Magnetic field effects on compact stars Mon.Not.Roy.Astron.Soc. 456 (2015) 2937-2945 Phys.Rev. D92 (2015) 8, 083006 Bruno Franzon Collaborators: V. Dexheimer, S. Schramm Frankfurt Institute for Advanced Studies Astrocoffee, January 2016
Plan of the talk ◮ Motivation ◮ Effects of magnetic field on the Equation of State ◮ Magnetized Neutron Stars: fully-general relativistic approach Langage Objet pour la RElativit´ e Naum´ eriquE (LORENE) ◮ Results ◮ Summary
Motivation: magnetic fields Neutron stars with a strong magnetic field: Earth: B ∼ 0 . 5 G Duncan and Thompson (1992), Thompson and Duncan (1996) . MR: B ∼ 10 3 G Atlas: B ∼ 10 20 G Typical NS: B s ∼ 10 12 G Magnetars: B s > 10 14 G
Motivation: magnetic fields Surface magnetic field and at the pole: B d = 3 . 2 × 10 19 � P ˙ P G Virial theorem: B c ∼ 10 18 G Origin? Duncan, Thompson, Kouveliotou
How to model highly magnetized stars Einstein Equation R µν − 1 2 Rg µν = 8 π GT µν Geometry Energy Content 1. Spherical: TOV 1. Matter: particles 2. Perturbation 2. Fields: magnetic 3. Fully-GR field
Magnetized EoS I. An extended hadronic and quark SU(3) non-linear realization of the sigma model that describes magnetized hybrid stars containing nucleons, hyperons and quarks. See, e.g. Hempel M. at all (2013); Dexheimer V., Schramm S. (2008, 2010) . II. The anomalous magnetic moment of the hadrons. III. Landau levels ν : � � 2 �� M ∗ 2 E ∗ k 2 i ν s = z i + + 2 ν | q i | B − s i κ i B i IV. Effect of B on the EoS: 1.8 1.6 B = 0 B = 9.4x18 18 G 1.4 1.2 p (fm -4 ) 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 12 14 ε (fm -4 )
Fully-General Relativistic Approach • Stationary neutron stars with no magnetic-field-dependent EoS were studied by Bonazzola (1993), Bocquet (1995). • magnetic fields effects in the EoS was presented in Chatterjee (2014), for a quark EoS . • Our case: nucleons, hyperons, mixed phase with quarks, AMM of all hadrons (even the uncharged ones): I. much more complex EoS II. much higher magnetization
Mathematical setup ◮ The energy-momentum tensor: Chatterjee at all. 2014 T µν = ( e + p ) u µ u ν + pg µν + m B ( b µ b ν − ( b · b )( u µ u ν + g µν )) + 1 � − b µ b ν + ( b · b ) u µ u ν + 1 � 2 g µν ( b · b ) µ 0 where m and B are the lengths of the magnetization and magnetic field 4-vectors. ◮ In the rest frame of the fluid: T µν = fluid + magnetization + field ( z direction ) e + B 2 0 0 0 2 µ 0 p − mB + B 2 0 0 0 T µν = 2 µ 0 p − mB + B 2 0 0 0 2 µ 0 p − B 2 0 0 0 2 µ 0
