Magnetic field effects on compact stars Mon.Not.Roy.Astron.Soc. 456 - - PowerPoint PPT Presentation

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

Earth: B∼ 0.5 G MR: B∼ 103 G Atlas: B∼ 1020 G Neutron stars with a strong magnetic field: Duncan and Thompson (1992), Thompson and Duncan (1996). Typical NS: Bs ∼ 1012 G Magnetars: Bs > 1014 G

Motivation: magnetic fields

Surface magnetic field and at the pole: Bd = 3.2 × 1019 P ˙ P G Virial theorem: Bc ∼ 1018 G Origin?

Duncan, Thompson, Kouveliotou

How to model highly magnetized stars

Einstein Equation

Rµν − 1

2Rgµν = 8πGTµν

Geometry

• 1. Spherical: TOV
• 2. Perturbation
• 3. Fully-GR

Energy Content

• 1. Matter: particles
• 2. Fields: magnetic

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:

E ∗

iνs =

• k2

zi +

• M∗2

i

+ 2ν|qi|B − siκiB 2

• IV. Effect of B on the EoS:

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 4 6 8 10 12 14 p (fm-4) ε (fm-4) B = 0 B = 9.4x1818 G

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)uµuν + pgµν +m B (bµbν − (b · b)(uµuν + gµν)) + 1 µ0

• −bµbν + (b · b)uµuν + 1

2gµν(b · b)

• 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) T µν =       e+ B2

2µ0

p−mB+ B2

2µ0

p−mB+ B2

2µ0

p − B2

2µ0

     

Mathematical setup

◮ Stationary and axisymmetric space-time, the metric is written

as: ds2 = −N2dt2 + Ψ2r2 sin2 θ(dφ − Nφdt)2 + λ2(dr2 + r2dθ2) where Nφ, N, Ψ and λ are functions of (r, θ).

◮ A poloidal magnetic field satisfies the circularity condition:

Aµ = (At, 0, 0, Aφ)

◮ The magnetic field components as measured by the observer

(O0) with nµ velocity can be written as: Bα = − 1

2ǫαβγσF γσnβ =

• 0,

1 Ψr2 sin θ ∂Aφ ∂θ , − 1 Ψ sin θ ∂Aφ ∂r , 0

• At, Aφ → Maxwell Equations. Static case : no electric field

3+1 decomposition of Tµν

◮ Total energy density (fluid + field):

Chatterjee at all. 2014 E = Γ2(e + p) − p +

1 2µ0 (BiBi) ◮ and the momentum density flux can be written as:

Jφ = Γ2(e + p)U + 1

µ0

m

B BiBiU

• .

◮ 3-tensor stress components are given by:

Sr

r = p + 1 2µ0 (BθBθ − BrBr) + 2m B BθBθ Γ2

Sθ

θ = p + 1 2µ0 (BrBr − BθBθ) + 2m B BrBr Γ2

Sφ

φ = p + Γ2(e + p)U2 + 1 2µ0

• BiBi + 2m

B (1 + Γ2U2) BiBi Γ2

• with Γ = (1 − U2)− 1

2 the Lorenz factor and U the fluid velocity

defined as:

U = Ψr sin θ N (Ω − Nφ)

◮ Remember: p = p (h, B), with h(r, θ) := ln

• e+p

mbnbc2

Field equations: our 4 unknowns N, Nφ, Ψ, λ

◮ Einstein equations: Rµν − 1 2Rgµν = 8πGTµν

∆3ν = 4πGλ2 E + Si

i

• + Ψ2r2 sin2 θ

2N2 (∂Nφ)2 − ∂ν∂(ν + β) ˜ ∆(Nφr sin θ) = −16πG Nλ2 Ψ Jφ r sin θ − r sin θ∂Nφ∂(3β − ν) ∆2[(NΨ − 1)r sin θ] = 8πGNλ2Ψr sin θ(Sr

r + Sθ θ )

∆2(ν + α) = 4πGλ2(E + Sφ

φ) + Ψ2r2 sin2 θ

2N2 (∂Nφ)2 − ∂ν∂(ν + β)

◮ Definitions:

ν = ln N, α = ln λ, β = ln Ψ ∆2 =

• ∂2

∂r2 + 1 r ∂ ∂r + 1 r2 ∂2 ∂2θ

• ∆3 =
• ∂2

∂r2 + 2 r ∂ ∂r + 1 r2 ∂2 ∂2θ + 1 r2 tan θ ∂ ∂θ

• ˜

∆3 = ∆3 −

1 r2 sin2 θ

E = E(PF) + E(EM) Si

i = S(PF) i i

+ S(EM) i

i

(i = r, θ and φ)

