Past exam of the formation of Structure in the Universe course of the University of Cambridge

2016 paper: Public original PDF, physical page 810. Student-answer availability for this paper remains unverified.

Original paper: Public original PDF, physical pages810–812; question pages811–812.

Write $A=\lambda \beta {\ell }_E$, where ${\ell }_E=L_{\mathrm{Edd}}/M_{\mathrm{BH}}$. For ionized hydrogen with Thomson scattering opacity, radiation force on one electron-proton pair balances gravity whenThe mass-loading coefficient $A$ has units of inverse time. With the stated neglect of returned and directly accreted gas, mass conservation givesAt a specified stellar mass, the gas-balance nullcline isA positive reservoir exists only below the limiting massEquality has zero gas, so this is the supremum of masses with a positive instantaneous equilibrium. For $\Phi ,ϵ,A>0$, the corresponding gas fraction isIt decreases from one to zero: the derivative is $-ϵ\Phi /[\Phi +(ϵ-A)M_s{]}^2<0$. This is a gas regulator with an outflow increasing with the stellar mass through the assumed black-hole scaling.

There is an important time-dependence qualification. A positive gas mass cannot remain exactly constant forever while the stellar mass continues to increase and $M_{\mathrm{BH}}=\beta M_s$ is maintained: differentiating ${\dot{M}}_g=0$ would require $AϵM_g=0$. The expression above is the instantaneous equilibrium, or a quasisteady approximation when gas relaxation is fast relative to stellar-mass evolution. The only exact fixed point of the coupled idealized equations is $(M_g,M_s)=(0,\Phi /A)$. It is not a sustained star-forming state.

Above the limiting mass, ${\dot{M}}_g<0$ for every nonnegative reservoir: the assumed Eddington-driven wind already demands more than the incoming supply, even before star formation consumes gas. The reservoir is depleted and star formation is quenched. The equations must cease to apply when gas runs out; one cannot continue to negative gas mass or power an indefinitely Eddington-luminous nucleus without fuel. At fixed $M_s>M_{s,\max }$ the depletion time from $M_{g,0}$ isAllowing $M_s$ to grow only strengthens the initial depletion tendency. If $\lambda \beta =0$, this particular feedback threshold is absent.

Distinguish the rest-mass supply ${\dot{M}}_{\mathrm{in}}$ from the actual black-hole mass growth. With the usual radiative-efficiency definition,At the Eddington limit the growth is exponential:The available time is the difference of the supplied ages, $\Delta t=650\mathrm{Myr}$; no new cosmological age-redshift relation is needed.

Using the printed ${\sigma }_e=6.6\times 10^{-29}{\mathrm{m}}^2$, together with $G=6.67430\times 10^{-11}{\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $m_p\simeq 1.67262\times 10^{-27}\mathrm{kg}$, $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$ and a Julian year of $31557600\mathrm{s}$, givesThereforeThis is about $13.09$ e-folds. The constants just listed are evaluation constants, not additional observational inputs asserted to be on the paper.

Sometimes an elementary growth convention identifies ${\dot{M}}_{\mathrm{BH}}$ directly with $L/(ϵc^2)$ and drops the radiated fraction of the incoming rest mass. That approximation instead gives $t_{\mathrm{Sal}}\simeq 44.69\mathrm{Myr}$ and $M_0\simeq 9.7\times 10^2{M}_{\odot }$. The first result retains the physically distinct supplied and retained masses; the convention must be stated because exponential growth magnifies this difference.

The main implication is already severe in the optimistic calculation: a low-mass stellar-remnant seed has insufficient time under the specified efficiency and strictly Eddington-limited accretion. A heavier seed, earlier formation, a different efficiency, mergers or episodes beyond that idealized luminosity limit can relax the inference. Any duty cycle below unity or sub-Eddington episodes require a larger initial mass within the adopted model. The reported high-redshift quasar motivating this calculation was identified spectroscopically at $z=7.085$; its published age estimate need not replace the ages supplied here.^[ref]

