G. Mark Voit et al- Modified-Entropy Models for the Intracluster Medium

8/3/2019 G. Mark Voit et al- Modified-Entropy Models for the Intracluster Medium

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arXiv:astro-ph/0205240v116May2002

ApJ, received 26 February 2002, accepted 15 May 2002

Preprint typeset using LATEX style emulateapj v. 14/09/00

MODIFIED-ENTROPY MODELS FOR THE INTRACLUSTER MEDIUM

G. Mark Voit1, Greg L. Bryan2, Michael L. Balogh3, Richard G. Bower3

ApJ, received 26 February 2002, accepted 15 May 2002

ABSTRACT

We present a set of cluster models that link the present-day properties of clusters to the processes thatgovern galaxy formation. These models treat the entropy distribution of the intracluster medium as itsmost fundamental property. Because convection strives to establish an entropy gradient that rises withradius, the observable properties of a relaxed cluster depend entirely on its dark-matter potential and theentropy distribution of its uncondensed gas. Guided by simulations, we compute the intracluster entropydistribution that arises in the absence of radiative cooling and supernova heating by assuming that thegas-density distribution would be identical to that of the dark matter. The lowest-entropy gas would thenfall below a critical entropy threshold at which the cooling time equals a Hubble time. Radiative coolingand whatever feedback is associated with it must modify the entropy of that low-entropy gas, changingthe overall entropy distribution function and thereby altering the observable properties of the cluster.Using some phenomenological prescriptions for entropy modification based on the existence of this coolingthreshold, we construct a remarkably realistic set of cluster models. The surface-brightness profiles, mass-

temperature relation, and luminosity-temperature relation of observed clusters all naturally emerge fromthese models. By introducing a single adjustable parameter related to the amount of intracluster gasthat can cool within a Hubble time, we can also reproduce the observed temperature gradients of clustersand the deviations of cooling-flow clusters from the standard luminosity-temperature relation.

Subject headings: cosmology: theory galaxies: clusters: general galaxies: evolution intergalactic medium X-rays: galaxies: clusters

1. introduction

The hot gas in clusters of galaxies seems like it shouldbe easy to understand. Because of the relatively low ratioof baryons to dark matter, the potential well of a clus-ter should be dark-matter dominated. The dynamicaltime within a cluster potential is shorter than a Hubble

time, so most clusters should be relaxed. Also, the cool-ing time of the vast majority of intracluster gas is longerthan a Hubble time. It would appear that cluster struc-ture ought to be scale-free, as long as the shape of a clus-ters potential well does not depend systematically on itsmass. If that were the case, then the global propertiesof clusters, such as halo mass, luminosity-weighted tem-perature, and X-ray luminosity, would scale self-similarly(Kaiser 1986). In particular, the gas temperature wouldscale with cluster mass as T M2/3 and the bolometricX-ray luminosity would scale with temperature as L T2in the bremsstrahlung-dominated regime above 2 keV.Indeed, numerical simulations that include gas dynamicsbut exclude non-gravitational processes such as radiativecooling and supernova heating produce clusters that obeythese scaling laws (e.g., Evrard, Metzler, & Navarro 1996;Bryan & Norman 1998; Thomas et al. 2001b).

Real clusters are not so simple. We have known for adecade that the observed luminosity-temperature relationis closer to L T3 (e.g., Edge & Stewart 1991), indi-cating that non-gravitational processes must influence thedensity structure of a clusters core, where most of the lu-minosity is generated (Kaiser 1991; Evrard & Henry 1991;

Navarro, Frenk & White 1995; Bryan & Norman 1998).The mass-temperature relation, on the other hand, seemedlike it ought to be more fundamental and less sensitive tonon-gravitational effects. Yet, observations collected overthe last few years indicate that this relation also disagreeswith both the scale-free predictions and simulations that

exclude non-gravitational processes (Horner, Mushotzky,& Scharf 1999; Nevalainen, Markevitch, & Forman 2000;Finoguenov, Reiprich, & Bohringer 2001; Xu, Jin, & Wu2001). These results derive mostly from resolved X-rayand temperature profiles coupled with the assumption ofhydrostatic equilibrium, but they do seem consistent withgravitational lensing measurements (Allen, Schmidt, &Fabian 2001).

Understanding the scaling properties of clusters is ofbroad importance because these scaling laws are integralto determination of cosmological parameters. For exam-ple, the number density of clusters in the local universereflects the amplitude of matter perturbations on 20Mpc scales (e.g., Henry & Arnaud 1991; White, Efs-

tathiou, & Frenk 1993; Viana & Liddle 1996; Eke, Cole &Frenk 1996; Kitayama & Suto 1997; Oukbir & Blanchard1997; Henry 2000). In order to measure this amplitude,one needs accurately measured cluster masses. Gravita-tional lensing measurements of cluster mass have grownrapidly in sophistication during the last few years, but X-ray determined cluster temperatures, converted to clustermasses using the mass-temperature relation, remain themost commonly used mass measurement for cosmologicalstudies. Thus, any inaccuracies in the mass-temperature

1 Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, [email protected] Physics Department, University of Oxford, Keble Road, Oxford OX1 3RH, UK, [email protected] Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK, [email protected], [email protected]

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VOIT ET AL. 3

2.1. Equilibrium Structure

The equilibrium structure of the intracluster mediumis determined by three things, the gravitational potentialof the clusters dark matter, the entropy distribution ofthe clusters gas, and the confining pressure at the outerboundary of the cluster. In order to solve for that struc-ture, one must integrate the equations of hydrostatic equi-

librium and gas mass conservation,dP

dr= GM(< r)

r2 (1)

dMgdr

= 4r2 , (2)

using the equation of state P = K , where is the adi-abatic index and K specifies the adiabat of the gas. Inthis system of equations, M(< r) is the total mass withinradius r, Mg is the gas mass within that radius, and theother symbols have their usual meanings. Throughout thispaper, we will assume that the gas mass is gravitationallynegligible and that = 5/3, as appropriate for an idealmonatomic gas.

In a sufficiently relaxed cluster, convection ensures thatK(r) monotonically increases with radius. Thus, if oneknows Mg(K), the entropy distribution of the intraclus-ter medium expressed in terms of the mass of gas withP 5/3 < K, then one can use the inverse relation K(Mg)to solve the equilibrium equations that determine the clus-ters structure. Because this crucial quantity K is soclosely related to specific entropy, we will often refer to itas the entropy of the gas, even though the standard ther-modynamic entropy per particle for an ideal monatomicgas is s = ln K3/2 + const.

2.1.1. Dimensionless Form

We can gain insight into the key parameters that control

the structure of the intracluster medium by investigating afamily of dimensionless models for clusters in hydrostaticand convective equilibrium. Because the virial radius ofa cluster lies close to the radius r200 within which themean matter density of the cluster is 200 times the crit-ical density cr, we elect to represent radii in terms ofr r/r200 and gas density in terms of /fb200, where200 = 200cr and fb = 0.02/mh2 is the fractional con-tribution of baryons to the total mass of the universe. Onecan then rewrite the equilibrium equations in dimension-less form as follows:

dP

dr= 2 M

r2 (3)

dfgdr = 3r2 , (4)

where M200 = (4/3)r3200200 is the virial mass, T200 =GM200mp/2r200 is the temperature in energy units1

of the corresponding singular isothermal sphere, P =P/[T200fb200(mp)

1] is the dimensionless pressure, M =M(< r)/M200 is the dimensionless mass within r, andfg = Mg/(fbM200) is the fraction of a clusters baryonsin the intracluster medium within radius r. The corre-sponding dimensionless entropy distribution is K(fg) =K/[T200(fb200)2/3(mp)1], and we will also make use

of the quantity T = P 1mpT1

200.

General solutions to these equations can be found fora few idealized cases. For example, if the potential is asingular isothermal sphere and the gas is also isothermal,then we have P r2, fg r, and K r4/3 for amonatomic ideal gas. If instead the mass profile followsthe form of Navarro, Frenk & White (1997; NFW here-

after), with M

[ln(1 + cr)

cr(1 + cr)1] where c is the

concentration parameter, then the equation of hydrostaticequilibrium becomes (see Wu, Fabian, & Nulsen 2000)

dP

dr= 2

ln(1 + c) c

1 + c

1d

dr

ln(1 + cr)

r

. (5)

Given an isentropic gas in which P5/3 = K = const.we thus obtain [r1 ln(1 + cr) + C]3/2, where C is aconstant of integration.

2.1.2. Boundary Conditions

A particular solution to the equilibrium equations canbe found by choosing P and fg at the origin and inte-grating outwards. The gas mass fraction at the origin isalways zero by definition, but there is some freedom in thechoice of P0 P(0). Any given choice of P0 correspondsto a unique ICM structure with a unique luminosity andtemperature profile, but which solutions are the physicalones?

Here is where the confining pressure comes into play.It is generally assumed that accreting matter confines theICM in the neighborhood of r200. The pressure at r200would then be determined by the ram pressure of the in-falling matter. Now suppose that K(Mg) has been modi-fied by non-gravitational processes. For example, heatingmight drive gas out of the cluster potential, pushing the ac-cretion shock to a radius beyond r200. Alternatively, cool-

ing and condensation might reduce the total gas mass ofthe intracluster medium so that the accretion shock movesinward. In either case the radius at which the integrationof hydrostatic equilibrium should terminate differs fromr200.

