% % Lecture presentation aid for the textbook: % % Laszlo P. Csernai: " Introduction to Relativistic Heavy Ion Collisions" % (John Wiley and Sons Ltd, Chicester, New York, Brisbane, Toronto, % Singapore, 1994; ISBN - 0-471-93420-8) % % Transparencies for Lecture 11 / Chapter 11 % % The following file is containing transparencies in plain LaTeX format. % Figures are usually not included, they can be pasted from the printed % textbook. Empty space is left for this purpose. Some LaTeX figures are % included in the files. 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Csernai: Inroduction to Relativistic Heavy Ion Collisions (Wiley, 1994)} \put(10,235){\parbox[t]{170mm}{\Large\sf \parskip=0.3cm #1}}\end{picture} }%end transparencyframe % \begin{document} \pagestyle{plain} % transparency =============================================================== %\setlength{\unitlength}{1mm} %}%end tr-page %\newpage % transparency ====================================== start p. 1 \transparencyframe{ \setcounter{chapter}{10} \vspace*{-2.3cm} \chapter{Connections of astrophysics and Heavy Ions} \B High energy heavy ion physics and astrophysics are strongly connected. Matter in HI collisions, ---- comparable to matter \\ \B in early universe,\\ \B in neutron or hybrid (neutron and quark) stars, and\\ \B in supernovae. \vspace*{-0.5cm} \section{Neutron and hybrid stars} \B Stars undergo evolution and nuclear burning,\\ -\ \ energy from graviataional collapse,\\ -\ \ when nuclear burning is over\\ -\ \ star collapses further to cold final state\\ -\ \ Final collapse stopped:\\ --\ \ \ \ \ \B by the pressure of the degenerated ($T\approx 0$) fermi gas\\ --\ \ \ \ \ \ \ pressure of the electrons \LT {\bf White Dwarfs}, \\ --\ \ \ \ \ \B by the degenerated fermi gas pressure of neutrons,\\ --\ \ \ \ \ \ \ \LT {\bf Neutron Stars},\\ --\ \ \ \ \ \B if the initial star is extermely massive the collapse\\ --\ \ \ \ \ \ \ is not stopped --- \LT {\bf Black Hole}. }%end tr-page \newpage % transparency ======================================= end of p. 1 \transparencyframe{ \vspace*{-1.2cm} \B The Neutron Stars are massive and dense, \LT\\ -\ \ the curvature of the space-time\\ -\ \ possible \ quark-gluon plasma core. \B Nuclear and astrophysics: connected by Einstein's general relativity,\\ -\ \ \ \ \ \ connects curvature \& energy momentum\\ -\ \ \ \ \ \ via Einstein's equation [2] \beq G_{\mu\nu} = - 8\pi T_{\mu\nu} \ \ \ \left( = - \frac{8\pi G}{c^4} T_{\mu\nu} \right) \ . \eeq{11e1} Here $G_{\mu\nu}= R_{\mu\nu} - {1 \over 2} g_{\mu\nu} R $ is Einstein's curvature tensor,\\ - \ \ \ a function of the space-time metric, $g_{\mu\nu}$,\\ - \ \ \ (it is usually given in terms of the Ricci tensor, $R_{\mu\nu}$,\\ - \ \ \ and the invariant scalar curvature, $R$).\\ $T_{\mu\nu}$ is the matter energy-momentum tensor. \setlength{\baselineskip}{11pt}{\normalsize\sf The gravitational constant, $G = 6.672 \times 10^{-8}$cm$^3$g$^{-1}$s$^{-2}$, and the speed of light, $c = 2.998 \times 10^{10}$cm/s, are the fundamental constants in the theory. The constant $G/c^2 =$ $ 7.423 \times 10^{-29}$cm/g$ = 1.325 \times 10^{-42}$fm\, c$^2$/MeV is also useful since $GM/c^2$ is having the unit of length. If we use so called gravitational units, $G=c=1$, both the time unit and the mass unit will be identical with the unit of length. } }%end tr-page \newpage % transparency ===================================== end of p. 2 \transparencyframe{ \vspace*{-0.8cm} Far from any massive object the space-time is smooth\\ -\ \ \ \LT metric tensor, $g_{\mu\nu} =$ diag$(1,-1,-1,-1)$. In empty space outside a massive static spherical star (of $R$ \& $M$)\\ Schwarzschild found a simple solution of Einstein's equation\\ All but the diagonal components of the metric vanish.