Differentially rotating neutron stars in general relativity

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Universit`

a di Pisa

Dipartimento di Fisica

Corso di Laurea Magistrale in Fisica Elaborato Finale

Differentially rotating neutron stars

in General Relativity

Relatore Dr. Ignazio Bombaci Candidato Francesco Niccoli 11/12/2017 Anno Accademico 2016/2017

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LIST OF FIGURES

2.1 Interior of a neutron star as proposed by P. Hansel,A.Y. Potekhin and D.G. Yakovlev in their book Neuton Stars 1. Distances values reported in figure, have the role to give a general idea on neutron star’s sizes . . . 11 2.2 Bi-logarithmic plot of the pressure as function of the total

energy density for various EoSs. The left panel is a zoom at the high density regime. . . 13 2.3 Profile of pressure and mass as function of the radial

coor-dinate from the centre of symmetry for a spherical NS de-scribed by a politropic EoS. The solid red line and the blue dot-dashed line have been obtained integrating the equations for the equlibrium in newtonian gravity and GR respectively for the same central pressure. . . 19 2.4 The panels are an illustrative example of non rotating stellar

structures among their stable branches for various EoSs. On the left is reported the central total energy density-gravitational mass plot with the corresponding configurations in the right panel where models are displayed as function of the circum-ferential equatorial radius. Sequences are interrupted once the upper limit mass has been reached. . . 19

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List of Figures

2.5 Representation of the dragging due to an object of angular velocity Ω as measured at infinity along the equatorial plane (θ = π/2 , uθ = 0). ω(r) is the angular velocity of the local inertial frame of radial coordinate r with respect to the grav-itational source. Clearly the frequency of the dragged frame goes to zero at great distances where the spacetime is flat. . . 21 2.6 Constant angular velocity sequences of uniform rotating

struc-tures. These curves move between the static configuration (solid black line) and the keplerian one (dashed black line) the latter being the upper limit on the rotation velocity be-yond which particles at the equator would be lost by the star. The EoS used for the plot is the TM1-2 . . . 24 2.7 Here are visible several sequences for a given value of the

dimensionless angular momentum for structures in rigid mo-tion for the TM1-2 EoS. In the inset is a particular of the plot where it is possible appreciate better the separation from sec-ular unstable configurations marked with the black dotted line i.e. turning-point line. The turning point for spheri-cal models coincide with the maximum value for the static sequence (blue point) while differs from the maximum mass model of the keplerian sequence (green point). The curves show models for constant dimensionless angular momentum j ≡ cJ/(GMJ2 ) . . . 24

2.8 The two plots illustrate sequences of non rotating models (solid blue line) and of structures at the mass-shed limit for rigidly rotating configurations (solid red line) together with curves of fixed rest masses (dashed black lines) and constant angular momenta (dotted black lines) for the TM1-2 EoS. . . 25 2.9 Contour plot of vertical sections of NSs in the static and rigid

rotation with same central total energy density for the TM1-2 EoS in the left and in the right panel respectively. The colors mark the variation of internal distribution of the total mass density while the x and y axis have spatial dimensions. Effects of rotations on the structures are strongly visible here, where the rotating model result in a more oblate geometry. . 26 2.10 Vertical section of NSs rotating differentially with same εcand

axis ratio for the TM1-2 EoS. The only variation in the inputs for the two plots is in the differential parameter ˆA−1 which measures 0.5 and 1 for the left and the right panel respec-tively. The colors evaluate the changing on the total mass density distribution. A remarkable aspect comes observing that now the configurations could have central total energy density different from the maximum one. . . 28

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List of Figures

2.11 Trends of rotational frequencies as function of the radial coor-dinate considered at the equatorial plane for objects rotating differentially at ˆA−1 = 1, 0.5 and 0.3 respectively solid, dashed and dot dashed black lines. In particular the plot highlights the decreasing in the difference between the central and the equatorial frequencies as parameter ˆA−1 moves to zero. . . . 29 2.12 From the left the two panels display curves of sequences

tak-ing advantage of the TM1-2 EoS in the central total en-ergy density-gravitational mass (εc/c2, Mg) and

circumferen-tial equatorial radius-gravitational mass (Re, Mg) planes

re-spectively. The models satisfy the requirements on the ob-served mass value, marked with a colored horizontal line, and on the maximum frequency detected the latter showed dis-playing sequences at angular velocities of 600 Hz, 800 Hz and 1000 Hz. . . . 31

3.1 Contour plots of differentially rotating objects of last obtain-able configuration at ˆA−1 = 1 (left panel) and of a mass-shedding model considered at ˆA−1= 0.3 (right panel). The x and y axis have spatial dimensions while the colors describe the total energy density distribution. In particular figure shows the discrepancy between the central and the maximum total energy density. . . 36 3.2 Differentially rotating models at the keplerian limit for the

ˆ

A−1 = 1 and ˆA−1 = 0.5 cases compared together with the mass-shed configurations for rigid structures, in the maximum total energy density-rest mass (εmax/c2, M0) plane for the

TM1-2 EoS. . . 37 3.3 Type A and type C models for the TM1-2 EoS are displayed

in the left and right panels respectively where are shown se-quences of rest masses as function of the Rperatio. On the left

it is possible appreciate the shift of the keplerian models to lower Rpe with respect to the maximum mass structures. On

the right the maximum mass model is expected at Rpe = 0

and coincides with the keplerian one. However RNS fails to converge at Rpe ∼ 0.22 so that we can obtain just a lower

limit for mass shed structures above ˆA−1_crit. . . 38 3.4 Sequences of fixed angular momenta are displayed here for

structures rotating differentially at ˆA−1 = 1 and ˆA−1 = 0.5 for the left and the right panels respectively, in the maximum total energy density-rest mass (εmax/c2, M0) plane making

use of the TM1-2 EoS. Black points locate the turning points while for comparison it has been plotted as solid black line the j = 0 curve i.e. the non rotating models. . . . 39

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List of Figures

3.5 The plot reports equation (3.4) (solid black line) and the turn-ing point lines for the ˆA−1 = 1 (blue points) and ˆA−1 = 0.5 (green points) cases in the gravitational mass-angular mo-mentum (Mg, j) plane. The match between the lines

individ-uates the threshold model of the remnant resulting from the merger of two NSs. . . 41 3.6 Curves of fixed angular momentum of differentially rotating

models at ˆA−1 = 0.5 are displayed with sequences of configu-rations in the uniform keplerian (solid black line) and static (thin black line) cases for the TM1-2 EoS and the threshold on the total rest mass for system of BNSs. Remnants above this limit do not exist stable. . . 42 3.7 The angular momentum of models at the keplerian

configura-tions (dashed black line) and at the turning point (blue filled cirlces) for rigid structures produced using the TM1-2 EoS, are displayed in the plot as function of the rest mass. The match between the two curves localizes the structure at the highest angular momentum over which the compact object could not live stable. . . 43 3.8 The three panels give the ”coordinates” in the rest

mass-angular momentum system of the theoretical model of the remnant that we expect could be found from the BNS re-sponsible of the GW170817 (red point), compared with the mass-shed sequence considered for parameters ˆA−1 = 0.3, 0.5 and 1 respectively from left to right, for the TM1-2 EoS. . . . 44 3.9 Contour plots of possible outcomes of the BNSs responsible

of the GW170817 signal for three distinct differential parame-ters. On the top is displayed the remnant at ˆA−1= 0.7 while on the left and on the right panels there are the ˆA−1= 1 and

ˆ

A−1 = 1.42 cases respectively. . . . 45 3.10 Trend of the angular momentum for the uniform keperian

sequence (black dashed line) and the uniform turning points (blue filled circles) as function of the rest mass. The red point corresponds to the remnant rotating differentially with coordinate in the plane of (2.95 MJ, 6.41434 GMJ2 /c), while

the green point marks the first possible stable structure in the rigid pattern for the same rest mass and has coordinate (2.95 MJ, 4.48777 GMJ2 /c). . . 46

