**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 *ηγMc* \_ 2. Some energy is radiated from a

spherically accreting MBH in the form of protons and neutrons. The power radiated as baryons is given as *ηpMc* \_ 2, since protons are more numerous than neutrons. The overall radiative efficiency of an MBH is

$$
\eta\_\Gamma = \eta\_\gamma + \eta\_p. \tag{47}
$$

#### *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):

$$\mathfrak{A} = \frac{\left(\mathbf{1}.43 \cdot \mathbf{1}0^{16} \text{ s}\right) \dot{M}}{M} = \frac{\dot{M}}{\left(\mathbf{1}0 \cdot \frac{\text{kg}}{\text{s}}\right) \mathbf{M}\_{18}},\tag{48}$$

where *M*<sup>18</sup> is the mass of MBH in units of 10<sup>18</sup> kg.

Below we will summarize some previous works calculating *ηγ* for 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 10�<sup>10</sup> to over *:*1 have been obtained for different parameters. The magnetic field greatly increases *ηγ* ([13], p. 34–35). For 10�<sup>4</sup> ≤ A ≤1, radiative efficiency can be as high as 0*:*1 if the flow is turbulent ([13], p. 35).

Detailed calculations of spherical accretion are presented in Ref. [14]. For a black hole of 2 � <sup>10</sup><sup>38</sup> kg, radiative efficiency starts growing almost from zero at <sup>A</sup> <sup>¼</sup> *:*02 and reaches *ηγ* <sup>¼</sup> *:*19 for <sup>A</sup> <sup>¼</sup> <sup>1</sup>*:*2. For a black hole of 2 � <sup>10</sup><sup>31</sup> kg, radiative efficiency starts growing almost from zero at A ¼ *:*5 and reaches *ηγ* ¼ *:*15 for A ¼ *:*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 � <sup>10</sup><sup>31</sup> kg and 2 � <sup>10</sup><sup>38</sup> kg are considered. Accretion rates between <sup>A</sup> <sup>¼</sup> <sup>7</sup> � <sup>10</sup>�<sup>3</sup> and <sup>A</sup> <sup>¼</sup> 2 are considered. In all cases, the efficiency stays within *ηγ* <sup>∈</sup> <sup>4</sup>*:*<sup>8</sup> � <sup>10</sup>�<sup>3</sup> , 7 � <sup>10</sup>�<sup>3</sup> . Notice, that all of the aforementioned studies considered black holes many orders of magnitude heavier than 10<sup>18</sup> kg. To obtain better results for MBH, more detailed studies for black holes within 10<sup>16</sup> kg–<sup>3</sup> � <sup>10</sup><sup>19</sup> kg are needed.

For black holes with accretion rates <sup>A</sup> <sup>∈</sup>ð Þ 1, 300 , the values of *ηγ* range from 10�<sup>6</sup> to 10�<sup>2</sup> ([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 *η<sup>p</sup>* is performed below. In order to calculate *η<sup>p</sup>* precisely, 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):

$$T(\mathcal{yr}\_s) = \frac{T\_s}{\mathcal{Y}},\tag{49}$$

where *rs* is the Schwarzschild radius and *Ts* ≈1012*<sup>o</sup> K*. When the distance from MBH is corresponding to *y*∈ð Þ 25, 50 Schwarzschild 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 *yrs* � � from the MBH center. We take the non-relativistic approximation valid for *y*≥2.

$$E\_p \left( \mathcal{y} r\_s \right) = -\frac{m\_p \text{ MG}}{\mathcal{y} r\_s} = -\frac{m\_p \text{ MG}}{\mathcal{y} \frac{2 \text{MG}}{c^2}} = -\frac{m\_p \text{ c}^2}{2 \text{y}},\tag{50}$$

where *mp* is the proton mass. Below, we express Eq. (50) in terms of Boltzmann constant *<sup>k</sup>* <sup>¼</sup> <sup>1</sup>*:*<sup>381</sup> � <sup>10</sup>�<sup>23</sup> <sup>J</sup>*=*K:

$$E\_p(yr\_s) = -\frac{m\_p}{2y} = -\frac{k}{y} \cdot \frac{m\_p}{2k} = -5.44 \cdot 10^{12} K \cdot \frac{k}{y} \approx -\frac{5.44 \ k \ T\_s}{y}.\tag{51}$$

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

$$f(E) = \frac{2}{\sqrt{\pi}} \sqrt{\frac{E}{kT}} \,\, \exp\left(-\frac{E}{kT}\right),\tag{52}$$

At any distance *yrs* � � from 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

$$\mathcal{F}\_{\varepsilon} = \int\_{5.44}^{\infty} f(E) dE = 0.012. \tag{53}$$

Nucleons escaping from a distance *yrs* � � from the MBH center carry excess kinetic energy. That energy is

$$\mathcal{F}\_t = \frac{kT\_s}{\mathcal{Y}} \int\_{5.44}^{\infty} (E - 5.44) f(E) dE = 0.009 \,\, \frac{kT\_s}{\mathcal{Y}} \,. \tag{54}$$

