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Ramjet acceleration of microscopic black holes within stellar material

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Abstract

In this work we present a case that Microscopic Black Holes (MBH) of mass \(10^{16}\text{ kg}\) to \(3 \times 10^{19}\text{ kg}\) experience acceleration as they move within stellar material at low velocities. The accelerating forces are caused by the fact that a MBH moving through stellar material leaves a trail of hot rarified gas. The rarified gas behind a MBH exerts 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 \(\mathscr {M}<\mathscr {M} _{0}\). The equilibrium Mach number \(\mathscr {M}_{0}\) depends on MBH mass and stellar material characteristics. Our calculations open the possibility of MBH orbiting within stars including Sun at Mach number \(\mathscr {M}_{0}\). At the end of this work we list some unresolved problems which result from our calculations.

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Correspondence to Mikhail V. Shubov.

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Appendices

Appendix A: Estimation of bounds on \(\eta _{{G}}\)

A.1 Minimum value of \(\eta _{{G}}\) for MBH moving at subsonic speed

We assume strictly subsonic regime with \(\mathscr{M} \le 0.8\). In Fig. 2, we describe the MBH passing through stellar material.

Fig. 2
figure2

Heat wave caused by MBH moving at a subsonic speed

The heated stellar material produced by MBH moving at subsonic speed consists of two regions. The first is the parabolic head region of hot gas surrounding the MBH. The second is the hot gas trail. The second region is denoted \(\mathscr {R}_{hg}\).

An important issue is the location of stellar material mass displaced by the heat wave. For an MBH travelling at supersonic speed, the displaced mass does not have time to move very far. This greatly decreases the effect of rarefication and thus \(r_{1}\) and \(\eta _{{G}}\). In our case of MBH travelling at subsonic speed, all of the extra mass of stellar material pushed out of the cylindrical region of hot gas is carried away by sonic density waves. These waves are spherical shells. Each shell’s center is a point at which the wave originated. Due to MBH’s subsonic velocity, the MBH is located inside of all the aforementioned spherical shell sound waves. Thus, these waves exert no net gravitational pull on the MBH.

The force exerted on MBH comes from the fact that hot rarified gas both within the head region and within \(\mathscr {R}_{hg}\) exerts lower gravitational pull on MBH than the dense stellar material in front of MBH. Recall, that the total accelerative force is called \(F_{r}\). Even though the head region contributes a small part of \(F_{r}\), we ignore its contribution.

We calculate the force exerted by \(\mathscr {R}_{hg}\). The region \(\mathscr {R}_{hg}\) can be approximated by a cylinder with radius \(r_{h}\). The cylinder starts at the distance at most \(r_{h}\) from the MBH. By taking the distance to be \(r_{h}\), we are estimating minimal value of the force. The region \(\mathscr {R}_{hg}\) is represented in cylindrical coordinates with MBH at the origin. The direction in which the MBH is travelling is \(-\hat{z}\). In cylindrical coordinates, \(\mathscr {R}_{hg}\) is given by

$$ \left \{ \begin{aligned} z & \in ( r_{h}, \infty ) \\ r & \in [ 0, r_{h} ) \end{aligned} \right . $$
(A.1)

The temperature distribution of gas in \(\mathscr {R}_{hg}\) is approximated by a uniform temperature \(KT\), where \(T\) is the temperature of surrounding material, and \(K>1\) is a constant. The pressure of gas in \(\mathscr {R}_{hg}\) is approximately the same as that of surrounding stellar medium. The density of the gas in \(\mathscr {R}_{hg}\) is \(\rho /K\), where \(\rho \) is the density of surrounding stellar material. The effective negative density \(\rho _{-}\) of the material in \(\mathscr {R}_{hg}\) is the difference of density of material in \(\mathscr {R}_{hg}\) and the density of surrounding material:

$$ \rho _{-}=\frac{\rho }{K}-\rho =-\rho \frac{K-1}{K}. $$
(A.2)

The force acting on a MBH due to rarefied region \(\mathscr {R}_{hg}\) is

$$ \textbf{F} = \iiint _{\mathscr {R}_{hg}} (\rho _{-} ) \textbf{g}(\mathbf{r}) dV =-\rho \frac{K-1}{K} \iiint _{ \mathscr {R}_{hg}} \textbf{g}(\mathbf{r}) dV, $$
(A.3)

where

$$ \textbf{g}(\mathbf{r})=-M G \frac{\mathbf{r}}{\|\mathbf{r}\|^{3}} $$
(A.4)

is the acceleration due to MBH gravity at point \(\mathbf{r}\). Substituting (A.4) into (A.3), we obtain

$$ \textbf{F} =-M G \rho \frac{K-1}{K} \iiint _{\mathscr {R}_{hg}} \frac{\mathbf{r}}{\|\mathbf{r}\|^{3}} dV. $$
(A.5)