A.1 Hierarchy on the codes through which have been developed the tables below. In particular from our modified version of RNS we got the parameters necessary to run rns and Lorene which are (εc, rp/re) and (Hc, Ωc/2π) respectively. . . 59

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List of Figures

A.2 Radius-mass graphs in which we plot non rotating sequences performed making use of Lorene (red solid line), rns (blue dot-dashed line) and RNS (blue dotted line) for the TM1-2 EoS. The right panel is a magnification of the left one around the maximum mass models and no sensible differences seem to appear from the use of a code in spite of one other. . . 59

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LIST OF TABLES

2.1 Rapid description of EoSs used in this section. For a deeper understanding of these models see the references reported. . . 17 2.2 Turning points for models of TM1-2 EoS in the uniform

ro-tating case. . . 27 2.3 Some important quantities used in the present pages with

corresponding definitions. . . 30 2.4 Properties of the maximum mass model of static sequeces at

various EoSs. . . 32 2.5 Properties of the maximum mass model of uniformly rotating

sequences for fixed frequency f = Ω/2π = 716 Hz at various EoSs. . . 32 2.6 Properties of the maximum mass model of uniformly rotating

sequences at the mass-shed limit at various EoSs. . . 32 2.7 Properies of differentially rotating models for fixed ratio /rpre

and central rest mass density ρc/c2 at various EoSs with

pa-rameters ˆA−1= 1, 0.5, 0.3. . . . 33 2.8 Summary of the findings concerning the maximum mass

mod-els of sequences at the static configuration, uniform rotation at f = 716 Hz, uniform rotation at the mass-shed limit and differential rotation for the TM1-2 EoS. . . 33

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List of Tables

3.1 Properties of three models at the mass-shed limit at uniform rotation and differential rotation with parameters ˆA−1 = 1 and ˆA−1 = 0.3 and compared with the static case, all con-sidered at the same maximum energy density for the TM1-2 EoS. . . 48 3.2 Properies of the maximum mass models for mass-shed

se-quences at uniform rotation and differential rotation with pa-rameters ˆA−1 = 1 and ˆA−1 = 0.3 and compared with the analogous of the static case, for the TM1-2 EoS. . . 48 3.3 Properies of the turning points with differential parameter

ˆ

A−1 = 0.5 for the TM1-2 EoS. . . . 48 3.4 Properies of the turning points with differential parameter

ˆ

A−1 = 1 for the TM1-2 EoS. . . . 49 3.5 Properies of theoretical remnants resulting form the merging

of the BNSs responsible of the GW170817 signal, at various differential parameters and compared with the rigidly rotat-ing strucuture at the mass-shed, considered at the same rest mass for the TM1-2 EoS. . . 49

A.1 Properies of the maximum mass models of static sequences for the TM1-2 EoS. . . 60 A.2 Properies of models considered at same central density/central

enthalpy and axis ratio/central frequency in the rigid case for the TM1-2 EoS. . . 60 A.3 Properies of models considered at same central enthalpy,

cen-tral frequency and differential parameter ˆA−1 = 1 for the TM1-2 EoS. . . 61 A.4 Properies of models considered at same central enthalpy,

cen-tral frequency and differential parameter ˆA−1 = 0.5 for the TM1-2 EoS. . . 61 A.5 Properies of models considered at same central enthalpy,

cen-tral frequency and differential parameter ˆA−1 = 0.3 for the TM1-2 EoS. . . 61

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List of Tables

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INTRODUCTION

The recent multi-signals detection from the merging of two neutron stars (NSs) has definitely sanctioned the beginning of a new era with which probe the nature of these objects [62]. NSs indeed are some of the densest struc-tures known in the Universe and for this reason represent an ideal laboratory where provide connection between nuclear physics and astrophysics.

However the interest toward this kind of stars has known a fluctuating story from their first formulation in 1934 by Baade and Zwicky [6, 7], with a new impetus arrived from the first pulsar signal’s detection thanks to Jocelyn Bell in 1967 [27]. NSs are indeed an open window on both nuclear physics and astrophysics representing a natural laboratory for the ultra-dense matter whose behavior is quite obscure still today. But an other intriguing aspect comes considering that these stars are the most compact besides Black Holes (BH). In effect NSs result to have mass of M ∼2MJ according to present

measure of NS masses, corresponding to a Schwarzschild’s radius of rs∼6 km

while their typical radius measures R ∼10 km. Therefore both oscillating NS as well as binary systems of NS are expected to emit detectable gravitational waves (GW) from which it would be possible to understand better several properties of these stars as the matter they are made of.

With the present work we aim to study the effects of differential rotation that is thought to be found both in newly born NS and in compact objects resulting from the merger of binary NSs (BNS). In particular, we will deal with the latter case in order to understand what kind of evolutionary pattern would be followed by structures resulting from the merger. The reasons be-hind this work are then motivated by the importance of GW which depend on the fate of binary systems and on the equation of state (EoS) adopted as well. The total mass (M1+M2) of the two merging NSs is the crucial

param-eter which control the final fate of the compact object formed in the BNS merger. In other words depending on the values of (M1+M2) one could have

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Introduction

(i) a prompt collapse to a BH, (ii) a delayed formation of a BH, or (iii) the formation of a stable NS. The fate followed by the structures will produce different GW signals in the post-merger phase. Thus differentially rotating post-merger compact objects with ”large mass” (larger than the maximum mass for rigidly rotating neutron stars, i.e. the so called hypermassive con-figurations) can be temporarily stabilized and then form a BH within the relaxation time of differential rotation to rigid-body rotation. The study of differentially rotating relativistic stars is thus of great importance for the interpretation and modeling of the GW signal from BNS mergers.

The method adopted for our calculations of remnants’ models is based on the assumption that the angular momentum would be conserved. It means that no dissipative phenomena are taken into consideration leading differentially rotating structures to rigid rotation maintaining a constant value of the angular momentum. At the moment several works have been developed using schematic EoSs and particularly polytropic EoSs. On the contrary we use ”realistic” EoSs of nuclear matter (namely EoSs derived using various approaches to describe strong-interacting quantum many-body systems) at T = 0 i.e. neglecting thermal effects.

Differential rotation of remnants is expected to support structures against gravitational collapse over timescales of several milliseconds to minutes [14, 73]. Hence our analysis moved considering ”snapshots” of models provided using two open source codes for the numerical relativity which are Lorene and RNS which have been widely used by various research groups working in numerical relativity.

Consequently the present pages are organized as follow: in chapter 1 the fluids’ behavior will be explained in the General Relativity context with the assumptions done of perfect and zero-temperature fluid ; a description of internal structures, EoSs of neutron stars and models affected by different rotating patterns is illustrated in chapter 2; chapter 3 deals with a systematic description on the effects that differential rotation has on the structures, taking advantage of the TM1-2 EoSs with a section devoted to the modeling of remnants with the properties deducible from the GW170817 event showing the possible fate of post merger objects. Two appendices have been included describing the comparison between the codes used, and the math involved in the present pages.

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CHAPTER

1 RELATIVISTIC TREATMENT

Extension of hydrodynamics laws to the Special Relativity case is required whether the fluid examined presents velocities close to that of light c to a reference observer as well as when internal energy and pressure are non-negligible with respect to the rest mass-energy density, meaning that par-ticles have relativistic velocities. Such a fluid is clearly found in NS whose central density could reach the value of (7 −13)ρ0 with ρ0the nuclear density.

Moreover NS are collapsed structure supported just by degenerate matter with typical radius close to the Schwarzschild radius of a star of same mass. Hence effects of General Relativity (GR) become non negligible and require to be involved for a proper description of these kind of stars.

In the following pages we present the laws of the perfect fluid model in the relativistic context, the approximations done and the formalism adopted.

1.1 Stress Energy Tensor

Whenever one has to face with the dynamics of a large amount of particles, it results natural to treat them as forming a continuos medium to which we will refer to as a fluid. All the necessary informations about the energy and the momentum of such a system are described through a tensor (0,2) or bilinear form namely the stress energy tensor T. It consists in a description of matter at a macroscopic level considered within a specific region of spacetime.

