Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 72 (Galaxies)

The five printed subparts are separated below. All use one well-mixed closed-box model of galactic chemical evolution with a stellar yield defined per permanently locked mass; the gross star formation rate must be distinguished from that locked-mass rate.

Write the gross star formation rate as $\psi =k{M}_g$, with $k>0$ constant. Use the instantaneous recycling approximation: a fraction $F$ returns promptly, while a fraction $1-F$ stays in long-lived stars and remnants. Assume $0\le F<1$, instantaneous mixing, and no galactic gas inflow or galactic outflow. If $M_s$ denotes the permanently locked stellar mass, mass conservation givesSet $\lambda =(1-F)k$. Integration, with $M_g(0)=M_0$, gives the exponential closed-box gas consumption lawThe locked mass is $M_s=M_0(1-e^{-\lambda t})$, so $M_g+M_s=M_0$ at every time. The decay time is $[(1-F)k{]}^{-1}$, longer than the gross gas-processing time $k^{-1}$ because returned gas can form subsequent generations. The formal $F=1$ limit retains all gas and locks no mass; a finite stellar yield defined per locked mass then needs separate treatment.

Let $Z$ be the well-mixed gas-phase metallicity, measured as a metal mass fraction. Removing gas into stars takes metals at rate $Z\psi$; prompt return restores existing metals at rate $ZF\psi$, together with newly synthesized metals at rate $Y(1-F)\psi$. Thus the closed-box model of galactic chemical evolution has metal budgetExpand the left side and use ${\dot{M}}_g=-(1-F)\psi$. The terms proportional to existing gas-phase metallicity cancel, leavingSince $\psi =k{M}_g$, initially pristine gas obeysThe stellar yield here is per permanently locked mass, so inserting another factor $1-F$ into $Y$ would double-count recycling. The approximation treats metals as a trace constituent and ignores finite stellar-return delays. Apply it while $Y\lambda t\ll 1$; the formal unlimited linear growth is not a physical prediction for almost entirely metallic gas.

A newly formed stellar population inherits the current gas-phase metallicity. Eliminate time between locked-mass growth and enrichment:For $Y>0$, integration from pristine gas gives $M_g=M_0{e}^{-Z/Y}$ and hence the closed-box metallicity distributionTo obtain numbers rather than mass, specify a fixed initial mass function and count a long-lived tracer population. Let $\eta$ be its number of surviving stars per unit locked stellar mass. At the final gas-phase metallicity $Z_f$,with zero density outside that interval. The cumulative count is $N(<Z)=\eta M_0(1-e^{-Z/Y})$ within the interval, and the normalized metallicity distribution function isThese equations do not use $\psi (t)$: any positive star formation rate gives the same distribution under the same chemical assumptions. If the tracer number is instead normalized per gross formed mass, replace $\eta$ by that normalization divided by $1-F$. Short-lived stars require an age-dependent survival factor and generally do not follow this number distribution. In a logarithmic metallicity distribution, $dN/d{\log }_{10}Z=(\ln 10)ZdN/dZ$; its interior peak is at $Z=Y$, whereas the distribution per linear interval decreases from $Z=0$.

Divide metal enrichment by gas depletion in the closed-box model of galactic chemical evolution:The logarithm appears on integrating from $M_0$ and $Z=0$:With the gas fraction of a galaxy $\mu =M_g/M_0$, this isIt gives $Z=0$ at $\mu =1$ and increasing gas-phase metallicity as gas is consumed, independently of the time dependence of the star formation rate. For this closed initially star-free system, $M_0=M_g+M_s$, so this initial-reservoir fraction equals the present gas-to-total baryonic fraction. That equality fails in an open system.

The qualitative prediction of galactic chemical evolution is useful: later stellar populations tend to inherit more enriched gas, and old stellar populations can preserve low stellar metallicity. The closed-box metallicity distribution, however, predicts many more metal-poor long-lived dwarfs in the solar neighbourhood than are observed. This is the G-dwarf problem, also apparent with other long-lived tracers. A single well-mixed box cannot describe the different age and abundance distributions of the galactic disk, galactic bulge, and stellar halo of the Milky Way. Nor need its exponentially declining star formation rate match the history of ongoing disk star formation.

