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Self-similar Spherical collapse Lecture Notes Max-Planck Institut f ¨ ur Astrophysik 5th October 2017 Xun Shi

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Self-similar Spherical collapse

Lecture Notes

Max-Planck Institut fur Astrophysik

5th October 2017

Xun Shi

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Contents

1 Introduction 1

2 The spherical collapse model in cosmology 22.1 Spherical collapse in an Einstein de Sitter (EdS) universe . . . . . . . . . . . . . . 22.2 Spehrical collapse in a flat ΛCDM universe . . . . . . . . . . . . . . . . . . . . . 32.3 Spherical collapse model: implications for halo profile . . . . . . . . . . . . . . . 5

3 Self-similar solutions: an introduction 7

4 Self-similar spherical collapse of dark matter 9

5 Bondi accretion 13

6 Self-similar collapse of cold gas onto a black hole 17

7 Self-similar collapse of self-gravitating gas 20

8 Self-similar collapse of gas and dark matter 26

9 Asymptotic profile of gravitational collapse: beyond self-similar solutions 29

A Forms of hydrodynamical equations 32A.1 Equation of mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32A.2 Equation of momentum conservation . . . . . . . . . . . . . . . . . . . . . . . . . 32A.3 Equation of energy conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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1 Introduction

Many structures in the Universe, such as galaxies and clusters of galaxies, form from accu-mulations of matter collapsing under gravitational attraction. During this gravitational collapse,non-linear structures form in the center of the gravitational field, such as dark matter halos embed-ding galaxies and galaxy clusters, and baryonic halos of galaxy clusters known as the intraclustermedium. These structures are concentrations of matter with relative universal profiles. How to un-derstand the emergence of these universal profiles? What physical factors determine their shapes?This lecture will present some efforts to answer these questions.

Numerical simulations have been a central tool in the study of these structures from the begin-ning. Early efforts in 1960s and 1970s already realised the existence of universal structures out ofgravitational collapse. Today, the profile shapes of these structures have been established in moredetailed simulations, and so far seem to be robust against the increasing resolution.

Despite the general success of numerical simulations, there are motivations for finding analyt-ical solutions to the gravitational collapse problem. The biggest motivation is to provide physicalintuitions accessible to our human minds. Besides this, analytical solutions provide an anchor tothe numerical solutions, in which numerical artifacts and limited resolution remain always an issue.Although it is in general hard to obtain analytical solutions in the non-linear regime of structureformation, the approximate spherical symmetry of the gravitational collapse of high density peaksmakes analytical efforts feasible.

The main analytical approach we shall introduce is the self-similar spherical collapse model byFillmore and Goldreich [1984], Bertschinger [1985]. The additional symmetry of self-similarityenables us to compute profile shapes in more complicated settings. We shall start introducing itsfoundation: the Gunn and Gott [1972] spherical collapse model, and then explain the motivationsfor searching for self-similar solutions to the spherical collapse problem. Then various self-similarspherical collapse models will be presented, roughly matching the sections of the Bertschinger[1985] paper. In between, Bondi accretion will be presented as a comparison. In the end we shalldiscuss how to go beyond self-similar spherical collapse to search for general asymptotic profiles ofgravitational collapse.

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2 The spherical collapse model in cosmology

It was already realised in the 1970s that in computation of structures on the physical scale ofgalaxies and above, cosmic expansion plays an important role. The emerging picture is that thegravitational attraction of initially over-dense regions cause the surrounding matter to deviate fromthe Hubble flow and collapse onto them, forming larger and larger collapsed structures. In theseminal Gunn and Gott [1972] paper “On the Infall of Matter Into Clusters of Galaxies and SomeEffects on Their Evolution”, James Gunn and Richard Gott first solved the evolution of sphericalshells of matter surrounding an over-density, and presented what is now referred to as the “sphericalcollapse model”.

2.1 Spherical collapse in an Einstein de Sitter (EdS) universe

We first consider the simplest case that was studied in the original Gunn and Gott [1972] paper:spherical collapse in an EdS universe (i.e. flat universe with cold matter only).

Consider a small positive spherical density perturbation to an otherwise homogeneous universewith mean density ρH(t). Of this mass concentration, consider a shell of radius r(t) encompassing amass Mi. Since the perturbation is initially small, ∆i = (Mi − Mi)/Mi 1, with Mi = 4πρH(ti)r3

i /3being the average mass within a radius ri = r(ti). The radius of the shell initially expands withthe Hubble flow, but the expansion will slow down, turn over and eventually collapse since it isgravitationally bound to the excess mass it encloses. As the radius evolve in time, it encloses thesame mass M, unless mass shells start to cross. The equation of motion of the shell is

d2rdt2 = −

GMi

r2 . (1)

This second-order ODE can be integrated once (by multiplying both sides by dr/dt to get the energyequation (

drdt

)2

=2GMi

r− K . (2)

The integral constant K = 8πGρH(ti)r2i ∆i/3 is determined from the initial conditions. Physically, it

describes the curvature caused by the over-density of the enclosed region.When we take K = 0, Eq. (2) describes the Hubble expansion of the EdS universe itself, i.e R

is proportional the cosmic scale factor a. In this case, it is easy to integrate once again to obtain thefamiliar a ∝ t2/3 expansion law of EdS universe

rEdS =12

(GMi)13 (6t)

23 . (3)

We see that in this case, r → ∞ when t → ∞, the shell never collapses.A finite positive K resulting from an initial over-density causes deviation from the Hubble flow.

In this case, solution to Eq. (2) is best expressed in parametric form

r(θ) =GMi

K(1 − cos θ) ,

t(θ) =GMi

K3/2(θ − sin θ) ,

(4)

where we’ve chosen the initial time t = 0 at r = 0 in accordance to the K = 0 case. Now we canexamine the property of this solution at some specific times: When θ = π, the radius of the shell

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reaches its maximum rta = 2GMi/K , at a time tta = πGMi/K3/2. The shell then turns around at this

point (thus the subscript ‘turn-around’), and collapses to r = 0 again at θ = 2π when tcoll = 2tta.Comparing rta to rEdS at the same time, we get

ρta

ρH=

r3EdS(tta)

r3ta

=

(3π4

)2

≈ 5.55 , (5)

i.e. at turnaround, the sphere is 5.55 times denser than the EdS universe in the mean. In reality,instead of collapsing to R = 0, the non-linear structure virializes at rvir determined by the virialtheorem Ekin = Epot/2, which yields

E =Epot(rvir)

2= −

GM2rvir

= Epot(rta) = −GMrta

,

(6)

i.e. rvir = rta/2. At rvir,ρvir

ρH=

r3EdS(tcoll)

r3vir

= 18π2 ≈ 178 . (7)

If we compute the matter density contract at tta and tcoll in linear theory, we will find the familiarnumbers δta,lin = 1.06 (‘linear density threshold for turnaround’) and δcoll,lin = 1.69 (‘linear densitythreshold for collapse’). Comparing these to the spherical collapse results 1+δta = 5.55 and 1+δcoll =

178, we can appreciate how fast structures grow when the perturbation gets non-linear, and how thissimple analytic model of spherical collapse casts light in the fully non-linear regime.

These results motivated the following paragidm, that we can look at the linear density contrastsof matter concentrations at an early cosmic time to know which will collapse at when: whenever thedensity contrast grows to 1.69 according to linear growth, the matter concentration shall collapse toa virialized object, with a characteristic over-density of 178.