\\ The line element for $r>R$ is \beq ds^2 = \left( 1 - {{2M}\over r} \right) dt^2 - \left( 1 - {{2M}\over r} \right)^{-1} dr^2 - r^2 d\theta^2 - r^2 sin^2 \theta d\phi^2 \ . \eeq{11e2} \B\B The metric functions in front of $dt^2$ and $dr^2$\\ -\ \ \ change by an infinitesimal amount\\ -\ \ \ over the distance between nucleons in a star\\ \LT in the local rest frame the matter properties and $T_{\mu\nu}$\\ --\ \ \ \ \ can be calcualted the same way as in the flat space-time.\\ --\ \ \ \ \ (\B\LT EOS from HI reactions is relevant for astrophysics.) \bigskip \B Estimate the mass and radius of a star\ \ \ near the black hole limit.\\ -\ The metric becomes singular at $r = 2M$,\\ -\ this radius is in the interior of the star\\ --\ \ \ \ \ \ \ \ \ \ \ \ \ \ where the Schwarzschild solution is not valid. }%end tr-page \newpage % transparency ======================================= end of p. 3 \transparencyframe{ \vspace*{-0.8cm} \B Estimate the features of a star with radius $R \approx 2 M$ (or larger)\\ --\ \ \ Assume nucleons compressed to hard core limit,\\ --\ \ \ just before phase transition to QGP, \ \ $n \approx 1$/fm$^3$.\\ \LT Then the number of nucleons in the star is $N=nV= n \times 4 \pi R^3 /3$,\\ --\ \ \ the mass is $M = n \ m_N\ 4 \pi R^3 / 3$, ($m_N = 939$MeV)\\ \B Now we estimate the radius of the star, from $R=2M$ we get $$ R = 9.79 \times 10^{18} \ {\rm fm} = 9.79 {\rm km} \ . $$ The number of nucleons in the star is $3.93\times 10^{57}$. The mass of the star: $$ M = {R\over 2} = 4.9 {\rm km} \ , $$ and in terms of the solar mass, $M_\odot = 1.5$ km, as \\ - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ $M=3.26 M_\odot$. We expect a slightly smaller mass and larger radius than the the values given by the Schwarzschild relation.\\ The typical size and mass of of a neutron star are \beq R \approx 10{\rm km}, \ \ \ {\rm and}\ \ \ M=2M_\odot. \eeq{11e3} }%end tr-page \newpage % transparency ========================================= end of p. 4 \transparencyframe{ \vspace*{-1.2cm} \B The density of the nucleons is quite large in the star,\\ -\ so charge$\ne 0$, the Coulomb force would overwhelm gravity. \\ \LT Thus the nucleons are mostly neutrons in such a star. \bigskip \B To calcualte the structure of a star more precisely\\ -\ we need the energy momentum tensor and the equation of state (EOS)\\ -\ Change of the metric is small between two neigbouring nucleons\\ \LT we can neglect gravitation \ \ \ when the EOS is calculated.\\ \LT Thus in the local rest frame $T_{\mu\nu}= $diag$(e,p,p,p)$. \B For a spherical, static star Einstein's equation takes a special form, the Tolman-Oppenheimer-Volkoff equation (TOV): \beqar {{dP}\over{dr}} &=& {{M e} \over {r^2}} ( 1 + {P\over e} ) \left( 1+ \frac{4 \pi r^3 P}{M} \right) \left( 1 - \frac{2M}{r} \right)^{-1} \ , \nonumber \\ {{dM}\over{dr}} &=& 4 \pi r^2 e \ . \eeqar{11e6} \setlength{\baselineskip}{11pt}{\normalsize\sf We can easily interpret these equations:\\ Let us consider a shell of matter in the star of radius $r$ and thickness $dr$. The second equation gives the mass energy in this shell. The pressure of matter exterior to the shell is $P(r)$ and interior to it $P(r) + dP(r)$. The left side of the first equation is the pressure difference or force force acting outward on unit surface of the shell, and the first term on the right hand side is the attractive force of gravity acting on a unit size portion of the shell by the mass interior to it. This term is present in Newton's theory aslo. The remaining three factors are the exact corrections for general relativity. So these equations express the balance of internal pressure and gravity. The equation of state, $p = p(e)$, completes the solvable set of equations, and provides the influence of the matter on the solution. } }%end tr-page \newpage % transparency ======================================= end of p. 5 \transparencyframe{ \vspace*{-0.5cm} \B This set of eqs. can be integrated from the origin:\\ -\ \ with initial condition, $M(r=0) = 0$,\\ -\ \ and arbitrary $e(r=0)=e(0)$, \\ -\ \ until the pressure, $P(r)$, becomes zero. [3] \\ \B That point, $R$, defines the radius of the star,\\ -\ \ and $M(R)$ its mass. \bigskip For the given EOS there is a unique relationship\\ -\ \ \ between the mass and central density, $e(0)$ (Fig. 11.1). \bigskip \LT So for each possible equation of state there is a family of stars,\\ -\ \ \ \ \ \ \ \ \ \ \ \ \ parameterized by the central density. }%end tr-page \newpage % transparency ======================================= end of p. 6 \transparencyframe{ \vspace*{-1.2cm} \vspace*{13.5cm} Figure 11.1 {\it Radial baryon density profiles, $n(r)$, for a pure neutron star and a hybrid star. The upper figure shows the density profile for a neutron star with central density of 2$n_0$, calculated from a so called ``Quadratic EOS'' with different compressibilities indicated. The lower figure displays a hybrid star profile with a central density of 10$n_0$ for the same nuclear EOS as above and and a QGP EOS with a bag constant $B^{1/4} = 165$MeV and $\alpha_s=0.4$. The discontinuities of $n(r)$ at the radii around 6-7.5 km reflect the first order phase transition between the quark core and the outer layer of neutron matter. From ref. [4] } }%end tr-page \newpage % transparency ======================================= end of p. 7 \transparencyframe{ \vspace*{-0.8cm} \B Each family has a maximum mass star, called the LIMITING MASS,\\ -\ \ and the central density, $e(0)$, of the limiting mass star\\ -\ \ is higher the softer the EOS. \B -\B \ The part of the curve for which the slope is POSITIVE\\ --\ \ \ \ \ \ \ \ \ \ \ corresponds to stable config.'s (Fig. 11.2). \B -\B \ For NEGATIVE slope, one can readily verify that the star is unstable to radial perturbations. }%end tr-page \newpage % transparency ======================================= end of p. 8 \transparencyframe{ \vspace*{-1.2cm} \vspace*{17cm} Figure 11.2 {\it The mass of hybrid (quark-neutron) stars as a function of the central density for different nuclear and QGP equations of state. Note that positive slope is required for stability against gravitational collapse. From ref. [4] } }%end tr-page \newpage % transparency ======================================= end of p. 9 \transparencyframe{ \vspace*{-0.8cm} \B If the EOS includes a phase transition to QGP\\ \LT \ \ \ \ the star has a special structure. The high density center is in the QGP phase,\ \ \ \ \ then moving outwards,\\ .\ \ \ \ \ \ we reach the phase transition density.\\ At this density the QGP and hadronic pressures are equal in eq., so\\ .\ \ \ \ \ \ we have the phase transition and\\ the DENSITY DROPS suddenly to the equilibrium hadronic density. (Fig. 11.1). \B This is a sharp density discontinuity at some radius like at the surface of the see on the Earth. \B There are a large number of attempts to study the structure of such hybrid stars. [1-4,5,7] The hybrid stars are smaller, but they have approximately similar masses as the neutron stars. \B The stars with mass beyond the maximum are unstable to collapse to black holes. (Fig. 11.2). It is in the limiting mass that a constraint on the EOS arises. Obviously an acceptable EOS must have a limiting mass at least as large as the largest observed mass. The masses of observed neutron stars are between $M = 1-2.5 M_\odot$. \B The stability of stars requires that the mass and the radius of the star should increase with increaseing central density. Furthermore the mass of the star should increase with increasing star radius. This latter requirement makes pure quark stars unstable against collapse to black holes according to most model calculations. }%end tr-page \newpage % transparency ======================================= end of p. 10 \transparencyframe{ \vspace*{-1.2cm} \subsection{Pulsars and neutron stars} \B Discovery of pulsars: 1967 \B By now about 400 pulsars have been observed. \B Period: milliseconds to seconds, \LT period of rotation Ordinary stars have magnetic fields and rotate.