Parameters for MBH passing through a sun-like star.

## 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

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 * Moving MBH experiences a net forward force.* This force is called

**force. The effect is illustrated in Figure 1.**MBH ramjet

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**,

**,**Radiative efficiency

**,**Accretion efficiency

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):

where

where

In Appendix A, a minimal value of

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

## 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

where

For an MBH traveling at Mach number

Since

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:

Using MBH speed change, we calculate the effective force as

The third force is accelerative. It is caused by matter rarefaction behind the moving MBH. This force is denoted by

where

The radius

where

The ** gas redistribution efficiency** is defined as

As we show later in this section, the ramjet force

The radiative power of the MBH passing through stellar material is

where ** radiative efficiency** of MBH and

where

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

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

where

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

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

Thus,

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

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

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

### 2.2 Conditions for supersonic MBH acceleration

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

The above equation shows that MBH accelerates if and only if

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

Recalling Eq. (4), and the fact that

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

where

As we show in Subsection 3.2,

Based on the above data, we can be almost certain that relation Eq. (29) does not hold for Mach numbers

### 2.3 Conditions for subsonic MBH acceleration

The tidal decelerating force acting on an MBH traveling through stellar material at Mach number

The above equation shows that MBH will accelerate if and only if

Rewrite Eq. (31) as:

Given that

where

Notice that

The Mach number for which an MBH settles into a stable intrastellar orbit is such that the net force acting on the MBH is

All three efficiencies in Eq. (35) are nonzero. Thus, Eq. (35) does have a solution

## 3. Estimation of gas redistribution, accretion and radiative efficiencies

### 3.1 The value of η G

In Appendix A.1, we prove that

for MBH traveling at subsonic speeds with

### 3.2 The value of η A

From Eq. (17), we estimate

where

In Eq. (38),

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

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

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

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

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

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

As we see,

Calculation of

### 3.3 The value of η Γ

Some energy is radiated from a spherically accreting MBH in the form of photons. The power radiated as photons is given as

#### 3.3.1 Gamma radiation from spherically accreting MBH

Accretion rate per unit MBH mass is one of the main factors determining

where

Below we will summarize some previous works calculating

Detailed calculations of spherical accretion are presented in Ref. [14]. For a black hole of

A model which considers separate ion and electron temperatures within accreting gas is given in Ref. [15]. Black hole masses between

For black holes with accretion rates

#### 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

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):

where

At this point, we calculate the depth of the potential well in which nucleons appear at a distance

where

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

At any distance

Nucleons escaping from a distance

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

Integrating Eq. (55) for

The value of

## 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):

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

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:

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:

where

where

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

In Eq. (63) above,

As we see from Eq. (63), the initial growth of an MBH within a star is slow. As the MBH gains mass with

The growth of intrastellar black holes has been considered by previous researchers [17]. 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 [18]. The final stages of stellar consumption by MBH may be responsible for long

## 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

where

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 [21]. 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

As mentioned in Subsection 3.3,

In Appendix A, we estimate a minimum value of

Accretion efficiency

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.

### A.1 Minimum value of η G for subsonic MBH

We assume strictly subsonic regime with

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

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

The gravitational force exerted by

Stellar gas temperature in

The rarefied gas region

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

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

The gas displaced by the MBH passage in

As we have mentioned earlier, the real force is greater or equal to the one calculated by approximating

Below,

where

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

Hence,

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

Calculation of

### A.2 Estimation of K

Recall, that the average temperature of the gas in the hot tail is

Below we estimate

where

where

According to data presented in ([22], 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):

where

where

As mentioned in Subsection 2.2, for average stellar material,

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

where

If

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

Using

where ^{3} kg/m^{3}, and ^{6} 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 10^{3} kg/m^{3}. Column 3 contains an estimated speed of a MBH arriving from a distance of thousands of solar radii. Column 4 contains

0.0 | 146 | 1.39 | |

0.1 | 82 | 1.33 | |

0.2 | 35 | 1.19 | |

0.3 | 12.3 | 1.06 | |

0.4 | 4.0 | 0.96 | |

0.5 | 1.35 | 0.87 | |

0.6 | 0.49 | 0.80 | |

0.7 | 0.185 | 0.74 | |

0.8 | 0.077 | 0.69 |

The Solar radius is

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