Reconstruction of Gas Temperature and Density Profiles of the Galaxy Cluster RX J1347.5-1145

Chin. J. Astron. Astrophys. Vol. 8 (2008), No. 6, 671-676Chinese Journal of(tp://ww.chjaa.org)Astronomy andAstrophysicsReconstruction of Gas Temperature and Density Profiles of theGalaxy Cluster RX J1347.5-1145*Qiang Yuan;2, Tong Jie Zhang1,3 and Bao-Quan Wang4! Department of Astronomy, Beijing Normal University, Beiing 100875, China2 Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy ofSciences, Beijing 100049, China; tjzhang @ bnu.edu.cn3 Kavli Institute for Theoretical Physics China, Institute of Theoretical Physics, Chinese Academy ofSciences (KITPC/ITP-CAS), Bejing 100080, China4 Department of Physics, Dezhou University, Dezhou 253023, ChinaReceived 2008 January 7; accepted 2008 February 28Abstract We use observations of Sunyaev-Zel dovich effect and X ray surface brightnessto reconstruct the radial profiles of gas temperature and density under the assumption of aspherically symmetric distribution of the gas. The method of reconstruction, first raised bySilk & White, depends directly on the observations of the Sunyaev Zel'dovich effect and theX-ray surface brightness, without involving additional assumptions such as the equation ofstate of the gas or the conditions of hydrostatic equilibrium. We applied this method to thecluster RX J1347.5-1 145, which has both the Sunyaev-Zel 'dovich effect and X-ray observa-tions with relative high precision. It is shown that it will be an effective method to obtain thegas distribution in galaxy clusters. Statistical errors of the derived temperature and densityprofiles of gas were estimated according to the observational uncertainties.Key words: X-rays: galaxies: clusters一cosmology: theory - cosmic microwave back-ground一galaxies: clusters: individual (RX J1347.5- -1 145)1 INTRODUCTIONGalaxy cluster is known to be the largest virial gravitational bound system in the universe. The stronggravitational potential heats the gas (Hydrogen and Helium) to be fully ionized with a very high temperature~ 107 - 10*K. The thermal electrons collide with ions and emit bremsstrahlung radiation, which is mainly inX-ray band, making galaxy clusters strong X-ray sources. The intensity of thermal brermsstrahlung radiationis approximately proportional to Te ' n? (Birkinshaw 1999). So by measuring the X-ray surface brightness,we can obtain information about the gas temperature and density distributions of the galaxy cluster. Oneof the most popular models is the so-called isothermal β model (Cavaliere & Fusco-Femiano 1976), inwhich the temperature distribution as a function of radius is assumed to be constant and the density profletakes the form ne(R) = ne0(1 + R2/R2)- -3/2. This simple model gives good approximations to manygalaxy clusters. However, more and more observations indicate that the temperature distribution may notbe a constant over the whole galaxy. Hughes et al. (1988) introduced a revised model with a constanttemperature in a ranger < rigo and a decrease at large radi煤化字/ has a truncationat a radius rIim. This form provides an adequate description中国煤-ther modificationon the β model has been proposed to deal with the existen:TYHCNMHGterofthecluster(Fabian et al. 1984). .* Supported by the National Natural Science Foundation of China.672Q. Yuan et al.In order to get rid of the degeneracy between To and The, further assumptions like the hydrostaticequilibrium condition or the polytropic equation of state (EOS) of the electron gas (Cowie et al. 1987;Xue & Wu 2000; Wu & Chiueh 2001), or the spectrometric analysis (Jia et al. 2006) are needed. However,the results are still model dependent. A realistic reconstruction is expected to come directly from the obser-vations.The inverse Compton scattring between the electrons in clusters and the cosmic microwavebackground (CMB) photons, namely the Sunyaev-Zel'dovich (SZ) effect (Zeldovich & Sunyaev 1969;Sunyaev & Zeldovich 1970a, b), provides