Magnetic field effects oncompact stars

Mon.Not.Roy.Astron.Soc. 456 (2015) 2937-2945

Phys.Rev. D92 (2015) 8, 083006

Bruno FranzonCollaborators: V. Dexheimer, S. Schramm

Frankfurt Institute for Advanced Studies

Astrocoffee, January 2016

Plan of the talk

I Motivation

I Effects of magnetic field on the Equation of State

I Magnetized Neutron Stars: fully-general relativistic approachLangage Objet pour la RElativite NaumeriquE (LORENE)

I Results

I Summary

Motivation: magnetic fields

Earth: B∼ 0.5 G

MR: B∼ 103 G

Atlas: B∼ 1020 G

Neutron stars with astrong magnetic field:Duncan and Thompson (1992),Thompson and Duncan (1996).

Typical NS: Bs ∼ 1012 GMagnetars: Bs > 1014 G

Motivation: magnetic fields

Surface magnetic field and atthe pole:

Bd = 3.2× 1019√

PP G

Virial theorem: Bc ∼ 1018 GOrigin?

Duncan, Thompson, Kouveliotou

How to model highly magnetized stars

Einstein Equation

Rµν − 12Rgµν = 8πGTµν

Geometry

1. Spherical: TOV2. Perturbation3. Fully-GR

Energy Content

1. Matter: particles2. Fields: magneticfield

Magnetized EoS

I. An extended hadronic and quark SU(3) non-linear realization ofthe sigma model that describes magnetized hybrid stars containingnucleons, 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

=

√k2zi +

(√M∗2

i + 2ν|qi |B − siκiB)2

IV. Effect of B on the EoS:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14

p (

fm-4

)

ε (fm-4

)

B = 0B = 9.4x18

18 G

Fully-General Relativistic Approach

• Stationary neutron stars with no magnetic-field-dependent EoSwere 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 EoSII. much higher magnetization

Mathematical setupI The energy-momentum tensor: Chatterjee at all. 2014

Tµν = (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 andmagnetic field 4-vectors.

I In the rest frame of the fluid:

Tµν = fluid + magnetization + field (z direction)

Tµν =

e+ B2

2µ00 0 0

0 p−mB+ B2

2µ00 0

0 0 p−mB+ B2

2µ00

0 0 0 p − B2

2µ0

Mathematical setup

I Stationary and axisymmetric space-time, the metric is writtenas:

ds2 = −N2dt2 + Ψ2r 2 sin2 θ(dφ− Nφdt)2 + λ2(dr 2 + r 2dθ2)

where Nφ, N, Ψ and λ are functions of (r , θ).

I A poloidal magnetic field satisfies the circularity condition:

Aµ = (At , 0, 0,Aφ)

I The magnetic field components as measured by the observer(O0) with nµ velocity can be written as:

Bα = −12εαβγσF 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

I Total energy density (fluid + field): Chatterjee at all. 2014

E = Γ2(e + p)− p + 12µ0

(B iBi )

I and the momentum density flux can be written as:

Jφ = Γ2(e + p)U + 1µ0

(mB B iBiU

).

I 3-tensor stress components are given by:

S rr = p + 1

2µ0(BθBθ − B rBr ) + 2m

BBθBθ

Γ2

Sθ θ = p + 12µ0

(B rBr − BθBθ) + 2mB

BrBrΓ2

Sφφ = p + Γ2(e + p)U2 + 12µ0

[B iBi + 2m

B (1 + Γ2U2)BiBiΓ2

]with Γ = (1− U2)−

12 the Lorenz factor and U the fluid velocity

defined as:

U =Ψr sin θ

N(Ω− Nφ)

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

e+pmbnbc2

)

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

I Einstein equations: Rµν − 12 Rgµν = 8πGTµν

∆3ν = 4πGλ2(E + S i

i

)+

Ψ2r 2 sin2 θ

2N2(∂Nφ)2 − ∂ν∂(ν + β)

∆(Nφr sin θ) = −16πGNλ2

Ψ

Jφr sin θ

− r sin θ∂Nφ∂(3β − ν)

