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COOLING OF NEUTRON STARS D.G. Yakovlev Ioffe Physical Technical Institute, St.-Petersburg, Russia 1. Formulation of the Cooling Problem 2. Superlfuidity and Heat Capacity 3. Neutrino Emission 4. Cooling Theory versus Observations • • • • Introduction Physical formulation Mathematical formulation Conclusions Ladek Zdroj, February 2008, Cooling theory: Primitive and Complicated at once BASIC PROPERTIES OF NEUTRON STARS Chandra image of the Vela pulsar wind nebula NASA/PSU Pavlov et al M ~ 1.4M SUN , U ~ GM 2 / R ~ 5 1053 erg ~ 0.2 Mc 2 g ~ GM / R 2 ~ 2 1014 cm/s 2 Composed mostly of closely packed neutrons 3M /(4 R3 ) 7 1014 g/cm3 ~ (2 3) 0 0 2.8 1014 g/cm3 standard density of nuclear matter Nb ~ M / mN ~ 1057 = the number of baryons R ~ 10 km OVERALL STRUCTURE OF A NEUTRON STAR Four main layers: 1. Outer crust 2. Inner crust 3. Outer core 4. Inner core The main mystery: 1. Composition of the core+ 2. The pressure of dense matter= The problem of equation of state (EOS) PHYSICAL FORMULATION OF THE COOLING PROBLEM Heat diffusion with neutrino and photon losses Equation of State in Neutron Stars: Main Principles 1. Equation of state (EOS) determines the pressure of the matter, P. 2. The neutron star matter is so dense that P is almost independen t of the temperatu re T and is determined by the mass density and the compositio n of the matter; one usually w rites P P ( ). 3. The mass density is defined as E / c 2 , where E [erg/cc] is the total energy density (including rest - mass energies of particles) . 4. It is commonly assumed that th e neutron star matter is in its lowest (ground, minimum - energy) sta te, which is equivalent to full thermo dynamic equilibriu m with respect t o all reaction channels (involving strong, Coulomb, and weak interactio ns). 5. One usually imposes the condition of electric neutrality of matter elements. 6. It is convenient to introduce the baryon number density nb and calculate E E (nb ); then P nb2 d( E / nb )/d nb . 7. The stiffness of the EOS is described by the adiabatic index d ln P / d ln (so that P ~ ). Mathematical Formulation of the Cooling Problem Equations for building a model of a static spherically symmetric star: { (1) (2) (3) (4) dP Gm Hydrostatic equilibrium: 2 dr r dm HYDROSTATIC STRUCTURE Mass growth: 4 r 2 dr Equation of state: P P( ) dS Thermal balance and transport: Q THERMAL EVOLUTION dt m m( r ) Neutron stars: Hydrostatic equilibrium is decoupled from thermal evolution. Effects of General Relativity : For a neutron star : rg R rg 2GM M 2 . 95 km 2 c M Sun ~ 0.3 one cannot neglect General Relativity Space-Time Metric Variables: t , r , , ds 2 c 2 dt 2 e 2 e 2 dr 2 r 2 d 2 Metric for a spherically -symmetric static star d 2 d 2 sin 2 d 2 (r ), (r ) ? r (r ) (r ) 0 t const, 1 Metric functions Radial coordinate In plane space r const, / 2, ds 2 dl 2 r 2 d 2 0 2 l 2 r Radial coordinate r determines equatorial length – «circumferential radius» dS r 2 sin d d = proper surface element 2 t const, dl e dr , const, const, 0 r r0 r0 l dr e r0 0 Proper distance to the star’s center 3 Periodic signal: dN cycles during dt dN dN Pulsation frequency e in point r d dt dN Frequency detected by a r 0 distant observer dt Determines gravitational redshift of r e (r ) d dt e , r signal frequency Instead of e 2 (r ) it is convenient to introduce a new function m(r): m(r) = gravitational mass inside a sphere with radial coordinate r 1 2Gm 1 2 c r dV 4 r 2 dr 1 2Gm / c r 2 = proper volume element HYDROSTATIC STRUCTURE 1 8G Einstein Equations Rik g ik R 4 Tik 2 c Rik Ricci curvature tensor; R Rii scalar curvature Tik ( P E ) ui uk P g