So what is the termination radius rmax at whichfgfbM200 equals the total uncondensed gas mass, and whatis the pressure at that radius? In a freely falling accretionflow with a constant mass flux, the density should varyas v1ff r

2, where vff r1/2

ln(1 + cr) is the free-fallvelocity from r 1 in an NFW potential. We thereforeassume that the accretion pressure scales with radius asPacc(r) r5/2

ln(1 + cr), and we normalize that pres-

sure to equal the unmodified NFW value (see

2.2) at

r = 1. Various choices of P0 will lead to pressure profilesthat intersect Pacc(r) at different values of rmax, each cor-responding to a unique value of fg(rmax) determined bya particular integration. The correct solution is the onethat gives the proper value for the uncondensed gas massat the radius where the gas pressure equals the accretionpressure.

In practice, the physically reasonable solutions are notparticularly sensitive to the outer boundary condition. Inthe models that follow, applying the boundary conditionP = Pacc at r = 1 would lead to similar results. How-ever, the value of fg derived at r = 1 using this boundary

1 Boltzmanns constant k is absorbed into T throughout the paper.


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4 MODIFIED-ENTROPY MODELS

condition is not physically significant and can even be in-consistent with models in which cooling and condensationsignificantly reduce the maximum value of fg. One exam-ple of a physically unreasonable boundary condition is re-quiring fg = 1 at r = 1. Such a requirement does not allowintracluster gas to expand beyond r200 even when heatingsubstantially raises the intracluster entropy. In the limit of

extreme heating, this condition therefore artificially booststhe central pressure, density, and temperature, leading toan unphysically large cluster luminosity.

2.1.3. Integrated Characteristics

Once a correct solution has been identified, one can com-pute two quantities that correspond to the dimensionlessluminosity and luminosity-weighted temperature:

L = 3

rmax0

2r2dr (6)

Tlum = 3L1

rmax0

T2r2dr . (7)

As long as the cooling function (T) remains sufficientlyconstant within the gas contributing the bulk of the lu-minosity, the actual luminosity-weighted temperature isTlum T200Tlum, and the bolometric X-ray luminosity is

L 43

r3200

npne

n2e(Tlum)L , (8)

where ne = (ne/)fb200 = 1.2 104 (M/0.33)1 cm3.Scaling to a 10 keV cluster at z = 0 gives

L (9.0 1043 h3 ergs1)

T20010 keV

3/2 M0.33

2

23(Tlum)L , (9)where 23(Tlum) = (Tlum)/(10

23 erg cm3 s1).

The emission-measure profile and emissivity-weightedtemperature profile are also of interest. Thus, we definethe following dimensionless analogs as functions of the pro-

jected radius r:

S(r) = 2

r2max

r2

0

2dl (10)

T(r) = 2S1

r2max

r2

0

T2dl , (11)

where l =

r r is the dimensionless distance along theline of sight.

2.2. Unmodified Solutions

Before exploring the consequences of modifying the en-tropy of intracluster gas, we need to know what the en-tropy distribution would be if it were not modified. Tosimplify matters, we will assume that the underlying darkmatter profile of the cluster is of NFW form with some con-centration parameter c. This parameter typically rangesfrom c 5 for hot clusters to c 10 for groups of galaxies(see 3.1). If the intracluster medium were collisionless,then its density profile would be identical to that of thedark matter. Thus, we define the unmodified entropy dis-tribution K0(Mg) of a cluster of concentration c to be thatof gas in hydrostatic equilibrium in the cluster potentialwith a density profile identical to that of the dark matter.

Fig. 1. Intracluster density profiles in dimensionless units. Thedotted line shows the matter density () of an NFW profile of con-centration c = 8 in units of the mean density within r200 as afunction of radius (r) in units of r200. The dashed line shows the

density profile of a simulated cluster whose dark matter halo is wellapproximated by an NFW halo with c = 8. Outside of r = 0.1,these curves closely correspond. The solid lines show how the runof density with radius changes as the entropy distribution of an un-modified c = 8 NFW halo is truncated at progressively larger values

of the dimensionless entropy Kc. For these cases, is the gas densityin units of the mean density of the unmodified gas. Notice that the

model with Kc = 0.1 is quite similar to the gas density distributionfrom the simulation.

Numerical experiments reveal that the entropy distri-bution of intracluster gas in clusters simulated withoutradiative cooling or supernova heating is quite similar tothe NFW form throughout most of the cluster but is el-evated above NFW within the clusters core. Figure 1shows the density distribution of a cluster simulated with

an adaptive-mesh refinement code (Norman & Bryan 1998;Bryan 1999) for the Santa Barbara cluster comparisionproject (Frenk et al. 1999). The dotted line labeledNFW shows an NFW density distribution with c = 8,which is very close to the underlying dark-matter distri-bution of the cluster. The dashed line labeled simula-tion shows the gas-density distribution from that samesimulation. Agreement is quite close outside 10% of thevirial radius, but the gas density levels off within that ra-dius, implying that the lowest-entropy gas has a somewhathigher entropy than assumed in our unmodified distribu-tion K0. However, this discrepancy at the low-entropyend of the distribution is inconsequential to the modified-entropy models that follow because the entropy of the dis-

crepant gas will always be subject to further modification.The unmodified distribution is also somewhat dis-

crepant with simulations at the high-entropy end. Assum-ing hydrostatic equilibrium and an NFW gas-density lawnear the virial radius leads to gas temperatures that are20-30% higher than those in our fiducial simulation. Thus,our unmodified entropy levels at large radii are slightlyhigher than what a numerical simulation would produce.However, in order to properly reproduce the density pro-file near the virial radius in a hydrostatic model, we mustretain these elevated entropy levels. This overestimate atthe clusters outskirts has virtually no effect on emissivity-weighted global quantities, such as L and Tlum, but if one


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VOIT ET AL. 5

is interested in the temperature of cluster gas near r200,then departures from hydrostatic equilibrium, which areoutside the scope of our models, must be taken into ac-count.

2.3. Entropy Modification

Modification of this baseline entropy distribution is in-

evitable because clusters simulated without radiative cool-ing are physically inconsistent. In general, the gas at thecenter of such a simulated cluster can radiate many timesits thermal energy within a Hubble time, resulting in clus-ter luminosities that vastly exceed those observed (e.g.,Muanwong et al. 2001; Bryan & Voit 2001). Condensa-tion and removal of the lowest-entropy gas from the intr-acluster medium must happen at some level, because thatis how the clusters galaxies form stars. Furthermore, themost massive of these stars must explode, resulting in su-pernova feedback. Condensed gas accreting onto an activegalactic nucleus can provide additional feedback (Heinz,Reynolds, & Begelman 1998; Kaiser & Alexander 1999;Quilis, Bower, & Balogh 2001; Reynolds, Heinz, & Begel-man 2002; Bohringer et al. 2002). All of these processeswill modify the entropy distribution of the intraclustermedium, but modeling them in detail is a daunting task.

Because the physics of feedback is so complex, we shalladopt a highly simplistic phenomenological approach toentropy modification. Instead of trying to model all theconsequences of heating and cooling we will restrict ourinvestigation to three qualitatively different modificationsof the entropy distribution K0: (1) truncation of the dis-tribution, (2) shifting of the distribution, and (3) radia-tive losses from the distribution. Each of these modifica-tions depends on a single entropy threshold parameter Kc,which we take to be the entropy at which the cooling timeof gas of temperature T200 equals the age of the universe.Thus, for each type of entropy modification, we obtain atwo-parameter family of modified-entropy models specifiedby a concentration c and a dimensionless entropy Kc.

As in Voit & Bryan (2001), we use an approximate cool-ing function for gas with a metallicity of one-third solarto determine the appropriate entropy threshold. Accord-ing to this approximation, the threshold for cooling within15 Gyr can be expressed as

Kc 135keVcm2

mp

ne

2/3 Tlum

2 keV

, (12)

with = 2/3 for T > 2keV and = 0 for T < 2 keV.Converting this threshold to dimensionless units yields

Kc 0.164 Tlum

Tlum2keV

1 M0.33

2/3. (13)

As long as Tlum 1 (see 2.5), then Kc 0.1 will beappropriate for the hottest clusters ( 10 keV) and K0.5 1.0 will be appropriate for the coolest ( 0.5 keV)groups.

2.3.1. Truncation

One simple yet physically motivated way to modify theentropy distribution of intracluster gas is to truncate it atKc (Bryan 2000; Voit & Bryan 2001). Mathematically, we

Fig. 2. Modified entropy distributions. The thick solid line

shows how dimensionless entropy K rises with fg, the fraction of aclusters baryons in the intracluster medium within radius r, for an

unmodified c = 8 NFW halo. Gas in the unmodified halo is assumedto be in hydrostatic equilibrium with a density profile identical tothat of the dark matter. The thin solid line illustrates the entropy

distribution KT(fg) that results when the unmodified distribution

is truncated at Kc = 0.3, corresponding to a horizontal translation

of the unmodified distribution. The dotted line shows KS(fg), the

unmodified distribution shifted vertically by Kc = 0.3. The long-and short-dashed lines show two different modifications designed tomimic radiative losses from the unmodified distribution (see 2.3.3).

