, so observing a flat segment does not by itself exclude an NFW halo.

The measured circular speed responds to the total gravitational potential, including stars and gas as well as dark matter. For components in the same plane,For spherical components this reduces to $G{\sum }_j{M}_j(<r)/r$; a disk contribution must instead be computed from its actual disk potential. Baryonic mass can supply much of the inner rotation and modify the halo through slow contraction or more impulsive evolution. Thus the appropriate comparison is with the combined mass model, not an isolated halo-only curve. At arbitrarily large radius a finite isolated galaxy must eventually have a declining circular speed; an observed flat segment is not a claim of flat rotation to infinity.

Use $M$ for the mass parameter throughout; the PDF uses $m$ in its formulas and $M$ in the subsequent prose for the same quantity. Set $a=r_j>0$. The Jaffe model density simplifies toIts enclosed mass isIt tends to $M$ as $r\to \infty$, verifying the total mass. It tends to zero at the center, so the cusp carries no extra point mass.

Spherical symmetry gives ${\Phi }^{\prime }(r)=GM(r)/r^2=GM/[r(r+a)]$. Integrating with $\Phi (\infty )=0$ givesThis also directly satisfies the Poisson equation for Newtonian gravity: differentiating $r^2{\Phi }^{\prime }=GMr/(r+a)$ gives $4\pi G{r}^2\rho (r)$. The circular speed isFor $r\ll a$, $v_c=\sqrt{GM/a}[1-r/(2a)+O((r/a{)}^2)]$ is approximately constant. For $r\gg a$, $v_c=\sqrt{GM/r}[1-a/(2r)+O((a/r{)}^2)]\propto r^{-1/2}$. Thus the finite-mass model has an inner nearly flat galaxy rotation curve and an outer Keplerian decline.

Inside the uniform sphere, $V_c^2=3{\sigma }^2{r}^2/r_0^2$. Thus, with $X=V_c/(\sqrt{2}\sigma )$, part (b) givesThe Gaussian integral needed for the lowered Maxwellian velocity distribution isSubstitution $v=\sqrt{2}\sigma y$ therefore yieldsSince $V_c<v_e$ throughout the core, the integrand is positive and so is $I(X)$ for $X>0$. Using $m{n}_0={\rho }_0$, part (a)'s massive-test-body drag magnitude isThe slow drift of a circular orbit is determined by torque, not by equating its changing circular speed to the tangential drag. Here $V_c=\omega r$, with constant $\omega =\sqrt{4\pi G{\rho }_0/3}$. The specific angular momentum $j=r{V}_c=\omega r^2$ obeysHence the positive local decay timescale isThere is a genuine factor-of-two discrepancy with the requested denominator $32{\pi }^2$. That denominator would give $V_c/(2D)$, which is the timescale for a flat galaxy rotation curve, not this uniform-density core. It also follows from the incorrect replacement ${\dot{V}}_c=-D$, which drops the $V_c\dot{r}$ term in $d(r{V}_c)/dt$. The literal signed definition $r/(2\dot{r})$ is negative for an inward drift; it equals $-{\tau }_{+}$. A positive decay time requires the absolute value or a minus sign.

This is the slow circular inspiral under tangential drag calculation. Its assumptions require $D$ small enough for nearly circular adjustment and a meaningful large Coulomb logarithm in stellar dynamics; the leading-log approximation is not uniform at $V_c\to 0$. Also, integrating the truncated distribution all the way to $v_e$ does not give exactly $n_0$. The model uses $n_0$ as the stated core normalization; exact self-consistency of a sharply truncated uniform core with that distribution is not implied.

An integral of motion is a function of phase space that is constant along a trajectory. Put $\varphi =-\psi +\mathrm{constant}$, so the stellar acceleration is $\dot{\mathbf{v}}=\nabla \psi$. In a time-independent spherical Newtonian gravitational potential,The second equality uses the radial direction of $\nabla \psi$. Thus both the binding energy and the full specific angular momentum vector are integrals of motion.