Mathematical setup ◮ Stationary and axisymmetric space-time, the metric is written as: ds 2 = − N 2 dt 2 + Ψ 2 r 2 sin 2 θ ( d φ − N φ dt ) 2 + λ 2 ( dr 2 + r 2 d θ 2 ) where N φ , N , Ψ and λ are functions of ( r , θ ). ◮ A poloidal magnetic field satisfies the circularity condition: A µ = ( A t , 0 , 0 , A φ ) ◮ The magnetic field components as measured by the observer ( O 0 ) with n µ velocity can be written as: 2 ǫ αβγσ F γσ n β = � � ∂ A φ ∂ A φ B α = − 1 1 1 0 , ∂θ , − ∂ r , 0 Ψ r 2 sin θ Ψ sin θ A t , A φ → Maxwell Equations . Static case : no electric field
3+1 decomposition of T µν ◮ Total energy density ( fluid + field ): Chatterjee at all. 2014 E = Γ 2 ( e + p ) − p + 2 µ 0 ( B i B i ) 1 ◮ and the momentum density flux can be written as: � m J φ = Γ 2 ( e + p ) U + 1 B B i B i U � . µ 0 ◮ 3-tensor stress components are given by: B θ B θ S r 2 µ 0 ( B θ B θ − B r B r ) + 2 m 1 r = p + Γ 2 B B r B r S θ 2 µ 0 ( B r B r − B θ B θ ) + 2 m 1 θ = p + Γ 2 B φ = p + Γ 2 ( e + p ) U 2 + � � B (1 + Γ 2 U 2 ) B i B i S φ 1 B i B i + 2 m 2 µ 0 Γ 2 with Γ = (1 − U 2 ) − 1 2 the Lorenz factor and U the fluid velocity defined as: U = Ψ r sin θ (Ω − N φ ) N � � e + p ◮ Remember: p = p ( h , B ), with h ( r , θ ) := ln m b n b c 2
Field equations: our 4 unknowns N, N φ , Ψ , λ ◮ Einstein equations: R µν − 1 2 Rg µν = 8 π GT µν + Ψ 2 r 2 sin 2 θ ( ∂ N φ ) 2 − ∂ν∂ ( ν + β ) ∆ 3 ν = 4 π G λ 2 � E + S i � i 2 N 2 ∆( N φ r sin θ ) = − 16 π G N λ 2 J φ ˜ r sin θ − r sin θ∂ N φ ∂ (3 β − ν ) Ψ ∆ 2 [( N Ψ − 1) r sin θ ] = 8 π GN λ 2 Ψ r sin θ ( S r r + S θ θ ) φ ) + Ψ 2 r 2 sin 2 θ ( ∂ N φ ) 2 − ∂ν∂ ( ν + β ) ∆ 2 ( ν + α ) = 4 π G λ 2 ( E + S φ 2 N 2 ◮ Definitions: ν = ln N , α = ln λ , β = ln Ψ � ∂ 2 ∂ 2 � ∂ r 2 + 1 ∂ 1 ∆ 2 = ∂ r + E = E ( PF ) + E ( EM ) r r 2 ∂ 2 θ � � ∂ 2 ∂ 2 ∂ r 2 + 2 ∂ 1 1 ∂ ∆ 3 = ∂ r + ∂ 2 θ + i = S ( PF ) i + S ( EM ) i S i (i = r , θ and φ ) r 2 r 2 tan θ r ∂θ i i ˜ 1 ∆ 3 = ∆ 3 − r 2 sin2 θ
Structure of the star ◮ Mass λ 2 Ψ r 2 × � � N ( E + S ) + 2 N φ Ψ( E + p ) Ur sin θ � M = sin θ drd θ d φ ◮ Circumferential Radius R circ = Ψ( r eq , π 2 ) r eq
Increasing of the mass due to the magnetic field and effect of EoS(B) and magnetization m 2.6 2.4 2.2 2 M g /M O • 1.8 TOV µ = 1.0x10 32 Am 2 - no EoS(B), no mag EoS(B), no mag 1.6 EoS(B), mag µ = 2.0x10 32 Am 2 - no EoS(B), no mag EoS(B), no mag 1.4 EoS(B), mag µ = 3.0x10 32 Am 2 - no EoS(B), no mag EoS(B), no mag EoS(B), mag 1.2 µ = 3.5x10 32 Am 2 - no EoS(B), no mag EoS(B), no mag EoS(B), mag 1 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 H c (c 2 ) B. Franzon, V. Dexheimer, S. Schramm, MNRAS, 456 (2015) 2937-2945 → Very small reduction of stellar masses due to magnetization (negative sign in T µν ). → Effect on the maximum mass through the effect on the equation of state is negligible .
