**A.1 Minimum value of** *η<sup>G</sup>* **for subsonic MBH**

We assume strictly subsonic regime with M≤0*:*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 R*hg* .

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

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 *Fr* exerted on MBH comes from the difference in density of ambient stellar material and hot rarefied gas within the tail region R*hg* as 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 R*hg* is calculated below. The region R*hg* can be approximated by a cylinder with radius *rh*. This cylinder starts at the distance at most *rh* from the MBH. By taking the distance to be *rh*, we are estimating the minimal value of the force. The region R*hg* is represented in cylindrical coordinates with MBH at the origin. The direction in which the MBH is traveling is �^*z*. In cylindrical coordinates, R*hg* is given by

$$\begin{cases} z \in (r\_h, \infty) \\ r \in [0, r\_h) \end{cases} \tag{A.1}$$

Stellar gas temperature in R*hg* is approximated by a uniform temperature *KT*, where *K* > 1 is a constant and *T* is the ambient temperature. Gas pressure within R*hg* is almost the same as ambient gas pressure. From temperature and pressure in R*hg* , it follows that the gas density in that region is *ρ=K*, where *ρ* is the ambient density. From the standpoint of gravitational interaction, **effective negative density** *ρ*� of the material in R*hg* can be defined as the difference between gas density in R*hg* and ambient gas density. Effective negative density is

$$
\rho\_- = \frac{\rho}{K} - \rho = -\rho \,\,\,\frac{K-1}{K} \,\,. \tag{A.2}
$$

The rarefied gas region R*hg* exerts the following force on MBH:

$$\mathbf{F} = \iint\_{\mathcal{R}\_{\mathbf{g}}} (\rho\_{-}) \mathbf{g}(\mathbf{r}) dV = -\rho \xrightarrow[K]{K-1} \iint\_{\mathcal{R}\_{\mathbf{g}}} \mathbf{g}(\mathbf{r}) dV. \tag{A.3}$$

**Figure 2.** *Heat wave caused by subsonic MBH.*

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

$$\mathbf{g}(\mathbf{r}) = -MG \frac{\mathbf{r}}{\left\|\mathbf{r}\right\|^3} \tag{A.4}$$

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

$$\mathbf{F} = -M\mathbf{G}\rho \parallel \frac{K-1}{K} \iiint \frac{\mathbf{r}}{\left\|\mathbf{r}\right\|^3} dV,\tag{A.5}$$

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

$$F\_r = -\mathbf{F} \cdot \hat{\mathbf{z}} = M\mathbf{G}\rho \quad \frac{K-1}{K} \iint\limits\_{\mathcal{R}\_{\text{th}}} \frac{\mathbf{r} \cdot \hat{\mathbf{z}}}{\left\lVert\mathbf{r}\right\rVert^3} dV = M\mathbf{G}\rho \quad \frac{K-1}{K} \iint\limits\_{\mathcal{R}\_{\text{th}}} \frac{\mathbf{z}}{\left\lVert\mathbf{r}\right\rVert^3} dV$$

$$\geq M\mathbf{G}\rho \quad \frac{K-1}{K} \int\_0^n \int\_{r\_1}^\infty \frac{\pi rz}{\left(\mathbf{z}^2 + r^2\right)^{3/2}} dz dr = \pi M\mathbf{G}\rho \quad \frac{K-1}{K} \int\_0^n \left[-\frac{r}{\sqrt{\mathbf{z}^2 + r^2}}\right]\_{z=r\_h}^{z=\infty} dr$$

$$= \pi M\mathbf{G}\rho \quad \frac{K-1}{K} \int\_0^n \frac{r}{\left\lvert r\_{h\_2} + r^2 \right\rvert} dr = \pi M\mathbf{G}\rho \quad \frac{K-1}{K} \left[\sqrt{n\_2 + r^2}\right]\_{r=0}^n$$

$$= \pi M\mathbf{G}\rho \quad r\_h \left(\frac{K-1}{K} \left(\sqrt{2} - 1\right)\right). \tag{A.6}$$

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

$$r\_1 \ge \frac{3}{2} \left(\sqrt{2} - 1\right) \xrightarrow[K]{K - 1} r\_h \approx 0.62 \xrightarrow[K]{K - 1} r\_h. \tag{A.7}$$

