1. COOLING OF NEUTRON STARS D.G. Yakovlev Ioffe Physical Technical Institute, St.-Petersburg, Russia Ladek Zdroj, February 2008, 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

2. Cooling theory: Primitive and Complicated at once

3. BASIC PROPERTIES OF NEUTRON STARS Chandra image of the Vela pulsar wind nebula NASA/PSU Pavlov et al Composed mostly of closely packed neutrons

4. 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)

5. Heat diffusion with neutrino and photon losses PHYSICAL FORMULATION OF THE COOLING PROBLEM

6. Equation of State in Neutron Stars: Main Principles

7. Mathematical Formulation of the Cooling Problem Equations for building a model of a static spherically symmetric star: Neutron stars: Hydrostatic equilibrium is decoupled from thermal evolution. HYDROSTATIC STRUCTURE THERMAL EVOLUTION

8. Space-Time Metric Metric for a spherically -symmetric static star Metric functions Radial coordinate In plane space 1 Radial coordinate r determines equatorial length – «circumferential radius» 2 Proper distance to the star’s center

9. 3 Periodic signal: dN cycles during dt Pulsation frequency in point r Frequency detected by a distant observer Determines gravitational redshift of signal frequency Instead of it is convenient to introduce a new function m(r): m(r) = gravitational mass inside a sphere with radial coordinate r = proper volume element

10. HYDROSTATIC STRUCTURE Einstein Equations for a Star Tolman- Oppenheimer- Volkoff (1939) Einstein Equations

11. Outside the Star

12. Non-relativistic Limit Gravitational potential

13. 1. Thermal balance equation: 2. Thermal transport equation Equations of Thermal Evolution +Q h Both equations have to be solved together to determine T(r) and L(r) Thorne (1977)

14. At the surface (r=R) T=T s Boundary conditions and observables =local effective surface temperature =redshifted effective surface temperature =local photon luminosity =redshifted photon luminosity

15. 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. Exact solution of transport and balance equations Is considered separately in the static plane-parallel approximation which gives the relation between T s and T b 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 (~100 m under the surface)

16. Degenerate layer Electron thermal conductivity Non-degenerate layer Radiative thermal conductivity Atmosphere. Radiation transfer THE OVERALL STRUCTURE OF THE BLANKETING ENVELOPE Nearly isothermal interior Radiative surface T=T F = onset of electron degeneracy Heat blanket z Z=0 Heat flux F T=T S T=T b T S =T S (T b ) ?  SEMINAR 1

17. 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 = proper volume element

18. 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

19. 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. REFERENCES