Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2013 / Paper 1 / 1

heating and cooling balance in the interstellar medium depends on density, ionization, radiation and shielding. Write net cooling per volume as $\mathcal{C}=n^2\Lambda (T)-n\Gamma$. Ultraviolet grain photoelectric heating is important in diffuse neutral gas; photoionization heats ionized regions, cosmic rays penetrate shielded gas, and shock waves from stellar winds and supernovae supply hot gas. Cooling includes collisionally excited atomic lines, fine-structure metal lines at lower temperatures, molecular lines in shielded clouds, and continuum radiation and dust emission where appropriate. The heating efficiency and cooling abundances matter as much as a nominal temperature.

A multiphase interstellar medium can contain a cold neutral medium near $10^2\mathrm{K}$, a warm neutral medium and warm ionized medium near $10^4\mathrm{K}$, a hot ionized medium near $10^6\mathrm{K}$, and still colder molecular clouds. Cold and warm atomic equilibrium branches can coexist at similar thermal pressure over a finite pressure interval; this is a calculated heating/cooling balance, not a requirement that all galactic gas have the same pressure.^[ref]

For net cooling per unit mass $\mathcal{L}=\mathcal{C}/\rho$, an isobaric perturbation at equilibrium obeys $c_p{\partial }_t\delta T=-(\partial \mathcal{L}/\partial T{)}_p\delta T$. Thus a negative constant-pressure cooling derivative is thermally unstable: a cool perturbation cools further and condenses. Sound crossing must be fast enough to maintain pressure balance; thermal conduction suppresses sufficiently short wavelengths. Supernova-driven motions, magnetic pressure, finite cooling times and interfaces make the actual medium dynamic, so equilibrium branches describe useful phases rather than a static collection of perfectly separated components.

differential rotation winds a material arm because $d\Omega /dr\ne 0$. A spiral density wave instead has a pattern speed different from the material speed: stars and gas can pass through its crests. Collective self-gravity, orbital resonances and shear can sustain or repeatedly amplify non-axisymmetric patterns; a visible spiral need not be a single permanent rigid material structure.

For a razor-thin gas disk, local tightly wound disturbances satisfyRotation stabilizes long wavelengths and pressure stabilizes short ones. Minimizing the axisymmetric right-hand side over $|k|$ gives $|k{|}_{\ast }=\pi G\Sigma /c_s^2$ andfor local axisymmetric stability. A collisionless stellar disk instead has $Q_{\ast }={\sigma }_R\kappa /(3.36G{\Sigma }_{\ast })$; the radial random velocity dispersion is not literally a gas sound speed.^[ref]

Local axisymmetric stability does not guarantee global disk stability. Gas and stars gravitate together, thickness modifies the short-scale force, and a disk with $Q>1$ may still have spiral or bar responses. Dissipation in gas can generate shock waves and concentrate star formation near a spiral crest; stellar random motions tend to reduce the collective response.

Galactic star formation is limited by how much gas becomes cold, shielded and self-gravitating, and by how long it remains in that state. A useful local estimate isThe star-formation efficiency per free-fall time ${ϵ}_{\mathrm{ff}}$ absorbs the fact that turbulence, magnetic support and stellar feedback prevent all eligible gas from collapsing in one free-fall time of a uniform sphere. Rotation and disk stability influence the supply of dense clouds, while gravitational collapse and gravitational fragmentation operate inside them.

For galaxy-averaged measurements, the Kennicutt–Schmidt law has ${\Sigma }_{\mathrm{SFR}}\propto {\Sigma }_g^N$, with the classic mixed spiral/starburst sample giving $N\simeq 1.4$. This empirical relation was obtained using H-alpha and far-infrared star-formation tracers together with atomic-hydrogen and CO gas measurements.^[ref] Dust attenuation, contamination by older stellar heating, the stellar initial mass function and the CO-to-molecular-gas conversion must be controlled. A disk-averaged total-gas exponent is not automatically a small-scale cloud law.

Young stars then photoionize gas, drive winds and eventually inject supernova energy and momentum. These processes can disrupt clouds, sustain random motions or expel gas; radiative cooling and fresh inflow rebuild the cold supply. Large-scale star formation is a regulated conversion of gas, rather than unimpeded free fall of the entire disk. A long gas-depletion time ${\Sigma }_g/{\Sigma }_{\mathrm{SFR}}$ is compatible with short galactic dynamical times in the dense fraction.

A dense cloud core first collapses when self-gravity overwhelms its effective support. While radiation escapes readily, compression can remain approximately isothermal and the Jeans mass decreases as density rises, encouraging gravitational fragmentation. Angular momentum usually leads to a disk rather than unrestricted spherical infall.

As the central optical depth increases, cooling becomes slower than compression and pressure produces a first hydrostatic core. molecular hydrogen dissociation near $2000\mathrm{K}$ absorbs compressional energy and lowers the effective adiabatic response, permitting a second collapse of a protostar. After dissociation the gas stiffens and a much more compact stellar core forms. An accretion shock and surrounding disk then convert infall energy into radiation. A small central core forms before most of the envelope arrives, as demonstrated in radiation-hydrodynamic collapse calculations.^[ref]

The isothermal inside-out collapse of a singular isothermal sphere model provides one idealized supply law, $\dot{M}\sim a_T^3/G$, but rotation, magnetic fields and an unsteady reservoir modify real accretion. Its luminosity scale is $L_{\mathrm{acc}}\sim GM\dot{M}/r_{\ast }$. Dust can absorb the emerging short-wavelength radiation and re-emit it in the infrared, so an embedded source need not have a directly visible stellar photosphere.

As envelope accretion declines, gravitational contraction becomes important on the Kelvin-Helmholtz cooling time $t_{\mathrm{KH}}\sim G{M}^2/(r_{\ast }L)$. deuterium-burning energy-generation rate can temporarily influence the radius; sustained core hydrogen burning eventually establishes the main sequence if the final mass is sufficient. A low-mass convective object approaches along a Hayashi-like track, while more massive objects need not share that track. accretion, contraction and nuclear burning are distinct energy sources, and a protostar is not generally a fixed-mass object.