Download Neutron star structure in strong magnetic fields

Neutron star structure
in strong magnetic fields
Debarati Chatterjee
luth, observatoire de paris,
meudon
Collaborators:
micaela oertel
jerome novak
neutron stars
with strong magnetic fields
•
high B pulsars, a few XDINSs and RRATs, having super
critical magnetic fields
Popov et al. (2006)
•
Soft Gamma-ray Repeaters (SGRs), Anomalous X-ray
Pulsars (AXPs)
•
Observations indicate common features ⇒
SGRs/AXPs : neutron stars with high surface field BS
•
P-Pdot, magnetic braking ⇒ B ~ 1015 - 1016 G
(Duncan & Thomson 1992; Thomson & Duncan 1993)
•
Direct measurements of the field (Ibrahim et al.)
•
Virial Theorem ⇒ Bmax ~ 1018 - 1019 G
aim of the study
Consistent neutron star models in a strong magnetic field
•
effects of magnetic field on the dense matter Equation of State
•
interaction of the electromagnetic field with matter (magnetisation)
Noronha and Shovkovy (2007), Ferrer et al. (2010), Paulucci et al. (2010),
Dexheimer, Menezes, Strickland (2012)
•
anisotropy of the energy momentum tensor caused by breaking of the
spherical symmetry by the electromagnetic field
•
to calculate the structure and observable properties of the neutron star
within General Relativistic framework
Bonazzolla, Gourgoulhon, Salgado, Marck (1993)
Bocquet, Bonazzola, Gourgoulhon, Novak (1995)
all the equations can be easily transformed.
1
Energy-momentum tensor for a fermion field coupled to a (claseffect
of
magnetic
field
on
dense
matter
Belinfante-Rosenfeld
tensor
[2,
1]
is
a
symmetrized
and
gauge
invariant
formulation
of
sical) electromagnetic field
αµν
canonical energy-momentum tensor. The idea is that we can add a function ∂α B
with
ν = −B µαν to the canonical tensor without changing the divergence such that
Within
this section, we are dealing with microscopic derivations. The relevant scales are such that
the metric can be assumed as (locally)
flat,
µν
µν i.e. we
αµνare working here with the Minkowski metric. Let
T = Θ + ∂α B
(8)
us start with
the Lagrangian,
including
the minimal
density
of a fermion
systemsubstitution,
in the presence of a magnetic field
• Lagrangian
mmetric, gauge invariant and divergence free. For the fermion 1field the Belinfante correction
µ
µν
L
=
ψ̄(x)(D
γ
−
m)ψ(x)
−
F
F
,
(1)
µ
µν
be chosen as
16π
1
αµν
α
µν
µ
να
ν
Bf = ψ̄ µν
({γ , σµ }ν + {γν ,matter
σ
} − {γ , σ αµ }) ψ , field
(9)
µ
where Dµ = i∂µ − eAµ and8 F = ∂ A − ∂ A , the field strength tensor of the electromagnetic field.
There
several ways to
for theare
electromagnetic
onederive an energy-momentum tensor from the Lagrangian:
αµν
BEM
= −F αµ Aν .
(10)
• The•canonical
energy-momentum
tensor,
derived
from
the
invariance
of
the
Lagrangian
microscopic energy-momentum tensor in a symmetrized and gauge invariant
formwith
ing torespect
the following
Belinfante-Rosenfeld
tensor
to translations
in space and
time. It is the conserved Noether current associated with
the symmetry of1 space-time translations.
It 1can be calculated from
1
T µν =
(F µα Fαν + g µν Fαβ F αβ ) + ψ̄(γ µ Dν + γ ν Dµ )ψ .
(11)
4π
4 µν � 2∂L
Θ =
∂ ν ϕ − g µν L ,
(2)
∂(∂µ ϕ)
fieldtensor is the
ϕ tensor
Einstein-Hilbert energy-momentum
appearing asmatter
source of the Einstein
ations. It is obtained by requiring that the action,
where the sum over ϕ indicates here the sum over all fields involved in the Lagrangian. In our
�
√
case this gives
Si = L −gd4 x
(12)
µν
µ ν
ν
µ
µ ν ρ
µν
Θ = (ψ̄γ ∂ ψ − (∂ ψ̄)γ ψ) − Fρ ∂ A − g L .
(3)
2
variant with respect to variations of the metric. This leads to
With the help of the Euler-Lagrange equations for the fermion fields,
−2 δµ √ µ
µν
τ = √ ((i∂ −( eA
−gL)
(13)
)γµ.− m)ψ = 0 ,
(4)
−g δgµν
andτ the
field
µν =electromagnetic
at space
T µν .
1
αβ
¯
β
µν
the
averaging
volume.
The
thermal
average
of
T
can then
all the equations can be easily transformed.
be written as, see Kapusta (1994),
!
! β
!
1field
1
1 Energy-momentum tensor for a fermion
coupled
to a dλ
(clas"T µν # =
DψDψ̄ exp(S̃)
d3 x T µν , (7
Z
Belinfante-Rosenfeld
tensor [2, 1] is afield
symmetrized and gauge βV
invariant
formulation of 0
sical) electromagnetic
αµν with
canonical energy-momentum tensor. The idea is thatwhere
we canthe
addpartition
a function
∂
B
α
function is given by
ν = −B µαν to the canonical tensor without changing the divergence such that
Within this section, we are dealing with microscopic derivations. The relevant
scales are such that
!
the metric can be assumed as (locally)
flat,
the Minkowski
metric.
Let
µν
µν i.e. we
αµνare working here with
Z
=
DψD
ψ̄
exp(
S̃)
,
(8
T = Θ + ∂α B
(8)
us start with
the Lagrangian,
including
the minimal
density
of a fermion
systemsubstitution,
in the presence of a magnetic field
• Lagrangian
and the1field
action
mmetric, gauge invariant and divergence free. For the fermion
the is
Belinfante correction
µ
µν
L = ψ̄(x)(Dµ γ − m)ψ(x) −
Fµν F !, β
(1)
!
be chosen as
16π
3
i
1
αµν
α
µν
µ
να
ν
αµ
S̃
=
dλ
d
x(L(λ,
x
(9
Bf = ψ̄ µν
({γ , σµ }ν + {γν ,matter
σ
} − {γ , σ }) ψ , field
(9) ) − µn̂) .
µ
where Dµ = i∂µ − eAµ and8 F = ∂ A − ∂ A , the field strength tensor
0 of the electromagnetic field.
There
several ways to
for theare
electromagnetic
onederive an energy-momentum tensor from the Lagrangian:
0
β
=
1/T
is
the
inverse
temperature,
λ
=
ix
, and the term
αµν
BEM
= −F αµ Aν .
(10)
µn̂
tothe
beinvariance
introduced
grand
canonical
• The•canonical
energy-momentum
tensor,
derived
from
of in
theinvariant
Lagrangian
microscopic
energy-momentum
tensor
in ahas
symmetrized
and gauge
formwithtreatment t
ing torespect
the following
Belinfante-Rosenfeld
tensor
average
particle
number
conservation.
to translations
in space and
time. It isguarantee
the conserved
Noether
current
associated
with The num
0
ber
density
operator
is
n̂
=
−i
ψ̄γ
Q̂ψ, where the operator Q̂
the symmetry of1 space-time translations.