a possible way to achieve this goal. The distortion of the CMBspectrum is related to the electron temperature and density, and will provide another equation of Te and ne.Combining the SZ effect and the X-ray observations, one can obtain the electron temperature and densityprofiles. This method was first suggested by Silk & White (1978), but has not yet been put into practice dueto the limited instrumental sensitivity and resolution in the detection of the temperature fluctuation of CMB.Up to the 1990s some preliminary observational results of the radial temperature ditribution were obtainedfor a few clusters (Birkinshaw et al. 1991; Birkinshaw & Hughes 1994), but it was still dificult to obtainprecise reconstructions of the temperature and density profles because the data points were too sparse anduncertain (Wu 2003).In this paper, we try with this method to acquire some preliminary information about the gas tem-perature and density distribution. For simplicity, we assume a spherically symmetric distribution of thegas in the clusters. We adopt the ACDM model with Sm = 0.27, S2A = 0.73, and Hubble constantHo= 71km s-1 Mpc-1 (Spergel et al. 2003). This paper is organized as follows: we describe the methodof reconstruction in Section 2. In Section 3 we apply this method to the galaxy cluster RX J1347.5-1145to reconstruct its of temperature and density profiles, together with an estimate of the uncertainties. Theconclusions and discussion are presented in Section 4.2 METHOD OF RECONSTRUCTIONThe fuctuation of the temperature of CMB due to the thermal SZ effect is (Birkinshaw 1999)△TcmB(R)TcMB= 9(x)u(R),(1)where g(x) = xcoth(x/2) - 4 with x = hv/kgTcmB the dimensionless frequency, y(R) is theComptonization parameter as a function of the two-dimensional skymap radius R. In the Rayleigh-Jeanslimit,x《1,g(x)= -2, so we have△TcmB(R)/TcmB = -2y(R). The Comptonization parameter isrelated to the electron temperature and density byrdry(R)= 2AyTE()o()-√r2- R(2)where Ay = kpσr/mec2, kB is the Bolzmann constant, σr the Thomson cross section, me the massof electron and C the speed of light. r denotes the three-dimensional spatial radius from the center of thegalaxy cluster, while R = dAθ with dA the angular diameter distance from the Earth to the cluster and 0 the(projected) angular separation. Te(r) and ne(r) represent the electron gas temperature and number densityas functions of the radius r, respectively.The X-ray surface brightness of thermal bremsstrahlung emission is1Se(R)= r(n+zy .242J)R 1220701()(0)声Vr2- R丽'(3)whereAx= m(n)2)peg,be = 2/(1 + X) wid中国煤化工1 Hydrogen massfraction, g≈1.2 the average Gaunt factor (this average lead:IHCNMH; in the bolometricemissivity, see e.g., Ettori 2000) and z the redshift of the c(1 + z)/kpTg]-exp[- Emax(1 + z)/kpTl} is a correction factor from the whole band of thermal bremsstrahlung emissionto the detector bands [Emin, Emax] (e.g.. for ROSAT, Emin = 0.1 keV and Emax = 2.4 keV).Temperature and Densily Profiles of the Galaxy Cluster RX J1347.5- -1145673Table 1 Parameters used in Equations (6) and (7)X-ray .Szo(ergs- I cm~ 2 arcmin~ 2) r(arcsec)βcx1.34x 10-10a4.29土0.106 0.535土0.003SZOry(arcsec)4.1+1:4x 10-456.3+12:00.89-0:20°Derived from the Chandra ACIS-S detector counts, seehttp://heasarc.nasa. gowTools/w3pimms.html."Note that the angular diameter distance is different from Allen et al. (2002)due to different cosmological models.Using Abel's integral equation, Equations (2) and (3) can be inverted to (Yoshikawa & Suto 1999)dRT.()m()=一[山(R)_一(4)dR VR2-r2'4(1+z)4 r" dSr(R)T!2()n<(r)C(Tc) =，-。