Prolonged relatively metal-poor galactic gas inflow supplies fuel while diluting the enrichment, allowing many stars to form after the earliest enrichment stage; pre-enrichment of the initial reservoir also removes a pristine tail. Galactic outflows lower the effective yield, but a constant well-mixed outflow alone still gives an exponential metallicity distribution function, and is not automatically a cure for the G-dwarf problem. Spatially varying star formation, radial transport and migration, and selection of local survivors complicate a comparison with one global box. A varying initial mass function or stellar yield changes the enrichment law. Finally, finite stellar lifetimes and Type Ia supernovae make iron enrichment delayed, so an observed iron-abundance distribution is not interchangeable with the instantaneous total-metal distribution. Gas exchange, spatially distinct populations, and delayed enrichment are essential qualifications to the closed-box prediction.

The stated kinematics do not, by themselves, fix a galaxy's intrinsic axis ratio. The requested curve is the usual simplified oblate-rotator flattening proxy; its additional geometric and streaming conventions should be explicit. Put $q=b/a=1-e$ and adopt the approximate moment prescriptionHere $v_p^2$ is the ordered support assigned to one in-plane component. Using ${\sigma }_z^2=(1-\beta ){\sigma }_x^2$ and solving givesIn this prescription, isotropic random motions give the oblate isotropic rotator curve; positive $\beta$ allows flattening with less rotation. A real solution requires $0\le e<1$, $\beta <1$, and $e\ge \beta$. Equality $e=\beta$ gives no rotation. This is a useful diagnostic approximation, not an exact identity for arbitrary density distributions.

To see what the tensor virial theorem actually requires, take a finite stationary axisymmetric system with mass-weighted constant velocity dispersions and azimuthal streaming speed $v_{\mathrm{az}}$. Since $u_x=-v_{\mathrm{az}}\sin \varphi$, the azimuthal average has $⟨u_x^2⟩=v_{\mathrm{az}}^2/2$. The $xx$ and $zz$ components giveand therefore, with the oblate tensor-virial shape factor $Q=W_{xx}/W_{zz}$,For similar oblate self-gravitating density surfaces, write $s=\sqrt{1-q^2}$. The ellipsoidal force coefficients, obtained by integrating the Newtonian force over those surfaces, areThe common radial integral in the stellar potential-energy tensor cancels, but these shape coefficients remain. At $q=1/2$, $Q\simeq 1.79$, rather than $1/q=2$. With $\beta =0$, the intrinsic azimuthal ratio is about $1.26$, whereas the printed simplified ratio is $1$. Recovering the latter uses both $v_p=v_{\mathrm{az}}/\sqrt{2}$ and $Q\simeq 1/q$. Thus, if the printed $v$ literally means intrinsic azimuthal speed, its claimed exact relation omits necessary model assumptions and a streaming normalization. The next sketch displays the requested approximate curve with that qualification.

The rotation-anisotropy diagram below uses the oblate-rotator flattening proxy for $\beta =0,0.2,0.4$. Each curve starts at $e=\beta$ and rises as $e$ increases. The plotted domains cover typical spheroid values, roughly $0\le e\lesssim 0.7$ and $0\le v/\sigma \lesssim 1.5$, with the upper axis extended to show the reference curve's rise. The shaded regions are schematic overlapping populations, not measurements or sharp classification boundaries.

Giant elliptical galaxies commonly have low ordered rotation for their observed flattening, lying below the oblate isotropic rotator reference. Anisotropic velocity dispersions and, in many cases, triaxial structure provide support. Lower-luminosity elliptical galaxies are more commonly rotating, disky systems near the isotropic reference. Classical galactic bulges of spiral galaxies often overlap this rotating-spheroid locus; disk-like bulges can be more strongly rotation supported. These are population trends with substantial overlap, not universal rules for individual galaxies.

Observations usually use projected ellipticity and a line-of-sight rotation measure such as peak speed divided by a central velocity dispersion. The intrinsic $e$ and global dispersions used in the derivation are different quantities. Inclination, apertures, and unresolved streaming can change both coordinates. Consequently a point below the reference suggests extra random-motion support, but does not uniquely measure $\beta$.

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.

For a concrete Milky-Way-like illustration, takeand isotropic tracer velocity dispersionWith local tracer density $\nu \propto e^{-R/R_d}$, the exponential-disk asymmetric drift then predictsThe sketch shows the cold gas following $V_c$, rising in the inner galaxy and becoming approximately flat, while the stellar curve sits below it. Only radii $R\ge 1\mathrm{kpc}$ are used for the stellar approximation; a central galactic bulge needs a separate model.

