high-energy astrophysics: white dwarfs, neutron stars and black holes

Aix Marseille Universit´e Master SPACE Program

Marko Shuntov

HIGH-ENERGY ASTROPHYSICS: WHITE DWARFS, NEUTRON STARS AND BLACK HOLES -M1/S2 PROJECT REPORT-

Supervisor: Carlo Schimd April, 2018 Marseille

Contents Contents

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1 Introduction

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2 White Dwarfs 2.1 Structure of White Dwarfs . . . . . 2.1.1 Equations of Structure . . . 2.1.2 Polytropic equation of state 2.1.3 Ideal Fermion Gas . . . . . 2.2 Results and Discussion . . . . . . .

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3 Neutron Stars 3.1 Tolman-Oppenheimer-Volkoff equations 3.1.1 Numerical Integration of TOV . 3.1.2 Equation of State . . . . . . . . . 3.2 Results and Discussion . . . . . . . . . .

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4 Conclusion A Appendix A.1 Python Codes . . A.1.1 Numerical A.1.2 Numerical A.1.3 Numerical

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. . . . . Solution Solution Solution

. . . . . . . . . . . . . . . . . of Lane-Emden Equation . . for the Ideal Fermi Gas Case TOV Equations . . . . . . .

References

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1

Introduction Compact objects are one of the most interesting objects in the universe, exhibiting extreme states of matter. Their structure can be studied by deriving and solving the equations of structure complemented by an equation of state describing the matter in the interior of the objects. The goal of this project is to study the structure of white dwarfs and neutron stars. White dwarfs will be studied in chapter 2 where their equations of structure will be derived, first using the limiting cases of a polytropic equation of state, and then using the full special relativistic EoS for degenerate Fermi gas at zero temperature. Plots relating the central density in the white dwarfs and their mass as well as the mass-radius relationship will be obtained and presented, followed by a discussion on the physical interpretation of the results. In chapter 3, neutron stars will be studied, such that the the Tolman-OppenheimerVolkoff equations describing the structure will be solved. An equation of state will be constructed as a combination of a white dwarf EoS for the low-density regime, and for the high-density regime an EoS describing matter from the crust to the quark matter domains in the core of the neutron star. The mass-density and mass-radius relationships will be obtained and presented, followed by a discussion on the physical interpretation of the results. All the numerical computations in this project will be done using Python programming language. The codes written for that purpose will be presented in the appendix.

2

White Dwarfs A star spends most of its time on the main sequence burning nuclear fuel which creates the pressure to counteract the pressure of gravity. When a star with mass similar to the Sun’s mass burns all of its nuclear fuel it starts to shrink even further and expels its outer regions of matter. What is left is a compact very dense and hot core - white dwarf. White dwarfs have typical masses of less or similar to M , typical radii of ∼ 10−2 R and mean densities of ≤ 107 g cm−3 .

2.1

Structure of White Dwarfs

The force that counteracts gravity in white dwarfs is the pressure of the degenerate electron gas. The force of gravity compresses the matter so tight that the electrons are forced to occupy all the lowest energy levels. Due to the Pauli exclusion principle only two electrons can occupy one energy level and thus they create the pressure outwards in order to fill all the available states; this quantum mechanical pressure counteracts the gravity and the white dwarf is stable.

2.1.1

Equations of Structure

The equations that describe the structure of a white dwarf at equilibrium are dM (r) = 4πr2 ρ(r), dr dP (r) GM (r)ρ(r) =− , dr r2 P = P (ρ),

(2.1a) (2.1b) (2.1c)

where the pressure will be expressed as a function of the density. In order to solve the equations of structure we need to use an expression for the pressure that the electrons in the white dwarfs supply which is calculated by ˆ 1 ∞ ve (p) p ne (p) d3 p, (2.2) P = 3 0 p where ve (p) = pc2 / p2 c2 + m2 c4 , m the mass of the electron and ne = 8πp3F /3h3 is the number density of electrons. The gas is considered to be at an effective zero temperature, allowing to approximate the Fermi-Dirac distribution function by a Heaviside step function Θ(pF − p). The equation then becomes ˆ pF 8π c2 p4 p P = 3 dp. (2.3) 3h 0 p2 c2 + m2 c4 3

2.1. STRUCTURE OF WHITE DWARFS

The matter density of the gas can be defined as the number density of electrons ne times the mas of a nucleon mN times the number of electrons per nucleons κ: ρ = κmN ne . A critical density ρcrit can be defined at which the electron gas becomes relativistic pF = mc).