Only the photon energy above the hydrogen ionization threshold initially becomes photoelectron kinetic energy. The stated mean therefore suppliesLet $\mathcal{R}$ be the local number of ionizations per volume per time. Photoionization equilibrium makes this equal to the recombination rate. If $⟨E_{\mathrm{rec}}(T)⟩$ is the mean kinetic energy removed per recombination, thermal balance reduces toNeither the ionization threshold nor the density remains in this event-by-event balance. The threshold energy is stored as atomic ionization energy and returned in recombination radiation; it is not a net thermal input. This separation between photoelectron heating and kinetic energy lost by recombining electrons is the relevant cooling balance.^[ref]

In the elementary mean-thermal-energy approximation, a captured electron removes $\frac{3}{2}{k}_{B}T$. This gives the concise estimateThat coefficient is a stated approximation, not a universal recombination identity. Real recombination weights the electron distribution by its velocity-dependent capture rate. If the chosen cooling model instead writes $⟨E_{\mathrm{rec}}⟩={\eta }_{\mathrm{rec}}(T)k_{B}T$, the precise answer is ${\eta }_{\mathrm{rec}}(T_{\mathrm{eq}})T_{\mathrm{eq}}=T_{\ast }$. The source supplies no recombination cooling coefficient or capture kernel from which a unique more accurate coefficient can be computed. Its emitted-photon average is also being used as the average of the absorbed ionizing photons; spectral filtering would require a different weighted average.

Use the supplied flat radiation-era background equations. Write $\overline{\rho }{R}^4=C$, with $C$ constant. On the expanding branch,Choosing the origin at the big bang givesThe supplied perturbation equation consequently becomesA trial $\Delta =t^p$ gives $p(p-1)+p=1$, so $p=\pm 1$. Thus both independent modes areThe growing mode is ${\Delta }_{\mathrm{grow}}\propto t\propto R^2$, and the other decays as $t^{-1}$. At sufficiently late times any nonzero growing-mode coefficient dominates while linearization remains valid.

The calculation follows the stated differential equation, including its coefficient $32\pi G/3$. The name Einstein-de Sitter ordinarily denotes a flat pressureless matter universe, whose growth law is different; here the explicit radiation scaling fixes the intended background. Nor does this scale-free equation describe every subhorizon radiation perturbation: including a pressure-gradient term introduces acoustic behavior. No wavelength-independent claim beyond the supplied model is needed.

For the binding energy of a truncated singular isothermal sphere, build the mass distribution from concentric shells. Its enclosed mass and mass element areThe interaction of a newly added shell with the interior contributes $-GM(r)\mathrm{d}M/r$. Counting each pair once,Eliminating $k$ givesThe central density is singular, but the mass and this energy integral are finite. The result assumes the stated truncation and no additional gravitating component.

The gravitational potential of a truncated singular isothermal sphere is a separate quantity from this shell-assembly energy allocation. With zero potential at infinity, both interior and exterior shells contribute:It matches $-G{M}_0/r_0=-2k$ at the edge. The equivalent total energy is $W=\frac{1}{2}\int \rho {\Phi }_{\mathrm{grav}}\mathrm{d}V$; the half prevents double counting. By contrast, $\int \rho [-{\Phi }_{\mathrm{grav}}]\mathrm{d}V$ counts the escape work of test parcels in a fixed potential and equals $-2W$. Confusing these quantities would change a local feedback threshold.

Introduce the constant $\nu >0$ by $\psi (r,t)=\nu {\rho }_g(r,t)$, and set $q=\gamma E_{\mathrm{SN}}$, the injected energy per unit mass of newly formed stars. Before a local wind switches on, no return or matter exchange impliesIn this independent-volume supernova quenching model, the accumulated injection per volume isFor any specified positive binding energy per unit remaining gas mass $B(r)$, the local threshold isGas conversion must be included in this comparison; replacing the integrated star formation by its initial rate times $t$ is only a short-time approximation.

For a quenching front in a fixed galactic potential, take $B(r)=-{\Phi }_{\mathrm{grav}}(r)=2k[1+\ln (r_0/r)]$. This treats the gravitating profile as fixed: newly formed stars initially replace gas at the same radius, and later changes of the potential caused by expelled gas are neglected. Gas first becomes unbound at the outer edge, atThereafter the inner edge of the quenched exterior region isThere is no quenched radius inside the galaxy before this onset. The remaining star-forming region is $0<r<r_{\mathrm{quench}}$; the front moves inward. The ideal logarithmically divergent central potential delays complete removal to infinite time in this fixed-profile model. It should not be interpreted as a physical unresolved central core.

