Accretion Onto Endoparasitic Black Holes at the Center of Neutron Stars

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Honors Projects Student Scholarship and Creative Work

2021

Accretion onto endoparasitic black holes at the center of neutron stars

Chloe B. Richards Bowdoin College

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Recommended Citation Richards, Chloe B., "Accretion onto endoparasitic black holes at the center of neutron stars" (2021). Honors Projects. 285. https://digitalcommons.bowdoin.edu/honorsprojects/285

This Open Access Thesis is brought to you for free and open access by the Student Scholarship and Creative Work at Bowdoin Digital Commons. It has been accepted for inclusion in Honors Projects by an authorized administrator of Bowdoin Digital Commons. For more information, please contact [email protected]. Accretion onto endoparasitic black holes at the center of neutron stars

An Honors Paper for the Department of Physics and Astronomy

By Chloe B. Richards

1 Introduction 1 1.1 Background ...... 1 1.2 Summary ...... 2 1.3 Contributions ...... 4

2 Bondi accretion 7 2.1 General treatment ...... 7 2.2 Minimum accretion ...... 17

3 Analytical Estimates of accretion rates and accretion times 21 3.1 Accretion rates ...... 21 3.1.1 Bondi accretion: m(ra) MBH ...... 24 3.1.2 Dynamical accretion: m(ra) MBH ...... 25 3.2 Eﬀects of stellar evolution . . . . .∼ ...... 27 3.3 Accretion times ...... 28 3.3.1 Without stellar evolution ...... 29 3.3.2 With stellar evolution ...... 30 3.4 Fiducial neutron star model ...... 31

4 Numerical Simulations 34 4.1 Initial data proﬁle ...... 34 4.2 Numerical evolution ...... 39 4.3 Diagnostics ...... 41 4.3.1 Black hole mass ...... 41 4.3.2 Growth of black hole ...... 43 4.3.3 Rest-mass ﬂux ...... 45

5 Accretion Results 48 5.1 Comparison with Bondi ﬂow ...... 48 5.2 Complete consumption ...... 51 5.3 Accretion rates ...... 53

6 Conclusion 55

2 A Cubic equation solution for as 57

ii B Integration of Eq. (3.30) 59

Acknowledgments 62

References 63

iii Chapter 1

Introduction

1.1 Background

In this thesis we consider the accretion onto a non-rotating, endoparasitic1 black hole residing at the center of a neutron star. We do so in order to extend previous work discussing the possibility of neutron stars as dark matter detectors (see e.g. Goldman and Nussinov (1989); de Lavallaz and Fairbairn (2010); Bramante and Linden (2014); Bramante and Elahi (2015); Capela et al. (2013); Bramante et al. (2018); East and Lehner (2019); Génoliniet al. (2020)). We consider neutron stars to be possible dark matter detectors by supposing that they capture dark matter in the form of tiny black holes. Such systems evolve until the neutron star collapses, meaning the endoparasitic black hole residing at the center will accrete all matter from the neutron star. Currently, one constraint on black holes as sources of dark matter arises from the existence of neutron star populations. The scenario above can be considered in at least two different ways, namely the neutron star captures a primordial black hole (PBH) or some other dark matter particles. In the former case, we consider the possibility that PBHs, which formed in the early universe (see Hawking (1971); Carr and Hawking (1974)) can be captured by neutron 1We define endoparasitic to describe a tiny body residing inside a much larger body

1 stars. After capture, the PBH resides at the center of the neutron star and accretes matter until the star completely collapses (see Hawking (1971); Markovic (1995)). In the latter case, we consider that dark matter particles captured by a neutron star may under favorable conditions, collapse to form a small black hole (see Goldman and Nussinov (1989); de Lavallaz and Fairbairn (2010); Bramante and Linden (2014); Bramante et al. (2018)). After the black hole has been either captured by the neutron star or forms inside the neutron star, the resulting system evolves until the black hole completely consumes the neutron star. A number of authors including Fuller et al. (2017); Takhistov et al. (2020); G´enoliniet al. (2020) have discussed observational signatures of such a process. Numerical simulations by East and Lehner (2019) consider three diﬀerent equations of state for both rotating and non-rotating neutron stars. Their simulations were restricted

