Equation of state for compact stars Lecture 1 - LUTh

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$J. Phys. A: Math. Gen. 32 (1999) - LUTH - Observatoire de Paris$

Ground state crust - T = 0 Ion-sphere model used - Γ ≫ 1. Terminology: ion sphere ≡ (spherical) unit cell, (spherical) Wigner-Seitz cell. Charge neutrality of the unit cell, under pressure P , ne = Zn N , P = Pe(ne, Z) + PL(n N , Z) , (31) where Pe is the electron pressure and PL is the “lattice” contribution (also called the electrostatic correction) resulting from the Coulomb interactions (PL = Pii). Spherical ”unit cell” per one nucleus in a OCP. At T = 0 enthalpy of the cell=Gibbs free energy (G = H − T S) of the cell =Gcell. Gcell(A, Z) = W N (A, Z) + WL(Z, n N ) + [Ee(ne, Z) + P ]/n N , (32) where W N is the energy of the nucleus (including rest energy of nucleons), WL is the lattice (Coulomb) energy per cell, and Ee is the mean electron energy density. Neglecting quantum and thermal corrections and the nonuniformity of the electron gas, we have WL = −CM Z 2 e 2 /rc, CM ≈ 0.9. (33) Pawe̷l Haensel (CAMK) EOS for compact stars Lecture 1, IHP Paris, France 30 / 54
Ground state crust - T = 0 The lattice (Coulomb) contribution to the pressure, Eq.(31), is thus PL = Pii = 1 3 WLn N . The Gibbs free energy per nucleon h = Gcell/A is just the baryon (=nucleon) chemical potential µb(A, Z) for a given nuclide. To determine the ground state at a given P , one has to minimize µb(A, Z) with respect to A and Z. To a very good approximation, a density jump, at which optimal values (A, Z) change into (A ′ , Z ′ ), is given by ∆ρ ρ ≈ ∆nb nb ≈ Z A A ′ − 1 . (34) Z ′ This equation follows from the continuity of the pressure P Pe. A sharp discontinuity in ρ and nb is a consequence of the assumed one-component plasma model. Detailed calculations of the ground state of dense matter by Jog & Smith (1982) show, that actually the transition between (A, Z) and (A ′ , Z ′ ) shells takes places through a very thin layer of a mixed state of these two species. However, since the pressure interval, where this mixed phase exists, is ∼ 10 −4 P , the approximation of a sharp density jump is quite adequate. Pawe̷l Haensel (CAMK) EOS for compact stars Lecture 1, IHP Paris, France 31 / 54
Page 1 and 2: Equation of state for compact stars
Page 3 and 4: Anticipation and prediction of neut
Page 5 and 6: EOS - an overview EOS of neutron st
Page 7 and 8: Plasma parameters Outer envelope: o
Page 9 and 10: Electrons - continued The density o
Page 11 and 12: ni = nN Ions The strength of the Co
Page 13 and 14: Ions - continued The scale-length t
Page 15 and 16: ρ − T diagram for the outer enve
Page 17 and 18: Pressure - electrons - main term (i
Page 19 and 20: Coulomb gas of ions (id)i Ideal cla
Page 21 and 22: N = N N = Ni = n N V Strongly coupl
Page 23 and 24: Melting - phase transition The soli
Page 25 and 26: Quantum zero-point vibrations and m
Page 27 and 28: Other corrections relevant to elect
Page 29: Ground state crust - T = 0 Notation
Page 33 and 34: Ground state crust - T = 0 - nuclei
Page 35 and 36: Matter should be in equilibrium wit
Page 37 and 38: Effective nucleon-nucleon interacti
Page 39 and 40: Nucleon density distribution in the
Page 41 and 42: Semi-classical - Thomas-Fermi metho
Page 43 and 44: Thomas-Fermi method - continued Neu
Page 45 and 46: Spherical unit cell radius rc, the
Page 47 and 48: Funny nuclei - ”nuclear pasta”
Page 49 and 50: Overall inner crust EOSs - SLy and
Page 51 and 52: Melting temperature Melting tempera
Page 53 and 54: Shear modulus of the crust Effectiv

Equation of state for compact stars Lecture 1 - LUTh

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