The local binding-energy allocation in galactic feedback also admits a different convention, because it calls $\varphi$ an energy of a volume element without specifying how the nonlocal self-gravitating energy is assigned to that element. If the intended prescription is instead the shell-assembly contribution from part (a),then $B_{\mathrm{shell}}=2k$ at every radius. The same time $t_{\mathrm{start}}$ applies everywhere: all shells quench simultaneously, and there is no continuously propagating radial front. This is a shell-energy toy model, not the escape potential including exterior shells.

Both results follow from explicit definitions. A convention assigning $\frac{1}{2}{\rho }_g{\Phi }_{\mathrm{grav}}$ locally replaces $B$ by $B/2$ in $Q(t)=B(r)$; it is again an energy allocation rather than test-parcel escape. Updating the potential after gas loss or choosing a fixed initial-gas binding budget instead of the remaining-gas budget would require an additional closure. The stated independent-volume assumptions do not uniquely choose between these. The displayed escape-potential result and shell prescription must therefore not be presented as interchangeable universal laws.

For the shallower initial density profile,Including outer shells, the initial gravitational potential isIts central value is finite. With the same remaining-gas escape criterion and fixed-potential approximation as above,This front is inside the galaxy only when $k^{\prime }{r}_0\le Q(t)\le 2k^{\prime }{r}_0$. ThusQuenching still moves from the outside inward, but now reaches the centre in a finite time. The contrast with the $r^{-2}$ escape model is the finite central potential, not a reversed gravitational force.

Under the alternative shell-energy quenching prescription, the conclusion is different:Here the quenched region is inside the front and grows from the centre outward, unlike the simultaneous quenching of the isothermal shell-allocation model. Its edge reaches $r_0$ at ${\nu }^{-1}\ln (1+k^{\prime }{r}_0/q)$. This change of direction is a consequence of the chosen local energy model: $GM(r)/r$ increases with radius for this profile, whereas the true binding potential including all outer shells always becomes shallower outward. The source does not identify which local allocation it intends; keeping the conventions explicit resolves the different conditional predictions without discarding the exterior-shell potential.

To test radial stability of an axisymmetric circular orbit, let $\Phi (r)$ be the time-independent gravitational potential per unit stellar mass in the midplane. Axial symmetry conserves the specific angular momentum $j=r^2\dot{\phi }$. The radial equation isA circular orbit at $r_0$ obeys $j_0^2=r_0^3{\Phi }^{\prime }(r_0)$. Choose the guiding circular orbit with the same conserved angular momentum as the perturbed star. Setting $r=r_0+x$ and expanding at fixed $j=j_0$ giveswhere the radial epicyclic frequency isHere ${\Omega }^2(r)={\Phi }^{\prime }(r)/r$ and $j_c=r^2\Omega$ refer to the family of circular orbits. They are differentiated as functions of radius only after the perturbation equation has been expanded at conserved stellar $j$.

For a radially stable circular orbit, ${\kappa }^2>0$, so to linear orderThus the orbit performs small simple harmonic radial oscillations about its guiding radius.

Symmetry alone is insufficient to guarantee stability: ${\kappa }^2<0$ gives exponentially growing displacement, and ${\kappa }^2=0$ is marginal. The intended statement assumes a stable unperturbed orbit.

For a Kepler potential $\kappa =\Omega$; for a flat circular-speed curve $\kappa =\sqrt{2}\Omega$; for solid-body rotation $\kappa =2\Omega$. Conservation of $j$ also gives $\dot{\phi }\simeq {\Omega }_0(1-2x/r_0)$, explaining the accompanying azimuthal epicycle rather than a purely radial motion in the inertial plane.