Structure of the star

◮ Mass

M =

• λ2Ψr2 ×
• N(E + S) + 2NφΨ(E + p)Ur sin θ
• sin θdrdθdφ

◮ Circumferential Radius

Rcirc = Ψ(req, π

2 )req

Increasing of the mass due to the magnetic field and effect of EoS(B) and magnetization m

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

Mg/MO

• Hc (c2)

TOV µ = 1.0x1032 Am2- no EoS(B), no mag EoS(B), no mag EoS(B), mag µ = 2.0x1032 Am2- no EoS(B), no mag EoS(B), no mag EoS(B), mag µ = 3.0x1032 Am2- no EoS(B), no mag EoS(B), no mag EoS(B), mag µ = 3.5x1032 Am2- no EoS(B), no mag EoS(B), no mag EoS(B), mag

• 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 × 1032 Am2. → It corresponds to a central enthalpy of Hc = 0.26 c2 (n = 0.463 fm−3). → The gravitational mass obtained for the star is 2.46 M⊙ for a central magnetic field of 1.62 ×1018 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

0.5 1 1.5 2 2.5 3 11 12 13 14 15 16

Mg/ MO

• Rcirc (km)

TOV µ = 1.0x1032 Am2 µ = 2.0x1032 Am2 µ = 3.0x1032 Am2 µ = 3.5x1032 Am2 MB = 2.20 MO

·

• 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

• f 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 MB = 2.20 M⊙

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 nBc(fm-3)

Bc(1018 G)

1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2 2.01 2.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mg/MO

• Bc(1018 G)

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rp/req

Bc(1018 G)

0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 µ (1032 Am2)

Bc(1018 G)

→ Change in behaviour for Bc ∼ 0.9 − 1.0 × 1018 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 × 1018 G at the center.

Population change for a star with MB = 2.20 M⊙

0.001 0.01 0.1 1 900 1100 1300 1500

Yi

B = 0

n p

star center

µ Λ d u s

900 1100 1300 1500

µ = 1.0x1032 Am2

0.001 0.01 0.1 1 900 1100 1300 1500

Yi µB (MeV)

µ = 2.0x1032 Am2

900 1100 1300 1500

µB (MeV)

µ = 3.5x1032 Am2

Mg/ MO

• Rcirc (km)

• 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 → Densities 105−9g/cm3 → 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

• bservations of distant supernovae”

Properties of White Dwarfs

→ But, motivated by observations

• f supernova that appears to be

more luminous than expected (e.g. SN 2003fg, SN 2006gz, SN 2007if, SN 2009dc), it has been argued that the progenitor of such super-novae should be a white dwarf with mass above the well-known Chandrasekhar limit: 2.0 - 2.8 Msun . → Several magnetized WDs discovered with surface fields of 105 − 109 G → For a typical white dwarf: Bmax ∼ 1013 G → It has been suggested that strongly magnetized white dwarfs can violate the Chandrasekhar mass limit significantly (Kundu, Mukhopadhyay 2012)

• The new mass limit could explain super-luminous Type Ia

supernovae from exploding white dwarfs

Mass-radius diagram for magnetized white dwarfs

2 4 6 8 10 12 14 16 18 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

WD mass (Msun) Radius (106 m)

Chandrasekhar white dwarfs Bc = 1.1x1011 Gauss Bc = 1.3x1012 Gauss Bc = 5.4x1012 Gauss Bc = 3.9x1013 Gauss

Franzon, B. ; Schramm, S. 2015, Physical Review D, 92, 083006

→ Magnetic field effects can considerably increase the star masses and, therefore, might be the source of superluminous SNIa.

Summary

• Self-consistent stellar model including a poloidal magnetic field
• Effects of the magnetic field on the equation of state, including

the magnetization.

• Leading contribution to the macroscopic properties of stars, like

mass and radius, comes from the pure field contribution of the energy-momentum tensor.

• Assuming that the magnetic field decays over time, stars would

not only become less massive and smaller over time, but also go through phase transitions to more exotic phases.

• Observables: distinct change in the cooling and stellar braking

index: in preparation.

• Magnetic field effects can considerably increase WD masses

Thank you!