−2 to relatively large black hole masses MBH/M 10 , where M is the neutron star mass. ≥ Through their simulations, they determined that the accretion rate follows the relation

2 M˙ BH M as suggested by Bondi (see Bondi (1952); see Shapiro and Teukolsky (2004) ∝ BH for a textbook treatment). Furthermore, their results concluded that the accretion rate was largely independent of the neutron star spin, which agrees with results from Kouvaris and Tinyakov (2014).

1.2 Summary

We extend previous work on the accretion process of tiny black holes residing at the center of neutron stars in several ways in this thesis. Bondi accretion for an isolated black hole, i.e. where the gravitational forces are dominated by the black hole, is well understood. To determine the Bondi accretion rate for our system, we consider a spacetime describing a black hole in the interior of a neutron star. Additionally, we determine such Bondi accretion for stiﬀ equations of state. In Chapter 2, we give a general treatment

2 of Bondi accretion as described by Bondi (1952); Shapiro and Teukolsky (2004). For stiff equations of state with Γ > 5/3, we find the accretion rate under a relativistic treatment (see Appendix G in Shapiro and Teukolsky (2004)) and identify an associated minimum accretion rate. We do so in order to determine the accretion rate for our neutron star system described by a stiff, polytropic equation of state with adiabatic index Γ = 2. In Chapter 3, we then adopt our relativistic Bondi accretion model to describe a non-rotating black hole residing at the center of a neutron star. We crudely integrate the resulting equations both including and ignoring stellar evolution in order to give estimates for the lifetime of our neutron star model. In Chapter 4 we perform numerical simulations in full general relativity, extending previous results of East and Lehner (2019). We focus on spherical neutron stars with non-rotating black holes at the center. Our system adopts a stiff, polytropic equation of state with adiabatic index Γ = 2, which is commonly used to describe neutron stars in numerical simulations. We implement our code using spherical polar coordinates (see Baumgarte et al. (2013, 2015)) and a logarithmic radial coordinate in order to resolve the vastly different length scales between the black hole mass and the neutron star mass. We extend the work of East and Lehner (2019) by considering black holes with mass ratios

−9 as small as MBH/M 10 . With such small masses, we extend into the mass range of ' PBHs and black hole dark matter candidates. Our mass range gives black hole masses comparable to that of dwarf planets. We compare our analytical estimates with our numerical results in Chapter 5. We find excellent agreement between our analytical Bondi flow and the numerical rest- mass flux. For larger black hole masses, we evolve the system until the neutron star is completely consumed and compare with our analytical results accounting for and ignoring stellar evolution. We present both our analytical Bondi accretion rates and our numerical accretion rates in Table 5.1. We summarize our findings for a large range of black hole masses in Fig. 1.1 for

3 both our analytical estimates and numerical results. In Fig. 1.1, the solid line represents the analytical accretion rate for our fiducial neutron star model from Eq. (3.42), and the dashed line represents the corresponding minimum accretion rate given by Eq. (3.43). We note that such a minimum accretion rate exists only for stiff equations of state with Γ > 5/3. Our numerical results are presented through two different measurements of the accretion shown as triangles and open circles. The triangles represent the growth of the black hole mass with respect to time. As shown in the Figure, we are only able to compute this accretion for “large” black hole masses because the accretion rate is proportional to the square of the black hole mass, so the accretion rate becomes exceedingly small and difficult to determine numerically from the growth of the black hole. We are able to compute the accretion rate by measuring rest-mass flux across the horizon of the black hole for all black hole masses as shown by the open circles and find excellent agreement with our analytical estimates. Furthermore, we confirm that Bondi accretion for a stiff equation of state accurately describes the accretion onto an endoparasitic black hole at the center of our neutron star model, and therefore, we are able to use these analytical expressions to determine the lifetimes of such neutron stars. Throughout this thesis we use geometrized units with G = 1 = c unless noted otherwise where G is the gravitational constant and c is the speed of light.

