2. BREMSSTRAHLUNG RADIATION

Transcription

1 . BREMSSTRAHLUNG RADIATION In gnral, whn discussing th proprtis of mission procsss, w nd to discuss basically th following aspcts: a) th total mittd powr by th singl charg; b) th spctrum of radiation from a singl charg; c) th radiation spctrum from an nsmbl of particls; d) radiation slf-absorption; ) polarization proprtis; f) th volution of th particl nrgy spctrum bcaus of th mission..1 Th classical limit. Small angl scattring. Th lctric dipol contribution. W will discuss first th phnomnon in th classical limit, in spit that somtims th photon nrgis will b comparabl to thos of intracting particls. So w will provid th rsults in th classical limit, which ar th usful ons in most cass as thy covr th rlvant domain of th paramtr spac, and thn w will discuss corrction factors of quantum origin (s discussion about th Gaunt factor). Anothr condition is to nglct, on th first instanc, a rlativistic tratmnt, that will b discussd latr in th chaptr (lss oftn w find astrophysical situations involving rlativistic Brmsstrahlung). Th Brmsstrahlung procss 1 involvs radiation mission from particl acclration by Coulomb intraction with anothr particl. Now lt us considr th main 1 Th Brmsstrahlung procss gos also undr th nam of fr-fr mission, bcaus it involvs nrgy lvls of fr particls. Brmsstrahlung mans radiation from an acclratd particl..1

2 contributions to th radiation fild, bcaus th situation for a ralistic astrophysical plasma may b quit complicatd in gnral. From what w hav sn in Sct.1.4.3, + + radiation from intraction of idntical particls (, p p ) is null in th dipol approximation, whil th quadrupol contribution, though dirnt from zro, maks ngligibl radiant nrgy. Intractions of mor than two particls can gnrat dipol and quadrupol trms dpnding on th kind of particls, but ar vry unlikly and rar, so thy do not contribut significantly to th radiation fild from astrophysical plasmas. So th rlvant contributions com from intractions of two particls with dirnt charg, that is intractions of lctrons and protons (or lctrons and ions in gnral). Not that in this cas th magntic dipol momnt is null (Appndix 1A), so w can just considr for this th lctric dipol componnt hr. Sinc th mass of th ion is much largr than that of th lctron, w can assum just th motion of th lattr and th ion at rst. Lt us assum also that th motion of th lctron is along a lin that corrsponds to th assumption that th motion is not violntly prturbd by th intraction (s figur abov). During th transit in th vicinity of th ion th lctron acclrats bcaus of th Coulomb forc and mits a puls of radiation according to th Larmor rlation. Lt us thn analyz th spctrum of photons producd. In th dipol limit, th lctric fild of th radiation is d E rad d d = sinθ cr with th orintation of th lctric fild as shown in th following graph [.1] Now w xploit th discussion dvlopd in Sct. 1.6 and rmmbr that th spctrum of th puls is givn by th Fourir transform.

3 dw 8π 8π 4 d( ω) ω d( ω) 3 3 dω = 3c = 3c [1.35] i.. it is proportional to th squar of th Fourir transform of th acclration of th dipol momnt. Following [1.34] w hav + ω d d ( ω) = vxp( iωt ) dt p. Lt us rsolv in a simplifid way this rlation and dfin now th collision tim as τ b. For ωτ >> 1 th intgrand oscillats rapidly and th intgral is small, v th contrary for ωτ << 1, th xponntial convrgs to 1, and th solution bcoms v for ωτ << 1 d ( ω) = πω 0 for ωτ >> 1 whr v is th vlocity impuls that th lctron acquirs whn passing clos by th ion. Th mittd powr is dw dω ωτ << 3 = 3π c ωτ >> v for 1 0 for 1 [.] v is asily In our assumd cas of a small prturbation of th lctron orbit, calculatd by intgrating th acclration from th Coulomb law. Assuming Z as th atomic numbr of th ion and th origin of tims t=0 whn th particl gts at th closst distanc from th ion, and considring only th acclration componnt orthogonal to th unprturbd trajctory as in th figur,.3