At $R=8\mathrm{kpc}$, $A\simeq 4.22$ and $\sigma =30\mathrm{km}{\mathrm{s}}^{-1}$, giving gas speed about $220\mathrm{km}{\mathrm{s}}^{-1}$, stellar speed about $211\mathrm{km}{\mathrm{s}}^{-1}$, and a lag about $9\mathrm{km}{\mathrm{s}}^{-1}$. These are representative scales for the requested schematic, not a fit to the Milky Way. Hotter old stellar populations would have a larger lag: doubling this dispersion at the same radius gives a stellar speed about $182\mathrm{km}{\mathrm{s}}^{-1}$. The specified radial laws and isotropy make the example reproducible while retaining the qualifications in the preceding derivation.

For a singular isothermal sphere, spherical integration of the mass density givesThe spherical shell theorem makes the radial attraction $GM(<R)/R^2$. Balancing it with circular centripetal acceleration yieldsThus the model has a flat galaxy rotation curve, resembling the outer galaxy rotation curves of many spiral galaxies. Central rotation curves usually rise, and not all real curves are flat throughout. A pressure-supported elliptical galaxy does not itself rotate at this test-particle circular speed.

The model has divergent central mass density, although finite enclosed mass at each finite radius, and an infinite total mass because $M(<R)\propto R$. Its potential is logarithmic, requiring an outer cutoff for a finite escape energy. A globally constant isotropic velocity dispersion, spherical symmetry, and one common profile for all gravitating components are further idealizations. Stellar light commonly has a quite different radial profile from the dark matter halo; real disks are flattened, and halos are neither exact isothermal spheres nor infinite. Tidal truncation also changes the distribution near the satellite boundary.

An orbital prescription is needed to fix the coefficient in a tidal radius. Adopt a low-mass satellite on a circular orbit of separation $D$, with $r_t\ll D$. The host singular isothermal sphere hasIn the corotating frame, subtract the host force on the satellite centre and add the differential centrifugal force. Along the line joining the centres, the outward effective acceleration to first order in displacement $r$ is $({\Omega }^2-{\Phi }_h^{\prime \prime })r=4{\sigma }_h^{2}r/D^2$. Equate this to the dwarf's inward acceleration at its edge:This gives the Jacobi truncation of two singular isothermal spheresIt is the distance to the radial escape saddle, not an exact spherical surface: the actual tidal lobe is distorted. For an eccentric orbit, separation alone does not specify an instantaneous effective angular speed or the stripping history; a pericentric estimate is commonly used with further assumptions.

A simpler order-of-magnitude convention equates mean internal and host densities, $m_d/r_t^3=M_h(D)/D^3$, giving $r_t\sim D{\sigma }_d/{\sigma }_h$. Equivalently it retains the host gradient without the circular-orbit centrifugal contribution. Both reproduce the scaling $r_t\propto D{\sigma }_d/{\sigma }_h$, but their coefficients differ by $\sqrt{2}$. The familiar point-mass factor three is not appropriate for this extended host, whose enclosed mass grows linearly with radius. The circular-orbit Jacobi tidal radius will be used consistently below, with the rough convention's numerical values also given.

Integrate the dwarf singular isothermal sphere only to its truncation radius:ConsequentlyThe dimensions are $(\text{length})(\text{speed}{)}^2/G$, as required for mass. At fixed host dispersion and separation the mass scales as ${\sigma }_d^3$, rather than the luminosity's fourth-power scaling used later. Under the rough equal-density tidal radius convention, the corresponding mass is $2D{\sigma }_d^3/(G{\sigma }_h)$, larger by $\sqrt{2}$.

Using the supplied approximate conversion factors, $D=50\times 10^3\times 3\times 10^{16}=1.5\times 10^{21}\mathrm{m}$, ${\sigma }_d=10^4\mathrm{m}{\mathrm{s}}^{-1}$, and ${\sigma }_h=3\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$. The circular-orbit Jacobi tidal radius isThe dwarf mass isTo one significant figure,The rough equal-density convention instead gives $r_t=1.67\mathrm{kpc}$ and $m_d=7.14\times 10^7{M}_{\odot }$, or $2\mathrm{kpc}$ and $7\times 10^7{M}_{\odot }$ to one significant figure. The unrounded results are retained for subsequent ratios. For the circular prescription $r_t/D\simeq 0.024$, so the local expansion is well controlled.

Use the classical Faber-Jackson relation $L\propto {\sigma }^4$ as the explicitly requested extrapolation. The supplied host fixes its normalization:With the mass derived using circular-orbit Jacobi truncation of two singular isothermal spheres, the dynamical mass-to-light ratio isHence to one significant figureThe rough equal-density truncation convention gives ${{\rm Y}}_d\simeq 6\times 10^2{M}_{\odot }/L_{\odot }$ instead. Both conclusions indicate a very large dynamical mass-to-light ratio compared with a normal old stellar population.

