**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 *Ft* is the tidal or gravitational drag. For a supersonic MBH, *Ft* is given by ([6], p. 8)

$$F\_t = -\frac{\mathbf{1}}{\nu\_0} P\_t = -\frac{4\pi (MG)^2 \rho}{\nu\_{0\_2}} \ln\left(\frac{r\_{\text{max}}}{r\_{\text{min}}}\right),\tag{3}$$

where *Pt* is the decelerating power produced by the drag force, *ρ* is the density of the surrounding medium, *r*max is the approximate distance from the MBH to the farthest location where the stellar material is consistent, and *r*min is the radius at which matter is initially unperturbed by the MBH radiation. In Eq. (3), *M* is the mass of the MBH. We take *<sup>r</sup>*max to be about 5 � <sup>10</sup><sup>7</sup> *<sup>m</sup>* for a Sun-like star. We take *<sup>r</sup>*min to be about 0.1 *m*. Hence,

$$
\ln \left( \frac{r\_{\text{max}}}{r\_{\text{min}}} \right) \approx 20. \tag{4}
$$

For an MBH traveling at Mach number M≲0*:*8, the gravitational drag can be given by the following formula (see [7], p. 5, [8], p. 69, [9], p. 8)

$$F\_t = -\frac{4\pi (MG)^2 \rho}{v\_{0\_2}} \left[ \frac{1}{2} \ln \left( \frac{1+\mathcal{M}}{1-\mathcal{M}} \right) - \mathcal{M} \right]. \tag{5}$$

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

$$F\_l = -\frac{4\pi (MG)^2 \rho}{v\_{0\_2}} \sum\_{n=1}^{\infty} \frac{\mathcal{M}^{2n+1}}{2n+1}.\tag{6}$$

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:

$$\frac{\partial p}{\partial t} = \frac{\partial}{\partial t}(Mv\_0) = \dot{M}v\_0 + M\dot{v}\_0 = \mathbf{0} \qquad \Rightarrow \qquad \dot{v}\_0 = -v\_0 \frac{\dot{M}}{M}.\tag{7}$$

Using MBH speed change, we calculate the effective force as

$$F\_m = M \frac{\partial v\_0}{\partial t} = -\dot{M}v\_0. \tag{8}$$

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, *r*<sup>1</sup> and *r*2. The radius *r*<sup>1</sup> is defined in terms of *Fr*. A sphere of gas directly behind the MBH having radius *r*1, and density *ρ=*2 would 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=*2 � *ρ* ¼ �*ρ=*2. The accelerating "ramjet" force *Fr* acting on the MBH expressed in terms of *r*<sup>1</sup> is:

$$F\_r = -\frac{MM\_\prime G}{r\_{1\_2}} = MG\frac{\frac{1}{2}\frac{4}{3}\pi r\_{1\_3}\rho}{r\_{1\_2}} = \frac{2}{3}\pi MG\rho r\_{1\_2} \tag{9}$$

where *Ms* is the effective negative mass of the sphere of rarefied gas.

The radius *r*<sup>2</sup> is defined in terms of the power *P* radiated by MBH passing through the stellar material. Imagine that power *P* is used to uniformly heat a cylinder of stellar material along the path of MBH. The MBH moves at speed *v*0. The radius *r*<sup>2</sup> is defined as the radius of the aforementioned cylinder for which the temperature of gas contained in it would double. Then the relation between *P* and *r*<sup>2</sup> is:

$$P = \text{(Mass heated per unit of time)} \cdot T \cdot \text{C}\_v = v\_0 \left(\pi r\_{2\_2} \rho \right) T \cdot \text{C}\_v = \pi v\_0 \rho T C\_v r\_{2\_2},\tag{10}$$

where *Cv* is the heat capacity of gas of stellar material at constant volume. The **gas redistribution efficiency** is defined as

$$
\eta\_G = \frac{r\_1}{r\_2}.\tag{11}
$$

As we show later in this section, the ramjet force *Fr* acting on MBH is proportional to *ηG*. The minimal value for *η<sup>G</sup>* for subsonic MBH is estimated in Appendix A. The radiative power of the MBH passing through stellar material is

$$P = \eta\_I c^2 \dot{M},\tag{12}$$

where *<sup>η</sup>*<sup>Γ</sup> is the **radiative efficiency** of MBH and *<sup>M</sup>*\_ is the mass accretion rate. For a supersonic MBH, the Bondi-Hoyle-Lyttleton accretion rate is ([10], p. 203)

$$
\dot{M}\_{\rm BH} = 4\pi r\_{b\_2} \rho \sqrt{v\_{\mathcal{O}\_2} + v\_{\mathcal{t}\_2}} \ , \tag{13}
$$