A protostar forms when a self-gravitating cloud fragment can contract faster than pressure support and energy loss can arrest it. A useful local scale is the Jeans length, ${\lambda }_J\sim c_s/\sqrt{G\rho }$, with characteristic Jeans mass $M_J\sim c_s^3/(G^{3/2}{\rho }^{1/2})$ and free-fall timefor an initially uniform sphere. Geometry, turbulence and magnetic support change the detailed threshold; none of these estimates asserts that every cloud contracts as a uniform sphere.

While cooling is efficient, compression is nearly isothermal. Then $M_J$ decreases as the density rises, allowing fragmentation. Rotation becomes increasingly important during contraction because a parcel conserving specific angular momentum $j$ encounters a centrifugal scale $r_c\sim j^2/(GM)$. The resulting accretion disc must transport angular momentum before matter can reach the central object. Infall, a dense accreting interior and an outer envelope can therefore coexist; a protostar is not initially an isolated fixed-mass star.

Increasing optical depth eventually slows radiative cooling. The central gas then becomes less compressible and can form a pressure-supported first core. Molecular hydrogen dissociation absorbs compressional energy and can trigger a second collapse before a compact stellar core is established. Subsequent accretion supplies a luminosity of orderThe retained fraction depends on shock and disc radiation, so this expression is an energy scale rather than a guaranteed luminosity all emerging from the stellar surface.

A central accreting core can be nearly hydrostatic on its short sound-crossing time while its mass, entropy and radius evolve on longer times. A low-mass object is often largely convective because of its opacity and steep radiative temperature gradient; deuterium burning can temporarily contribute to support and moderate contraction. Neither convection nor the timing of deuterium ignition is universal across all masses and accretion histories.

When the main accretion phase ends, gravitational contraction remains an energy source. The virial theorem explains how contraction both heats the interior and supplies radiation: for an ideal-gas-supported object with negligible surface-pressure correction, $2U+W\simeq 0$, so only part of the released binding energy appears as increased thermal energy. The corresponding Kelvin-Helmholtz time isFully convective pre-main-sequence evolution approximately follows a Hayashi track, with decreasing radius and luminosity at comparatively weakly changing effective temperature. Objects developing radiative interiors can follow Henyey tracks. Eventually sustained core hydrogen burning balances the luminosity and the star reaches the main sequence. Massive objects can ignite hydrogen while substantial accretion continues, so “accretion ends, then hydrogen ignites” is not a universal ordering.

A useful structural distinction is therefore between an accretion-powered protostar and a largely fixed-mass pre-main-sequence contraction model. An infrared-bright envelope or disc traces obscured formation and reprocessing of radiation; a schematic evolutionary track should not replace the actual mass- and entropy-dependent stellar-structure problem.

An encounter between galaxies or clusters mainly couples their collective gravitational fields; it need not involve physical impacts between individual stars. Direct stellar collision rates depend on number density, relative speed and gravitational focusing. For two stars the collision cross-section is approximatelyIn an ordinary galaxy, the small geometrical stellar filling factor makes such impacts rare even while the galaxy undergoes a major encounter. Dense stellar cores are an important exception, not a reason to treat all galaxies as collisional fluids.

Tidal forces depend on the gradient of the perturber's field. For a distant fast encounter of mass $M_p$, impact parameter $b$ and speed $V$, the impulse scale for a star at displacement $r$ isA more specific result illustrates the geometry. Put the straight-line perturber trajectory in the $(x,y)$ plane along $y$, with closest approach at $x=b$. In the tidal and impulse limits,for an initially spherical, nonstreaming target with randomly sampled internal phases. The cross term averages to zero, and $⟨x^2+z^2⟩=2⟨r^2⟩/3$. Heating, expansion and tidal stripping preferentially affect weakly bound outer stars. A slow encounter can be adiabatically shielded rather than impulsive; a penetrating or strongly resonant encounter needs more than this distant straight-line approximation.