This paragidm then enabled theoretical estimates of the number density of collapsed halos as afunction of mass and cosmic time by studying the statistics of linear density peaks, e.g. the Press-Schechter theory and the excursion-set theory. The numbers of the critical linear and non-linearover-densities, of course, depend on cosmology.

2.2 Spehrical collapse in a flat ΛCDM universe

In a ΛCDM universe, the cosmology constant Λ enters the dynamical equation and affects theevolution of the spherical shells [e.g. Mo et al., 2010]:

d2rdt2 = −

GMi

r2 +Λr3, (8)

with the corresponding energy equation being(drdt

)2

=2GMi

r+

Λr2

3− K . (9)

We use the initial mass encompassed by the shell Mi in the energy equation to keep in mind that thisequation is valid only when the enclosed mass does not change, i.e. before shell-crossing. Again,these two equations also describe the expansion of the background universe. For a Ωm + ΩΛ = 1 flat

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universe, the curvature K = 0.

With another term added to the equations, the dynamical behavior of the shell is now controlledby three parameters:

• the mass / size of the over-dense region

• the curvature K of the over-dense region, which depends on the over-density of the regionwith respect to the mean matter density

• the value of the cosmological constant which sets a a typical time scale in the evolution of theuniverse.

Thus it now makes more sense to rewrite the equations in their simplest forms to reduce the numberof control parameters, and to reveal the physical essence of the system. Shi [2016a] did this basedon early works of Peebles [1984] and Eke et al. [1996]. It is found that the quantities should be bestscaled in the following way:

scaled cosmological constant w = Λr3i /(6GMi) = Ω−1

m0 − 1

scaled time I =

(8πGρ0w

3

) 12

t

scaled cosmic scale factor u = w13 a

scaled radius y = w13 r/ri

scaled curvature κ = Kri/(2GMi)

scaled over-density β = κ/κmin .

(10)

Here κmin = 3w13 /2

23 is the minimum scaled curvature of a perturbation for it to be able to collapse

(before the cosmological constant takes over and makes collapsing impossible), and ri = r(ai)/ai

is the comoving size of the shell at the initial time when the cosmic scale factor a = ai. Since thedensity fluctuations are small in the early universe, δ(ri) 1, the mass of the overdense region isMi = 4πρ0r3

i /3 to a good approximation, with ρ being the mean matter density of the universe, andsubscript ‘0’ indicates today (a = 1).

With the scaled quantities, the relation between time and cosmic scale factor is

I =23

sinh−1(u

32

), (11)

the dynamical equation (8) and the energy equation (9) become

d2ydI2 = −

M2Mi

1y2 + y , (12)

and (dydI

)2

= y−1 + y2 − 3β/223 . (13)

Instead of three parameters governing equations (8) and (9), the the simplified equations have onlyone control parameter: the scaled over-density β! The whole family of solutions can then be shown

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

u

0.0

0.1

0.2

0.3

0.4

0.5

0.6

y

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

log(β)

Figure 1: Spherical collapse trajectories (scaled radius y as a function of scaled cosmic scale factoru) in an ΛCDM universe (dashed lines) for shells with different scaled initial overdensities β (shownwith different colors). The turn-around locations of the trajectories are marked as the upper solidline, whose analytical expression is given by Eqs. 14. The locations where the shells collapse to halftheir turn-around radius are shown as the lower solid line.

in one figure (Fig. 1). One see that in the early universe when u → 0, the expansion and collapseof the shells are symmetric, as in the case of an EdS universe. Then the dynamic effect of thecosmological constant starts to be important at u ≈ 1, slowing down the collapse. The scaled turn-around radius (upper colored line in Fig. 1) has also an analytical expression as a function of thescaled over-density β,

y∗ = 223 β

12 sin

(13

sin−1(β−

32

)). (14)

2.3 Spherical collapse model: implications for halo profile

The spherical collapse model requires that the enclosed mass within the shell is constant. In thecase of secondary collapse onto a singularity e.g. a black hole that sucks all the mass falling ontoit, the model follows the evolution of the shell all the way to the center. In the case of secondarycollapse onto a small density perturbation to the homogeneous background universe, no such sin-gularity exists, shells of collisionless dark matter expand again after collapsing to the origin, crossother shells of matter, and the shells collectively form a (more or less) virialized halo.

Although not able to describe the formation of the halo, we can get a very rough guess of thehalo density profile using the spherical collapse model by assuming that each shell artificially stopsat, say, the effective virial radius where equation (6) holds. The resulting profile depends on the massdistribution of the initial perturbation. Suppose ∆i ≡ ∆ρi/ρ = ∆Mi/Mi ∝ M−εi ∝ r−3ε

i (∆Mi is theexcess mass compared to the universal mean within a region Ri enclosing mass Mi), this gives a halodensity profile with a slope of −9ε/(1 + 3ε) in an EdS universe1. For a top-hat density perturbation

1This can be derived by combining rvir ∝ Mi/K and Kri/2GMi ∝ ∆i.

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at the origin that corresponds to ε = 1, ρ ∝ r−9/4. This shall be compared to the results of theself-similar spherical collapse.

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3 Self-similar solutions: an introduction

The spherical collapse model is restricted to the collapse of a single shell and cannot deal withthe relaxed structure formed out of shell-crossing in a rigorous way. The self-similar sphericalcollapse model by Fillmore and Goldreich [1984] and Bertschinger [1985], on the other hand, cansolve the equations after shell-crossing by looking for a unique self-similar solution of them.

Self-similar solutions to differential equations are those which are invariant under certain “stretch-ing” transformations. A famous example in astrophysics is the Sedov-Taylor solution of sphericalblast wave2, which is used to describe a phase (the “Sedov phase”) of supernova remnant evolution:Consider the expansion of the radius r of a supernova blast wave with time t, given the energy of thesupernova explosion E and the density ρ of ambient interstellar medium. The only dimensionlessgroup that can be constructed with these quantities is Et2/r5ρ, suggesting (from the Π-theorem) thatr ∝ t2/5E1/5ρ−1/5 - the stretching transformation in this case. Inserting this relation to the equationsof motion, the profiles / history of the gas expansion can be derived.

In the early days, self-similar solutions were much sought after in various fields of physics be-cause their construction led to the simplification of PDEs into ODEs, and often solving the systemwas only possible in the later case. With the advances of numerical computation methods, it is nolonger difficult to solve the original PDE, and sometimes the PDEs are even easier to solve thanthe ODEs since the latter often contain singularities. Nevertheless, self-similar solutions remainsignificant due to a special property of it realized with time, mostly by the Russian school includ-ing Kolmogorov, Barenblatt and Zel’dovich. This special property is that these solutions are notsimply particular cases, but rather, they describe the asymptotic behaviour of a wide class of situa-tions. The situations asymptotic to the self-similar solutions when the dependence on the details ofthe initial/boundary conditions has already disappeared, but still the system is very far away fromits “dead end” - the final equilibrium state. Thus, self-similar solutions are usually “intermedi-ate asymptotics”. This intermediate asymptotic property is exactly the one we are looking for inunderstanding the universal profiles!