These lines correspond to a reduction of K at each point of the un-modified distribution, followed by a horizontal translation so that

K(0) = 0.

define the truncated distribution to be

KT(fg)

K0(fg + fc) , (14)

where fc is defined by K0(fc) = Kc. Such a modificationcorresponds to removing all the gas with a cooling timeless than a Hubble time and ignoring the effects of coolingon the rest of the intracluster medium. This type of mod-ification can viewed either as an extreme form of cooling,in which all the gas below the critical threshold condenses,or an extreme form of heating, in which all the gas belowthe threshold is heated to very high entropy (K 1) andconvects beyond the virial radius of the cluster. Figure 1shows some gas density profiles resulting from differentvalues of Kc. The thin solid line labeled KT in Figure 2shows the truncated entropy distribution for c = 8 andKc = 0.3.

Notice that the maximum value of fg in these trun-cated models is less than unity because some of the gashas been removed from the intracluster medium. Thethick solid line in Figure 3 illustrates how the fractionfg,max = 1 fc of a clusters baryons that remain in theintracluster medium depends on Kc. For hot clusters thisfraction is 90%, but for cool groups it drops below 50%.Nevertheless, the outer radius rmax of the intracluster gasremains close to r200, even when fg,max drops below 0.5.Figure 1 shows the reason. As the entropy threshold fortruncation rises, the central entropy of the cluster alsorises. The resulting density profile is therefore much flat-ter for large values of K, enabling a smaller amount of gas


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Fig. 3. Maximum radius and baryon content of the intraclustermedium in modified-entropy models. The thick solid line shows howfg,max, the fraction of a clusters baryons residing in the intraclus-

ter medium, depends on the entropy threshold Kc in the modified

distributions KT, KR, and KG3. The other lines show how rmax,

the maximum radius of the intracluster medium, depends on Kc ineach of these modified distributions.

to fill a similar volume. Note also that the density profileof a modified-entropy model with c = 8 and Kc = 0.1 isquite similar to that of a c = 8 cluster simulated withoutcooling and feedback, suggesting that the cooling thresh-old may have only a modest effect on the structure of thehottest clusters.

2.3.2. Shift

Another way to modify intracluster entropy is to adda constant term Kc to the unmodified entropy distribu-tion (Voit & Bryan 2001). This type of modification is aconvenient way to mimic preheated models for clusterformation that inject a fixed amount of entropy per par-ticle into the intergalactic medium at some early time. Inthis case, the modified entropy distribution is defined by

KS(fg) K0(fg) + Kc . (15)The dotted line in Figure 2 shows this distribution forc = 8 and Kc = 0.3. Because entropy has been addedbut no gas has been removed, the outer radius for mod-els with a shifted entropy distribution extends beyond theusual virial radius (see Figure 3). Qualitatively, this cor-responds to heating that inflates the intracluster medium,driving the accretion shock to larger radii.

Comparing the structure of shifted models to truncatedmodels reveals interesting similarities. Figure 4 depictsthe dimensionless density profiles for Kc = 0.1, 0.3, and1.0 in clusters with c = 8. The run of density with radiusis virtually identical for the shifted and truncated modelswhen Kc = 0.1 and is still quite similar when Kc = 0.3.Thus, for most of the interesting range of entropy thresh-olds, there is little observable difference between shiftedmodels and truncated models. However, as Kc rises to1.0, the density profile of the shifted model becomes no-ticeably flatter than that of the truncated model.

Fig. 4. Density profiles for modified entropy models with c = 8and Kc = 0.1, 0.3, and 1.0. Solid lines show (r) for truncated

models (KT), dotted lines represent shifted models (KS), and thelong- and short-dashed lines represent two types of radiative-loss

model (KR and KG3). The density structure of truncated and

shifted models is almost indistinguishable for small values of Kc.The radiative-loss models have the elevated central densities char-acteristic of cooling-flow clusters.

Similarity between the truncated and shifted models atthe lower entropy thresholds arises because the unmodifiedentropy K0 depends approximately linearly on fg for small

K. Thus, shifting the distribution by Kc and truncating itat Kc both produce similar modified entropy distributions

and therefore similar density, temperature, and pressuredistributions in the inner regions of the cluster. That iswhy entropy modification by heating and cooling can havesimilar effects on the luminosity and temperature of clus-ters as long as they establish similar values of the minimumentropy.

2.3.3. Radiative Losses

Observations show that many clusters contain gas witha cooling time less than a Hubble time (Fabian 1994), mo-tivating us to consider slightly more complicated modi-fications to the entropy distribution. Radiative coolingreduces the the specific entropy of a gas parcel according

tods

dt=

d ln K3/2

dt= mpnpne(T)

T. (16)

This entropy equation can be rewritten to eliminate thedependence on gas density:

dK3/2

dt=

npne

2

(mp)

1/2 T1/2(T) . (17)

In general, the rate at which entropy declines will de-pend on the prevailing hydrodynamic conditions, stan-dard examples being isobaric cooling or isochoric cooling.


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Fig. 6. Best-fitting -model parameters for truncated modified-entropy models. Each line gives the best-fitting (, rc) pair for aparticular value of the concentration c. The points on each line showbest fits for five different values of the entropy threshold. The pro-

gression Kc = 1.0, 0.5, 0.3, 0.2, 0.1 runs from the upper left to lowerright in each case. Because the best fits depend somewhat on thefitted interval, we depict best fits over the interval 0 .03 < r < 0.3with solid triangles and solid lines and best fits over the interval0.05 < r < 1.0 with empty squares and dotted lines.

Fig. 7. Best-fitting -model parameters for shifted modified-

entropy models. Each line gives the best-fitting (, rc) pair for aparticular value of the concentration c. The points on each line showbest fits for five different values of the entropy threshold. The pro-

gression Kc = 1.0, 0.5, 0.3, 0.2, 0.1 runs from the upper left to lowerright in each case. Because the best fits depend somewhat on thefitted interval, we depict best fits over the interval 0 .03 < r < 0.3with solid triangles and solid lines and best fits over the interval0.05 < r < 1.0 with empty squares and dotted lines.

The reason that raising the core entropy of gas in anNFW potential has only a modest effect on its luminosity-weighted temperature involves both the gas density profile

Fig. 8. Dimensionless temperature profiles. Solid lines show

how temperature (T) depends on radius (r) in models with trun-

cated entropy distributions (KT). Dotted lines show how tempera-ture depends on radius in models with shifted entropy distributions

(KS). Raising the entropy threshold Kc increases the temperatureat all radii for both kinds of entropy modification. This increase isslightly larger for shifted models because the extra gas at large radiileads to slightly higher central pressures.

and the dark-matter density profile. Extra entropy flat-tens the density profile of intracluster gas, diminishing therelative luminosity of gas in the core. The characteristicradius of gas contributing the bulk of the clusters lumi-nosity therefore moves outward. Because the shape of anNFW halo is steeper than isothermal at r > 1/c, the char-

acteristic temperature of gas within such a halo declinesoutside that radius (see Figure 8). Thus, the rise in gas

temperature resulting from an increase in Kc is largelyoffset by an increased contribution to the total luminos-ity from lower-temperature gas at larger radii. This ef-fect is most pronounced in the shifted models, which havesubstantial amounts of relatively cool gas beyond r = 1.In those models, Tlum actually declines as Kc approachesunity.

The large central temperatures in the shifted and trun-cated cases also seem somewhat unrealistic in light of ob-servations showing either level or radially increasing tem-perature gradients within cluster cores (e.g., Arnaud et al.2001a,b; Allen et al. 2001). In order to compare these tem-

peratures more directly with observations, we must lookat T, the luminosity-weighted temperature along variouslines of sight through a cluster. Figure 10 illustrates howT depends on projected radius r in clusters with c = 8,Kc = 0.3, and four different kinds of entropy modification.Projected temperatures at r < 0.1 in both the shiftedand truncated models are still quite high (T 2T200).However, the core temperatures in the radiative-loss mod-els look more like those in real clusters (see 3.2.2), risingwith radius in the innermost regions and peaking between1.2T200 and 1.6T200 at 0.05r200 in these particular mod-els.


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VOIT ET AL. 9

Fig. 9. Dimensionless luminosity-weighted temperatures (Tlum)

as a function of entropy threshold (Kc) for halos of concentration

c = 4, 6, 8, 10, and 12. Solid lines depict truncated models (

KT);dotted lines depict shifted models (KS). The dependence ofTlum on

Kc is rather modest over the interesting range 0.1 < Kc < 1.0, es-pecially for larger values of c. This effect arises because a larger pro-

portion of the emission from high-Kc clusters comes from r > 1/c,where temperatures are lower than in the core. This outward shiftof surface brightness largely offsets the rise in temperature at allradii evident in Figure 8.

Fig. 10. Dimensionless luminosity-weighted line-of-sight tem-

perature T as a function of projected radius r. The solid anddotted lines representing truncated and shifted models, respectively,both begin at rather high temperature ( 2T200) at small radii andproceed to decline monotonically with radius. The long- and short-dashed lines representing radiative-loss models (see 2.3.3) begin atlow temperature at r = 0 and rise to a maximum at 0.05r200, inbetter accord with observed clusters (see 3.2.2).

