Turbulent Heating in the X-ray Brightest Galaxy Clusters · Turbulent small-scale regime: teddy

Turbulent Heating in the X-ray Brightest Galaxy Clusters

SnowCluster, Snowbird, Utah, March 15-20, 2015

Irina ZhuravlevaKIPAC, Stanford University

E. Churazov, A. Schekochihin, S. Allen, P. Arevalo, A. Fabian, W. Forman, M. Gaspari, E. Lau, D. Nagai,

S. Nelson, I. Parrish, J. Sanders, A. Simionescu, R. Sunyaev, A. Vikhlinin, N. Werner

Turbulent dissipation in AGN feedback

Possible channeling mechanisms:shocks, sound waves (Randall et al. 11; Fabian et al. 06)

turbulent dissipation (Churazov et al. 02; Fukita et al. 04; Banerjee et al 14) turbulent mixing (Kim & Narayan 03)

cosmic rays (Chandran & Dennis 06; Pfrommer et al. 13)

radiative heating (Ciotti & Ostriker 01; Nulsen & Fabian 00) etc.

energy release => bubbles => ? => heating

Definitions and assumptions

1

L

1

lO

1

ldiss

1

�mfp

1

⇢i

Kinetic energy

Magnetic energy

k-5/3

k

unmagnetizedmagnetizedk⏊-5/3

kr-3

stratified turbulence

isotropic turbulence

assumptions:no magnetic fieldsno plasma effects


Kolmogorov constant

gas mass density

Kolmogorov 41 Sreenivasan et al. 95 Kaneda et al. 03 Dennis & Chandran 05

Qturb = K0⇢V 3l

lvelocity amplitude

on scale l

cluster in HE: V=0 disturbed cluster: V≠0

Is there any way to probe gas dynamics using currently-available data?

δρ —> V?

How do density perturbations scale with the velocity field?

homogeneous box

stratified atmosphere

δρ∝M2from Bernoulli's equation

(solenoidal motions)

?

g-modes in stratified atmospherestratification in clusters:

gradient of S, ρ

NBV

~g

low-entropy gas

high-entropy gas small and slow perturbations => g-modes (internal or gravity waves)

surface gravity waves

g-modes in stratified atmosphere!2 = N2 k

2?k2

N =

rg

�Hs

Hs = (d lnS/dr)�1

k2 = k2? + k2r

Dispersion relation:

Brunt-Väisälä frequency:

Entropy scale height:

Wavenumber:

~r?

g-modes are confined inside the cluster core => interact => non-linear => turbulent motions

Balbus & Soker, 1990

—>

g-modes in stratified atmosphere

~r?

in “⊥” direction in “r” direction

“pancake” turbulenceon large scales V is dominated by V⊥

Waite & Bartello 2006

if ω << N (stratification is important) => k⊥ << kr

!2 = N2 k2?k2

Buoyancy-dominated regime of motions

ΔrL

grav

ity

Vr << V⊥ ~ V

ΔrL

≈ ωNBV

Turbulent eddy at injection scale L :

V = NBV �r

gravity provides V - Δr relation

stir the gas slowly with ω<<NBV

V? = L! ⇠ V

Gas displacement and density contrastS(r)

r

slow displacement Sb=const

Sb

Sb

gas in pressure equilibrium Pb=P

Δr

Pb=SbρbᵞP=(Sb+δS)ρᵞ

density contrast after (slow) gas displacement: �⇢

⇢=

1

�

�S

S⇡ 1

�

�r

H

entropy gradient gives δρ - Δr relationentropy scale height

�⇢

⇢=

1

�

�r

Hs

V = NBV �r

NBV =cs

�p

HsHp

�⇢

⇢= ⌘

V

cs

⌘ =

rHp

Hs⇠ 1

Buoyancy-dominated regime of motions

valid on large, buoyancy-dominated scales

entropy gradient:

gravity:

the system adjusts so that η ~ 1

Turbulent small-scale regime: teddy << tNObukhov-Corrsin approach:

Obukhov 1949; Corrsin 1951

the spectrum of the passive scalar follows the spectrum of the velocity field in the inertial range

hot gas gas entropy (density)

any perturbation in ICM

�⇢k⇢

= ⌘Vk

csη~1 is set on large, buoyancy-dominated scales

Verifying the coefficient ηAMR cosmological simulations, NR runs, relaxed clusters

Kravtsov et al. 99;03; Nagai et al. 07a; Nelson et al. 14

⌘ = 1± 0.3

V

ρ

sample averaged

Zhuravleva et al. 2014a

hydro simulations: η ~1 w/o conductionGaspari et al. 2014

Nature of fluctuations in Perseus

1) sound waves (Fabian et al. 03; 06; Sanders et al. 07)

2) stratified turbulence (Zhuravleva et al. 14; 15)

Nature of fluctuations in Perseus: stratified turbulence?