Practically, a key procedure of finding self-similar solutions is to find the appropriate stretchingtransformation. In the Sedov blast wave example and in many other cases, the stretching transforma-tion can be derived from dimension analysis - a powerful tool on its own. Here we give other twocommon examples: First, the period of a pendulum has to be T ∝

√`/g with ` being the length of

the rope and g the gravitational acceleration, because it is the only way of forming the right dimen-sion. For an other example, the expansion law of an EdS universe can be derived with dimensionanalysis: In such a homogeneous universe without curvature or other scale-generating physics, themean density is the only drive for the expansion dynamics. From the only dimensionless groupt√

Gρ, we have ρ ∝ t−2, and thus a ∝ t2/3.While studying many examples of self-similar solutions in e.g. fluid dynamics and the propaga-

tion of flames, Barenblatt and Zel’dovich realized that there are problems for which the power-lawindex of the “stretching” transformation can be pinned down only while solving the differentialequations as an eigenvalue problem. Based on this they distinguished type-I and type-II self-similarsolutions. The self-similar spherical collapse belongs to the easier type-I solutions where dimensionanalysis is sufficient.

Numerous examples of problems solvable with self-similar solutions are given in the classical

2An anecdote made this solution famous even outside the scientific community: After the first explosion of an atomicbomb - the Trinity test in New Mexico in 1945, a series of pictures of the explosion along with size scale and time stampswere released and published in popular magazines in 1947. Reading these on the other side of the Atlantic ocean, G. I.Taylor was able to estimate the power released by the explosion, still classified at the time, based on this solution.

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books: Courant and Friedrichs [1948], Sedov [1959], Zel’dovich and Raizer [1967], Barenblatt[1996]. In astrophysics, apart from the Fillmore and Goldreich [1984] and Bertschinger [1985]articles on self-similar spherical collapse that is relevant to the formation of dark matter halos andintracluster medium, classical literature include Ostriker and McKee [1988] on astrophysical blastwaves, Whitworth and Summers [1985] on self-similar collapse of molecular clouds, and those onthe homology equations in stellar structures (e.g. the Kippenhahn et al. 2012 textbook). Otherexamples include self-similar solutions of structures of accretion disk and giant gaseous planets.

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4 Self-similar spherical collapse of dark matter

We now sketch the studies of Fillmore and Goldreich [1984] and Bertschinger [1985] on theself-similar spherical collapse of dark matter (in general, a self-gravitating, collisionless) in an EdSuniverse. The dynamical equation has the same form as in the spherical collapse model:

d2rdt2 = −

GMr2 . (15)

However, after shell-crossing, the mass enclosed by the shell M is no longer the initial mass Mi,but should instead depend on the density profile of the central object, which also evolves with time.This introduces an explicit time-dependence on the r.h.s. and makes the equation hard to solveanalytically. Moreover, the mass profile of the central object that enters on the r.h.s. depends on theinstant

It turns out that the symmetry of self-similarity can simplify the situation to the degree thatthe equation can be solved with numerical integration. Self-similarity suggests that trajectories ofdifferent shells are the same, and so do the density profiles at different epochs, when they are scaledto the characteristic scales. Thus, density profile at any time, which can be constructed by puttingall shells together, can be constructed based on the trajectory shape of a single shell. To allowself-similarity, we restrict ourselves to a situation where the central object has a power-law massgrowth M ∝ as ∝ t2s/3, which implies a power-law mass profile of the initial over-dense peak∆Mi/Mi ∝ M−εi with ε = 1/s.

An apparent choice of the characteristic length and time scale of a shell is its turn-around radiusand time, r∗ and t∗. Thus we define dimensionless radius λF = r/r∗ and time τ = t/t∗, as Fillmoreand Goldreich [1984] did. The trajectories of all shells should lie on a single curve in the τ - λF plane.The mass profile M(r, t) on the other hand, has the apparent characteristic length of the radius of theshell which is turning around rta at the moment t. This means, if we define λ = r/rta, the mass profileshape should depend only on the single parameter λ, instead of on radius and time independently.This motivates us to write M(r, t) = MtaM(λ) with Mta = 4πρHr3

ta/3, where the mean matter densityof the EdS universe ρH = 1/(6πGt2). Let us work out the expression of λ in terms of λF and τ: Inthe EdS universe we are considering, rta ∝ tδ with δ = 2(1 + s/3)/3 which can be easily found usingthe spherical collapse model3. Thus, λ = λFτ

δ. Physically, this rta ∝ tδ proportionality is governedby the law of energy conservation, and can be derived also from heuristic dimensional analysis.

With our stretching transformation defined, now we can simply the dynamical equation (15) tobe

τ2−δ d2λF

dτ2 = −29M(λ)λ2 , (16)

whereM(λ) is contributed by all the shells which have already turned around i.e. τ′ > 1 and withcurrent radius λ′(τ′) < λ. ThusM(λ) has the expression

M (λ) =

∫ Mta

0

dMi

MtaH

[λ − λ′(τ′)

]=

∫ ∞

1

dτ′

τ′1+2s/3 H[λ − λ′(τ′)

].

(17)

The equations (16) and (17) can now be solved with boundary conditions at turn-around: λF(1) = 1and dλF(1)/dτ = 0. The inter-dependency between mass-profile and shell trajectory now expresses

3from rta ∝ M1/3i /∆, tta ∝ ∆

−2/3i , and ∆i ∝ M−1/s

i

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2 4 6 8 10 12 14

τ

0.0

0.2

0.4

0.6

0.8

1.0

λF

s=0.5

s=1

s=2

s=3

s=5

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ξ

0.0

0.2

0.4

0.6

0.8

1.0

λ

s=0.5

s=1

s=2

s=3

s=5

Figure 2: The trajectories of self-similar infall of dark matter. In the upper panel, the radii of theaccreted shells normalized to their turn-around radius, i.e. λF = r/r∗, and the time normalizedto their turn-around time, i.e. τ = t/t∗. In the lower panel, the radii of the accreted shells arenormalized to the current turn-around radius, i.e. λ = r/rta, and are plotted against the logarithmictime ξ = ln(t/t∗).

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Table 1: Dimensionless radial position, velocity and acceleration expressed with different variables(taken from Shi [2016b]).

Variables r, t λ = r/rta, ξ = ln(t/t∗) λF = r/r∗, τ = t/t∗ V = vt/rta, λ = r/rta

Radial position 1rta(t) r λ τ−δλF λ

Velocity trta(t)

drdt

dλdξ + δλ τ1−δ dλF

dτ V

Acceleration t2rta(t)

d2rdt2

d2λdξ2 + (2δ − 1) dλ

dξ + δ (δ − 1) λ τ2−δ d2λFdτ2 (V − δλ)V ′ + (δ − 1)V

10-2 10-1 100

λ

10-1

100

101

102

103

104

105

106

ρ/ρH

s=0. 5

s=1

s=2

s=3

s=5

Figure 3: Nondimensional dark matter density profile from self-similar spherical collapse.

itself as the integral in Eq. (17). At this point, we can use iterative method to solve these equations:assume an initial functional form ofM (λ), integrate (16) to obtain λ(τ) to update the form ofM (λ).Note that this iterative method is in principle also possible for the original dynamical equation (15),however one needs to iterate over M(r, t) rather than a one-dimensional function. Doing the self-similar transform has removed one dimension.