It
can
be
calculated
from
1
1
µ ν
T µν =
(F µα Fαν + g µν Fαβ F αβ ) + ψ̄(γ
D + γ νthe
Dµ )ψ
.
(11)
associates
number
density of
the particle species a with
4π
4 µν � 2∂L
µν . µ represents the associated chemical poten
its charge
aL ,
Θ =
∂ ν ϕ − gQ
(2)
∂(∂µtial.
ϕ)
fieldtensor is the
ϕ tensor
Einstein-Hilbert energy-momentum
appearing asmatter
source of the Einstein
The thermal average of the energy-momentum tensor i
ations. It is obtained by requiring that the action,
where the sum over ϕ indicates here the sum over all fields involved in the EoS
Lagrangian. In our
�
then given by (see Appendix A for details of thematter
calculations
√
case this gives
Si = L −gd4 x
(12)
µν
µ ν
µν
pure fermionic
µν
µ
ν
ν
µ
µ
ν
ρ
µν
"T
#
=
(ε
+
p)
u
u
+
p
g
of the
= (ψ̄γ
∂ ψ − (∂ ψ̄)γ ψ) − Fρ ∂ A − g L .
(3)
• thermodynamicΘaverage
2
1 ν τµ
µ
τν
variant with energy-momentum
respect to variations tensor
of the metric. This leads to
(F
M
+
F
M
) magnetisation
+
τ
τ
With the help of the Euler-Lagrange equations for the fermion2 fields,
µν
−2 δµ √ µ
µν
1
g
µα
ν
αβ
τ = √ ((i∂ −( eA
−gL)
)γµ.− m)ψ = 0−, (F Fα + (13)
F
F
) . (4)
(10
αβ
field
−g δgµν
µ0
4
T. Elghozi, M. Oertel,
andτD.
the
fieldJ. Novak, arXiv:1410.6332
µν C.,
at space
=electromagnetic
T µν .
The first two terms on the right hand side can be identified
1
αβ
¯ β
effect of magnetic field on dense matter
numerical resolution
•
The structure equations of neutron stars are obtained by solving Einstein’s field
equations
• In the 3+1 Formalism, solving the Einstein’s equations (system of 2nd order PDEs)
are reduced to integration of a system of coupled 1st order PDEs subject to certain
conditions:
Bonazzolla, Gourgoulhon, Salgado, Marck (1993)
- 6 evolution equations for the extrinsic curvature
- 1 Hamiltonian constraint equation
- 3 momentum constraint equations
• The formulation has been employed to construct a numerical code (LORENE)
using spectral methods
Langage Objet pour la RElativité NumériquE
•The code has been extended to include coupled Einstein-Maxwell equations
describing rapidly rotating neutron stars with a magnetic field
Bocquet, Bonazzola, Gourgoulhon, Novak (1995)
• To incorporate magnetisation one must modify the inhomogeneous Maxwell
equations
.1.
we can write Mαβ =N�αβµν∂r
m u = xF
∂rαβ with the scalar x = (m · b)/(b · b). The inhomogeneous
! ϕ
"
elliptic partial diffe
t equation),
Maxwell equation (Maxwell-Ampère
1 ∂A
j t− Ωj ϕ=∂A
(εεϕ+ =
p)f (A
)
.
(33)
ε(nϕ )
(3.67)
defined in Eq. (16)
b
+N
(17b)
Eθ =
N
∂θ
∂θ
σα
σ
σα
∇
F
=
4πj
+
∇
M
, be(43)
α
α
p
=
p(n
),
(3.68)
free
Under these two assumptions, the Lorentz
force term
b
2
∂A
1
ϕ
∆
ν
=
4πGA
3
comes
(17c)
B
=
rtransformed
2
can
then
be
to
give
3
Hydrodynamic
equations
for
the
stationary
and
axisymmetric
ilibrium can be derived
sin θ density
∂θ
where nb is the baryonBr
number
in the fluid frame.
)
*
∂M
−∂ν
ρ
ϕ ∂Aϕ t ∂Aϕ
momentum,
1
The equations
of
motion
are
the
energy-momentum
conservation
law
(1.36)
:
=
−
(ε
+
p)
,
(34)
F
j
=
j
−
Ωj
case expressed
iρ free
Bθ =
−
σα∂xi 1
σ
∇
F
=
(4πjfree
+∂x
Fiσα ∇α x) .(17d)
(44)
y-momentum tensor:
B sin θ ∂rα
ϕ
˜
1µ− x
∆3 (N r sin θ)
∇
T
=
0
The hydrodynamic(25)
equationswith
can be
derived
from the conservation
of energy Fand
momentum,
The
homogeneous
Maxwellαµequation
= 0 expressed (3.69)
[µν;λ]
Within
theseenergy-momentum
equations we have tensor:
distinguished
between a free current and the currents responsable
+ Aϕ (r,θ) fulfilled,
as vanishing divergence
of the
(Faraday-Gauss) is automatically
when taking the
for
magnetostatic
equilibrium
(from
the
conservation
of
energy
and momentum):
• Equations
for
the
magnetisation,
in
the
lines
of
the
derivation
of
the
macroscopic
from the
and the baryon
number
conservation
law:
M (r,for
θ) =
−
(x)dx.
(35) Maxwell equations
form
in Eq.
(3)
the
tensor Ffµν
. The inhomogeneous
∆
2 [(N B − 1)
µν
microscopic
ones
in
Jackson’s
book
on
electrodynamics.
A
problem
is,
however,
that
in
a
neutron
star
0
x
β στ
∇
T
=
0
(42)
µ
Maxwell
equation
(Gauss-Ampère)
in
presence
of
external
Fστ ∇ F
, (26)
µ it is not completely clear how to define correctly the
2
we generally have homogeneous matter and
forb ume
∇µ (n
) = 0.
(3.70)
=
8πGN
A
Br
2µ0
The last
term
can
bethe
written
in terms
of the magnetic
magnetic
field
(∇
covariant
derivative
associated
µ is
currents.
Inµ the present
model
they 2will
be
self-consistently
fromrecover
the magnetic field.
µdetermined
where ∇µ is now thefree
covariant
derivative.
Upon
projection
on
the
hypersurface
Σ
,
we
formally
t
field
b
in
the
FRF
as
(with
b
=
b
b
):
µ
with
the
metric
(16)),
uid contribution
to the
They
could
arise
from
a
charged
fluid,
i.e.
if
protonsexpanding
and electrons
not haveorthogonally
the same velocity.
Inserting
the
perfect
fluid
form
(1.37)
into
Eq.
(3.69),
anddid
projecting
to = 8
Inhomogeneous
Maxwell
equations:
•
for
the
fermionic
part
Eq.
(5.8)
of
Ref.
[8].