2.1.2

Polytropic equation of state

We can now look at the two extreme cases of the electron gas being non-relativistic and ultra-relativistic. Non-relativistic limit (ρ ρcrit =⇒ cpF mc2 ) In the non-relativistic limit equation 2.3 becomes P = Knr ρ5/3 , where Knr

h2 = 20m

2/3 5/3 3 1 . π κmN

(2.4)

(2.5)

Ultra-relativistic limit (ρ ρcrit =⇒ cpF mc2 ) In the ultra-relativistic limit equation 2.3 becomes P = Kur ρ4/3 , where Kur =

hc 8

1/3 4/3 3 1 . π κmN

(2.6)

(2.7)

Thus, equation 2.1c can be written in a polytropic form P = Kργ . Introducing n = 1/(γ − 1) the density ρ in terms of the central density ρ0 can be rewritten as ρ = ρ0 Θn (r). (1+1/n) n+1 Thus the pressure will be P = Kρ0 Θ (r). Using 2.1a and 2.1b and the introduced variables the Lane-Emden equation can be obtained 1 d 2 dΘ ξ = −Θn , (2.8) ξ 2 dξ dξ q 1/n+1 which is a second order differential equation, where ξ = r/α and α = (n + 1)Kρ0 /(4πG) The initial (boundary) conditions apply ρ(0) = ρ0 and P (R) = 0. In terms of the quantities figuring in the Lane-Emden equation these conditions become Θ(ξ) = 1 and Θ0 (ξ) = 0. This equation needs to be numerically solved for the two cases. In the numerical integration scheme there appears a problem of singularity at ξ ≈ 0 and in order to account for it we do a Taylor expansion of the equation around zero for which we get 1 n 4 ξ , Θ(ξ) = 1 − ξ 2 + 6 120 1 n Θ0 (ξ) = − ξ + ξ 3 . 3 30

(2.9a) (2.9b)

Using these two equations we can evaluate the function in a small interval around zero up to the point which we chose to be ξ0 = 0.00012 as it is shown in figure 2.1. Thus we obtain the new initial conditions for equation 2.8 Θ(ξ0 ) and Θ0 (ξ0 ).

4


Figure 2.1: Solutions of the Lane-Emden equation around zero obtained with Taylor expansion. This way the new inital conditions are obtained for 2.8 for Θ(ξ0 = 0.00012) and Θ0 (ξ0 = 0.00012). At this scale, the difference between the two polytropes is negligible (blue line for γ = 5/3 and orange line for γ = 4/3 are superimposed.

With these new initial conditions the Lane-Emden equation 2.8 is solved using a numerical integration scheme. For that purpose, 2.8 can be rewritten as a system of two first order differential equations dΘ(r) −Θ0 = 2 , dξ ξ dΘ0 (ξ) = Θ(ξ)n ξ 2 . dξ

(2.10a) (2.10b)

This system of two first order differential equations is numerically solved using the RungeKutta 2nd order scheme in which equations 2.10a and 2.10b are written δξ Θ0i , 2 ξi2 δξ = Θ0i + Θni ξ 2 , 2 ξi+1 + ξi = , 2 Yi+1/2 = Θi − δξ 2 , ξn+1/2

Yi+1/2 = Θi −

(2.11)

Zi+1/2

(2.12)

ξi+1/2 Θi+1

n 2 Θ0i+1 = Θ0i + δξZi+1/2 ξi+1/2 .

(2.13) (2.14) (2.15)

The solution of for Θ(ξ) for the two polytropes (n = 3/2 for the non-relativistic and n = 3 for the ultra-relativistic) is presented on figure 2.2. Given the fact that we defined the density in terms of Θ as ρ = ρ0 Θn (r), and that at the surface of the star ρ(R) = 0 we can retrieve the value for the radius by finding ξ∗ for which Θ(ξ∗ ) = 0. This value ξ∗ can be obtained by finding the argument for which the function |Θ(ξ)| has the minimum (is closest to zero) figure 2.3. The values for ξ∗ found are 5


Figure 2.2: Solutions of the Lane-Emden equation for the two cases of the non-relativistic and ultra-relativistic limit.

Figure 2.3: Finding the argument ξ∗ for which Θ(ξ∗ ). The method yields results that are precise up the 4th decimal point.

6


Figure 2.4: White dwarf radius as a function of the central density for the two cases of nonrelativistic limit (blue lines) and ultra-relativistic limit (orange lines) for two different chemical compositions of κ = 2 with solid lines (composition of He, C, O2 ) and κ = 2.33 with dashed lines (composition of Fe). Data from measurements of white dwarf radii are also included from Camenzind 2007.

ξ∗ = 3.65354 for n = 3/2,

ξ∗ = 6.89652 for n = 3,

(2.16)

which when compared to the references Camenzind 2007 yield a satisfying precision. Thus, for the radius of the white dwarf we have s 1/n+1 (n + 1)Kρ0 R = αξ∗ = ξ∗ , (2.17) 4πG whereas for the mass we have ˆ R ˆ 2 3 M= 4πr ρdr = 4πα ρ0 0

ˆ

ξ∗

2

n

3

ξ Θ dξ = −4πα ρ0

0

M = 4π

(n + 1)K 4πG

0

(3/2)

ξ∗

d dξ

(3−n)/2n 2 ξ∗ |Θ0 (ξ∗ )|.