Fig. 11. Dimensionless luminosity-weighted temperatures for

the standard radiative-loss model (KR). Comparision of these

Tlum(Kc) relations with those in Figure 9 shows that emission from

the core gas lowers the luminosity-weighted temperature by 30%,relative to the shifted and truncated models.

Fig. 12. Dimensionless luminosity-weighted temperatures for

generalized radiative-loss model KG3 with = 3. Solid lines

show the Tlum(Kc) relations for truncated models, and long-dashedlines show the corresponding relations for the radiative-loss models.Luminosity-weighted temperature is 10% smaller in these gener-alized radiative-loss models. The difference is less dramatic than inthe = 3/2 case because the amount of gas below Kc is smaller,resulting in somewhat higher core temperature and somewhat lowercore density and luminosity.


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Cool gas in the cores of clusters with entropy modi-fied through radiative losses can significantly affect theluminosity-weighted temperature if the fraction of gas be-low Kc is relatively large. Figure 11 shows that Tlumfor a given concentration c and entropy threshold Kc is30% lower in the standard radiative-loss model (KR)than in the corresponding truncated model. This effect

is also present but not nearly so strong in the generalizedradiative-loss model with = 3, where Tlum is 10% lowerthan in the truncated models (see Figure 12).

2.6. Entropy Modification and Cluster Luminosity

All of our schemes for entropy modification substantiallylower the luminosities of clusters. Figure 13 shows how di-mensionless luminosity L depends on the entropy thresh-old Kc for clusters with c = 8. In all cases, luminositydeclines by over an order of magnitude as Kc rises from0.1 to 1. This decline is shallowest for the shifted modelbecause the additional gas at large radii contributes sig-nificantly to the total luminosity when Kc 1. Note alsothat L is over twice as large in the standard radiative-losscase (KR) because a large proportion of the luminosityin such models comes from the dense core. Luminosityenhancement in the = 3 radiative-loss model is moremoderateonly a few tens of percent greater than in thetruncated case.

Analyzing how L depends on Kc leads to some impor-tant insights into how the cooling threshold determines theluminosity-temperature relation of clusters and groups.

Figure 14 shows that L K3/2c is a good approximationto the L-Kc relation for truncated models in the range0.1 < Kc < 1.0. This scaling reflects the asymptotic slopeof the NFW density profile. Notice that the outer por-tions of the density profiles in Figure 1 track the underlyingNFW profile of the dark matter, and the inner portions arenearly isentropic with K Kc. One can therefore approxi-mate the density profile of a truncated model by joining anisentropic density profile to an NFW profile at the radiusrK at which the density of the unmodified NFW profile

is K K3/2c . The analytical isentropic solution from 2.1.1 tells us that [r1 ln(1 + cr)]3/2 within rK , sothe luminosity from the isentropic gas scales 2K r3K. Theluminosity from the NFW portion of the profile scales inthe same way for the radii of interest. Thus, for larger val-ues of r, where the density of the NFW profile approaches

r3, we have L K K3/2c .This particular scaling ofL with Kc has some very inter-

esting consequences. Section 2.1.3 showed that the phys-ical luminosity of a cluster scales as L r32002crL. Inorder to substitute physical quantities for L, we need to

recognize that K3/2c K3/2c T3/2200 cr and that K3/2c

T1/2tH, where tH is the age of the universe. We there-fore obtain

L T5/2lum T3lum (H tH)1 , (20)

where H is the Hubble constant at time tH.

Fig. 13. Dependence of dimensionless luminosity L on the en-

tropy threshold Kc for clusters with c = 8. The decline of L with

increasing Kc is strong for all types of entropy modification. The ef-fect is stronger in the truncated (KT) and radiative-loss (KR, KG3)

models than in the shifted model (KS) because the latter has more

gas at large radii, which contributes significantly to L when theentropy threshold is high.

Fig. 14. Dimensionless luminosity L as a function of entropy

threshold Kc in truncated (KT) models for clusters with concen-tration c = 4, 6, 8, 10, and 12. The cluster luminosities in these

models sharply decline as Kc increases. For 0.1 < Kc < 1.0 this

decline approximately scales as L K3/2c .


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VOIT ET AL. 11

The following three features of this expression are par-ticularly noteworthy:

The power-law slope it predicts for the X-rayL Tb relation is very close to the observedslope of b 2.6 2.9 (e.g., Markevitch 1998;Arnaud & Evrard 1999). Because the halos oflow-temperature groups have higher concentrationsthan high-temperature clusters (see 3.1), weexpect Tlum to decline by a factor 1.3 as Tlumrises by an order of magnitude. This dependenceon concentration steepens the power-law slopeimplied by equation (20) to b 2.8.

The L-T relation does not depend on the form ofthe cooling function, which scales as T1/2 inthe free-free cooling regime at T > 2 keV and as T1/2 in the line-cooling regime at T < 2 keV.If the entropy threshold in real clusters did notdepend on , then we would expect a distinctsteepening of the L-T relation as T rises through

2 keV, which is not observed. Equation (20)

is indepedent of because the dimensionlessgas mass within rK scales K r3K , and thusremains approximately constant in an NFW halofor ri > 1/c. The cooling time of that gas is tH,so its physical luminosity scales

M200T200t1H .This scaling law also applies to the entire cluster,because the luminosity of gas outside rK scales inlockstep with gas inside rK . Applying the relation

M200 T3/2200 H1 thus leads back to equation (20),except for the Tlum factor.

Finally, the L-T relation implied by equation (20)changes very little with time, again in agreement

with observations (Mushotzky & Scharf 1997;Donahue et al. 1999; Della Ceca et al. 2000;Borgani et al. 2001b). That happens because themass of a cluster of a given temperature declines H1 with increasing redshift. According to theL M200T200t1H scaling, the resulting declinein the intracluster gas mass compensates almostprecisely for the drop in the cooling thresholdowing to the shorter cooling time tH.

This scaling breaks down at the low-temperature end,where Kc 1, because the entire intracluster (or in-tragroup) medium in the truncated and shifted casesbecomes effectively isentropic. Then, the gas den-

sity determined by the cooling threshold is T /tH,and the luminosity from within r200 scales as L T

7/2lum

2T3/2

lum (HtH)2H1. Thus, we should expect the

L-T relation for low-temperature halos to be steeper thanfor clusters, roughly L T4.5 for T1/2, and weshould expect the luminosity of ob jects at a given temper-ature to increase with time.

In order to make further progress, we will need to un-derstand how halo concentration depends on halo mass.

3. modeling real clusters

The previous section explored the properties of dimen-sionless modified-entropy models for clusters depending

on two parameters, the halo concentration parameter cand an entropy-threshold parameter Kc. As in Voit &Bryan (2001), we argued that the crucial entropy thresh-old should be the entropy level at which intracluster gaswould cool in a Hubble time. Now we will use a set of re-lations between halo concentration and halo mass to con-struct modified-entropy models for real clusters. In these

models, the halo mass M200 determines the concentrationparameter c. The mass and halo concentration then set thetemperature of the cluster, and this temperature sets thelevel of the entropy threshold Kc. Cluster properties aretherefore determined primarily by the overall cosmologicalmodel, because it fixes the global baryon fraction, the rela-tion between halo mass and concentration, and the Hubbletime that governs Kc(T). Some of a clusters properties,such as the temperature of the innermost gas, also dependon the chosen scheme for entropy modification. The prop-erties of the resulting modified-entropy models turn outto be remarkably similar to those of observed clusters andgroups.

The section begins by defining the M200-c relations we

will use and comparing them to some of the scarce data.Then, we explore the structure of the intracluster mediumin our modified-entropy cluster models, showing that theobserved relationship between and Tlum may be morethan just a selection effect and demonstrating how en-tropy modification can lead to temperature profiles similarto those observed in both the inner and outer regions ofclusters. We go on to compare the mass-temperature re-lation from our modified-entropy models to observations,showing that the models reproduce the observational re-sults of Nevalainen et al. (2000) and Finoguenov et al.(2001). We also compare the LX-Tlum relation from ourmodels to cluster data, again showing excellent agreement.Much of the dispersion of this relation, in both the models

and the data, stems from differences in the amount of gasbelow the cooling threshold, and we conclude with somespeculations about what governs these differences.

3.1. Concentration Parameter

The concentration of a dark matter halo is determinedby its formation history. Early formation generally leads toa denser core and thus a higher concentration. Low-masshalos, which tend to collapse earlier in time, are thereforeexpected to be somewhat more concentrated than high-mass halos. Because a halos collapse and merger historydetermines its concentration, the relation between concen-tration and halo mass depends on the underlying cosmo-logical model, specified by the matter density

M, the

dark-energy density , the normalization 8 of the per-turbation spectrum, and the shape of that spectrum, oftenparameterized by (Bardeen et al. 1986; Sugiyama 1995).In this paper, we will restrict our attention to currently-favored CDM models, in which the resulting clusters turnout to be thoroughly consistent with observations.

