Open access peer-reviewed chapter - ONLINE FIRST

Ramjet Acceleration of Microscopic Black Holes within Stellar Material

Written By

Mikhail V. Shubov

Reviewed: January 10th, 2022Published: March 22nd, 2022

DOI: 10.5772/intechopen.102556

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Solar Wind - Space, Humans and CommunicationsEdited by Yann Chemin

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Solar Wind - Space, Humans and Communications [Working Title]

Dr. Yann H. Chemin

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Abstract

In this work, we present a case that Microscopic Black Holes (MBH) of mass 1016kg–3×1019kg experience acceleration as they move within stellar material at low velocities. The accelerating forces are caused by the fact that an MBH moving through stellar material leaves a trail of hot rarefied gas. The rarefied gas behind an MBH exerts a lower gravitational force on the MBH than the dense gas in front of it. The accelerating forces exceed the gravitational drag forces when MBH moves at Mach number M<M0<1. The equilibrium Mach number M0 depends on MBH mass and stellar material characteristics. Our calculations open the possibility of MBH orbiting within stars including the Sun at Mach number M0. At the end of this work, we list some unresolved problems which result from our calculations.

Keywords

  • primordial black holes
  • microscopic black holes
  • ramjet acceleration
  • accretion
  • intrastellar orbits

1. Introduction

In the research presented in the works [1, 2, 3], it has been suggested that Primordial Black Holes make up a significant fraction of dark matter. Microscopic Black Holes (MBH) can also be formed within stars by coalescence of dark matter composed of weakly interacting massive particles [4, 5]. According to the plot in ([3], p. 14), considerations other than stellar capture constrain the masses of MBH as a dark matter to the range of 1016kg5×1021kg.

Up to now, researchers believed that all MBH captured by a star would be slowed down within stellar material until they settle in the stellar center [1, 2]. In the present work, we explore the possibility of MBH accelerating during their passage through stellar matter at low Mach numbers. As MBH passes through matter, it accretes material at a rate we denote Ṁ. Some of the mass accreted by MBH is turned into energy. This energy escapes the MBH in the form of protons and gamma rays. These rays heat the surrounding material, causing its rarefaction. The rarefied material behind the moving MBH exerts a lower gravitational pull on the MBH than the dense material in front of it. Moving MBH experiences a net forward force.This force is called MBH ramjetforce. The effect is illustrated in Figure 1.

Figure 1.

MBH passage through matter.

Figure 2.

Heat wave caused by subsonic MBH.

The conditions under which MBH accelerates within the stellar material are derived in this work. In order to define these conditions, three efficiencies must be defined. These are gas redistribution efficiency, radiative efficiency, and accretion efficiency. Gas redistribution efficiency, ηG, is the ratio of the accelerating force caused by gas rarefaction behind the MBH to the theoretical maximum of such force. The exact definition starts at paragraph containing Eq. (9) and ends with a paragraph containing Eq. (11). Radiative efficiency, ηΓ, is the ratio of the total power radiated by MBH to the power Ṁc2of the mass falling into MBH. It is expressed in Eq. (12). Accretion efficiency, ηA, is the ratio of the actual and the zero-radiation mass capture rates. It is defined in Eq. (16).

We show that in the case of MBH moving through stellar material at supersonic (supersonic MBH) speed, the condition for MBH acceleration is given in Eq. (29):

N=ηAηΓηG2T641041+M23/2,E1

where T6is the temperature of the stellar material in millions Kelvin. Even though we do not have precise values for efficiencies ηA, ηΓ, and ηG, we are almost certain that for supersonic MBH, condition Eq. (1) is never met. In the case of MBH moving through stellar material at a subsonic speed (subsonic MBH), the condition for MBH acceleration is given in Eq. (33):

N=ηAηΓηG2T69107M3FMηA,E2

where Mis the Mach number and FMηA>0.11is given in Eq (34). Supersonic MBH always experiences deceleration within stellar material. Subsonic MBH experiences acceleration when the Mach number exceeds M0(the equilibrium Mach number) and deceleration when the Mach number is below M0. Eventually the MBH settles into an intrastellar orbit with Mach number M0. The value of M0can be obtained by solving Eq. (2) as an equality.

In Appendix A, a minimal value of ηGfor subsonic MBH is estimated. Estimating ηGfor supersonic MBH remains an open problem. Calculating the values of ηAand ηΓalso remain open problems. As we discuss later in this work, different theorists obtained different results for ηΓ.

We briefly outline the content of the present chapter. In Section 2, we calculate forces acting on MBH. We also derive conditions for MBH acceleration at subsonic and supersonic speed. In Section 3, we present estimates for ηAand ηΓ. In Section 4, we present an empirical discussion of possible behaviors of MBH within stellar material. In Section 5, the problems remaining after this work are briefly described.