ρ0

2 dΘ ξ dξ dξ (2.18)

The radius and the mass of the white dwarf depend on the central density and also on the chemical composition κ in K. Figures 2.4 and 2.5 show the white dwarf radius and mass as a function of the central density for the both limits of the non-relativistic and the ultra-relativistic and for two different chemical compositions κ = 2 (composition of He, C, O2 ) and κ = 2.33 (composition of Fe) Camenzind 2007. We can derive an expression for the highest mass of the white dwarfs by looking at the ultra-relativistic case where n = 3 and the mass doesn’t depend on the central density; replacing for all the constants we obtain 2 2 M = MCh = 1.457 M , (2.19) κ 7


Figure 2.5: White dwarf mass as a function of the central density for the two cases of nonrelativistic limit (blue lines) and ultra-relativistic limit (orange lines) for two different chemical compositions of κ = 2 with solid lines (composition of He, C, O2 ) and κ = 2.33 with dashed lines (composition of Fe). Data from measurements of white dwarf masses are also included from Camenzind 2007.

which is the Chandrasekhar mass for white dwarfs. MCh depends on the chemical composition of the white dwarfs, as it can be seen from 2.5, and represents the highest mass a white dwarf can have not to collapse in a neutron star.

2.1.3

Ideal Fermion Gas

We now consider the general case for the equation of state for the electron gas given by 2.3 where the only approximation made is that of non-interacting fermions at an effective zero temperature. In order to obtain the structure of the white dwarfs in this general case, we need to solve for the pressure in 2.3 and use it in 2.1a and 2.1b, after what we have to obtain a differential equation and solve it. Solving the integral in 2.3 gives P = KΦ(x),

(2.20)

where

i p 1 h p x 1 + x2 (2x2 /3 − 1) + ln x + 1 + x2 , 8π me c2 h pF K= , Λe = , x= . Λ3e 2πme c me c The Fermi momentum pF can be related to the matter density by 1/3 1/3 3 h 3 , thus x = c1 ρ1/3 , c1 = . pF = h ρ 8πκmn me c 8πκmn Φ(x) =

(2.21)

Combining 2.1a and 2.1b we have 1 d r2 dr

r2 dP ρ dr

8

= −4πGρ.

(2.22)


We need to express the pressure in terms of the density and so we make the following chain derivatives dP dP dx dP dx dρ = = , (2.23) dr dx dr dx dρ dr where we express the radius derivative of the pressure in terms of the radius derivative of the density; the derivatives dP/dx and dx/dρ can be calculated using the quantities defined above. Doing that we obtain the differential equation   5 −1/3 dρ  Kc1 1 d  2 ρ rq = −ρ. (2.24) 3 2 36π G r dr 1 + c2 ρ2/3 dr 1

Introducing the changes Kc51 , α = 36π 3 G 2

ξ=

r , α

we obtain the differential equation we use to solve for the white dwarf structure   −1/3 ρ dρ  1 d  2 ξq = −ρ. 2 ξ dξ 1 + c2 ρ2/3 dξ

(2.25)

1

To solve it numerically we have to get equation 2.25, which is a second order differential equation, into a two first order equations by defining the term in the brackets as −z q dρ ρ1/3 = −z 2 1 + c21 ρ2/3 dξ ξ dz = ξ2ρ dξ

(2.26a) (2.26b)

which has to be written in a numerical algorithm which we chose to be Runge-Kutta 2nd order. From the system 2.26a-2.26b we can obtain the radius of the white dwarf by taking the value of ξ for which the density reaches zero R = αξ∗ ,

(2.27)

whereas for the total mass we have ˆ R ˆ M= 4πr2 ρdr = 4πα3 0

ξ∗

ξ 2 ρ(ξ)dξ,

(2.28)

0

using 2.25

ˆ M = 4πα

3

ξ∗

0

dz dξ, dξ

(2.29)

thus, for the total mass of the white dwarf we get M = 4πα3 z(ξ∗ ), which is the relation we use to obtain the total mass of the white dwarf.

9

(2.30)

2.2. RESULTS AND DISCUSSION

Figure 2.6: White dwarf mass as a function of the central density for the ideal Fermi gas EoS (solid blue line) along with the two cases of non-relativistic limit (orange dashed) and ultra-relativistic limit (green dashed) chemical compositions of κ = 2 (composition of He, C, O2 ). Data from measurements of white dwarf radii are also included from Camenzind 2007.