Given a generic observer, say O, with 4-velocity ~u the informations car-ried by the stress energy momentum as measured by O are:

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Chapter 1. Relativistic Treatment • energy density: E = T(~u, ~u) = Tµνuµuν; (1.1) • momentum density: pα= − 1 cTµνu ν _⊥µ α; (1.2)

• energy flux vector :

φα = − 1 cTµνu µ_⊥ν α; (1.3) • stress tensor : Sij = Tµν ⊥µα⊥νβ; (1.4)

where ⊥ is the tensor (1,1) orthogonal to ~u and defined as:

⊥α_β= δα_β+ uαuβ. (1.5)

Note that since the stress-energy tensor is symmetric the matter momentum density and the energy flux vector must coincide, i.e. ~p = φ.

1.1.1 Perfect Fluid

From now on we will restrict ourselves to the case of the perfect fluid model. It has been used by Oppenheimer and Volkoff and Tolman in their works on NS [59, 80] and still today represents a fundamental assumption for several studies.

The perfect fluid model ignores effects deriving both from viscous matter and heat conduction and is determined by the expression:

Tµν = (ε + P )uµuν+ P gµν, (1.6)

where ~u is now the field of the 4-velocities of the matter i.e. the 4-velocity of an observer comoving (or fluid-comoving observer ) with the fluid while ε and P are scalar fields representing respectively the energy density (related to the mass density through the relation ε = ρc2) and the pressure both as measured in the fluid frame.

Consider now an arbitrary observer O0 with 4-velocity ~u0. Retrace the

equations of the components of the stress-energy tensor (1.1), (1.2) and (1.4) within the model for the perfect fluid.

As for the energy density E we obtain:

E = T(~u0, ~u0) = (ε + P )(~u · ~u0)2+ P g(~u0, ~u0)

= Γ2(ε + P ) − P . (1.7)

Last equality is easily get substituting the Lorentz’s factor Γ = −~u · ~u0 µ ν

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Chapter 1. Relativistic Treatment

surprised by the presence of Γ2 factor in spite of Γ as one would expected from a superficial reading. Indeed E is the energy density i.e. energy per unit volume and as consequence it is required an adjunctive Lorentz’s factor in order to take into account of length’s contraction in the fluid’s motion.

In a similar way, the momentum density in O0’s frame is found trough

the relation: ~ p = −1 cT(~u0, ⊥u0) = − 1 c(ε + P )(~u · ⊥u0)(~u · ~u0) + P g(⊥u0, ~u0) = Γ2ε + P c2 U .~ (1.8)

Full passages omitted here are found in appendix B. The term ~U = (~u/Γ − ~

u0) is the relative velocity and has a null value if ~u0= ~u i.e. for an observer

comoving with the fluid while ⊥u0 is the orthogonal projector of ~u0 already encountered in equation (1.5).

Finally the stress tensor. As seen, its components could be calculated as: Sij = T(~ei, ~ej) = (ε + P )(~u · ~ei)(~u · ~ej) + P~ei· ~ej = P δij + Γ2 ε + P c2 UiUj. (1.9)

Even in this case we have given the final result for brevity but all passages are in appendix B. Remark that δij = ~ei · ~ej and (~ei) is the orthonormal

basis of the hyperplane to the rest frame of O.

It is particularly interesting retrace the results above from the reference system of a comoving observer. In this case the fluid and the observer are described by the same worldline meaning that their 4-velocities coincide. Hence since ~u0 = ~u we obtain Γ = −~u · ~u = 1 for the Lorentz factor and

consequently a null value for the relative velocity ~U = 0. At this point we easily get:

E = ε, pi = 0, Sij = P δij. (1.10)

A remarkable finding is represented by the stress tensor. It is of the form Sij = P δij showing the isotropy of pressure for a comoving observer in the

perfect fluid model.

1.2 Thermodynamics Relations

A crucial aspect on NS theory is given by the internal relations on thermo-dynamics quantities. As we will see all these informations are found in the equation of state (EoS) whose microphysical model is a direct consequence of the assumption on the matter composition of the fluid.

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Suppose to deal with a system composed by N particles of different types and let us indicate with na the number density in the rest frame i.e.

the proper number density and with ma the relative mass of the individual

particle of species a. Then the total energy density can be written in the form: ε = N X a=1 manac2+ εint (1.11)

where the first term in the right hand side is the proper rest mass-energy density ε0 = ρc2 = PNa=1manac2 while with εint we refer to the internal

energy which contains contributions both from the potential energy due to interactions between particles and the kinetic energy. A different way to write equation (1.11) is given by the expression:

ε = ρ(c2+ e) , (1.12)

with e the specific internal energy and ρ the rest-mass density. Going further the total energy density ε results in a dependency of the entropy density s of the system and on the proper number density of particles of species a and we will refer to it as EoS or equation of state of the fluid

ε = ε(s, n1, · · · , nN) . (1.13)

From the concept of equation of state, we define the temperature and the chemical potential:

T ≡ ∂ε ∂s na (1.14) µa≡ ∂ε ∂na s,nb6=a (1.15)

and inserting (1.11) in the partial derivate for µa we obtain:

µa= mac2+ ∂εint ∂na s,nb6=a = mac2+ µinta . (1.16)

µaas derived above is the relativistic chemical potential and takes distances

from the classical case, sayµ_ea, for the mass term mac2 so that µa= mac2+

e µa.

Now consider both (1.14) and (1.15). Unifying them together we reach the expression: dε = T ds + N X a=1 µadna. (1.17)

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d(εV ) and making use of (1.17) we arrive to

d U = d(εV ) = εdV + V dε = V h T ds + N X a=1 µadna i + εdV . (1.18)

Finally after some algebra we get

d U = T dS + ε − T s − N X a=1 µana dV + N X a=1 µadNa, (1.19)

and compared with the first law of thermodynamic

d U = T dS − P dV +

N

X

a=1

µadNa (1.20)

we arrive to the equation below:

P = −ε + T s + N X a=1 µana −→ ε + P = T s + N X a=1 µana. (1.21)

What just reached put together in an unique expression thermodynamic quantities and shows how these are related to each other.

Let us move now to specific cases. We have already stressed on the fact that NS are astrophysical objects whose internal part is made of degenerate nucleons and no nuclear reactions take place inside meaning that it has been achieved the equilibrium between the constituents (or said differently that reaction rates are zero). Therefore it can be deduced the number density of particles of species A through the relation na= Yan for a given value of the

fraction YA once n is known. Actually the latter is nothing but the baryon

number per unit volume as measured by a comoving observer (proper baryon number density). Hence since the particles fraction Yaexhibits a dependence

on n this model of perfect fluid is characterized by an EoS in the form

ε = ε(s, n) (1.22)

and is called simple fluid.

Moreover we are allowed to introduce another approximation which con-sists in neglecting the temperature. This assumption is valid when the Fermi energy of nucleons is higher than the thermal one and it is largely satisfied in a NS1. In this case the proper energy density results in a function of the

1

This is surely true for the majority of their life but as for the ultimate phase of the merger of a binary system and for the instants immediately after it becomes a nontrivial argument. However the T = 0 hypothesis will be maintained for all the rest of the present work.

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proper baryon density only and we will refer to it as barotropic fluid or cold catalyzed matter :

ε = ε(n) and so P = P (n) . (1.23)

Therefore, equation (1.21) reduces to (T = 0):

ε + P = µn (1.24)

and the baryon chemical potential µ will coincide2 with the enthalpy per baryon, whose definition is:

h ≡ ε + P

n c2 . (1.25)

Once h is known it is possible deduce the specific enthalpy hs since they are

simply related through the expression h = hsmb with mb = 1.66 10-24g the

unified atomic mass unit or baryonic matter for brevity as we will refer to it. Therefore from equation (1.25):

hs=

ε + P n mbc2

. (1.26)

Another quantity of interest and that will be used later for the rotating structure is the pseudo enthalpy H. At zero temperature it is defined as the natural logarithm of the specific enthalpy:

H ≡ ln hs = ln " ε + P n mbc2 # . (1.27)

However a different expression for H can be found differentiating the specific enthalpy n mbc2hs= ε + P as follow:

dn(mbc2hs) + n mbc2dhs= dε + dP

= dn(hsmbc2) + dP .