Disc rotation and orbital orientation matter. Prograde passages can drive stronger coherent distortions than corresponding retrograde passages. Tidal bridges, long tails, asymmetric arms and shells can record the encounter. Restricted gravitational calculations already reproduced prominent bridge and tail structures without requiring individual stellar collisions; full self-consistent models additionally follow the gravity of the responding discs and haloes.^[ref]

A first high-speed passage need not bind the pair. Capture or merging requires orbital energy and angular momentum to be redistributed into internal motions, escaping material, gas radiation or the surrounding halo. Dynamical friction transfers orbital energy to a gravitational wake and can bring a bound satellite inward. In a major merger, the potential changes on orbital times and violent relaxation redistributes stellar energies; phase mixing then smooths fine orbital structure. These are collective collisionless processes, distinct from slow two-body relaxation.

Gas behaves differently from stars: gas streams can shock, dissipate energy, lose angular momentum and fuel a central starburst or active nucleus. Gas-poor mergers are correspondingly more nearly dissipationless. Outcomes depend on mass ratio, orbital angular momentum, relative velocity, halo structure and gas fraction: they range from a modest flyby and tidal debris to disruption of a satellite or a merged remnant. Slow encounters in a group are generally more favorable to merging than otherwise comparable very fast unbound passages, but no single speed or mass-ratio threshold follows without an explicit structural model.

Galaxy formation combines gravitational assembly of dark-matter haloes with dissipative baryonic physics. Primordial density fluctuations supply a statistical seed distribution. In linear theory, each Fourier mode evolves through a growth factor and a scale-dependent transfer function; the power spectrum describes the variance by scale. Radiation pressure, free streaming and the time of horizon entry influence the small-scale transfer, so one should not replace the spectrum by a single universal growing power of time.

When fluctuations become nonlinear, overdense regions collapse, virialize and build larger structures through accretion and mergers. Cold dark matter permits a broad hierarchy of bound haloes. A spherical-collapse threshold and a mass-variance prescription, such as the Press-Schechter formalism, organize an approximate halo abundance; departures from spherical collapse and correlated environments limit its literal interpretation.

Gas falling into a halo is compressed and may be shock-heated toward a virial scaleUnlike collisionless dark matter, it can radiate away thermal energy. A rough cooling time iswhere the denominator specifies a chosen optically thin astrophysical cooling function convention. Cooling is effective for condensation when this time is short compared with the relevant dynamical or supply time. Atomic transitions, molecular processes, ionization state, density and metal abundance matter. A stable virial shock is not obligatory for all accretion: cold dense inflows can supply gas in regimes where shock-heated gas would cool rapidly.

The distinction between a gravitationally assembled massive halo and a radiatively condensing luminous component is fundamental. In the classic two-stage picture, non-dissipative clustering builds potential wells while cooling allows gas to concentrate much further than the halo component.^[ref]

Angular momentum inherited from tidal torques and later accretion helps gas settle into a rotationally supported disc after cooling; disc size depends on angular-momentum retention and transport. Locally self-gravitating gas fragments and forms stars. Stellar radiation, winds and supernovae then heat gas, drive turbulence, enrich it with heavy elements and can expel it from shallow potential wells. Metal enrichment can also accelerate later cooling, so feedback is more than a single positive or negative energy term.

Galaxy growth is usefully expressed as a gas-budget competition:Here $\mathcal{R}$ is the promptly returned stellar mass fraction; unlike the simplification in Question1, a general evolutionary model need not set it to zero. The equation separates newly supplied, converted, expelled and recycled gas rather than assuming that halo mass uniquely fixes stellar mass.

Accreting central black holes can provide radiative and mechanical feedback. Heating a halo, ejecting gas or maintaining a hot atmosphere can reduce fresh fuel; removal of a satellite's gas supply and ram pressure in a surrounding medium provide additional environmental routes. Quiescent galaxy assembly can still add stellar mass through dry mergers, so cessation of star formation is not identical to cessation of all mass assembly.

Mergers and disc instabilities can rearrange morphology and trigger bursts, while smooth accretion can rebuild discs. Cosmological expansion changes the supply and merger environment; accelerated late-time expansion also slows linear growth. A predictive model must connect these mechanisms to the stellar mass function, gas fractions, star-formation histories, sizes, chemical abundances and clustering. Those observables test different pieces of the assembly-and-feedback picture; none alone implies a one-to-one instantaneous relation between galaxy luminosity and total halo mass.