The system 2.26a-2.26b is solved numerically using the Runge-Kutta 2nd order scheme in which our system is written ! 1/3q ρi 1 2/3 R = ρi − δξ zi 2 1 + c21 ρi , (2.31a) 2 ξi 1 Z = zi + δξ ξi2 ρi , (2.31b) 2 ! q R1/3 (2.31c) ρi+1 = ρi − δξ Z 2 1 + c21 R2/3 , ξi+1/2 2 zi+1 = zi + δξ ξi+1/2 R . (2.31d) The initial conditions are: at the center of the white dwarf ρ(0) = ρ0 is the initial central density and for z(0) = 0. We start the integration from ξ ≈ 10−5 and use a sufficiently large number of integration steps to conserve stability. Rigorously, one would do a Taylor expansion of ρ and z to obtain the initial conditions at ξ ≈ 10−5 , but we can argue that starting at this value for ξ the difference in the initial conditions is in the order of ≈ 10−6 and we chose to neglect it as at the end it yields expected results consistent with the references Camenzind 2007.

2.2

Results and Discussion

We solve the equations such that we construct an array of initial central densities to serve as initial conditions ρ0 and we integrate our system of equation for each ρ0 . We find ξ∗ in

10


Figure 2.7: White dwarf radius as a function of the central density for the ideal Fermi gas EoS (solid blue line) along with the two cases of non-relativistic limit (orange dashed) and ultrarelativistic limit (green dashed) chemical compositions of κ = 2 (composition of He, C, O2 ). Data from measurements of white dwarf radii are also included from Camenzind 2007.

the same way described in the section 2.1.2 (fig. 2.3). Using ξ∗ we can obtain the radius and mass for the white dwarfs eq: 2.27, 2.30. Figure 2.6 shows the white dwarf mass as a function of the central density using the general equation of state for an ideal Fermi gas (blue solid line). The low central density end the curve conforms to the non-relativistic regime of the Lane-Emden equation, and the high central density end asymptotically approaches the Lane-Emden ultra-relativistic regime and the Chandrasekar limit. The full (special)relativistic treatment of the equation of state gives an accurate representation of the behavior of the mass with respect to the central density. The asymptotic behavior of the mass as it tends towards MCh for further increasing densities is not what is observed in nature; as for higher densities the general relativistic effects have to be taken into account, giving rise to the TOV equations of stellar structure (to be discussed in the next chapter). However, in the density range of gravitationally stable white dwarfs, the general relativistic effects introduce minor changes which can be neglected. Furthermore, corrections due to the electrostatic (Coulomb) interaction between the ions and electrons can be introduced in the fermion EoS in order to obtain an even more accurate curve. This effect derives from the nonuniform local distribution of charge, such that positive charge is concentrated in ions which causes the average electron-ion separation to be smaller than the one between electrons. The electrons thus feel an attractive electric potential that reduces the pressure for a given density. Another correction that can be introduced is due to the inverse beta-decay for higher densities. Figure 2.8 shows the mass-radius relationship for white dwarfs of two different chemical composition: κ = 2 composition of He, C, O2 and κ = 2.33 for iron rich cores. Higher mass white dwarfs have more compact sizes of order of magnitude of a few thousand kilometers. Data from measurements of white dwarfs is plotted which conforms to the obtained massradius curves.

11


Figure 2.8: White dwarf Mass-radius relationship for two different chemical compositions of κ = 2 with blue line (composition of He, C, O2 ) and κ = 2.33 with orange line (composition of Fe). Data from measurements of white dwarf masses are also included from Camenzind 2007. The presented plots conclude our discussion on the structure of white dwarfs and on the work done on this part of the project. As it was mentioned there are further corrections that can be included to obtain more realistic models. An extension of the work on this part of the project would be to include the corrections deriving from electrostatic interaction and the inverse beta decay as well as including cores composed of different elements. It is also worth mentioning that we were considering white dwarfs in thermodynamical equilibrium, which is another idealization; in reality white dwarfs cool down, especially soon after they were formed - something which in a more complete model should also be included.

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Neutron Stars Neutron stars are end states for massive stars that had more than 6 solar masses but were not massive enough (less then 20 M ) to collapse in black holes. Stars of such masses after burning their nuclear fuel explode in a supernova that blows away the outer regions leaving behind a core which is unable to counteract the gravitational force and collapses into a neutron star. During the gravitational collapse, the density pressure and temperature grow so high that an inverse beta decay occurs and protons and electrons combine to form neutrons and a flood of neutrinos that escape the star. Densities reach up from 4.3 × 1011 g/cm3 and from this point pressure from degenerate neutron gas due to the Pauli exclusion principle counteracts the gravitational collapse. What is left is a neutron star with a layered structure as shown in figure 3.1. For densities up to ρd = 4.3 × 1011 g/cm3 (called neutron drip) the outer crust consists of a lattice of atomic nuclei and Fermi liquid of relativistic degenerate electrons. This is essentially white dwarf matter. At the neutron drip all bound states available in the nuclei for neutrons are filled, and neutrons start leaking out by Pauli exclusion principle. Above the neutron drip density and up to a so called transition density ρtr = 2.67 × 1014 g/cm3 the degenerate neutron gas exists in the inner crust. Above ρtr all the nuclei are dissolved into their nucleons and due to the extreme conditions other particles like hyperons exist. Neutron stars are compact objects with a high surface gravitational potential such that they enter the regime of general relativity. The structure of the neutron stars is described by the differential equations of structure that include the general relativistic treatment - the so called Tolman-Oppenheimer-Volkoff (TOV) equations supplemented by an appropriate equation of state that describes the state of the matter inside the neutron stars. In this part of the project we will examine the structure of neutron stars. We will solve the equations of structure, the Tolman-Oppenheimer-Volkoff equations, using an appropriate equation of state.