4 w gt 1 Z 1 Z b a = sina m r = mb + vt b + vt Z bdt Z v = = = + + a dt 3/ m [.3] ( b + vt) mbv having changd variabl from t to x = bt v (th intgral bcoms just a factor ). As a function of th impact paramtr b, th mittd spctrum of th puls is dw ( b) dω 8Z for v /, v / b 3π cmv b 0 for bv/ ω, ω v/ b 6 b0 ω ω 0 3 = [.4] This xprssion dos not giv us th dtails of th function, but just its asymptotic bhavior. An important point to not hr is th indpndnc of th puls spctrum on frquncy ω, a rsult that will imply a similar gnral proprty of Brmsstrahlung (both thrmal and non-thrmal). To not also th (invrs quadratic) dpndncs on mass, impact paramtr that ar rathr obvious and that on vlocity, this lattr coming from th duration of th impact, that is vry short whn th vlocity is high). Th nxt task is to dtrmin th flux from a st of particls in a uniform mdium with lctron dnsity n and ion dnsity n i and for particls with a fixd vlocity v. Th lctron flux will b n v and th dirntial ara πbdb. Pr unit volum and tim (bcaus w hav an infinit train of packts, s our discussion of th tim dpndnc of th spctrum in Sct. 1.5), w will hav dw (v) max ( ) v b dw b nni π dωdvdt bmin dω = bdb, whr th limits in b ar th minimum and maximum ctiv impact paramtrs contributing significantly to th radiation fild. In practic, th dtaild knowldg of th W(b) function is not ndd and knowldg of th asymptotic bhavior is nough. From [.3] and [.4] 6 6 dw 16 b db 16 b ln dωdvdt 3c m v b 3c m v b [.5] max max = nnz 3 i = nnz 3 i b min min.4

5 . Th Gaunt factor Th logarithmic trm in [.5] is an important on bcaus it includs all corrction factors with rspct to our so far simplifid tratmnt, including th quantum corrctions trms. Lt us s it in som dtail. Th b max trm is that at which th asymptotic rsult for dw/dω is no mor valid and th contribution to radiation is small. W can thn obviously tak b max v ω =. As for th lowr limit b min, it has to account for two factors. a) Th first on is our approximation of a linar wakly prturbd trajctory, and this can b st by comparison of th Coulomb potntial and kintic nrgy: mv Z b = Z b min 1. mv b) Th scond factor coms from th considration that if b is too small, th scattring cannot b tratd any mor as a classical procss. From th uncrtainty principl, x p h, putting x b and p mv, w gt b = h. min mv Whn b< b min th procss cannot b tratd any mor as a classical on, and th 1 quantum corrctions bcom important. Whn b >> b, th classic tratmnt min min 1 holds valid, sinc w can show that th numbr of scattrings with b< b min ar numrically ngligibl. This happns whn Z h Z Z 7 cm >> v << 10 / sc m v mv h (corrsponding to a tmpratur of T 1000 K ). Whn approximation can b obtaind by putting Not that, in any cas, th ratio ( ) max b min min = b. min v>> Z h, a good ln b b falls undr th logarithmic oprator, so it cannot mak a larg ct on th calculation of Brmsstrahlung mission. Thn w can xprss [.5] as dw dωdvdt 16 6 nnz g 3 3c m v i = [.6] (v, ω).5