Eugene’s GIMP simulations

Linné FLOW Centre

Δr ~HV/cs talk by E. Churazov

cross-spectra analysis of SB fluctuations in different energy bands:isobaric fluctuations (gas sloshing, turbulence, g-modes)

Amplitude of density fluctuations in Perseus

IX /Z

n2e✏(T )dl /

Zn2edl

0.5-3.5 keV0.37 0.53 0.68 0.84 1 1.2 1.3 1.5 1.6 1.8 1.9

3’

Gas Fluctuations in the Perseus Cluster 3

0.000000 0.000006 0.000031 0.000129 0.000522 0.002076

3’

0.44 0.58 0.71 0.84 0.98 1.1 1.2 1.4 1.5 1.6 1.8

3’

Figure 2. Left: Chandra mosaic image of the Perseus Cluster in 0.5�3.5 keV band. The units are counts/s/pixel. Right: residual imageof the cluster (the initial image divided by the best-fitting spherically-symmetric ��model of the surface brightness), which emphasizesthe surface brightness fluctuations present in the cluster. The point sources are excised from the image. Black circles show a set of annuliused in the analysis of fluctuations. The width of each annulus is 1.50 (⇡ 31 kpc). The outermost annulus is at distance 10.50 (⇡ 218kpc) from the center. Both images are slightly smoothed with a 300 Gaussian.

0 0.0007 0.0021 0.0049 0.01 0.022 0.044 0.088 0.18 0.35 0.7

3’0.00e+00 2.89e+05 5.80e+05 8.70e+05 1.16e+06 1.45e+06

3’

Figure 3. Left: shape of the combined Chandra PSF within the field of view of the Perseus Cluster. The random positions of individualPSFs are used (see Section XX for details). Right: the combined exposure map in seconds slightly smoothed with a 300 Gaussian.

c� 2014 RAS, MNRAS 000, 1–16


A23D(k) = 4⇡P3D(k)k3

Units are the same of the variable in real space

�⇢k⇢

⇠ 7� 15%outside central 30 kpc: on scales 6-30 kpc

8 Zhuravleva et al.

Figure 8. Left: amplitude of one-component velocity of gas motions versus wavenumber k = 1/l, measured in a set of radial annuli (seelegend) in the Perseus Cluster. The velocity is obtained from the amplitude of density fluctuations, shown in Fig. 7, using relation 5. Thecolor-coding and notations are the same as in Fig. 7. The slope of the amplitude for pure Kolmogorov turbulence (Kolmogorov 1941),k�1/3, is shown with dash line. Right: radial profile of one-component velocity amplitude measured at certain length scales written inthe legend.

kpc (see Table 1). The velocity amplitudes quantitativelymatch our expectations of typical velocities in the ICM fromvarious observational constraints (see e.g. Churazov et al.2004; Schuecker et al. 2004; Werner et al. 2009; Sanders,Fabian, & Smith 2011; de Plaa et al. 2012; Sanders & Fabian2013, and references therein) and numerical simulations (seee.g. Norman & Bryan 1999; Dolag et al. 2005; Iapichino &Niemeyer 2008; Lau, Kravtsov, & Nagai 2009; Vazza et al.2011; Miniati 2014, and references therein). Even though theSB fluctuations analysis gives us reasonable constraints onstatistical properties of the velocity field in the ICM, themethod should be calibrated with the direct velocity mea-surements with X-ray calorimeter on-board Astro-H obser-vatory (Takahashi et al. 2014).

It was recently shown that in the cores of Perseus andVirgo clusters, where the cooling time is shorter than theHubble time, the heating of the gas due to dissipation ofturbulence is su�cient to o↵set radiative cooling losses (Zhu-ravleva et al. 2014b). Accounting for this fact, it is straight-forward to estimate the Ozmidov scale lO of the turbulenceusing only thermodynamic properties of the Perseus Clus-ter and compare it with the scales we are probing with ourmeasurements. Knowing the Brunt-Vaisala frequency in the

cluster atmosphere, N =

rg

�Hs, through the acceleration

of gravity g and the entropy scale height Hs =

„dlnS

dr

«�1

,

and the density-normalized dissipation rate ", the Ozmidovscale is

lO = N�3/2"1/2 = N�3/2

„Qcool

⇢

«1/2

, (6)

where we assumed " ⇠ the cooling rate Qcool = neni⇤n(T ),normalized by the gas density ⇢. Here ne and ni are the num-ber densities of electrons and ions, respectively, and ⇤n(T )is the normalized gas cooling function (Sutherland & Dopita1993).

Fig. 9 shows the radial profile of the Ozmidov scale lOand a range of scales we are probing in each annulus (hatchedregions in Fig. 8). One can see that lO is within the intervalof scales we are probing at each distance from the centerwithin the cluster core. This means that the necessary re-quirement for the proportionality coe�cient between densityand velocity ⌘1 ⇠ 1 is satisfied, namely the Ozmidov scale islower than the injection scale of turbulence (assuming thatwe are probing velocity PS within the inertial range). Notice,that we do not show lO at R < 20 kpc since the measuredgas entropy is flat towards the center, leading to lO !1.

It is also interesting to compare scales, on which veloc-ities of gas motions were measured, with the Kolmogorov(dissipation) scale

lK =⌫

3/4kin

(Qcool/⇢)1/4, (7)

where ⌫kin =⌫dyn

⇢is the kinematic viscosity, which is ob-

tained through the dynamic viscosity ⌫dyn for an ionizedplasma without magnetic field. Fig. 9 shows the Kolmogorovscale lK as well as the mean free path for comparison. TheKolmogorov scale is significantly below the scales we are

c� 2014 RAS, MNRAS 000, 1–16

Velocity Power Spectrum in Perseus

• V higher towards the center —> power injection from the center • larger V on smaller k —> consistent with cascade turbulence • 70 km/s < V1,k < 200 km/s on scales 6-30 kpc (within central 200 kpc)

