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Dynamics of Massive Shells Ejected in a Supernova Explosion

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a r X i v :a s t r o -p h /9404055v 1 22 A p r 1994Alberta-Thy-07-94Dynamics of Massive Shell ejected in a Supernova ExplosionDar´ıo N´u ˜n ez ∗,H.P.de Oliveira †Theoretical Physics Institute,University of Alberta,Edmonton,Alberta T6G 2J1,Canada.e-mail nunez@phys.ualberta.ca,oliveira@phys.ualberta.ca ∗Instituto de Ciencias Nucleares,UNAM,CU,A.P.70-543,M´e xico,D.F.04510,M´e xico.e-mail nunez@roxanne.nuclecu.unam.mx†Universidade do Estado do Rio de Janeiro,Instituto de Fisica,R.Sao Francisco Xavier,524,Maracana,CEP 20550,Rio de Janeiro.AbstractAn expanding shell is accelerated outward by radiation from the remnant star and slowed by ram pressure and accretion as it plows into the interstellar medium.We set up the general relativistic equations of motion for such a shell.In the non-relativistic limit,the reduce to the standard Ostriker-Gunn equations.Furthermore,it is shown that the motion equation can be integrated in general,reducing in this way the study of the dynamics of a shell to a set of four coupled first order differential equations,solvable up to quadrature.Subject headings:Shells,dynamics,explosion.PACS number:97.60LTypeset using REVT E X11.INTRODUCTIONThe study of the dynamics of a supernova remnant,a plerion,has been greatly developed since the works of Ostriker and Gunn(1971),where they modeled the dynamics of this remnant as a thin shell moving in a background which inside has a radiating mass and outside is dust.Supposing spherical symmetry,they started with the Newton’s second law for describing the acceleration of the shell.The force terms were introduced essentially by hand.This is the equation of motion:M s d2R2M S)M Sd t)2,a friction term known as the”snow plow”effect(Oort1946).Thefirst term represents the gravitational and self-gravitational force,the second a driving force due to the radiation and the third a decelerating one due to the interaction with the medium.The dynamical study is completed with an equation for the change of the mass of the shell(usually due only to the dust that the shell collects while moving in the medium)and finally an equation for the change of the shell’s internal energy,proportional to the radiation emitted by the star.This system of equations has been analysed in many ways.Chevalier(1977)and Ostriker &Mc Kee(1988)have analysed the kinetic equation of the shell to show the different stages of the evolution when each of the terms in turn becomes dominant,while Sato(1988a)and Sato&Yamada(1991)have applied this set to the study of the SN1987A in particular. Most of the studies use numerical methods to obtain the change in the luminosity or in the mass with respect to the change in the radius of the remnant.2.SHELLS IN GENERAL RELATIVITYOn the other hand,the study of the dynamics of a shell separating two backgrounds in the context of General Relativity has developed in a powerful and direct formalism since the firsts works of Werner Israel(1966),and has been applied to cosmology,mainly to inflation (Berezin1987),and to modeling the dynamics of the border between two regions in different states,like bubbles in water,or between two given spaces(Sato1988b).Nevertheless,it has not been widely used in the study of the dynamics of the plerion.In the context of General Relativity the equation of motion for the shell is obtained from Einstein’s equations and from the junction conditions for the metric tensor and itsfirst derivative.We will work in spherically symmetric spaces,described byds2=−e2ψf dv2−2b eψdr dv+r2dΩ2,(2) where f andψare in general functions of the null coordinate v and of the radius r,and b=±1depending whether the null coordinate is advanced or retarded.The kinetic equation for the shell in this case are the following,with our conventions following MTW(Misner, Thorne,&Wheeler1973),nαδuαR2+8πp,(3)σ(nαδuαδτ|−)=2{−[Tαβnαnβ]+p(F++F−)where nαis the4-vector normal to the shell,uαis the4-velocity vector,τis the proper time,M=4πR2σis the proper mass of the shell,σbeing the surface energy density of the shell,and R is the radius.A function in square brackets stands for the difference of that function in the region outside,denoted by+,with the region inside the shell,denoted by −,and evaluated on the shell.Finally p is the pressure of the shell,and F±is given byF±=f eψ.(8) The motion equation(3)is then rewritten as[¨R+BR2+8πp,(9)whereB=f,r2f,v eψ˙v2.(10)Now,from the definition of F,equation(5),we see that˙R¨R2F,(11)3so,after some manipulations,the motion equation(9)can be expressed asdR )−˙M2F]−8π˙R p=0.(12)The equation for the rate of change of the mass,equation(6),in the spherical case can be expressed as˙M2+ψ,r F˙R]−8π˙R p,(13) Substituting this last equation in the motion equation(12),we obtain thatdR)+[2B˙R−˙f2f,v eψ˙v2−ψ,r F˙R]=0.