Within their bright stellar bodies, typical giant elliptical galaxies have optical dynamical mass-to-light ratios of order a few to ten in solar units, with variations in band, initial mass function, and dark matter. Large outer apertures can give substantially larger ratios. Many dwarf spheroidal galaxies have much larger inferred ratios, from tens to hundreds and, for some very faint systems, still larger values. Thus the computed ratio is plausible as the scale of a very dark-matter-dominated faint dwarf, not as a purely stellar ratio. Dynamical inference depends on member identification, binary motions, equilibrium, and the assumed mass profile.

The stipulated Faber-Jackson relation is not a reliable universal extrapolation down to every dwarf spheroidal galaxy. Also, the fundamental plane of elliptical galaxies contains radius and surface brightness as well as velocity dispersion; the two-variable Faber-Jackson relation is a projection of that information, not an interchangeable definition of the full plane.

The host mass inside $D$ follows from the same singular isothermal sphere:Using the supplied host luminosity givesThis treats essentially all the stated optical light as contained in that aperture; a measured luminosity profile would refine that convention. Keeping consistent unrounded quantities,Thus the dwarf ratio is about twenty times the parent ratio; the rough equal-density tidal convention would give a factor thirty.

There is no logical contradiction in two objects sharing $L\propto {\sigma }^4$ but having different mass-to-light ratios. A virial theorem scaling gives $M\propto {\sigma }^{2}R$, so that $M/L\propto R/{\sigma }^2$. Equal ratios additionally require homologous relevant radii scaling as $R\propto {\sigma }^2$. Here the satellite's tidal radius scales as ${\sigma }_d$ in the fixed host field, not as ${\sigma }_d^2$. Equating the two ratios would require $r_t=D({\sigma }_d/{\sigma }_h{)}^2\simeq 56\mathrm{pc}$, far smaller than the derived tidal radius.

The heuristic derivation of the Faber-Jackson relation with common mass-to-light ratio and common surface brightness made auxiliary structural assumptions; the empirical relation alone cannot enforce them. Furthermore, the aperture dependence of mass-to-light ratio matters: a satellite's whole tidal mass and a host's $50\mathrm{kpc}$ halo mass are not homologous half-light-aperture masses. Dark matter need not follow stellar light, and a dwarf spheroidal galaxy is not automatically a scaled giant elliptical galaxy. The model calculates total gravitating mass: reading its density literally as stars alone would imply an implausibly large purely stellar ratio, exposing the limits of that interpretation.

The relevant comparison is continuing hierarchical galaxy formation in the Lambda-CDM model against an idealized monolithic galaxy formation model in which both formation and assembly were essentially complete early. Each observation below concerns a population or a fossil record, rather than a currently merging pair. None individually proves that every galaxy formed through the same route.

Fossil substructure in stellar halos. Old stars in the Milky Way's stellar halo include coherent groups in orbital-integral space, and external halos contain extended debris and shells. Phase mixing can erase spatial concentration while retaining correlations of orbital energy or angular momentum. Such stellar streams, when linked to a disrupted dwarf progenitor, record accretion of an already formed stellar population. A smooth single collapse followed by quiet passive evolution has no equally direct mechanism for creating several dynamically and chemically distinct accreted populations. This supports a composite assembly history, without requiring that every halo star was accreted or counting a current merging pair as evidence.

Growth of the quiescent population. The galaxy red sequence contains old stellar populations, yet its inferred stellar mass density grows between approximately redshift one and the present. Pure fading of an already fixed early population changes luminosities and colours, but cannot increase the mass in that population. Quenching adds newly passive systems, while subsequent assembly combines older stellar populations into larger galaxies. The observed evolution therefore requires demographic evolution after the putative completed collapse epoch. It is consistent with hierarchical galaxy formation plus quenching, although red-sequence growth alone does not distinguish assembly from all possible histories of in-situ star formation; stellar-mass estimates and survey selection must be treated consistently.

Evolution of galaxy morphology. Distant galaxy samples contain a larger prevalence of irregular and peculiar structures than a simple non-evolving extrapolation of the local population. Settled discs and smooth spheroids become common outcomes as time proceeds. Continued gas supply, dynamical interactions, and internal rearrangement in hierarchical galaxy formation naturally allow this galaxy morphological evolution. A model in which structure became fixed during one early collapse and then only stellar light faded has difficulty producing the population change. Rest-frame wavelength, resolution, and surface-brightness selection can affect the observed classifications, and clumpiness is not unique evidence for a galaxy merger: unstable gas-rich discs can also be irregular.