1.3 Contributions

This thesis is based on collaborative work that resulted in the two publications Richards et al. (2021b) and Richards et al. (2021a). I would like to highlight my primary contributions:

• Derived the Bondi accretion for stiﬀ equations of state in Section 2.1

• Determined data describing our ﬁducial neutron star model presented in Table 3.2

• Performed all numerical solutions described in Chapter 4

4 9 10 4 10 Black hole growth 7 10 6 Rest-mass flux 10 Bondi accretion 8 105 10 Minimum accretion ] 10 103 r

10 y / H * B

12 1 M

10 [ M 10 H * 6 B 10 1 10 14 10 M

3 10 16 10 106 7 10 5 10 18 10 6 × 10 4 10 3

10 8 10 6 10 4 10 2 MBH/M

˙ ? Figure 1.1: Comparison of analytical and numerical accretion rates MBH for various black holes with initial mass MBH(0) for our fiducial neutron star model with Γ = 2. We express the accretion rate in geometrized units on the right-hand side of the plot and we express it in units of solar masses per year on the left-hand side. The solid blue line represents the analytical Bondi accretion rate given by Eq. (3.42). The numerical results for the accretion rate are shown by red triangles and open green circles, representing the growth of the black hole mass and the rest-mass flux over the black hole horizon respectively. The dashed orange line represents the minimum accretion rate for a Γ = 2 polytrope as given by (3.43). (See also Table 5.1 for a detailed listing of these results.) In the inset, we show −3 the all numerical and anayltical results for one particular black hole MBH/M 10 . we notice the excellent agreement between the analytical Bondi accretion for stiff' equations of state and numerical rest-mass flux. The numerical accretion given by black hole growth differs slightly from the analytical estimate as it includes thermal energy in addition to rest-mass.

5 • Determined numerical data for our numerical simulations presented in Table 5.1

6 Chapter 2

Bondi accretion

We use Bondi accretion to consider our model of a black hole embedded in a neutron star. Newtonian Bondi accretion as outlined in Bondi (1952) describes the steady-state, spherically symmetric ﬂow of gas onto a point mass. We assume the gas to be governed

Γ by a polytropic equation of state P = Kρ0 , where K is a constant and Γ is the adiabatic index. In this chapter, we will a derive a general solution for all adiabatic indices in the range 1 < Γ < 3. We note that the Newtonian treatment breaks down for Γ > 5/3 so that we need to employ a relativistic treatment. Furthermore, we will specify our result for Γ = 2, which is an adequate approximation for the equation of state of a neutron star and therefore allows us to consider accretion onto a black hole at the center of a neutron star.

2.1 General treatment

To consider relativistic spherical Bondi ﬂow onto an endoparisitic black hole, we follow Michel (1972) to ignore the inﬂow’s self-gravity. Rather than adopting the Schwarzschild metric in its usual form,

2M 2M −1 ds2 = 1 dt2 + 1 dr2 + r2dΩ2, (2.1) − − r − r

7 to describe the geometry of spacetime, we will allow for the general form

−1 κ e−2Dκ ds2 = e2D dt2 + 1 dR2 + R2dΩ2. (2.2) − − R − R

In the above, M is the mass of our black hole and D and κ are constants of integration. As described in Appendix A in Richards et al. (2021a), we adopt the generalized Schwarzschild metric given by (2.2) to re-derive the Bondi accretion equations for a black hole at the center of a neutron star. We deﬁne a local asymptotic observer to be one who is far from the black hole, but well within the radius of the neutron star. For the purposes of this chapter, a local asymptotic observer is identical to an asymptotic observer at inﬁnity because we are considering an isolated black hole, i.e. without a neutron star. The local asymptotic observer becomes important, however, in Chapter 3 as we then consider a black hole inside a neutron star. We note that we recover the Schwarzschild metric (2.1) for D = 0 and κ = 2M.