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

$$r\_b = \frac{MG}{v\_{t\_2} + v\_0}.\tag{14}$$

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

$$\dot{M}\_{BH} = 4\pi r\_b^2 \rho v\_0 = \frac{4\pi (\mathcal{MG})^2 \rho}{\left(v\_0^2 + v\_s^2\right)^{3/2}}.\tag{15}$$

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

$$
\dot{M} \approx \frac{4\pi (\mathbf{MG})^2 \rho\_r}{\left(v\_0^2 + v\_{sr}^2\right)^{3/2}},\tag{16}
$$

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

where *vsr* is the sound speed at the accretion radius and *ρ<sup>r</sup>* is the density at the accretion radius. Recall the **accretion efficiency** *η<sup>A</sup>* is the quotient of actual and zeroradiation mass capture rates:

$$\eta\_A = \frac{\dot{M}}{\dot{M}\_{BH}} \approx \left(\frac{\upsilon\_{0\_2} + \upsilon\_{s\_2}}{\upsilon\_{0\_2} + \upsilon\_{sr\_2}}\right)^{3/2} \frac{\rho\_r}{\rho}. \tag{17}$$

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

$$
\pi v\_0 \rho T C\_v r\_{2\_1} = \eta\_{\Gamma} c^2 \dot{M}.\tag{18}
$$

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

$$4\pi v\_0 \rho T C\_v r\_{\mathfrak{Z}\_2} = \eta\_\Gamma c^2 \eta\_A \frac{4\pi (\mathcal{MG})^2 \rho}{\left(v\_{\mathfrak{O}\_2} + v\_{\mathfrak{s}\_2}\right)^{3/2}}.\tag{19}$$

Thus,

$$r\_2 = 2\sqrt{\eta\_A} \sqrt{\frac{\eta\_\Gamma c^2}{T\mathcal{C}\_v}} \frac{\mathcal{M}\mathcal{G}}{\left(v\_{0\_2} + v\_{s\_2}\right)^{3/4} \sqrt{v\_0}} = 2\sqrt{\frac{\eta\_A \eta\_\Gamma c^2}{T\mathcal{C}\_v}} \frac{\mathcal{M}\mathcal{G}}{v\_{0\_2}} \left(1 + \frac{v\_{s\_2}}{v\_{0\_2}}\right)^{-3/4}.\tag{20}$$

Substituting Eq. (20) into Eq. (11), we obtain an expression for *r*1:

$$r\_1 = 2\eta\_G \sqrt{\frac{\eta\_A \eta\_\Gamma c^2}{T C\_v}} \frac{\mathcal{M}G}{v\_{0\_2}} \left(1 + \frac{v\_{s\_2}}{v\_{0\_2}}\right)^{-3/4}.\tag{21}$$

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

$$F\_r = \frac{2}{3}\pi \mathcal{M} G \rho r\_1 = \left[\frac{4}{3}\eta\_G \sqrt{\frac{\eta\_A \eta\_\Gamma c^2}{T C\_v}} \left(1 + \frac{v\_{s\_2}}{v\_{0\_2}}\right)^{-3/4}\right] \frac{\pi (\mathcal{M} G)^2 \rho}{v\_{0\_2}}.\tag{22}$$

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

$$\begin{split} F\_m &= -\dot{M}v\_0 = -\eta\_A \dot{M}\_{BH} v\_0 = -\eta\_A \quad \frac{4\pi (MG)^2 \rho}{\left(v\_{0\_2} + v\_{s\_2}\right)^{3/2}} \quad v\_0 \\ &= -4\eta\_A \left(1 + \frac{v\_{s\_2}}{v\_{0\_2}}\right)^{-3/2} \frac{\pi (MG)^2 \rho}{v\_{0\_2}}. \end{split} \tag{23}$$