∆
(ν
+
α)
2
akrecognise
and M. the
Oertel
n
usual
Then,
in princple
one-fluid
used
is(1.30)],
noνlonger
valid
and∂b
two fluid-velocities
should be
" the
the
fluid 4-velocity
u
[via
getthe
the
relativistic
Euler
equation:
x
x model
νµ ⊥
ν here
νµ we
µνthe 1projector
µ given
µ by
As
mentioned
before,
for
isotropic
media,
the
magnetisation
is
aligned
with
magnetic
field
and
∇
F
=
j
+
∇
M
,
(18)
Fµνnotion
∇i F of=fluid(b
,
µ µ ∇i b − free
µi u ) = b∇i b = m
m free currents. In the
µ b ubecomes
ν∇
i It could arise, too, from a
introduced.
The
rest frame bthen
problematic, ∂x
too.
µ
µ
ν
2µ
µ
0
0
0
we
can
write
Mto
=
with the
scalar
x =
(m
·µb)/(b ·Gourgoulhon,
b). The inhomogeneous
αβ
αβµν m u = xFpotential
µ problematic
ssion
is the
same
asthe
in �electromagnetic
Bonazzolla,
Marck (1993)
ar velocity
Athis
through
an
arbitrary
ϕµ,∇
zero
net charge in the αβ
FRF,
butp)u
seems
very
with
energy.
(ε +
u
+
(δ
+
u
u
)∇
0.
(3.71)
(36) Salgado,
µ α
α
µ p = Coulomb
α
iequation (Maxwell-Ampère
can
then
be
transformed
to
give
Maxwell
equation),
with the same nota
gnetic (B •) Einstein-Maxwell
function
called
theexpression
current function:
equations
Wef ,now
obtain
from
the
(11), and the definition (13).x
αβ
αβ 1
βν
β στ
he case of rigid rotation
Finally, E,(45)
Ji ,
∇
T
=
∇
T
−
F
j
−
F
∇
F
.
Now the Gibbs-Duhem
relation
at
zero
temperature
states
that
σµ
σ
σµ
Thus,
this
last
term
cancels
with
its
counterpart
in
α
α
στ
f
reeν
σα
σ
σα
f
ϕ
t
∇µ=F(ε
=p)f
(µ
j free ,+ F (33)
∇µ x)
(19)
8π .
∇Ωj
=+
4πj
+ϕ∇
(43) 3+1 decomp
j
−
(A
)
.
αF
α0M
ntegral of the following
free
called
1 − x (27) keeps
Eq. (32) and the first integral
exactly
the
same
Bocquet,
Bonazzola,
Gourgoulhon,
Novak
(1995)
We can recognize here the usual Lorentzdpforce
term,
arising
from
free
currents.
In
the
absence
of
(for definitions,
= nb dµ,
(3.72) see
form
as
without
magnetisation:
This
equation
canthe
expressed
terms
of the two nonintegral
ofthese
fluid
motion
: expression
Under
two
assumptions,
Lorentz
term
can then •befirst
transformed
to give
magnetisation,
the
isbe
the
same asforce
ininRef.
[8]. beEq. (10) describing
µ
ρ
vanishing
components
of
A
,
with
the
Maxwell-Gauss
equacomes
I am
thusln
looking
for
a first
integral
expression
(forthanks
rigid rotation
Ωincluding
is constant)
− Fiρ j free
h(r, θ)chemical
+ ν(r,
θ) potential,
−
ln Γ(r, of
θ)µthe
+:=
Mfollowing
(r,
θ) =b .const.
(37)
where µ
is the
baryon
dε/dn
Moreover,
to field,
the first
law ofmag
be derived
1
σα
σ in 3.25σα
Jérôme, I tion
suppose
that=the
factor
Nfree
of∇Ref.
Bonazzola
et al,
factor µofiseverything
amd
that , as
in
)
*
∇
F
(4πj
+ Fe.g.
. for
(44)
∂M
∂A
α
α x) [45]
details),
equal
to
the
enthalpy
Thermodynamics
at
zero
temperature
(see
ϕ
axisymmetric
statio
ρ
ϕ
t
expressed
xdoes
=# −not
(ε2+
,
(34)
Fiρ j book,
− Ωj 11−
$ p) itt directly?:
%
&
Eric’s
Σ
give
free =thej projection
Fµν ∇i F µν = per
0 .(27)
t
i
i
ϕ
∂x
∂x
baryon
h
defined
by
2
m tensor:
× µ0 A gtt j free + gtϕ j free + ∂At ∂x
∆3 A t =
E
=
Γ
(ε + p
�
�
x − 1 between a free εcurrent
Within these equations
we
have
distinguished
and
the
currents
responsable
+
p
1 ∂p
x
*
Numerical
with
ρ
µν
of
enthalpy3.4
per baryon
for
neutron
star+ ∂ν h−:=∂ ln Γ −
• In terms
2 resolution
gral, one
introduces
the
.
(3.73)
1
(25)
(�
+
p)
F
j
−
F
∇
F
=
0
.
(46)
iρMaxwell
µν i
B
free
for the magnetisation, in the lines of the derivation
equations
from the
ϕ
2
2the
i of∂x
i macroscopic
i bϕ
n
+
(1
�
+
p
∂x
∂x
8π
+
−
N
r
sin
θ∂A
∂N
tives. It can be shown
t the library lorene, usAϕ
(r,θ) solved with
2
2µ
The
equations
have
been
0
N
microscopic
ones
in Jackson’s
book on electrodynamics.
A problem "
is, however, that in a neutron star
and
current
function
!
M
(r, θ)(3.72)
=methods
− as dpto=solve
(x)dx.
(35) differential
agnetic field, theThus
logaµ in the FRF as 2
2f
welast
may
n
dh,
hence
ingrewrite
spectral
Poisson-like
partial
The
term
can be
written
in
the
Eric’s
notation
with
the
magnetic
field
b
b
Jϕthe= Γ (ε + p
B
we
generally
have
homogeneous
matter
and
for
me
it
is
not
completely
clear
how
to
define
correctly
0
2
2
ϕ 2
ϕ
στ
epresents
again
a
first
r sin
θ (N ) ∂Aϕsystem
∂N
− 1 + in 2the
, (26)
equations appearing
Einstein-Maxwell
(22),
free
currents.
In
the
present
model
they
will
be
determined
self-consistently
from
the magnetic field.
1 * 2
N
1
at end, let us first note
µν ∇α p
µ methods,
ν
The last(20)
term
can(21).
be written
in Fterms
ofabout
magnetic
=the
nbµb ∇
+ (47)
A
and
ForFµν
more
details
these
seeα b ,
αbh.
∇
=
b
∇
−
b
u
∇
u
=
b∇
ϕ
α
µ
α
µ
ν
α
They
could
arise
from
a
charged
fluid,
i.e.
if
protons
and
electrons
did
not
have
the
same
velocity.
µ ∂Aϕ ) ∂ (β − ν)
µ0
magnetic field infield
beta− (∂A
2N
2b2 &
t +
bµ in the
as (with
=
bNovak
e.g.FRF
Grandclément
The
code
follows
the
µ b ): (2009).
tion to the
!
"
magnetostatic equilibrium
(without magnetisation)
3
(17c)
Br =
!