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2. Forces acting on an MBH passing through matter

2.1 Total force acting on an MBH

Three forces act on a black hole, which passes through stellar material. The first force denoted by Ftis the tidal or gravitational drag. For a supersonic MBH, Ftis given by ([6], p. 8)

Ft=1v0Pt=4πMG2ρv02lnrmaxrmin,E3

where Ptis the decelerating power produced by the drag force, ρis the density of the surrounding medium, rmaxis the approximate distance from the MBH to the farthest location where the stellar material is consistent, and rminis the radius at which matter is initially unperturbed by the MBH radiation. In Eq. (3), Mis the mass of the MBH. We take rmaxto be about 5107mfor a Sun-like star. We take rminto be about 0.1 m. Hence,

lnrmaxrmin20.E4

For an MBH traveling at Mach number M0.8, the gravitational drag can be given by the following formula (see [7], p. 5, [8], p. 69, [9], p. 8)

Ft=4πMG2ρv0212ln1+M1MM.E5

Since M<1, Eq. (5) can be rewritten in the form of a converging series

Ft=4πMG2ρv02n=1M2n+12n+1.E6

The second force is drag caused by mass acquisition. As the MBH passes through stellar material, it consumes mass that was formerly at rest. MBH momentum does not change as a result of mass acquisition. Change of MBH speed can be calculated from conservation of momentum:

pt=tMv0=Ṁv0+Mv̇0=0v̇0=v0ṀM.E7

Using MBH speed change, we calculate the effective force as

Fm=Mv0t=Ṁv0.E8

The third force is accelerative. It is caused by matter rarefaction behind the moving MBH. This force is denoted by Fr. In order to estimate Fr, we need two radii, r1and r2. The radius r1is defined in terms of Fr. A sphere of gas directly behind the MBH having radius r1, and density ρ/2would cause the MBH to experience accelerative force Fr. The sphere of rarefied gas behind an MBH acts as a sphere with a negative density of ρ/2ρ=ρ/2. The accelerating “ramjet” force Fracting on the MBH expressed in terms of r1is:

Fr=MMsGr12=MG1243πr13ρr12=23πMGρr1,E9

where Msis the effective negative mass of the sphere of rarefied gas.

The radius r2is defined in terms of the power Pradiated by MBH passing through the stellar material. Imagine that power Pis used to uniformly heat a cylinder of stellar material along the path of MBH. The MBH moves at speed v0. The radius r2is defined as the radius of the aforementioned cylinder for which the temperature of gas contained in it would double. Then the relation between Pand r2is:

P=MassheatedperunitoftimeTCv=v0πr22ρTCv=πv0ρTCvr22,E10

where Cvis the heat capacity of gas of stellar material at constant volume.

The gas redistribution efficiencyis defined as

ηG=r1r2.E11

As we show later in this section, the ramjet force Fracting on MBH is proportional to ηG. The minimal value for ηGfor subsonic MBH is estimated in Appendix A.

The radiative power of the MBH passing through stellar material is

P=ηΓc2Ṁ,E12

where ηΓis the radiative efficiencyof MBH and Ṁis the mass accretion rate. For a supersonic MBH, the Bondi-Hoyle-Lyttleton accretion rate is ([10], p. 203)

ṀBH=4πrb2ρv02+vs2,E13

where rbis the Bondi radius, and vsis the sound speed in the stellar material. The Bondi radius is ([10], p. 203)

rb=MGvs2+v02.E14

Substituting Eq. (14) into Eq. (13), we obtain

ṀBH=4πrb2ρv0=4πMG2ρv02+vs23/2.E15

The actual mass capture rate is considerably smaller. The radiative heating of the gas surrounding MBH increases its temperature. This increases the gas sound speed and decreases gas density. Thus, the actual mass capture rate is

Ṁ4πMG2ρrv02+vsr23/2,E16

where vsris the sound speed at the accretion radius and ρris the density at the accretion radius. Recall the accretion efficiencyηAis the quotient of actual and zero-radiation mass capture rates:

