Anisotropic EoS and stellar structure equations for magnetized compact stars

A. Pérez MartínezInstituto de Cibernética Matemática y Fisica (ICIMAF),

La Habana CUBA

In collaboration withDaryel Manreza Paret, Universidad de la Habanaand Jorge E Horvath IAG-USP Sao Paulo Brazil

December 19, FortLauderdele, USA

Outline

Introduction /Motivation

1. EoS dependent on the magnetic field: anisotropies.-Magnetized Strange Stars/Magnetized White Dwarfs

2. Static equilibrium structure equations: TOV equations

-Magnetized Strange Stars MSSs-Magnetized White Dwarfs MWDs

3.Static anisotropic structure equilibrium equation:cylindrical metric- Magnetized Strange Stars MSSs-Magnetized White Dwarfs MWDs4. Conclusions and perspective of the work

Introduction

In Astrophysics there are Magnetars with 10 15 G at the surface inside B goes up 1018 G

Early Universe huge magnetic field would explain intergalactic field 10-6 G

It is an open problem the origin of these field but they play a profound role in the stellar/cosmic physics

Heavy ions colliders, strong magnetic field at least locally up to

Magnetic field is present in all scale of the Universe

2 1810m G

2410 G

….focus in Astrophysical scenarios: Compact objects…

Looking for more realist models of CO which could explain some

observables:

---maximum values of Masses of NS/SSs

-PSR J1614-2230, Demorest et al Nature, 467, 1081 2010

-PSR J0348+04232 J Antoniadis et al Science, 2013,

---discard/approve magnetized WD with Super Chandrasekhar

masses proposed by Das, U., & Mukhopadhyay, B. 2013, Phys Rev

Lett, 110, 071102

---upper limits for the values of the magnetic field in CO

max 1.97M M

max 1.44M M

Introduction/Motivation

max 2.01M M

To do these tasks:

EoS in presence of magnetic field

¨MICROSCOPIC PHYSICS¨

but also to explore

structure stellar equilibrium equations and the

¨MACROSCOPIC CONSEQUENCES¨

Introduction

We have been devoted to study “microscopic physics”

- Quantum magnetic collapse Masud Chaichian, et al PRL84, 5261 (2000)

-Neutron stars. Magnetic collapse of a neutron gas: Can magnetars

indeed be formed? APM HPR and H JMC Eur Phys J C 29 111-123 (2003)

-Magnetized Quark Stars: MSQM and MCFL.

R.Gonzalez Felipe A.P.M, H.Perez Rojas H.Mosquera Cuesta PhyRevC 77,015807 (2008). R.Gonzalez Felipe A.P.M J Phys G 36(7):075202. (2009)

M. Benghi Pinto, S.S. Avancini, A. P. M, C. Providencia.Phys Rev C (2009) 79,035807. R.Gonzalez Felipe, D.Manreza Paret and A.P.M., Eur.Phys.J. A47:1

(2011).

-EoS the inclusion of AMM moment, and the insignificant effect!

E. Ferrer, V de la Incera, D.Manreza Paret, A.P.M, A.Sanchez Phys Rev D 91:085041 (2015).

Introduction

Anisotropic pressures dueto the presence of constantB in z-direction1

0 0 0

0 0 0

0 0 0

0 0 0

a

b

P

PT

P

2

8

B

fPB

2

8

B

f fP MB

B

More realistic EoS to take into account the magnetic field B

( ( , , ))P f T B 1Quantum magnetic collapse. M.C, S. S. M, C. M, APM, and HPR, Phys. Rev. Lett.84 (2000) 5261-5264 Magnetic collapse of a neutron gas: Can magnetars indeed beformed?APM H.PR and H JMC Eur Phys J C 29 (2003) 111-123

Main result: Anisotropies due to B

Anisotropic EoS/matter

4

1 *3

20

( , , ) ln det ( )(2 )

f f

f f f

pl

d e B dpB T G p

4 3* ( ,0, 2 , )f fp ip e Bl p 0,1,2,...l

1 *[ ]f fG det p m

( ,0,0) ( , , )vac est

f f fB B T

2 2

3 2 | |f f fp m e B l

we start from the thermodynamical potential for charge fermion system

Anisotropic EoS/matter

( ,0,0) ( , ,0)vac st

f f fB B

In astrophysical scenario T<<TF, so the degenerate limit is considered (T=0)

320

2

( ,0,0) | |4

ln6

fvac

f f

l c

B Bp

B

e BB d

2

020 0

( , , n4

0) lmax

Flf f f fst

f f

F

l f f f

l f

d e B ppB

2

2

0

2

2 2

3

0

2

2

2

2

l

f

l

F

f

p m eBl

p m eBl

m eBl

MSQM in Bag ModelEoS+stellar chemical equilibrium conditions

0

2 3 0

3 0.

d s

u e d

u d s e

u d s B

N N N N

N N N n

2

2

2

8

8

8

( ) ,

,

( B ,

B

)

B

B

f f f

f

B

f

f

B

f f

bag

bag

b

f

ag

B

B

N

P

BMB

P

f

f

N

MB

EoS+stellar chemical equilibrium conditions MWD

/ 2

e pN N

A Z

2

2

2

,8

,8

,8

e e e N e

e

e e

AN m N

P

M

B

Z

B

PB

B

e

e

N

MB

Z atomic number A baryon number.