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 *η* <sup>∗</sup> *<sup>p</sup>* as 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 *η<sup>p</sup>* <*η* <sup>∗</sup> *<sup>p</sup>* . We estimate *η* <sup>∗</sup> *<sup>p</sup>* as

$$\frac{d\eta\_p^\*}{d(lny)} = \frac{\mathcal{F}\_\epsilon(y)}{m\_p c^2} = \frac{0.009 \text{ } kT\_s}{m\_p c^2} \frac{1}{y} \approx \frac{8 \cdot 10^{-4}}{y}.\tag{55}$$

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

$$
\eta\_p^\* \approx \int\_{y=2}^{\infty} \frac{8 \cdot 10^{-4} \text{ dy}}{y} = 4 \cdot 10^{-4} \text{.} \tag{56}
$$

The value of *η<sup>p</sup>* depends 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 *η<sup>p</sup>* higher than the value of *η* <sup>∗</sup> *<sup>p</sup>* estimated in Eq. (56). At this point, precise efficiencies are unknown.

### **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):

$$
\Delta E\_{\rm pass} = (2.0 \cdot 10^{19} \text{ J}) \text{M}\_{18\_{2}}.\tag{57}
$$

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

$$
\Delta E\_{\text{orbit}} = \frac{G M\_{\text{Sun}} M\_{\text{MBH}}}{1 \text{ AU}} = \left( 8.9 \cdot 10^{26} \text{ J} \right) \text{M}\_{18}.\tag{58}
$$

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:

$$N = \frac{\Delta E\_{\text{orbit}}}{\Delta E\_{\text{pass}}} \approx 4.5 \cdot 10^7 \left(M\_{18}\right)^{-1}. \tag{59}$$

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:

$$
\dot{M} = \frac{4\pi (\mathcal{MG})^2 \rho\_r}{v\_{sr\_\odot}},
\tag{60}
$$

*Ramjet Acceleration of Microscopic Black Holes within Stellar Material DOI: http://dx.doi.org/10.5772/intechopen.102556*

where *ρ<sup>r</sup>* is the density at Bondi radius and *vsr* is the sound speed and Bondi radius. Sound velocity within the gas is proportional to *T*0*:*<sup>5</sup> . Gas density is proportional to *T*�<sup>1</sup> . Hence,

$$
\dot{M} = \frac{4\pi (MG)^2 \rho}{v\_{\text{3}}} \left(\frac{T}{T\_r}\right)^{2.5},\tag{61}
$$

where *ρ*, *vs*, and *T* are the density, sound speed, and temperature of stellar material, while *Tr* is the temperature at Bondi radius. In the Solar center, the density is <sup>1</sup>*:*<sup>5</sup> � <sup>10</sup><sup>5</sup> kg m3 and the sound speed is 5*:*<sup>1</sup> � <sup>10</sup><sup>5</sup> <sup>m</sup> <sup>s</sup> ([11], p. 378). Substituting the above into Eq. (61), we obtain the accretion rate

$$\dot{M} = 6.3 \times 10^4 \text{ s} \frac{\text{kg}}{\text{s}} \left( \frac{T}{T\_r} \right)^{2.5} M\_{18}^2 = \frac{2 \times 10^{18} \text{ kg}}{\text{Million years}} \left( \frac{T}{T\_r} \right)^{2.5} M\_{18}^2. \tag{62}$$

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

$$\frac{d}{dt}(\ln M\_{18}) = \frac{\dot{M}\_{18}}{M\_{18}} = \frac{\dot{M}}{M} = \frac{2M\_{18}}{\text{Million years}} \quad \left(\frac{T}{T\_r}\right)^{2.5}.\tag{63}$$

In Eq. (63) above, *T* and *Tr* are 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 *M*<sup>18</sup> >100, all emitted radiation is absorbed by the accreting gas and *T* ≈*Tr*. Then the star is consumed by MBH over several millennia.

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 *γ*-ray pulses [19, 20].

### **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

$$\mathcal{N} = \frac{\eta\_A \eta\_\Gamma \eta\_{G\_2}}{T\_6} \gtrsim \begin{cases} 4 \cdot 10^{-4} \left( 1 + \mathcal{M}^{-2} \right)^{3/2} & \text{for supersonic MBH} \\ 9 \cdot 10^{-7} \mathcal{M}^3 \mathfrak{F}(\mathcal{M}, \eta\_A) & \text{for subsonic MBH} \end{cases} \,, \tag{64}$$

where Fð Þ M, *η<sup>A</sup>* is given in Eq. (34). *T*<sup>6</sup> is the temperature of the stellar material in millions Kelvin. The **gas redistribution efficiency** *ηG*, **radiative efficiency** *η*Γ, and **accretion efficiency** *η<sup>A</sup>* are 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 [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 *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 efficiencies** *η*<sup>Γ</sup> of accreting MBH can not be determined at this point. Most advanced theories give results, which vary by several orders of magnitude. Values ranging from 10<sup>10</sup> to 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 *η<sup>G</sup>* for subsonic MBH. The exact calculation of *η<sup>G</sup>* is a remaining problem. It would involve extensive theoretical work and simulations using gas dynamics and radiation-matter interaction.

Accretion efficiency *η<sup>A</sup>* for both subsonic and supersonic MBH is given by Eq. (45) in terms of *Tr*—the temperature at the Bondi radius. The calculation of *Tr* is a remaining problem. Exact calculation of *Tr*, and *η<sup>A</sup>* would 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.
