Mass, Radius and Moment of Inertia of Neutron Stars · Riccardo Belvedere, Jorge A. Rueda, Remo Ruﬃni Mass, Radius and Moment of Inertia of Neutron Stars Core-Crust transition layer

Mass, Radius and Moment of Inertia of Neutron Stars

Riccardo Belvedere, Jorge A. Rueda, Remo Ruffini

Sapienza Universita di RomaICRA & ICRANet

X-Ray Astrophysics up to 511 KeVFerrara, Italy, 14-16 September 2011

Riccardo Belvedere, Jorge A. Rueda, Remo Ruffini Mass, Radius and Moment of Inertia of Neutron Stars

Standard approach to neutron stars (a)

In the standard picture the structure of neutron stars is given by:

Core:mainly neutrons and a small presence of protons and electrons in β-equilibrium

local charge neutrality ne = np in order to close the system of equations ⇒ a priorineglecting of the electromagnetic interactions

ρ ≥ ρ0, where ρ0 ≈ 2.7 × 1014g cm−3 is the nuclear saturation density for ordinarynuclei

Crust:inner crust: nuclei lattice (Coulomb lattice) in a background of electrons and neutrons;

ρ0 ≤ ρ ≤ ρdrip , where ρdrip ≈ 4.33 × 1011g cm−3 is the neutron-drip density

outer crust: nuclei lattice in a white-dwarf-like material (free electrons); ρ ≤ ρdrip

(Note that the core-crust transition happens when the uniform phase becomesenergetically favorable with respect to the non-uniform Coulomb lattice)


Standard approach to neutron stars (b)

The system of equation for such an object is given by local charge neutrality andβ-equilibrium in addition to the Einstein equations:

dP

dr= −

G(E + P)(M + 4πr3P)

r2(1− 2GMr

),

dM

dr= 4πr2E,

where M = M(r) is the mass enclosed at the radius r and the energy-density E andthe pressure P are given by

E =∑

i=n,p,e

Ei , P =∑

i=n,p,e

Pi ,

being Ei and Pi the single fermion fluid contributions

Ei =2

(2π)3

∫ PFi

0ǫi (p) 4πp

2dp,

Pi =1

3

2

(2π)3

∫ PFi

0

p2

ǫi (p)4πp2dp.


New approach to neutron stars

We follow a new description for neutron stars:

Core:

degenerate gas of neutrons, protons and electrons in β-equilibrium

global charge neutrality Ne = Np ⇒ the electromagnetic interactions are taken intoaccount

gravitational-electrodynamical-strong-weak equilibrium ensures the constancy of thegeneral relativistic Fermi energy EF

i of all particles species

ρ ≥ ρ0 as in the standard picture

Crust:

inner crust: disappears in favor of a tiny transition layer (details in the next slides)

outer crust: as in the standard description, nuclei lattice in a white-dwarf-like material(free electrons); ρ ≤ ρdrip


Core: Quantum Hadrodynamical Model (a)

How describe electromagnetic interaction between electrons and protons plus nuclearinteraction???

We follow the so-called Walecka model, in which the nucleons interact through aYukawa coupling, so that the strong interaction is modeled by a meson-exchange ofthe σ, ω and ρ virtual meson-fields.

σ=isoscalar meson fields→ attractive long-range nuclear force

ω=massive vector field→ repulsive short-range nuclear force

ρ=massive isovector field→ surface effects modeling a repulsive nuclear force


Core: Quantum Hadrodynamical Model (b)

The total Lagrangian density of the system has to take into account gravitational,strong, weak and electromagnetic interactions and is given by

L = Lg + Lf + Lσ + Lω + Lρ + Lγ + Lint,

Lg = −R

16πG,

Lγ = −1

4FµνF

µν ,

Lσ =1

2∇µσ∇

µσ − U(σ),

Lω = −1

4ΩµνΩ

µν +1

2m2

ωωµωµ,

Lρ = −1

4RµνR

µν +1

2m2

ρρµρµ,

Lf =∑

i=e,N

ψi (iγµDµ −mi )ψi ,

Lint = −gσσψNψN − gωωµJµω − gρρµJ

µρ

+ eAµJµγ,e − eAµJ

µγ,N ,


Core: Einstein-Maxwell-Dirac Equations in Spherical Symmetry (a)

We consider non-rotating spherically symmetric neutron stars, so we introduce the spacetimemetric

ds2= e

ν(r)dt

2− e

λ(r)dr

2− r

2dθ

2− r

2sin

2θdϕ

2.