A spiral density wave is an organized density and gravitational-potential perturbation through which stars and gas can move; it is not generally a permanently labelled set of stars. Differential rotation winds a material arm rapidly. In a wave description, a perturbation has phase $m\varphi +\int kdr-\omega t$, with pattern speed ${\Omega }_p=\omega /m$ distinct from the local stellar angular speed.

For a thin fluid disk the local tightly wound dispersion relation isRotation and pressure resist self-gravity. A collisionless stellar disk requires its stellar response function instead of simply replacing $c_s$ by a dispersion; the familiar local axisymmetric stability parameter is $Q_{\star }={\sigma }_R\kappa /(3.36G\Sigma )$. This is distinct from the gas $Q_g=c_s\kappa /(\pi G\Sigma )$.

Waves exchange angular momentum at Lindblad resonances and near corotation radius. Swing amplification can strengthen a leading disturbance as shear turns it into a trailing one. Bars or companions can force a coherent spiral; recurrent self-excited disturbances also occur. Thus a single indefinitely steady wave is one model, not a description established for every spiral galaxy. Gas compression and shocks along arms can organize star formation, while gravitational torques redistribute angular momentum. Arm brightness, recent star formation and stellar mass density need not trace precisely the same pattern.^[ref]

Supermassive black holes are compact galactic nuclei with masses typically of millions to billions of solar masses. Their gravitational influence is measured from the motion of surrounding matter; accretion luminosity is evidence of feeding, not a necessary property of every black hole. Keplerian stellar orbits near the Galactic Centre and molecular-maser rotation curves in other nuclei provide particularly direct compact-mass constraints. A rough black-hole sphere of influence is $r_{\mathrm{infl}}\sim G{M}_{\bullet }/{\sigma }_{\star }^2$; resolving it helps dynamical mass estimates.

Accretion converts binding energy into radiation,The latter is the Eddington luminosity for isotropic radiation opposed by electron-scattering opacity, not an absolute luminosity ceiling for every geometry. The retained black-hole mass grows at $(1-\eta ){\dot{M}}_{\mathrm{in}}$. Sustained feeding can produce an active galactic nucleus, whereas gas-poor nuclei may be faint. Loss of gas angular momentum is a central feeding problem.

Black-hole mass correlates with classical-bulge stellar velocity dispersion through the M-sigma relation. Accretion-powered radiation, winds and jets can heat or remove gas, linking nuclear growth to galaxy evolution. Such correlations do not establish one universal feedback history, and disks and pseudobulges do not follow the same tight relations as classical bulges. Seed formation, accretion and mergers all contribute; seed masses and feeding histories remain model dependent.^[ref]

Galactic star formation requires gas to collect into cold dense regions that can become self-gravitating. Disk-scale gravity, turbulent motions, shear, pressure and magnetic fields affect this collection; formation then proceeds within molecular clouds. A useful local parametrization iswhere the star-formation efficiency per free-fall time summarizes inefficient conversion rather than requiring every cloud to collapse at once.

Across galactic disks the empirical Kennicutt–Schmidt law relates star formation rate surface density to gas surface density, ${\Sigma }_{\mathrm{SFR}}\propto {\Sigma }_g^N$. A slope near $1.4$ is a familiar integrated total-gas fit, while molecular-gas relations in ordinary disks can be closer to linear; the slope and normalization depend on tracer, averaging scale and environment. The gas-depletion time $t_{\mathrm{dep}}=M_g/\Psi$ is the corresponding fuel-consumption scale, not a prediction that inflow and recycling cease.

Massive stars supply ionizing radiation, winds and supernova energy and momentum. This stellar feedback can disperse clouds, sustain turbulence and drive outflows, regulating rather than simply extinguishing all future star formation. Fresh accretion and recycled gas replenish the reservoir. UV and recombination-line emission trace young stars, while infrared emission traces absorbed and re-radiated light; extinction, stellar populations and the assumed initial mass function affect inferred rates. A galactic rate reflects supply, cloud formation and feedback together, rather than only the free-fall time of its densest gas.^[ref]

In a matter-dominated Einstein-de Sitter universe, the growing linear density contrast is proportional to the scale factor. When $|\delta |\sim 1$, linear evolution no longer describes collapse. An overdense region initially expands more slowly than the background, reaches cosmological turnaround, and then contracts under self-gravity.

