言 200
む o ←
ω
」 Φ〉
電 N oo
o ←
100 ら
1 5 10 152025 「0．1 0．2 0．5 1 2 Radius（arcmin） Radius（γ1r200）
β
Fig．7．6：Entropy（a）and equilibration七ime scale（b）of A 1689（Kawaharada， e七a1．2010）．
（a）：entropy pro丘1e（black diamond：Suzaku， black solid line：fitted model）．（b）：τei profile
（diamonds）compared withτelapsed（black solid line）．
Table 7．1：The best fit parameters of entropy profile mode1．
r＜0．5r200 γ ＞0．5γ・200
A1413 0．90±0．10 0．97±0．45 A2204 1．03±0．06 0．45土0．27 A1795＊ 0．71±0．06 0．33土1．7 PKSO745−191† 0．93±0．04 0．02土0．12 A1689‡ 1．08±0．12 −0．43±1．30
＊The observed data is from Bautz et a1．（2009）．
†The observed data is from George et a1．（2008）．
‡The observed data is from Kawaharada， et al．
（2010）． Only this case， we fit 5＝50十5r午in r＞0．5r200 because its slope is clearly negative．
7．4．DIFFERENCE BE TWEEN E1沼CTRON．AND∫ON TEMPERATσRE8 107
electron process（ちe）and about 45 times longer than ion−ion relaxation time（ちi）．
According to Fox＆Loeb（1997）， Takizawa（1998）， and Rudd＆Nagai（2009）， the electron−ion timescale including contributions from both protons and He2＋is estimated
as（Spitzer 1956）い2・・×・・8yr
i （エ／108K）3／2
ﾅi／10−3cm−3）（ln A／40）， （72）
where ln A is the Coulomb logarithm． We simply assume that ions are initially heated through accretion shocks at r200． In the post−shock region， ions achieve thermal equilib−
rium with a timescale ofτii after this hea七ing． The ion temperature 7］will then be signifi−
cantly higher than the elec七ron temperature Z二． Eventually， thermal energy is transferred from ions to electrons through Coulomb collisions， and 7もwill be equal to男after七he relaxation timeτei．
In figure 7．2（b）， figure 7．3（b）， figure 7．4（b）， figure 7．5（b）， and figure 7．6（b）， we
show老ei estimated from the data for A1413， A2204， PKSO745−191， A1795， and A1689
respectively In the region ou七side of r〜0．9 r200，ちi is almost Gyr order except for PKSO745−191 because this object is suspected for wrong backgroulld subtraction in George et a1．（2008）． It means that if merging gas within Gyr order， the outer regions are not equilibrium still now． Then we exactly observe the gas condition to equilibration in the OUter regiOnS．7．4 Difference between Electron and Ion Tempera一 tures
Fox＆Loeb（1997）were the first to investigate the two−temperature nature of the ICM．
Takizawa（1998）showed that in a one−dimensional numerical simulation there existed a
significant difference be七ween the electron and ion temperatures， which will affect theentropy profile and the inferred gravita七ional mass． Recentlyi Rudd＆Nagai（2009）
reported the results of simulations which indicated that the temperature difference had
amaximum of about 30％at r200． We will examine here a possible deviation between
electron and ion temperatures． These studies can help us understand how the cluster gas obtains hydrostatic equilibrium over large volumes．We de丘ne七he average gas temperature as，
隅品一ηe ﾇ男， （73）
which will change over a typical electron−ion equilibration timescale，ちi． We estimate the average gas temperature，え7ピニSηi／3， by assuming a single power−law with 7ニ1．1 for the radial entropy profile， normalized in the cluster inner regions where 7］＝7二because the relaxa七ion times are much shorter there． Figure 7．7（c）shows the ratio of the observed
electron temperature to七he estimated average gas temperature， where we have adopted
108 CHAPTER 7． D工SCσSS∫ON
（a）Entropy （b）Entropy／rl・1
0 8 N
口
山
o o
ζ〜」 （N
d
o o
、「0．1 0．2 0．5 1 2 0．1 0．2 0．5 1 2 Radius（r／らoo） Radius（r／r200）
（C）た工／ん7三品 N
←
詔
トごめ
へ
』匂 o
へ
o
…｝ ．・1／
0．1 0．2 0．5 1 2 Radius（γ／r200）
Fig．7．7：（a）Entropy pro且1es（black diamond；Suzaku， grey diamond：XMM−Newton，
black solid line：丘tted model to Suzaku in 7L 20 ， black dashed line；6tted model to XMM−Newton in O ．5−7 ， grey solid cross：PKSO745−191、 grey dotted cross：A1795， grey diamond：A1689）．（b）Entropy Ilormalized toぐ×r1・1 pro61e．（c）現／7三径pro且les compared