(1.28)

where we have used µ = (∂ε/∂n)s,nand the fact that at T = 0, µ = h. Then

we obtain:

n mbc2dhs= dP −→ dhs=

dP n mbc2

. (1.29)

Now, in order to get the logarithm of the specific enthalpy we perform the ratio: dhs hs = d ln hs= dP n mbc2 n mbc2 ε + P = dP ε + P , (1.30)

from which we easily get the pseudo enthalpy:

dH ≡ d ln hs−→ H ≡ ln hs=

Z P

0

dP

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Let us recast then equations (1.31) and (1.27) together:

H = Z P 0 dP ε + P = ln " ε + P n mbc2 # . (1.32)

The analytical form and the integrated expression just reached are com-pared in appendix A where is discussed the agreement between them when tabulated values for the pressure, energy density and the baryonic num-ber density are taken into consideration and a perfect equality cannot be achieved then.

1.3 The general relativity conservation laws and

the equation of fluids

In order to solve numerically the problem of a selfgravitating perfect fluid, we introduce here two fundamental conservation laws from which will be possible determine the unknowns of our system.

First of all consider the well known cotracted Bianchi identity ∇µ(Rαµ−

1/2Rgαν) = 0. It is easy to recognize here the Einstein field equation that

gives us the vanishing of the divergence of T for an isolated fluid:

∇µTαµ= ∇µTµα= gµν∇νTµα= 0 , (1.33)

and states for the local conservation of energy and momentum. In addition making use of the perfect fluid model we can rewrite (1.33) as follow:

∇µTµν = ∇µ[(ε + P )uµuν+ P gµν] = 0 . (1.34)

Another conservation law can be derived as consequence of the hypoth-esis of isolated fluid. In this case the flux of the baryon 4-current ~J = n~u through a closed hypersurface Σ must vanish leading to the expression:

0 = Z _∂n ∂tdV = − Z ∂V n~udΣ = − Z V ∇ · (n~u)dV (1.35) from which ∇ · (n~u) = ∇µ(nuµ) = ∇µJµ= 0 , (1.36)

i.e. the general-relativistic conservation of rest mass.

Then we have 5 equation (4 from (1.33) or (1.34) and 1 from (1.36)) for the 6 variables n, ε, P, uµ (one of the components of the 4-velocity comes from the normalization ~u · ~u = uµuµ = −1). The missing relation required to

close the system is given by the EoS.

Moreover it could be instructive to decompose the energy momentum conservation law in its separated parts i.e. the momentum-conservation

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and the energy-conservation equations. Consider then the divergence of the stress-energy tensor in the perfect fluid model and the fact that ∇g = 0:

∇T = [uµ∇µ(ε + P ) + (ε + P )∇µuµ]uα+ (ε + P )gανuµ∇µuν+ ∇αP = 0 .

(1.37) Now projecting it in the direction orthogonal to ~u:

(ε + P )gανuµ∇µuν = −∇αP − (uµ∇µP )uα. (1.38)

This expression is the relativistic extension of the classical Eulerian equation with aα= uµ∇µuα the fluid acceleration.

Let us back to equation (1.37) and consider now its projection along ~u’s field line. This procedure gives the scalar equation:

uµ∇µε = −(ε + P )∇µuµ. (1.39)

Finally what we have obtained are the three and one components respec-tively of equation (1.37).

1.4 3+1 formalism

With the purpose of solving numerically the equations above, it has been developed a conservative formulation known as the 3+1 formalism or ADM formalism. It is an approach to GR based on the slicing of the 4-dimension spacetime into 3-dimension spacelike surfaces. In this context is possible to formulate the problem of resolution of Einstein equations as a Cauchy problem with constraints [57].

More specifically within this formalism the spacetime results in an union of hypersurfaces (Σt)t∈R i.e. a sequence of 3 dimensional sub-manifold

spaces (or rather a time slices succession) whose tangent vectors are all spacelike:

M = [

t∈R

Σt

such that ΣtT Σt0 = 0 ⇔ t 6= t0. Σ_tare then spacelike hypersurfaces, which implies to have a unique unit time normal vector ~n and it is future oriented, so that

~

n · ~n = −1 .

Hence this approach lead to a separation between the spatial and time co-ordinate namely a 3+1 decomposition. Note that the gradient of coco-ordinate time t, ~∇t, and ~n must be colinear since they are both vectors normal to Σt

so that:

~

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N is called lapse function. Using the time coordinate expressed by x0 = t and the x1, x2, x3 as for spatial coordinates, we obtain a system in which the generic line element in a 3+1 decomposition take the following form:

ds2 = −(N2− β_iβi)dt2+ 2βidxidt + γijdxidxj. (1.41)

Here γij are the components of the induced metric on Σt(broadly speaking,

metric’s g restriction on Σt), calculated as γij = gij + ninj and βj the

components of the shift vector i.e. the displacement of lines of constant spatial coordinates from the normal vector of the hypersurfaces and given by

~

∂t= N ~n + ~β . (1.42)

An interesting implication concerns the inertial observers or Eulerian ob-server (it will be resumed in section 2.4, here we are interested on conse-quences in using the 3+1 formalism) defined as those whose 4-velocity is given by nµ_{. Actually they are not at rest for a reference system at infinity}

but locally they do not feel of angular momentum (from which the alterna-tive name of Zero Angular Momentum Observer, ZAMO ).

In particular with the present formalism and still indicating with Γ ≡ −uµ_n

µ = N ut the Lorentz factor, the components of the 4-velocity ~v of

the fluid as measured by a ZAMO can be written as

vi= 1 N ui ut + β i _v i = ui Γ = ui N ut, (1.43)

Equation (1.43) should be compared to that for the special relativity to which reduces for N = 1 and βi = 0.

Finally in order to show better the split of the 4-velocity uµ of a fluid into the temporal and spatial terms, after some calculations it is possible write (for detailed passages see [69])

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CHAPTER

2 EQUATION OF STATE &

NEUTRON STAR MODELS

The study of models of compact objects moves considering two different sets of inputs. They concern the dynamical properties of the fluid such as the rotation and the microphysics of the local matter element whose infor-mations are found within the EoS. The determination of extreme-density matter properties on which is based the EoS of a NS, is an hard challenge of nuclear physics. The extreme conditions reached in these stars indeed are not reproducible in laboratory at the moment so that various models have been developed which predicts even the appearance of new fermions or bosons condensates, deconfined quarks or simply pure nucleonic matter.

As for the rotating models we deal with rigid structures and differentially rotating stars since the latter is expected to be found in newly born NSs and in the phases immediately after the merger of a binary neutron stars (BNS). The main effect of the rotation is given by the centrifugal force which stands against the gravitational pull producing more massive structures than if they were non-rotating. Therefore a dissertation on stable models is required in order to understand the conditions under which the post merger object of a binary system would experience the collapse into BH or not.

The chapter describes then the outputs coming from the use of various EoSs moving from an ideal polytropic to a ”realistic” one and the effects on models due by different rotating patterns in a full general-relativistic treatment.

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Chapter 2. Equation of state & Neutron star models

Figure 2.1: Interior of a neutron star as proposed by P. Hansel,A.Y. Potekhin and D.G. Yakovlev in their book Neuton Stars 1. Distances values reported in figure, have the role to give a general idea on neutron star’s sizes

2.1 EOS

In order to close the system of equations required for the building of equilib-rium structures we have introduced the EoS which relates thermodynamics quantities to each other. In particular they are based on a specific model of interacting particles and for this reason several EoSs have been put forward. In effect the interior of a NS is still an open argument being the densest object known with uncertainties beginning at the ultra-dense regime i.e. at ρ ≥ ρ0 with ρ0= 2.8 1014g cm-3 the nuclear density.