ηA=ṀṀBHv02+vs2v02+vsr23/2ρrρ.E17

Equating the power from Eqs. (10) and (12), we obtain

πv0ρTCvr22=ηΓc2Ṁ.E18

Substituting Eqs. (15) and (16) into Eq. (18), we obtain

πv0ρTCvr22=ηΓc2ηA4πMG2ρv02+vs23/2.E19

Thus,

r2=2ηAηΓc2TCvMGv02+vs23/4v0=2ηAηΓc2TCvMGv021+vs2v023/4.E20

Substituting Eq. (20) into Eq. (11), we obtain an expression for r1:

r1=2ηGηAηΓc2TCvMGv021+vs2v023/4.E21

At this point, we calculate the second and the third forces acting on the MBH. The first one is given in Eq. (3) for a supersonic MBH and in Eq. (5) for a subsonic MBH. Substituting Eq. (21) into Eq. (9), we obtain

Fr=23πMGρr1=43ηGηAηΓc2TCv1+vs2v023/4πMG2ρv02.E22

Substituting Eqs. (15) and (16) into Eq. (8), we obtain

Fm=Ṁv0=ηAṀBHv0=ηA4πMG2ρv02+vs23/2v0=4ηA1+vs2v023/2πMG2ρv02.E23

2.2 Conditions for supersonic MBH acceleration

The total force acting on a supersonic MBH is obtained by summing Eqs. (3), (22) and (23):

F=Ft+Fm+Fr=πMG2ρv024lnrmaxrmin4ηA1+vs2v023/2+2ηGηAηΓc2TCv1+vs2v023/4.E24

The above equation shows that MBH accelerates if and only if F>0, i.e.

ηGηAηΓc2TCv1+vs2v023/4>2lnrmaxrmin+2ηA1+vs2v023/2.E25

In this subsection we estimate conditions under which the MBH passing through matter accelerates, i.e., Eq. (25) holds. This condition can be rewritten as

ηAηΓηG2>4TCvc2lnrmaxrmin+ηA1+vs2v023/221+vs2v023/2E26

Recalling Eq. (4), and the fact that ηA<1, we rewrite the estimate to Eq. (26) as

ηAηΓηG21.7103TCvc21+vs2v023/2=1.7103TCvc21+M23/2.E27

The heat capacity at the constant volume of a monatomic gas is

Cv=3R2ma,E28

where mais the average molar mass of the gas, and Ris the gas constant R=8.314JmoloK.Typical stellar material consists of monatomic gas with an average particle mass of 0.62amu([11], p. 378). Hence, the heat capacity at constant volume for stellar material is Cv=2.01104JkgoK. Thus, Eq. (27) can be rewritten as

N=ηAηΓηG2T641041+M23/2.E29

As we show in Subsection 3.2, ηAis very small if the temperature of the gas at Bondi radius is high. As we discuss in Subsection 3.3, different calculations of ηΓin previous works yield different results, yet all of them are below 0.1. For supersonic MBH, ηGshould rapidly decrease with increasing Mach number. We have only qualitative arguments regarding the values of ηG. Stellar matter behind the MBH, which is displaced by a heat wave, remains within the Mach cone. Its gravitational pull can not be much lower than the pull of the unaffected matter in front of MBH. As the MBH Mach number increases, the cone becomes narrower. The difference of gravitational pull between matter in front of MBH and behind MBH decreases. Hence, ηGdecreases as well. Calculation of ηGis beyond the scope of this work. The solar gas temperature exceeds T6=4for radius under 0.5 Solar radii [11].

Based on the above data, we can be almost certain that relation Eq. (29) does not hold for Mach numbers M>1, thus a supersonic MBH can not accelerate. Very extensive analysis is needed in order to rigorously prove this assertion. Such analysis is beyond the scope of this work. It may be beyond the scope of any previous work on black hole accretion.

2.3 Conditions for subsonic MBH acceleration

The tidal decelerating force acting on an MBH traveling through stellar material at Mach number M0.8is given by Eq. (6). The total force acting on MBH is obtained by summing Eqs. (6), (22), and (23):

F=Ft+Fm+Fr=πMG2ρv024n=1M2n+12n+14ηA1+vs2v023/2+2ηGηAηΓc2TCv1+vs2v023/4=πMG2ρv024n=1M2n+12n+14ηA1+1M23/2+2ηGηAηΓc2TCv1+1M23/4.E30

The above equation shows that MBH will accelerate if and only if F>0or

ηGηAηΓc2TCv1+M23/4>2n=1M2n+12n+1+2ηA1+M23/2.E31

Rewrite Eq. (31) as:

ηAηΓηG2>4TCvc21+M23/2n=1M2n+12n+1+ηA1+M23/22=4TCvc2M31+M23/2n=0M2n2n+3+ηA1+M23/22.E32

Given that Cv=2.01104JkgoK, we rewrite Eq. (32) as

ηAηΓηG2T69107M3FMηA,E33

where

FMηA=1+M23/2n=0M2n2n+3+ηA1+M23/22.E34

Notice that FMηA>.11.