Zhuravleva et al., 2015, MNRAS, submitted

Vk = cs�⇢

⇢


Kolmogorov constant

gas mass density

Kolmogorov 41 Sreenivasan et al. 95 Kaneda et al. 03 Dennis & Chandran 05

Qturb = K0⇢V 3l

lvelocity amplitude

on scale l

we measure Vl and each scale l => can obtain Qturb


Perseus cluster Virgo cluster

H(k) = CH⇢V 31,kk

C = neni⇤n(T )cooling rate:

heating rate:

locally: cooling ~ heating

AGN —> Bubbles —> Turbulent dissipation —> HeatZhuravleva et al. 2014b


If we assume Cooling~Heating, then:

Dissipation of turbulence with M~0.15 is sufficient to balance cooling of the gas in cores

Mk ⇡ 0.15⇣ ne

10�2 cm�3

⌘1/3✓

cs1000 km/s

◆�1 ✓ l

10 kpc

◆1/3

What happens outside cool cores?

tcool

theat

tHubble

R

tour expectations:

cooling ~ heating

What happens outside cool cores?

Work in progress

XMM-Newton, Perseus

dissipation of gas motions on timescales ~ cluster age

Chandra

Hubble time

1. Bias in the normalization of the amplitude of density fluctuations from the Δ-variance method: ~1.3 (Perseus) and 1.2 (Virgo);

2. Division of cluster into “perturbed” and “unperturbed” components: small-scale power is robust;

3. Conversion between 2D to 3D power spectrum: < 20%; 4. Accuracy of the Poisson noise subtraction, PSF correction, unresolved

point sources correction - robust regions; 5. Stochasticity of density fluctuations - broad annuli; 6. Weighting scheme used to calculate the amplitude of fluctuations:

<20%; 7. We cannot prove that we see turbulence; 8. η scatter is ~ 30%, not sure that it is the same in the range of scales -

Ozmidov scale is within the range of scales we are probing; etc.

➣ final uncertainty in the heating rate is ~ a factor of 3➣ Astro-H is needed to calibrate density/velocity relation

Uncertainties (incomplete list)

M A N Y

Summary• relaxed clusters • subsonic motions • simplest approach

V measurements on different scales: • Perseus: 70 km/s < V1,k < 200 km/s on 6 - 30 kpc (within ~ 200 kpc) • Virgo: 40 km/s < V1,k < 90 km/s on 2 - 10 kpc (within ~40 kpc)

AGN-feedback: • turbulence dissipation is sufficient to offset cooling locally at each r • AGN —> Bubbles —> Turbulent dissipation —> Heat

Nature of fluctuations in Perseus: dominated by isobaric fluctuations (gas turbulence, sloshing, g-modes)

Astro-H (end 2015), Athena (2028), Smart-X (?): • verification of the linear relation, importance of microphysics • nature of fluctuations

�⇢k⇢

= ⌘V1,k

cs

Velocity Power Spectrum in Virgo

0.000010 0.000016 0.000042 0.000145 0.000555 0.002178

3’

0.38 0.53 0.69 0.84 1 1.2 1.3 1.5 1.6 1.8 1.9

3’

0.763 0.847 0.932 1.02 1.1 1.19 1.27 1.36 1.44 1.53 1.61

3’ typical values:40 km/s < V1,k < 90 km/s on scales 2 - 10 kpc

Zhuravleva et al. 2014b, Arevalo et al. in prep.

Known unknowns in ICM(incomplete list)

• velocity field (turbulence, gas sloshing) • transport processes (viscosity, diffusion, convection,

conduction) • heating of ICM • cosmology: non-thermal pressure • particle acceleration • amplification of mag. fields • origin of radio halos

crucially important for gastrophysics, plasma physics, cosmology

proj

ectio

n

A2D / A3D

r1

NN is a number of fluctuations of the certain size along the line of sight

f2D!3D is smaller for large-scale fluctuations (left) than for small-scale fluctuations (right)

ℓ=1/k

P2D / P3D1

L⇒A2

2D = 2⇡P2Dk2

A23D = 4⇡P3Dk3

A3D / A2D

pN N =

L

l

P2D of SB —> P3D of density fluctuations

proj

ectio

nL

simulated clustercentral 200 kpc

Churazov et al. 12; Zhuravleva et al. 12

Perseus: 70 km/s < V1,k < 200 km/s on scales 6-30 kpc M87: 40 km/s < V1,k < 90 km/s on scales 2 - 10 kpc

NGC4636: V<100 km/s (RS) (Werner, IZ et al. 2009)

NGC 5044: 300 (RS) < V < 950 (width) km/s (de Plaa, IZ et al. 2012) NGC 5813: 100 (RS) < V < 670 (width) km/s (de Plaa, IZ et al. 2012)

A1835: V< 274 km/s (width) (Sanders et al. 09)

62 cores of clusters: V < 700/500 km/s (width) (Sanders et al. 11)

44 cores of clusters: V< 500 km/s (width) (Pinto et al. 2015)

Measured velocities (so far):


Adiabatic fluctuations (sound waves, shocks):

Isobaric fluctuations (gas sloshing, turbulence, g-modes):

⇢ T P↑, ↑ ↑⇒

⇢ T P↑, ↓ ~const⇒

T range in Perseus

F / n2✏(T )

independent of T

increases with T

soft band: density hard band: T-dependent

Work in progress

adiabatic

isothermal

isobaric

amplitude is larger in hard band (3.5-7.5 keV)

amplitude is larger in soft band (0.5-3.5 keV)

need to compare the PS of SB fluctuations in both bands


V 1,k, k

m/s

Do we see Kolmogorov slope of the spectrum?