The Collisionless Boltzmann equation isFor a stationary galactic distribution function, this says $df/dt=0$ on every characteristic, so $f$ is constant on each stellar orbit. Locally choose a complete set of orbit labels $I_j$, themselves integrals of motion; constancy along the characteristic means $f=F(I_1,\dots ,I_s)$. Conversely, for any such differentiable $F$, the chain rule gives $df/dt={\sum }_j{F}_{,j}{\dot{I}}_j=0$. This proves Jeans theorem in its orbit-invariance form. It does not claim that just two particular integrals classify every orbit in an arbitrary potential. For the specified spherical model, $F(E,L)$ is a rotationally invariant stationary choice.

For the velocity integration, write $v_r=v\cos \eta$, $v_{\perp }=v\sin \eta$, $L=rv\sin \eta$, and set $k=r/r_a$. At fixed position the velocity-dependent factor isUse ${\int }_0^{\infty }{v}^2{e}^{-A{v}^2/(2{\sigma }^2)}dv=\sqrt{\pi /2}{\sigma }^3{A}^{-3/2}$ and integrate over $\chi$. The mass density isSince ${\rho }_c=\rho (0)={\rho }_1{e}^{\psi (0)/{\sigma }^2}$,These integrals use the stated untruncated galactic distribution function over all velocities; an additional escape-energy cutoff would change the result.

In local orthogonal velocity components the same exponent is $-[v_r^2+(1+k^2)(v_{\theta }^2+v_{\varphi }^2)]/(2{\sigma }^2)$. Each component is a centred Gaussian distribution. The one-dimensional identity $\int v^2{e}^{-A{v}^2/(2{\sigma }^2)}dv/\int e^{-A{v}^2/(2{\sigma }^2)}dv={\sigma }^2/A$ yieldsIn particular the velocity-anisotropy parameter is ${\beta }_{\mathrm{an}}=1-({\sigma }_{\theta }^2+{\sigma }_{\varphi }^2)/(2{\sigma }_r^2)=r^2/(r_a^2+r^2)$. This is Gaussian angular-momentum suppression in a stellar distribution.

The radial laws also require self-consistency through the Poisson equation for Newtonian gravity:For $r\ll r_a$, outside a sufficiently small core, the model is almost isotropic and isothermal. Seek its scale-free envelope with $\rho =A/r^2$. Its distribution-density relation gives ${\psi }^{\prime }=-2{\sigma }^2/r$, and the Poisson equation for Newtonian gravity gives $A={\sigma }^2/(2\pi G)$. Thus the intended intermediate singular isothermal sphere behaviour iswhere a core scale is $r_c=\sigma /\sqrt{4\pi G{\rho }_c}$.

The stated $r^{-2}$ law cannot be the actual central limit of a regular finite-density model. For finite ${\rho }_c$ and $\psi (0)$, regularity and the Poisson equation for Newtonian gravity instead giveand $v_c^2=(4\pi G{\rho }_c/3)r^2+O(r^4)$. A broad intermediate $r^{-2}$ region exists only if $r_c\ll r_a$; the radius $r_a$ alone does not ensure it.

For the outer radially anisotropic isothermal halo asymptotics, put $s=\log (r/r_a)$, $y=[\psi (r)-\psi (0)]/{\sigma }^2$ and $\lambda =4\pi G{\rho }_c{r}_a^2/{\sigma }^2$. At $r\gg r_a$ the density relation and Poisson equation becomeThe slow outer balance is $y_s\sim -\lambda e^y$, so $d(e^{-y})/ds\sim \lambda$, giving $e^y\sim 1/(\lambda s)$. Its consistency check is $y\sim -\log (\lambda s)$, for which $y_{ss}/y_s\sim -1/s\to 0$. ThereforeThe reference radius $r_{\ast }$ changes only subleading logarithmic terms. The circular speed falls as $(\log r{)}^{-1/2}$, rather than remaining exactly flat.

Physically, the model has an almost isotropic central region and, if the scales separate, a flat-rotation isothermal envelope. Beyond $r_a$, suppression of large specific angular momentum leaves predominantly radial orbits: radial velocity dispersion remains $\sigma$, while transverse dispersion falls as $r^{-1}$. The outer density acquires a logarithmic suppression and the galaxy rotation curve declines slowly. The enclosed mass still grows approximately as $r/\log r$, so this untruncated model has infinite total mass.