Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2016

Solution

Distinguish the rest-mass supply

\dot{M}_{in}

from the actual black-hole mass growth. With the usual radiative-efficiency definition,

L = ϵ \dot{M}_{in} c^{2}, \dot{M}_{BH} = (1 - ϵ) \dot{M}_{in} .

At the Eddington limit the growth is exponential:

\frac{M ˙ _{BH}}{M _{BH}} = \frac{1 - ϵ}{ϵ} \frac{4 π G m _{p}}{σ _{e} c} = \frac{1}{t _{Sal}}, t_{Sal} = \frac{ϵ}{1 - ϵ} \frac{σ _{e} c}{4 π G m _{p}} .

The available time is the difference of the supplied ages, $Δ t = 650 Myr$ ; no new cosmological age-redshift relation is needed.

Using the printed

σ_{e} = 6.6 \times 1 0^{- 29} m^{2}

, together with

G = 6.67430 \times 1 0^{- 11} m^{3} k g^{- 1} s^{- 2}

m_{p} ≃ 1.67262 \times 1 0^{- 27} kg

c = 299792458 m s^{- 1}

and a Julian year of

31557600 s

, gives

t_{E} = \frac{σ _{e} c}{4 π G m _{p}} ≃ 446.94 Myr, t_{Sal} ≃ 49.66 Myr .

Therefore

M_{BH, 0, m i n} = (2 \times 1 0^{9} M_{⊙}) e^{- 650/49.66} ≃ 4.1 \times 1 0^{3} M_{⊙} .

This 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}_{BH}

directly with

L / (ϵ c^{2})

and drops the radiated fraction of the incoming rest mass. That approximation instead gives

t_{Sal} ≃ 44.69 Myr

and

M_{0} ≃ 9.7 \times 1 0^{2} M_{⊙}

. 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]

c

Solution

Only the photon energy above the hydrogen ionization threshold initially becomes photoelectron kinetic energy. The stated mean therefore supplies

E_{heat} = ⟨ h ν ⟩ - h ν_{Lye} = k_{B} T_{*} .

Let

R

be the local number of ionizations per volume per time. Photoionization equilibrium makes this equal to the recombination rate. If

⟨ E_{rec} (T)⟩

is the mean kinetic energy removed per recombination, thermal balance reduces to

R k_{B} T_{*} = R ⟨ E_{rec} (T)⟩, ⟨ E_{rec} (T_{eq})⟩ = k_{B} T_{*} .

Neither 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 estimate

T_{eq} ≃ \frac{2}{3} T_{*} if ⟨ E_{rec} ⟩ = \frac{3}{2} k_{B} T .

That 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_{rec} ⟩ = η_{rec} (T) k_{B} T

, the precise answer is

η_{rec} (T_{eq}) T_{eq} = T_{*}

. 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.

d

Solution

Use the supplied flat radiation-era background equations. Write

\overline{ρ} R^{4} = C

, with

C

constant. On the expanding branch,

\dot{R}^{2} = \frac{8 π GC}{3 R ^{2}}, R \dot{R} = \frac{8 π GC}{3} .

Choosing the origin at the big bang gives

R \propto t^{1/2}, H = \frac{R ˙}{R} = \frac{1}{2 t}, \overline{ρ} = \frac{3}{32 π G t ^{2}} .

The supplied perturbation equation consequently becomes

\ddot{Δ} + \frac{1}{t} \dot{Δ} = \frac{Δ}{t ^{2}} .

A trial

Δ = t^{p}

gives

p (p - 1) + p = 1

, so

p = \pm 1

. Thus both independent modes are

Δ (t) = C_{+} t + C_{-} t^{- 1} .

The growing mode is $Δ_{grow} \propto t \propto R^{2}$ , 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 π 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.

2

a

Solution

For the binding energy of a truncated singular isothermal sphere, build the mass distribution from concentric shells. Its enclosed mass and mass element are

M (r) = \int_{0}^{r} 4 π s^{2} ρ (s) d s = \frac{2 k}{G} r, d M = \frac{2 k}{G} d r, M_{0} = \frac{2 k r _{0}}{G} .

