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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 $R^{-3}$ 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.

Galactic disk flaring is an outward increase of vertical thickness. A constant vertical velocity dispersion in an approximately thin-sheet stellar Newtonian gravitational field gives $h\propto 1/{\Sigma }_{\ast }$. An exponential galactic disk then predicts $h\propto e^{R/R_d}$ for a gas tracer. Inside a thick stellar layer the potential is instead approximately harmonic, and the tracer width is ${\sigma }_z/{\nu }_z$ with vertical epicyclic frequency ${\nu }_z$.

Use cylindrical coordinates and tracer density $\nu (R,z)$. A steady axisymmetric Jeans equation, with no mean radial or vertical flow, isWrite $\overline{v_R^2}={\sigma }_R^2$, $\overline{v_{\varphi }^2}={\bar{v}}_{\varphi }^2+{\sigma }_{\varphi }^2$, and $V_c^2=R{\partial }_R\Phi$. Rearrangement gives the stellar asymmetric drift equationFor an isotropic velocity ellipsoid, all three dispersions equal $\sigma$ and the cross stress vanishes. Therefore the required general result isThe positive root is for a prograde galactic disk. A declining radial stress gives $A>0$: random motions supply some support, so mean stellar rotation is slower than cold-gas circular rotation. For $A{\sigma }^2\ll V_c^2$, $V_c-{\bar{v}}_{\varphi }\simeq A{\sigma }^2/(2V_c)$. The condition $\sigma <V_c$ alone is insufficient; one also needs $A{\sigma }^2\le V_c^2$.

There is no universal numerical coefficient expressible from $V_c$ and $\sigma$ alone. For constant velocity dispersion and $\nu \propto R^{-p}$, $A=p$. Thus $p=2$ gives the often-used estimate ${\bar{v}}_{\varphi }=\sqrt{V_c^2-2{\sigma }^2}$, while $p=1$ gives a different answer with the same $V_c$ and $\sigma$. For an exponential galactic disk tracer with $\nu \propto e^{-R/R_d}$ and ${\sigma }^2\propto e^{-R/R_{\sigma }}$, the exponential-disk asymmetric drift has $A=R(R_d^{-1}+R_{\sigma }^{-1})$.

Real galactic disks generally have anisotropic velocity dispersions, often ${\sigma }_R>{\sigma }_{\varphi }>{\sigma }_z$. In the small-random-motion epicyclic motion limit for a flat galaxy rotation curve, ${\sigma }_{\varphi }^2/{\sigma }_R^2\simeq 1/2$, so even the non-gradient term matters. The velocity ellipsoid can tilt, and bars or spiral structure can violate steady axisymmetry. Age-dependent galactic disk heating makes the lag population dependent. Gas has finite turbulent and thermal stresses too, though a cold gas disk often approximates $V_c$ much better than a hot stellar population.

Measure the logarithmic surface brightness slope in resolved radial annuli after accounting for instrumental blurring and foreground emission. Between a halo core and its outer boundary, dark-matter decay radiation predicts $d\ln I/d\ln R=-1$, whereas dark-matter annihilation radiation predicts $-3$; doubling radius therefore reduces the respective surface brightness by factors $2$ and $8$. These are quantitative templates to compare with independently mapped stellar and gaseous components.

An exponential galactic disk has $I_{\ast }(R)\propto e^{-R/R_d}$, so $d\ln I_{\ast }/d\ln R=-R/R_d$ and $I_{\ast }(2R)/I_{\ast }(R)=e^{-R/R_d}$. At $R=5R_d$ the latter is about $0.0067$, much faster decline than either halo template, and the projected emission should also be flattened along the galactic disk. A galactic bulge described by a Sérsic profile has $d\ln I_{\ast }/d\ln R=-(b_n/n)(R/R_e{)}^{1/n}$; for $n=4$ and $b_4\simeq 7.67$, this is about $-3.4$ at $10R_e$ and continues to steepen, whereas a pure halo power law keeps a fixed slope.