Deformation due to the magnetic field → The maximum mass for the value µ = 3 . 5 × 10 32 Am 2 . → It corresponds to a central enthalpy of H c = 0.26 c 2 ( n = 0 . 463 fm − 3 ). → The gravitational mass obtained for the star is 2 . 46 M ⊙ for a central magnetic field of 1.62 × 10 18 G . → The ratio between the magnetic pressure and the matter pressure in the center for this star is 0.793 .
Mass-Radius Diagram for different fixed magnetic moments µ 3 TOV µ = 1.0x10 32 Am 2 µ = 2.0x10 32 Am 2 µ = 3.0x10 32 Am 2 2.5 µ = 3.5x10 32 Am 2 M B = 2.20 M O · 2 M g / M O • 1.5 1 0.5 11 12 13 14 15 16 R circ (km) B. Franzon, V. Dexheimer, S. Schramm, MNRAS, 456 (2015) 2937-2945 → Effects of the magnetic field into the equation of state and the magnetization are also included . → The gray line shows an equilibrium sequence for a fixed baryon mass of 2 . 2 M ⊙ . → The full purple circles represent a possible evolution from a highly magnetized neutron star to a non-magnetized and spherical star.
Global Quantities for a star with fixed M B = 2 . 20 M ⊙ 0.6 2.02 2.01 0.55 2 0.5 1.99 n B c (fm -3 ) 1.98 M g /M O • 0.45 1.97 1.96 0.4 1.95 0.35 1.94 1.93 0.3 1.92 0.25 1.91 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 B c (10 18 G) B c (10 18 G) 4 1.1 3.5 1 µ (10 32 Am 2 ) 3 0.9 2.5 r p /r eq 0.8 2 0.7 1.5 0.6 1 0.5 0.5 0 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 B c (10 18 G) B c (10 18 G) → Change in behaviour for B c ∼ 0 . 9 − 1 . 0 × 10 18 G . At this point, the magnetic force has pushed the matter off-center and a topological change to a toroidal configuration can take place Cardall (2001). → The ratio between the polar and the equatorial radii can reach 50 % for a magnetic field strength of ∼ 1 × 10 18 G at the center.
Population change for a star with M B = 2 . 20 M ⊙ 1 d B = 0 star center n u s p µ = 1.0x10 32 Am 2 0.1 Y i 0.01 3 TOV µ = 1.0x10 32 Am 2 µ = 2.0x10 32 Am 2 µ = 3.0x10 32 Am 2 2.5 µ Λ µ = 3.5x10 32 Am 2 M B = 2.20 M O · 0.001 1 2 M g / M O • 900 1100 1300 1500 900 1100 1300 1500 1.5 µ = 2.0x10 32 Am 2 µ = 3.5x10 32 Am 2 0.1 1 Y i 0.5 11 12 13 14 15 16 R circ (km) 0.01 0.001 900 1100 1300 1500 900 1100 1300 1500 µ B (MeV) µ B (MeV) B. Franzon, V. Dexheimer, S. Schramm, MNRAS, 456 (2015) 2937-2945 → As one increases the magnetic field, the particle population changes inside the star. → These stars are represented in MR diagram by the full purple circles. → Younger stars that possess strong magnetic fields might go through a phase transition later along their evolution, when their central densities increase enough for the hyperons and quarks to appear.
Properties of White Dwarfs → The sizes are the size of the planet Earth 10 5 − 9 g / cm 3 → Densities → Typical composition : C and/or O → Gravity is balanced by the electron degeneracy pressure → The masses are up to 1.4 Msun, the Chandrasekhar limit Progenitors of Type Ia supernovae : Chandrasekhar White Dwarfs
Standard candles EXPANSION OF THE UNIVERSE 2011 Saul Perlmutter Brian P. Schmidt Adam G. Riess ”for the discovery of the accelerating expansion of the Universe through observations of distant supernovae”
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