The interaction of a newly added shell with the interior contributes

- GM (r) d M / r

. Counting each pair once,

W = - \int_{0}^{r_{0}} \frac{GM ( r )}{r} d M (r) = - \frac{4 k ^{2}}{G} \int_{0}^{r_{0}} d r = - \frac{4 k ^{2} r _{0}}{G} .

Eliminating

k

gives

W = - \frac{G M _{0}^{2}}{r _{0}} .

The 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:

Φ_{grav} (r) = - G [\frac{M ( r )}{r} + \int_{r}^{r_{0}} \frac{d M ( s )}{s}] = - 2 k [1 + ln \frac{r _{0}}{r}], 0 < r \leq r_{0} .

It matches

- G M_{0} / r_{0} = - 2 k

at the edge. The equivalent total energy is

W = \frac{1}{2} \int ρ Φ_{grav} d V

; the half prevents double counting. By contrast,

\int ρ [- Φ_{grav}] d V

counts the escape work of test parcels in a fixed potential and equals

- 2 W

. Confusing these quantities would change a local feedback threshold.

b

Solution

Introduce the constant

ν > 0

ψ (r, t) = ν ρ_{g} (r, t)

, and set

q = γ E_{SN}

, the injected energy per unit mass of newly formed stars. Before a local wind switches on, no return or matter exchange implies

ρ_{g} (r, t) = ρ_{0} (r) e^{- ν t}, ρ_{s} (r, t) = ρ_{0} (r) (1 - e^{- ν t}) .

In this independent-volume supernova quenching model, the accumulated injection per volume is

u_{SN} (r, t) = \int_{0}^{t} γ E_{SN} ψ (r, t^{'}) d t^{'} = q ρ_{0} (r) (1 - e^{- ν t}) .

For any specified positive binding energy per unit remaining gas mass

B (r)

, the local threshold is

u_{SN} = ρ_{g} B (r) ⟺ Q (t) \equiv q (e^{ν t} - 1) = B (r) .

Gas 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) = - Φ_{grav} (r) = 2 k [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, at

t_{start} = ν^{- 1} ln (1 + \frac{2 k}{q}) .

Thereafter the inner edge of the quenched exterior region is

r_{quench} (t) = r_{0} exp [1 - \frac{q}{2 k} (e^{ν t} - 1)], t \geq t_{start} .

(5)

There is no quenched radius inside the galaxy before this onset. The remaining star-forming region is

0 < r < r_{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

ϕ

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),

ϕ_{shell} = - \frac{G M _{0} ( r )}{r} ρ_{g} = - 2 k ρ_{g},

(6)

then

B_{shell} = 2 k

at every radius. The same time

t_{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} ρ_{g} Φ_{grav}

locally replaces

B

B /2

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.

Figure 1.
Original comparison of quenching thresholds for two initial density slopes, distinguishing escape potential from shell-energy allocation
. **Details**.

c

Solution

For the shallower initial density profile,

M_{0} (r) = \frac{k ^{'}}{G} r^{2}, M_{0} = \frac{k ^{'} r _{0}^{2}}{G} .

Including outer shells, the initial gravitational potential is

Φ_{grav} (r) = - G [\frac{M _{0} ( r )}{r} + \int_{r}^{r_{0}} \frac{d M _{0} ( s )}{s}] = - k^{'} (2 r_{0} - r) .

Its central value is finite. With the same remaining-gas escape criterion and fixed-potential approximation as above,

r_{quench} (t) = 2 r_{0} - \frac{q}{k ^{'}} (e^{ν t} - 1) .

This front is inside the galaxy only when

k^{'} r_{0} \leq Q (t) \leq 2 k^{'} r_{0}

. Thus

t_{start} = ν^{- 1} ln (1 + \frac{k ^{'} r _{0}}{q}), t_{all} = ν^{- 1} ln (1 + \frac{2 k ^{'} r _{0}}{q}) .

Quenching 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:

B_{shell} (r) = \frac{G M _{0} ( r )}{r} = k^{'} r, r_{quench}^{shell} (t) = \frac{q}{k ^{'}} (e^{ν t} - 1) .