6 having dfind g 3 b (v, ω) ln max π b min as th fr-fr Gaunt factor, dpndnt on th lctron nrgy and th frquncy of th mittd photon..3 Thrmal Brmsstrahlung From an astrophysical point of viw, th most intrsting cas is that of th thrmal Brmsstrahlung, that is whn th particl vlocity distribution is thrmal. In such cas, th probability that an lctron has vlocity in 3 v distribution d d is givn by th Maxwllian d d v= xp( v / ) v=4p v xp( v / ) v E/ kt 3 3 dp d m kt d m kt d having assumd an isotropic distribution of vlocitis. If w now intgrat [.6] ovr vlocitis, w gt dw dw ( T, ω) v min dv dt dω = dv dt dω m kt d (v, ω) v xp( v m / kt ) d v v xp( v / ) v 0 Th intgral has bn confind to a minimal vlocity v min bcaus w nd to produc photons of nrgy h, which is mad only by particls having 1 v m h, so that 1/ v min = ( hn m), and dω = πd. By inclusion of rlation (.6) it turns out that th rlvant intgral is prfctly analytic (dnominator is solvabl numrically), and w finally gt: 5 6 dw p p = n n T Z xp( h kt ) g 3 n dvdt dn 3mc 3km or in th CGS systm dw. dvdt dn 1/ i., [.7] 38 1/ Z nnit xp( hn kt ) g [ rg / cm / s / Hz].6

7 whr g is th Gaunt factor avragd ovr vlocitis. W rport in th figur blow (takn from Ribicki & Lightman) various approximations for this quantity in dirnt domains of th plan of photo frquncy and plasma tmpratur. W s that for 10-4 <u<1, th Gaunt factor varis from 1 to 5 (u=h/kt). For u>>1 th Brmsstrahlung missivity dcays xponntially and th Gaunt factor, though larg, is no mor important. Anothr rprsntation of analytic xprssions for th Gaunt factor in dirnt rgims of th v T plan is rportd in th figur blow. Not that all what concrns valus of h kt xponntially cuto thr. > is of limitd intrst bcaus th Brmsstrahlung flux is.7

8 Th frquncy dpndnc of th Gaunt factor in th astrophysically intrsting rgion (small angl) is rathr wak, of ordr of g 0. v up in th figur of th optically thin Brmsstrahlung spctrum blow.. This dpndnc shows In th optically thin limit, th Brmsstrahlung spctrum can b rprsntd as an almost flat xponntially cuto function lik in th figur (including Gaunt factor):.8

9 X-ray mission from th Cntaurus clustr of galaxis dtctd by proportional countrs on HEAO1-A xprimnt. Simulatd X-ray spctrum by a hot plasma with solar mtal abundanc..9

10 X-ray mission from a complx sourc including a starburst componnt of stllar origin and a slfabsorbd AGN. Mor in gnral, if w hav a non-isotropic non-maxwllian distribution of particls, w nd to intgrat [.6] ovr th actual distribution of vlocitis, with th appropriat Gaunt factors. In such cass w talk of non-thrmal Brmsstrahlung, s blow an application in Sct..6. Rlvant xampls of high-nrgy spctra of cosmic sourcs immdiatly idntifid as fr-fr missions ar rportd in th abov figurs. In th prvious two cass th sourcs ar hot plasmas prsnt in rich galaxy clustrs (s Sct..7 blow). Th lattr is a mor complicatd cas including plasma mission in a star-forming galaxy plus an AGN contribution at photon nrgis largr than 4 kv. Polarization proprtis of fr-fr mission. For thrmal plasmas, lik thos considrd hr, th motions of th particls in th mitting rgion ar compltly random. Consquntly, this fr-fr mission is compltly un-polarizd, a fatur that can disntangl among dirnt intrprtations of X-ray obsrvations of X-ray astrophysical sourcs (again, s Sct. 3.9)..10

11 X-ray mission spctrum by a plasma coolr than that in prvious figurs. Th prominnc of lin mission compard to th continuum is vidnt, as wll as th xponntial cuto at lowr nrgis. Assumd to b a solar abundanc plasma..4 Cooling of a plasma through Brmsstrahlung mission If w intgrat [.7] ovr all frquncis, w hav in th CGS systm th bolomtric fr-fr missivity: dw rg = Z n nt g 3 dvdt cm s. [.8] 7 1/ i B with gb th Gaunt factor avragd ovr frquncis. It coms out that 1.1< g B < 1.5. Th rat of nrgy loss pr unit volum for a plasma of solar abundanc (Z 1.35) can thn b approximatd as 7 1/ rg nt 3 cm s W can now compar [.8] with th thrmal nrgy dnsity of th plasma. For this w assum to considr an astrophysical plasma dominatd by hydrogn, such that n n i : ρ = kt ntot = kt ( n + ni ) 5 10 nt [.9] and th fr-fr cooling tim-scal is obtaind:.11