3.1

Tolman-Oppenheimer-Volkoff equations

In general, the internal structure of any spherically symmetric non-rotating star in hydrostatic equilibrium is described by the TOV equations. The low-surface potential cases of main sequence stars and white dwarfs are well described by the Newtonian equations. Neutron stars have surface potential of order of magnitude ∼ 10−1 and need to be treated relativistically. The TOV equations are derived from the theory of general relativity and are given by dM (r) = 4πr2 ρ, dr dp(r) (ρ + p)(M + 4πr3 p) =− , dr r(r − 2M )

13

(3.1a) (3.1b)

3.1. TOLMAN-OPPENHEIMER-VOLKOFF EQUATIONS

Figure 3.1: The internal structure of a neutron star. where ρ is the mass-energy density. The equations are written in geometrized units where G = 1 and c = 1. An equation of state p = p(ρ) relates the mass-energy density and the pressure enabling the system to be solved. The boundary conditions are M (r = 0) = 0 and p(r = 0) = p0 .

3.1.1

Numerical Integration of TOV

Our goal is to obtain a plot relating the central density in a neutron star to its total mass, as well as a plot relating the total mass to the radius of the neutron star. For that purpose, the TOV equations need to be solved numerically while utilizing an equation of state (discussed in the next section). They need to be solved for a set of j number of central densities ρ0 serving as initial conditions p0 = p(ρ0 ), thus obtaining j different solutions. For that purpose, we rewrite the TOV equations 3.1 in the RK4 numerical scheme k1M = 4πri2 ρi k2M k3M k4M

(3.2a) 2

= 4π(ri + δr/2) ρ

(3.2b)

= 4π(ri + δr/2)2 ρ

(3.2c)

2

= 4π(ri + δr2) ρ

(3.2d)

(ρi + pi,j )(Mi,j + 4πri3 pi,j ) ri (ri − 2Mi,j ) [ρi,j + (pi,j + δk1p /2)][Mi,j + 4π(ri + δr/2)3 (pi,j + δk1p /2)] k2p = − (ri + δr/2)[(ri + δr/2) − 2Mi,j ] [ρi,j + (pi,j + δk2p /2)][Mi,j + 4π(ri + δr/2)3 (pi,j + δk2p /2)] k3p = − (ri + δr/2)[(ri + δr/2) − 2Mi,j ] [ρi,j + (pi,j + δk3p )][Mi,j + 4π(ri + δr)3 (pi,j + δk3p )] k4p = − (ri + δr)[(ri + δr) − 2Mi,j ] δr M Mi+1,j = Mi,j + (k1 + 2k2M + 2k3M + k4M ) 6 δr p pi+1,j = pi,j + (k1 + 2k2p + 2k3p + k4p ) 6 k1p = −

14

(3.2e) (3.2f) (3.2g) (3.2h) (3.2i) (3.2j)

3.1. TOLMAN-OPPENHEIMER-VOLKOFF EQUATIONS

where i is the index for the integration step, δr is the integration step size, and each initial condition (central density - new NS) is indexed by j. ρi,j is the mass-energy density for j NS in the i integration step or at ri radius, and the way it is obtained will be discussed in the next section. The solution of these equation gives the mass and pressure profiles as a function of the radius of the neutron stars.

3.1.2

Equation of State

An equation of state is essential in the solution of TOV equations because it relates the pressure to the density. Extreme conditions of matter exist in neutron stars and in its description, effects from different theories have to be taken into account; thus, it is difficult to obtain a simple EoS equation. Most often an EoS for the interior of the NS is provided in a table of values from which interpolation can be later performed. We need an EOS for the range of densities found in the interior of neutron stars: from zero at the surface and up to 1017 g/cm3 in the core. For the low density regime, in the outer crust of the NS, the white dwarf equation of state discussed in chapter 2, section 2.1.3 can be used (figure 3.2) P = KΦ(x), where Φ(x) =

(3.3)

i p 1 h p x 1 + x2 (2x2 /3 − 1) + ln x + 1 + x2 , 8π K=

me c2 , Λ3e

Λe =

h , 2πme c

x=

(3.4)

pF . me c

The Fermi momentum pF can be related to the matter density by pF = h

1/3 3 ρ , 8πκmn

thus

x = c1 ρ1/3 ,

c1 =

h me c

3 8πκmn

1/3 ,

where κ = 2.33 is used (iron rich cores). When entering the inner crust and then the core an appropriate EoS describing the matter beyond the neutron drip has to be used. For that purpose we use an EOS presented by Baym et al. 2017 called Quark-Hadron-Crossover 2018 (QHC18), which covers the EoS from the crust to quark matter domains. The physics behind QHC18 is discussed in the Baym et al. 2017 review paper. The EoS is presented in figure 3.2. The table for the QHC18 EoS is given from ρ ≈ 105 g/cm3 up to ρ ≈ 1016 g/cm3 . The WD EoS overlaps at the QHC18 starting point with a negligible difference.