k⊥,kr

E⊥(k⊥) E⊥(kr)

k⊥-5/3 kr-5/3kr-3

kO

Ozmidov scale: kO=N-3/2ε1/2

Ozmidov 1992


Figure 9. Characteristic length scales present in the PerseusCluster at di↵erent distances R from the center. Colored shadedregions: range of scales, on which we calculate the PS of densityfluctuations and velocity PS and on which our measurements areleast a↵ected by systematic and statistical uncertainties (hatchedregions in Fig. 7 and Fig. 8). Solid curve: the Ozmidov scale lOof turbulence in the core of Perseus, obtained assuming local bal-ance between turbulent heating and radiative cooling (relation 6).We do not plot lO in central ⇠ 20 kpc, since the measured gasentropy is flat there, leading to lO ! 1. Dash curve: the Kol-mogorov (dissipation) scale lK for unmagnetized plasma (relation7). Dot-dash curve: the cluster mean free path. See Section XXfor discussion.

probing, which justifies even better our assumption aboutthe inertial range of scales. In reality, the situation can bemore complicated. For example, the presence of magneticfields could violate our assumptions and arguments, how-ever we neglect these e↵ects here, considering the simplestcase as a 0th order approach to the problem.

We cannot claim, of course, that the structures seen inthe SB are due to turbulence only. For example, there aremany bubbles of relativistic plasma in the Perseus core (seee.g. Pedlar et al. 1990; Boehringer et al. 1993; Churazov etal. 2000; Fabian et al. 2000), which may contribute to thesignal. The question is whether their contribution is dom-inant in considered regions. Various sharp fronts (bubbleedges) would give the slope of the amplitude k�1/2, which ishard to discriminate from the Kolmogorov slope k�1/3 ac-counting for the uncertainties of our measurements and theassumption used. However the contribution of the bubblesto the signal can be easily seen if we repeat the analysis ex-cluding the know bubbles from the image of Perseus. Fig.10 shows the one-component velocity amplitude calculatedfrom such image. Notice, that the velocity amplitude de-creases by a factor ⇠ 1.6 � 1.2 depending on scale in thecentral 1.5 arcmin. This is expected since bubbles in this re-gion are particularly prominent and occupy a vast fractionof the volume. In the 1.5 � 3 arcmin annulus, this factor is

⇠ 1.15 over the whole range of scales. While outside 3 ar-cmin, the exclusion of bubbles from the analysis does notchange the measured velocity amplitude.

Sound waves (Fabian et al. 2006; Sternberg & Soker2009), mergers and gas sloshing (Markevitch & Vikhlinin2007) might also contribute to the observed density fluc-tuation spectra. Unsharp masking of the Perseus Clusterrevealed quasi-spherical structures (“ripples”) in the SB,which have been interpreted as isothermal sound waves(Fabian et al. 2000, 2003). An alternative interpretation isstratified turbulence (Zhuravleva et al. 2014a; Brethouweret al. 2007), which arises naturally is cluster atmospheres,where rough estimated give Froude number Fr ⇠ 0.3 � 1(Zhuravleva et al. 2014a). In this case, the radial size ofeach “ripple” is determined by HV/cs, where H is the char-acteristic scale height and V is the velocity amplitude. Inorder to make a firm conclusion about the nature of suchstructures, the cross spectra of SB fluctuations in di↵erentenergy bands have to be measured, which is the scope of ourfuture work. Our preliminary calculations show that the fluc-tuations are preliminary isobaric. Therefore, even if bubblesand sound waves are present in the cluster core, fluctuationsdue to turbulence and gas sloshing are dominating the signalin the measured PS. At the very least, measured turbulentvelocities can be treated as an upper limit.

5.3 Gas clumping

By gas clumping we will mean any deviations of the gasdensity and temperature isosurfaces from the equipotentialsurfaces3. This definition includes both: large-scale inhomo-geneities (e.g. induced by gas sloshing, perturbations of po-tential) and small-scale clumpiness (e.g., caused by gas tur-bulence, motions of galaxies and subhalos etc). That is tosay, gas clumping is a scale-dependent characteristic and,being measured on di↵erent scales, can be used to constrainphysics of the ICM (this is especially useful for comparisonobservations and implemented physics in numerical simula-tions of clusters).

The gas clumping factor is usually defined as

C =hn2

eihnei2

, (8)

where hi denotes the mean (sometimes median) inside spher-ical shell. Clumping of gas in cluster outskirts is of particu-lar interest for cosmologists, since it can bias the hydrostaticmass (Lau, Kravtsov, & Nagai 2009) and SZ measurementsof the Compton parameter (Khedekar et al. 2013). Numer-ical simulations predict clumping < 1.05 in central 0.5r500,⇠ 1.1 � 1.4 at r500 and up to 2 at r200 (see e.g. Roncarelliet al. 2013; Zhuravleva et al. 2013; Vazza et al. 2013; Nagai& Lau 2011; Mathiesen, Evrard, & Mohr 1999).