For a stellar halo with outer luminosity density proportional to $r^{-p_{\ast }}$, the projection of a spherical power-law emissivity gives $I_{\ast }\propto R^{1-p_{\ast }}$. Illustratively, $p_{\ast }=3$--$4$ gives projected slopes $-2$--$-3$: then $I_d/I_{\ast }\propto R^{p_{\ast }-2}$ rises strongly outward, but annihilation can resemble the steeper stellar template. Gas-related gamma emission must be compared with observed gas columns and the cosmic ray distribution; it is not always proportional to optical light.

A shallow, extended, approximately spherical $R^{-1}$ component is particularly distinctive; an $R^{-3}$ component needs additional evidence. Fit the stellar, gas and halo templates together over a broad radial range, including morphology and emission spectra. A measured slope alone cannot uniquely identify dark matter, since stellar or gas profiles can coincide with a halo slope over a limited interval.

In the thin-sheet regime, the gas scale height is $h_g(R)={\sigma }_g^2(R)/(2\pi G{\Sigma }_{\ast }(R))$. Hence radius-independent gas velocity dispersion givesFor an exponential galactic disk, ${\Sigma }_{\ast }(R)={\Sigma }_{\ast ,0}{e}^{-R/R_d}$, soThe gas therefore exhibits galactic disk flaring as stellar surface density falls outward. Stellar galactic disks, by contrast, commonly have roughly constant thickness over their main optical region; applying the same scale height relation to the stars requires ${\sigma }_{z,\ast }^2(R)\propto {\Sigma }_{\ast }(R)$, or ${\sigma }_{z,\ast }\propto e^{-R/(2R_d)}$ for an exponential galactic disk. This comparison does not follow by imposing a radially constant stellar velocity dispersion: that extra assumption would predict stellar galactic disk flaring too. The constancy used in the vertical calculation is constancy with height locally, not a mandatory constancy with radius.

For a spherical system, use the spherical shell theorem with shell mass $dm=4\pi \rho (s)s^{2}ds$. Inner shells contribute $-Gdm/r$ to the gravitational potential, while an outer shell contributes its constant interior potential $-Gdm/s$. With $\Phi (\infty )=0$, assuming the required integrals converge,On differentiation, the two terms containing $\rho (r)r$ cancel. Hence ${\Phi }^{\prime }=GM(r)/r^2$, where $M(r)=4\pi {\int }_0^r\rho (s)s^{2}ds$. Radial balance for a circular orbit gives $v_c^2=r{\Phi }^{\prime }=GM(r)/r$.

For a razor-thin axisymmetric astrophysical disk, the element of mass is $dm=\Sigma (s)sdsd{\varphi }^{\prime }$, where $\Sigma$ is the surface density. Superposing Newtonian gravitational potentials therefore givesAxisymmetry permits $\varphi =0$. Unlike the spherical case, an exterior annulus generally exerts a radial force: enclosed mass does not determine a disc rotation curve.

For the Legendre expansion of thin-disk gravity, split the radial integral at $s=R$. In the inner part the kernel expands in $s/R$, while in the outer part it expands in $R/s$. The angular integrals of odd Legendre polynomials vanish, and those of even degree $n$ equal $2{\alpha }_n$, whereIntroduce $I_n(R)={\int }_0^R\Sigma (s)s^{n+1}ds$ and $J_n(R)={\int }_R^{\infty }\Sigma (s)s^{-n}ds$. The gravitational potential becomesWhen differentiating each bracket, the moving-limit terms cancel: $R^{-(n+1)}{I}_n^{\prime }=\Sigma (R)$ and $R^n{J}_n^{\prime }=-\Sigma (R)$. Since $v_c^2=R{\Phi }^{\prime }$, this givesThe zeroth term has ${\alpha }_0=\pi$ and $M(R)=2\pi I_0$. Separating it provesThe inner correction is inward, while the exterior correction is outward. For a smooth surface density, the original potential singularity at coincident points is integrable. Its radial force is understood through a symmetric Cauchy principal value or a vanishing-thickness regularization; the paired interior and exterior terms above retain the cancellation at $s=R$. This avoids treating the two singular local force contributions separately.