In fact, as Bertschinger [1985] found out, the dynamical equation can be further simplified to be

d2λ

dξ2 + (2δ − 1)dλdξ

+ δ (δ − 1) λ = −29M(λ)λ2 . (18)

with ξ = ln(τ) (see Table. 4 for the expressions of radius, velocity and acceleration in terms ofvarious variables), and the mass profile has an analytical expression (again thanks to self-similarity):M(λ) =

∑i

(−1)i−1 exp[−(2/3) ξi

], where ξi is the value of ξ at the i-th point where λ = λ(ξ).

The alternating signs in this expression are related to adding/subtracting the contribution from darkmatter streams moving inward/outward at radius λ (see lower panel of Fig. 2). The 2s/3 factor inthe exponent is related to the fact that the mass enclosed in a certain radius λ grows with time asM(λrta, t) ∝ t2s/3.

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After the first passage through the center, the shells expand again out to a radius now called the“splashback radius”, fall back again and continue to oscillate. Different shells of matter interactintensively during their oscillations, causing dynamical relaxation. As a result, the apapsis distancetypically shrinks (upper panel of Fig. 2). Relaxation occurs especially evidently during the firstoscillations, causing the splashback radius to be significantly smaller than the turn-around radius ofthe shells. At a mild mass accretion rate s < 3/2, the the apapsis distance of a shell asymptotesto a certain Eulerian radius, but at s > 3/2, the oscillation of a shell never ceases to shrink in itsamplitude.

The power-law slope of the inner mass profile M ∝ rΥ is well studied by Fillmore and Gol-dreich [1984]. The mass accretion rate is related to the logarithmic slope ε of the initial massperturbation δMi/Mi ∝ M−εi with s = 1/ε. In terms of s, the Fillmore and Goldreich [1984] resultreads4

Υ =

3s/(s + 3) if s ≤ 3/21 if s ≥ 3/2 .

(19)

The critical mass accretion rate is s = 3/2. For small accretion rates s < 3/2, the apapsis of theshell trajectories approaches a constant with time (see upper panel of Fig. 2). Thus, in this case,the mass in the inner halo is dominated by shells which collapsed at early times, whose turn-aroundradii are small, and the gravitational potential in the inner halo does not grow with time. In this case,the inner density profile has a slope of Υ − 3 = −9/(s + 3) = −9ε/(1 + 3ε), the same power-lawslope as derived at the end of Sect. 2! This is natural in the sense that the shells do stop at a certainfraction of their turn-around radius when s < 3/2, as is assumed in Sect. 2.

For large accretion rates s > 3/2, on the other hand, the apapsis of the shell trajectories neverceases to shrink as more shells contribute to the mass and gravitational potential in the inner halo.In this case, it turns out that the inner density slope equals −2, irrespective of s 5.

Apart from the inner power-law slopes, the self-similar profiles show features such as the densitycaustics at the apapsis of the shells. In reality, when the matter possesses inhomogeneities andangular momentum, most caustics would be smoothed out, with an exception of the outmost one,whose location is the “splashback radius”. The self-similar model gives a full analytical descriptionof the profile shape around the splashback radius, i.e. the transition between single-stream andmulti-stream regimes of the dark matter envelope.

4See Sect. III of Fillmore and Goldreich [1984] for a derivation.5In self-similar spherical collapse, this −2 is the shallowest slope of the inner density profile. The −1 inner slope of the

NFW profile is most likely due to non-negligible angular momentum, which is not considered in this basic formulation ofspherical collapse. On the other hand, the steepest power-law density slope is −3 when s = 0. In the outskirts of the darkmatter halo, the density profile steepens further and deviate from a power-law shape. Additional steepening can also becaused by a decrease in the mass accretion rate with time, which would occur in general in the late times of a dark energydominated universe.

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5 Bondi accretion

Being collisional, the gravitational collapse of gas behaves rather differently than that of darkmatter. To get a feeling of gas dynamics and to serve as a comparison to the self-similar solutions,we now present the classical Bondi [1952] accretion model, which describes the spherical collapseof a polytropic gas onto a black hole or neutron star.

We seek steady state (i.e. ∂/∂t = 0, ∂/∂r = d/dr), spherically symmetric solutions to theequations of mass and momentum conservation (Eq. 48 and 51). with the steady state and sphericalsymmetry ansatz, the two equations reduce to

dρdr

+2r

+1v

dvdr

= 0 , (20)

vdvdr

= −∇Φ −1ρ

dpdr

, (21)

with Φ = −GM/r in this case, M being the mass of the central object. This set of differentialequations for variables ρ, p and v is closed by an equation of state

d ln p = γd ln ρ , (22)

which effectively describes how the energy of the system changes. These three differential equationscan then be solved given the boundary conditions. Alternatively, the steady-state equations (20) and(21) can be integrated to obtain the integral forms

M = 4πr2ρv = const. (23)

v2

2+

c2s

γ − 1−

GMr

=c2∞

γ − 1. (24)

The latter being the Bernoulli’s equation. Here we have chosen the boundary condition to be zerovelocity gas at r = ∞ with p∞ and ρ∞, and sound speed c∞ =

√γp∞/ρ∞. The original Bondi [1952]

paper presented a thorough study of the solutions of the steady state spherical accretion problemwith the integral equations in their dimensionless forms. The presentation is full of mathematicaland physical beauty. Here we try to give a taste of it without actually solving the equations.

Using the expression of the sound speed c2s = dp/dρ, and eliminating dρ/dr in (20) and (21), we

obtain the “Bondi equation”

12

(1 −

c2s

v2

)ddr

v2 = −GMr2

[1 −

2c2s r

GM

](25)

describing how the specific kinetic energy changes with the radius. Inspecting this equation, wecan already see the qualitative behaviors of its solutions. Most importantly, there exists a criticalpoint in the solution space where the right hand side of the Bondi equation equals zero, i.e. whenGM/rcrit = 2c2

s . At this point, the left hand side must vanish too, which means either the mattermoves with the local speed of sound v2 = c2

s , or the velocity has an extrema dv2/dr = 0.This critical point divides the solutions into three families: (i) The flow passes through the

critical point with v2 = c2s . This give two specific solutions of transonic flows (blue and red lines

in Fig. 4). We are most interested in the Bondi accretion solution, where the flow is at rest at largeradius, and finally falls onto the central object with supersonic speed. The opposite case, where thevelocity is subsonic at small radii and supersonic at large radii, represents Eugene Parker’s 1958

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Figure 4: Solutions of steady-state, spherically symmetric flow with equation of state γ = 7/5.

solution of the transonic solar wind. (ii) The flow passes through the critical point with dv2/dr = 0(the gray solid lines in Fig. 4). In this case, the flow stays either subsonic or supersonic everywhere.(iii) The flow cannot reach the critical point (the gray dashed lines in Fig. 4). These are a physicallyunrealistic family of solutions.

The control parameter distinguishing the three family of solutions is the integral constant of thecontinuity equation (20), or physically, the mass accretion rates M. By evaluating the Bernoulli’sequation at the critical point, we can find the critical mass accretion rate

Mcrit = π

(2

5 − 3γ

) 5−3γ2(γ−1) G2M2ρ∞

c3∞

= 4παBr2gρ∞c∞ , (26)

with rg = GM/c2∞ being the gravitational radius (a.k.a. Bondi radius) within which gravity of the

central object is much felt, and the pre-factor αB vary from e3/2/4 ≈ 1.12 for γ = 1 and 1/4 forγ = 5/3.