Observations of the gas clumping are challenging. In

3 Less general definitions are also used in the literature. For ex-ample, sometimes clumping is referred to gas clumps and subha-los, which are significantly denser and cooler than ambient gasand often have their own dark matter halo, neglecting inhomoge-nenities of smaller amplitude on larger scales.

c� 2014 RAS, MNRAS 000, 1–16

Do we probe turbulence within the inertial range?

dissipation scale (for unmagnetized plasma) < scales we are probing

k⊥, k

P(k)

kinj

dominated by buoyancy

buoyancy not important

kdisinertial range

�⇢k⇢

= ⌘Vk

cs⌘ ⇠ 1

Power spectrum V ~ power spectrum ρ

Zhuravleva, Churazov, Schekochihin et al. 2014a

Gas dynamics in simulations

ICM turbulence within r500:• driven by mergers and central AGN • typical V ~ few 100s km/s, sound speed ~ 1000-1300 km/s • mostly subsonic (in relaxed clusters), M<0.5 • nearly incompressible • ~Kolmogorov slope -5/3 of E(k) • up to 30 % non-thermal pressure contribution

k

E(k)

kinj kdiss

inertial range of scales

k-5/3

⟳

⟳⟳ ⟳energy cascade

Cluster Turbulence 5

1 Mpc

Fig. 2. The large scale velocity field on a thin slice though the center of cluster SBshown overtop the logarithm of gas density (image, contours). The maximum velocityvector is 2090 km/s. The image is 6.4 Mpc on a side.

vation (Navarro, Frenk & White 1995; Bryan & Norman 1998a) that the meancluster temperatures were, on average, about 0.8 of its virial value.

Thus we see that the gas has not completely virialized and sizable bulkmotions exist. Since the mean entropy profile increases with increasing radius,the halo is globally stable, so this turbulence must be driven by external massesfalling into the cluster and damped by viscous heating. The turbulence amplitudein the halo appears to be roughly compatible with this explanation since thedriving timescale — approximately the Hubble time — is slightly larger thanthe damping timescale which is essentially the crossing time. Moreover, σ2 seemsto drop (and T approaches Tvir) as r → 0 and the crossing time decreases. Wediscuss this point further in the last section.

Figure 2 shows the chaotic flowfield on a slice through the center of cluster

Cluster Turbulence 7

100 kpc

Fig. 3. The velocity field on a thin slice in the inner 600 kpc of cluster SB. Themaximum velocity vector is 520 km/s.

turbulence in strongest in the outskirts of the cluster and weaker in the core.Due to the declining temperature profile in cluster halos, the turbulence is foundto be mildly supersonic (M ∼ 1.6) near rvir , decreases rapidly to M ∼ 0.5 at∼ 1

3rvir , and thereafter declines more slowly to M ∼ .3 in the core.Here we argue that infrequent major mergers cannot sustain the observed

level of turbulence in the core. It is known from simulations of decaying turbu-lence in a box that the turbulent kinetic energy decays as t−η where t is measuredin units of the dynamical time. The exponent η depends weakly on the natureof the turbulence, but is around 1.2 for compressible, adiabatic, hydrodynamicturbulence (Mac Low et al. 1998). The time for a sound wave to propagate fromthe center of the cluster SB to a radius .01, .1, 1 × rvir is .014, .173, 3.1 Gyr,respectively. The cluster underwent a major merger at z = 0.4, or 5.2 Gyr ear-lier. Taking the sound crossing time as the dynamical time, we predict thatfluid turbulence induced by the major merger at z = 0.4 would have decayed to.006, .017, .56 of its initial value by z = 0.

Several possibilities suggest themselves to account for the high fluid velocity

Kolmogorov spectrum

Bryan et al. 1999

Vmax=2020 km/s

Vmax=520 km/s


0.000000 0.000006 0.000031 0.000129 0.000522 0.002076

3’

0.44 0.58 0.71 0.84 0.98 1.1 1.2 1.4 1.5 1.6 1.8

3’

Figure 2. Left: Chandra mosaic image of the Perseus Cluster in 0.5�3.5 keV band. The units are counts/s/pixel. Right: residual imageof the cluster (the initial image divided by the best-fitting spherically-symmetric ��model of the surface brightness), which emphasizesthe surface brightness fluctuations present in the cluster. The point sources are excised from the image. Black circles show a set of annuliused in the analysis of fluctuations. The width of each annulus is 1.50 (⇡ 31 kpc). The outermost annulus is at distance 10.50 (⇡ 218kpc) from the center. Both images are slightly smoothed with a 300 Gaussian.

0 0.0007 0.0021 0.0049 0.01 0.022 0.044 0.088 0.18 0.35 0.7

3’0.00e+00 2.89e+05 5.80e+05 8.70e+05 1.16e+06 1.45e+06

3’

Figure 3. Left: shape of the combined Chandra PSF within the field of view of the Perseus Cluster. The random positions of individualPSFs are used (see Section XX for details). Right: the combined exposure map in seconds slightly smoothed with a 300 Gaussian.

c� 2014 RAS, MNRAS 000, 1–16

PS calculations using modified Δ-variance method

(Arevalo et al. 2012) Poisson noise

initial PS

PSF PS

Next step: P2D of SB -> P3D of density


0.37 0.53 0.68 0.84 1 1.2 1.3 1.5 1.6 1.8 1.9

3’

0.000000 0.000006 0.000029 0.000122 0.000492 0.001955

3’

statistical approach to probe fluctuations: power spectrum of SB fluctuations


soft band: 0.5-3.5 keV hard band: 3.5-7.5 keV

Let’s compare the amplitudes of fluctuations by calculating their auto- and cross-spectra