Since the very large number of fermions (∼ 1057), we can adopt the mean-field approximation, sothat the full system of general relativistic constitutive equations become

e−λ(r)

(

1

r2−

1

r

dλ

dr

)

−1

r2= −8πGT

00 ,

e−λ(r)

(

1

r2+

1

r

dν

dr

)

−1

r2= −8πGT

11 ,

V′′+

2

rV

′

[

1 −r(ν′ + λ′)

4

]

= −4πe eν/2

eλ(np − ne),

d2σ

dr2+

dσ

dr

[

2

r−

1

2

(

dν

dr+

dλ

dr

)]

= eλ[∂σU(σ) + gsns ] ,

d2ω

dr2+

dω

dr

[

2

r−

1

2

(

dν

dr+

dλ

dr

)]

= −eλ(

gωJ0ω − m

2ωω

)

,

d2ρ

dr2+

dρ

dr

[

2

r−

1

2

(

dν

dr+

dλ

dr

)]

= −eλ(

gρJ0ρ − m

2ρρ

)

,

EFe = e

ν/2µe − eV = constant, (1)

EFp = e

ν/2µp + Vp = constant,

EFn = e

ν/2µn + Vn = constant.


Core: Einstein-Maxwell-Dirac Equations in Spherical Symmetry (b)

T 00 = E + (E2/8π) + U(σ) + (1/2)Cωn

2b+ (1/2)Cρn

23

T 11 = −P + (E2/8π) + U(σ)− (1/2)Cωn

2b+ (1/2)Cρn

23

Jω0 = (nn + np)eν/2 = nbu0; Jρ0 = (np − nn)eν/2 = n3u0; J

ch0 = (np − ne)eν/2 =

nchu0

EFn = EF

p + EFe

mN = mN + gsσ; me = me

Vp = gωω + gρρ+ eV

Vn = gωω − gρρ

ne = e−3ν/2

3π2 [V 2 + 2me V −m2e(e

ν − 1)]3/2,whereV ≡ eV + Ee

ns = 〈ψNψN〉 =2

(2π)3

∑

i=n,p

∫

d3kmNǫi (p)

e−λ(r) = 1−

2GM(r)r

+ Gr2E2(r) = 1−2GM(r)

r+

GQ2(r)

r2


Core: Einstein-Maxwell-Dirac Equations in Spherical Symmetry (c)

In order to integrate the equilibrium equations we need to fix the parameters of thenuclear model, namely, fixing the coupling constants gs , gω and gρ, and the mesonmasses mσ , mω and mρ. Conventionally, such constants are fixed by fittingexperimental properties of nuclei. Usual experimental properties of ordinary nucleiinclude saturation density, binding energy per nucleon (or experimental masses),symmetry energy, surface energy, and nuclear incompressibility. In the following tablewe present selected fits of the nuclear parameters. In particular, we show the followingparameter sets: NL3, NL-SH, TM1, and TM2.

NL3 NL-SH TM1 TM2mσ (MeV) 508.194 526.059 511.198 526.443mω (MeV) 782.501 783.000 783.000 783.000mρ (MeV) 763.000 763.000 770.000 770.000gs 10.2170 10.4440 10.0289 11.4694gω 12.8680 12.9450 12.6139 14.6377gρ 4.4740 4.3830 4.6322 4.6783g2 (fm−1) -10.4310 -6.9099 -7.2325 -4.4440g3 -28.8850 -15.8337 0.6183 4.6076c3 0.0000 0.0000 71.3075 84.5318

Table: Selected parameter sets of the σ-ω-ρ model.


Core-Crust transition layer (a)

the Coulomb potential energy inside the core of the neutron star is found to be∼ mπc

2 and related to this there is an internal electric field of order ∼ 10−14Ec

where Ec is the critical field for vacuum polarization.

at the core-crust boundary the continuity of all general relativistic particle Fermienergies guarantee a self-consistent matching of the core and the crust: acore-crust transition surface of thickness ≥ λe = ~/(mec) ∼ 100fm (electronscreening scale) is developed in which an overcritical electric field appears.

the continuity of the electron Fermi energy force the variation of electronchemical potential at the core-crust boundary to be of orderµe(core)−µe(crust)∼ eV (core). mπc

2.

therefore we obtain a suppression of the so-called inner crust of the neutron starif µe(crust)∼ µe(core)-eV(core).25 MeV, which is approximately the value of theelectron chemical potential at the neutron drip point.