The spherical top-hat collapse model supplies a useful idealization: pressureless shells obey $\ddot{r}=-GM/r^2$. At formal collapse, the linearly extrapolated growing mode isThis is a linear extrapolation, not the true finite nonlinear density at that instant. In the ideal energy-conserving spherical model, virial equilibrium gives a virial radius about half the turnaround radius; its mean density at the collapse time is $18{\pi }^2\simeq 178$ times the cosmic matter density.

Real perturbations are nonspherical: tides and anisotropic collapse produce sheets, filaments and haloes. Collisionless dark matter develops shell crossing and phase mixing rather than radiatively losing its orbital energy. Baryonic gas can shock, cool and collect within the resulting potential wells. Hierarchical merging then assembles larger systems where small-scale fluctuations have sufficient initial power. Press-Schechter formalism connects a collapse threshold with the statistics of the smoothed initial field, but halo abundances and internal structure need more than the single spherical model. These ideas apply after perturbations become nonlinear; one must not extend pressureless subhorizon equations through all radiation-era or horizon-scale conditions.^[ref]

With constant isothermal sound speed $a_T$, the equation of state is $p=a_T^2\rho$. Spherical hydrostatic equilibrium and mass continuity requireTry $\rho =A/r^2$. With no central point mass, $M(r)=4\pi Ar$, and hydrostatic balance becomes $2a_T^{2}A=4\pi G{A}^2$. The positive-density branch is thereforeThe constant external pressure fixes the truncation radius $r_s$ through $p(r_s)=p_0$, givingThis is the pressure-truncated singular isothermal gas sphere, one hydrostatic solution, not a claim that every isothermal cloud has a singular centre. Its central density diverges, but its enclosed mass tends to zero as $r\to 0$.

For inside-out collapse of a singular isothermal sphere, the expansion front is at $r_f=a_{T}t$. The original mass swept by this front isThus the only time-independent accretion scale is $a_T^3/G$. The front-sweeping estimate has coefficient two, but gas takes time to fall from the front to the object; it is not an exact central accretion coefficient. Writefor fixed mean particle mass $\mu m_H$. Determining the dimensionless coefficient $m_0$ requires the collapse solution; it is not fixed by front propagation alone and is not needed for the requested temperature scaling.^[ref]

Well inside the front, where the point mass dominates and infall is approximately pressureless and locally steady, $|v_r|\simeq \sqrt{2G{M}_{\star }/r}$. Conserving the radial mass flux gives the free-fall density profile around a growing point mass:The inner slope is $-3/2$, versus $-2$ in the undisturbed outer cloud. For constant ${\dot{M}}_{\star }$ and negligible initial object mass, $M_{\star }\propto t$ and the inner normalization varies as $t^{-1/2}$. These are inner asymptotes, not an assertion of an abrupt exact power-law join at the front.

At distance $r$, one grain intercepts power through its projected area and emits through its full surface:The blackbody dust sublimation radius follows by setting $T_d=T_{\mathrm{sub}}$:Here the supplied luminosity is $3.8\times 10^{37}\mathrm{W}$. The grain radius cancels because both cross-sections scale as $a^2$. The temperature exterior to this boundary scales as $T_d\propto r^{-1/2}$. This estimate assumes isotropic illumination, no shielding, and unit absorption/emission efficiency as specified; real grain emissivities can change it. The numerical evaluation uses the external conversion values ${\sigma }_{\mathrm{SB}}\simeq 5.6704\times 10^{-8}\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-4}$ and $1\mathrm{pc}\simeq 3.0857\times 10^{16}\mathrm{m}$.^[ref]

The dust-cooled equilibrium is thermally stable under the stated constant irradiation, with fixed dust properties and dust-to-gas ratio. At a fixed position, absorbed power per grain is independent of its temperature, while emission is $4\pi a^2{\sigma }_{\mathrm{SB}}{T}^4$. A positive temperature perturbation increases cooling, causing a restoring energy loss; a negative perturbation reduces cooling, allowing reheating.