"0
2
Under
these
two
Br sin θ3.3∂θ Magnetostatic
∂ can
ln Γ then
1 ∂pfield ∂ν
equilibrium
ρ &αβµν uβ bα
3.4
Num
F
=
(11)
ntum tensor.
bµ is nonzero. The electromagnetic
tensor
be
µν
introduces the comesln
(ε +Inp)order to obtain
+ this
− firsti integral,
− Fiρ jone
h(r,∂(ν
θ) ++
−∂ν
free
i
i
1
∂A
ϕ
ε
+
p
∂x
∂x
∂x
1
1
µα
ν
µ
ν
ν
µ
µν
tromagnetic
expressed
terms
ofThe
bµ. as
(Gourgoulhon
(2012))
equations
for
magnetostatic
equilibrium
beassociated
derived
per baryon
and
its derivatives.
Itcancan
be shownhere with
B
(17d)
F Fα + ψ̄(γ
Dθ + =
γ in
D−
)ψ + genthalpy
L
(6)
−
The equatio
with
the
Levi-Civita
tensor
&,
the
x
B
sin
θ
∂r
µν
µ0
2
ϕ
ρ
˜
from
the
conservation
of
energy
and
momentum,
expressed
Fµν ∇
= the
0 .(27)
that, even in the
presence
of−the magnetic
logai F field,
∆
r
sin
=−
Fiρhowjθ)
β
α
3 (N
Minkowski
metric.
The
above
expression,
Eq.
(11),
is,
ing
spectral
free =
2µ
F
=
&
u
b
(11)
0
αβµνdivergence
asµνvanishing
of the
energy-momentum
tensor:
The contribution
homogeneous
Maxwell
equation
F
= be
0 in
Magnetic
field
effects
neutron
3 a
[µν;λ]
rithm
enthalpy
per
baryon
represents
again
a first
term represents the well-known
ofofthethe
ever,
more
general
and
can
employed
withstars
any
metric.
equations
µν
3.4
Numeri
−
In the
order
obtain
thishere
first
integral,
onethe
introduces
the
µν
(Faraday-Gauss)
isfrom
automatically
fulfilled,
when
taking
g L .field
(6)and thewith
with
agnetic
second
arising
integral
of the
fluid
equations.
To
that
end,
let
us
first
note
theterm,
Levi-Civita
tensor
&, toassociated
with
the
If we
assume
in
addition,
that
the
medium
is
isotropic
and
∇ T = 0.
(25)
(20) and (2
µνand µits derivatives. It can be shown
enthalpy
per
baryon
field,
agrees with Eq.
(36) in
etthat
al.
form
inFerrer
Eq. (3)
for(2010)
thethetensor
F
. Eq.
The
inhomogeneous
for
neutron
star
case
with
ahowmagnetic
field
in
betaThe
equations
∆2 [(N
Bfield,
−
1)
r sin θ
that
the
magnetisation
is
parallel
to
the
magnetic
the
Minkowski
metric.
The
above
expression,
(11),
is,
e
Einstein-Hilbert
energy-momentum
tensor.
b
is
nonzero.
The
electromagnetic
field
tensor
can
then
be
µ
e.g.
Grandc
hozi,
J.
Novak
and
M.
Oertel
This
can
be
detailed
as
:
that,
even in theatpresence
of the magnetic
field, the
M
that
indeed
ways
toanevaluate
the
energybution
of the
ingalgorithm
spectral
me
equilibrium
zero
temperature,
the
enthalpy
alogafuncMaxwell
equation
(Gauss-Ampère)
in in
presence
ofbµcan
external
magnetisation
tensor
be
writtenisas
ever,
general
and
can beand
employed
with
any
fermion
fieldboth
coupled
tomore
electromagnetic
expressed
terms
ofmetric.
as
(Gourgoulhon
(2012))
2
p
x
=
8πGN
A
Br
sin
θ
rithm
of
the
enthalpy
per
baryon
represents
again
a
first
β
στ
αβ
αβ
βν
f
ree
m
tensor
are
equivalent.
equations
app
ng
from
the
tion
of
both
baryon
density
and
magnetic
field
F
∇
F
,
(26)
∇
T
=
∇
T
−
F
j
−
magnetic
field
(∇
is
the
covariant
derivative
associated
στ
µ
α
µ
ν
by
If we assume
in
addition,
that
the
medium
isthrough
isotropic
and
βα α
f
modificatio
β
angular
velocity
to
the
electromagnetic
potential
A
,
an
arbitrary
2µ
integral
of
the
fluid
equations.
To
that
end,
let
us
first
note
ϕ
M
=
&
u
m
(12)
0
µν
αβµν
F
=
&
u
b
(11)
etoordinate
we
are
interested
in
studying
the
structure
of
Rest
Frame,
assuming
perfect
conductor,
E
=
0
The
last
ter
(20)
and
(21).
µν
αβµν
• In the Fluid
al.i (2010)
with
the
metric
(16)),
i
that
the
magnetisation
is
parallel
to
the
magnetic
field,
the
ε
+
p
i.e.
depend
c
(E
)
and
magnetic
(B
)
function
f
,
called
the
current
function:
µ
for the
starbcase
with a magnetic
field in beta1assuming
αβ neutron
µαmacroscopic
ν
µ
ν isotropic
ν µ medium,
µνthethat
nthe
length
scales,
we
need
to
calcub Grandclém
in 8πGA
the FR
=
µ
.
(28)
h
=
h(n
,
b)
=
e.g.
∆
+
α)
=
b
2 (νfield
where
T
represents
the
perfect-fluid
contribution
to
the
magnetisation
is
aligned
with
the
magnetic
field
energy•
with
the
magnetisation
four-vector
F
+
ψ̄(γ
D
+
γ
D
)ψ
+
g
L
.
(6)
f
magnetisation
tensor
can
be
written
as
in these
pa
n 3.1.α
nb tensor
with
the Levi-Civita
&, associated
with
thepres
and
temperature,
the enthalpy
is a func- here algorithm
1equilibrium
νµ
νt at zero
νµone can
ϕ
2
hermodynamic
average of the microscopic
energyenergy-momentum
tensor;
recognise
the
usual
j −
= (ε
(Aϕ ), . The above
(33)
∇µ F
=Ωj
jbaryon
+α+
∇p)f
(18)
µM
free
x
x is,
been
shown
µν
Minkowski
metric.
expression,
Eq.
(11),
howtion
of
both
density
and
magnetic
field
β
−
(∂ν
Hence
we
have
µ
m
tensor,
Eq.
(6).
It
is
assumed
in
the
following
modification
o
F
∇
F
=
0M
b
.
(13)
m
=
Lorentz
force
term,
too,
arising
from
free
currents.
In
the
µν
i
µ
µ
=
&
u
m
(12)
structure
of well-known contribution of µν
epresents the
the αβµν
2µ0 the
fluid in
µemployed
#the general
0%
ever, more
and
can
with any
metric.
