Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 3

Let $R$ be a characteristic radius enclosing mass $M$, and let $\sigma$ denote a one-dimensional velocity dispersion. Gravity and random-motion support give a relation $GM/R=\eta {\sigma }^2$, where the dimensionless structure coefficient $\eta$ depends on the chosen radius and density profile. A crossing-based galactic dynamical time is thereforeThe scaling follows by equating acceleration scales ${\sigma }^2/R$ and $GM/R^2$, not by identifying a complete orbital period with a crossing time. An explicit scale-free example checks the coefficient: constant circular speed and spherical symmetry give $M(<r)=v_c^{2}r/G$ and $\rho =v_c^2/(4\pi G{r}^2)$. With isotropic constant velocity dispersion, the Spherical Jeans equation becomesso $v_c^2=2{\sigma }^2$, $M(<R)=2{\sigma }^{2}R/G$, and $R/\sigma =GM/(2{\sigma }^3)$. This is the singular isothermal sphere scaling, valid over an interior radial range; a literal infinite model has no finite total mass, and a hard finite boundary needs a pressure term or an outer transition. A free-fall time of a uniform sphere or orbital period changes the numerical factor. The subsequent estimates use the nominal scale $GM/{\sigma }^3$, keeping these order-unity structural uncertainties explicit.

Using the given constants, the mass is $2\times 10^{41}\mathrm{kg}$ and the velocity dispersion is $2\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$. The nominal galactic dynamical time isThus the collapse clock is a few tens of millions of years, of order $10^8$ years, with $R/\sigma \simeq 3\times 10^7$ years in the interior singular isothermal sphere convention. The nominal radius $GM/{\sigma }^2\simeq 3.5\times 10^{20}\mathrm{m}\simeq 12\mathrm{kpc}$ is a plausible galactic scale. A full stellar orbit takes several crossing times, giving order $10^8$ years near a galactic disc radius, whereas the age of the Milky Way is of order $10^{10}$ years. Many orbital passages therefore fit into the lifetime of the galaxy, as expected; a galactic dynamical time need not equal either its full orbital period or its age.

Putting the whole assumed gas reservoir into stars during the nominal galactic dynamical time givesAn order-unity shorter collapse clock would raise this to a few thousand solar masses per year. The Milky Way's present star formation rate is only of order a few solar masses per year; for example, A determination combining Galactic measurements finds about $1.7M_{\odot }{\mathrm{yr}}^{-1}$ with a specified initial mass function. The extreme rapid-collapse rate is therefore roughly three orders of magnitude larger, resembling an intense short-lived starburst galaxy phase rather than ongoing ordinary disc star formation. This estimate assumes all the quoted mass can form stars; it would be smaller if some mass were nonbaryonic dark matter or if the conversion efficiency were below unity.

For an isolated final system with isotropic one-dimensional velocity dispersion $\sigma$, the random kinetic energy is $K=\frac{1}{2}M⟨v^2⟩=\frac{3}{2}M{\sigma }^2$. The virial theorem gives $W=-2K$ and total energy $E_f=K+W=-K$. Starting from a diffuse, almost unbound gas reservoir with $E_i\simeq 0$, conservation of energy therefore requires the radiative energy of virialized gravitational collapseIt is the decrease in total energy, not the whole final magnitude $|W|$, that is lost; half of the binding-energy magnitude remains as kinetic energy. Assuming radiation carries most of this loss, the average luminosity during the nominal galactic dynamical time isusing $L_{\odot }\simeq 4\times 10^{26}\mathrm{W}$. This is comparable in order of magnitude to the stellar luminosity of a characteristic $L_{\ast }$ galaxy in the Schechter function, often of order $10^{10}{L}_{\odot }$ with a band-dependent definition. The equality is an energy-scale comparison: ordinary present starlight is powered mainly by stellar nucleosynthesis, whereas this estimate concerns gravitational energy lost during formation. Appreciable initial binding energy, mass loss or external stresses alter the coefficient, and the finite-boundary singular isothermal sphere cannot be used as an isolated exact virial equilibrium without its boundary term.

The assumed central black hole mass is $M_{\bullet }\sim 10^8{M}_{\odot }\simeq 2\times 10^{38}\mathrm{kg}$. A mass alone does not fix radiated energy: specify a formation mechanism and its radiative efficiency of black-hole accretion. For radiatively efficient accretion, if a fraction $ϵ$ of the supplied rest-mass energy escapes and the remainder increases the black hole mass, then $M_{\bullet }=(1-ϵ)M_{\mathrm{supplied}}$ andTaking the customary order-of-magnitude efficiency $ϵ\sim 0.1$ givesThe coeval accretion phase would be a quasar-scale source, hundreds of times brighter than a typical present $L_{\ast }$ galaxy. Compared directly with the preceding gravitational-collapse estimate,Despite its small mass fraction, black-hole accretion extracts a fraction of rest-mass energy, while galactic collapse releases only an energy scale $M{\sigma }^2\ll M{c}^2$. The numerical luminosity is conditional: a radiatively inefficient accretion flow, direct collapse or assembly by mergers need not radiate this amount over the galactic clock, and the question supplies no unique efficiency.