(14)Finally,using the formula for˙v,equation(8),can be shown that the expression inside the squared brackets is zero!,sodR)=0,(15) which implies that[F]+MR =C in the second motion equation(4).In this way we have shown that equation(16)is a first integral of the general motion equation of a thin shell with spherical symmetry.It proves convenient to introduce a function m=m(u,r)such thatf±=1−2m±R,(18)where the function m is defined as m=m+−m−,and in some cases can be identified with the gravitational bining thefirst integral equation(16),with equation(18),can2r,where m1,2and e1,2 stand for the gravitational masses and electrical charges in both sides of the shell.They defined m=m2−m1=constant as the gravitational mass of the shell and identified this quantity as the conserved total energy of the shell.4be obtain an expression for F on either side of the shell in terms of the proper mass and the gravitational massF±=m2R.(19)The squared of this last equation,recalling equation(17),results in˙R2=(mR +M22M˙R2−M(m−+MR.(21)The above equation simply states that the sum of the rest energy,the kinetic energy,the mutual potential energy and the gravitational self-energy of the shell,respectively in the rhs,is not constant,but varies according with the quantity m.In this way,the dynamics of a shell moving between any two backgrounds,with spherical symmetry reduces to a set offirst order equations,namely,the radial motion equation(20), the balance equation for the proper mass,equation(13),the motion equation for the null coordinate,equation(8),a state equation for the shell,giving a relation between the pressure p and the energy density,σ,and for specifying the kind of backgrounds between which the shell moves,we have the Einstein’s equations,which for this spherical case are(Barrabes& Israel1991)m,v=4πr2T v r;m,r=−4πr2T v v;ψ,r=4πr T rr,(22) where Tµνis the stress energy tensor,specified by the distribution of matter and energy in the chosen background space.Despite the above characteristic concerning the dynamics of massive shells,it will be worth while to write down conveniently the second order motion equation in order to obtain the Newtonian limit,and show how equation(1)is recovered.After a direct calculation starting from equation(20),we obtain:M¨R=−M(m−+MR2+˙m−M˙R−˙M F−F+Note that thefirst term in the rhs is nothing else than the relativistic generalization of the gravitational interaction,and the remaining terms are related with the variation of m−, m and M.In fact,we have obtained the relativistic version of the second order equation analysed by Ostriker and Gunn(1971)in studying the dynamics of remnants of supernovas modelled by thin spherical shells.To show in a clear way the role played by the three last terms of the above equation,let us consider that the shell separates two Vaidya spacetimes: the interiorfilled with outgoing radiation and the exterior with ingoing radiation.Thus, after taking the Newtonian limit of equation(23),we arrive to the following expression:M¨R∼=−M(m−+MR2+8πR p+4πR2(q−−q+)(24)where q−and q+are the energy density of the radiation at each side of the shell and p is the pressure of the shell.Therefore,we have recovered,in this simple model,the terms that describe the effect of the pressure of radiation inside the cavity and the snow plow effect (compare the last term of this equation with the correspondent of the Ostriker’s equation). Also,we have an extra term related to the pressure of the shell,since Ostriker has apparently considered pressure-less shell.However,a more realistic model is realized in considering the exterior spacetime characterized by dust,like in the Friedman-Walker universe,in order to taken into account more properly the interstellar medium outside the shell.We want to stress the fact that the second order equation,equation(23)obtained from thefirst order one,equation(20)is completely equivalent to the two second order equations obtained from the Einstein’s equations,equations(3,4)but has a more tractable form and the different terms are easier to identify.Finally,if instead of the proper time of the shell,τ,is used the time of an exterior or an interior observer,v+,v−,and then consider the radius of the shell as a function of v, R=R(v),so that˙R=R,v˙v,after some manipulations can be shown that the motion equation for the radius of the shell is given byR,v±=eψ±A[b A+η(m g2R)],(25)withη=±,for the two possible solutions to the algebraic equation and withA2=(m gR+M2REFERENCESBarrab`e s,C.,&Israel,W.1991,Phys.Rev D,43,1129Berezin,V.A.,Kuzmin,V.A.,&Tkachev,I.I.1987,Phys.Rev.D,36,2919Chase,J.E,1972,Ph.D thesis,University of Alberta,Canada.Chevalier,R.A.1977,Astroph.and Space Science Lib.,66,proceedings(Supernovae,ed. Schramm D,Reidel Pub.Co.)De la Cruz,V.,&Israel,W.1967,Il Nuovo Cimento,LI A,3,744Israel,W.1966,Il Nuovo Cimento,44B,1——.1967,48B,463Lake,K.1979,Phys.Rev.D,19,2847Misner,C.W.,Thorne,K.S.,&Wheeler,J.A.1973,Gravitation,(S.Fco.W.H.Freemann and Co.)Oort,J.H.1946,MNRA,106,159Ostriker,J.P.,&Gunn,J.E.1971,ApJ,164,L95Ostriker,J.P.,&McKee,C.F.1988,Rev.Mod.Phys.,60,1Sato,H.1988a,Prog.of Theo.Phys.,80,96Sato,H.1988b,Prog.of Theo.Phys.,80,96Sato,H.,&Yamada,Y.1991,Prog.of Theo.Phys.,85,541N´u˜n ez,D.,&de Oliveira,H.P.,in preparation7。