$the expression
$ isbe
Under
these
twoover
assumptions,
Lorentz
force
term
bens
that the electromagnetic fields
are constant
ε
+
p
i.e.
depending
absence
of
magnetisation,
the
same
as
in
um
$in=∂n
can
then
be transformed
tolngive
to calcucdfield
and the second
term,
arising
from
the
=
µ$$that
(28)the
h=
h(n∂h
∂
h
1
∂h
with
same
notations
b . ∂b
b , b)
µν
change.
b
If
we
assume
addition,
the
medium
is isotropic
and
with
the
magnetisation
four-vector
comes
$
ging volume.
The
thermal
average
of
T
can
then
As
we
shall
see,
the
dependence
of
the
different
equations
in
these
partia
Bonazzola
et
al.
(1993).
n
=
+
.
(29)
Modified
inhomogeneous
Maxwell
equations:
b
•
i
from
the
express
$
$
i
i
i
opic
energyequilibrium
can
be(36)
derived
agrees
with
Eq.
in
Ferrer
et
al.
(2010)
Finally,
E,
J
,
S
a
∂x
h
∂n
∂x
∂b
∂x
The
m
i
b
1
that
the
magnetisation
is
parallel
to
the
magnetic
field,
the
j in
b
nbnow
n as, see Kapusta (1994),
As
in
Bocquet
et
al.
(1995),
in
the
case
of
rigid
rotation
σµ
σ
σµ
been
shown
on
the
magnetisation
can
be
reduced
to
a
dependence
)
*
x
∂M
∂A
Thus, this
ρ
ϕ have
tj free ϕ
Hence
we
∇
F
=m
(µ
+=F− (ε∇+µ x)
. ,(13) (19)
and
momentum,
expressed
µ
0µ
he
following
ndeed
both
ways
to
evaluate
the
energyb
.
=
p)
(34)
F
j
=
j
−
Ωj
(1995)
com
µ
called
3+1
decompositio
iρ
magnetisation
tensor
can
be
written
as
magnetisation
(Ω
constant
across
the
star),
a
first
integral
of
the
following
free
the
fluid
in
th
on
quantity
can
conveniently
be
comi# scalar
i $x, which
1
−
x
!
!
!
%
$
∂x
∂x
β
µ
Eq.
(32)
and
the
0
ergy-momentum
tensor:
In
addition,
the
following
thermodynamic
relations
are
valid
1
1
onstant
over
sor are equivalent.
$ ∂n
µν
needed
vari
(for definitions,
see
e.g.
expression
∂ lnishsought
1 in
∂hthe
∂h $$ the
∂b energy-momentum
change.
b
puted
FRF.
First,
tensor
can
β
α
=µν
DψDψ̄ exp(This
S̃) equation
dλ d3 xcan
T
,
(7)
$
form
as
without
be
expressed
in terms
of iequations
the
two
= assumptions
+
.
(29)
under
Mµν
= non&∂x
(12)most
with
αβµν
! present
"
Z then in studying
$
$
can
i of the
i u m Eq. (10) describing
As(25)
we shall
the
dependence
different
are
interested
structure
ofthe
(instead
of
0.βV
0 thesee,
∂x
h
∂n
∂x
∂b
The
a
pe
be
rewritten
in
the
following
way
b
µ
b
n
∂p
∂ν
∂
ln
Γ
1
$
b
ρ
first
integral
of
fluid
motion
:
vanishing
components
of
A
,
with
the
Maxwell-Gauss
equa•
+ A+ϕ (r,θ)to−a dependence
(εcan
+
− νFiρ j freeµν
croscopic
calcuon
the we
magnetisation
reduced
$ p)now be
amplitude
ln
h(r,
θ) + νi
(1995)
comes
1
∂p
field, including
magnetis
ithe ∂x
i
i
e partition length
functionscales,
is given
byneed to ∂h
µν
µ four-vector
with
magnetisation
ε
+
p
∂x
∂x
$
= θ) the
(30)
T conveniently
= thermodynamic
+ p)be
u u
+relations
p (35)
g
tion
Mwhich
(r,
=can
−following
f(ε
(x)dx.
In addition,
are valid
$
odynamic average
of
energyon the
themicroscopic
scalar quantity
x,
coma
needed
variabl
!
∂nb b
nb ∂n0b
"x
#computed
axisymmetric
stationary
x
µν
x
β
στ
under
the 2energy-momentum
present
assumptions
$
% Fµµν
F & bµ
=µ. ν0 .(27)
$ # the
$ can
1−tensor
1 µν
i=
FστZ
∇= FIt DψD
(26)
−
1 First,
Eq.
(6).
is, puted
assumed
in, the
following
ν∇
in
the
FRF.
(instead
ofcode
one
by the
(13)
m
t
ϕ
µ
xsor,
T µν
,
(7)
ψ̄
exp(
S̃)
(8)
2µ
$
$
&
+
−b
b
+
(b
·
b)u
u
+
g
(b
·
b)
0
2µ0
$
2
µ
A
g
j
+
g
j
+
∂A
∂x
∆3 At = The last
∂ε
∂p×
0
tt
tϕ
t
µ
free
be written
terms
of the$ magnetic
0
free
µin
$ term
E2(31)
= Γamplitude
+ p) −
µ
$ =can
in pt
0−
t the electromagnetic
fields are constant
over
a(εbi-dimens
m
)
=
m(=
m
.
be rewritten
in x
the
following
way
1
∂p
µ
µ− 1∂h
2
µ
3.4
Numerica
$
$
$
In order
to obtain
first
integral,
the
=as (with
(30)
∂bν nb one introduces
fieldµνb in2∂b
theµ $FRF
b = bthis
* follow
µ b ):
x the
t-fluid
to average
the
computed
and
µ dependence
µofν the µν
b
olume.
The
thermal
of
T
can
then
used,
1
As
we
shall
see,
different
equations
ction iscontribution
∂n
n
∂n
b
b
b
µν
µ
ν
µν
it
can
be
shown
that
B
•
(b
b
−
(b
·
b)(u
u
+
g
))
.
(14)
+
b
ϕ
2
2
ϕ
enthalpy
baryon
and its derivatives.
It can be shown
+
(1 + co
2x
p) u uN
g per
$+r psin
$
canKapusta
recognise
The
equations
ha
θ∂A
∂N
µ
by
the
code
! β (1994),
!the Tusual = (εx+ −
t
0
ensure
therm
see
on
the
magnetisation
can
now
be
reduced
to
a
dependence
x
∂b
2∂p
2µ0 and it
$ we
$ the
And
obtain
for
the
of
logarithm
of the per baryon
µν
µ&
µderivative
ν
"
#
∂ε
In
order
to
obtain
a∇first
integral,
let
us
have
ab =
look
at the
that,
even
in
the
presence
of
the
magnetic
field,
the enthalpy
logaF
∇
b
u
)
=
b∇
m
,
3In the
i
from
free
currents.