Work in progress


Gas fluctuations in Perseus(based on Chandra 1.4 Ms observations)

• modest amplitude of density fluctuations: 7-15% on scales 6-30 kpc

• predominantly isobaric fluctuations (preliminary)

• velocity: 70 km/s < V1,k < 200 km/s on scales 6-30 kpc


tcool

=3

2

nkT

n2⇤(T )

Hubble time

Basic concept of AGN feedback:radiative cooling of gas => accretion rate onto SMBH => energy release (bubbles) => ? => dissipation of released energy

Churazov et al. 00, Omma & Binney 04, McNamara et al. 07, Fabian et al. 12; Birzan et al. 12

Lea 76; Cowie & Binney 77; Fabian 77

Astro-H (Jan 2016) soft X-ray spectrometer (SXS)

T rise upon each incident X-ray photon, achieving ~5eV energy resolution

X-ray microcalorimeter array of 6x6 pixels: ~ 5 eV spectral resolution, ~ 1.7’ spatial resolutions, 3’x3’ field of view in the 0.3-12 keV band

What’s next?

At a given R an interval Leff ~ R contributes to the line flux (and width)

Observed σ(R) ≈ structure function (Leff)

Zhuravleva et al. 2012a

small-scale motions: σ large-scale motions: V

rough concentric rings “ripples”

isothermal sound waves Δr ~ time variability of AGN

Fabian et al. 2003, Fabian et al. 2006

Very deep Chandra observation of the Perseus cluster 423

Figure 8. Temperature, density, pressure and pressure variation profiles. The thick line shows the ripples from the unsharp mask image. The top left-hand side

figure shows the profile in the NE direction, top right-hand side shows E direction, bottom left-hand side shows S direction and bottom right-hand side shows

NW direction. The dashed line in the NE profiles indicates the position of the shock front.

that mixing takes place with larger masses of unseen cold gas which

radiates much of the thermal energy in yet unseen bands.

An issue which could be very important for shock propagation

in the inner ICM is the presence of a relativistic plasma (cosmic

rays and magnetic field) in the inner core of the Perseus cluster.

This is evident here from the synchrotron emission seen as the radio

‘mini-halo’ (Pedlar et al. 1990; Gitti, Brunetti & Setti 2002) and

the inverse Compton emission seen as a hard X-ray flux compo-

nent (Sanders et al 2005; it appears as the 16-keV component in

Fig. 10). In the collisionless conditions relevant to the shock, it may

be possible that the relativistic plasma soaks up the energy, leaving

the gas isothermal. Indeed, it could be repeated shocks from the

bubbles which re-accelerates the relativistic particles. They could

redistribute the energy to larger radii, serving to transport some of

C⃝ 2005 The Authors. Journal compilation C⃝ 2005 RAS, MNRAS 366, 417–428

P jumps, no T jumps

Nature of fluctuations in Perseus: sound waves?

M2

r2500

O2

O1

M4

M3

M1

C4

C3

C2

C1

C0

Astro-H SXS observations of the Perseus Cluster

12 Zhuravleva et al.

3’

Figure 12. Top: underlying models of “unperturbed” component of the Perseus Cluster. From left to right: spherically-symmetrical��model (default choice), patched ��models with � = 80 arcsec (removes large-scale asymmetry), � = 30 arcsec and � = 10 arcsec.Bottom: residual images of fluctuations in Perseus obtained from the initial images divided by the underlying model (top panels). Thesmaller the � the smaller structures included to the model and the less structures present in the residual image.

PS of density fluctuations is a power-law k�↵, the convolu-tion of the image PS with the Mexican Hat filter will give(see relation A8 in Arevalo et al. 2012)

Pmh(kr) /Z

k�↵

„k

kr

«4

e�2(k/kr

)2dk, (10)

where kr is the characteristic wavenumber. While the con-volution of the PS of the image, with removed large-scalepart smoothed with a Gaussian, is

Pmh,�(kr) /Z

k�↵

„k

kr

«4

e�2(k/kr

)2“1� e�2⇡2k2�2”2

dk.(11)

The ratio of both will give us an estimate of a wavenum-ber, on which the amplitude will be suppressed dependingon the size � of the window function used for the under-lying model of the SB distribution. For the KolmogorovPS, k�11/3, the suppression of the amplitude obtained us-ing patched ��model relative to the amplitude measuredusing a spherically-symmetric model is ⇠ 20 per cent onk[arcsec�1]⇡ 0.8/�[arcsec]. This is roughly consistent withwhat we see in Fig. 13. For example, the underlying patchedmodel with � = 20 arcsec gives a 20 per cent di↵erencein the amplitudes of density fluctuations on kchar ⇡ 0.04arcsec�1. On k < kchar the “patched” amplitude is stronglysuppressed, while on k > kchar the amplitudes remain almostthe same. The fact that we are roughly consistent with thepredictions means that the measured amplitude of densityfluctuations on scales not a↵ected by the model is dominatedby the presence of fluctuations on these scales and not by

the power leakage from the larger scales. The net conclu-sion is that the amplitude of density fluctuations measuredon scales < 30 kpc is almost not a↵ected by the choice ofthe underlying model, unless the cluster potential is verydisturbed, which is unlikely in relaxed clusters.