Several versions of the concentration parameter c canbe found in the literature, each defined with respect toa radius r within which the mean halo density is cr.Here we will define c with respect to r200, in order to beconsistent with the models developed in the previous sec-tion; thus, c c200. Figure 15 shows various relationsbetween halo mass and c200 that can be found in the lit-


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Fig. 15. Concentration parameter (c200) as a function ofhalo mass (M200). Solid triangles give concentrations derived

from Chandra observations of real clusters by Allen et al. (2001).Empty squares give the concentrations of clusters simulated by Eke,Navarro, & Frenk (1998) in a CDM cosmology with 8 = 1.1. Solidlines show c200 values predicted by models S0.9, S1.2, and S1.6 fromEke et al. (2001), for a CDM cosmology in which 8 = 0.9, 1.2,and 1.6, respectively, for a power spectrum with shape parameter = 0.2. Dashed lines show c200 given by models 0.1 and 0.5from Eke et al. (2001), for which = 0 .1 and 0.5, respectively, and8 = 0.9. The dotted line labeled TN shows the c200 values fromthe fit of Tozzi & Norman (2001) to the NFW prescription for aCDM cosmology with 8 = 1.1.

erature. The Chandra observations of Allen et al. (2001)suggest that concentrations corresponding to c200 4 6are appropriate for massive ( 1015 M) clusters. Concen-trations of groups are less certain because they are more

model-dependent, and the dispersion in c200 is expectedto increase at lower halo masses (Afshordi & Cen 2002).In this section, we will adopt the halo mass-

concentration relations predicted by models S0.9 and S1.2from Eke, Navarro, & Steinmetz (2001), which are con-sistent with the Chandra data on massive clusters. Bothmodels are derived from N-body simulations of CDMcosmologies with M = 0.3, = 0.7, and h = 0.65.The power spectrum in each model has a shape parame-ter = 0.2, but the normalization of the power spectrumdiffers: 8 = 0.9 in model S0.9 and 8 = 1.2 in model S1.2.The higher normalization in model S1.2 leads to halos thatare about 30% more concentrated. Figure 16 shows the de-pendence ofKc on halo mass that follows from model S1.2;

the entropy thresholds for model S0.9 are very similar.

3.2. Cluster Structure

The cluster models we analyze in this section are com-pletely determined by the assumed relationship betweenhalo mass and concentration (either S0.9 or S1.2) and the

chosen scheme for entropy modification (KT, KS, KR,

or KG3). For each halo mass, we compute the bolomet-ric X-ray luminosity LX , luminosity-weighted temperatureTlum, X-ray surface-brightness profile SX(r), and pro-

jected temperature profile T(r). Unlike in 2.1.3, eachintegral includes a cooling function (T) drawn from the

Fig. 16. Dimensionless entropy threshold (Kc) determinedby cooling as a function of halo mass M200. The lines show theentropy thresholds for truncated models (solid line), shifted mod-

els (dotted line), and radiative-loss models with = 3/2 (short-dashed line), and = 3 (long-dashed line), assuming the halo mass-concentration relation from model S1.2. The corresponding thresh-olds for model S0.9 are very similar. Above 1014 h1 M, differ-ent entropy-modification schemes produce slightly different dimen-sionless thresholds because the temperature Tlum associated with agiven halo mass is slightly different. Equations (12) and (13) give

the entropy threshold Kc and dimensionless entropy threshold Kc,respectively, as functions of temperature.

models of Raymond & Smith (1977) for gas with 30%solar metallicity. We will first focus on the spatially re-solved properties of these model clusters, showing howtheir surface-brightness profiles and temperature gradientscompare with those of observed clusters.

3.2.1. Surface-Brightness Profiles

Modified-entropy models of groups tend to have flattersurface-brightness profiles than hot clusters, in accordancewith observations (e.g., Ponman et al. 1999; Horner etal. 1999). Because the surface-brightness profiles of bothreal clusters and our modified-entropy models are well de-scribed by -models with SX [1+(r/rc)2]3+1/2 insideof 0.3 r200 (see 2.4), we compare our models with thedata by comparing the best-fitting -model parameters.Figure 17 shows how and rc depend on Tlum in trun-cated modified-entropy models with relation S1.2 and inthe cluster data of Finoguenov et al. (2001). The modelsreproduce the observed range of both parameters and the

observed rise in each parameter with increasing tempera-ture. However, the observed dispersion in each parameterat a given temperature is quite large. Wu & Xue (2002)find the same kind of temperature dependence in theircluster models, based on the work of Bryan (2000), whichare very similar to the truncated models developed in thispaper.

Some of the dispersion in and rc at a given tempera-ture may stem from selection effects. Because the surface-brightness profiles of modified-entropy clusters continuallysteepen with radius, unlike those of -models, the best-fitting and rc depend somewhat on the range of radiibeing fit. Figure 17 shows that the best-fitting values ofand rc both decline as the surface-brightness threshold of


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VOIT ET AL. 13

Fig. 17. Best-fitting -model parameters as a function ofluminosity-weighted temperature Tlum. In a -model, X-ray sur-face brightness declines with radius as SX [1 + (r/rc)

2]3+1/2,

where rc is the core radius and determines the asymptotic slope.Triangles show measured values of (upper panel) and rc (lowerpanel) compiled by Finoguenov et al. (2001). Solid lines showthe values of these parameters in the best fits to truncated mod-

ified entropy models (KT) over the radial range 0.03 < r < 0.3,where r r/r200. Dotted lines show the best-fitting values over therange 0.05 < r < 1.0. The other lines show best fits above surface-brightness thresholds of 2 1015 erg cm2 s1 arcmin2 (short-dashed lines), 21014 erg cm2 s1 arcmin2 (long-dashed lines),and 2 1013 erg cm2 s1 arcmin2 (dot-dashed lines).

the fit rises from 2 1015 erg cm2 s1 arcmin2 through2 1014 erg cm2 s1 arcmin2 to 2 1013 ergcm2 s1arcmin2. This effect is more severe for groups than forclusters because their overall surface brightness is smaller.

The figure also shows that fits over the radial range 0.05 5 1015 erg cm2 s1 arcmin2 produces relationsthat track the lower envelope of the group data. Unlike thecluster measurements, the data on groups have not beencorrected for the effects of gas below the cooling threshold,which could counterbalance the surface-brightness bias byraising LX and lowering Tlum (see Figure 27). We there-fore suspect that the dispersion in the L-T data of Helsdon& Ponman (2000) arises from a combination of differingsurface-brightness biases and differing amounts of gas be-low the cooling threshold.

Reproducing the behavior of clusters and groups withsignificant amounts of gas below the cooling threshold re-quires an additional parameter. In our models, that pa-rameter is , which characterizes the slope of the entropydistribution function below the cooling threshold. Fig-ure 28 shows the relationship between Tlum and LROSAT,the ROSAT-band (0.1-2.4 keV) luminosity, in the datafrom Markevitch (1998) and in models with differing val-ues of . Triangles show data corrected to remove the

Fig. 25. Relation between bolometric X-ray luminosity LXand luminosity-weighted temperature (Tlum). Solid triangles showmeasurements of clusters with insignificant cooling flows compiled

by Arnaud & Evrard (1999). Open squares show cooling-flow cor-rected measurements by Markevitch (1998). Solid circles show groupdata from Helsdon & Ponman (2000). Because the cluster data arenot strongly influenced by cooling flows, we compare them to mod-els without radiative losses. The solid and dotted lines show theLX-Tlum relations derived from truncated (KT) and shifted (KS)models, respectively, and the concentration parameter relation S1.2(see Figure 15). The dot-dashed line shows the relation derived

from model KT using the concentration parameter relation S0.9.The models assume a standard CDM cosmology with M = 0.33, = 0.67, and b = 0.02 h

2, and a Hubble parameter of h = 0.65has been applied to both the models and the data.

Fig. 26. Relation between LX and luminosity-weighted tem-perature (Tlum) at different surface-brightness levels. Because tem-perature declines with radius at large distances from the center ofour group models, the limiting surface brightness affects the ob-served mass-temperature relation. The solid line shows the LX-

Tlum relation for model KT, S1.2. The other lines show howthat relation changes when temperature and luminosity are mea-sured within regions where the bolometric surface brightness inerg cm2 s1 arcmin2 exceeds 11015 (dotted), 21015 (short-dashed), and 5 1015 (long-dashed). The data are the same asin Figure 25. Some of the observed dispersion below the solid linecould stem from this surface-brightness bias.


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Fig. 2 7. Relation between LX and luminosity-weightedtemperature (Tlum) above a surface-brightness threshold of 5 1015 erg cm2 s1 arcmin2. The solid and dotted lines show the

relation for truncated and shifted models, respectively. The short-dashed and long-dashed lines show the relations for radiative-loss

models KR and KG3, respectively. The data are the same as inFigure 25. The presence of gas below the cooling threshold near thecenters of the radiative-loss models lowers the central temperatureand raises the central luminosity. Much of the dispersion to the up-per left of the solid and dotted lines near 1 keV could be due tocooling flows, which have not been corrected for in the group data.