Family (i), (ii), and (iii) correspond to critical, sub-critical and super-critical mass accretionrates, respectively. For those accretion flows starting at large radii with small velocities and manageto reach the center, the flow either stays subsonic all the time and have sub-critical accretion rates,or is transonic and have the critical mass accretion rate. Bondi speculated that the critical flow,now called the Bondi accretion solution, is favored by nature. It has a unique mass accretion rateM = Mcrit, that can be determined by the central mass and the boundary conditions at r = ∞! Thiscritical accretion rate is the most quoted result in the Bondi paper. It represents an explosive massgrowth and is used to describe e.g. the growth of early seed Black Holes and their luminosity.

Let us inspect further the physics of this transonic Bondi accretion. First, note that the 2ndterm on the r.h.s. of Eq. (25) arises from the r2 term in the continuity equation as a “geometricalcompression” due to the spherical geometry. Without this geometrical compression (e.g. when

14

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matter falls in a plan-parallel fashion), there would be no critical point, and the steady-state flow cannever be transonic. Physically, this geometrical compression acts like a nozzle, efficiently focusingthe flow, converting potential energy to kinetic energy in the initial subsonic regime. Then, whenthe flow comes closer to the central object, at some point (close to rg) the gravitational potentialenergy starts to take over the thermal energy. Beyond this point, if the gas stays subsonic, thenthe further compression of the gas will channel the potential energy more and more to the thermalenergy instead of kinetic energy, the sound speed shoots up and the flow becomes more and moresubsonic, like in the case of family (ii) subsonic accretion flows. However, if the flow transits to besupersonic at the critical point, there is not sufficient time for the sound wave to heat up the flowsubstantially, and the flow can keep accelerating6.

10-3 10-2 10-1 100 101

rc2∞/GM

10-3

10-2

10-1

100

101

102

103

104

105

ρ/ρ∞

Bondi accretion

Parker wind

γ=1. 0

γ=1. 2

γ=1. 4

γ=1. 6

Figure 5: Density profiles of the transonic solutions of steady-state, spherically symmetric flow withvarious equation of state γ.

What about the density and temperature profiles for the steady-state solutions, to which lessattention has been paid? For the Bondi accretion solution, we get a density profile transiting fromthe constant density at r → ∞ to a power law profile ρ ∝ r−1.5 in the center, irrespective of theequation of state (Fig. 5)! This is actually no surprise: in the inner supersonic region, pressure force

6Very similar phenomena occur in the case of gas passing through a de Laval nozzle (a tube that is pinched in themiddle), and that of gas going across a mountain range. See the Thorne and Blandford [2017] textbook for a beautifuldescription of the former. For the later, the important dimensionless number is not the Mach number like in the Bondisolution, but is the Froude number.

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is negligible, and the flow approaches the “free fall asymptotic”

v ∝ −

√2GM

r∝ r−1/2 ,

ρ ∝M

4πvr2 ∝ r−3/2 ,

(27)

that describes the asymptotic structure of a non-self-gravitating free falling gas. Note that, althoughthe Bondi-accretion flow approaches free fall within the sonic point, overall speaking, the accretionis much slower than a free fall. The accreted matter spends a long time at large radii in the sub-sonic region, because sound waves propagating inwards tend to establish hydrostatic equilibrium,reducing the acceleration.

Finally, we must note that the Bondi solution is not fully consistent: with the matter inflow andthe growth of the central object, the flow cannot be stationary. To be approximately correct, the massgrowth rate of the central object has to be moderate, and thus accreting gas has to be hot.

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6 Self-similar collapse of cold gas onto a black hole

The symmetry of self-similarity, i.e. the invariance under a “stretching transformation”, is moregeneral than the symmetry of steady-state i.e. the invariance under a translation in time, and inprinciple includes the latter. When the character radius is chosen to be rc ∝ tα with α → 0, theself-similar solution approaches a steady-state solution. However, one cannot reproduce the Bondisolution with a self-similar framework for two reasons: the inconsistency within the Bondi solution,and that the “hot” gas already possess a typical temperature scale and so would break the self-similarity.

Here, instead, we consider another physically interesting setup: the spherical collapse of gas ini-tially following the Hubble flow onto a sink at the origin (a black hole) under gravitational attractionto a small density perturbation. Like in the case of self-similar spherical collapse of dark matter, weconsider an initial density perturbation of gas with a power-law shape ∆i ∝ M−εi ∝ M−1/s

i . We con-sider a cold gas whose pressure can be neglected, so as not to break the self-similarity. The wholedynamical process is then fully described by the spherical collapse model with

d2rdt2 = −

GMi

r2 . (28)

The situation is equivalent to the self-similar spherical collapse of dark matter, with the massprofile in the dynamical equation (18) being replaced byM(λ) = exp

[−(2/3) ξ1

], i.e. only using the

first root of λ = λ(ξ) since shells do not go across the origin in this case.Here we present a more intuitive approach using the spherical collapse model in its parametric

form Eqs. (4). Each shell of matter is labeled by the mass Mi it encloses, and the curvature of itK ∝ M2/3

i ∆i. From the parametric expression of the time evolution, we can work out the phases (i.e.the θ values) of the shells at a particular time as a function of their enclosed mass, and then with theother equation the positions of them. Keeping in mind that the profiles are self-similar when scaledto values at turn-around, we again define

M ≡M

Mta, λ ≡

rrta

, V ≡t

rta

drdt, D ≡

ρ

ρH, (29)

and then, the dimensionless profiles can be expressed in parametric forms

M =

(3π4

)2 (θ − sin θ

π

)−2/(3ε)

,

λ =1 − cos θ

2

(M

Mta

)1/3+ε

,

V =π

2sin θ

1 − cos θ

(M

Mta

)1/3−ε/2

,

D =9π2

(2 + 6ε)(1 − cos θ)3 − 9ε sin θ(θ − sin θ)(1 − cos θ)

(M

Mta

)−3ε

.

(30)

HereMta = (3π/4)2 ≈ 5.55 is the value ofM at turn-around (θ = π), and according to howM isdefined, it is the mean over-density at turn-around.

The resulting density and velocity profile is very similar to the Bondi solution in shape in theinner region. In the limit λ → 0 (θ → 2π), we have ρ ∝ λ−3/2 and V ∝ λ−1/2, i.e. again the “freefall asymptotic” like in the Bondi accretion. Here, however, this is because the black hole massdominates in the central region.

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10-2 10-1 100 101

λ

10-1

100

101

102

103

D

ε= 0. 2

ε= 0. 5

ε= 1. 0

ε= 2. 0

ε= 5. 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

λ

2

1

0

1

2

V

ε= 0. 2

ε= 0. 5

ε= 1. 0

ε= 2. 0

ε= 5. 0

0.0 0.5 1.0 1.5 2.0 2.5 3.0

λ

0

5

10

15

20

25

30

M

ε= 0. 2

ε= 0. 5

ε= 1. 0

ε= 2. 0

ε= 5. 0

Figure 6: The nondimensional density, velocity, and mass profiles of cold self-similar infall onto ablack hole for initial mass excess with various power-law slope ε.

18

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The black hole mass is just the mass of the shells that has already collapsed. By the self-similarity set-up, it grows in proportional to Mta, the average mass within rta. This suggests MBH ∝

t3δ−2 ∝ t2/3ε , a power-law growth instead of the explosive growth in Bondi accretion.