aRVe-p:(5)Thus, given the observational distributions of y(R) and Sr(R), we can easily find the electron temperatureand number density distributions from Equations (4) and (5).3 APPLICATION TO CLUSTER RX J1347.5-1145RX J1347.5-1145 is a highly X-ray luminous and dynamically relaxed galaxy cluster with redshiftz = 0.451. Both ROSAT and Chandra have measured the X-ray emission of this cluster with high pre-cision (Schindler et al. 1997; Allen et al. 2002). The X-ray surface brightness profile can be parameterizedfollowing the conventional β model asR2'-38cx +1/2S:(R)=So (1+-好昭)(6)where dA is the angular diameter distance of the cluster. For our adopted cosmological model, we finddA = 1185 Mpc. Using the 18.9 ks Chandra ACIS exposure, Allen et al. (2002) gave a detailed X-rayimage of this cluster in the energy band 0.3 - 7.0keV. After subtracting the south east excess, the X-raysurface brightness can be well described by Equation (6).Detections of the SZ efect at different frequencies in this cluster were published by several groups(Pointecouteau et al. 2001; Komatsu et al.2001; Reese et al.2002). An empirical formula similar to the βmodel can also be used to pararmeterize the observational Comptonization parameter of the SZ effect y(R),y(R)= y0o(+品R2-3βew/2+1/2We use the ftting parameters from Pointecouteau et al. (2001). The parameters in Equations (6) and (7) arelisted in Table 1.Substituting y(R) and Sz(R) in Equations (4) and (5) by Equations (6) and (7), we can obtainTc(r)ne(r) =1 (0(38-y/2-1/2)_ T33ry/2)-3ey/2VπAydAOcyT(3ey/2+1/2)(1+aa晗)= Cy,中国煤化工(8)- -3βcz:(!0(m(l)c(r.) = 4√元(1+2) S0(3.-1/MHCNMHG_AxdAOcxr(3Bex+1/2) \^ ' d旺= Cx.(9)674Q. Yuan et al.ThenT。(r)3/2/C(To) = Cg/Cx,(10)ne(r) = Cy/Te(r).(11)From Equation (10) we know that, ifθcx = θcy, βcx = βcy, the temperature should be independent ofthe radius, as expected from the isothermal β model. If βcu > βcr, the temperature will decrease at largeradius; while for Pcy < βcx, the temperature will increase in the outer region of the cluster, which seems tobe unreasonable.It is easy to calculate Te(r) and ne(r) using the parameters given in Table 1. To obtain the uncertaintiesof the derived temperature and density profiles, we run a Monte-Carlo (MC) sampling of the parametersaccording to their uncertainties. The parameter is thought to be Gaussian distributed peaked at the centralvalue with width of 1σ error-bar. However, we note that the 1σ error-bars are not the same in the“+" and“一”directions. So the distribution of a parameter is the combination of two Gaussian functions, connectedat the peak point. For example, for a parametera = ao-02, the widthofa> anisσ1 anda < ao is O2,with the connecting condition kx1/σ1 = k2/σ2. The probability distribution function of parameter a can bewritten as告exp(- (0)只)a> ao,p(a)=(12); exp(-202L) a< ao.Here, k1 and k2 can be derived according to the normalization condition j p(a)da = 1. Some cut conditionsare adopted in the MC sample. Equations (8) and (9) requirepey > 1/3 and Bca > 1/6; the other parametersare required to be greater than zero. Even so, some of the parameter combinations make Equation (10)unresolved. These cases were also discarded.The average values and variances of the temperature and density of the MC sampling are shown inFigure I. Shaded regions are the 1x and 2x sample variances as represented by the士1σ and土2σ er-rors. The results of previous authors are also shown for comparison. The solid crosses are the depro-jected temperature profiles from the Chandra X-ray spectroscopy (Allen et al. 2002), the dotted crossesare the derived results from the SZ effcct and X-ray data using a method slightly different from this work(Kitayama et al.2004), and the dot-dashed line is the extrapolated result from the Chandra spectroscopyobservations (Schmidt et al. 2004). It is shown in this figure that our derived results are roughly consistentwith the other results at 2σ level. For the inner region of the cluster, our results show an under-estimation ofthe temperature. This might be due to the systematic errors of the observations (especially the SZ measure-ments). From Equations (10) and (11) we know that a larger 9yo or a smaller Sxo may result in a larger centraltemperature. From Pointecouteau et al. (2001) it was shown that different ftting models could indeed givedifferent yo. The density profile, as shown in the