12 [.10] In som mor dtail, how a hot plasma cools down can b wll undrstood by considring this figur and compar it to th spctrum of a synthtic sourc with hottr tmpratur ( kv) shown prviously. It is vidnt that th lin mission bcoms progrssivly mor important at lowr tmpraturs, whil th continuum mission is dominant at high T. At high T th cooling is via continuum mission, at lowr T via lin mission. Th divid lin is, abov which fr-fr cooling dominats..5 Radiativ transfr and Brmsstrahlung slf-absorption It is usful to brifly discuss hr th cts of fr-fr slf-absorption, which givs us th occasion to mntion gnralitis on th radiativ transfr in a mdium. Lt us first rport th radiativ transfr quation. Whn radiation crosss mattr, part of th radiant nrgy will b addd to th fild, and part will b subtractd..5.1 Radiation intnsity Lt us first dfin th radiation intnsity (or surfac brightnss) as i.. th amount of nrgy flowing pr unit ara, tim, solid angl and frquncy. known to b indpndnt of th distanc from th obsrvr in a flat (Euclidan) univrs if thr is no rlativistic motion btwn sourc and obsrvr. This coms is.1

13 immdiatly by considring two arbitrary surfacs at a givn distanc R, such that th flowing nrgis across thm ar qual: 1 de = I da dt dω d = de = I da dt dω d. Sinc frquncis and tims do not chang, and solid angls hav th sam scaling with R, d da R Ω = 1 and dω 1 = da R, also th two intnsitis will b th sam. Th intnsity and surfac brightnss is indpndnt on sourc distanc. A dirnt situation applis, of cours, if thr is motion btwn th sourc and th obsrvr, or in th cas of a curvd xpanding univrs. As wll known, in an xpanding univrs th radiation intnsity scals with th cosmological rdshift as 4 I (1 + z)..5. Th radiativ transfr quation Lt us dfin th mission and absorption coicints. Th mission coicint or missivity is dfind as th radiant nrgy producd pr unit volum, tim, solid angl and frquncy dw = de = j dvdtdw d. Th absorption coicint is instad dfind as th (fractional) loss of nrgy whn crossing a givn layr of mattr of dpth ds: with α 1 [ cm ] di = α I ds =. This quation has a trivial solution as I = I0xp( αds) = I0xp( τ), whr τ αds is th mdium s optical dpth. Th coicint α is simply rlatd to th particl cross sction αn = nσ n = ρσn / m with obvious maning of th symbols. Not that undr th trm absorption w man to includ both th tru absorption and stimulatd mission, as in both cass th variation is proportional to th intnsity I : th nt ct on th radiation fild may b positiv or ngativ according to th dominanc of tru absorption or stimulatd mission. W can incorporat all cts on th radiation fild in a formally concis xprssion di ds = α I + j [.11] In gnral w dfin as thrmal radiation that mittd by mattr in thrmal quilibrium. Black body radiation corrsponds instad to a mor rstrictiv.13