Integration Procedure The integration of the TOV equations 3.2 are done such that an array (of size j) of initial values for the central densities is created {ρ0 } in the range of ρ0 = 2.7 × 105 g/cm3 up to ρ = 1.2 × 1016 g/cm3 . This range includes WD central densities, meaning that TOV equations will also be solved for white dwarfs. TOV equations are integrated with a step that changes with every iteration in a logarithmic law starting from δr ≈ 10−13 km and going to 0.1 km. This is chosen in order to preserve numerical stability at the start. For every new iteration the density at r is interpolated from the EoS by passing p(r) obtained in the previous iteration. The interpolation is linear. The actual code in Python is presented in the appendix A.1.2.

15


Figure 3.2: The equation of state utilized: for low densities the WD EoS (green line) and for high densities the Quark-Hadron-Crossover EoS (blue circles). The Baym-Bethe-Pethick EoS is also presented (in orange crosses) for comparison.

Initial Conditions and Singularities The initial conditions for the pressure are obtained by extrapolating from our EoS p0 = p(ρ0 ). The TOV equations have a singularity at r = 0 which is a problem in the integration. The rigorous way to solve this problem is to Taylor expand the equations around zero and compute M and p at some initial value of r very close to zero. The approach taken in our solution is such that we take a very small value r0 = 10−9 km, and argue that this is close enough to zero, so that ρ(r0 ) = ρ0 (our initial central density) and M (r0 ) = 4/3πr03 ρ0 are valid. This way we have avoided the singularity and established our initial conditions. Finding the Zero Values The pressure at the surface of the star is equal to zero. We are interested in finding the value of r = R for which p(r = R) = 0 and this will be the radius of the star; conversely at this point the total mass of the star is obtained M (r = R). We call these the zero values (since p = 0). The zero values are found such that we integrate the equations only if the condition p ≥ 0 is met, and we save the last value for which this condition is met. This point alone is very close to the real zero since a small enough integration step is used. The real zero can be found simply by extrapolating. Note on the Units The TOV equations 3.1 are written in geometrized units. To be consistent, we convert all quantities in geometrized units, and in kilometers (justified by a more intuitive reading while debugging). The conversion of the quantities is well documented in the code presented in the appendix A.1.2.

16


Figure 3.3: Compact objects (WD+NS) mass as a function of their central density.

3.2

Results and Discussion

We present the solution of TOV equation as a relation between the mass of the compact object and the central density in figure 3.3, and as mass-radius relationship in figure 3.4. For the low density regimes the familiar behavior of white dwarfs mass is obtained. WD mass peaks at around some critical density of ρc ≈ 109 g/cm3 , and the Chandrasekhar mass for WD is obtained. The value of MCh,wd ≈ 1 M is reached, which is slightly lower than that in the WD specific treatment as seen in the previous discussion. This could be due to the effect of the EoS and some approximations made. Stability of the configuration can be defined as 1 dM >0 dρ0 dM 0 : RHO = rho [ i , j ] − d x i /2 ∗ ( zed [ i , j ] ∗ ( rho [ i , j ] ∗ ∗ ( 1 / 3 ) / x i [ i ] ∗ ∗ 2 ) ∗ s q r t (1+ c1 ∗∗2∗ rho [ i , j ] ∗ ∗ ( 2 / 3 ) ) ) ZED = zed [ i , j ] + d x i /2 ∗ ( x i [ i ] ∗ ∗ 2 ∗ rho [ i , j ] ) x i 2 = ( x i [ i +1]+ x i [ i ] ) /2 rho [ i +1, j ] = rho [ i , j ] − d x i ∗ ( ZED ∗ (RHO∗ ∗ ( 1 / 3 ) / x i 2 ∗ ∗ 2 ) ∗ s q r t (1+ c1 ∗∗2∗RHO∗ ∗ ( 2 / 3 ) ) ) zed [ i +1 , j ] = zed [ i , j ] + d x i ∗ ( x i 2 ∗∗2 ∗ RHO ) else : xi_zero [ j ] = xi [ i ] arg_zero [ j ] = i n t ( i ) zed_zero [ j ] = zed [ i , j ] break #################################################################### ###### c a l c u l a t i o n o f M and R ####### Radii = xi_zero ∗ alpha R a d i i = R a d i i /Rsun M = 4∗ p i ∗ a l p h a ∗∗3 ∗ zed_zero M = M/Msun