Nevertheless we are able to determine in a very general way the structure of a model as showed in figure 2.1.

Moving from the outside to higher densities and neglecting the atmosphere, one encounters the outer crust. It consists in a solid region which extends for 8 g cm-3_{≤ ρ ≤ ρ}

ND = 4 1011g cm-3. The latter is the neutron drip density

namely the limit over which no neutrons can be added to the nuclei and start to produce a free neutron gas. Since the electron Fermi energy grows with the density the electron capture is then favored giving a gradual increasing of neutrons in the nuclei until the bottom of the outer crust where the drip begin. Moreover the atoms are completely ionized and the electrons result in a relativistic degenerate gas which contributes primarily to the pressure.

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In this regime, the EoS can be derived from experimental data and semiem-pirical nuclear mass formulae.

Subsequently is found the inner crust. It is formed of electrons, free neu-trons and neutron rich nuclei and is characterized by a density which ranges in the interval ρN D ≤ ρ . 0.5ρ0= 1.4 1014g cm-3 with ρ0 the nuclear density.

In this region the nuclei pass to stretched geometries and disappear at the limit with the core.

We will not dwell here in details regarding the core since as we have already stressed it is model dependent. Some formulations divide it into 2 more parts which are the outer core (0.5ρ0 <

∼ ρ <∼ 2ρ0) and the inner

core (2ρ0 <

∼ρ). The most relevant feature is given by the absence of nuclei

which have left the place to neutrons, protons, electrons and muons in the so called npeµ composition and possibly other particles. The nature of the matter here is indeed determined by the conditions on the electric charge neutrality and the β equilibrium which brings to the formation of heavy particles whenever a certain threshold on the density is reached.

Within this work we limited ourselves to NSs whose core would be formed of pure nucleonic matter npeµ. However for further readings on more exotic EoSs currently performed we send to [?, ?, 20, 21, 25, 41, 81].

Future gravitational wave (GW) signals from BNS will give important informations on the nature of this stars and the matter they are made of [17, 68].

2.1.1 Polytrope EoS

Let us retrace now paragraph 1.2 on the thermodynamics relations and consider a polytrope as an example of barotropic fluid. Such an EoS is given by

ε(n) = mbc2n +

k γ − 1n

γ _(2.1)

with mb ' 1.66 10-24g the baryon mass, γ the adiabatic index and k a constant

depending on the system considered.

Since for a barotropic fluid the temperature is zero (ε manifests a dependency on n only, from which T = dε/ds = 0) we get ε + P = µn and taking into account the definition µ = dε/dn, from (2.1) with some algebra we arrive to the following expression:

P (n) = knγ. (2.2)

It represents an analytic EoS with k and γ depending on the assumed stellar composition and for its simplicity is largely used in testing codes for the numerical relativity or as starting point for various studies.

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Chapter 2. Equation of state & Neutron star models 1014 ₁₀15

/c

2

_{[ g/cm}

3

_]

1031 1032 1033 1034 1035 1036 1037 1038

P [

d

yn

/cm

2

]

BBB2 SLy4 Akmal FPS TM1-2 104 ₁₀8 ₁₀12 ₁₀16

/c

2

_{[ g/cm}

3

_]

1013 1016 1019 1022 1025 1028 1031 1034 1037

neutron drip

core-crust line

Figure 2.2: Bi-logarithmic plot of the pressure as function of the total energy density for various EoSs. The left panel is a zoom at the high density regime.

2.1.2 Realistic EoS

Actually great uncertainties still exist on what NSs are made of as conse-quence of the high densities reached hardly achievable in laboratory and for this reason several models have been proposed leading then to different outcomes regarding macroscopic configurations. Hence with realistic EoS one refers to tabulated quantities thermodynamically consistent, based on various microphysics effects and on the particles involved, and that an ana-lytical expression alone could not provide. However theories could result to be increasingly less accurate for densities overcoming ρ0.

Figure 2.2 exhibits the total energy density-pressure profile for values ranging from the core to the crust for several EoSs in the right panel, where vertical dashed lines locate the neutron drip and the core-crust borders. The left panel is a particular of the previous one for the high density regime in which differences are more appreciable.

The study of nuclear matter beyond the drip line has been performed starting from the relativistic mean-field (RMF) theory and the contribution from Walecka’s model [82]. In his work the interactions between nucleons are described through the exchange of the two mesons σ and ω with the free parameters (the couplings between both the fields and the nucleons) fixed in order to obtain again well known nuclear properties, namely the binding energy per nucleon of symmetric nuclear matter (SNM) (E0 = 16 MeV) and

the saturation density (n0 ∼ 0.15 fm−3). Going further we have even

consid-ered the interactions with the isovector ρ meson which plays a fundamental role as for the symmetry energy in finite nuclei [19, 77].

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The advantages coming from the use of an EoS based on the RMF theory are due to the capacity of the latter to satisfy the few informations at our disposal regarding the extreme conditions we are interested in, i.e. the:

1. causality, vs/c = |dP/dε|1/2< 1 with vs the speed of sound in matter ;

2. microscopic stability dP/dε ≥ 0 ;

3. saturation properties of nuclear matter .

Many RMF models have been developed in the years which try to describe the nuclear matter so that NSs became automatically a natural laboratory from which verify the reliability of the assumptions formulated.

As for the present pages we worked primarily with an EoS whose core is nothing but the TM1 [77] considered with a smaller quartic omega term and that is denoted as TM1-2 [67]. In particular the two mentioned EoS present same properties at the saturation even though the latter results in a stiffer trend at supra-densities but still within the limit coming from the constraints deduced from the heavy ion flows [29].

In particular we have followed a RMF approach where neutrons and pro-tons interact through the exchange of the scalar and vector mesons, namely the σ and the ω, and the isovector meson ρ. Such a system is described mak-ing use of a non linear finite range RMF given by the lagrangian density ([34] for more details):

LN L = Lnm+ Lσ+ Lω+ Lρ (2.3) with Lnm= ψ(iγµ∂µ− M − gσσ − gωγµωµ− gρ 2γ µ_~_ρ µ~τ )ψ, (2.4) Lσ = 1 2(∂ µ_σ∂ µσ − m2σσ2) − g2 3σ 3₋g3 4 σ 4_, _(2.5) L_ω = −1 4F µν_F µν+ 1 2m 2 ωωµωµ+ c3 4(g 2 ωωµωµ)2, (2.6) L_ρ= −1 4B~ µν_B_~ µν+ 1 2m 2 ρ~ρµ~ρµ. (2.7)

Above Lnmindicates the kinetic part of the nucleons together with the terms

expressing the interactions between the nucleons and the mesons σ, ω and ρ while the Lj are the lagrangian density components for the free and

self-interacting terms of the meson j = σ, ω, ρ of rest masses mj. The mass of

the nucleon is indicated with M and Fµν and ~Bµν are the antisymmetric

field tensors, respectively given by:

Fµν = ∂νωµ− ∂µων, (2.8)

~

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Taking advantage of the lagrangian density just written within the mean-field approximation in which the meson mean-fields are considered as classical and indicated with σ, ω0, ~ρ0 and from the Euler-Lagrange equations, we arrive

to the expressions:

m2_σσ = gσρs− g2σ2− g3σ3, (2.10)

m2_ωω0= gωρ − c3gω(gωω0)3, (2.11)

m2_ρρ₀= gρ

2 . (2.12)

At this point consider the lagrangian density (2.3) again. Since the total energy density and the pressure can be calculated respectively as ε = T00

and P = Tii/3, we obtain1: ε = X j=n,p,e,µ εkin_j +1 2m 2 σσ2+ g2 3 σ 3₊g3 4σ 4₋1 2m 2 ωω20− c3 4(g 2 ωω02)2 −1 2m 2 ρρ20+ gωω0n + gρ 2 nn3 (2.13) P = X j=n,p,e,µ P_jkin−1 2m 2 σσ2− g2 3σ 3₋g3 4σ 4₊1 2m 2 ωω20+ c3 4(g 2 ωω02)2+ 1 2m 2 ρρ20. (2.14) where εkin_j = ν 2π2 Z k_Fj 0 k2(k2+ m2_j)1/2dk (2.15) P_jkin= 1 3 ν 2π2 Z k_Fj 0 k4dk (k2_{+ m}2 j)1/2 . (2.16)

The ν above stands for the degeneracy which measures 2 for the system considered and kFj are the Fermi momentum of particle j. Moreover, mj=n,p = M∗ here are the baryons effective mass while indicate the

lep-tons’s rest masses in the other cases. Finally the n and n3 that appear in

(2.13) are the number density obtained as follow:

n = hψγ0ψi = np+ nn n3 = hψγ0τ3ψi = np− nn. (2.17)

For what concern the external stellar regions we made use of BPS +HP 94 [13, 46, 47] for the outer crust and SLy4 for the inner crust [32] in the form provided by Lorene and available in the directory Eos tables of the present code.