#### **2.2 Conditions for supersonic MBH acceleration**

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

$$\begin{split} F &= F\_{\text{I}} + F\_{m} + F\_{r} \\ &= \frac{\pi (MG)^{2} \rho}{v\_{0\_{2}}} \Bigg[ -4 \ln \left( \frac{r\_{\text{max}}}{r\_{\text{min}}} \right) - 4 \eta\_{\text{A}} \Bigg( 1 + \frac{v\_{\text{s}\_{2}}}{v\_{0\_{2}}} \Bigg)^{-3/2} + 2 \eta\_{\text{G}} \sqrt{\frac{\eta\_{\text{A}} \eta\_{\text{I}} c^{2}}{T \text{C}\_{\text{v}}}} \Bigg( 1 + \frac{v\_{\text{s}\_{2}}}{v\_{0\_{2}}} \Bigg)^{-3/4} \Bigg]. \end{split} \tag{24}$$

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

$$\eta\_G \sqrt{\frac{\eta\_A \eta\_I c^2}{T C\_v}} \left( \mathbf{1} + \frac{v\_{s\_2}}{v\_{0\_2}} \right)^{-3/4} > 2 \ln \left( \frac{r\_{\text{max}}}{r\_{\text{min}}} \right) + 2 \eta\_A \left( \mathbf{1} + \frac{v\_{s\_2}}{v\_{0\_2}} \right)^{-3/2} . \tag{25}$$

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

$$
\eta\_A \eta\_{\Gamma} \eta\_{G\_2} > \frac{4T \mathbf{C}\_v}{c^2} \left[ \ln \left( \frac{r\_{\text{max}}}{r\_{\text{min}}} \right) + \eta\_A \left( 1 + \frac{v\_{s\_2}}{v\_{0\_2}} \right)^{-3/2} \right]^2 \left( 1 + \frac{v\_{s\_2}}{v\_{0\_2}} \right)^{3/2} \tag{26}
$$

Recalling Eq. (4), and the fact that *η<sup>A</sup>* < 1, we rewrite the estimate to Eq. (26) as

$$
\eta\_A \eta\_\Gamma \eta\_{G\_2} \gtrsim \mathbf{1}.7 \cdot \mathbf{10}^3 \cdot \frac{T \mathbf{C}\_v}{c^2} \left( \mathbf{1} + \frac{\nu\_{s\_2}}{\nu\_{0\_2}} \right)^{3/2} \\
= \mathbf{1}.7 \cdot \mathbf{10}^3 \cdot \frac{T \mathbf{C}\_v}{c^2} \left( \mathbf{1} + \mathcal{M}^{-2} \right)^{3/2}. \tag{27}
$$

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

$$C\_v = \frac{\Im R}{2m\_a},\tag{28}$$

where *ma* is the average molar mass of the gas, and *R* is the gas constant *<sup>R</sup>* <sup>¼</sup> <sup>8</sup>*:*<sup>314</sup> *<sup>J</sup>* mol*<sup>o</sup> K* � �*:* Typical stellar material consists of monatomic gas with an average particle mass of 0*:*62 amu ([11], p. 378). Hence, the heat capacity at constant volume for stellar material is *Cv* <sup>¼</sup> <sup>2</sup>*:*<sup>01</sup> � <sup>10</sup><sup>4</sup> *<sup>J</sup>* kg*<sup>o</sup> <sup>K</sup>*. Thus, Eq. (27) can be rewritten as

$$\mathcal{N} = \frac{\eta\_A \eta\_\Gamma \eta\_{G\_2}}{T\_6} \gtrsim 4 \cdot 10^{-4} \left(1 + \mathcal{M}^{-2}\right)^{3/2}. \tag{29}$$

As we show in Subsection 3.2, *η<sup>A</sup>* is very small if the temperature of the gas at Bondi radius is high. As we discuss in Subsection 3.3, different calculations of *η*<sup>Γ</sup> in previous works yield different results, yet all of them are below 0*:*1. For supersonic MBH, *η<sup>G</sup>* should 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, *η<sup>G</sup>* decreases as well. Calculation of *η<sup>G</sup>* is beyond the scope of this work. The solar gas temperature exceeds *T*<sup>6</sup> ¼ 4 for 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 M≲0*:*8 is given by Eq. (6). The total force acting on MBH is obtained by summing Eqs. (6), (22), and (23):