For an exponential galactic disk, write $\Sigma (s)={\Sigma }_0{e}^{-s/R_d}$, so $M_{\infty }=2\pi {\Sigma }_0{R}_d^2$ and $M(R)=M_{\infty }[1-(1+R/R_d)e^{-R/R_d}]$. At large $R$, the missing mass and exterior-ring terms are exponentially small. The leading nonspherical interior term is $k=1$, with ${\alpha }_2=\pi /4$ and $I_2(\infty )=6{\Sigma }_0{R}_d^4$. Consequently the exponential-disk Keplerian asymptotic isThe positive leading correction shows that the rotation curve approaches the Keplerian limit from above. The finite-order large-radius expansion is sufficient here; an infinite moment expansion need not converge for a disk extending to infinity.

A useful special example is the Mestel disk, with surface density $\Sigma (s)=C/s$ for $s>0$, $C>0$. It has $M(R)=2\pi CR$. For every positive even $n$,All the nonspherical corrections cancel, givingThus the Mestel disk has a perfectly flat galaxy rotation curve.

This example has infinite total mass and a singular central surface density. The absolute gravitational potential cannot be set to zero at infinity, but the radial force exists as a limit of disks with increasing outer cutoff. Potential differences are $\Phi (R)-\Phi (R_0)=v_M^2\log (R/R_0)$. The earlier potential integral is therefore interpreted up to a radius-independent divergent constant for this example; the force calculation remains valid. Truncating the Mestel disk gives a more physical finite system but changes the exact flat curve near its edges.

Put $x=r/r_s$. The enclosed mass of the Navarro--Frenk--White profile isSince $y/(1+y{)}^2=(1+y{)}^{-1}-(1+y{)}^{-2}$,The circular-orbit condition $v_c^2/r=GM(r)/r^2$ then gives the galaxy rotation curve

For $x\ll 1$,and thereforeFor $x\gg 1$,soThe NFW curve thus rises from the center, peaks, and eventually declines slowly. Typical spiral-galaxy rotation curves remain approximately flat over the observed outer disk, although an NFW halo combined with baryons can approximate such a plateau over a finite interval.

At the maximum, $x_m=2.16$, so the scale radius of a Navarro--Frenk--White profile isWriting $f(x)=\log (1+x)-x/(1+x)$, the characteristic density of a Navarro--Frenk--White profile isWith $G=4.3009\times 10^{-6}\mathrm{kpc}(\mathrm{km}{\mathrm{s}}^{-1}{)}^2{M}_{\odot }^{-1}$ and $f(2.16)=0.4670$, this is

An NFW cusp has $\rho \propto r^{-1}$ and hence $v_c\propto r^{1/2}$. Many dwarf and low-surface-brightness galaxies instead favor a nearly constant-density core, for which $M(r)\propto r^3$ and $v_c\propto r$; after matching the outer speed, this gives the more slowly rising observed inner curve. This is the cusp--core problem. Repeated burst-driven gas outflows can fluctuate the central potential and transfer orbital energy to dark matter, while self-interacting dark matter provides another possible core-forming mechanism. Beam smearing and noncircular motions are observational systematics but do not explain every case.

For an exponential galactic disk,soSince $7\mathrm{kpc}\simeq 2.3R_d$ is near the disk maximum, assigning the full $220\mathrm{km}{\mathrm{s}}^{-1}$ there givesIf the disk mass is increased while the observed total curve is fixed, the halo contribution must decrease in the inner galaxy. An NFW fit therefore needs a lower characteristic density, a larger scale radius and lower concentration, or a correlated combination of these changes.

A maximum disk model raises the stellar mass-to-light ratio as far as the rotation curve permits, conventionally making the disk contribute about $85\%$ of the speed near $2.2R_d$. High stellar surface densities, population-synthesis mass-to-light ratios, microlensing optical depths, and fast bars that have suffered little dynamical-friction braking can support a large disk contribution. Directly measured vertical stellar dispersions often favor submaximal disks, especially in low-surface-brightness galaxies, while stability and cosmological halo constraints may also disfavor an excessively massive disk.

The disk--halo degeneracy follows fromOver a finite radial range, raising the stellar mass-to-light ratio and lowering or broadening the halo can leave the same sum. The degeneracy can be reduced by measuring the disk surface density dynamically from vertical velocity dispersion and scale height, or by constraining the stellar mass-to-light ratio with resolved populations, colors, and a specified initial mass function. Strong or weak gravitational lensing, gas-layer flaring, stellar streams, and rotation data far beyond the optical disk provide further independent halo constraints.