The gas displaced by the MBH passage in \(-\hat{z}\) direction retains cylindrical symmetry. This symmetry implies that the net force on the MBH will act only in \(-\hat{z}\) direction. Thus, (A.5) can be further simplified to

$$\begin{aligned} F_{r} &=-\textbf{F} \cdot \hat{z} =M G \rho \frac{K-1}{K} \iiint _{ \mathscr {R}_{hg}} \frac{\mathbf{r}\cdot \hat{z}}{\| \mathbf{r}\|^{3}} dV \\ & =M G \rho \frac{K-1}{K} \iiint _{\mathscr {R} _{hg}} \frac{z}{ (z^{2}+r^{2} )^{3/2}} dV \\ & \ge M G \rho \frac{K-1}{K} \int _{0}^{r_{h}} \int _{r_{h}}^{\infty } \frac{\pi r z}{ (z^{2}+r^{2} )^{3/2}} dz dr \\ & =\pi M G \rho \frac{K-1}{K} \int _{0}^{r_{h}} \biggl[ - \frac{r}{\sqrt{z^{2}+r^{2}}} \biggr]_{z=r_{h}}^{z=\infty } dr \\ &= \pi M G \rho \frac{K-1}{K} \int _{0}^{r_{h}} \frac{r}{\sqrt{r _{h}^{2}+r^{2}}} dr \\ & = \pi M G \rho \frac{K-1}{K} \bigl[ \sqrt{r _{h}^{2}+r^{2}} \bigr]_{r=0}^{r_{h}} \\ &=\pi M G \rho r_{h} \biggl( \frac{K-1}{K} ( \sqrt{2}-1 ) \biggr). \end{aligned}$$
(A.6)

As we have mentioned earlier, the real force is greater or equal to the one calculated by approximating \(\mathscr {R}_{hg}\) by (A.1). Recall (3.7):

$$ F_{r}=\frac{2}{3} \pi M G \rho r_{1}. $$
(A.7)

From (A.6) and (A.7), it follows that

$$ r_{1} \ge \frac{3}{2} ( \sqrt{2}-1 ) \frac{K-1}{K} r_{h} \approx 0.62 \frac{K-1}{K} r_{h}. $$
(A.8)

Below, \(r_{h}\) is estimated in terms of \(r_{2}\). The power needed to heat the gas trail is

$$\begin{aligned} P_{{T}} &= (\text{Mass heated per unit of time} ) \cdot ( \text{Temperature} ) \cdot C_{p} \\ &=v_{0} \biggl(\pi r_{h}^{2} \frac{\rho }{K} \biggr) \bigl((K-1)T \bigr) \biggl(\frac{5}{3} C_{v} \biggr) \\ & = \frac{5(K-1)}{3 K} \pi v_{0} \rho T C_{v} r_{h}^{2}, \end{aligned}$$
(A.9)

where \(P_{{T}}\) is the thermal power and \(C_{p}\) is the heat capacity of the gas at constant pressure. Recall that for monatomic gas, \(C_{p}=\frac{5}{3} C_{v}\). The total power radiated by the MBH must be greater, since some of it goes into the production of the sound waves. Even though determining the thermal efficiency of the sonic boom production by a MBH moving through the stellar material is beyond the scope of this work, it can be assumed that it does not exceed 20% for subsonic MBH. Hence, the total radiative power of MBH is

$$ P \le \frac{2(K-1)}{K} \pi v_{0} \rho T C_{v} r_{h}^{2} $$
(A.10)

Recall (3.8):

$$ P=\pi v_{0} \rho T C_{v} r_{2}^{2}. $$
(A.11)

Combining (A.10) and (A.11), we obtain

$$ \frac{2(K-1)}{K} \pi v_{0} \rho T C_{v} r_{h}^{2} \ge \pi v_{0} \rho T C_{v} r_{2}^{2}. $$
(A.12)

Hence,

$$ r_{2} \le r_{h}\sqrt{ \frac{2(K-1)}{K}}. $$
(A.13)

Dividing (A.8) by (A.13), we obtain an estimate for the thermal efficiency for an MBH moving at subsonic speed

$$ \eta _{{G}}=\frac{r_{1}}{r_{2}} \ge .44 \sqrt{ \frac{K-1}{K}}. $$
(A.14)

Calculation of \(K\) is beyond the scope of this work. Some considerations regarding the value of \(K\) are presented in Appendix A.3.