1_{The local inertial frames in the gravitational field of a NS are sufficiently extensive in} order to allow the matter they are made of to act as it was considered in a flat spacetime.

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2.2 β-equilibrium

The equilibrium of a cold neutron star is translated into the demand for the charge neutrality and for energy minimization. In particular, for the final state we want the equilibrium for the weak processes

p + e−←→ n + νe (2.18)

n → p + e−+ νe, (2.19)

which is equivalent to impose same reaction velocities.

Let us consider then the neutrinos-free matter case. The charge neutrality implies: X i n+_b i+ n + li =X i n−_b i+ n − li , (2.20)

being n the number density while b and l stand for baryons and leptons respectively.

As for the chemical potential:

µn= µp+ µe, (2.21)

that express the β-equilibrium condition.

At this point we take into account the threshold at which the muons start to be formed and so the following processes:

e−→ µ−+ νµ+ νe

p + e−←→ n + νe.

(2.22)

Still in the neutrino free matter case we require the vanishing of the chemical potential µνl = µνl = 0 with l standing for both e, µ so that we can set the system:      µn− µp= µe µµ= µe np = ne+ nµ (2.23)

with the latter equation required for the charge neutrality condition and alternatively given by nj = k3_F_j/(3π2).

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Table 2.1: Rapid description of EoSs used in this section. For a deeper under-standing of these models see the references reported.

Equation Of State Composition Reference

BBB2 npeµ [10]

AP R npeµ [1]

F P S npeµ [56, 65]

SLy4 npeµ [32]

T M 1-2 npeµ [67, 77]

2.3 Models of Neutron Stars

A proper description of the equilibrium of a NS could not be done with-out the general relativistic treatment. From observations the masses of these stars reach the value of ∼ 2MJ at most so that the corresponding

Schwarzschild radius would be of rs = 2GM/c2 ' 6 km. However the

typi-cal radius seems to measure R = 10 ÷ 14 km namely twice the Schwarzschild radius. Said differently the compactness parameter of a NS arrive to range in the interval rs/R ∼ 0.2 ÷ 0.5 while for white dwarfs it is of rs/R ∼ 10-4.

NSs are then the most compact objects known in the universe after BH and general relativistic effects cannot be neglected.

2.3.1 Non Rotating Neutron Stars

First of all we retrace the work by Oppenheimer & Volkoff and Tolman concerning the non-rotating configurations [59, 80]. Let us consider then a gravitational field characterized by central symmetry. The most general expression for the metric able to satisfy this property is:

gµν =     −e2φ ₀ ₀ ₀ 0 e2ψ 0 0 0 0 r2 0 0 0 0 r2sin2θ     , (2.24)

which induces the infinitesimal line element: ds2 = gµνdxµdxν

= −e2φc2dt2+ e2ψdr2+ r2dθ2+ r2sin2θdφ2 . (2.25)

From the expression just written and using the stress energy tensor of a perfect fluid (1.6), it is possible obtain the analogous hydrostatic equilibrium equation for the GR (TOV from now on) and that we give below omitting the passages, in a suggestive post-newtonian form together with the continuity

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Chapter 2. Equation of state & Neutron star models equation: dP (r) dr = − Gε(r)m(r) c2_r2 1 +P (r) ε(r) 1 +4πr 3_{P (r)} m(r)c2 1 −2Gm(r) c2_r −1 (2.26) dm(r) dr = 4πr 2_{ε ,} _(2.27)

where m(r) is the mass embedded in a sphere of radius r. Moreover note that the classical form for the equilibrium structures is found again from (2.26) in the limit P ε and 2Gm(r)/c2 r.

Another fundamental relation can be deduced calculating the Ricci curva-ture tensor from the Einstein equation and that leads to:

dφ dr = −1 ρc2 dP dr 1 + P ρc2 −1 = −1 ε + P dP dr . (2.28)

The latter, unified with (2.27) and (2.26), form the system to be solved in order to build the configuration of the star and the missing relation required to close it, is provided by the EoS.

Effects of the curvature of the spacetime are visible in figure 2.3 where are shown the profiles of the pressure and the mass as function of the radial distance to the centre of symmetry of a single object considered in the new-tonian gravity (solid red line) and in GR (blue dot-dashed line) too. The ”correction factors” to the newtonian physics of equation (2.26) are all pos-itive determining then a sort of magnification on the gravity which causes the lowering of the curves. Going further let us integrate again the equations for the stability this time repeating every single cycle increasing the central value for the total energy density in order to get a sum of models which will form a line at a given EoS. A remarkable finding is showed in figure 2.4 where we plotted curves on total energy density-gravitational mass (ρc, Mg)

and circumeferential equatorial radius-gravitational mass (Re, Mg) planes

and consists on the existing of a limiting mass over which structures cannot be found stable. The nature of a maximum mass in the sequences of cold objects comes observing that thermal pressure is zero by definition. Hence variations on the central density imply changes on the other macroscopic quantities and on the pressure and the energy density as well. However the corresponding raising of the gravitational pull is not followed by a likewise rate on the pressure which is related to the energy density through the speed of sound v_s2 = dP/dε c2. Therefore since no signals travel faster than the light, the pressure cannot assume an arbitrary value but will result in a maximum in order to conserve the causality.

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Chapter 2. Equation of state & Neutron star models 0 2 4 6 8 10 12 r [ Km ] 0.0 0.5 1.0 Mg [ M ] General Relativity Newtonian Gravity 0 2 4 6 8 10 12 r [ Km ] 0 2 4 P [ d yn / cm 3 ] 1e19 General Relativity Newtonian Gravity

Figure 2.3: Profile of pressure and mass as function of the radial coordinate from the centre of symmetry for a spherical NS described by a politropic EoS. The solid red line and the blue dot-dashed line have been ob-tained integrating the equations for the equlibrium in newtonian grav-ity and GR respectively for the same central pressure.

0.5 1.0 1.5 2.0 2.5 3.0 3.5 c/c2 [ g/cm3 ] 1e15 0.5 1.0 1.5 2.0 Mg [ M ] TM1-2 Akmal BBB2 FPS SLy4 9 10 11 12 13 14 15 16 17 Re [ Km ] 0.5 1.0 1.5 2.0 FPS Akmal BBB2 BPAL12 SLy4

Figure 2.4: The panels are an illustrative example of non rotating stellar struc-tures among their stable branches for various EoSs. On the left is reported the central total energy density-gravitational mass plot with the corresponding configurations in the right panel where models are displayed as function of the circumferential equatorial radius. Se-quences are interrupted once the upper limit mass has been reached.

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2.3.2 Stability of neutron stars configurations

The resolution of the TOV equation leads to configurations that could not represent a satisfactory model. In effect equilibrium does not signify stability meaning that the star calculated could be unstable for oscillations and in this case would not be of any astrophysical interest. Actually the determination of stability is a specialized topic and we will not enter in details but we give here the findings coming from the works on pulsations [31, 57, 79].