14 condition of prfct quilibrium btwn mattr and radiation. In th formr cas it can b shown that w can dfin a sourc function dpndnt on mattr tmpratur T and photon frquncy and indpndnt on th gomtry of th sourc, such that S j = B( T) = a c 3 h ( h ) xp 1 kt (Kirchho law) In trms of this sourc function, th transfr quation can b writtn as For a black body radiation w hav [.1] di I S, dτ αds dτ = + =. [.13] di 0 I S B( T) dτ = = =. For a thrmal radiation w can apply th Kirchho law and writ, for th missivity in th fr-fr cas j = dw 1 α B ( T) dvdtd 4π = and invrting this on w gt th fr-fr absorption coicint, from [.6 and.1] α 6 1/ 4 π 1/ 3 hn kt 1 = nnt i Z n g For For 3mhc 3km h >> kt an hn << kt an n - 3 (1 ) [cm ] (Win rgion) (Ryligh-Jans rgion) Th nt important ct is that th fr-fr absorption gts largr at lowr frquncis, whil at high photon frquncis th mdium is optically thin.. [.14].5.3 Formal solution of th transfr quation and th total fr-fr spctrum Lt us now addrss how th full spctrum of a thrmal fr-fr sourc will look lik. This implis rsolving formally th radiativ transfr quation. To this nd w dfin τ A I τ and B S. By multiplying [.13] by τ, w gt.14

15 or A ( τ ) di d τ τ = A+ B = = dτ dτ 1 1 d( A) d ( τ ) τ τ d( A) τ + A = A τ dτ dτ dτ 0 ( ) d( A) d A A= A+ B = B dτ dτ τ A( τ) = A(0) + B( τ') dτ', and substituting th trms A and B τ τ τ I ( τ ) = I xp( τ ) + xp( τ ' τ ) S ( τ ) dτ ' [.15] 0 0 Th physical intrprtation of this is obvious: th intnsity is partly xtinguishd by absorption in th path from τ = 0 to τ = τ, and is partly incrasd from th intgratd contribution of th sourc function along th sam path. Assuming that th sourc function is constant S = const, thn th solution gts n I ( τ ) = I + S (1 ) = S + ( I S ) τ τ τ 0 0 which at larg optical dpths implis I For, n lik in th following figur. S. So for τ >> 1: j I = S = hn << kt j const, w hav α I and th ovrall spctrum looks Hr ar two xprssions for th fr-fr opacity. From [.14] in th Win rgion: α 8 1/ nnt i Z n g [cm ] =. In th Rylight-Jans rgim: [.16] α 3/ nnt i Z n g [cm ] =..15

16 .6 Rlativistic Brmsstrahlung It is worthwhil just to mntion a compltly dirnt approach to trat th cas in which th kintic nrgis of particls ar so larg to bcom rlativistic, as a consqunc for xampl of a vry high plasma tmpratur. This is basd on th mthod of virtual quanta (introducd by von Wizsackr & Williams in 1934). This is still valid in a non-quantum mchanical rgim, although th complt tratmnt would rquir th full nvironmnt of th quantum lctrodynamics. W thn considr th collision of an lctron with an ion. In th currnt cas, it is usful to rvrs th rfrnc systm by considring th lctron at rst and a rlativistically moving ion, as shown in th abov figur. W can asily show, basd on th rlativistic Lorntz transformation of th E and B filds, that th lctric fild gnratd by th ion onto th lctron is an.m. puls travlling in a plan orthogonal to th dirction of rlativ vlocity. This radiation puls is thn Compton scattrd by th lctron and dtctd by th obsrvr back in th ion fram (ssntially two Lorntz transformations ar ndd). Th situation is illustratd in th following figur, whr w show th two rfrnc systms, K' is that of th lctron at rst and th moving ion Z, K is back to th ion fram..16

17 In th ion fram, th E and B filds can b writtn as qx qy qz Ex = E 3 y = E 3 z = 3 r r r B = 0 B = 0 B = 0 x y z [.17] whil th Lorntz transform implis that going into th lctron fram (with obvious maning of th symbols): E ' = E B ' = B [.18] E ' = γ( E + β B) B ' = γ( B + β E) W nd to apply this transformation to th filds in [.17] to gt th filds in th lctron fram. Considring th gomtry sktchd in th figur abov, w hav still 3 3/ in th K' fram r ( γ v t b ) of transit to minimum distanc, w gt = + and assuming th origin of tims t at th tim qx γ vt Ex' = Ex = = q B ' x = r r' qγ y' qγβ z' Ey' = B ' 3 y = 3 r' r' qγ z' qγβ y' Ez' = B 3 z = 3 r' r'.17

