Galactic disk

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.

A stellar galactic disk component with greater vertical extent and typically larger random velocity dispersion than a thin galactic disk. In the Milky Way it contains an important old population. Its stars are not defined simply by being in a geometrically sharp slab or by all sharing one age.

An elongated overdensity in a galactic disk, commonly traced by gas, young stars and star formation. An arm pattern need not rotate at the same rate as individual stars.

A collective perturbation of surface density and gravitational potential in a galactic disk. Material can cross its spiral-shaped crests; the pattern angular speed need not equal the local material angular speed. A tightly wound fluid approximation has $(\omega -m\Omega {)}^2={\kappa }^2+c_s^2{k}^2-2\pi G\Sigma |k|$, while a collisionless stellar disk requires its own response function.

For tightly wound perturbations in a locally uniform, razor-thin gas disk, continuity equation and rotating-fluid momentum equations couple the surface-density perturbation to radial and azimuthal velocities. The sheet potential is $\delta \Phi =-2\pi G\delta \Sigma /|k|$. Eliminating these perturbations gives the displayed dispersion relation. radial epicyclic frequency supplies the rotation term, the sound-speed term resists short waves, and self-gravity supplies the negative term. For an axisymmetric mode $n=0$, use ${\omega }^2$ directly; the pattern speed $\omega /n$ is undefined.

Two radii with different orbital frequencies develop a relative angular displacement $\Delta \varphi =\Delta \Omega t$. A spiral made of a fixed set of stars therefore winds increasingly tightly. A density pattern need not follow that material winding because matter can pass through it.

A self-gravitating disk can transiently amplify a leading disturbance as differential rotation shears it into a trailing disturbance. Its strength depends on disk self-gravity, random motions and disturbance wavelength. Amplification does not by itself establish one indefinitely steady spiral.

For a perturbation phase $m\varphi +\int kdr-\omega t$, a constant-phase pattern rotates at ${\Omega }_p=\omega /m$. At corotation the local orbit angular speed equals this value. One observed galaxy can contain several patterns with distinct angular speeds and lifetimes.

A thin galactic disk is a stellar component whose vertical extent is small compared with its radial size. Small vertical velocity dispersion and substantial specific angular momentum are characteristic. Its survival constrains cumulative galactic disk heating, but its stars need not all have the same formation epoch.

Galactic disk heating increases the random velocities of stars through encounters with molecular clouds, nonaxisymmetric gravitational disturbances, and other perturbations. Older stellar populations usually have larger velocity dispersions and greater vertical thickness, although birth conditions and migration also affect the relation.

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 ${\nu }_z$.

A plane-symmetric, self-gravitating stellar population with constant vertical velocity dispersion satisfies the vertical Jeans equation and the Poisson equation for Newtonian gravity. With ${\Phi }^{\prime }(0)=0$, their solution isIts surface density is $\Sigma =2{\rho }_0{z}_0$, so $z_0={\sigma }_z^2/(\pi G\Sigma )$. Its outer density tail is exponential with scale height $z_0/2$; its central profile is not exponential.

For a self-gravitating isothermal slab with $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ and $H^2={\sigma }^2/(2\pi G{\rho }_0)$, hydrostatic equilibrium gives the displayed Newtonian gravitational potential. A test particle has constant in-plane velocity and conserved vertical energy $E_z={\dot{z}}^2/2+\Phi (z)-\Phi (0)$. Its vertical motion is periodic for every finite $E_z>0$, with turning height $H\mathrm{arcosh}\exp [E_z/(2{\sigma }^2)]$. Near the plane it is a harmonic oscillator of squared frequency $4\pi G{\rho }_0$; at large height the force tends to the constant $2\pi G\Sigma$, where $\Sigma =2{\rho }_{0}H$. The ideal infinite slab has no finite vertical escape energy.

For an isothermal gas slab, a horizontal Fourier mode has $\delta \rho =-ik\rho {\xi }_x-(\rho {\xi }_z{)}_z$, $W=c_s^2\delta \rho /\rho +\varphi$, ${\omega }^2{\xi }_x=ikW$, ${\omega }^2{\xi }_z=W_z$ and $({\partial }_z^2-k^2)\varphi =4\pi G\delta \rho$. Multiply momentum by $\rho {\mathbf{\xi }}^{\ast }$ and integrate by parts, then use the Poisson equation for Newtonian gravity. Vanishing pressure-work and gravitational boundary terms give the displayed real quadratic identity, with $I=\int \rho |\mathbf{\xi }{|}^{2}dz>0$. Thus every admissible squared eigenfrequency is real; a negative value gives gravitational growth rather than an oscillatory instability.

For mass density $\rho ={\rho }_0{\mathrm{sech}}^2(z/H)$ with $H^2=c_s^2/(2\pi G{\rho }_0)$, a zero-frequency gravitational mode with $k\ne 0$ obeys $W=0$. Its potential satisfies ${\varphi }_{zz}+[2{\mathrm{sech}}^2(z/H)/H^2-k^2]\varphi =0$. The substitution $\tau =\tanh (z/H)$ gives the associated Legendre differential equation of degree one and order $|k|H$. Solutions decaying at both infinities exist only at $|k|H=1$, when $\varphi \propto \mathrm{sech}(z/H)$. Longer horizontal wavelengths are unstable. Neutral mass-preserving relabeling displacements and the uniform vertical translation at $k=0$ are distinct from this fragmentation threshold.