For heat capacity $C>0$ of a grain together with its thermally coupled gas,so the perturbation decays. The thermal stability of externally heated blackbody dust relies on remaining below sublimation with the grain population intact; destruction, variable illumination or finite dust-gas equilibration can change the response. An isobaric parcel with a fixed grain-to-gas ratio has the same restoring sign when heating and cooling are compared per particle.

First specify the perturbation: take fixed gas density, fixed ionic abundance and the same temperature-independent heating, so thermal stability requires $d\mathcal{C}/dT>0$ at an equilibrium. Let $\theta =\Delta E/k_B$, $n_e$ be the collider density and $n_A=n_1+n_2$ the atomic density. A two-level optically thin balance isThe critical density of an atomic transition is approximately $A_{21}/q_{21}$. At low density, each rare collisional excitation is followed by an escaping photon, soThe isochoric thermal stability of a collisionally excited line therefore givesAt $T=2\theta$ the linear restoring coefficient vanishes; the cooling maximum is marginal to first order and is not robust two-sided stability.

At high density, collisional de-excitation competes successfully with radiative decay, and collisions establish local thermodynamic equilibrium level populations. Detailed balance gives $n_2/n_1=g{e}^{-\theta /T}$, where $g=g_2/g_1$. The cooling is thenThus the high-density fixed-density branch is stable for finite $T$, becoming weakly restoring as the upper fraction saturates at high temperature. It is incorrect to retain the low-density factor $T^{-1/2}$ in the radiative cooling law once populations are thermalized.^[ref]

The isobaric thermal stability of a two-level coolant is a distinct question. If the intended heating is a fixed rate per particle, then $n\propto T^{-1}$ and stability tests the cooling per particle. The low-density boundary becomes $T<2\theta /3$; the high-density cooling per particle remains proportional to $f_u$, so remains stable. If instead heating is constant per volume, the isobaric low-density boundary is $T<2\theta /5$, and the high-density stability condition is $\theta (1-f_u)/T>1$. The source gives no pressure constraint or explicit heating normalization: the boxed $2\theta$ result is the fixed-density answer, with these conditional alternatives stated to avoid asserting a universal boundary.

For constant inflow $\Phi$ and efficiency $ϵ>0$, with no recycling or outflow, mass conservation gives the constant-inflow gas regulator solution:The equilibrium $M_{g,\mathrm{eq}}=\Phi /ϵ$ is stable: a gas-mass perturbation obeys $\delta {\dot{M}}_g=-ϵ\delta M_g$. At late times the star formation rate approaches $\Psi =\Phi$, while the gas reservoir approaches a constant.

Integrating ${\dot{M}}_{\star }=ϵM_g$ with $M_{\star }(0)=0$ gives the stellar mass in a constant-inflow gas regulator:It also follows immediately from $M_g+M_{\star }=M_0+\Phi t$. The gas mass equilibrates; the stellar mass generally keeps growing. With $\Phi =0$, these reduce to exponential gas depletion and $M_{\star }=M_0(1-e^{-ϵt})$.

A constant inflow is implicit in the requested steady asymptote, not automatic for an arbitrary prescribed rate. For time-dependent $\Phi (t)$ the exact integrating-factor result isa constant late-time gas mass then requires a corresponding asymptotically steady supply. Both printed mark groups remain in this single unlettered continuation of part(a).

Let $D$ be the separation of galaxy centres and $x$ a small displacement from the satellite centre along the line of centres. Expanding the central point-mass field gives the radial differential accelerationInside a uniform-density satellite, the restoring self-gravity isComparing the two at the same displacement yields the differential-gravity disruption limit of a uniform satellite:The satellite radius cancels because both accelerations are linear in $x$. This is the instantaneous tidal-gravity estimate requested by the given criterion, not a full rotating Roche calculation. Including centrifugal terms for a circular corotating orbit replaces the radial coefficient two by three, giving $D_{\min }\simeq [9M/(4\pi \rho ){]}^{1/3}$ while preserving the required scaling. An eccentric orbit requires checking the stronger tide near periapsis, and extended density profiles require a more detailed restoring-force calculation.