6.2 Non-Gaussianity of the surface brightnessfluctuations

The first-order statistics, like the PS, characterize com-pletely only Gaussian fields. Clearly, SB fluctuations arenot perfectly Gaussian, however the question is to whatextent. Fig. 14 shows histograms of the SB fluctuations ineach annulus and the whole image of the Perseus Cluster.Distributions of the fluctuations relative to a spherically-symmetric ��model show noticeable deviations from theGaussian statistic. These deviations are mostly due to thepresence of coherent large-scale structures clearly seen in thecluster residual image. However, if we remove these struc-tures from the field (by using patched models of the under-lying SB), the shape of distributions becomes significantlycloser to the Gaussian. Central 0 � 1.5 arcmin region is anexception since many sharp and very bright fluctuations arepresent there. Of course, higher-order statistics like bispec-trum and trispectrum can provide additional useful informa-tion, however, this is out of our consideration in this paper.On scales we are probing, ⇠ 5 � 30 kpc the PS statistics isessential and provides most information.

c� 2014 RAS, MNRAS 000, 1–16

Two major problems: “perturbed” and “unperturbed” components

Imod

= I�

S�

[Ix

/I�

]

I� Imod

,� = 8000 Imod

,� = 3000 Imod

,� = 1000

Two major problems: “perturbed” and “unperturbed” components


Figure 13. Amplitude of density fluctuations in 3 � 4.5 arcminannulus in the Perseus Cluster obtained by using spherically-symmetrical ��model as the underlying one (COLOR) and moreflexible models shown in Fig. 12. Notice, that the removal of theglobal cluster asymmetry (gray curves) does not a↵ect the am-plitude measurements on scales we are probing using the defaultmodel (COLOR). Also, notice that even if amplitude suppres-sion is present on scales, which are included to the underlyingmodel (as expected), the amplitude on smaller scales, which arenot a↵ected by the model, remains almost the same. This meansthat there is not power leakage from the larger scales and, indeed,fluctuations on these scales are present in the cluster.

6.3 Smoothing of the cluster image whilecalculating the power spectrum

When calculating the PS using modified ��variancemethod, we e↵ectively smooth the exposure-corrected resid-ual image of the cluster S�[counts/(exp ⇤ Imod)] to filtercertain scales (see Section 2.2 in Arevalo et al. 2012). How-ever, this operation might produce a lot of noise in someregions, where the SB and/or the total exposure are low.Therefore, instead, we first smooth the counts and exposureimages and then divide them, i.e. S�[counts]/S�[exp⇤Imod].In ideal situation (high SB, high, homogeneously-covered ex-posure), both ways should give the same results. In the caseof Perseus a systematic uncertainty is ⇠ 1�2 per cent on themeasured amplitude of density fluctuations in some regionson certain scales within the range we are probing.

6.4 Bias in the ��variance method

The normalization of the PS obtained through the��variance method may be biased slightly, depending onthe slope of the PS (Appendix B in Arevalo et al. 2012). Theapproximations of the initial PS of SB fluctuations with apower-law functions give the slopes ⇠ �2��3.4. Therefore,the normalization of the measured amplitude of density fluc-tuations is on average overestimated by ⇠ 10� 20 per centand slightly higher ⇠ 23 per cent in 1.5� 3 arcmin region.

Figure 14. Histograms of surface brightness fluctuations in eachannulus in the Perseus Cluster and in the whole central region0�10.5 arcmin (the bottom right panel). Blue: distribution of fluc-tuations relative to spherically-symmetric ��model. Red: fluctua-tions are relative to patched ��model with � = 80 arcsec (i.e. thelarge-scale fluctuations are removed from the analysis). Dashedcurves show the the Gaussian distributions. Notice, that fluctu-ations on scales we are probing in this work are reasonably welldescribed by the Gaussian distribution.

6.5 Inhomogeneous exposure coverage

The exposure map is not uniform and the brightness of thecluster itself varies across each annulus as seen in Fig. 2and 3. Both might bias the measured amplitude of den-sity fluctuations. By default, when calculating the ampli-tude by averaging an image after applying a filter that se-lects perturbations with a given scale, we use the most uni-form weighting scheme, w / 1. The measured amplitude isthen more sensitive to the largest fluctuations (/ area). Theleast uniform scheme, but at the same time the most opti-mal for the reduction of Poisson noise requires weights to bew / texpImod, where texp is the exposure map and Imod isthe underlying model of the surface brightness. In this case,those parts of the cluster that have higher numbers of countswould have larger weights. We also experimented with theweight w / texp, which evaluates the amplitude of the SBfluctuations, which more sensitive to the deepest-exposurefluctuations. Table 2 shows the amplitudes measured usingweight / texpImod and / texp. The uncertainty does not ex-ceed 30(?) per cent everywhere except for the ?? regions ofthe cluster. [Q to EC]

6.6 Conversion from 2D to 3D power spectrum

We convert the two-dimensional PS of the SB fluctua-tions P2D into the three-dimensional PS of density fluctua-tions P3D following procedure described in Churazov et al.

c� 2014 RAS, MNRAS 000, 1–16

ksupp.[1/00] = 0.8/�[00]

ksupp.=0.01 arcsec-1




removal of large-scale asymmetry does not change the result amplitude of fluctuations on intermediate scales due to the presence of fluctuations on

these scales (not a leakage of power from the larger scales)