µνN
i F $ = = (bµm(=
i b −m
µµbmuν)∇
i
i
$
ing
spectral
meth
a2be
bi-dimension
=
− for$ax,magnetic
.which∂xcan
(31)
i
"
10 .!enthalpy
1
µ quantity
µn̂)
tities
(p(h,
b
µrithm
ν$ 2 (9)
µ
ν
µν
2µ
µ
on
the
scalar
conveniently
com!S̃ = (8)dλ d x(L(λ,
! β x !) −
0
It
is
obvious
that
field
pointing
in
z-direction
of
the
enthalpy
per
baryon
represents
again
a
first
For
the
neutron
star
case,
with
magnetic
field,
beta
equilibrium
and
zero
temperature,
∂b
∂b
+
−b
b
+
(b
·
b)u
u
+
g
(b
·
b)
J
=
Γ
(ε
+
p)
U
pression
is
the
same
as
in
B
equations
appea
ϕ
µb
0
1
2
2
ϕ 2
ϕ nb
used,
following
(36)
2
#
(
%
The
fre
puted
FRF.
First,
the
energy-momentum
tensor
can
θfluid
(N'in
) the$ ∂A
∂N
this
expression
reduces
to
the
well-known
form(20)
with
magϕ To
integral
the
equations.
that
end,
let
us
first
note
DψDψ̄ exp(S̃)
dλ d3µx0 −
T µν 1, +
(7)2 r ofsin
$
*
(
and
(21).
F
1
N
0 the ∂
ensure
thermod
$
$
from
expression
(11),
and
the
definition
(13).
Z the inverse temperature,
ε
+
p
2
r
And
we
obtain
for
the
derivative
of
the
logarithm
of
the
ln
h
1
∂p
∂p
∂n
∂b
x
0
b
is
λ
=
ix
,
and
the
term
the
EoS,
th
netisation,
see
e.g.
al.
(2010).
Neglecting
effect
rewritten
in
the
following
way
µbe
µν
that
for=the
neutron
star$case
with
aFerrer
magnetic
field
inµbbetaper baryon +
for neutron
star
magnetic
$ =et−
+ the
A B
h
=
h(n
,
b)
=
.
n the case•ofenthalpy
rigid rotation
e.g.
Grandcléme
+
m
b
(bµ bνenthalpy
−i (bwith
· b)(u
uν cancels
+ gfield
))with
.
(14)
ϕ term
Thus,
this
last
its
counterpart
in
tities
(p(h,
b),lε
$
$
i
i
µ
n
0 and
−
(∂A
+
2N
∂A
)
∂
(β
−
ν)
∂x
ε
+
p
∂n
∂x
∂b
∂x
otion
be
introduced
in
grand
canonical
treatment
to
b
Ω
the
equilibrium
and
at
zero
temperature,
the
enthalpy
is
a
funcof
magnetisation,
i.e.
taking
x
=
0,
it
agrees
with
the
stant
ϕ
b
µ
0
b
nb µν
function
is following
given
st integral
the
µν
µ ν
algorithm
presen
of by
logarithm
of(32)
enthalpy
•ofderivative
(
Eq.
and
the
first
integral
(27)
keeps
exactly
the
same
!
"
'
#
(
%
The
free p
T
=
(ε
+
p)
u
u
+
p
g
$
$
!
"
ϕtionnum1
e average!(9)
particle number conservation.NThe
of
both
baryon
density
and
magnetic
field
dard
MHD
form,
see
e.g.
Gourgoulhon
(2012).
0)).
Once
r
θth
modification
of
∂A
1 pointing
∂Anet
"
#
$ϕ ∂n
$ last step.∂bIt would
ϕassumed
Note
that
we
have
zero
charge
in
the
be
interesting
to
se
∂
ln
h
1
∂p
∂p
∂b
1
E
E
S
=
p
+
form
as
without
magnetisation:
It
is
that
for
a
magnetic
field
in
z-direction
b
0obvious
the
EoS,
the
cu
r
$
$
−2
+
,
(20)
ty"operator
is
n̂
=
−i
ψ̄γ
Q̂ψ,
where
the
operator
Q̂
1
1
=
+
−
m
As
already
pointed
out
e.g.
by
Potekhin
&
Yakovlev
− mi ε i+µp ν. $
(32)
global
quan
µ νi
µν
2µ0 net
i.e.
depending
o
Z = ρ DψDψ̄ exp(
S̃)
, Lorentz
iterm
i $form
r ∂x(8)
∂r=
rbe
tan
θ∂n
∂θ
force
can
included
automatically
in
the
first
integral
if
a
nonzero
ch
−b
b
+
(b
·
b)u
u
+
g
(b
·
b)
ε
+
∂x
∂b
∂x
Γ
this
expression
reduces
to
the
well-known
with
magΩ
and
the
loga
ε
+
p
∂x
∂x
b+
=
µ
.
(28)
h
=
h(n
,
b)
=
"v
bb
nbb
s the
species
a ν(r,
with
(2012),
there
has
been
some
confusion
in
the
literature
about
− number
Fiρ j free density of the particle
µ
2
the
star’s
in
these
partial
ln
h(r,
θ)
+
θ)
−
ln
Γ(r,
θ)
+
M
(r,
θ)
=
const.
(37)
0
n
b
2xOnce
! Neglecting
" effectρ
the Ferrer
corresponding
current
is assumed.
θ the e
nd the term
0)).
netisation, with
see e.g.
et al. (2010).
the
given by
magnetostatic equilibrium
(with magnetisation)
p [MeV/fm3]
magnetic field dependent equation of state
0.004
250
L200
pairing
150
100
3
GP !
=−
(ψ̄Pη C ψ̄ T )(ψ T C P̄η ψ) ,
4 η=1
x
p [MeV/fm3]
.
B = 10 G
The evaluation of the matter pressure and energy density
B = 1019 G
400
(EoS) for different models of neutron star matter in the
350
presence of a magnetic field can be found in many papers in
300
the (recent) literature, see e.g. Noronha & Shovkovy (2007);
250
Rabhi et al. (2008); Ferrer et al. (2010); Rabhi & Providen200
cia (2011); Strickland et al. (2012); Sinha et al. (2013). Basi150
cally, charged particles become Landau quantized (Landau
100
& Lifshitz (1960)) in the plane perpendicular to the mag50
netic field. For our numerical applications, we will employ
0
the quark
model quantisation
in the MCFLinphase
to describe
the neu- to the magnetic 0field200 400 600 800 1000 1200 1400 1600
the direction
perpendicular
• Landau
3
! [MeV/fm
]
tron starThe
interior.
Let
us
now
briefly
summarise
the
main
(e)(c)
13
(p)(c)
20
critical field for electrons is Bm
= 4.4 × 10 G, and for protons it is Bm
∼ 10 G
•
characteristics of this model.
in MCFL
phase
(NoronhaFigure
and Shovkovy
Paulucci
2011)phase for different
• Example:
1. EoS 2007,
of quark
matter et
in al.
MCFL
The
effect of aQuark
strongMatter
magnetic
field on
quark mat3) + Pairing
fields interaction of NJL-type
ter was• extensively
studiedMIT
earlier
many
authors,
massless 3-flavor
Bag by
model
(with
B = 60see
MeV/fmmagnetic
Oertel
e.g. Gatto & Ruggieri (2013); Ferrer & de la Incera (2013)
and references therein. Here, we employ a simple massless
three-flavor
0.008
500 MIT bag model, supplemented with a pairing
µB = 1200 MeV
B
=
0
interaction
to include the possibility of colour
18
µ
450of NJL-type
0.007
B = 1500 MeV
B = 10 G
19
superconductivity
in Gthe colour-flavor locked state similar
B = 10
400
0.006
to the model
in
Noronha
&
Shovkovy
(2007);
Paulucci
et
al.