with the simulated result by Rudd＆Nagai（2009）．
75．MA88 E8TZMATτON TO」B200 109
ηi＝0．92η，（including He2＋）for a fully ionized gas with X＝0．7 andγ＝0．28．
Temperature difference betweenエand隅品is even larger than the simulation sample result（Rudd＆Nagai 2009）．
The rapidエdecrease in七he cluster outer regions can be explained by either the ICM not being in hydrostatic equilibrium or by di任erences betweenエand男． We can determine which interpretation is correct if we could direc七ly estimate Zi from the line width． This measurement should be made possible in the near future using the microcalorimeters on
the ASTRO−H mission（Takahashi et al．2008）．
7．5 Mass Estimation to r200
We calculated the gravita七ional mass of A1413 and A2204 to r200 assuming spherical symmetry and hydrostatic equilibrium． From numerical simulations， these assumptions
are valid within〜2r200 except for the core region at r＜0．3 r200， where cooling and heating of AGN give significan七efFects on the physical state of七he gas（Roncarelli e七 al．2006；Borgani et a1．2006）． Previous X−ray studies mainly showed gravitational mass within r500 because of ins七rumental limitations． In this sec七ion， we determine七he mass profile in the ou七er region of A1413．Assuming hydrostatic equilibrium， the total integrated gravitational mass， M＜R， within the 3−dimensional radius R is given by（Fabricant et a1．1980）
』一一議
（74）一鵬ぱ篇＋鵠・ （7．5）
where（7 is the gravitational constant，μis the mean molecular weight of the gas and mp is the pro七〇n mass． We derive the above temperature and gas density pro丘les us−
ing the observed projected temperature and surface brightness pro丘1es． We use the projected temperature direc七b but discuss the validity of this assumption below． We
calculate七he gas density from七he normaliza七ion of the ICM spectral fit by taking in七〇 account the projec七ion efFect． Theαρec normalization parameter is de丘ned as Norm＝10−14∫πeηH∂γ／（4π（1十z）2D1）cm−5， with、DA七he angular diameter distance to the
source． We estimated the de−projectedηeηH values assuming spherical symmetry and a
constant temperature in each allnular region as described in Kriss et al．（1983）， and thenassumedηe＝1．2ηH（excluding He2＋）as described above．
Allowing for the possibility ofエ≠71， we consider two cases for T：the electron
temperature and the average gas temperature． We show七he integrated mass profiles in
丘gure 7．8（b）and丘gure 7．9（b）based・n賦． These pr・丘les are・btained with・ut Using any particular model since we calculate the derivatives by dif［erencing the temperatures and densities of a（U acent radial bins． The integrated mass within 13 ．2±5：1， which encom−passes r200（14 ．8）is（6．6±2．3）×1014．MO for A1413 which is about 7 times larger than gas
110 CHAPTER 7． Dl旧CUS8∫ON mass，0．93士0．67×1014Mo． The 30％difference in the七empera七ures propagates almost
directly to the mass difference． Our mass determina七ion for A1413 agrees with tha七〇f Vikhlinin et al．（2006）， but not with Pointecou七eau et a1．（2005）． These masses imply an overdensity within r200 with respect to the critical density of 177土47 and 132±47， where the errors are only from the mass errors．In the above mass estimation， we assumed that the observed projected七emperature is the 3−dimensional value at七he observed radius． We need to examine the systematic error caused by this assumption． In the following we denote the true 3−dimensional temperature of the ICM by 75d， which varies with radius． We derive the七emperature from the spectral fit as a weighted mean of different七emperatures projected alollg the line of sight． Often the prolected temperature is defined as the emission−weighted temperature Zもw，