Characteristic scales in Perseus Gas Fluctuations in the Perseus Cluster 9

Figure 9. Characteristic length scales present in the PerseusCluster at di↵erent distances R from the center. Colored shadedregions: range of scales, on which we calculate the PS of densityfluctuations and velocity PS and on which our measurements areleast a↵ected by systematic and statistical uncertainties (hatchedregions in Fig. 7 and Fig. 8). Solid curve: the Ozmidov scale lOof turbulence in the core of Perseus, obtained assuming local bal-ance between turbulent heating and radiative cooling (relation 6).We do not plot lO in central ⇠ 20 kpc, since the measured gasentropy is flat there, leading to lO ! 1. Dash curve: the Kol-mogorov (dissipation) scale lK for unmagnetized plasma (relation7). Dot-dash curve: the cluster mean free path. See Section XXfor discussion.

probing, which justifies even better our assumption aboutthe inertial range of scales. In reality, the situation can bemore complicated. For example, the presence of magneticfields could violate our assumptions and arguments, how-ever we neglect these e↵ects here, considering the simplestcase as a 0th order approach to the problem.

We cannot claim, of course, that the structures seen inthe SB are due to turbulence only. For example, there aremany bubbles of relativistic plasma in the Perseus core (seee.g. Pedlar et al. 1990; Boehringer et al. 1993; Churazov etal. 2000; Fabian et al. 2000), which may contribute to thesignal. The question is whether their contribution is dom-inant in considered regions. Various sharp fronts (bubbleedges) would give the slope of the amplitude k�1/2, which ishard to discriminate from the Kolmogorov slope k�1/3 ac-counting for the uncertainties of our measurements and theassumption used. However the contribution of the bubblesto the signal can be easily seen if we repeat the analysis ex-cluding the know bubbles from the image of Perseus. Fig.10 shows the one-component velocity amplitude calculatedfrom such image. Notice, that the velocity amplitude de-creases by a factor ⇠ 1.6 � 1.2 depending on scale in thecentral 1.5 arcmin. This is expected since bubbles in this re-gion are particularly prominent and occupy a vast fractionof the volume. In the 1.5 � 3 arcmin annulus, this factor is

⇠ 1.15 over the whole range of scales. While outside 3 ar-cmin, the exclusion of bubbles from the analysis does notchange the measured velocity amplitude.

Sound waves (Fabian et al. 2006; Sternberg & Soker2009), mergers and gas sloshing (Markevitch & Vikhlinin2007) might also contribute to the observed density fluc-tuation spectra. Unsharp masking of the Perseus Clusterrevealed quasi-spherical structures (“ripples”) in the SB,which have been interpreted as isothermal sound waves(Fabian et al. 2000, 2003). An alternative interpretation isstratified turbulence (Zhuravleva et al. 2014a; Brethouweret al. 2007), which arises naturally is cluster atmospheres,where rough estimated give Froude number Fr ⇠ 0.3 � 1(Zhuravleva et al. 2014a). In this case, the radial size ofeach “ripple” is determined by HV/cs, where H is the char-acteristic scale height and V is the velocity amplitude. Inorder to make a firm conclusion about the nature of suchstructures, the cross spectra of SB fluctuations in di↵erentenergy bands have to be measured, which is the scope of ourfuture work. Our preliminary calculations show that the fluc-tuations are preliminary isobaric. Therefore, even if bubblesand sound waves are present in the cluster core, fluctuationsdue to turbulence and gas sloshing are dominating the signalin the measured PS. At the very least, measured turbulentvelocities can be treated as an upper limit.

5.3 Gas clumping

By gas clumping we will mean any deviations of the gasdensity and temperature isosurfaces from the equipotentialsurfaces3. This definition includes both: large-scale inhomo-geneities (e.g. induced by gas sloshing, perturbations of po-tential) and small-scale clumpiness (e.g., caused by gas tur-bulence, motions of galaxies and subhalos etc). That is tosay, gas clumping is a scale-dependent characteristic and,being measured on di↵erent scales, can be used to constrainphysics of the ICM (this is especially useful for comparisonobservations and implemented physics in numerical simula-tions of clusters).

The gas clumping factor is usually defined as

C =hn2

eihnei2

, (8)

where hi denotes the mean (sometimes median) inside spher-ical shell. Clumping of gas in cluster outskirts is of particu-lar interest for cosmologists, since it can bias the hydrostaticmass (Lau, Kravtsov, & Nagai 2009) and SZ measurementsof the Compton parameter (Khedekar et al. 2013). Numer-ical simulations predict clumping < 1.05 in central 0.5r500,⇠ 1.1 � 1.4 at r500 and up to 2 at r200 (see e.g. Roncarelliet al. 2013; Zhuravleva et al. 2013; Vazza et al. 2013; Nagai& Lau 2011; Mathiesen, Evrard, & Mohr 1999).

Observations of the gas clumping are challenging. In

3 Less general definitions are also used in the literature. For ex-ample, sometimes clumping is referred to gas clumps and subha-los, which are significantly denser and cooler than ambient gasand often have their own dark matter halo, neglecting inhomoge-nenities of smaller amplitude on larger scales.

c� 2014 RAS, MNRAS 000, 1–16

lO=NBV-3/2(Qturb/ρ)1/2

lK=νkin3/4 (Qturb/ρ)-1/4

Ozmidov scale:

Kolmogorov scale:

Coherence and regression

C =P12pP1P2

y = R · x

R = ↵P12

P1/ f2

f1

Regression:

isobaric fluctuations (gas sloshing, turbulence, g-modes) are dominating the signal

Soft bandHard band

Work in progress

adiabatic

isothermalisobaric

see talk by E. Churazov

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