Evolution of sizes at fixed stellar mass. Massive high-redshift galaxies, including concentrated systems, are often more compact than comparable populations at low redshift. Galaxy size evolution implies structural growth beyond passive luminosity fading. In particular, deposition of accreted stars at large radii in minor galaxy mergers can enlarge the outer stellar body while preserving an old central stellar population; gas accretion can grow discs from the inside outward. A completed early collapse with a fixed stellar structure does not naturally supply those new outskirts. Changes in sample membership and expansion associated with stellar mass loss also contribute, so this is evidence requiring continued structural evolution rather than a unique quantitative measurement of a merger rate.

Diffuse intracluster stars. Intracluster light and extended outer envelopes show that some stars no longer belong to individual cluster galaxies. Their irregular distribution and substructure retain evidence of redistribution within an assembling cluster. Tidal stripping and disruption of member systems supply stars to the common environment, as expected when smaller bound systems are incorporated into a growing hierarchy. Passive evolution of independent galaxies formed in isolated early collapses cannot itself move their stars into the space between galaxies. Early stellar formation followed by later stripping remains possible, but admitting that later dynamical assembly already departs from the strict completed-collapse picture. This evidence concerns the cumulative stellar distribution, not merely the existence of an occasional present-day merging pair.

Interpret the model comparison at the exam's 2008 epoch. A simulation's failure depends on its treatment of baryons, resolution, and adopted stellar feedback; it is not a theorem that the Lambda-CDM model cannot produce the observation under any physical implementation. The following were important tensions for then-standard realizations, with different implications for monolithic galaxy formation.

Old, alpha-enhanced massive spheroids and downsizing. The most massive quiescent elliptical galaxies tend to contain older stellar populations and elevated alpha element abundances relative to iron, indicating that much of their star formation occurred early and rapidly. Prompt core-collapse enrichment followed by shutdown before substantial delayed iron enrichment accounts for the abundance pattern. Models that spread a massive galaxy's star formation over many late progenitor-merger episodes can make its stars too young or insufficiently alpha enhanced. A rapid early dissipative collapse naturally gives a short formation interval and is therefore more immediately consistent with this particular observation. However, downsizing in galaxy formation is about stellar formation, not necessarily final assembly: late dry galaxy mergers can assemble already old alpha-enhanced stars. Early quenching and an appropriate chemical-enrichment model can consequently reconcile hierarchical assembly with the fossil record.

Thin, high-angular-momentum stellar discs. The existence of extended thin galactic disks, including systems with small galactic bulges, challenged simulations that produced compact central stellar concentrations or overly thick discs. Premature gas cooling into small progenitors, transfer of specific angular momentum to the halo, and repeated dynamical perturbations could cause the galactic angular-momentum catastrophe; satellite encounters also increase galactic disk heating. A comparatively quiet dissipative collapse with approximate conservation of baryonic specific angular momentum more readily produces an extended rotating disc, and avoiding subsequent strong perturbations helps retain thinness. Nevertheless, rapid collapse alone does not explain a disc's prolonged star formation and gas replenishment. Hierarchical histories with quieter late accretion, better feedback, delayed cooling, and dissipative disc regrowth can reduce these tensions; a problem in early implementations is not an unavoidable property of the cosmological framework.

Cores inferred in low-mass galaxies. Resolved inner galaxy rotation curves in a number of low-surface-brightness and dwarf systems favoured shallower central mass densities than the cusps of collisionless cold-dark-matter simulations. For a roughly constant-density dark-matter core, $M(<r)\propto r^3$ and $V_c\propto r$; an inner Navarro--Frenk--White profile has $\rho \propto r^{-1}$, hence $M(<r)\propto r^2$ and $V_c\propto r^{1/2}$. The differing slopes explain the cusp--core problem. Beam smearing, inclination, and noncircular motions complicate inference, but do not make the issue merely a normalization mismatch. A simple early dissipative collapse does not automatically solve it: baryonic contraction can deepen the central concentration. Potential fluctuations from gas loss and stellar feedback can instead redistribute dark matter and modify a cusp. Thus this observation tests the joint halo-and-baryon model rather than preferentially vindicating the old collapse picture.

Rapid formation of the stars and hierarchical assembly of the final galaxy are compatible histories. The comparison must keep those clocks separate, and the historical modelling tensions should not be presented as universal failures of all later simulations.