Below, *rh* is estimated in terms of *r*2. The power needed to heat the gas trail is

$$\begin{split} P\_T &= \text{(Mass heated per unit of time)} \cdot (\text{Temperature}) \cdot \text{C}\_p \\ &= v\_0 \left( \pi r\_{h\_2} \frac{\rho}{K} \right) ((K-1)T) \left( \frac{\text{5}}{\text{3}} \text{C}\_v \right) = \frac{\text{5} (K-1)}{\text{3K}} \pi v\_0 \rho T \text{C}\_v r\_{h\_2}, \end{split} \tag{A.8}$$

where *PT* is the thermal power. For monatomic gas, *Cp* <sup>¼</sup> <sup>5</sup> <sup>3</sup>*Cv*. Some of the power *P* radiated by the MBH goes into the production of the sonic waves, hence *P*> *PT*. Accurate calculation of *PT=P* is 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=P*≳0*:*8. Using this data, we estimate the total radiative power of MBH:

$$P \le \frac{2(K-1)}{K} \pi v\_0 \rho T C\_v r\_{h\_2} \tag{A.9}$$

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

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

$$\frac{2(K-1)}{K} \pi \upsilon\_0 \rho T C\_{\upsilon} r\_{h\_2} \ge \pi \upsilon\_0 \rho T C\_{\upsilon} r\_{2\_1}.\tag{A.10}$$

Hence,

$$r\_2 \le r\_h \sqrt{\frac{2(K-1)}{K}}.\tag{A.11}$$

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

$$
\eta\_G = \frac{r\_1}{r\_2} \ge .44 \quad \sqrt{\frac{K-1}{K}}.\tag{A.12}
$$

Calculation of *K* is beyond the scope of this work. Some considerations regarding the value of *K* are 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 *T* is 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 radius** *r<sup>γ</sup>* is the average distance traveled by a photon or another energy-carrying particle from PBH before being absorbed by stellar material. **Minimal hot tail radius** *rmh* is the minimal radius the hot tail can have regardless of *K*.

Below we estimate *r<sup>γ</sup>* and *rmh*. The radiation radius is

$$
\sigma\_{\gamma} = \frac{\mathfrak{S}\_{\gamma}}{\rho\_p} = \frac{\mathfrak{S}\_{\gamma}}{\mathbf{1}\mathbf{0}^3 \underset{m^3}{\text{kg}} \ \rho\_{3p}},\tag{A.13}
$$

where S*<sup>γ</sup>* is the planar density of material through which an energy carrying particle has to travel before being absorbed by stellar material. The density *ρ<sup>p</sup>* is an average density of the material over the path of the energy-carrying particle, and *ρ*3*<sup>p</sup>* is the same density in 103kg*=*m3. The value of S*<sup>γ</sup>* is inversely proportional to average absorption cross-section of the energy-carrying particles:

$$\mathfrak{S}\_1 = \frac{1\text{ kg}}{1000\text{ N}\_A\text{ amu}} \cdot \frac{1}{10^{-28}\text{ }\sigma} = \frac{17\stackrel{\text{kg}}{\text{m}^2}}{\sigma\text{ in barn}},\tag{A.14}$$

where *σ* is 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

$$r\_{\gamma} = \frac{\mathfrak{S}\_{\gamma}}{\rho\_p} = \frac{0.017 \text{ m}}{\rho\_{3p} (\sigma \text{ inbarn})}. \tag{A.15}$$

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):

$$r\_{mh} = \sqrt{\frac{3P\_T}{5\pi\nu\_0\rho T C\_v}},\tag{A.16}$$

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

$$P\_T = \eta\_h P = \eta\_h \eta\_\Gamma \eta\_A \frac{4\pi (\mathbf{MG})^2 c^2 \rho}{\left(\upsilon\_{0\_2} + \upsilon\_{s\_2}\right)^{3/2}} = \eta\_h \eta\_\Gamma \eta\_A \frac{4\pi (\mathbf{MG})^2 c^2 \rho}{\upsilon\_{0\_3} \left(1 + \mathcal{M}^{-2}\right)^{3/2}},\tag{A.17}$$

where *η<sup>h</sup>* ≈0*:*8 is 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

$$r\_{mh} \approx \mathbf{1.5} \ \sqrt{\eta\_{\Gamma} \eta\_{A}} \frac{\text{MGc} \left(\mathbf{1} + \mathcal{M}^{-2}\right)^{-3/4}}{v\_{0\_{2}} \sqrt{T C\_{v}}} \tag{A.18}$$