lower panel of Figure 1, is consistent with the results inSchmidt et al. (2004) forr < 500 kpc. As pointed out in Allen et al. (2002), the surface brightness showeda steepening with increasing radius, which means an increase of the β parameter. A broken power law withβ = 0.54 changing to 0.78 at r = 487 kpc could well describe the observations. We adopt a uniform βpararmeter here, so for large radii our results are somewhat high. According to Equation (10), a larger βcxwill also lead to a higher temperature at large radius.4 CONCLUSIONS AND DISCUSSIONUsing the observational data of the SZ efect and X-ray surface brightness, we applied the method suggestedby Silk & White (1978) to the galaxy cluster RX J1347.5-1 145, to reconstruct its gas temperature and den-sity profles. This is a direct way to obtain the temperature and density profiles of galaxy clusters, withoutadditional theoretical assumptions. However, the quality of the observational data strongly affect the recon-struction results. Our attempt on cluster RX J1347.5-1 145, which has both the X-rav and sz measurementswith relative high precision, demonstrates the effectiveness 0中国煤化工sults show similarbehaviors as the previous studies. It indicates that there is a:he cluster, but thecentral value we derived is lower than the others. Poor qualiYHCNMH(2ncertainties of theSZ effect data might be responsible for this discrepancy. It should be noted that the model parameters of theX-ray surface brightness and SZ effect were derived from finite area images around the center of the cluster,Temperature and Density Profiles of the Galaxy Cluster RX J1347.5-114567525] 201- Average. A02KO45---S0410102Fig.1 Derived profiles of temperature (upper panel) and density (lower panel) together with their uncer-tainties. Also shown are the results of some previous studies, see the text for explanations.then extrapolated to large radi. This may result in additional uncertainty (see the discussion in Section 3).When high quality measurements of the SZ effect become available in the future, it will be a powerful toolto study the intrinsic gas distribution independent of any theoretical arguments.Acknowledgements This work was partly supported by the National Science Foundation of China(Grants No.10473002), the Scientific Research Foundation for the Returmed Overseas Chinese Scholars,State Education Ministry and the Scientific Research Foundation for Undergraduate of Beijing NormalUniversity.ReferencesAllen S. W, Schmidt R. W., Fabian A. C.. 2002, MNRAS, 335, 256Birkinshaw M, 1999, Phys. Rep, 310, 97Birkinshaw M, Hughes J. P., 1994, ApJ, 420, 33Birkinshaw M, Hughes J. P., Amaud K. A., 1991, ApJ, 379, 466Cavaliere A., Fusco Femiano R, 1976, A&A, 49, 137Cowie L. L.. Henriksen M., Mushotzky R., 1987, ApJ, 317, 593中国煤化工Etori S.. 2000, MNRAS, 311, 313MHCNMHGFabian A. C., Nulsen P. E. J.. Canizares C. R., 1984, Nature, 310,Hughes J P, Yamashita K.. Okumura Y. et al..1988, ApJ, 327, 615Jia s. M.. Chen Y, Chen L.. 2006, Chin. J. Astron. Astrophys. (ChJAA), 6, 181676Q. Yuan et al.Kitayama T, Komatsu E., Ota N. et al, 2004, PASJ, 56, 17Komatsu E., Matsuo H., Kitayama T. et al., 2001, PASJ, 53, 57Pointecoutcau E., Giard M., Benoit A. et al, ApJ, 552, 42Reese E. D.. CarlstromJ. E., Joy M. et al, 2002, ApJ, 581, 53Schindler S.. Hattori M.. Neumann D. M. et al, 1997, A&A, 317, 646Schmidt R. W., Allen s. W, Fabian A. C., 2004, MNRAS, 352, 1413Silk J., White s. D. M., 1978, ApJ, 226, LI03Spergel D. N, Verde L.. Peiris H. V. et al, 2003, ApJS, 148, 175Sunyaev R. A., Zeldovich Y. B., 1970a, Ap&SS, 7,3Sunyaev R. A., Zeldovich Y. B., 1970b, Ap&SS,7,20Wu X.-P, 2003, in Astronomical Society of the Pacific Conference Series, Vol. 301, Astronomical Society of the PacificConference Series, ed. s. Bowyer, C.-Y. Hwang, 365Wu X.-P.,. Chiueh T, 2001, ApJ, 547, 82.Xue Y.J, Wu X.-P, 2000, A&A, 360, L43Yoshikawa K., Suto Y, 1999, ApJ, 513, 549Zeldovich Y. B., Sunyaev R. A., 1969, Ap&SS, 4, 301中国煤化工MYHCNMHG