Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 1

Assume a steady, locally plane-symmetric galactic disk, with negligible radial tilt term in the vertical Jeans equation. Write $\rho (z)$ for the stellar mass density, and $K_z={\partial }_z\Phi$ for the upward gradient of the Newtonian gravitational potential; the actual vertical gravitational acceleration is $-K_z$. Constant vertical velocity dispersion givesIntegrating the plane-parallel Poisson equation for Newtonian gravity from the symmetry plane gives $K_z(z)=4\pi G{\int }_0^z\rho (z^{\prime })d{z}^{\prime }$. Outside most of the mass, this tends to $2\pi G\Sigma$ for $z>0$, where the surface density includes both sides of the galactic disk. Treating this field as constant and integrating the vertical Jeans equation yieldsThis exponential is the thin-sheet approximation or the outer tail, not the exact profile at every height. To see the distinction without an additional assumption, set $\rho ={\rho }_0{e}^{-u}$ with $u=(\Phi -\Phi (0))/{\sigma }^2$. The Poisson equation for Newtonian gravity becomes $u^{\prime \prime }=(4\pi G{\rho }_0/{\sigma }^2)e^{-u}$, with $u(0)=u^{\prime }(0)=0$, and is solved bySince $\Sigma =2{\rho }_0{z}_0$, the exact self-gravitating isothermal slab has $z_0={\sigma }^2/(\pi G\Sigma )$. At large $|z|/z_0$, ${\mathrm{sech}}^2(z/z_0)\sim 4e^{-2|z|/z_0}$, recovering precisely the boxed scale height; near the plane its slope vanishes, whereas an exponential has a cusp.

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.

The thin-sheet field used above requires the gas to sample heights beyond most of the stellar mass, while the stellar galactic disk still dominates vertical gravity and is approximately locally plane-parallel. In inner regions a cold gas layer can lie inside the thicker stellar layer, where reflection symmetry gives $K_z\simeq {\nu }_z^{2}z$ rather than a constant field. The vertical Jeans equation, or gas hydrostatic equilibrium, then givesHere $H_g$ is the Gaussian width and ${\nu }_z$ is the vertical epicyclic frequency; in a locally flat galaxy rotation curve, ${\nu }_z^2\simeq 4\pi G{\rho }_{\mathrm{total}}(R,0)$. A stellar layer of approximately fixed thickness then gives $H_g\propto {\Sigma }_{\ast }^{-1/2}$, rather than the inverse surface density law.

At large radii, the dark matter halo or the gas's own gravity can exceed the dwindling stellar restoring force, so replacing the total force by $2\pi G{\Sigma }_{\ast }$ fails. A nearly spherical dark matter halo instead gives ${\nu }_z\simeq v_c/R$ near its plane, and $H_g\simeq {\sigma }_{g}R/v_c$ for a flat galaxy rotation curve, much slower than exponential galactic disk flaring. Warps, radial gradients across a thick gas layer, non-equilibrium disturbances, and height-dependent velocity dispersion also invalidate the locally plane-parallel steady isothermal approximation. The inverse stellar surface-density law applies only while its force and geometric assumptions apply.

Measure the number-density profile $n(z)$ and vertical velocity dispersion ${\sigma }_z(z)$ of a tracer stellar population. The vertical Jeans equation estimates the restoring field fromThe second term accounts for the tilt of the velocity ellipsoid; an exponential tracer with constant velocity dispersion and negligible tilt gives $K_z={\sigma }_z^2/h$ above the plane. The tracer need not supply all the gravitating mass. To infer the dynamical surface density from vertical stellar motions, integrate the axisymmetric Poisson equation for Newtonian gravity:For a nearly flat galaxy rotation curve with a small radial correction, this reduces to $K_z/(2\pi G)$. Estimate stellar surface density independently from counts, the initial mass function and faint objects, and add the gas surface density and stellar remnants. The residuallimits the allowed unseen material within the same height interval. A fitted dark matter halo contributes too: measurements at several heights help distinguish its extended density from a thin dark component in the galactic disk. Under a specified decomposition, the disc dark-matter fraction is the inferred dark-disc surface density divided by the total disc surface density, with uncertainty or an upper limit if the residual is consistent with zero. Selection effects, disequilibrium, the tilt term and uncertain baryonic counts must enter the error budget; an excess alone does not establish that all of it is in the galactic disk.