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L.-S. The 1

L.-S. The 1


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L.-S. The et al.: CGRO/OSபைடு நூலகம்E observations of the Cassiopeia A SNR
the extrapolated ux of OSSE measurement in the same range of energy is 7.3 10?11 ergs cm?2 s?1 . The power-law spectrum proposed by Asvarov et al. (1989, Eq.(2) in The et al. 1995) tted to OSSE data between 0.04{0.25 MeV does not produce a good t ( 2 = 1.183 for D.O.F.= 488) shown in Figure 1 as a longdashed line. If we allow the overall intensity parameter to vary the result improves to 2 = 1.138 with intensity of (10.7 2.4) 10?4 cm?2 s?1 MeV?1 at 0.1 MeV. If we relax both the power-law index and the intensity in 2. Observations the t, we obtain the t shown in Figure 1 as a solid line After the rst OSSE Cas A observation (VP34) reported whose parameters are shown in Table 2. by The et al. (1995), we obtained three more observations. These OSSE pointing strategies were very similar to VP34. The duration and number of detectors in each viewing period are listed in Table 1. The OSSE spectral analysis technique subtracts background measured in o set pointings of the detectors from the source pointings (Johnson et al. 1993). The quadratically interpolated background estimates are subtracted from each source spectrum to obtain a di erence spectrum for each 2-minute source pointing. The average spectrum of each of the four OSSE detectors is obtained for each viewing period. To t the data, a single photon model is convolved with an instrument response function for each detector for each observation, and each resulting count rate model is compared with each observed count rate spectrum. A detailed description of OSSE performance and spectral data analysis procedures can be found in Johnson et al. (1993).
Linear sigma model at finite temperature

Linear sigma model at finite temperature


2 is the order of this transition, is it first–order with latent heat, or second–order, or maybe a crossover between the A. The potential at one–loop 27 phases? Lattice calculations suggest that when we consider two massless quarks, the transition is second order B. One loop mass correction 27 and the same suggest other approaches based on effective models. If we consider three massless flavours of quarks, C. The Keldysh contour 28 the transition is probably first–order [3, 6]. The aim of this work is to study the chiral phase transition. D. The pion self–energy graphs 28 1. The pion self–energy graph with one thermal pion28 Chiral symmetry breaking is a necessary ingredient for low energy hadron physics, since unbroken chiral symmetry results in massless baryons, without parity doubled 2. The pion
On the equilibrium of a charged massive particle in the field of a Reissner-Nordstrom black