Core-Crust transition layer (b)

At the transition-point from the core to the crust of the neutron star, themean-field-approximation for the meson-fields is not valid any longer and therefore fullnumerical integration of the meson-field equations of motion, taken into account allgradient terms, must be performed. Since ,as can be shown numerically, the metricfunctions hold almost constant on the core-crust transition layer and then have theirvalues at the core-radius eνcore ≡ eν(Rcore) and eλcore ≡ eλ(Rcore), the system ofequations can be reduced to

V ′′ +2

rV ′ = −eλcoreeJ0ch ,

σ′′ +2

rσ′ = eλcore [∂σU(σ) + gsns ] ,

ω′′ +2

rω′ = −eλcore

[

gωJ0ω −m2

ωω]

,

ρ′′ +2

rρ′ = −eλcore

[

gρJ0ρ −m2

ρρ]

,

eνcore/2µe − eV = constant ,

eνcore/2µp + eV + gωω + gρρ = constant ,

µn = µp + µe + 2 gρρe−νcore/2 .


Core-Crust transition layer (c)

0 50 100 150 2000

500

1000

1500

2000

Hr-RcoreLΛΣ

E

Ec0 50 100 150 200

10-410-310-210-1

0 50 100 1500

50

100

150

200

250

300

Hr-RcoreLΛΣ

E

Ec0 50 100 150

10-510-410-310-210-1

Figure: Left:gρ 6= 0.Right : gρ = 0

Left region: on the proton-profile we can see a bump due to Coulomb repulsion while theelectron-profile decreases. Such a Coulomb effect is indirectly felt also by the neutrons due tothe coupled nature of the system of equations. However, the neutron-bump is much smallerthan the one of protons and it is not appreciable due to the plot-scale.

Central region: the surface tension due to nuclear interaction produces a sharp decrease ofthe neutron and proton profiles in a characteristic scale ∼ λπ . Moreover, it can be seen aneutron skin effect, analogous to the one observed in heavy nuclei, which makes the scale ofthe neutron density falloff slightly larger with respect to the proton one.

Right region: smooth decreasing of the electron density similary to the behavior of theelectrons surrounding a nucleus in the Thomas-Fermi model


Crust, two different EoS (a)

Due to the neutrality of the crust, the structure equations to be integrated in the crustare the Tolman-Oppenheimer-Volkoff equations.

In order to see the effects of the microscopic screening on the structure of theconfiguration we will consider two equations of state for the crust:

the locally neutral case (uniform approximation) used for instance byChandrasekhar in the study of white dwarfs

the equation of state due to Baym, Pethick and Sutherland (BPS), which is byfar the most used equation of state in literature for the description of the neutronstar crust


Crust, two different EoS (b)

In the uniform approximation, both the degenerate electrons and the nucleonsdistribution are considered constant inside each cell of volume Vws. This kind ofconfiguration can be obtained only imposing microscopically the condition of localcharge neutrality

ne =Z

Vws

. (2)

The total pressure of the system is assumed to be entirely due to the electrons, i.e.

P = Pe =2

3 (2π~)3

∫ PFe

0

c2p24πp2√

c2p2 +m2ec

4dp, (3)

while the total energy-density of the system is due to the nuclei, i.e. E=(A/Z)mNne ,where mN is the nucleon mass.


Crust, two different EoS (c)

In the BPS model, we have a point-like nucleus surrounded by uniformly distibutedelectrons.The sequence of the equilibrium nuclides present at each density in the BPS equationof state is obtained by looking for the nuclear composition that minimizes the energyper nucleon for each fixed nuclear composition (Z ,A).

P = Pe +1

3WLnN ,

E

nb=

WN +WL

A+

Ee(nbZ/A)

nb,

where the electron energy-density is given by

Ee =2

(2π)3

∫ PFe

0

√

p2 +m2e4πp

2dp,

and WN(A,Z) is the total energy of an isolated nucleus given by the semi-empiricalformula

WN = mnc2(A− Z) +mpc

2Z − bA,

with b being the Myers and Swiatecki binding energy per nucleon. The lattice energyper nucleus WL is given by

WL = −1.819620Z2e2

a.