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7 Self-similar collapse of self-gravitating gas

Now we consider the self-similar collapse of a self-gravitating baryonic gas initially cold andexpanding with the Hubble flow. As there exists no black hole at the center, the gas will accumulateand establish pressure gradient. The infalling gas will first abruptly decelerate at a shock (the so-called “accretion shock”), converting the infall kinetic energy to pressure, and then further mildlydecelerate by the pressure gradient of the established gas halo, so that they come to rest at the center.Unlike dark matter, shell-crossing does not occur for the collisional gas.

The dynamics of the gas shells obey the equation of motion

d2rdt2 = −

GMr2 −

∂p∂r

, (31)

or in dimensionless form,

d2λ

dξ2 + (2δ − 1)dλdξ

+ δ (δ − 1) λ = −29M(λ)λ2 −

P′

D, (32)

with P = p/(ρHr2

ta/t2), and other parameters defined in the same way as in previous sections. Once

again, δ = 2(1 + s/3)/3 is the power-law index of the growth of the turn-around radius with time,determined by the mass growth rate s.

0 5 10 15 20

τ

0.0

0.2

0.4

0.6

0.8

1.0

λF

γ= 5/3

γ= 4/3

γ= 1. 3

Figure 7: Trajectories of self-similar spherical collapse of baryonic gas with various gas equation ofstate γ, for mass accretion rate s = 1. The radius (y-axis) and time (x-axis) are normalised to theturn-around radius and time of the shell.

Before the accretion shock, the trajectories of the shells of cold gas are identical to those of thedark matter (since p = 0). After / within the accretion shock, it is convenient to use the hydrody-namical equations (a.k.a. equations of motion of the gas) to solve for the gas density, pressure and

20

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0 5 10 15 20

τ

0.0

0.2

0.4

0.6

0.8

1.0

λF

s= 0. 5

s= 1

s= 2

s= 3

s= 5

Figure 8: Trajectories of self-similar spherical collapse of baryonic gas with an adiabatic equationof state γ = 5/3, for various mass accretion rates s.

velocity profiles:

dρdt

= −ρ

r2

∂r

(r2v

),

dvdt

= −GMr2 −

∂p∂r

,

ddt

(pρ−γ

)= 0 ,

∂M∂r

= 4πr2ρ .

(33)

Applying the self-similarity ansatz to the gas, the dimensionless form of this set of equationsreads

[V − δλ] D′ + DV ′ +2DVλ− 2D = 0 ,

[V − δλ] V ′ + (δ − 1)V = −29M

λ2 −P′

D,

[V − δλ](

P′

P− γ

D′

D

)= −2(γ − 1) − 2(δ − 1) ,

M′ = 3λ2D .

(34)

They are solved with the outer boundary conditions at the accretion shock given by the shock jump

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10-2 10-1 100

r/rta

10-1

100

101

102

103

104

105

106

D

10-2 10-1 100

r/rta

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

0.2

V

10-2 10-1 100

r/rta

100

101

102

103

104

105

106

107

P

s= 0. 5

s= 1

s= 2

s= 3

s= 5

10-2 10-1 100

r/rta

10-2

10-1

100

101

M

10-2 10-1 100

r/rta

10-4

10-3

10-2

10-1

100

K

10-2 10-1 100

r/rta

10-1

100

101

T

Figure 9: Nondimensional radial profiles of the self-similar spherical collapse of baryonic gas withγ = 5/3: gas density (upper left), velocity (upper middle), pressure (upper right), mass (lower left),entropy (lower middle), and averaged temperature (lower right). Different colors indicate differentmass accretion rate s.

conditions

V2 =γ − 1γ + 1

[V1 − δλsh] + δλsh ,

D2 =γ + 1γ − 1

D1 ,

P2 =2

γ + 1D1 [V1 − δλsh]2 ,

M2 =M1 .

(35)

The pre-shock velocity V1 and density D1 are given by the pre-shock gas velocity and mass profilesonce the shock radius λsh is known. The shock radius λsh is in turn determined by requiring thesolutions of Eq. (34) to satisfy the inner boundary conditions V(λ→ 0)→ 0 andM(λ→ 0)→ 0.

The self-similar spherical collapse solution satisfy a few integrals of motion. Apart from pro-viding checks for the numerical solution, they are valuable for understanding the shapes of the gasprofiles.

• Mass integralTo keep self-similarity, the fraction of gas mass within a radius λ should stay constant withtime. This gives a relation between the gas mass and its flux [Bertschinger, 1983],

MMta

=4πr2ρ(λvta − v)

4πr2taρtavta

. (36)

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10-4 10-3 10-2 10-1 100

r/rta

10-1100101102103104105106107108109

10101011

D

10-4 10-3 10-2 10-1 100

r/rta

10-6

10-5

10-4

10-3

10-2

10-1

100

101

−V

10-4 10-3 10-2 10-1 100

r/rta

101102103104105106107108109

10101011101210131014

P

s= 0. 5

s= 1

s= 2

s= 3

s= 5

10-4 10-3 10-2 10-1 100

r/rta

10-2

10-1

100

101

M

10-4 10-3 10-2 10-1 100

r/rta

10-2

10-1

100

K

10-4 10-3 10-2 10-1 100

r/rta

100

101

102

103

T

Figure 10: Profiles of self-similar spherical collapse of baryonic gas with γ = 4/3.

Using the definitions of the dimensionless quantities, and that

ρta =Mta

4πr3ta

d ln Mta

d ln t

(d ln rta

d ln t

)−1

=Mta

4πr3ta

2s3δ

(37)

to express Eq. (36) in terms of the dimensionless quantities, it is

M = −92sλ2DV , (38)

with V(λ) = V(λ) − δλ being the velocity in the frame comoving with r = λrta.

• Entropy integralWe re-write the continuity and adiabatic equations of the gas (the first and the third equationsof the fluid equations Eq. 34) as

(ln λ2DV)′ = −3δ − 2

V,

(ln P)′ − γ(ln D)′ = −2(γ + δ − 2)

V.

(39)

In this form, it is straightforward to derive that K(λ2DV

)ζ=const, where K is the adiabatic

invariant (also referred to as the entropy, in particular in the study of intracluster medium)defined as K = P/Dγ, and the power-law index

ζ = −2(γ + δ − 2)

3δ − 2= −

23−

1s. (40)

Together with Eq. (39), this gives KMζ =const., suggesting that the entropy actually providesa good Lagrangian coordinate! This is not surprising because the entropy is generated from

23

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the kinetic to thermal energy conversion at the accretion shock, and thus keeps a record ofhow much mass already lies in the collapsed object.

• Energy conservationThe two integrals of motion above are local ones which is valid at each radius. A globalintegral of motion is apparently the total energy within the shock radius. The dimensionlessspecific kinetic, thermal and gravitational energies averaged within the accretion shock are,respectively,

t =

∫ M2

0

V2

2dM ,

u =

∫ M2

0

P/Dγ − 1

dM ,

w = −29

∫ M2

0

M

λdM .

(41)

From energy conservation, we must have

t + u + w = E(λs) . (42)

The total dimensionless energy contained with in the shock radius E(λs) can be determinedfrom the initial conditions, noting that the dimensional energy E = MiK/2, as given by Eq. (2).