Now we consider the accretion rate M˙ at the critical radius rs as described by hydrodynamic spherical accretion in Eq. (14.3.4) of Shapiro and Teukolsky (2004),

˙ 2 M = 4πρ0susrs , (2.3)

where ρ0 is the rest mass density and u describes the radial component of the ﬂuid

r 4-velocity u u . Here, the critical radius rs reduces to the transonic point in a New- ≡ − tonian limit. In the relativistic limit, we assume that the ﬂow is subsonic asymptotically and therefore must past through the critical radius rs (see Appendix G in Shapiro and Teukolsky (2004)). For the remainder of this thesis, any variable with the subscript s is evaluated at the critical radius rs. Furthermore, when referencing the asymptotic region where r , variables will be marked with a subscript (for example, a∞ is the → ∞ ∞ asymptotic sound speed).

8 To avoid singularities in M˙ , we deﬁne the critical ﬂow velocity us as

as 2−1/2 us = 1 + 3as a∞ (2.4) a∞

and critical radius rs as 2 2 a∞ 1 + 3as M rs = 2 2 (2.5) as 2 a∞

as shown in Eq. (G.17) of Shapiro and Teukolsky (2004). Here we deﬁne the sound speed a in terms of pressure P and density ρ

2 dP dP ρ0 a = = , (2.6) dρ s dρ0 s ρ + P

where derivatives are taken at constant entropy. We assume a polytropic equation of state

Γ P = Kρ0 , (2.7)

where K is a constant and Γ the adiabatic index. The adiabatic index Γ parameterizes the stiﬀness of the equation. For Γ 5/3, increasing density ρ0 results in an increased ≥ pressure P , so for the purposes of this discussion, we will refer to all Γ 5/3 as describing ≥ stiﬀ equations of state. The sound speed is related to rest-mass density ρ0 by

ΓKρΓ−1 a2 = 0 . (2.8) 1 + ΓKρΓ−1/(Γ 1) 0 −

We notice that for ﬁnite densities ρ0, the sound speed must be smaller than a maximum

value amax, a2 < a2 = Γ 1. (2.9) max −

We evaluate Eq. (2.8) at the critical radius rs and in the asymptotic region where

r to ﬁnd a relationship between the rest-mass density ρ0s and the rest-mass density’s → ∞

9 asymptotic value ρ0∞,

2/(Γ−1) 2 1/(Γ−1) as Γ 1 a∞ ρ0s = − − 2 ρ0∞. (2.10) a∞ Γ 1 a − − s

Inserting (2.4), (2.5), and (2.10) into (2.3) we now ﬁnd

2 ˙ M M = 4πλGR 2 ρ0∞a∞, (2.11) a∞ where (5−3Γ)/(Γ−1) 2 1/(Γ−1) 2 3/2 as Γ 1 a∞ (1 + 3as) λGR − − 2 (2.12) ≡ a∞ Γ 1 a 4 − − s is a dimensionless “accretion rate eigenvalue”.

We note that the relationship between the sound speeds as and a∞ in the second

term of (2.12) dictates the limiting value of GR as a∞ 0. For soft equations of state, → λGR reaches a non-zero value as a∞ 0 as shown in Figure (2.1). However, for stiﬀ → equations of state, λGR 0 as a∞ 0 in order to maintain a ﬁnite accretion rate as → → given by (2.11).