The Mach number for which an MBH settles into a stable intrastellar orbit is such that the net force acting on the MBH is 0. It can be estimated by solving an equation derived from Eq. (33):

N=ηAηΓηG2T6=9107M3FMηA.E35

All three efficiencies in Eq. (35) are nonzero. Thus, Eq. (35) does have a solution M0. An MBH traveling in stellar material accelerates when its Mach number is below M0and decelerates when its Mach number is above M0. Thus, an MBH traveling within a star is bound to settle into a stable intrastellar orbit. In order to calculate M0from Eq. (35), one must know gas redistribution, accretion and radiative efficiencies. In the next section, we present preliminary estimates for the three aforementioned efficiencies.

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3. Estimation of gas redistribution, accretion and radiative efficiencies

3.1 The value of ηG

In Appendix A.1, we prove that

ηG=r1r2.44K1KE36

for MBH traveling at subsonic speeds with M<0.8. Kis the ratio of average temperature in the hot gas trail and ambient temperature of stellar material. A possible route for estimating Kis outlined in Appendix A.2.

3.2 The value of ηA

From Eq. (17), we estimate ηAas

ηAv02+vs2v02+vsr23/2ρrρ,E37

where vsris the sound speed at the accretion radius and ρris the density at the accretion radius. Given that gas density is proportional to its pressure divided by temperature, we obtain

ηAv02+vs2v02+vsr23/2ρrρ=v02+vs2v02+vsr23/2PrTPTr=v0/vs2+1v0/vs2+vsr/vs23/2PrTPTr=M2+1M2+vsr/vs23/2PrTPTr.E38

In Eq. (38), Tis the ambient temperature of stellar material, and Tris the temperature of stellar material at Bondi radius. The ambient pressure is P, and pressure at Bondi radius is Pr. Notation Pcan not be used for pressure, as it is already used for power. For a gaseous medium, the sound velocity is proportional to the square root of temperature. Thus,

vsrvs2=TrT.E39

Substituting Eq. (39) into Eq. (38), we obtain the approximation

ηAM2+1M2+Tr/T3/2PrTPTr.E40

The pressure within the immediate vicinity of MBH should be approximated by the sum of gas pressure and dynamic pressure:

Pr=P+ρv022=P+ρvs2M22.E41

Notice that the gas density at the Bondi radius must be equal to or lower than the density of unperturbed gas. Hence, the approximation in Eq. (41) above works only when

PrTPTr<1,E42

The relation between pressure and sound velocity in a monatomic ideal gas is ([12], p. 683):

P=ρvs2γ=35ρvs2.E43

Substituting Eq. (43) into Eq. (41), we obtain the pressure ratio

PrP=35ρvs2+ρvs2M2235ρvs2=1+56M2.E44

Substituting Eqs. (44) and (42) into Eq. (40), we obtain

ηAM2+1M2+Tr/T3/2min11+56M2TTr.E45

As we see, ηAis a rapidly increasing function of the Mach number and a rapidly decreasing function of Tr/T. For subsonic MBH and for all cases where Tr/TM2, Eq. (45) can be approximated as

ηA1+56M2M2+13/2TTr5/2.E46

Calculation of Trremains an unsolved problem.

3.3 The value of ηΓ

Some energy is radiated from a spherically accreting MBH in the form of photons. The power radiated as photons is given as ηγṀc2. Some energy is radiated from a spherically accreting MBH in the form of protons and neutrons. The power radiated as baryons is given as ηpṀc2, since protons are more numerous than neutrons. The overall radiative efficiency of an MBH is

ηΓ=ηγ+ηp.E47

3.3.1 Gamma radiation from spherically accreting MBH

Accretion rate per unit MBH mass is one of the main factors determining ηγ. This rate should be expressed as a multiple of the Eddington accretion rate ([13], p. 51):

A=1.431016sṀM=Ṁ70kgsM18,E48

where M18is the mass of MBH in units of 1018kg.

Below we will summarize some previous works calculating ηγfor spherical accretion. Spherical accretion on black holes have been studied theoretically, with different theories producing different values of radiative efficiency ηγ([13], p. 25–55). Radiative efficiencies ranging from 1010to over .1have been obtained for different parameters. The magnetic field greatly increases ηγ([13], p. 34–35). For 104A1, radiative efficiency can be as high as 0.1if the flow is turbulent ([13], p. 35).

Detailed calculations of spherical accretion are presented in Ref. [14]. For a black hole of 21038kg, radiative efficiency starts growing almost from zero at A=.02and reaches ηγ=.19for A=1.2. For a black hole of 21031kg, radiative efficiency starts growing almost from zero at A=.5and reaches ηγ=.15for A=.12. MBH was not considered.