On the equilibrium of a charged massive particle in the field of a Reissner-Nordstrom black

a rXiv:g r-qc/68139v 131Aug26On the equilibrium of a charged massive particle in the field of a Reissner-Nordstr¨o m black hole D.Bini a ,b ,A.Geralico b ,R.Ruffini b ,c a Istituto per le Applicazioni del Calcolo “M.Picone”,CNR I-00161Rome,Italy b International Center for Relativistic Astrophysics -I.C.R.A.University of Rome “La Sapienza”,I-00185Rome,Italy c Physics Department,University of Rome “La Sapienza”,I-00185Rome,ItalyThe study of a massive charged particle in equilibrium in a Reissner-Nordstr¨o mblack hole background presents a variety of conceptual issues still widely open after more than twenty years of research,ranging from the classical aspects of general relativity to the quantum aspects of black hole tunneling processes (see e.g.Parikh and Wilczek[1]).The problem of the interaction of a charged particle,neglecting its mass con-tribution,with a Reissner-Nordstr¨o m black hole was addressed by Leaute and Linet[2].They extended previous results obtained in the case of a Schwarzschild spacetime by Hanni[3],Cohen and Wald[4],Hanni and Ruffini[5]and Linet [6]himself.Their study was done in the testfield approximation neglecting the backreaction both of the mass and of the charge of the particle on the background electromagnetic and gravitationalfields.We here approach the complete problem of a massive charged particle of mass m and charge q at rest in thefield of a Reissner-Nordstr¨o m black hole with mass M and charge Q.The full Einstein-Maxwell system of equations are solved taking into account the backreaction on the backgroundfields due to the presence of the charged massive particle.The source terms of the Einstein equations contain the energy-momentum tensor associated with the particle’s mass,the electromagnetic energy-momentum tensor associated with the back-groundfield as well as additional interaction terms,first order in m and q.Such terms are proportional to the product of the square of the charge Q of the background geometry and the mass m of the particle(∼Q2m)and to the product of the charges of both the particle and the black hole(∼qQ).These terms originate from the“electromagnetically induced gravitational pertur-bation”[7].On the other hand,the source terms of the Maxwell equations contain the electromagnetic current associated with the particle’s charge as well as interaction terms proportional to the product of the black hole’s chargeQ and the mass m of the particle(∼Qm),originating the“gravitationally induced electromagnetic perturbation”[8].We summarize here the main results based on thefirst order perturbation approach formulated by Zerilli[9]using the tensor harmonic expansion of both the gravitational and electromagneticfields.Details will be found in[10].The Reissner-Nordstr¨o m black hole metric is given byds2=−f(r)dt2+f(r)−1dr2+r2(dθ2+sin2θdφ2),f(r)=1−2Mr2,(1)with associated electromagneticfieldF=−QM2−Q2=M±Γ.We consider the case|Q|≤M and the region r>r+outside the outer horizon.The“extreme”charged hole corresponds to|Q|=M.The particle is assumed to be at rest at the point r=b on the polar axis θ=0.The only nonvanishing components of the stress-energy tensor and of the current density are given byT part 00=m2πb2δ(r−b)δ(cosθ−1),(3)and the combined Einstein-Maxwell equations are thus˜Gµν=8π T partµν+˜T emµν ,˜Fµν;ν=4πJµpart,∗˜Fαβ;β=0.(4)The quantities denoted by a tilde refer to the total electromagnetic and grav-itationalfields,tofirst order of the perturbation˜gµν=gµν+hµν,˜Fµν=Fµν+fµν,˜T em µν=14˜gµν˜Fρσ˜Fρσ ,˜Gµν=˜Rµν−10=e2ν 2K′′−2r K′−4 1r W −2λeνr4−4Qeν˜f01rW′− ν′+2r2(W−K)−2Q2e−νW r2, 0=K′′+ ν′+2r W′+ ν′′+ν′2+2ν′r4+4Qe−νr2,0=˜f01′+2r2−Q2(l−1)(l+2),H0=H2≡W and eν=f(r)is Zerilli’s notation;a prime denotes differentiation with respect to r.The quantitiesA00=8√2l+12√2l+1M b−Q2,(8)involving the black hole and particle parameters as well as their separation distance b.This condition coincides with the equilibrium condition for a test particle of mass m and charge q in thefield of a Reissner-Nordstr¨o m black hole given by Bonnor[12].There he simply considered the classical expressionfor the equation of motion of the particlemUα∇αUβ=qFβµUµ,(9) with4-velocity U=f(r)−1/2∂t,neglecting all the feedback terms,and obtained Eq.