WD composed by carbon

Structure equilibrium equations

Structure equations: TOV

Space-time static and spherically symmetric

2 2 ( ) 2 2 ( ) 2 2 2r rds e dt e dr r d

from Einstein equation

13

2 2 2

( ) ( ) ( ) 4 2 ( )1 1 1

( ) ( )

dp G m r r p r r p Gm r

dr c r r m r c r

2

2

4 ( )dm r r

dr c

TOV

with the boundary conditions

and the EOS determine

.

8G GT

(0) 0, ( ) 0M P R

P(0)=Pc

, , , T diag P P P and isotropic ; 0T

Mass Radius relations using

TOV for

magnetized/SSs/WDs

,P P

Mass-radius relation/MSSs

Mass-radius relation/MWDs

,P P

…but maximum mass for WDs are

1.44 M

Anisotropic estructure equations

…first attempt to obtain static

structure stellar equilibrium equations

for anisotropic B-dependent EoS

arXiv:1407.2280, RAA 2015 Vol.15 No.7,975–985 doi:10.1088/1674–

4527/15/7/005

arXiv:1501.04619, RAA Vol. 15 No. 10, 1735–1741 doi: 10.1088/1674–

4527/15/10/010

.

Anisotropic stellar structure equations

2 2 2 2 2 2 2 22

2 2

eds e dt e dr d e dz

e r

and with the main approximation that all variables depend only on r

8G GT Consider

energy-momentum with , , , PT d Piag P

following the procedure of

Trendafilova and S. A Fulling 2011 EurJPhys 32 1663/1677 devoted to obtain static solutions of Einstein equation with cyllidrical symmetry for vacuum.

; 0T

2

2

2

( ) ( ),

4 ( 2 ) ( ) ,

4 ( 2 ) ( ) ,

14 ( ) ( )

P E P P P

e E P Pr

e E P Pr

e P Er

( ), ( )P f E P f P

Six quantities to determine , , , , ,P P E

Anisotropic stellar structure equations

0

0

2

0 0 0 0 0

2

0 0 0 0 0

0

0 0

10 ( 2 )( 2 )

21

0 ( 2 )( 2 )2

P P

P P E r r

P P E r r

Initial conditions:

Anisotropic stellar structure equations

Metric coefficients/MSSs

Pressures versus equatorial r/MSSs

04 ( 2 ) .

RTM

re P P drR

0 1 2 3

0 1 2 3( )TM g T T T T dV

( 2 )TM re P P dV

We obtain the total Mass per unit length, not the total Mass!!!

Due to our assumption that , , depend only on r so

Tolman Mass

To calculate the total mass of the stars we have to use the definition of mass given by Tolman 1939

then, which information can be

extract for our model?

Imposing to determine the equatorial radius of

the star and with this M/L we illustrate

-Magnetized Strange Stars

-Magnetized White Dwarfs

0P R

M

R

The important thing is to obtain a bound for the maximum magnetic field that supportthe MSS stars!!! B=1.8 1018 Gcloser the value obtained by Virial theorem

Imposing 0P R

versus perpendicular radius for SS

First glance shows

1.44M

R MR

M

RVersus perpendicular radius

but this value is not associated tomaximum mass of the stars!!

We obtain a threshold for the magnetic field thatsupport A WD around B=1013 G the same obtained byVirial theorem

1. We have obtained an “anisotropic stellar

structure equations” based on cylindrical metric

compatible with anisotropic produced by the

magnetic field in the EoS.

2. The main approximation of our model is to

consider that metric coefficients only depend on r.

The price that we pay is that we can not compute

total mass of the stars.

Conclusions

3. We have obtained a MAXIMUM VALUE of the magnetic

field that can support

Magnetic Strange Stars/Magnetic White Dwarfs

MSS

MDWs

4. Our model rule out Super Chandrasekhar MWD because

their existence depend on to consider values of the

magnetic field higher than and beyond to this value

there are no stable configurations of MWD.

181.8 10 G

1310 G

Conclusions

Our future work is addressed to improve the

structure anisotropic equation taking into

account the dependency on z, and angles in

order to obtain the total mass of the stars.

We are open to suggestions to improve the

model.

Future works

Thank you, Gracias!!!