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!!!