Results (a)

0 2 4 6 8 10ρ(0)/ρnuc

0.5

1.0

1.5

2.0

2.5

3.0

M/M

⊙

NL3NL−SHTM1TM2

10.5 11.5 12.5 13.5 14.5R(km)

0.5

1.0

1.5

2.0

2.5

3.0

M/M

⊙

NL3NL−SHTM1TM2

0.5 1.0 1.5 2.0 2.5 3.0M/M⊙

0.10

0.15

0.20

0.25

0.30

0.35

GM/(c

2R)

NL3NL−SHTM1TM2

10.5 11.5 12.5 13.5 14.5R (km)

0.10

0.15

0.20

0.25

0.30

0.35

GM/(c2R)

NL3NL−SHTM1TM2

Figure: Top:Mass-central mass-density relation (left panel) and mass-radius relation (right panel),in the BPS-EoS

case. The mass has been plotted in solar masses, the mass density in nuclear mass density ρnuc = 2.7 × 1014 g

cm−3, and the star radius is given in km. Bottom: compactness of the star GMcore/(c2Rcore) as a function of

the star mass Mcore (left panel) and as a function of the star radius Rcore (right panel), for the BPS-EoS


Results (b)

0.5 1.0 1.5 2.0 2.5 3.0M/M⊙

2

4

6

8

MNO−C

crust

(10−5M⊙)

NL3NL−SHTM1TM2

0.5 1.0 1.5 2.0 2.5 3.0M/M⊙

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

∆R

NO−C

crust

(km)

NL3NL−SHTM1TM2

0.5 1.0 1.5 2.0 2.5 3.0M/M⊙

1

2

3

4

5

MBPS

crust(10−5M⊙)

NL3NL−SHTM1TM2

0.5 1.0 1.5 2.0 2.5 3.0M/M⊙

0.1

0.2

0.3

0.4

0.5

0.6

∆R

BPS

crust(km)

NL3NL−SHTM1TM2

Figure: Top: mass versus compactness and crust-thickness versus compactness for crust withoutCoulumb-interactions. Bottom: mass versus compactness and crust-thickness versus compactness for crust withCoulumb-interactions.


Crust comparison (a)

Equilibrium Nuclei Below Neutron Drip

Nucleus Z ρmax (g cm−3) ∆ R1 (km) R.A.1(%) ∆ R2 (km) R.A.2(%)56Fe 26 8.1 × 106 0.0165 7.56652 × 10−7 0.0064 6.96927 × 10−7

62Ni 28 2.7 × 108 0.0310 0.00010 0.0121 0.0000964Ni 28 1.2 × 109 0.0364 0.00057 0.0141 0.0005484Se 34 8.2 × 109 0.0046 0.00722 0.0017 0.0068382Ge 32 2.2 × 1010 0.0100 0.02071 0.0039 0.0198380Zn 38 4.8 × 1010 0.1085 0.04521 0.0416 0.0438478Ni 28 1.6 × 1011 0.0531 0.25635 0.0203 0.2530576Fe 26 1.8 × 1011 0.0569 0.04193 0.0215 0.04183

124Mo 42 1.9 × 1011 0.0715 0.02078 0.0268 0.02076122Zr 40 2.7 × 1011 0.0341 0.20730 0.0127 0.20811120Sr 38 3.7 × 1011 0.0389 0.23898 0.0145 0.24167118Kr 36 4.3 × 1011 0.0101 0.16081 0.0038 0.16344

Table: ρmax is the maximum density at which the nuclide is present;∆ R1, ∆ R2 and R.A.1(%),R.A.2(%) are rispectively the thickness of the layer where a given nuclide is present and theirrelative abundances in the outer crust for two different case, one with Mcore = 2.6M⊙ andRcore = 12.8Km, and one with Mcore = 1.4M⊙ and Rcore = 11.8Km.


Crust comparison (b)

7.56652´10-7

0.000100257

0.0005736960.00722231

0.02071010.0452611

0.256348

0.0419303

0.0207749

0.207296

0.238978

0.160805

Fe56 Ni63 Ni64 Se84 Ge82 Zn80 Ni78 Fe76 Mo124 Zr122 Sr120 Kr118

6.96927´10-7

0.0000929709

0.0005356470.00682903

0.01982980.0438411

0.253049

0.0418346

0.0207599

0.208114

0.241673

0.163439

Fe56 Ni63 Ni64 Se84 Ge82 Zn80 Ni78 Fe76 Mo124 Zr122 Sr120 Kr118

Figure: Relative abundances of chemical elements in the outer crust of neutron stars for the twodifferent cores: Mcore=2.6M⊙, Rcore=12.8 Km and Mcore=1.4M⊙, Rcore= 11.8 Km .