18 Prhaps intrsting to not, ths quations may wll b drivd from th Linard- Wichrt potntials in [1.0] (s Ribicki & Lightman). If w now insrt th Lorntz transform to th coordinats (only th x coordinat transforms), and assuming that th z coordinat is null (no loss of gnrality for this), w finally gt in th fram [.19] having nglctd th primd symbols. In th ultra-rlativistic cas, th filds display a strong maximum for, whr is th timscal of th duration of th transit. Th fild has instad a shiftd maximum, and in any cas. Th dtaild bhavior of th filds sn by th lctron ar rportd in th figur blow and in th not. Th bhavior of E y is immdiatly found from [.19]. Th maximum of E y is givn by, and th FWHM of th.m. puls is immdiatly found by stting th function qual to half this maximum:,. As for th maximum of E x it is calculatd from th drivativ of [.19]. W can rwrit, such that th maximum corrsponds to th minimum of th dnominator, that w can gt from th drivativ w.r.t. st to 0:. Th valu of E x at th maximum is givn by substituting this in [.19]:..18

19 To calculat th mittd spctrum of th.m. puls in th rfrnc fram of th lctron, w nd to tak th Fourir transform of th main componnt, th E y in [.19], and xprss it in trms of th modifid Bssl function K 1 of ordr 1, that is: 1 3/ E ( ω) E y = qγb ( γ v t + b ) xp( iωt ) dt p q bω bω pbv γv γv = K1 [.0] This intgral coms from a chang of variabl from tim t to t' = tγ v/b x, aftr having collctd b from parnthsis, thus obtaining that th intgral abov bcoms xp( / v)(1 ) 3/ iωbx γ + x dx that can b solvd in trms of th Bssl function. Th mittd spctrum in th rfrnc fram of th lctron thn gts aftr [.33 Sct.1], rintroducing th primd symbols, ω ω 1 dw ' ' ' ' ' [ / / ] ( ) c q b b ω rg cm Hz = c E = K dω' da' π b v gv gv This is a vry important and gnral rsult concrning th mission of rlativistic particls. Th spctrum of th puls is cuto at ω > γv/b, which is simply th rciprocal of th timscal of duration t 0 (s not ). In th limit in which th scattring procss in K' can b dscribd via th classical Thomson cross sction (i.. ), in th ultra-rlativistic cas, bcaus hω ' << mc th scattring in K' is symmtric ω = γω '(1 + β cos θ ') ω γω ', and considring that nrgis and frquncis transform in th sam way from K' to K, for th singl puls w hav (b'=b dos not transform): 6 dw ( b) dw ' dw ' 8Z bω b σ T K 5 1 ω = = == dω dω' dω' da' 3πb c m γ c γ c Now w nd to intgrat ovr all b valus, but thr is no gnral analytic solution to this. Thr xist howvr an asymptotic solution for low frquncis, ω << γv/bmin, whr bmin = h / mc is th minimum impact paramtr st by U.P. Thn summing up now all intractions happning in th unit volum, which is achivd as w hav don in Sct..3 by rplacing th vlocity with a constant c, w finally gt for a st of monochromatic lctrons with nrgy γ mc in th low-frquncy limit:.19

20 . For a thrmal distribution of hot rlativistic lctrons and aftr intgration ovr frquncis, th volum missivity bcoms: [.1] By comparison with [.8], it is mphasizd hr th incidnc of th rlativistic corrctions, that bcom important as soon as th plasma tmpratur gts comparabl to or in xcss of. Brmsstrahlung Gamma Rays from our Galaxy Th obsrvd widsprad mission by th Galaxy in gamma rays (s figur blow) is indd partly intrprtd as rlativistic Brmsstrahlung by high nrgy cosmic ray lctrons bombarding th dius Galactic intrstllar mdium..0