350
0.005
(2011), 300
(15)
2 0
where C =
50 iγ γ is the charge conjugation matrix. The
quark spinors
ψaα carry colour a = (1, 2, 3) and flavour
0
200 P̄400
1000 the
1200considered
1400 1600 pairα = (s, d, u) 0indices.
γ 0 Pη†800
γ 0 , and
η = 600
3
[MeV/fm ]5 abη $αβη , i.e. we only
ing matrix is given by (Pη )!ab
αβ = iγ $
take pairing in antisymmetric channels into account. FolFigure
1. EoS
quark matter
in MCFL
for different
lowing the
sameofscheme
as in Noronha
& phase
Shovkovy
(2007),
we
computed
the EoS
of quark matter
the MCFLonly
phase,
effects
offields
Landau
quantization
becomeinnoticeable
•magnetic
−2
using
GPof =
5.15
GeV , Λ = 1 GeV and a bag constant
19G.
for
fields
~ 10
Bbag = 60 MeV/fm3 .
The EoS
0.008 for different constant magnetic field values is
µB = 1200 MeV
displayed
in
Fig.
(1).
The
effect
of the magnetic
field starts
µM.
MeV
0.007
D. C., T. Elghozi,
Oertel,
J. Novak,
arXiv:1410.6332
B = 1500
to become significant only at very large fields (B ! 1019 G).
0.003
0.002
0.001
0
-0.001
100
1000
10000
B[1015 G]
Figure 2. Magnetisation divided by magnetic field as a function
of magnetic Magnetisation
field strength in the
MCFL phase.
negligible
•
• de Haas-van Alphen oscillations
i, j, . . . are used for spatial indices only, whereas Greek ones
α, µ, . . . denote the spacetime indices.
Within the theory of general relativity for the gravi-
maximal deformation due to magnetic field
8
BP = 8.16 x1017 G
D. Chatterjee, T. Elghozi, J. Novak and M. Oertel
Figure 3. Magnetic field lines (left) and enthalpy isocontours (right) in the meridional plane (x, z), for the static star configuration,
with a gravitational mass of 2.22M! and a polar magnetic field of 8.16 × 1017 G. The stellar surface is depicted by the bold line. In the
right figure, solid lines represent positive enthalpy isocontours, dashed lines negative ones (no matter).
Magnetic field lines and enthalpy isocontours in the meridional (x, z) plane for static configuration for
Bpolar=8.16x1017G, MG=2.22 Msol (including magnetic field effect in EoS)
• Stellar configurations strongly deviate from spherical symmetry
• Upon increasing magnetic field strength, the shape of the star becomes more and more
2.24
8.0!10
-4
EoS(B), no M
EoS(B), M
elongated, finally reaching toroidal shape
2.16
(B),no M)
2.2
6.0!10
-4
4
maximum gravitational
mass
Magnetic field effects in neutron stars
for
a
given
magnetic
moment
RESULTS AND DISCUSSION
15
G)
We computed models of fully relativistic neutron stars with
a poloidal magnetic field, employing the EoS described in
2.4Sec.(2.2), and a constant current function (33) f (x) = f .
600
0
As shown in Bocquet et al. (1995), the choice of other curno EoS(B), no M
EoS(B), no M
rent functions for f would not alter the conclusions. Vary500
x
EoS(B), M
2ing f allowed us to vary the intensity of the magnetic
field,
0
max
G
as measured for instance by the value of theMradial
com400
ponent at the star’s pole (polar magnetic field), or by the
1.6magnetic moment. The variation of the central enthalpy has
a direct influence on the star’s masses (MB and MG ), al300
though they depend on the rotation frequency
magnetic
31 and
2
10 Am
1.2field strength, too. To demonstrate pure magnetic
31
2
field ef2x10 Am
200
31
2
fects on the neutron star configurations, we
first
computed
3x10 Am
31
2
4x10 A m
static neutron stars.
31
2
m
100
0.8
The first point to emphasise is that, 5x10
as 31itAhas
already
2
6x10 A m
been illustrated, e.g. in Bocquet et al. (1995);
et al.
31Cardall
2
7x10 A m
(2001), the stellar configurations can strongly deviate from
0
0
100
200
50
150
0.4spherical symmetry due to the anisotropy of the energy30
2
0.1
0.2
0.3
0.4
0.5
Magnetic moment (10 A m )
2 of a non-vanishing electromomentum tensor in presence
Hc (c )
magnetic field. As an example we show in Fig. (3) the magnetic field lines and the enthalpy profile in the (r, θ)-plane
Figure 4. Polar magnetic field as a function of magnetic moment
32 enthalpy and magnetic field
Gravitational
mass
varies
with
central
log
•
for a configuration with a magnetic moment of 3.25 × 10 A
for constant current function and baryonic mass 1.6 M! with and
m2 and a•baryon
mass
of
2.56
M
.
These
values
correspond
! determined by different values
without
magnetic
field dependence
andconstant
magnetisation (see text
Static configurations
of central
log-enthalpy
along
17
to a polarsequences
magnetic field
of 8.16 ×10dipole
G and
a gravitational
for details).
of magnetic
moment
mass of 2.22 M! . The asymmetric shape of the star due to
gravitational
mass MGmax was
by parabolic interpolation
• Maximum
the Lorentz
forces exerted
by the electromagnetic
field determined
on
the fluid •isPlot
evident
from magnetic
the figures.field
Upon
increasing theto the values of magnetic moment for a neutron
of polar
corresponding
magnetic field strength the star’s shape becomes more and
star of MB =reaching
1.6 Msola toroidal shape, see Cardall
more elongated, finally
field
dependence
of the
and magnetisation.
The relation
D. C.,
T. Elghozi,
M.EoS
Oertel,
J. Novak, arXiv:1410.6332
et al. (2001). However, our code is not able to treat this
between the magnetic moment and the polar magnetic field
change of topology and the configuration shown in Figs. 3
changes only slightly depending on the baryon mass of the
Polar Magnetic Field (10
MG ( Msol )
7
Figure 3. Magnetic field lines (left) and enthalpy isocontours (right) in the meridional plane (x, z), for the static star configuration,
with a gravitational mass of 2.22M! and a polar magnetic field of 8.16 × 1017 G. The stellar surface is depicted by the bold line. In the
right figure, solid lines represent positive enthalpy isocontours, dashed lines negative ones (no matter).
effect of EoS(B) and M
2.24
8.0!10
-4
EoS(B), no M
EoS(B), M
(no EoS(B),no M)
2.2
4.0!10
max
2.12
-4
-4
max
/MG
MG
max
(Msol)
2.16
6.0!10
" MG
2.08
-4
EoS(B), M
EoS(B), no M
no EoS(B), no M
2.04
2
2.0!10
0
50
100
30
2
Magnetic moment (10 Am )
150
200
0.0
0
50
100
30
2
Magnetic moment (10 Am )
150
200
The 36.cases:
•Figure
Neutron star maximal mass (left panel) and relative difference in this mass among three models, as a function of magnetic
moment. The three models correspond to the possibility or not of including of magnetisation term x (“M” or “no M”), and to the
(i)
without magnetic field dependence in EoS, without magnetisation : no EoS(B), no M
magnetic field dependence or not of the EoS (“EoS(B)” or “no EoS(B)”).