$$\begin{split} F &= F\_{t} + F\_{m} + F\_{r} \\ &= \frac{\pi (MG)^{2} \rho}{v\_{0\_{2}}} \Bigg[ -4 \sum\_{n=1}^{\infty} \frac{\mathcal{M}^{2n+1}}{2n+1} - 4\eta\_{A} \Bigg( 1 + \frac{v\_{s\_{2}}}{v\_{0\_{2}}} \Bigg)^{-3/2} + 2\eta\_{G} \sqrt{\frac{\eta\_{A} \eta\_{\Gamma} c^{2}}{TC\_{v}}} \Bigg( 1 + \frac{v\_{s\_{2}}}{v\_{0\_{2}}} \Bigg)^{-3/4} \Bigg] \\ &= \frac{\pi (MG)^{2} \rho}{v\_{0\_{2}}} \Bigg[ -4 \sum\_{n=1}^{\infty} \frac{\mathcal{M}^{2n+1}}{2n+1} - 4\eta\_{A} \bigg( 1 + \frac{1}{\mathcal{M}^{2}} \Bigg)^{-3/2} + 2\eta\_{G} \sqrt{\frac{\eta\_{A} \eta\_{\Gamma} c^{2}}{TC\_{v}}} \Bigg( 1 + \frac{1}{\mathcal{M}^{2}} \Bigg)^{-3/4} \Bigg] .\end{split} \tag{30}$$

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

$$\eta\_G \sqrt{\frac{\eta\_A \eta\_\Gamma c^2}{T C\_v}} \left( 1 + \mathcal{M}^{-2} \right)^{-3/4} > 2 \sum\_{n=1}^\infty \frac{\mathcal{M}^{2n+1}}{2n+1} + 2 \eta\_A \left( 1 + \mathcal{M}^{-2} \right)^{-3/2}. \tag{31}$$

Rewrite Eq. (31) as:

$$\begin{split} \eta\_{A}\eta\_{\Gamma}\eta\_{G2} &> \frac{4TC\_{v}}{c^{2}} \left( \mathbf{1} + \mathcal{M}^{-2} \right)^{3/2} \left[ \sum\_{n=1}^{\infty} \frac{\mathcal{M}^{2n+1}}{2n+1} + \eta\_{A} \left( \mathbf{1} + \mathcal{M}^{-2} \right)^{-3/2} \right]^{2} \\ &= \frac{4TC\_{v}}{c^{2}} \mathcal{M}^{3} \left\{ \left( \mathbf{1} + \mathcal{M}^{2} \right)^{3/2} \left[ \sum\_{n=0}^{\infty} \frac{\mathcal{M}^{2n}}{2n+3} + \eta\_{A} \left( \mathbf{1} + \mathcal{M}^{2} \right)^{-3/2} \right]^{2} \right\}. \end{split} \tag{32}$$

Given that *Cv* <sup>¼</sup> <sup>2</sup>*:*<sup>01</sup> � <sup>10</sup><sup>4</sup> *<sup>J</sup> kg<sup>o</sup> <sup>K</sup>*, we rewrite Eq. (32) as

$$\frac{\eta\_A \eta\_\Gamma \eta\_{G\_2}}{T\_6} \gtrsim 9 \cdot 10^{-7} \mathcal{M}^3 \mathfrak{F}(\mathcal{M}, \eta\_A),\tag{33}$$

where

$$\mathfrak{F}(\mathcal{M}, \eta\_A) = \left(\mathbf{1} + \mathcal{M}^2\right)^{3/2} \left[\sum\_{n=0}^{\infty} \frac{\mathcal{M}^{2n}}{2n + 3} + \eta\_A \left(\mathbf{1} + \mathcal{M}^2\right)^{-3/2}\right]^2. \tag{34}$$

Notice that Fð Þ M, *η<sup>A</sup>* >*:*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):

*Magnetosphere and Solar Winds, Humans and Communication*

$$\mathcal{N} = \frac{\eta\_A \eta\_\Gamma \eta\_{G\_2}}{T\_6} = \mathfrak{P} \cdot \mathbf{10}^{-\sf T} \mathcal{M}^3 \mathfrak{F}(\mathcal{M}, \eta\_A). \tag{35}$$

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 M<sup>0</sup> and 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 M<sup>0</sup> from Eq. (35), one must know gas redistribution, accretion and radiative efficiencies. In the next section, we present preliminary estimates for the three aforementioned efficiencies.

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