In particular it has been shown that stable models must provide the so called stability criterion:

dM dρc

> 0 . (2.29)

This expression is a necessary but not sufficient condition so that the oppo-site dM/dρc< 0 brings structures to collapse inevitably [49, 85].

Similarly it has been obtained a relation for the stability concerning sequences on the mass-radius plane where stable configurations could be individuated whenever dM/dR < 0. The determination of stability on the mass-radius graph is quite a bit more complicated and we just give here the results.

2.4 Rotating configurations

A very remarkable feature of NSs consists on their rotation velocity. From observations on pulsars we know that the fastest observed has a frequency of 716 Hz [50] so that the static geometry of previous section will be abandoned now in favor of stationary and axisymmetric spacetimes in order to describe this class of objects. In particular we will deal both with uniform and differential rotations, restricting ourselves to circular motions around the rotation axis.

Therefore consider the metric tensor in spherical quasi-isotropic coordinate [54]: gµν=    

−eα+β _{+ ω}2_r2_sin2_θeα−β ₀ ₀ _−2ωr2_sin2_θeα−β

0 e2λ 0 0

0 0 r2e2λ 0

−2ωr2_sin2_θeα−β ₀ ₀ _r2_sin2_θeα−β

    , (2.30) and let us rewrite the infinitesimal line element of spacetime for a rotating structure:

ds2= gµνdxµdxν

= −eα+βdt2+ eα−βr2sin2θ(dφ − ωdt)2+ e2λ(dr2+ r2dθ2) , (2.31) where α, β, ω and λ are functions of r and θ. In particular note that the

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Figure 2.5: Representation of the dragging due to an object of angular velocity Ω as measured at infinity along the equatorial plane (θ = π/2 , uθ=

0). ω(r) is the angular velocity of the local inertial frame of radial

coordinate r with respect to the gravitational source. Clearly the frequency of the dragged frame goes to zero at great distances where the spacetime is flat.

We have already seen the consequences of a spacetime deformed by a self-gravitating object in spherical symmetry, however when the rotation is taken into account an additional effect comes out consisting in a circular dragging of the environment around and not similarities can be found in newtonian mechanics. Let us introduce two observers, say O and O0, at infinity with

respect to a general relativistic source and suppose O starting move radially to it. O0 at infinity will see the other one gain an angular momentum while

in O rest frame the motion result to be still radial.

Such an observer is the so called eulerian observer or Zero Angular Mo-mentum Observer (ZAMO) already introduced in section 1.4. Worldines of an eulerian observer are orthogonal to the hypersurfaces Σt, with 4-velocity

given by ~n.

From (1.40) it is possible to correlate the proper time of a ZAMO to coordinate t of the spacetime by writing:

dτ = N dt. (2.32)

It may be clearer now why N is called lapse function.

Due to the action of the gravitational field, ZAMOs are clearly acceler-ated observers but as consequence of being orthogonal to Σt they are

non-rotating. Hence, they will not suffer of Coriolis’ force nor centrifugal one which will appear for non inertial reference frames instead [44].

Retrace the line element for rotating structures above, and consider the second equation of (1.43) which give the fluid velocity relative to a ZAMO:

v = ul N ut = 1 N dl dt. (2.33)

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Chapter 2. Equation of state & Neutron star models ds, so that dl2 dt2 = e α−β_r2_sin2_{θ Ω}2_{+ ω}2_{− 2ωΩ = e}α−β_r2_sin2_{θ Ω − ω}2 . (2.34)

Ω ≡ uφ/ut is nothing but the angular velocity as measured at infinity. At this point we easily get:

v = e−βr sin θ Ω − ω . (2.35)

As for the 4-velocity of the fluid we have:

uα = e−(α+β)/2Γ1, 0, 0, Ω. (2.36)

2.5 Equilibrium equations

The equations required for rotating structures at the equilibrium are a bit more complicated with respect to the non-rotating case and different ap-proaches could be followed [69, 76]. We start considering the Einstein equa-tions that coherently with the metric adopted above, are given by

∇2(βeα/2) = Sν(r, µ) , (2.37) ∇2+1 r∂r− 1 r2µ∂µ αeα/2= Sα(r, µ) , (2.38) ∇2+2 r∂r− 2 r2µ∂µ ωe(α−2β)/2= Sω(r, µ) , (2.39)

where µ ≡ cos θ and ∇2 _{is the Laplacian in spherical polar coordinate and}

with ν we indicate ν = (α + β)/2. The right hand side of these expressions are given in [54]. The equation of hydrostationary equilibrium follows from the projection of the conservation of the stress-energy tensor normal to the 4-velocity (δcb+ ucub)∇aTab = 0 and is written as:

dP dr + (ε + P ) " dν dr + 1 1 − v2 −vdv dr + v2_dΩ/dr Ω − ω # = 0 . (2.40)

Moreover whenever the EoS is barotropic we can get a first integral of motion which takes the form:

Z P 0 dp ε + P − ln Γ + Z Ω Ωc F (Ω)dΩ = ν pole = α + β 2 pole, (2.41) H − ln Γ + Z Ω Ωc F (Ω)dΩ = ν pole = α + β 2 pole, (2.42)

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F (Ω) = uφutis known as J-law or rotation law and Ωc the angular velocity

at the rotation axis.

No analytical solutions are known for the resolution of equations (2.37) (2.38) (2.39) and for this reason they need to be solved numerically.

Finally some relevant quantities can be calculated once a structure reaches the equilibrium configuration for stationary axisymmetric spacetimes:

Mb ≡ Z ρΓ√γd3x Baryon mass (2.43) Mg ≡ Z (−2Tt_t+ Tµ_µ)N√γd3x Gravitational mass (2.44) Eint≡ Z ρΓε√γd3x Internal energy (2.45) T ≡ 1 2 Z ΩTt_φN√γd3x Kinetic energy (2.46)

W ≡ T + Eint+ Mb− M < 0 Gravitational binding energy (2.47)

β ≡ T /|W | Kinetic on gravitational energy (2.48)

J ≡ Z

Tt_φN√γd3x Angular momentum (2.49)

(2.50)

with γ the determinant of the components γij already encountered in 1.4

2.5.1 Uniform Rotation

Models with Ω = const are said to rotate uniformly or to be in rigid motion so that ∇Ω = 0 reducing equation (2.42) into:

H − ln Γ = α + β

2 . (2.51)

Effects of rotation on structures are illustrated in figure 2.6 and 2.7. We should not be surprised to see an increasing on the masses for gradually larger frequencies. Indeed the more the rotation intensity is the more the centrifugal force which will compensate the gravitational pull leading to more massive stars. However both frequencies and central densities cannot take arbitrary values. There exists a limit beyond which structures would mass-shed known in literature as mass-shedding frequency or Keplerian fre-quency. A stable star cannot be found over being the Kepler angular velocity the limit at which structures of given central density would start to lose mass at the equator along their geodesic.