A model which considers separate ion and electron temperatures within accreting gas is given in Ref. [15]. Black hole masses between2×1031kgand 2×1038kgare considered. Accretion rates between A=7103and A=2are considered. In all cases, the efficiency stays within ηγ4.81037103. Notice, that all of the aforementioned studies considered black holes many orders of magnitude heavier than 1018kg. To obtain better results for MBH, more detailed studies for black holes within 1016kg3×1019kgare needed.

For black holes with accretion rates A1,300, the values of ηγrange from 106to 102([16], p.10). The state-of-the-art results have a lot of uncertainty.

3.3.2 Proton and neutron radiation from spherically accreting MBH

Gas accreting toward MBH experiences great compression, which causes adiabatic heating. Hot gas reaches temperatures of tens to hundreds of billion degrees Kelvin. As a result, some protons and neutrons which have excess energy escape the gravitational well around MBH. A very rudimentary estimation of ηpis performed below. In order to calculate ηpprecisely, we would need to perform an extensive Monte Carlo simulation. This simulation would have to take into account proton motion and collisions.

During accretion, the electron gas is much colder than the proton gas. Average temperature of proton is approximated by ([14], p. 17, [15], p. 323):

Tyrs=Tsy,E49

where rsis the Schwarzschild radius and Ts1012oK. When the distance from MBH is corresponding to y2550Schwarzschild radii and the gas temperature is 20–40 billion Kelvin, the nuclei split into protons and neutrons.

At this point, we calculate the depth of the potential well in which nucleons appear at a distance yrsfrom the MBH center. We take the non-relativistic approximation valid for y2.

Epyrs=mpMGyrs=mpMGy2MGc2=mpc22y,E50

where mpis the proton mass. Below, we express Eq. (50) in terms of Boltzmann constant k=1.381×1023J/K:

Epyrs=mpc22y=kympc22k=5.441012oKky5.44kTsy.E51

Like particles of any gas, protons and neutrons within accreting gas should have Maxwell energy distribution:

fE=2πkTEkTexpEkT,E52

At any distance yrsfrom the MBH center, some nucleons have sufficient kinetic energy to escape from the gravitational potential well of MBH. The energy depth of that well is given by Eq. (51). The fraction of nucleons capable of escaping is

Fe=5.44fEdE=0.012.E53

Nucleons escaping from a distance yrsfrom the MBH center carry excess kinetic energy. That energy is

Fe=kTsy5.44E5.44fEdE=0.009kTsy.E54

The energy given in Eq. (54) above is the quotient of the excess energy of ejected nucleons to the total number of nucleons, including the ones not ejected.

Define ηpas the quotient of the kinetic energy of nucleons ejected from accreting material to the rest energy of all nucleons. Many ejected nucleons lose energy in collisions, and some return to MBH. Thus, the final energy radiated from MBH as nucleon radiation is ηp<ηp. We estimate ηpas

dηpdlny=Feympc2=0.009kTsmpc21y8104y.E55

Integrating Eq. (55) for y>2, we obtain

ηpy=28104dyy=4104.E56

The value of ηpdepends on the fraction of the nucleons which are slowed down by accreting gas and returning to MBH. An extensive study and simulation may yield the value of ηphigher than the value of ηpestimated in Eq. (56). At this point, precise efficiencies are unknown.

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4. Possible modes of interaction of MBH with a star

In this section, we discuss the behavior of a Primordial Black Hole (PBH) which is captured into an orbit that intersects a star. Every PBH discussed here is an MBH, since it is microscopic. Not every MBH is a PBH, since some MBH are not primordial.

PBH ejection from a star-intersecting orbit is the first mode of PBH-star interaction. Any MBH or PBH on a star-intersecting orbit moves within stellar material with supersonic speed. Thus, it experiences deceleration within stellar material. Such MBH or PBH can be ejected from its orbit only by gravitational interaction with the star’s planets. In our opinion, such ejections are not rare. Reasoning follows.

Kinetic energy loss of an MBH on a single intrastellar passage is (see Appendix B):

ΔEpass=2.01019JM182.E57

The energy needed to drop the apogee of an elliptic MBH orbit around a sun-like star to 1 Astronomical Unit is

ΔEorbit=GMSunMMBH1AU=8.91026JM18.E58

Dividing Eq. (58) by Eq. (57), we obtain the number of times an MBH has to pass through a star in order for its orbit apogee to descend to 1 AU:

N=ΔEorbitΔEpass4.5107M181.E59

During this number of passes, the gravity of satellites of a star may throw an MBH off the orbit.