(5)

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}

ν^{- 1} ln (1 + k^{'} 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.

d

Solution

To test radial stability of an axisymmetric circular orbit, let

Φ (r)

be the time-independent gravitational potential per unit stellar mass in the midplane. Axial symmetry conserves the specific angular momentum

j = r^{2} \overset{φ}{˙}

. The radial equation is

\overset{r}{¨} = \frac{j ^{2}}{r ^{3}} - Φ^{'} (r) = - \frac{d Φ _{eff}}{d r}, Φ_{eff} (r) = Φ (r) + \frac{j ^{2}}{2 r ^{2}} .

A circular orbit at

r_{0}

obeys

j_{0}^{2} = r_{0}^{3} Φ^{'} (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}

gives

\overset{x}{¨} = [- \frac{3 j _{0}^{2}}{r _{0}^{4}} - Φ^{''} (r_{0})] x + O (x^{2}) = - κ^{2} x + O (x^{2}),

where the radial epicyclic frequency is

κ^{2} = Φ^{''} (r_{0}) + \frac{3}{r _{0}} Φ^{'} (r_{0}) = 4 Ω_{0}^{2} + r_{0} \frac{d Ω ^{2}}{d r}_{r_{0}} = \frac{1}{r ^{3}} \frac{d j _{c}^{2}}{d r}_{r_{0}} .

Here

Ω^{2} (r) = Φ^{'} (r) / r

and

j_{c} = r^{2} Ω

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,

κ^{2} > 0

, so to linear order

x (t) = C cos (κ t) + D sin (κ t) .

Thus the orbit performs small simple harmonic radial oscillations about its guiding radius.

Figure 1.
Original linear epicycle calculation for stable circular motion with a flat circular-speed curve
. **Details**.

Symmetry alone is insufficient to guarantee stability:

κ^{2} < 0

gives exponentially growing displacement, and

κ^{2} = 0

is marginal. The intended statement assumes a stable unperturbed orbit.

For a Kepler potential

κ = Ω

; for a flat circular-speed curve

κ = 2 Ω

; for solid-body rotation

κ = 2Ω

. Conservation of

j

also gives

\overset{φ}{˙} ≃ Ω_{0} (1 - 2 x / r_{0})

, explaining the accompanying azimuthal epicycle rather than a purely radial motion in the inertial plane.

3

a

Solution

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,

λ_{J} \sim c_{s} / Gρ

, with characteristic Jeans mass

M_{J} \sim c_{s}^{3} / (G^{3/2} ρ^{1/2})

and free-fall time

t_{ff} = \frac{3 π}{32 Gρ}

for 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 order

L_{acc} \sim \frac{GM M ˙}{R} .

The 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,

2 U + W ≃ 0

, so only part of the released binding energy appears as increased thermal energy. The corresponding Kelvin-Helmholtz time is

t_{KH} \sim \frac{G M ^{2}}{R L} .

Fully 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.

b

Solution

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 approximately

σ_{coll} = π (R_{1} + R_{2})^{2} [1 + \frac{2 G ( m _{1} + m _{2} )}{( R _{1} + R _{2} ) v _{\infty}^{2}}], t_{coll} \sim (n_{⋆} σ_{coll} v_{\infty})^{- 1} .

In 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

Δ v \sim \frac{G M _{p} r}{b ^{2} V} .

A 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,

Δ v = \frac{2 G M _{p}}{b ^{2} V} (x, 0, - z), Δ E = \frac{4 G ^{2} M _{p}^{2} M ⟨ r ^{2} ⟩}{3 b ^{4} V ^{2}}

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.

c

Solution

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 scale

k_{B} T_{vir} \sim \frac{μ m _{p} V _{c}^{2}}{2} .

Unlike collisionless dark matter, it can radiate away thermal energy. A rough cooling time is

t_{cool} \sim \frac{\frac{3}{2} n _{tot} k _{B} T}{n _{e} n _{i} Λ ( T , Z )},

where 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:

\dot{M}_{g} = \dot{M}_{in} - (1 - R) ψ - \dot{M}_{wind} + \dot{M}_{reaccrete} .