• Virial theoremAnother global integral of motion is provided by the virial theorem. The virial theorem canbe derived from the Euler equation in Eq. (34) as(

252δ2 −

252δ + 3

) ∫λ2dM

=2t + 3(γ − 1)u + w

− 3λ3s D2V2

[V2 +

7δ − 32

λs

]−

6γ − 1

λ3s D2V2

2 .

(43)

The l.h.s. is d2I/dt2/2 where I is the moment of inertia. The two boundary terms on ther.h.s. can be interpreted as originating from the flux of inertia through the boundary and thepressure at the boundary respectively 7. The second boundary term −6λ3

s D2V22/ (γ − 1) can be

also expressed as −3P2λ3s .

The thermal and gravitational energy are the dominating terms in the virial theorem. Theother terms are (in decreasing order of their magnitudes): the second boundary term, theacceleration of the moment of inertia, the first boundary term, and the kinetic energy.

Let us now examine the asymptotic behavior of the profiles at λ→ 0. Assume

D(λ→ 0) ∝ λαD , (44)

then from Eq. (38) we haveM(λ→ 0) ∝ λαD+3 , (45)

7The second boundary term is missing in the virial theorem expression in Bertschinger [1985]. It is larger than thefirst boundary term in magnitude.

24

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and from Eq. (39),P(λ→ 0) ∝ λ2αD+2 . (46)

From the first equation in (34), we derive that αD = λ−2/δ = −9/(s + 3), if the central velocity satisfyadditional constraints V ′(λ→ 0) 1 and V(λ→ 0) λ. Again, this is exactly the same power-lawindex of the density slope as the spherical collapse case assuming shells stop at a fixed fraction oftheir turn-around radius, and the same power-law slope of the self-similar spherical collapse of darkmatter when the accretion rate is small s < 3/2! Same here, this holds only when shells of matter doasymptote to a radius at a fixed fraction of their turn-around radius. This means, only for a not too bigmass accretion rate s, and a not too small gas equation of state γ. At s = 1 which is the top-hat initialoverdensity case considered in Bertschinger [1985], the critical gas equation of state is found to beγ = 4/3. For a softer equation of state, i.e. γ ≤ 4/3, the additional central velocity constraints canno longer be satisfied, the gas density profile steepens with decreasing γ for all mass accretion rates(Fig. 10). Numerically, the long term behavior of the trajectory becomes increasingly sensitive to theprecise value of the shock radius λs. A settling solution with the trajectory asymptotes to a particularr > 0 is hard to find, a slight numerical deviation causes the radius of the shell to rapidly drop tothe center, i.e. even if settling solution exists, it is less stable. Also, the settling radius becomesextremely small already at γ = 4/3 (Fig. 7), corresponding to a very steep central density profile,and suggesting that the majority of the mass within the shock radius is packed in a tiny central region.Physically, this means that the system is more and more prone to a central gravitational collapseat smaller γ, which would lead to the formation a black hole unless other stabilizing mechanismscome to rescue. This is reminiscent of Chandrasekhar’s results on the stability of self-gravitatingpolytropic spheres, where the critical equation of state is also 4/3.

At this point, it is interesting to mention the McCrea [1956] effort in inserting a stationary shockto the Bondi solution to provide a “settling solution” with v(r = 0) = 0 when matter falls onto e.g.a neutron star with a surface. There, a stationary shock is possible for 1 < γ < 3/2. For a stifferequation of state, no stationary setting solution exists, since the shock must propagate outward8.

Another noteworthy feature in the self-similar gas profiles is the effective gas equation of stateΓ = (∂ ln P/∂ ln r)/(∂ ln ρ/∂ ln r). Different from the “real” gas equation of state γ which describeshow the gas reacts to a compression / expansion, the effective equation of state Γ describes thestructure of the gas halo. For a stationary solution like the Bondi solution, Γ = γ, but this is not thecase for a self-similar solution. Here, in fact, Γ is in general not a constant, although it varies notmuch with radius due to the approximate power-law shapes of the profiles. The reason why Γ < γ inself-similar solutions is because the mass growth is self-consistently considered. Recall the entropyintegral, the later-accreted shells develop larger entropy, so that the entropy profile has a positiveslope ln K/∂ ln ρ < 0, and thus P = Kργ has an overall less steep dependency on ρ .

8This is not mentioned in the original MaCrea paper, but follows from his analysis as Bertschinger noted.

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8 Self-similar collapse of gas and dark matter

The last case we shall consider is the self-similar collapse of gas and dark matter together. Inthis case, the form of the dynamical equations after accretion shock resembles those in the case ofself-similar spherical collapse of gas alone, however here we need to distinguish the total gravitatingmassM and the gas massMgas:

dρdt

= −ρ

r2

∂r

(r2v

),

dvdt

= −GM

r2 −1ρ

∂p∂r

,

ddt

(pρ−γ

)= 0 ,

Mgas

∂r= 4πr2ρ .

(47)

For simplicity, we show the case with Ωb = 0, i.e. assume that the mass is fully dominatedby the collisionless dark matter. The mass profileM is then given for the dark matter only case inSect. 4. Since it is very similar to the gas mass profile in Sect. 7, the resulting profiles are mostlyvery similar to that in Sect. 7, and do not vary much with Ωb.

10-2 10-1 100

λ

10-1100101102103104105106

D

s=0.5

s=1

s=2

s=3

s=5

10-2 10-1 100

λ

−1.5

−1.0

−0.5

0.0

V

10-2 10-1 100

λ

100

101

102

103

104

105

106

P

10-2 10-1 100

λ

10-2

10-1

100

101

M

10-2 10-1 100

λ

10-3

10-2

10-1

K

10-2 10-1 100

λ

100

101

102

103

104

105

106

∆m

Figure 11: Gas halo profiles from self-similar spherical collapse of non-self-gravitating gas (solidlines). Different colors indicate different mass accretion rate s. In the left panels, the density andmass profiles of dark matter are plotted as the color dashed lines for comparison. In the lower rightpanel, averaged overdensity of 200, 500, and 2500 are marked as the dashed, dash-dotted, and dottedlines to show the corresponding positions of commonly used radii r200, r500 and r2500. An adiabaticequation of state γ = 5/3 is taken for the gas.

The characteristic features emerging from the self-similar solution are the splashback and ac-cretion shock for the dark matter and gas respectively. They both mark the position of density jumpswhich give physical boundaries to the halo. From Fig. 12 one can clearly read out the splashbackradius and the accretion shock radius. Both the splashback radius and the shock radius decrease

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0.0 0.5 1.0 1.5 2.0 2.5 3.0

ξ

0.0

0.2

0.4

0.6

0.8

1.0

λ

s=0.5

s=1

s=2

s=3

s=5

2 4 6 8 10 12 14

τ

0.0

0.2

0.4

0.6

0.8

1.0

λF

s=0.5

s=1

s=2

s=3

s=5

Figure 12: Upper panel: Trajectories of self-similar spherical infall of dark matter (dashed lines)and gas (solid lines) starting from turn-around, shown in the λ-ξ plane. Lower panel: Trajectories ofself-similar infall of and gas shown in the λF-τ plane. The gray dashed line in the lower panel showsthe trajectory for the dark matter till its first core-passage as a comparison. Both are for gas with anequation of state γ = 5/3.