\LT \\ Collapse: $R \approx$ 10$^6$km $\longrightarrow \ 10$ km,\LT \\ Rotation frequency and field are scaled up\\ \ (conservation of angular momentum and magnetic flux.) \bigskip \B \LT rotation frequency observed in the pulsars. }%end tr-page \newpage % transparency ======================================= end of p. 11 \transparencyframe{ \vspace*{-1.2cm} \subsection{Supernova explosions} \B Maximum mass of the neutron and hybrid stars (NS): $1-2.5M_\odot$,\\ 1) massive stars (smooth burning, collapse) will not form a NS\\ \ -- above 5-20$M_\odot$ or larger, collapse into black holes\\ 2) smaller stars end up as White Dwarfs\\ \LT \B Accepted way to produce a NS: the supernova explosion\\ - the outer crust of the star is blown away by an explosion\\ - the internal core collapses to a NS \B Cause of such an explosion:\\ - a) shock wave propagating outwards\\ - -\ \ after a collapse of the core to a dense degenerated state,\\ - b) intense neutrino flux arising from the collapse\\ - -\ when the protons are converted into neutrons. \B Any constraint on the nuclear EOS at present is uncertain from SNs. \bigskip \setlength{\baselineskip}{11pt}{\normalsize\sf Only a small part of supernova remnants contains a neutron star and very little observational information on the explosion mechanism is available even in the best studied case (SN 1987A). We know that some supernovas leave neutron stars behind and that the kinetic energy of the ejecta is typically around 10$^{51}$erg. It is possible that only those stars can explode by the prompt shock wave mechanism which possess a significant amount of angular momentum initially. Nevertheless, there is little doubt that soft (supra-nuclear) equations of state favor prompt explosions, at least for non-rotating models. } }%end tr-page \newpage % transparency ======================================= end of p. 12 \transparencyframe{ \vspace*{-1.2cm} \section{Implications on the early universe} Einstein's eq. (\ref{11e1}) \LT \\ no static solution for the whole homogeneous and isotropic universe. \setlength{\baselineskip}{11pt}{\normalsize\sf Initially this was thought to be a problem of the theory and Einstein did introduce a small extra repulsive term the so called {\em cosmological constant} to the Einstein equation, $\Lambda \ g_{\mu\nu}$, added to the energy momentum tensor. } \setlength{\baselineskip}{18pt}{\Large\sf \B Later Friedmann: time dependent solution --- agrees with observations\\ \LT Universe: "Big bang" a singularity at time zero,\\ - then it is expanding.\\ For a closed universe the metric of the space time is \beq ds^2 = dt^2 - R^2(t) \left[ \frac{ dr^2 + r^2 ( d\Theta^2 + \sin^2\Theta d\phi^2) } { 1+ r^2/4} \right] . \eeq{11e7} Curvature of universe, $R(t)$: from Einstein eq. + EOS HI physics \LT experimantal information on the EOS\\ This EOS influences the early development of the universe.\\ hase transitions \LT to rapid expansion, to the {\em inflation}. HI reactions: relevant in the first few seconds of the universe. (see Table \ref{11e1}, [1MeV = 1.16045~$\times 10^{10} \ ^o$K if $k_B=1$]). } }%end tr-page \newpage % transparency ======================================= end of p. 13 \transparencyframe{ \vspace*{-0.6cm} \begin{center} \begin{tabular}{|l|r|r|r|r|} \hline\hline time &Temperature&Temperature& \\ & [$^o$K] & [MeV] & \\ \hline 10$^{-6}$ s & 10$^{15}$ & 10$^5$ & QGP \\ 10$^{-4}$ s & 10$^{12}$ & 100 & QGP hadronization \\ 1 - 2 s & 10$^{10}$ & 1 & $e\bar{e}$ annihilation \\ 1.5 min & 10$^9$ & 0.1 & $d$ formation \\ 10$^5$years & 10$^3$ & 0.1 eV & background radiation \\ 10$^{10}$years& 3 & 10$^{-3}$ eV & Today\\ \hline\hline \end{tabular} \end{center} Table 11.1 {\it Time and temperature scale of the early universe} \bigskip \B Thus the hadronization of QGP: between a few seconds \& a minute \B Entropy per baryon charge $\approx$ 