Fig. 28. Relation between ROSAT (0.1 2.4 keV) X-ray lumi-nosity LROSAT and luminosity-weighted temperature (Tlum). Opensquares show uncorrected measurements from Markevitch (1998),and solid triangles connected to those squares show results for thosesame clusters after correcting for a central cooling flow. (Errorbars representing temperature uncertainty 0.5 keV have been sup-pressed for legibility.) The solid line shows the LROSAT-Tlum rela-

tion for truncated models, the long-dashed line depicts model KG3( = 3), and the short-dashed line depicts model KR ( = 3/2).Dotted lines connect models for identical halo masses, showing howthe LROSAT-Tlum relation for a given halo mass shifts to higherluminosity and lower temperature as declines.

effects of gas below the cooling threshold, and the squaresconnected to those triangles show the uncorrected data.While the corrected data cluster around the line repre-senting truncated models, many of the uncorrected datapoints lie closer to the line representing radiative-loss mod-els with = 3. A few uncorrected data points stray evenfarther from the truncated models, toward models with

< 3.Notice that the lines connecting data points with thelargest corrections are generally parallel to the lines con-necting modified-entropy models with identical halo mass.This agreement suggests that the parameter adequatelycharacterizes the offset in LX-Tlum space owing to intr-acluster gas below the cooling threshold. An interestingtest of these radiative-loss models would be to comparethe value of inferred from the luminosity-temperatureoffset to that implied by the inner temperature gradient.

3.5. What governs ?

The modified-entropy models we have constructed sug-gest that most of the observable properties of clusters de-

pend on only two parameters. The halo mass M200 deter-mines a sort of cluster main sequence with well-definedM500-Tlum and LX-Tlum relations. Deviations from thoserelations and the severity of the inner temperature gradi-ent both depend on a second parameter, , related to theamount of intracluster gas below the cooling threshold,but what physical processes determine the value of ?

One is tempted to interpret in terms of a mass coolingrate MX ; however, XMM-Newtonobservations show littleevidence for the low-temperature line emission expected inthe standard cooling-flow picture (Peterson et al. 2001).Thus, instead of viewing as simply a measure of cooling-flow strength, we would like to suggest a wider range ofpossibilities:

Entropy history. Our simplistic schemes forentropy modification treat clusters as if they wereassembled very early in time, with some standardinitial entropy distribution. In fact, the entropyhistory of the intracluster medium is likely to befar more complex, influenced in differing ways atvarious times by merger shocks, radiative cooling,and feedback. The dispersion seen in could stemfrom differences in the entropy history of clusters,with some evolutionary paths leading to abundantgas below the cooling threshold and other pathsleading to very little. If this is the case, then aclusters value of would be an important window

into its past. Incomplete relaxation. Chandra observations of

several clusters have revealed cold fronts thoughtto be evidence of low-entropy gas sloshing aroundin the cores of those clusters (Markevitch et al.2000; Vikhlinin, Markevitch, & Murray 2001).Our models assume that the lowest-entropy gasin a cluster has already settled into hydrostaticequilibrium at the clusters center. Some ofthe apparent dispersion in could be due todifferences in the degree to which low-entropy gashas settled within cluster cores following the lastmajor merger.


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VOIT ET AL. 19

Recent feedback. Cooling and condensation oflow-entropy gas at the center of a cluster is likelyto lead to star formation and perhaps enhancedactivity in the nucleus of the central galaxy(Fabian 1994). Both star formation and radio jetscan provide feedback, imparting entropy to thecluster core that inhibits further condensation

(Bohringer et al. 2002). Perhaps these feedbackprocesses control the value of .

Thermal conduction. The presence of significanttemperature gradients within the cores of someclusters has often been cited as evidence thatmagnetic fields strongly suppress electron thermalconduction in clusters (Fabian 1994). RecentChandra observations of some particularly sharpcold fronts have reinforced that supposition (Ettori& Fabian 2000). However, these observationsconstrain conduction in only one dimension and donot necessarily rule out conduction in the othertwo dimensions. Furthermore, recent theoreticalwork suggests that magnetic fields might notsuppress conduction as effectively as previouslybelieved (Malyshkin & Kulsrud 2001; Malyshkin2001; Narayan & Medvedev 2001). It is possiblethat the magnetic field geometry in a clusters coreplaces a lower limit on the value of .

Investigating all of these possibilities is beyond the scopeof this paper, but in the next section we explore how onemight self-consistently calculate the intracluster entropydistribution within a hierarchical merging scenario.

4. coupling entropy, evolution, and feedback

We have shown that some very simple prescriptions forthe entropy distribution in present-day clusters lead to aremarkably realistic set of cluster models. However, theastrophysics that leads to these entropy distributions iscertainly more complicated than we have assumed. Un-derstanding how the the physics of structure formation,cooling, and feedback determine the intracluster entropydistribution will require a more complete theory based onhierarchical merging. This section outlines a few of thebasic issues involved in constructing such a theory.

4.1. Entropy Histories

Considering the entropy history of a single gas parcelaffords some insight into how radiative cooling establishesan entropy threshold (Voit & Bryan 2001). Suppose that

the parcel begins with entropy K1 in a halo of tempera-ture T1 at some early time t1. In a hierarchical mergingscenario, the parcels halo will eventually merge with an-other halo at time t2. Merger shocks will then raise theparcels entropy to some new value K2, and as that parcelsettles within the new halo, it will approach a new tem-perature T2, similar to the characteristic temperature ofthat new halo. Each subsequent merger will also raise theparcels temperature and entropy. Thus, the trajectoryof a parcel through the entropy-temperature plane can beschematically represented by a sequence of points like thatin Figure 29.

The dashed lines in Figure 29 show how the entropythreshold associated with cooling rises with time until

it achieves its present value, indicated by the solid line.As long as a parcels trajectory through the entropy-temperature plane remains above the cooling threshold,then radiative cooling and subsequent feedback will notsignificantly affect that parcels entropy. However, anyparcel that spends a large fraction of time below thatthreshold will be subject to substantial entropy loss, con-

densation, and whatever feedback ensues. Feedback cancontinue as long as there are gas parcels below the coolingthreshold, but once cooling and feedback have eliminatedall gas below the cooling threshold, feedback must cease.Thus, the threshold for entropy modification establishedby both cooling and feedback corresponds to the coolingthreshold defined by the Hubble time tH.

Although cooling sets the threshold for entropy modifi-cation, the model we are describing is not a cooling-flowmodel. Hierarchical structure formation will ensure thatmergers disperse the products of condensation throughoutthe final cluster. Stars and galaxies form out of the lowest-entropy gas long before the cluster itself forms. Hence,much of the low-entropy gas that would eventually find

its way into the cluster core in a simulation with no radia-tive cooling decouples from the intracluster medium beforeit can participate in a centrally-focused cooling flow. Fi-nally, for reasons we will describe next, a certain amountof feedback is needed to prevent the incipient intraclustermedium from overcooling at high redshift.

Fig. 29. Entropy-temperature diagram showing the entropy his-tory of a gas parcel and the evolution of the cooling threshold with

time. Solid and dashed lines show the locus in the T-T n2/3e plane

at which the cooling time tc of a gas parcel equals 15 Gyr (solidline), 10 Gyr (short-dashed line) and 5 Gyr (long-dashed line). Thislocus moves upward through the diagram as the universe ages. Thedots connected by dotted lines schematically illustrate the entropyhistory of a gas parcel as described in the text. As long as theparcels trajectory through the entropy-temperature plane remainsabove the cooling threshold, it will not substantially cool. However,both cooling and feedback triggered by that cooling will inevitablymodify the entropy of gas parcels that find themselves below thecooling threshold.


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4.2. Overcoming Overcooling

Overcooling is one of the classic problems that modelsof hierarchical structure formation must overcome (Cole1991; White & Frenk 1991; Blanchard, Valls-Gabaud, &Mamon 1992; Balogh et al. 2001). If radiative coolingwere allowed to proceed unchecked as structure formed,then a substantial percentage ( 20%) of the universes

baryons would have been locked into condensed objectsbefore the era of cluster formation. Because the fractionof uncondensed baryons in present-day clusters is 90%,feedback processes like supernova heating are presumed toinhibit cooling and condensation of high-redshift baryons(White & Rees 1978; Cole 1991; White & Frenk 1991).

In our models, overcooling manifests itself as an entropythreshold Kc that is much higher than the characteris-tic entropy of an unmodified high-redshift halo. To illus-trate this effect, let us assume that the progenitor haloof a present-day cluster has a temperature T(z) that de-pends on redshift. The entropy threshold for that halois then Kc(z) {T1/2(z) [T(z)] tH(z)}2/3, and the char-acteristic entropy within r200 in the unmodified halo is

K200(z) T(z)H4/3. The ratio of these two quantitiesin the high-redshift limit is thus

KcK200

[T(z)]

T(z)

2/3[H(z)tH(z)]

2/3H2/3(z) . (22)

Because H(z) rises and T(z) declines with increasing red-shift, this ratio must exceed unity at early times.

