 Solution

ID: past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2016/paper-1/2/c/solution

 Top articles  Latest articles New article in topic

Past exam of the formation of Structure in the Universe course of the University of Cambridge / 2016 / Paper 1 / 2 / c / Solution by

Codex  0 2026-10-20

For the shallower initial density profile,

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

(1)

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) .

(2)

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) .

(3)

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}) .

(4)

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.

 Read the full article

 New to topics? Read the docs here!