(8)as the equilibrium condition.The coincidence of these results is quite surprising,since our gravitational and electromagneticfields including all the feedback terms are quite different from those used by Bonnor.If the black hole is“extreme”,then from Eq.(8)follows that also the particle must have q/m=1,and equilibrium exists independent of the separation. In the general non-extreme case Q/M<1there is instead only one position of the particle which corresponds to equilibrium,for any given value of the charge-to-mass ratios of the bodies.In this case the particle charge-to-mass ratio must satisfy the condition q/m>1.We now give the general expression for both the perturbed gravitational and electromagneticfields in closed analytic form by summing over all multipoles of the Zerilli expansion[10].The perturbed metric is given byd˜s2=−[1−¯H]f(r)dt2+[1+¯H][f(r)−1dr2+r2(dθ2+sin2θdφ2)],(10) where¯H=2m¯D,(11) with¯D=[(r−M)2+(b−M)2−2(r−M)(b−M)cosθ−Γ2sin2θ]1/2.(12) Note that in the extreme case Q/M=q/m=1this solution reduces to the linearized form of the well known exact solution by Majumdar and Papapetrou [13,14]for two extreme Reissner-Nordstr¨o m black holes.The asymptotic massmeasured at large distances by the Schwarzschild-like behaviour of the metric of the whole system consisting of black hole and particle is given byM eff=M+m+E int,(13) where the interaction energy turns out to beE int=−m 1− 1−Mr3M r−Q2¯D− M(b−M)+Γ2cosθ+[(r−M)(b−M)−Γ2cosθ]Q2¯D2[(r−M)−(b−M)cosθ], Eθ=q M r−Q2¯D3sinθ.(15) The total electromagneticfield tofirst order of the perturbation is then˜F=− Q2 1−r M b−Q21−Mbr (r−M)(b−M)−Γ2cosθbr.(18)The second and third terms in the bracketed expression of(17)represent the“gravitationally induced”and“electromagnetically induced”electrostatic potential respectively and the equilibrium condition(8)has been conveniently used.The Zerilli’s procedure of expansion of both the gravitational and electro-magneticfields in tensor harmonics is largely used in the literature to study linear perturbations of spherically symmetric spacetimes due to some exter-nal source.We have given here the analytic solution for a problem which has raised much interest and discussions for many years.We have obtained closed form expressions for both perturbed metric and electromagneticfield due to a charged massive particle at rest in thefield of a Reissner-Nordstr¨o m black hole,taking advantage of the static character of the perturbation as well as of the axial symmetry of the configuration.The infinite sum of multipoles converges to an analytic form.In addition to its theoretical significance,this result can become an important tool in testing the validity of numerical investigations addressing the dynamics of many body solutions in Einstein-Maxwell systems.References[1]M.K.Parikh,F.Wilczek,Phys.Rev.Lett.85(2000)5042[2] B.Leaute,B.Linet,Phys.Lett.58A(1976)5[3]R.Hanni,Junior Paper submitted to the Physics Department of PrincetonUniversity,1970(unpublished)[4]J.Cohen,R.Wald,J.Math.Phys.12(1971)1845[5]R.Hanni,R.Ruffini,Phys.Rev.D8(1973)3259[6] B.Linet,J.Phys.A:Math.Gen.9(1976)7[7]M.Johnston,R.Ruffini,F.J.Zerilli,Phys.Lett.B49(1974)185[8]M.Johnston,R.Ruffini,F.J.Zerilli,Phys.Rev.Lett.31(1973)1317[9] F.J.Zerilli,Phys.Rev.D9(1974)860[10]D.Bini,A.Geralico,R.Ruffini,in preparation[11]T.Regge,J.A.Wheeler,Phys.Rev.108(1957)1063[12]W.B.Bonnor,Class.Quant.Grav.10(1993)2077[13]S.M.Majumdar,Phys.Rev.72(1947)390[14]A.Papapetrou,Proc.R.Irish Acad.51(1947)191。

Coherent Nuclear Radiation

Coherent Nuclear Radiation

3 2 1
Abstract The main part of this review is devoted to the comprehensive description of coherent radiation by nuclear spins. The theory of nuclear spin superradiance is developed and the experimental observations of this phenomenon are considered. The intriguing problem of how coherence develops from initially incoherent quantum fluctuations is analysed. All main types of coherent radiation by nuclear spins are discussed, which are: free nuclear induction, collective induction, maser generation, pure superradiance, triggered superradiance, pulsing superradiance, punctuated superradiance, and induced emission. The influence of electron-nuclear hyperfine interactions and the role of magnetic anisotropy are studied. Conditions for realizing spin superradiance by magnetic molecules are investigated. The possibility of nuclear matter lasing, accompanied by pion or dibaryon radiation, is briefly touched.