**3.2 The value of** *η<sup>A</sup>*

From Eq. (17), we estimate *η<sup>A</sup>* as

$$
\eta\_A \approx \left(\frac{v\_{0\_2} + v\_{s\_2}}{v\_{0\_2} + v\_{sr\_2}}\right)^{3/2} \frac{\rho\_r}{\rho},\tag{37}
$$

where *vsr* is the sound speed at the accretion radius and *ρ<sup>r</sup>* is the density at the accretion radius. Given that gas density is proportional to its pressure divided by temperature, we obtain

$$\begin{split} \eta\_{A} & \approx \left(\frac{\boldsymbol{v}\_{\text{O}\_{1}} + \boldsymbol{v}\_{\text{s}\_{1}}}{\boldsymbol{v}\_{\text{O}\_{1}} + \boldsymbol{v}\_{\text{r}\_{1}}}\right)^{3/2} \frac{\boldsymbol{\rho}\_{r}}{\boldsymbol{\rho}} = \left(\frac{\boldsymbol{v}\_{\text{O}\_{1}} + \boldsymbol{v}\_{\text{s}\_{1}}}{\boldsymbol{v}\_{\text{O}\_{1}} + \boldsymbol{v}\_{\text{r}\_{2}}}\right)^{3/2} \frac{\mathbf{P}\_{r}}{\mathbf{P}} \frac{T}{T\_{r}} = \left(\frac{(\boldsymbol{v}\_{\text{O}}/\boldsymbol{v}\_{\text{s}})^{2} + \mathbf{1}}{\left(\boldsymbol{v}\_{\text{O}}/\boldsymbol{v}\_{\text{s}}\right)^{2} + \left(\boldsymbol{v}\_{\text{r}}/\boldsymbol{v}\_{\text{s}}\right)^{2}}\right)^{3/2} \frac{\mathbf{P}\_{r}}{\mathbf{P}} \frac{T}{T\_{r}} \\ &= \left(\frac{\boldsymbol{\mathcal{M}}^{2} + \mathbf{1}}{\left(\boldsymbol{\mathcal{M}}^{2} + \left(\boldsymbol{v}\_{\text{s}}/\boldsymbol{v}\_{\text{s}}\right)^{2}\right)}\right)^{3/2} \frac{\mathbf{P}\_{r}}{\mathbf{P}} \frac{T}{T\_{r}} .\end{split} \tag{38}$$

In Eq. (38), *T* is the ambient temperature of stellar material, and *Tr* is the temperature of stellar material at Bondi radius. The ambient pressure is **P**, and pressure at Bondi radius is **P***r*. Notation *P* can 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,

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

$$\left(\frac{v\_{sr}}{v\_s}\right)^2 = \frac{T\_r}{T}.\tag{39}$$

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

$$
\eta\_A \approx \left(\frac{\mathcal{M}^2 + 1}{\mathcal{M}^2 + T\_r/T}\right)^{3/2} \frac{\mathbf{P}\_r}{\mathbf{P}} \frac{T}{T\_r} \,. \tag{40}
$$

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

$$\mathbf{P}\_r = \mathbf{P} + \frac{\rho v\_{0\_2}}{2} = \mathbf{P} + \rho v\_{s\_2} \, \, \frac{\mathcal{M}^2}{2} \,. \tag{41}$$

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

$$\frac{\mathbf{P}\_r}{\mathbf{P}\_s} \frac{T}{T\_r} < 1,\tag{42}$$

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

$$\mathbf{P} = \frac{\rho v\_{s\_2}}{\gamma} = \frac{\mathbf{3}}{\mathbf{5}} \quad \rho v\_{s\_2}. \tag{43}$$

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

$$\frac{\mathbf{P}\_r}{\mathbf{P}} = \frac{\frac{3}{5} \quad \rho v\_{s\_2} + \rho v\_{s\_2} \ \frac{\mathcal{M}^2}{2}}{\frac{3}{5} \quad \rho v\_{s\_2}} = \mathbf{1} + \frac{5}{6} \mathcal{M}^2. \tag{44}$$

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

$$\eta\_A \approx \left(\frac{\mathcal{M}^2 + 1}{\mathcal{M}^2 + T\_r/T}\right)^{3/2} \min\left\{1, \left(1 + \frac{5}{6} \,\,\mathcal{M}^2\right)\frac{T}{T\_r}\right\}.\tag{45}$$

As we see, *η<sup>A</sup>* is 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=T* ≫M<sup>2</sup> , Eq. (45) can be approximated as

$$
\eta\_A \approx \left( 1 + \frac{5}{6} \right. \left. \mathcal{M}^2 \right) \left( \mathcal{M}^2 + 1 \right)^{3/2} \left( \frac{T}{T\_r} \right)^{5/2} . \tag{46}
$$

Calculation of *Tr* remains an unsolved problem.