While λGR describes the accretion rate eigenvalue in general for all solutions, we can simplify it by considering it in the Newtonian limit, where as 1 and a∞ 1, meaning the sound speed in both the inner and outer asymptotic regions are signiﬁcantly smaller than the speed of light c = 1. Under this treatment, Eq. (2.12) reduces to the

Newtonian accretion eigenvalue λs,

(5−3Γ)/(Γ−1) 1 as λs = . (Newtonian) (2.13) 4 a∞

From Shapiro and Teukolsky (2004) (14.3.14), we also have

a 2 1/2 s = (Newtonian) (2.14) a∞ 5 3Γ −

10 4.0 = 4/3 = 5/3 3.5 = 2 3.0 = 2.5 = 2.9 2.5 = 1.001 R

G 2.0

1.5

1.0

0.5

0.0

0.0 0.2 0.4 0.6 0.8 1.0 a2

Figure 2.1: Solutions for accretion rate eigenvalue λGR as given by Eq. (2.12) for various 2 values of Γ, as a function of a∞. Although we focus on stiff equations of state, we include λGR for Γ < 5/3 to show the discontinuity in accretion eigenvalues in the Newtonian limit 2 a 1. For soft equations of state Γ < 5/3 the accretion eigenvalues λGR approach the ∞ Newtonian values λs shown by the dots (see Eq. (2.15) as a∞ 0), whereas for Γ 5/3 → ≥ λGR approaches zero in that limit. This result for stiff equations of state is necessary in order to maintain a finite accretion rate M˙ when considering stiff equations of state Γ 5/3 in the limit of low asymptotic sound speed a2 1. ≥ ∞

11 in the Newtonian limit, which we can insert into (2.13) to obtain the Newtonian accretion eigenvalues 1 2 (5−3Γ)/2(Γ−1) λ = , (Newtonian) (2.15) s 4 5 3Γ − which are given by dots in Figure (2.1). We conﬁrm this result for the Newtonian accretion eigenvalues with Eq. (14.3.17) and Table 14.1 from Shapiro and Teukolsky (2004). As we can see, the Newtonian treatment breaks down for adiabatic indices

Γ > 5/3. We note that the Newtonian accretion eigenvalues λs depend on only Γ, while the relativistic accretion eigenvalues λGR also depend on the critical and asymptotic sound speeds as and a∞ respectively. Therefore, we require a relativistic treatment for stiﬀ equations of state with Γ > 5/3 which we will describe in the following. To solve for our accretion rate M˙ we consider the relationship between the sound speeds in the inner and outer asymptotic regions

a2 2 a2 2 (1 + 3a2) 1 s = 1 ∞ (2.16) s − Γ 1 − Γ 1 − −

as shown in Eq. (G.30) in Shapiro and Teukolsky (2004), which we now write as a cubic

2 equation for as

a4 a2 a2 a6 + s (7 6Γ) + s (3Γ2 8Γ + 5) + ∞ (2Γ 2 a2 ) = 0, (2.17) s 3 − 3 − 3 − − ∞

We write this equation in the form

x3 + Ax2 + Bx + C = 0 (2.18)

12 2 for x = as and identify the coeﬃcients

1 A = (7 6Γ) 3 − 1 B = (1 Γ) (5 3Γ) (2.19) 3 − − a2 C = ∞ (2Γ 2 a2 ), 3 − − ∞ noting that all three coeﬃcients A, B, and C are real. We solve the cubic equation (2.18) in detail in Appendix A and ﬁnd that while

there are three solutions x1, x2, and x3 given by (A.5), only x3 gives a physically viable solution. As an aside, we note that we can also rewrite Eq. (2.16) as a quadratic equation

2 for a∞ 1/2 a2 = Γ 1 (Γ 1)2 + 3 a6 + Aa4 + Ba2 . (2.20) ∞ − ± − s s s

For most applications, however, we consider a∞ as given and therefore solve the cubic

equations (2.18) for as. As mentioned above, Eq. (2.9) gives a condition for determining all physical solutions for Γ 2. However, we must impose causality explicitly for Γ > 2