In both cases we obtain as average nuclear composition 10535 Br

Using these value for < A > and < Z > in the Chandrasekhar approximation weobtain:

Mchandracrust ∼ 1.18MBPS

crust and ∆Rchandracrust ∼ 0.95∆RBPS

crust


Observational constraints

8 10 12 14 16 18R (km)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

M/M

NL3NLSHTM1TM2

Figure: Constraints on the mass-radius relation given by J. E. Trumper and the theoreticalmass-radius relation presented in this slides. The solid line is the upper limit of the surface gravityof XTE J1814-338, the dotted-dashed curve corresponds to the lower limit to the radius of RXJ1856-3754, the dashed line is the constraint imposed by the fastest spinning pulsar PSRJ1748-2246ad, and the dotted curves are the 90% confidence level contours of constant R∞ of theneutron star in the low-mass X-ray binary X7. Any mass-radius relation should pass through thearea delimited by the solid, the dashed and the dotted lines and, in addition, it must have amaximum mass larger than the mass of PSR J1614-2230, M = 1.97 ± 0.04M⊙.


Magnetic field and momentum of inertia (a)

The upper limit on the magnetic field of a pulsar, obtained by requesting that therotational energy loss due to the dipole field be smaller than the electromagneticemission of the dipole, is given by

B =

(

3c3

8π2

I

R6PP

)1/2

,

where P and P are the rotational period and the spin-down rate of the pulsar whichare observational properties, and the moment of inertia I and the radius R of theobject are model dependent properties.


Magnetic field and momentum of inertia (b)

The general relativistic moment of inertia of the star can be obtained from the integral

I =8π

3

∫

R

0r4(E + P)e

λ−ν2 dr,

where the effects due to rotation and deformation are neglected.Due to the smallness of the rotation rates of these pulsars as compared to the maximum rotation rate of a neutronstar, the effects of rotation can be obtained by using the Hartle-Thorne formalism which approximately describesthe gravitational field of rotating stars up to second order in the angular velocity and up to first order in thequadrupole deformations. Within this formalism, the moment of inertia is given by

I =8π

3

∫

R

0r4(E + P)e

λ−ν2

ω

Ωdr,

where ω is the angular velocity of the fluid relative to the local inertial frame and Ω is the uniform angular velocityof the star. The function ω is obtained from the differential equation

1

r4

d

dr

(

r4jdω

dr

)

+4

r

dj

drω = 0 ,

where j(r) = exp−(λ + ν)/2, being λ(r) and ν(r) the metric functions of the corresponding non-rotatingconfiguration.

1.0 1.5 2.0 2.5M/M

0.2

0.4

0.6

0.8

1.0

1.2

1.4

I(1045gcm

2)

nonrotatingHartleThorne

Figure: Total momentum of inertia I as a function of the total mass M, for the NL3 parametrization.


Implications on the Magnetic-dipole model of Pulsar

High-Magnetic Field Pulsar B/Bc

J1846-0258 1.11J1819-1458 1.14J1734-3333 1.18J1718-3718 1.68J1847-0130 2.14

Table: Magnetic fields of the overcritical high-magnetic field pulsars assuming a canonical neutronstar of M = 1.4M⊙, R = 10 km and I = 1045 g cm2.

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4M/M⊙

0.4

0.6

0.8

1.0

1.2

B/B

c

J1819−1458J1734−3333J1718−3718

J1846−0258J1847−0130

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4M/M⊙

0.3

0.4

0.5

0.6

0.7

B/B

cF.D.

J1819−1458J1734−3333J1718−3718

J1846−0258J1847−0130

Figure: Magnetic field B, in units of the critical field Bc = m2e c

3/(e~) ∼ 4.42 × 1013 G, versus the total massof the star, for the NL3 parametrization without and with rotation effects (left and right panel respectively). Inparticular, we plot the magnetic fields of the high-magnetic field pulsars of previous table.


End of the talk

Thank you


Documents

Mass, Radius and Moment of Inertia of Neutron Stars · Riccardo Belvedere, Jorge A. Rueda, Remo Ruﬃni Mass, Radius and Moment of Inertia of Neutron Stars Core-Crust transition layer