21 Obsrvd and prdictd γ-ray spctra in dirnt rgions of th Galaxy. Th Galactic rgions ar indicatd in th panls. Th obsrvations ar shown by vrtical bars. Th prdictd contributions of π 0 dcays, invrs Compton scattring (IC) and rlativistic Brmsstrahlung (brmss), as wll as th infrrd xtragalactic background componnt, ar shown by dirnt lin symbols. [Strong t al. 004]. Th intrprtation is that ths ultra-rlativistic galactic cosmic ray lctrons collid with th gnral intrstllar mdium of th Galaxy and mit by rlativistic Brmsstrahlung. Othr componnts of th Galactic spctrum ar Invrs Compton mission by rlativistic lctrons and th nutral pion π 0 dcay. In collisions btwn high nrgy particls and nucli of atoms and molculs of th ISM, pions of all chargs, π +, π 0 and π ar producd. Th positiv and ngativ pions dcay into positiv and ngativ muons and nutrinos, th muons dcaying in turn into positrons and lctrons with rlativistic nrgis. Th lattr may mak a contribution to th low.1

22 nrgy lctron spctrum and th prdictd prsnc of positrons provids a dirct tst of th importanc of th pion production mchanism in intrstllar spac. Th nutral pions π 0 dcay into two γ -rays. In proton proton collisions, th cross-sction for th production of a pair of high nrgy γ -rays is roughly th gomtric siz of th proton, σ γ 10 4 cm. Th spctrum of γ -rays producd in such collisions has a broad maximum cntrd about 1000 MV and is a charactristic signatur of th nutral pion dcay procss. Th π 0 rst mass nrgy is 135 MV, so that th obsrvd photon pak at 1000 MV producd by π 0 dcay indicats avrag kintic nrgis of ths particls around 1 GV..7 Applications to fr-fr missions by hot intra-clustr plasmas W rport in th following som applications of th thory of th Brmsstrahlung procss in th prvious Scts. to th obsrvations of X-ray missions from th cors of clustrs of galaxis, a discovry dating back to th UHURU mission (1971), and confirmd by all subsqunt X-ray spac obsrvatory missions (among othrs, th Einstin Obsrvatory, Aril V, EXOSAT, GINGA; SAX, XMM Nwton, Chandra). X-ray sourcs associatd to th cors of galaxy clustrs ar vry numrous in th local Univrs, making about half of th low-z X-ray sourc population, th othr half bing Activ Galactic Nucli (AGNs). At highr-z AGNs bcom quit mor numrous, whil clustrs mor rar. Ths hug X-ray missions from clustrs ar du to th prsnc of normous amounts of high-tmpratur plasma, mitting via optically thin fr-fr. Exampls of such sourcs at both low and high-rdshifts ar rportd in th following figurs..

23 .3

24 .7.1 Physical paramtrs of th hot intra-clustr plasma X-ray obsrvations provid us with xtnsiv data on th hot intraclustr plasmas (as wll as on plasmas in many othr astrophysical nvironmnts), suicint to driv all rlvant physical paramtrs. First of all, X-ray imaging, as shown in th prvious figurs, allows us to masur th xtnt of th plasma distribution. In clustrs of galaxis, th mission is pakd at th clustr barycntr and smoothly dcrass towards th outr rgions. Howvr, w can considr th plasma to covr th whol clustr xtnt of about X 1 Mpc radius. Assuming that th cosmic distanc of th clustr is known from rdshift masurmnts (ths can b asily achivd from optical spctroscopy, or vn from X-ray spctroscopy using.g. th numrous atomic lins dtctd by X-ray spctrographs, as in th figurs in Scts..3), th volum V occupid by th plasma is immdiatly obtaind: ( ) V = 4π X 3 4(310 ) 10 cm. Thn w can xprss th total X-ray luminosity from fr-fr mission by th plasma as L = Vε [.5] whr th fr-fr missivity is givn by [.9]: X 7 1/ rg.4 10 nt 3 cm s.. [.6] Now th plasma tmpratur T hr is asily masurd from X-ray spctroscopy, ithr looking at th shap of th continuum comparing it with th xponntial cuto of th fr-fr mission q. [.7], or vn mor prcisly by considring th X-ray lin ratios compard with thortical prdictions basd on ionization quilibrium (s 7 8 Sct. 4). Plasma tmpraturs of about T K com out to b typical. In turn, L X is immdiatly obtaind from th bolomtric X-ray flux masurmnt intgrating ovr th clustr ara, togthr with th clustr luminosity distanc d L : 7 1/ 74 rg 4π fx dl = LX = V nt 10 3 cm s Thn this analysis brings us th so far unknown quantity, th plasma particl dnsity n 3 cm. Dnsitis as low as n cm ar indicatd..4