10$^9$-10$^{10}$, \LT\\ the mid rapidity region of highest energy HI collisions\\ at the CERN - LHC or the BNL - RHIC, are relevant. \B Possible signatures of the quark-hadron transition:\\ - STRANGELETS as candidates for the dark matter\\ - consequences in cosmological nucleosynthesis [8] \setlength{\baselineskip}{11pt}{\normalsize\sf At the time of the QGP hadronization at $\approx 10^{-6} - 10^{-4}$ s, the horizon mass was less than a solar mass, so fluctuations arising from the hadronization of QGP are unlikely to affect directly anything larger than a solar mass size system. Consequently the galaxy cluster structure or the galaxy structure is not likely to be connected with the QGP phase transition [8]. } }%end tr-page \newpage % transparency ======================================= end p. 14 \transparencyframe{ \vspace*{-1.2cm} \subsection{Strangelets} \B In QGP hadronozation: strange quark nuggets / strangelets may be formed\\ - if in QGP not only $u$ and $d$, but also $s$ quarks are present\LT\\ - - the Pauli exclusion effects will be decreased in dense matter\LT\\ - - high density of strange quarks may be present in a final state\\ - - - if the Fermi level is above the strange quark mass. \B In QGP hadronization: s and $\bar{s}$ may be separated [9]\\ - may happen in the early universe \& in HI collisions\\ - total strangeness charge is zero for the total system,\\ - - but it is not necessarily so in both phases separately. \B If the system has a net (positive) baryon charge\\ - strangeness and anti-strangeness will be separated in rehadronization\\ - \LT strangeness will be retained in the QGP\\ - \& hadronic phase will carry more anti-strange quarks. \B Assume that the reaction rate is sufficiently fast \\ - to maintain thermal, mech., and chem. eq. during ph-tr. \LT\\ - - the system follows the Maxwell construction closely, and\\ - - the extra entropy production is small.\\ \LT The thermodynamical intensives should be equal during the ph.-tr.: $T_H=T_{QGP}$, $P_H = P_{QGP}$, $\mu^{(B)}_H = \mu^{(B)}_{QGP}$ and $\mu^{(s)}_H = \mu^{(s)}_{QGP}$. \B The last condition indicates that\\ - not the strangeness density but\\ - the corresponding chemical potential is continuous in the transition. }%end tr-page \newpage % transparency ======================================= end p. 15 \transparencyframe{ \vspace*{-0.5cm} Given the total energy, volume, strangeness, and baryon charge for the combined system of the two phases the intensives can be determined. It is important to emphasize that even if the total strangeness is zero, the strange chemical potential and the net strangeness in the two phases will not be zero in phase equilibrium. \B For vanishing total strangeness\\ in pure QGP phase: the strangeness chemical potential is zero, but\\ in pure hadronic phase: it is not, if our system has a net baryon charge. In ref. [9] the development of strangeness abundance was calculated in the Bjorken model assuming phase equilibrium and adiabatic expansion:\\ The ratio of net strangeness density versus baryon density in the plasma was increasing during the collision and it reached 0.4-0.5 by 4-5fm/c time. The excess strangeness in the remaining plasma droplets will then contribute to massive strange objects, which are larger than conventional strange hadrons, i.e. to massive strange objects, strangelets or strange nuggets. \B Some suggested observable candidates are: multistrange hadrons with baryon charge B$\ge$2, the H-dibaryon ($uuddss$), or multistrange hypernuclei [10]. Even if the reaction rate is not sufficient to establish chemical equilibrium exactly at each stage of the hadronization [11,12], the hadronization will tend to develop a strangeness, anti-strangeness separation, although (usually) somewhat smaller than in a chemical equilibrium process. The END }%end tr-page % transparency ================================================ end p. 16 \end{document}