We can use extended Press-Schechter theory (Bond etal. 1991; Bower 1991) to approximate T(z) and esti-mate the temperature, entropy, and redshift scales atwhich overcooling is problematic. First, to demonstratethe principle with simple scaling relations, we consider aSCDM (M = 1) universe with a power spectrum givenby an n =

2 power law. In that case, the characteris-

tic mass scale evolves like M (1 + z)6, and assumingthat M T3/2(1 + z)3/2 yields T(z) (1 + z)3. Atlow redshifts, where T is high, we have T1/2 andKc/K200 (1 + z)2. However, at high redshifts, whenthe progenitor temperature is smaller, we have T1/2and Kc/K200 (1 + z)4. Thus, the progenitor of a halowith Kc/K200 0.1 at z 0 would have Kc/K200 1 atz 1 2.

Figure 30 illustrates this effect more precisely for aCDM universe seeded by a power spectrum with shapeparameter = 0.17. The dashed line shows K200(z) fora halo that reaches T = 5keV at z = 0. The solid lineshows the entropy threshold Kc in that evolving halo for

a metallicity of one-third solar. Feedback must impart anentropy 30keVcm2 to prevent overcooling at z 3.4.3. Interplay between Feedback and Merging

Feedback can solve the overcooling problem, but it in-troduces another, more subtle problem into our frame-work. Our modified-entropy models assume that gas abovethe cooling threshold follows the unmodified distributionK0(Mg). However, the need for feedback at early timesimplies that a large proportion of intracluster gas has ex-perienced entropy modification at least once in its history.Somehow, the process of cluster formation needs to restorethe unmodified distribution to gas above the threshold atz = 0.

11 2 3 4 5

10

100

1000

Fig. 30. Entropy evolution with redshift. The dashed lineshows how the characteristic entropy K200 of the progenitor halo

of a 5 keV cluster at z = 0 evolves with redshift in a CDM uni-verse with shape parameter = 0.17. The solid line shows how thecooling threshold Kc associated with that halo depends on redshift.Because K200 falls below the cooling threshold at z 3, feedback isneeded to prevent the majority of the baryons associated with theprogenitor from condensing and forming stars.

Exactly how that happens is unclear. Consider whathappens in a merger between two halos containing isother-mal gas at temperatures T1 and T2 that collide at velocity

v. Merger shocks with Mach number Mi vT1/2i willpropagate through each halos gas, raising the entropy ofthat gas by some more-or-less uniform factor depending onMi. This mechanism could potentially amplify the effectsof early feedback, which must raise the intrahalo entropyto at least Kc to avoid overcooling. Subsequent shockswould then boost the entropy even further, along a pathparallel to the dashed line in Figure 30. If this mechanismdominates, then very little gas should remain near Kc atz 0.

However, early feedback is likely to diminish the effectsof mergers, particularly in gas with low entropy. If feed-back raises all the gas in the progenitor halo of a clus-ter to Kc, then the intrahalo medium will be nearlyisentropic, with high-temperature gas at the center of thehalo and lower-temperature gas at the outskirts (see Fig-ure 8). When mergers occur, Mach numbers associatedwith lower-entropy gas in the halos core will therefore be

lower than those associated with higher-entropy gas nearthe virial radius. Hence, merger shocks will act to steepenany entropy gradient that exists. Dynamical friction mayalso add to this effect, because the cores of merging halostend to lose orbital energy before they collide and merge.Another effect to consider is the quasi-continuous accre-tion of gas associated with small halos, for which the Machnumber is likely to be quite high. This type of accretionwill enhance the amount of high-entropy gas in the finalcluster. Thus, it seems possible that the modified-entropymodels we have developed can successfully be linked withhierarchical structure formation, although many details re-main to be worked out.


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4.4. Excess Energy

Several recent analyses of similarity breaking in groupsand clusters have cast the problem in terms of energyrather than entropy (e.g., Wu et al. 1998, 2000; Lloyd-Davies, Ponman, & Cannon 2000; Loewenstein 2000;Bower et al. 2001; Lloyd-Davies et al. 2002). For ex-ample, if one knows what the configuration of the intra-

cluster gas would be in the absence of non-gravitationalheating and cooling processes, then one can opt to definethe excess energy of a clusters gas to be the differencein total energy between that unmodified configuration andthe actual configuration. Depending on what is assumedabout the unmodified configuration, these excess energiescan range from 0.3 1.0 keV for groups (e.g., Lloyd-Davies et al. 2000, 2002) to over 2 keV for clusters (e.g.,Wu et al. 2000).

The results of this paper indicate that some combinationof heating and cooling is responsible for similarity break-ing because cooling is needed to explain the core entropiesof clusters and groups and heating is needed to preventovercooling at early times. Therefore, we prefer to com-

pute excess energy in a way that explicitly accounts for theseparate contributions of cooling and heating. First, we as-sume that the baseline state of a cluster of mass M200 is atruncated model from which cooling and condensation hasremoved a fraction f of the lowest-entropy gas. In otherwords, the initial configuration of the intracluster mediumhas an entropy floor equal to K0(f) and contains a frac-tion 1 f of the clusters baryons. Then, we assume thatthe final state of the cluster is a shifted model with thesame proportion of baryons but an entropy floor equal tothe cooling threshold, so that it obeys the observed M-Tand L-T relations. Formally, the entropy distribution ofthe final state is K(fg) = K0(fg)+ Kc(Tlum) with an outerboundary where fg = 1

f. Because the amount of hot

baryonic gas in both models is the same, the amount ofheat input required to explain the cluster scaling relationsafter a fraction f of the intracluster gas has condensedequals the energy input needed to convert the initial con-figuration to the final configuration, including work doneat the outer boundary of the intracluster medium.

Figure 31 shows the energy difference between these con-figurations as a function of halo mass for different valuesof the condensed baryon fraction f. To facilitate com-parisons with other treatments of excess energy, we di-vide this energy difference by the total number of parti-cles in an unmodified cluster (fbM200/mp), giving thequantity in units of keV per particle. When the con-densed baryon fraction is small (f = 0.02), the excess en-ergy needed to account for the scaling relations of massiveclusters ( 1015 M) approaches the 3 keV level. How-ever, this large amount of excess energy depends heavilyon the assumed form of the unmodified entropy distribu-tion, which is likely to be a poor approximation for thesepurposes. The large difference in the excess energy of mas-sive clusters between models with f = 0.02 and f > 0.05indicates that a large amount of energy is needed to lift thelowest-entropy gas out of the very center of the gravita-tional potential. Yet, Figure 1 shows that the unmodifiedintracluster density distribution is unlikely to be as cen-trally condensed as we have assumed. Even in the absence

Fig. 31. Excess energy required to overcome the cooling thresh-old for different values of the condensed baryon fraction (f). At

each halo mass M200, the quantity [Kc(Tlum), f] is the energy in-put per particle needed to transform the truncated entropy distribu-

tion KT(f), from which a fraction f of the lowest-entropy baryons

have been removed, into a shifted distribution KS = K0 + Kccontaining the remaining fraction 1 f of the baryons. Thistransformation corresponds to first removing gas without addingheat energy, then adding enough heat to reproduce the L-T andM-T relations. The energy input required to produce these rela-tions for plausible condensed-baryon fractions of f 0.1 0.2 is 0.4 0.7 keV particle1 at group scales and is negligible for hotclusters.

of cooling, the true unmodified profile probably corre-sponds more closely to a truncated model with Kc 0.05 0.1, which substantially reduces the excess energyburden. Thus, when calculating excess energies for clus-

ters, one must make sure that the baseline model accu-rately represents the cluster configuration in the absenceof heating.

The curve with f = 0.1 is likely to be a more realis-tic representation of the heat input needed to explain thescaling properties of massive clusters for two reasons: (1)this value off roughly corresponds to the fraction of clus-ter baryons that have condensed into stars, and (2) whenf 0.1, truncation eliminates the innermost part of theentropy distribution that is discrepant with simulations.At 1015 M, this excess energy curve suggests that only afraction of a keV per particle is needed to explain clusterscaling relations. That is because a shifted model with anentropy shift determined by the cooling threshold is virtu-ally identical to a truncated model with the innermost fof gas removed. Likewise, the mass scales at which curveswith larger values off intercept the zero-energy level alsoindicate cases in which the truncated and shifted modelsare virtually identical, requiring no excess heating to pro-duce the observed scaling relations. In the limiting caseof zero supernova heating (e.g., Bryan 2000, Muanwong etal. 2001), these intercept points show the fraction of con-densed baryons implied by the L-T relation as a functionof halo mass.

If the condensed baryon fractions of groups are similarto those of clusters, then 0.30.7 keV of excess energy isneeded to explain the M-T and L-T relations at 1 keV.


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The main difference between the initial configuration andthe final configuration of groups with f 0.1 0.25 isthat the gas in the final configuration is considerably moreextended, with substantially more gravitational potentialenergy. Because models with higher f contain less in-tragroup gas, they are less extended, and less energy isneeded to change their configuration. We note that the

excess energy needs of groups are comparable to estimatesof the supernova energy released over the course of thegroups history (e.g., Ponman et al. 1999, Loewenstein2000; Pipino et al. 2002), which may mean that no othersource of energy is needed to explain their scaling proper-ties, if that energy efficiently heats the intragroup medium.