Effects of rotations are also visible in fig 2.9 where models have been computed both for static and rotating configurations at same central condi-tions. It shows the contour of a vertical section where the colors mark the

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Chapter 2. Equation of state & Neutron star models 0.25 0.50 0.75 1.00 1.25 1.50 1.75 c/c2 [ g/cm3 ] 1e15 0.0 0.5 1.0 1.5 2.0 2.5

M

g

[ M

]

k

Static Configurations

f = 600Hz f = 800Hz f = 1000Hz f = 1200Hz f = 1300Hz

Figure 2.6: Constant angular velocity sequences of uniform rotating structures. These curves move between the static configuration (solid black line) and the keplerian one (dashed black line) the latter being the upper limit on the rotation velocity beyond which particles at the equator would be lost by the star. The EoS used for the plot is the TM1-2

0.25 0.50 0.75 1.00 1.25 1.50 1.75 c/c2 [ g/cm3 ] 1e15 0.0 0.5 1.0 1.5 2.0 2.5

M

g

[ M

]

k

Static Configurations

J = 1 J = 1.5 J = 2 J = 2.5 J = 3 J = 3.5 J = 4 J = 4.5 1.4 1.5 1.6 1.7 1.8 1.9 1e15 2.2 2.4 2.6 2.8

Figure 2.7: Here are visible several sequences for a given value of the dimensionless angular momentum for structures in rigid motion for the TM1-2 EoS. In the inset is a particular of the plot where it is possible appreciate better the separation from secular unstable configurations marked with the black dotted line i.e. turning-point line. The turning point for spherical models coincide with the maximum value for the static sequence (blue point) while differs from the maximum mass model

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Chapter 2. Equation of state & Neutron star models 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 c/c2 [ g/cm3 ] 1e15 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Mg [ M ]

M

b

2.59 M

b

2.87

Non rotating stars Keplerian uniform rotation

j = 0.568146 j = 3.17102 12 14 16 18 20 22 Re [ km ] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Mg [ M ]

Figure 2.8: The two plots illustrate sequences of non rotating models (solid blue line) and of structures at the mass-shed limit for rigidly rotating con-figurations (solid red line) together with curves of fixed rest masses (dashed black lines) and constant angular momenta (dotted black lines) for the TM1-2 EoS.

variation on the total mass density distribution. Even for extremely rotating objects the maximum density coincides always with the central one as for uniform rotations.

Differently from the static case, equilibrium configurations for rotating (and especially for rapidly rotating) stars cannot be identified in the same way. Making use of figure 2.7 we illustrate sequences of a given angular momentum J and display the turning point too i.e. the union of maximum mass models for sequences of a fixed J identified by (∂M/∂εc)J = 0 [37].

This route represents a way through which is possible locate rotating models dynamically unstable and that result to be situated at higher rest mass densities with respect to the turning point line. Such a criterion illustrates that a turning point represents a limit at which structures become secularly unstable and where models give a null derivative in the central density for the quantities Mg, Mb and J as well. Indeed for cold objects the variation

on the entropy has not consequences on the energy of the system so that stationary points of fixed quantities must coincide to each other as showed in figure 2.8.

Therefore the turning point is a condition for secular stability for barotropic stars and is represented as a right edge in a (M, εc) plane over which

struc-tures result to be dynamically unstable. However from recent numerical simulations a new stability criterion has been traced replacing the turning point in the hierarchy for studies on collapsing NSs, the neutral-stability line [78]. The limit performed inhabits at smaller central rest mass densities with

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Chapter 2. Equation of state & Neutron star models 20 10 0 10 20 x [ Km ] 20 15 10 5 0 5 10 15 20 y [ K m ] 0 1 2 3 4 5 6 /c 2 [g /cm 3] 1e14 10 0 10 x [ Km ] 15 10 5 0 5 10 15 y [ K m ] 0 1 2 3 4 5 6 /c 2 [g /cm 3] 1e14

Figure 2.9: Contour plot of vertical sections of NSs in the static and rigid rotation with same central total energy density for the TM1-2 EoS in the left and in the right panel respectively. The colors mark the variation of internal distribution of the total mass density while the x and y axis have spatial dimensions. Effects of rotations on the structures are strongly visible here, where the rotating model result in a more oblate geometry.

respect to those of the turning point, meaning that stars found between the two lines could be unstable for secular instability. For this reason the turn-ing point result in a sufficient condition only. Nevertheless it continues to be a fundamental estimation for the dynamical unstable cold NSs and we will refer to it in the following pages.

More specifically we give in table 2.2 the values concerning the turning points for the TM1-2 EoS that we calculated using the RNS code with an accuracy of 10e-7 on the various quantities. The bottom string is simply the maximum mass model for non rotating sequences. Going further we also return in table 2.4 2.5 2.6 the maximum models respectively for non rotating and uniformly rotating at f = 716 Hz and at the mass shed limit configurations for the EoSs already displayed in table 2.1.

In particular we collect the findings for the TM1-2 alone in the additional table 2.8 separated from the others and from which we observe that the uniform rotation would affect structures with an increase on the mass of ∼ 19.29 % at most with respect to the static case (we get similar results for the other EoSs considered here).

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Table 2.2: Turning points for models of TM1-2 EoS in the uniform rotating case.

εc/c2 Mg Mb Re Ωc β J I rp/re [g/cm3] [MJ] [MJ] [km] [Hz] T /|W | [GMJ2 /c] [g cm2] 1.65281e+15 2.72 3.14 16.8391 8510.92 0.1167290 4.87164 5.03051e+45 0.565332 1.65736e+15 2.69 3.10 15.7125 8328.68 0.1096560 4.60000 4.85394e+45 0.631098 1.67775e+15 2.61 3.01 14.8549 7851.88 0.0931515 4.00000 4.47712e+45 0.698431 1.69257e+15 2.56 2.96 14.4376 7446.18 0.0813525 3.60000 4.24895e+45 0.738360 1.71663e+15 2.49 2.88 13.9192 6716.49 0.0633678 3.00000 3.92547e+45 0.795935 1.73709e+15 2.45 2.83 13.6163 6127.97 0.0512262 2.60000 3.72880e+45 0.833721 1.76649e+15 2.39 2.76 13.2269 5060.19 0.0335406 2.00000 3.47357e+45 0.889352 1.78446e+15 2.35 2.72 13.0133 4218.29 0.0227726 1.60000 3.33346e+45 0.923643 1.80201e+15 2.31 2.68 12.7712 2769.70 0.00954985 1 3.17307e+45 0.967416 1.81393e+15 2.28 2.64 12.6030 0 0 0 – 1 2.5.2 Differential Rotation

Whenever a NS is considered in the moments immediately after the merger of a binary system or at its birth from a supernovae (the so called proto-neutron stars), the uniform rotation assumption is not appropriate to de-scribe the system and a new pattern is required thus. In these cases the rigid motion cannot be taken into consideration so that investigations along years have been moved considering the differential rotation in which the angular velocity presents a non trivial dependency on the radial distance from the rotation axis and the angle subtened. The importance covered by this new rotation profile lies on the fact that as we will see models could stand more efficiently against the gravitational pull giving rise on more massive objects which could experience then a delayed collapse to a BH.

Let us retrace here the integral of motion (2.42) paying attention on the fact that ∇Ω has not a null value now. Therefore what we obtain is:

H − ln Γ + Z Ω

Ωc

F (Ω0)dΩ0 = α + β

2 . (2.52)

In line with previous works the J-law used in the following pages is charac-terized by a well known profile given by Komatsu [54,55], the One-parameter rotation law (to distinguish from the Three-parameter rotation law recently formulated [16]): F (Ω) = A2(Ωc− Ω) = (Ω − ω)r2sin2θe−2β 1 − (Ω − ω)2_r2_sin2_θe−2β (2.53)

Above A is a new parameter with dimension of a length and has been rescaled here in terms of the equatorial radius re:

ˆ A = A

re

Differentially rotating neutron stars in general relativity

Universit`

a di Pisa

Differentially rotating neutron stars

in General Relativity

CONTENTS

LIST OF FIGURES

LIST OF TABLES

INTRODUCTION

CHAPTER

1

RELATIVISTIC TREATMENT

1.1

Stress Energy Tensor

1.2

Thermodynamics Relations

1.3

The general relativity conservation laws and

the equation of fluids

1.4

3+1 formalism

CHAPTER

2

EQUATION OF STATE &

NEUTRON STAR MODELS

2.1

EOS

/c

[ g/cm

]

P [

d

yn

/cm

]

/c

[ g/cm

]

neutron drip

core-crust line

2.2

β-equilibrium

2.3

Models of Neutron Stars

2.4

Rotating configurations

2.5

Equilibrium equations

M

[ M

]

Static Configurations

M

[ M

]

Static Configurations

M

2.59

M

2.87

_{[ g/cm}

_]

_{[ g/cm}

_]