A.1.3

Numerical Solution TOV Equations

In this section the complete Python code written to solve the TOV equations is presented. %p y l a b from s c i p y . i n t e r p o l a t e import i n t e r p 1 d ####### d e f i n i t i o n o f a l l t h e c o n s t a n t s ( i n c g s )############ G = 6 . 6 7 e−8 #G r a v i t a t i o n a l Constant ( c . g . s . ) h = 6 . 6 2 6 e −27 #Planck Constant ( e r g . s ) c = 3 e10 #Speed o f l i g h t (cm/ s ) me = 9 . 1 1 e −28 #Mass o f e l e c t r o n ( g ) mh = 1 . 6 7 e −24 #Mass o f n u c l e o n s ( g ) Msun = 1 . 9 8 9 e33 #Mass o f t h e sun ( g ) Rsun = 695700 ∗ 1 . 0 e5 #cm kappa = 2 #No . o f e l e c t r o n s p e r n u c l e o n ( He , O, C)

21

A.1. PYTHON CODES

kappa2 = 2 . 3 3 #f o r wd o f i r o n r i c h c o r e s ############################################################# ################## c o n v e r s a t i o n f a c t o r s ##################### c d e n s = G/ c ∗∗2 ∗ 1 . e10 #km^−2 i n g e o m e t r i z e d c p r e s = 1 . 6 0 2 1 7 e33 ∗ G/ c ∗∗4 ∗ 1 . e10 #km^−2 1 . 6 0 2 1 7 e33 t o g e t i n c g s and G/ c ∗∗4 i n g e o m e t r i z e d u n i t s Msuncm = 1 . 4 7 6 #km i n g e o m e t r i z e d u n i t s ############################################################# ################## l o a d e o s : wd + hqc18 ##################### eos_wd_hqc18 = l o a d t x t ( ’eos_wd_HQC18 . dat ’ ) wd_hqc18_rho = eos_wd_hqc18 [ : , 0 ] # i n km^−2 geometrized wd_hqc18_pres = eos_wd_hqc18 [ : , 1 ] # i n km^−2 geometrized ###i n t e r p o l a t o r wd_hqc18_interp_p = i n t e r p 1 d ( wd_hqc18_rho , wd_hqc18_pres , kind = ’ l i n e a r ’ ) wd_hqc18_interp_r = i n t e r p 1 d ( wd_hqc18_pres , wd_hqc18_rho , kind = ’ l i n e a r ’ ) # the i n v e r s e i n t e r p ############################################################## ##### c r e a t e an a r r a y rho_ndrip = 2 . 7 5 e11 ∗ rho_nucl = 2 . 6 7 e14 ∗ #we d e f i n e i n g cm^−3

o f i n i t i a l c e n t r a l d e n s i t i e s ########### cdens #now i n km^−2 cdens #now i n km^−2 and c o n v e r t t o g e o m e t r i z e d

rho_low = 2 . 7 e5 ∗ c d e n s rho_upp = 1 . 2 e16 ∗ c d e n s

# the lower i n i t i a l density f o r the s t a r s # t h e upper i n i t i a l d e n s i t y f o r t h e s t a r s

Nr = 200 # number o f i n i t i a l c e n t r a l d e n s i t i e s rho_0 = l o g s p a c e ( l o g 1 0 ( rho_low ) , l o g 1 0 ( rho_upp ) , Nr ) ############################################################### ####### p l o t t i n g o f t h e data and t e s t o f i n t e r p o l a t o r ######### l o g l o g ( wd_hqc18_rho/ cdens , wd_hqc18_pres/ c p r e s , l a b e l = ’WD+HQC18␣ data ’ ) l o g l o g ( rho_0/ cdens , wd_hqc18_interp_p ( rho_0 ) / c p r e s , ’−−o ’ , m a r k e r s i z e = 2 . 5 , l a b e l = ’ I n t e r p o l a t e d ’ , alpha = 0 . 7 ) # x l a b e l ( ’ d e n s i t y (km$^{−2}$ ) ’ ) x l a b e l ( ’ $ D e n s i t y $ ␣ ( g /cm␣ $^{−3}$ ) ’ ) # y l a b e l ( ’ p r e s s u r e (km$^{−2}$ ) ’ ) y l a b e l ( ’ $ P r e s s u r e $ ␣ (MeV/fm$^{−3}$ ) ’ ) legend () ################################################################ ############## TOV SOLVER : RK4 SCHEME ######################### Ni = 100000 # integration points mass = z e r o s ( ( Ni , Nr ) ) p r e s = z e r o s ( ( Ni , Nr ) )