25 Finally, th total X-ray luminosity by optically-thin Brmsstrahlung from a plasma with ths paramtrs can b chckd to b: 1/ 7 n T LX = V rg / sc a hug luminosity in spit of th low particl dnsity, mostly bcaus of th normous volums involvd. Th total baryonic mass in th clustr plasma is 74 n M X = Vn / M 4 10 somwhat largr than th baryonic mass condnsd in stars and galaxis in th clustr cor (s also Sct. 4 blow). or.7. Fr-fr optical dpth Knowing th basic physical paramtrs of th intraclustr plasmas, w can now immdiatly vrify if our assumption about th optical thinnss of th fr-fr mission is corrct. Basd on th xprssion for th absorption coicint in th Rylight-Jans rgim [.16] α 3/ nt Zn g [cm ]., [.30] lt us first considr what happns in th intra-clustr plasmas. W hav: α. 3/ n T Z n [cm ] corrsponding to an optical dpth for fr-fr slf-absorption of [.31] l = dl l [.3] τ α α an normously small numbr! (Not for comparison that th Thomson scattring optical dpth by th sam lctrons is much largr, of th ordr of 1 to 10%, as w will s in Sct. 7)..5

26 W also finally not that thr ar many astrophysical nvironmnts within which th fr-fr dpth can gt much highr. Not th strong dpndnc of α on both th plasma tmpratur, dnsity and th photon frquncy. All ths gt much dirnt valus for xampl in plasmas in HII galactic rgions: much lowr tmpraturs, lowr frquncis and highr dnsitis, such that, in th avrag intrstllar mdium in our Galaxy: 3/ T Z n α. n [cm ] τ. α 10 pc Howvr, in th Orion nbula w hav n 700 particls pr cm, and th optical dpth bcoms unity τ 1 at 1 GHz. [.33].7.3 Astrophysical and cosmological implications of hot intra-clustr plasmas As a mattr of fact, hot intra-clustr plasmas includ an normous amount of dius baryonic mattr in clustrs of galaxis, as discussd in Sct. 4. This maks mor baryons than condnsd in stars and galaxis, and so it constituts a fundamntal componnt of th cosmos. An intrsting aspct to considr about it is th thrmodynamical history of this hug rsrvoir of thrmal nrgy, whn was it brought to such high tmpraturs. An important aspct is thn to considr th cooling tim of ths plasmas. Basd on [.10] w hav, in th optically thin limit which applis hr: t r 1 10 [yrs] >> 1 1/ 11 n T = [sc] t H [.34] 10 about a factor 100 largr than th Hubbl tim ( th yrs ). Th cooling tim may bcom comparabl to th Hubbl tim, or vn slightly lowr, in th innr cor rgions of cool clustrs. This is an important conclusion: th hot plasma kps mmory of all hating procsss happnd sinc th formation of th clustr, and is a kind of intgratd rcord..6

27 Anothr considration is about th dpndnc of th X-ray fr-fr missivity on th plasma dnsity I X n. This maks a dirnt spatial dpndnc than that of galaxis, for which IO ngal n In conclusion, th X-ray surfac brightnss is much mor pakd at th clustr cntr than th galaxy distribution obsrvabl in th optical, and this implis a much asir dtctability of clustr mission in X-rays: X-rays mak a wll-dfind smooth continuum mission whil galaxis ar an nsmbl of point sourcs, and in addition th cntrally pakd mission is asily dtctabl vn in noisy maps..7