The bottom line here is that calculations of the heatinput needed to account for the scaling relations of clus-ters and groups depend critically on what is assumed tohappen in the absence of heat input. Most calculationsof excess energy have assumed that the intracluster gasdensity obeys either a simple polytropic equation of state(e.g., Wu et al. 2000; Loewenstein 2000) or a -modeldensity profile (e.g., Bower et al. 2001; Lloyd-Davies et

al. 2002) whose parameters are adjusted according to theamount of heat input. The modified-entropy models dis-cussed in this paper allow for cooling and condensation ofthe lowest-entropy gas to occur before heat is added. Fig-ure 31 demonstrates that the amount of heating neededto explain the scaling relations depends strongly on theamount of cooling that occurs prior to heat input and thatrealistic amounts of precooling substantially reduce therequired amount of non-gravitational heating.

5. conclusions

A realistic family of models for clusters in hydrostaticand convective equilibrium can be constructed using the

Navarro, Frenk, & White (1997) density profile and somerudimentary prescriptions for how radiative cooling andfeedback induced by that cooling modify the entropy dis-tribution of intracluster gas. The models presume that theintracluster entropy distribution in the absence of coolingand feedback would be identical to that of gas that takeson the same density distribution as the dark matter asit settles into the clusters potential wellan assumptionsupported by numerical simulations. Because the lowest-entropy gas within that distribution can cool within a Hub-ble time, its entropy must somehow be modified. Thethreshold entropy in our prescriptions for entropy modi-fication corresponds to the entropy at which the coolingtime of intracluster gas equals the age of the universe. The

prescriptions themselves include truncation of the entropydistribution at the threshold entropy, shifting of the en-tropy distribution by adding the threshold entropy to theentire distribution, and a qualitative implementation ofradiative cooling.

Exploring the properties of dimensionless models de-pending only on the halo concentration c, the dimension-less entropy threshold Kc, and the prescription for entropymodification reveals that:

Removal of low-entropy gas acts to flatten thecore density profile, regardless of whether coolingor heating eliminates that low-entropy gas (seeFigures 1 and 4).

The observable properties of clusters dependmore critically on the threshold entropy than onthe mode of entropy modification. For example,truncation of the intracluster entropy distributionat a given value of Kc and shifting of the entropydistribution by that same amount lead to verysimilar pressure, density, and temperature profiles

within the clusters virial radius (see Figures 4 and8).

Within about 30% of the virial radius, thesurface-brightness profiles of truncated and shiftedmodels are very similar to -models (see Figure 5).Raising the entropy threshold tends to lower thebest-fitting value of , while raising the haloconcentration tends to decrease the core radius(see Figures 6 and 7).

All of our prescriptions for entropy modificationtend to augment the luminosity-weighted temper-ature of a cluster as the entropy threshold rises.

However, that temperature increase is relativelysmall because the increased central entropydecreases the luminosity of high-temperature gasin the core. Thus, modifiying the entropy of acluster shifts the bulk of a clusters luminosityto larger radii, where gas temperatures tend tobe smaller, mitigating the effects of the entropyincrease (see 2.5).

All of our prescriptions for entropy modificationreduce the X-ray luminosity of a cluster as theentropy threshold rises. In the high-threshold limit,

dimensionless luminosity scales as L K3/2c ,owing to the asymptotic r3 scaling of theunmodified NFW density distribution at largeradius (see 2.6). When luminosity scales withthe entropy threshold in this way, the L-T relationshould not flatten below 2 keV, where linecooling dominates free-free emission, and therelation should also evolve little with time. Theobserved L-T relation shares both of these features.Furthermore, this scaling of luminosity with theentropy threshold leads to an L-T relation slightly

steeper than L T5/2lum, with the extra steepeningcoming from the tendency for low-temperaturehalos to be more concentrated, also in agreementwith observations.

Adopting relations between halo concentration c andhalo mass M200 that are drawn from simulations and sup-ported by observations enables us to generate models forreal clusters that reproduce many of their observable prop-erties:

Fitting -models to our modified-entropy clustersyields and core-radius values that are similar tothose of observed clusters. Our models reproducethe observed tendency for to be lower in low-temperature halos, a behavior that arises becausethe entropy threshold determined by cooling has amuch larger impact on low-temperature halos. Ourmodels also reproduce the observed relationship


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between core radius and temperature, implyingthat rc 0.1 r200, regardless of temperature.Despite the flattening of the density profile inlow-temperature halos, the core radius remainsnear 0.1 r200 because of the increased haloconcentration (see Figures 6 and 7). The largedispersion observed in and rc could arise from

a number of effects, some involving observationalsystematics such as surface-brightness bias and therange of radii in the fit, others involving physicaldifferences such as cluster-to-cluster variations inthe amount of gas below the cooling threshold (seeFigures 17 and 18).

The temperature gradient at small radii in ourmodel clusters depends sensitively on the amountof gas below the cooling threshold, closely relatedto the parameter in our radiative-loss models.In our most extreme radiative-loss model, with = 3/2, the temperature gradient at 0.3 r200is very similar to the universal temperature

gradient observed by Allen et al. (2001). As theparameter rises to infinity, the limit in whichno gas lies below the cooling threshold, the innertemperature gradient flattens and then reverses,monotonically decreasing with radius in the large limit (see Figure 19).

All of our models have a negative temperaturegradient at large radii because the underlyingpotential is steeper than isothermal at thoseradii. In order to compare our models withobservations, we compute the effective polytropicindex eff d ln P/d ln as a function of radius. Atthe virial radius, we find eff 1.1 1.2, but thebehavior of eff at smaller radii depends stronglyon the nature of entropy modification. In modelswith no gas below the cooling threshold, thecluster core is nearly isentropic, with eff 5/3.In models with significant amounts of gas belowthat threshold, we find eff 1 within the coreradius, a consequence of the positive temperaturegradient (see Figure 20). Values of eff evaluatedat 0.2 r200 agree with observations, within the largeobservational uncertainties (see Figure 21).

The mass-temperature relations derived fromour truncated and shifted modified-entropymodels, which have no gas below the cooling

threshold, agree well with those derived fromcluster observations that have been corrected forthe presence of cooling flows (see Figure 22). Theslope of this relation is steeper than the M T3/2expectation from self-similar scaling becauselow-temperature halos are more concentrated,leading to a slightly higher temperature for a givenmass (see Figure 9), a feature these models sharewith those of Komatsu & Seljak (2001). Clustersof a given temperature are less massive thanthose in simulations without cooling and feedbackbecause entropy modification owing to theseprocesses shifts all of our model clusters to highertemperature. However, the agreement between

observations and our models does not necessarilyimply that this M-T relation is without problems,as both the observational determinations and themodels assume that the intracluster medium is anideal gas in hydrostatic equilibrium, which maynot be the case. Furthermore, this relation doesnot apply to clusters that have not been corrected

for cooling flowsthe temperature normalizationat a given mass in our radiative-loss models can beup to 30% lower, depending on the value of (seeFigures 11 and 12).

The luminosity-temperature relations derivedfrom truncated and shifted models also agree wellwith those derived from cooling-flow correctedobservations of clusters (see Figure 25). Whilethere are no adjustable parameters in these models,the baryon-to-dark-matter ratio does depend onour assumed value of M = 0.33. Thus, thisagreement can be taken as additional evidence fora low value of M.

Gas below the cooling threshold both raises the X-ray luminosity and lowers the luminosity-weightedtemperature of our model clusters. Hence, the L-Trelations derived from our radiative-loss models lieabove and to the left of the truncated and shiftedmodels in the L-T plane. Changing leads toa displacement almost perpendicular to the L-Trelations themselves (see Figure 28). Correctingfor cooling-flow emission leads to a very similardisplacement in L-T space, suggesting that theL-T relation for uncorrected clusters may dependalmost entirely on the two parameters M200 and. It will be interesting to see whether values of

derived from displacement in L-T space agree withthose implied by the inner temperature gradient.

The modified-entropy models we have derived can ac-count for many of the observed properties of present-dayclusters, but they are obviously too simplistic to tell ushow those clusters got to be the way they are. Somehow,hierarchical structure formation generated intracluster en-tropy distributions similar to our simple prescriptions forentropy modification. Cooling, feedback, and merging allhave important roles to play. The consistency of our mod-els with the data strongly suggests that radiative coolingsets the level of the intracluster entropy threshold and sub-stantially reduces the amount of heat input needed to ex-

plain the observed scaling relations. However, radiativecooling unchecked by feedback would lead to overcoolingof intrahalo gas at z 23 (see 4.2). At least some feed-back is needed to keep 90% of intergalactic baryons ingaseous form, as observed. Mergers following an episode offeedback can potentially amplify the entropy generated bythat feedback, when low-density gas associated with low-temperature halos is shocked. However, the overall impactof mergers on intracluster entropy remains unclear. Deter-mining just how cooling, mergers, and feedback conspireto produce the present-day entropy distribution of clus-ters will require high-resolution numerical simulations andsemi-analytical modeling focusing on the intergalactic en-tropy distribution and how it evolves.


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We acknowledge Megan Donahue and Don Horner forhelpful conversations and Trevor Ponman and StefanoBorgani for comments on the original manuscript. MLB

is supported by a PPARC rolling grant for extragalacticastronomy and cosmology at the University of Durham.

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G. Mark Voit et al- Modified-Entropy Models for the Intracluster Medium