As mentioned in Subsection 2.2, for average stellar material, *Cv* <sup>¼</sup> <sup>2</sup>*:*<sup>01</sup> � <sup>10</sup><sup>4</sup> *<sup>J</sup> kg<sup>o</sup> K*. From Eq. (A.18), we obtain

$$r\_{mh} \approx (0.21 \text{ m}) \ \sqrt{\eta\_{\Gamma} \eta\_{A}} \ \left(1 + \mathcal{M}^{-2}\right)^{-3/4} \text{M}\_{18} \ v\_{\theta\_{-2}} \ T\_{\theta\_{-1/2}}.\tag{A.19}$$

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

$$\mathcal{R}\_{\boldsymbol{\gamma}} = \frac{r\_{\boldsymbol{\gamma}}}{r\_{m\boldsymbol{h}}} \approx 0.8 \frac{v\_{\boldsymbol{\delta}\_{2}} \sqrt{T\_{6}}}{\rho\_{3\boldsymbol{p}} M\_{18}(\boldsymbol{\sigma} \text{ in barn}) \sqrt{\eta\_{\boldsymbol{\Gamma}} \eta\_{\boldsymbol{A}}}} \approx \frac{\sqrt{v\_{6} \cdot T\_{6}}}{\rho\_{3\boldsymbol{p}} M\_{18}(\boldsymbol{\sigma} \text{ in barn}) \sqrt{\eta\_{\boldsymbol{\Gamma}} \eta\_{\boldsymbol{A}}}},\quad \text{(A.20)}$$

where *v*6*<sup>s</sup>* is the sound velocity in 10<sup>6</sup> m*=*s.

If R*<sup>γ</sup>* ≪ 1, 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, *K* ≫ 1. For R*<sup>γ</sup>* ≫ 1, thermal energy is dissipated over a very large gas volume. This gas volume is heated only by a small margin, thus 0 <*K* � 1 ≪ 1.

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

Using *<sup>r</sup>*min <sup>¼</sup> <sup>0</sup>*:*<sup>1</sup> *<sup>m</sup>* and *<sup>r</sup>*max <sup>¼</sup> <sup>5</sup> � <sup>10</sup><sup>7</sup> *<sup>m</sup>* to express Eq. (3) in numerical terms we obtain:

$$F\_t = \frac{4\pi (MG)^2 \rho}{\nu\_{\text{O}\_2}} \ln\left(\frac{r\_{\text{max}}}{r\_{\text{min}}}\right) = \left(1.12 \cdot 10^9 \text{ N}\right) \frac{M\_{18\_2} \ \rho\_3}{\nu\_{\text{O}\_2}},\tag{B.1}$$

where *ρ*<sup>3</sup> is density in 10<sup>3</sup> kg/m3 , and *v*<sup>6</sup> is velocity in 10<sup>6</sup> 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<sup>3</sup> kg/m<sup>3</sup> . Column 3 contains an estimated

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


#### **Table 1.**

*Parameters for MBH passing through a sun-like star.*

speed of a MBH arriving from a distance of thousands of solar radii. Column 4 contains *Ft* for *M*<sup>18</sup> ¼ 1 (**Table 1**).

The Solar radius is *<sup>R</sup>*<sup>⊙</sup> <sup>¼</sup> <sup>6</sup>*:*<sup>96</sup> � <sup>10</sup><sup>8</sup> *<sup>m</sup>*. Thus, we estimate the energy loss of a MBH passing through the center of a Sun-like star:

$$
\Delta E = \int\_{-R\_{\odot}}^{R\_{\odot}} F\_t \, d\mathbf{x} = \left(2.0 \cdot 10^{19} \,\, J\right) \mathbf{M}\_{18\_2}.\tag{\text{B.2}}
$$