∆2[(NΨ− 1)r sin θ] = 8πGNλ2Ψr sin θ(S rr + Sθθ )

∆2(ν + α) = 4πGλ2(E + Sφφ ) +Ψ2r 2 sin2 θ

2N2(∂Nφ)2 − ∂ν∂(ν + β)

I Definitions:ν = ln N, α = ln λ, β = ln Ψ

∆2 =

(∂2

∂r2 + 1r∂∂r

+ 1r2

∂2

∂2θ

)∆3 =

(∂2

∂r2 + 2r∂∂r

+ 1r2

∂2

∂2θ+ 1

r2 tan θ∂∂θ

)∆3 = ∆3 − 1

r2 sin2 θ

E = E (PF ) + E (EM)

S ii = S

(PF ) ii + S

(EM) ii (i = r , θ and φ)

Structure of the star

I MassM =

∫λ2Ψr 2 ×

[N(E + S) + 2NφΨ(E + p)Ur sin θ

]sin θdrdθdφ

I Circumferential RadiusRcirc = Ψ(req,

π2 )req

Increasing of the mass due to the magnetic field andeffect 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/M

O•

Hc (c2)

TOVµ = 1.0x10

32 Am

2- no EoS(B), no mag

EoS(B), no magEoS(B), mag

µ = 2.0x1032

Am2- no EoS(B), no mag

EoS(B), no magEoS(B), mag

µ = 3.0x1032

Am2- no EoS(B), no mag

EoS(B), no magEoS(B), mag

µ = 3.5x1032

Am2- no EoS(B), no mag

EoS(B), no magEoS(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 Tµν).→ Effect on the maximum mass through the effect on the equation ofstate 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 centralmagnetic 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 magneticmoments µ

0.5

1

1.5

2

2.5

3

11 12 13 14 15 16

Mg/ M

O•

Rcirc (km)

TOV

µ = 1.0x1032

Am2

µ = 2.0x1032

Am2

µ = 3.0x1032

Am2

µ = 3.5x1032

Am2

MB = 2.20 M

O·

B. Franzon, V. Dexheimer, S. Schramm, MNRAS, 456 (2015) 2937-2945

→ Effects of the magnetic field into the equation of state and themagnetization are also included.→ The gray line shows an equilibrium sequence for a fixed baryon massof 2.2M.

→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 = 2.20M

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

nB

c(f

m-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/M

O•

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

r p/r

eq

Bc(1018

G)

0

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 (

10

32 A

m2)

Bc(1018

G)

→ Change in behaviour for Bc ∼ 0.9− 1.0× 1018 G. At this point, themagnetic force has pushed the matter off-center and a topologicalchange 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 = 2.20M

0.001

0.01

0.1

1

900 1100 1300 1500

Yi

B = 0n

p

star center

µ Λ

du

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

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 M

O·

B. Franzon, V. Dexheimer, S. Schramm, MNRAS, 456 (2015) 2937-2945

→ As one increases the magnetic field, the particle population changesinside 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 PerlmutterBrian P. SchmidtAdam G. Riess

”for the discovery of the accelerating expansion of the Universe through

observations of distant supernovae”

Properties of White Dwarfs

→ But, motivated by observations of supernova that appears to bemore luminous than expected (e.g. SN 2003fg, SN 2006gz, SN 2007if,SN 2009dc), it has been argued that the progenitor of such super-novaeshould be a white dwarf with mass above the well-known Chandrasekharlimit: 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 canviolate the Chandrasekhar mass limit significantly (Kundu,Mukhopadhyay 2012)

• The new mass limit could explain super-luminous Type Iasupernovae from exploding white dwarfs

Mass-radius diagram for magnetized white dwarfs

0

2

4

6

8

10

12

14

16

18

0 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, includingthe magnetization.• Leading contribution to the macroscopic properties of stars, likemass and radius, comes from the pure field contribution of theenergy-momentum tensor.• Assuming that the magnetic field decays over time, stars wouldnot only become less massive and smaller over time, but also gothrough phase transitions to more exotic phases.• Observables: distinct change in the cooling and stellar brakingindex: in preparation.• Magnetic field effects can considerably increase WD masses

Thank you!