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from the turn-around radius with an increasing mass accretion rate s. These two radii follow eachother closely for various accretion rates (indicated by the color of the lines) when the gas is adiabatic(γ = 5/3). It is remarkable that this behavior, which was discovered in adiabatic hydrodynamicalsimulations for galaxy clusters [Lau et al., 2015], is already captured by the self-similarity solutions.This alignment arises from the dependencies of the accretion shock and the splashback radius on theaccretion rate, which are the same in direction and similar in degree, but different in physics. Forthe gas, it is because the higher energy and momentum of the inflowing gas associated with a highermass accretion rate. For the dark matter, it is due to a more significant halo growth during the timebetween the splashback and when the matter at splashback was accreted. However, this holds trueonly for a gas with an equation of state γ ≈ 5/3 [Shi, 2016b], which is approximately the equationof state of the intracluster medium outside the core region. The accretion shock position shrinks forgas with a softer equation of state, i.e. for gas halos of less massive galaxy groups and ellipticalgalaxies, where gas cooling is more efficient.

In Sect. 7 we mentioned that for self-gravitating gas with an equation of state γ < 4/3, thedensity profiles steepen, and the system is prone to central collapse. For gas collapsing with darkmatter, however, this does not occur. In general, the inner gas density slope depends on both controlparameters γ and s. The gas profiles are most similar to those of dark matter at γ ≈ 5/3, exceptfor a difference at large accretion rates s > 3/2. There, the trajectories of the non-self-gravitatinggas still settles at a finite Eulerian radius within its shock radius, and the inner mass profile continueto steepen, in contrast to the dark matter saturating at a slope of 1. All these suggest that the darkmatter has a stabilising effect on the gas.

The intracluster medium is well-described by an adiabatic gas equation of state of 5/3, sincecooling is inefficient at the high temperature and low density regime it resides. At γ = 5/3, thestructure of the non-self-gravitating gas is reasonably well described by an effective equation ofstate Γ ≈ 1.1 − 1.2. At the same time, the power-law slope of the entropy profile also lies within arather narrow range ∼ 1.1 − 1.2. These are both confirmed by the X-ray observations.

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9 Asymptotic profile of gravitational collapse: beyond self-similar so-lutions

We have applied self-similarity analysis to various spherical collapse scenarios: that of colli-sionless dark matter and collisional gas, self-gravitating and not; with and without a central blackhole; and for various mass accretion rates and gas equations of state. The ansatz of self-similarityenables us to obtain analytical solutions even in the most difficult case: multi-stream regime of darkmatter collapse.

The self-similar solutions yield the only detailed analytical description so far of the envelopesformed under gravitational collapse. They exhibit rich phenomena in their dependence on the controlparameters: mass accretion rate and gas equation of state. For example, the different behaviors ofshell trajectories and the corresponding density profile at s > 3/2 and s < 3/2, γ > 4/3 and γ < 4/3,have enriched our understanding of the underlying dynamics. This is hardly possible with numericalsimulations alone. As another example, the existence of accretion shock for the gas and splashbackfor the dark matter, as well as their apparent alignment in radial position, are already captured bythe self-similar solutions and are only recently also discovered in numerical simulations.

On the other hand, there is still a huge gap between the clean and idealised picture from self-similar solutions to the messy but more realistic picture from numerical simulations. The self-similaransatz imposes a very strong symmetry. To be invariant under the “stretching transformation”,the resulting solutions contain little features. Thus, heuristically, one may regard the self-similarsolutions as a special kind of “least informative” solutions which keep global integrals of motion andlocal equilibrium conditions. This may explain why self-similar solutions are usually intermediateasymptotics. However, at the same time, the self-similar ansatz limits the degree of reality, andstrongly restricts the situations we can deal with. For example, no physical scales can be introducedby considering the existence of dark energy, non-power-law mass growth history, gas cooling, etc.For our general purpose: understanding the asymptotic profiles of gravitational collapse seen inobservations and simulations, we would also like to more realistic situations with richer physics.

Shi [2016a] made a first step in generalising the self-similar solutions of spherical collapse.There, the self-similar spherical collapse of dark matter is generalised to a ΛCDM universe. Thetrick is to apply the trajectory-mass profile iteration to the generalised dynamical equation to ex-plicitly look for a more general asymptotic solution M(λ, a), rather than a self-similar M(λ). Aslong as the actual mass profile changes slowly enough with the scale factor a, i.e. an intermediateasymptotic profile exists, the iterations converge fast, leading us to the asymptotic profile. The ideaof using iterative methods to look for intermediate asymptotic may apply generally to situations nolonger obey self-similarity. Finding intermediate asymptotic in more general situations will make afurther step in bridging the gap between analytic understanding and numerical simulation results.

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A. Whitworth and D. Summers. Self-similar condensation of spherically symmetric self-gravitatingisothermal gas clouds. MNRAS, 214:1–25, May 1985. doi: 10.1093/mnras/214.1.1.

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A Forms of hydrodynamical equations

A.1 Equation of mass conservation

The equation of mass conservation, or the continuity equation, describes that the change of theamount of matter equals the divergence of its flux in the absence of “sources” or “sinks”. In Eulerianform (applies to a fixed volume), it reads

∂ρ

∂t+ ∇ · (ρv) = 0 . (48)

Using the material derivative of density Dρ/Dt = ∂ρ/∂t + v · ∇ρ, one finds its Lagrangian form(applies to a fixed mass)

DρDt

+ ρ (∇ · v) = 0 . (49)

In spherical coordinates and for a spherically symmetric quantity f (i.e. f does not have θ or φdependence), the divergence operator in (48) and (49) reduces to

∇ · f =1r2

∂(r2 f

)∂r

=∂ f∂r

+2r

f . (50)

A.2 Equation of momentum conservation

The equation of momentum conservation is often called the “equation of motion” since it de-scribes how the motion of the matter is changed by the forces applied to it. In Lagrangian form, itreads

DvDt

=∂v∂t

+ v · ∇v = −∇Φ −∇pρ. (51)

Here we consider only the gravitational force and the pressure gradient force. Other forces suchas magnetic force and centrifugal force can be added to the right hand side of the equation. For aspherical distribution of mass, the derivative of the gravitational potential Φ can also be written as

∇Φ =GM(< r)

r2 . (52)

The Eulerian form of the equation of momentum conservation reads

∂t(ρv) + ∇ · Π = −ρ∇Φ , (53)

where Π is the momentum flux density (or another name: the fluid pressure tensor)

Π = ρvv + pI . (54)

One can derive the Eulerian form (53) and Lagrangian form (51) one from the other making use ofthe continuity equation (48).

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A.3 Equation of energy conservation

The equation of energy conservation is

∂t

(ρv2

2+ e + ρΦ

)+ ∇ ·

[v(ρv2

2+ e + p + ρΦ

)]= 0 . (55)

Inserting the continuity and momentum conservation equations, this simplifies to

DeDt

+ (e + p)∇ · v + ρ∂Φ

∂t= 0 . (56)

One can easily verify that a polytropic gas with Dp/p = γDρ/ρ and e = p/(γ − 1) satisfies auto-matically the energy conservation equation with stationary gravitational potential i.e. ∂Φ/∂t = 0.Thus, in many cases, it is sufficient to substitute the energy conservation equation with a polytropicequation of state.

33