Settling of MBH into an intrastellar orbit is the second mode of MBH-star interaction. One possibility of MBH entering an intrastellar orbit is an MBH is a capture by a star. Another possibility is MBH production at the star center by coalescence of dark matter [4, 5]. Such MBH would be accelerated until it settles in an intrastellar orbit.

Consumption of a host star by an MBH is the third mode of MBH-star interaction. The evolution of an intrastellar MBH depends on its growth rate. An intrastellar MBH moves at low subsonic speed, hence its mass growth rate can be approximated by Eq. (15) which holds for a stationary MBH:

Ṁ=4πMG2ρrvsr3,E60

where ρris the density at Bondi radius and vsris the sound speed and Bondi radius. Sound velocity within the gas is proportional to T0.5. Gas density is proportional to T1. Hence,

Ṁ=4πMG2ρvs3TTr2.5,E61

where ρ, vs, and Tare the density, sound speed, and temperature of stellar material, while Tris the temperature at Bondi radius. In the Solar center, the density is 1.5×105kgm3and the sound speed is 5.1×105ms([11], p. 378). Substituting the above into Eq. (61), we obtain the accretion rate

Ṁ=6.3×104kgsTTr2.5M182=2×1018kgMillionyearsTTr2.5M182.E62

Dividing both sides of the above equation by mass, we obtain

ddtlnM18=Ṁ18M18=ṀM=2M18MillionyearsTTr2.5.E63

In Eq. (63) above, Tand Trare gas temperatures of ambient matter and at Bondi radius respectively. A low mass MBH is unlikely to experience significant growth over the lifetime of the host star. Determining exact MBH and host star characteristics for which the host star is consumed remains an open problem.

As we see from Eq. (63), the initial growth of an MBH within a star is slow. As the MBH gains mass with M18>100, all emitted radiation is absorbed by the accreting gas and TTr. Then the star is consumed by MBH over several millennia.

The growth of intrastellar black holes has been considered by previous researchers [18]. As a black hole consumes a star, it obtains the star’s angular momentum and becomes a rapidly rotating black hole. As a rotating black hole absorbs matter, it radiates two jets along its axis [19]. The final stages of stellar consumption by MBH may be responsible for long γ-ray pulses [20].

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5. Conclusion and remaining problems

In this work, we have demonstrated that MBH passing through stellar material experiences acceleration rather than deceleration as long as

N=ηAηΓηG2T641041+M23/2forsupersonicMBH9107M3FMηAforsubsonicMBH,E64

where FMηAis given in Eq. (34). T6is the temperature of the stellar material in millions Kelvin. The gas redistribution efficiencyηG, radiative efficiencyηΓ, and accretion efficiencyηAare defined in Introduction and Section 2.

MBH in stellar material experiences deceleration at supersonic speed. Subsonic MBH either accelerates or decelerates until it reaches equilibrium Mach number calculated from (??) and settles into a stable intrastellar orbit.

If the Universe contains MBH, many or most of them may exist in intrastellar orbits within stars. Some MBH may be orbiting within the Sun. Some of these MBH may be PBH captured by stars. We do not know how frequent is stellar capture of PBH. The calculation of this frequency is one of the many open problems generated by this work. Different PBH masses as well as star and planetary system characteristics will have to be considered in this calculation.

Other MBH may be generated within stellar centers. According to some theories, most Dark Matter consists of Weakly Interacting Massive Particles (WIMPs). Within stellar centers, WIMPs may coalesce into MBH [4, 5]. These MBH would experience acceleration until they settle into intrastellar orbits.

Several detectable effects may be produced by MBH on intrastellar orbits. Some Type 1a supernovas may be triggered by these MBHs [4]. Some MBHs may be on an intrastellar orbit within Sun. These MBH produce very low-frequency sonic waves. These waves are detectable by helioseismology—study of vibrations of Solar photosphere.

Only very low frequency sound can travel long distances in any gas. Sound with a frequency of a few millihertz or lower can travel from the Solar center to the Solar surface [22]. From the data presented in Ref. ([11], p. 378) we calculate that the orbital period of an MBH on an intasolar orbit is at least 800 s. This shows that acoustic waves produced by MBH rich Solar surface. Hence, these waves can be detected.

As mentioned in Subsection 3.3, radiative efficienciesηΓof accreting MBH can not be determined at this point. Most advanced theories give results, which vary by several orders of magnitude. Values ranging from 1010to 0.1 have been obtained so far. We do not know which theory is correct. If one or more MBH orbiting within the Sun is detected, then true values of radiative efficiencies will be obtained from observation.