# i n i t i a l i z a t i o n o f mass and p r e s s a r r a y s

x i 0 = 1 . e−9 # start i n km xi up = 50000 # end i n km x i = l o g s p a c e ( l o g 1 0 ( x i 0 ) , l o g 1 0 ( xiup ) , Ni ) # d e f i n e r a d i u s ’ i n t e g r a t i o n ’ points dxi = xi [ 1 ] − xi [ 0 ] p r i n t ( ’ the ␣ f i r s t ␣ dxi ’ , dxi ) p r e s _ z e r o = z e r o s ( Nr ) x i _ z e r o = z e r o s ( Nr ) arg_zero = z e r o s ( Nr ) mass_tot = z e r o s ( Nr )

# check t h e s t e p # i n i t i a t e arrays f o r f i n d i n g the z e r o s

22

A.1. PYTHON CODES

f o r j i n r a n g e ( 0 , Nr ) : mass [ 0 , j ] = 4/3∗ p i ∗ x i 0 ∗∗3∗ rho_0 [ j ] p r e s [ 0 , j ] = wd_hqc18_interp_p ( rho_0 [ j ] ) print ( j ) p r i n t ( ’ mass [ 0 , ’ , j , ’ ] ’ , ’%e ’ % mass [ 0 , j ] ) p r i n t ( ’ p r e s [ 0 , ’ , j , ’ ] ’ , ’%e ’ % p r e s [ 0 , j ] ) f o r i i n r a n g e ( 0 , Ni −1) : i f pres [ i , j ] > 0: d x i = x i [ i +1] − x i [ i ] rho = wd_hqc18_interp_r ( p r e s [ i , j ] ) k1_m k2_m k3_m k4_m

= = = =

4∗ p i 4∗ p i 4∗ p i 4∗ p i

∗ ∗ ∗ ∗

xi [ ( xi ( xi ( xi

i ]∗∗2 [i] + [i] + [i] +

∗ rho d x i / 2 ) ∗∗2 ∗ rho d x i / 2 ) ∗∗2 ∗ rho d x i ) ∗∗2 ∗ rho

k1_p = −(rho + p r e s [ i , j ] ) ∗ ( mass [ i , j ] + 4∗ p i ∗ x i [ i ] ∗ ∗ 3 ∗ p r e s [ i , j ] ) / ( x i [ i ] ∗ ( x i [ i ] − 2∗ mass [ i , j ]) ) k2_p = −(rho + ( p r e s [ i , j ] + d x i ∗k1_p / 2 ) ) ∗ ( mass [ i , j ] + 4∗ p i ∗ ( x i [ i ] + d x i / 2 ) ∗∗3 ∗ ( p r e s [ i , j ] + d x i ∗ k1_p / 2 ) ) / ( ( x i [ i ] + d x i / 2 ) ∗ ( ( x i [ i ] + d x i / 2 ) − 2∗ mass [ i , j ] ) ) k3_p = −(rho + ( p r e s [ i , j ] + d x i ∗k2_p / 2 ) ) ∗ ( mass [ i , j ] + 4∗ p i ∗ ( x i [ i ] + d x i / 2 ) ∗∗3 ∗ ( p r e s [ i , j ] + d x i ∗ k2_p / 2 ) ) / ( ( x i [ i ] + d x i / 2 ) ∗ ( ( x i [ i ] + d x i / 2 ) − 2∗ mass [ i , j ] ) ) k4_p = −(rho + ( p r e s [ i , j ] + d x i ∗k3_p ) ) ∗ ( mass [ i , j ] + 4∗ p i ∗ ( x i [ i ] + d x i ) ∗∗3 ∗ ( p r e s [ i , j ] + d x i ∗k3_p ) ) / ( ( x i [ i ] + d x i ) ∗ ( ( x i [ i ] + d x i ) − 2∗ mass [ i , j ]) ) mass [ i +1, j k3_m + p r e s [ i +1 , j k3_p +

] = mass [ i , j ] + d x i ∗ (k1_m + 2∗k2_m + 2∗ k4_m) /6 ] = p r e s [ i , j ] + d x i ∗ ( k1_p + 2∗k2_p + 2∗ k4_p ) /6

else : # thet_zero [ j ] = thet [ i , j ] pres_zero [ j ] = pres [ i , j ] xi_zero [ j ] = xi [ i ] arg_zero [ j ] = i n t ( i ) mass_tot [ j ] = mass [ i , j ] break #################### end TOV s o l v e r ############################# ############### p l o t i n g t h e o b t a i n e d r e s u l t s #################### subplots () s e m i l o g x ( rho_0/ cdens , ( mass_tot /Msuncm) ) x l a b e l ( ’ $ D e n s i t y $ ␣ ( gcm␣ $^{−3}$ ) ’ ) y l a b e l ( ’$M/M_{\ odot } $ ’ ) m i n o r t i c k s _ on ( ) legend () subplots () s e m i l o g x ( xi_zero , ( mass_tot /Msuncm) ) x l a b e l ( ’ $Radius$ ␣ (km) ’ ) y l a b e l ( ’$M/M_{\ odot } $ ’ ) m i n o r t i c k s _ on ( ) legend ()

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