(ii) with magnetic field dependence in EoS, without magnetisation: EoS(B), no M
(iii) with magnetic field dependence in EoS, with magnetisation: EoS(B), M
gravitational
mass and radius
was found
to decrease with increase in magnetic
Maximal
gravitational
mass is an increasing functionpactness
of magnetic
moment
•the
moment. This is understandable from the centrifugal forces
GM
G
of
magnetic
field
dependence
of the by
EoS
and
the force
magnetisation
arecenter, increas• The effects of inclusion
exerted
the
Lorentz
on
matter
at
the
C=
,
(38)
Rcirc
c2
negligible, contrary to the
claims
of several previous works
ing with increasing magnetic moment, i.e. magnetic field, see
where Rcirc is the circumferential equatorial radius (see
Bonazzola et al. (1993)). We studied the behaviour of the
e.g. the discussion in Cardall et al. (2001). Again the lines
D.
C., T. Elghozi,
Oertel,
J. without
Novak,magnetisation
arXiv:1410.6332
corresponding
to the M.
cases
with and
0.2
rotating configurations
Magnetic field effects in neutron stars
9
2.3
!=0
!=700 Hz
2.25
31
max
(Msol)
2.2
2
0.3
2
Hc (c )
0.4
2.15
MG
10 Am
31
2
2x10 Am
31
2
3x10 Am
31
2
4x10 A m
31
2
5x10 A m
31
2
6x10 A m
31
2
7x10 A m
2.1
2.05
0.5
2
0
50
100
30
2
Magnetic moment (10 Am )
150
200
l mass as a function of central logFigure 8. Maximum gravitational mass as function of magnetic
constant curves of magnetic dipole momoment for static (0 Hz) and rotating (700 Hz) configurations,
Observed magnetars are
(P ~ s)and magnetic field dependence in
configurations.
withslowly
inclusionrotating
of magnetisation
the EoS.
•
• We chose a sequence of neutron stars rotating at 700 Hz, close to fastest known rotating pulsar
(716 Hz)
withany
magnetic
momentcounterpart for the moment,
not have
realistic observed
• Maximum mass increases
we perform this investigation mainly for curiosity. As
magnetic field dependence of EoS again found to be negligible
• Effect of magnetisationandand
no EoS(B), no M
EoS(B), no M
obtained in the static case, the maximum gravitational mass
was found to increase with the magnetic dipole moment M.
In particular, we chose a sequence ofD.
neutrons
stars rotating
C., T. Elghozi,
M. Oertel,
J. Novak, arXiv:1410.6332
Hc (c )
Magnetic moment (10 Am )
Figure 6. Neutron star maximal mass (left panel) and rela
Figure 8. Maximum gravitational mass as function of m
models
correspond
to rotating
the possibility
o
moment for
static (0 Hz) and
(700 Hz) configu
with inclusion of magnetisation and magnetic field depen
magnetic field dependence
or not of the EoS (“EoS(B)” or “
the EoS.
Figure 5. Gravitational mass as a function of central logmoment.
enthalpy Hc , along seven constant curves
of magneticThe
dipolethree
moment M for non-rotating configurations.
compactness
0.2
Compactness MG/Rcirc
0.19
0.18
0.17
not have any realistic observed counterpart for the m
and we perform this investigation mainly for curios
obtained in the static case, the maximum gravitation
the gravitational mass
and radius
was found to increase with the magnetic dipole mom
no EoS(B), no M
In particular, we chose a sequence of neutrons stars r
EoS(B), no M
GM
at 700 Hz, close
to G
the frequency of the fastest known
EoS(B), M
C
=
,
ing pulsar, which
Rcircrotates
c2 at 716 Hz (Hessels et al. (20
Fig. (8) we see the same behaviour for both cases: th
mass increases with the magnetic field and, al
where Rcirc is the imal
circumferential
equatorial radius
it is not shown in the figure, the effects of magnet
or inclusion
the magneticthe
fieldbehaviour
are very small, of
as
Bonazzola et al. (1993)).
Weof studied
non-rotating case.
compactness of a neutron
star of baryon mass 1.6 M!
magnetic moment, as illustrated in the Fig. (7). The c
5
In this work, we developed a self-consistent approach
termine the structure of neutron stars in strong m
0
100
200
50
150
30
2
Magnetic moment (10 A m )
fields, relevant for the study of magnetars. Startin
the microscopic Lagrangian for fermions in a magnet
we derived a general expression for the energy-mom
Figure 7. Compactness as a function of magnetic moment for
tensor of one fluid in presence of a non-vanishing
neutron
star
with
baryon
mass
1.6
M
with
and
without
magnetic
We studied the behaviour of compactness of a !neutron star with baryon mass
1.6 with
magnetic
field. magnetic
Due to themoment
perfect conductor assum
field dependence and magnetisation (see Fig.6).
the electric field vanishes in the fluid rest frame, and
The compactness was found to decrease with increase in magnetic moment
fore only magnetisation and the magnetic field depe
of the equation
of statemoment
enter the final results. Eq
Centrifugal forces exerted by the Lorentz force on matter increases with increasing
magnetic
able and the main effect arises from the purely electromagfor the star’s equilibrium are obtained as usual fr
The influence of magnetic
field
dependence
of
EoS
and
magnetisation
are
negligible
netic part already included in Bocquet et al. (1995).
conservation of the energy-momentum tensor cou
Maxwell and Einstein equations. This consistent der
we purely
computed
rotating configurations
The main effect arisesFinally,
from the
electromagnetic
part along a
shows in particular that, as claimed by Blandford &
sequence of constant magnetic dipole moments. For the moquist (1982), the equilibrium only depends on the t
ment the observed magnetars all rotate very slowly with peD.
C.,
T.
Elghozi,
M. Oertel, J. Novak, arXiv:1410.6332
dynamic equation of state and magnetisation explicit
riods of the order of seconds, see Mereghetti (2013), mainly
enters Maxwell and Einstein equations. This should
because the strong magnetic fields induce a very rapid spin0.16
•
•
•
•
•
CONCLUSIONS
summary
• In this work, we developed a self-consistent approach to determine the structure of
neutron stars in strong magnetic fields, relevant for the study of magnetars
• Taking as an example the EoS of quark matter in MCFL phase, we investigated the
effect of inclusion of magnetic field dependence of the EoS and magnetisation
•
In particular, it was found that the equilibrium only depends on the thermodynamic
EoS and magnetisation explicitly only enters Einstein-Maxwell equations
• In contrast to previous studies, we found that these effects do not significantly
influence the stellar structure, even for the strongest magnetic fields considered
• The difference arises due to the fact that in previous works isotropic TOV equations
were used to solve for stellar structure, whereas magnetic field causes the star to
deviate from spherical symmetry considerably