ik energy - momentum tensor ( E c 2 , u i 4 - velocity, g ik metric tensor) Einstein Equations for a Star { dP Gm P 4r 3 P 2Gm 1 2 (1) 2 1 2 1 2 dr r c mc rc dm (2) 4r 2 dr d 1 dP P 1 2 2 dr c dr c (4) P P( ) (3) 1 1 TolmanOppenheimerVolkoff (1939) Outside the Star The stellar surface: r R circumferential star radius at P( R) 0. Gravitational stellar mass: m( R) M . At r R : e 2 e 2 1 rg / r and one comes to the Schwarzschild metric: ds 2 c 2 dt 2 (1 rg / r ) dr 2 / (1 rg / r ) r 2 (d 2 sin 2 d 2 ). Gravitational redshifts of signals from the surface: () 1 rg / R ( R). One often introduces the baryon mass of the star : M b N b mb , N b total number of baryons; mb characteri stic baryon mass. Generally : M b M The difference : M M b M ~ 0.2 M Sun the binding energy. The radius R R / 1 rg / R the apparent radius ( R R). Non-relativistic Limit ( P c 2 ; r 3 P mc2 ; Gm rc 2 ) Gm dP 2 ; r dr dm 4 r 2 ; dr 1 dP d 2 c dr dr 4 G 1 d 2 d r 2 2 c dr r dr (r )c 2 Gravitational potential Equations of Thermal Evolution Thorne (1977) 1. Thermal balance equation: +Qh 2. Thermal transport equation Both equations have to be solved together to determine T(r) and L(r) Boundary conditions and observables At the surface (r=R) T=Ts =local effective surface temperature =redshifted effective surface temperature =local photon luminosity =redshifted photon luminosity HEAT BLANKETING ENVELOPE AND INTERNAL REGION To facilitate simulation one usually subdivides the problems artificially into two parts by analyzing heat transport in the outer heat blanketing envelope and in the interior. The boundary: r Rb , b ~ 109 1011 g/cc (~100 m under the surface) The interior: r Rb , b Exact solution of transport and balance equations The blanketing envelope: Rb r R, b Is considered separately in the static plane-parallel approximation which gives the relation between Ts and Tb Requirements: • Should be thin • No large sources of energy generation and sink • Should serve as a good thermal insulator • Should have short thermal relaxation time THE OVERALL STRUCTURE OF THE BLANKETING ENVELOPE SEMINAR 1 Z=0 T=TS Non-degenerate layer Radiative thermal conductivity Radiative surface T=TF = onset of electron degeneracy Heat flux F Degenerate layer Electron thermal conductivity Heat blanket Atmosphere. Radiation transfer T=Tb b ~ 109 1011 g cm 3 Nearly isothermal interior TS=TS(Tb) ? z ISOTHERMAL INTERIOR AFTER INITIAL THERMAL RELAXATION In t=10-100 years after the neutron star birth its interior becomes isothermal Redshifted internal temperature becomes independent of r Then the equations of thermal evolution greatly simplify and reduce to the equation of global thermal balance: =redishifted total neutrino luminosity, heating power and heat capacity of the star dV 4 r 2 dr 1 2Gm / c r 2 = proper volume element CONCLUSIONS ON THE FORMULATION OF THE COOLING PROBLEM • We deal with incorrect problem of mathematical physics • The cooling depends on too many unknowns • The main cooling regulators: (a) Composition and equation of state of dense matter (b) Neutrino emission mechanisms (c) Heat capacity (d) Thermal conductivity (e) Superfluidity • The main problems: (a) Which physics of dense matter can be tested? (b) In which layers of neutron stars? (c) Which neutron star parameters can be determined? Next lectures REFERENCES N. Glendenning. Compact Stars: Nuclear Physics, Particle Physics, and General Relativity, New York: Springer, 2007. P. Haensel, A.Y. Potekhin, and D.G. Yakovlev. Neutron Stars 1: Equation of State and Structure, New York: Springer, 2007. K.S. Thorne. The relativistic equations of stellar structure and evolution, Astrophys. J. 212, 825, 1977.