Evolution of Neutron Stars and Observational Constraints

The structure and evolution of neutron stars is discussed with a view towards constraining the properties of high density matter through observations. The structure of neutron stars is illuminated through the use of several analytical solutions of Einstein’s equations which, together with the maximally compact equation of state, establish extreme limits for neutron stars and approximations for binding energies, moments of inertia and crustal properties as a function of compactness. The role of the nuclear symmetry energy is highlighted and constraints from laboratory experiments such as nuclear masses and heavy ion collisions are presented. Observed neutron star masses and radius limits from several techniques, such as thermal emissions, X-ray bursts, gammaray flares, pulsar spins and glitches, spin-orbit coupling in binary pulsars, and neutron star cooling, are discussed. The lectures conclude with a discusson of proto-neutron stars and their neutrino signatures. The participation at this summer school was partially supported by the HISS Dubna program of the Helmholtz association and by the US Department of Energy Grant DE-FG02-ER40317. References 1. Lattimer, J.M. & Prakash, M., Science 304 (2004) 536-542 2. Lattimer, J.M. & Prakash, M., Phy. Rep. 442 (2007) 109-165 3. Steiner, A.W., Prakash, M., Lattimer, J.M. & Ellis, P., Phy. Rep. 411 (2005) 325-375 © Owned by the authors, published by EDP Sciences, 2010 DOI:10.1051/epjconf/20100703001 EPJ Web of Conferences 7, 03001 (2010) This is an Open Access article distributed under the terms of the Creative Commons Attribution-Noncommercial License 3.0, which permits unrestricted use, distribution, and reproduction in any noncommercial medium, provided the original work is properly cited. Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20100703001 Neutron Star Structure James M. Lattimer lattimer@astro.sunysb.edu Department of Physics & Astronomy Stony Brook University J.M. Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.1/64 Credit: Dany Page, UNAM J.M. Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 – p.2/64 EPJ Web of Conferences


Maximum Possible Density in Stars
The scaling from the maximally compact EOS yields A virtually identical result arises from combining the maximum compactness constraint (R min 2.9GM/c 2 ) with the Tolman VII relation Newtonian Roche model for rotation Numerical calculations show R p is nearly constant for arbitrary rotation Evaluate at equator: 51 3 sin(θ/3) .Consistent with 0 < K /MeV < 300 Dimensional analysis: General Relativistic analysis using Buchdahl's solution:  The Pressure of Neutron Star Matter Expansion of cold nucleonic matter energy near n s and isospin symmetry x = 1/2: Beta Equilibrium:

Nuclear Symmetry Energy
The density dependence of E sym (n) is crucial.Some information is available from nuclei (for n < n s ).Heavy ion collisions have potential for constraining it for n > n s .
It is common to expand E sym (n) as N s is the number of excess neutrons associated with the surface, is the asymmetry of the nuclear bulk Uuid, and μ n is the neutron chemical potential.
From thermodynamics,

Nuclear Structure Considerations
Information about E sym can be extracted from nuclear binding energies and models for nuclei.For example, consider the schematic liquid droplet model (Myers & Swiatecki): Fitting binding energies results in a strong correlation between S v and S s , but not deTnite values.

Schematic Dependence
Nuclear Hamiltonian: Lagrangian minimization of energy with respect to n (symmetric matter): Liquid Droplet surface parameters: ) , 2.0 (p = 1)  Beta equilibrium composition: Radiation Radius: Globular Cluster Sources The Neutron Star Crust Hydrostatic equilibrium in the crust: Crustal moment of inertia

Analysis-Samuelsson & Andersson
Include crust elasticity in a relativistic context of axial oscillations; calculate both fundamental and overtone frequencies using the Cowling approximation: neglect dynamical nature of spacetime (i.e., perturbations of gravitational Teld).Ignores magnetic Telds and superUuidity and is thus insensitive to core structure.

Pulsar Glitches
Pulsars occasionally undergo glitches, when the spin rate "hiccups".Each glitch changes the angular momentum of the star by ΔJ = I liquid ΔΩ.
The glitches are stochastic, but total angular momentum transfer is regular.
A leading model is that as the crust slows due to pulsar's dipole radiation, the interior acquires an excess differential rotation.The acquired excess is limited: J ≤ ΩI crust .Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 -p.56/64 Proto-Neutron Star Evolution In the diffusion approximation, Uuxes are driven by density gradients: λ ν and λ i E 's are mean free paths for number and energy transport, respectively.n ν (E ν ) and i (E ν ) are the number and energy density of species i = e, μ at neutrino energy E ν .J.M. Lattimer, Helmholtz International Summer School, Dubna, 18 July 2008 -p.57/64 EPJ Web of Conferences

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We can combine with the Trst law of thermodynamics to obtain the rate of change of the total lepton number and the entropy: There are two main sources of opacity: 1. ν-nucleon absorption.Affects only e−types.
Mean free paths for these processes are approximately: Diffusion approximation Number transport dominated by degenerate electron neutrino absorption Energy transport dominated by all-Uavor neutrino scattering Initial Hux F ν (R, 0) = 2.5 × 10 42 neutrinos cm −2 s −1 This ignores the larger initial neutrino Uux originating from the hot, shocked mantle, about 10 39 cm−2 s −1 . J.M.