In Appendix A, we estimate a minimum value of ηGfor subsonic MBH. The exact calculation of ηGis a remaining problem. It would involve extensive theoretical work and simulations using gas dynamics and radiation-matter interaction.

Accretion efficiency ηAfor both subsonic and supersonic MBH is given by Eq. (45) in terms of Tr—the temperature at the Bondi radius. The calculation of Tris a remaining problem. Exact calculation of Tr, and ηAwould involve extensive theoretical work and simulations using gas dynamics, radiation energy transport, and magnetohydrodynamics.

This work is purely theoretical. Nevertheless, helioseismological observations may eventually provide evidence of an MBH orbiting in an itrasolar orbit. This observation may open possibilities to obtain additional knowledge in many branches of physics. Knowledge in any branch of physics may lead to unforeseeable technological advances in the future.

RSunρ3v6Ft/M182
0.01461.3985109N
0.1821.3351109N
0.2351.1928109N
0.312.31.0612.3109N
0.44.00.964.9109N
0.51.350.872.0109N
0.60.490.800.86109N
0.70.1850.740.38109N
0.80.0770.690.18109N

Table 1.

Parameters for MBH passing through a sun-like star.

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A.1 Minimum value of ηGfor subsonic MBH

We assume strictly subsonic regime with M0.8. In a diagram below, we illustrate the MBH passing through stellar material.

The heated stellar material produced by subsonic MBH consists of two regions. The first region is the parabolic head region of hot gas surrounding the MBH. The second region is the hot gas trail, denoted by Rhg.

An important issue is the location of stellar material mass displaced by the heat wave. If MBH has subsonic speed, then sonic density waves carry away all of the displaced mass. Sonic waves are shaped as expanding spherical shells. Each shell is centered at the point of wave origin. A subsonic MBH can not outrun shells expanding at the speed of sound. Hence, all of the expanding shells contain advancing MBH inside them. As a result, these shells containing displaced matter exert no net gravitational force on MBH.

Accelerative force Frexerted on MBH comes from the difference in density of ambient stellar material and hot rarefied gas within the tail region Rhgas well as the head region as shown on Figure 2. We simplify calculation by ignoring the head region, which does provide a small propulsive force.

The gravitational force exerted by Rhgis calculated below. The region Rhgcan be approximated by a cylinder with radius rh. This cylinder starts at the distance at most rhfrom the MBH. By taking the distance to be rh, we are estimating the minimal value of the force. The region Rhgis represented in cylindrical coordinates with MBH at the origin. The direction in which the MBH is traveling is ẑ. In cylindrical coordinates, Rhgis given by

zrhr0rhEA.1

Stellar gas temperature in Rhgis approximated by a uniform temperature KT, where K>1is a constant and Tis the ambient temperature. Gas pressure within Rhgis almost the same as ambient gas pressure. From temperature and pressure in Rhg, it follows that the gas density in that region is ρ/K, where ρis the ambient density. From the standpoint of gravitational interaction, effective negative densityρof the material in Rhgcan be defined as the difference between gas density in Rhgand ambient gas density. Effective negative density is

ρ=ρKρ=ρK1K.EA.2

The rarefied gas region Rhgexerts the following force on MBH:

F=RhgρgrdV=ρK1KRhggrdV.EA.3

In Eq. (A.3) above, acceleration due to MBH gravity at point ris

gr=MGrr3EA.4

Substituting Eq. (A.4) into Eq. (A.3), we obtain

F=MGρK1KRhgrr3dV,EA.5

The gas displaced by the MBH passage in ẑdirection retains cylindrical symmetry. This symmetry implies that the net force on the MBH will act only in ẑdirection. Thus, Eq. (A.5) can be further simplified to

Fr=Fẑ=MGρK1KRhgrẑr3dV=MGρK1KRhgzz2+r23/2dVMGρK1K0rhrhπrzz2+r23/2dzdr=πMGρK1K0rhrz2+r2z=rhz=dr=πMGρK1K0rhrrh2+r2dr=πMGρK1Krh2+r2r=0rh=πMGρrhK1K21.EA.6

As we have mentioned earlier, the real force is greater or equal to the one calculated by approximating Rhgby Eq. (A.1). Eqs. (A.6) and (9), we obtain

r13221K1Krh0.62K1Krh.EA.7

Below, rhis estimated in terms of r2. The power needed to heat the gas trail is

PT=MassheatedperunitoftimeTemperatureCp=v0πrh2ρKK1T53Cv=5K13Kπv0ρTCvrh2,EA.8

where PTis the thermal power. For monatomic gas, Cp=53Cv. Some of the power Pradiated by the MBH goes into the production of the sonic waves, hence P>PT. Accurate calculation of PT/Pis beyond the scope of this work. Nevertheless, for subsonic MBH, we are certain that no more than 20% of MBH heating power is consumed by making sonic waves. Therefore, PT/P0.8. Using this data, we estimate the total radiative power of MBH:

P2K1Kπv0ρTCvrh2EA.9

Substituting Eq. (A.9) into Eq. (10), we obtain

2K1Kπv0ρTCvrh2πv0ρTCvr22.EA.10

Hence,

r2rh2K1K.EA.11

Dividing Eq. (A.7) by Eq. (A.11), we obtain

ηG=r1r2.44K1K.EA.12

Calculation of Kis beyond the scope of this work. Some considerations regarding the value of Kare presented in Appendix A.2.

A.2 Estimation of K

Recall, that the average temperature of the gas in the hot tail is KT, where Tis the temperature of the surrounding stellar material. In order to make any inference on the value of K, we introduce two radii and calculate their ratio. Radiation radiusrγis the average distance traveled by a photon or another energy-carrying particle from PBH before being absorbed by stellar material. Minimal hot tail radiusrmhis the minimal radius the hot tail can have regardless of K.

Below we estimate rγand rmh. The radiation radius is

rγ=Sγρp=Sγ103kgm3ρ3p,EA.13

where Sγis the planar density of material through which an energy carrying particle has to travel before being absorbed by stellar material. The density ρpis an average density of the material over the path of the energy-carrying particle, and ρ3pis the same density in 103kg/m3. The value of Sγis inversely proportional to average absorption cross-section of the energy-carrying particles:

S1=1kg1000NAamu11028σ=17kgm2σinbarn,EA.14

where σis the absorption cross-section. Given that most interactions are scattering, effective absorption cross-section has to be calculated. Substituting Eq. (A.14) into Eq. (A.13), we obtain

rγ=Sγρp=0.017mρ3pσinbarn.EA.15

According to data presented in ([17], pp. 41–42), cross-section per amu decreases with photon energy. For 10 keV photon, it is 0.55 barn For 1 MV photon, it is 0.18 barn. For 50 MV photon, it is 0.023 barn.

The minimal hot tail radius can be obtained from Eq. (A.8):

rmh=3PT5πv0ρTCv,EA.16

where PTis the part of MBH power used to produce heat rather than the sound wave. Substituting Eqs. (15)(17) into Eq. (12) we obtain

PT=ηhP=ηhηΓηA4πMG2c2ρv02+vs23/2=ηhηΓηA4πMG2c2ρv031+M23/2,EA.17

where ηh0.8is the fraction of MBH radiative power which goes into heating the stellar medium rather than producing a sonic wave. Substituting Eq. (A.17) into Eq. (A.16), we obtain

rmh1.5ηΓηAMGc1+M23/4v02TCvEA.18

As mentioned in Subsection 2.2, for average stellar material, Cv=2.01104JkgoK. From Eq. (A.18), we obtain

rmh0.21mηΓηA1+M23/4M18v62T61/2.EA.19

Combining Eq. (A.15) and Eq. (A.19), we obtain the ratio

Rγ=rγrmh0.8v62T61+M23/4ρ3pM18σinbarnηΓηAv6T6maxv6v6s3/2ρ3pM18σinbarnηΓηA,EA.20

where v6sis the sound velocity in 106m/s.

If Rγ1, then gas close to MBH is heated to a great temperature. This gas expands before it has time to diffuse its heat. The expanded gas must remain hot in order to balance the outside pressure. In that case, K1. For Rγ1, thermal energy is dissipated over a very large gas volume. This gas volume is heated only by a small margin, thus 0<K11.

B. Estimation of a MBH kinetic energy loss on passage through a sun-like star

Using rmin=0.1mand rmax=5107mto express Eq. (3) in numerical terms we obtain:

Ft=4πMG2ρv02lnrmaxrmin=1.12109NM182ρ3v62,EB.1

where ρ3is density in 103 kg/m3, and v6is velocity in 106 m/s. Below, we tabulate several parameters for a MBH passing through a sun-like star. We use the density data from Solar interior given in [11]. Column 1 contains the fraction of Solar radius. Column 2 contains the gas density in 103 kg/m3. Column 3 contains an estimated speed of a MBH arriving from a distance of thousands of solar radii. Column 4 contains Ftfor M18=1(Table 1).

The Solar radius is R=6.96108m. Thus, we estimate the energy loss of a MBH passing through the center of a Sun-like star:

ΔE=RRFtdx=2.01019JM182.EB.2

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Written By

Mikhail V. Shubov

Reviewed: January 10th, 2022Published: March 22nd, 2022