A.2 Maximum value of \(\eta _{{G}}\) for MBH moving at any speed

The absolute maximum for \(\eta _{{G}}\) is obtained if the stable region of hot gas denoted \(\mathscr {R}_{hg}\) starts right at MBH. Such configuration is impossible, but it is useful for calculating the absolute maximum. It is shown in Fig. 3.

Fig. 3
figure3

Gas configuration for maximal \(\eta _{{G}}\)

Recalling (A.6), we evaluate the force

$$\begin{aligned} F_{r} &=-\textbf{F} \cdot \hat{z} =M G \rho \frac{K-1}{K} \iiint _{ \mathscr {R}_{hg}} \frac{\mathbf{r}\cdot \hat{z}}{\| \mathbf{r}\|^{3}} dV \\ & =M G \rho \frac{K-1}{K} \iiint _{\mathscr {R} _{hg}} \frac{z}{ (z^{2}+r^{2} )^{3/2}} dV \\ & \le M G \rho \frac{K-1}{K} \int _{0}^{r_{h}} \int _{0}^{\infty } \frac{ \pi r z}{ (z^{2}+r^{2} )^{3/2}} dz dr \\ & =\pi M G \rho \frac{K-1}{K} \int _{0}^{r_{h}} \biggl[ - \frac{r}{\sqrt{z^{2}+r^{2}}} \biggr]_{z=0}^{z=\infty } dr \\ &=\pi M G \rho \frac{K-1}{K} \int _{0}^{r_{h}} dr = \pi M G \rho r _{h} \frac{K-1}{K}. \end{aligned}$$
(A.15)

The real force is less or equal to the one calculated by approximating \(\mathscr {R}_{hg}\) by Fig. 3. From (A.15) and (A.7), it follows that

$$ r_{1}= 1.5 \frac{K-1}{K} r_{h}. $$
(A.16)

Dividing (A.16) by (A.13), we obtain an upper limit for the thermal efficiency

$$ \eta _{{G}}=\frac{r_{1}}{r_{2}} \le 1.06 \sqrt{ \frac{K-1}{K}}. $$
(A.17)

We conjecture that, the real values of \(\eta _{{G}}\) are much lower, and they decrease with Mach Number. We would estimate \(\eta _{{G}}\) as

$$ \eta _{{G}} \approx \frac{0.8}{M} \sqrt{\frac{K-1}{K}} \quad \text{for MBH travelling at Mach } \ge 2, $$
(A.18)

but until there are rigorously derived and proven results, it remains an open problem.

A.3 Considerations regarding the value of \(K\)

Recall, that the average temperature of the gas in the hot tail is \(K T\), where \(T\) is the temperature of the surrounding stellar material. The exact calculation of \(K T\) is beyond the scope of this work, and possibly beyond the scope of modern science. Nevertheless, here are some general observations on \(K\).

We introduce two new constants. Radiation radius \(r_{{ \gamma }}\) is the average distance travelled by a photon or another energy-carrying particle from PBH before being absorbed by stellar material. Minimal hot tail radius \(r_{{mh}}\) is the minimal radius the hot tale can have regardless of \(K\).

If \(r_{{\gamma }} \ll r_{{mh}}\), 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, \(K \gg 1\). If \(r_{{\gamma }} \gg r_{{mh}}\), then heat is dissipated over a large volume of gas. That gas volume is much larger than the gas volume which can be significantly heated by MBH radiative power. Thus, the heating has to be by a small margin. Thus, \(0 < K-1 \ll 1\).

At this point we introduce some calculations for estimating \(r_{{\gamma }}\) and \(r_{{mh}}\). The radiation radius is

$$ r_{{\gamma }}=\frac{\mathfrak{S}_{{\gamma }}}{\rho _{{p}}}, $$
(A.19)

where \(\mathfrak{S}_{{\gamma }}\) is the planar density of material through which an energy carrying particle has to travel before being absorbed by stellar material. Recall that absorbtion of any radiation in any medium is proportional to planar density. The density \(\rho _{{p}}\) is an average density of the material over the path of the energy-carrying particle. The value of \(\mathfrak{S}_{{\gamma }}\) is inversely proportional to average absorbtion cross-section of the energy-carrying particles:

$$ \mathfrak{S}_{1} = \frac{1\ \text{kg}}{1000 N_{A}\ \text{amu}} \cdot \frac{1}{10^{-28} \sigma } =\frac{16.6\ \frac{\text{kg}}{\text{m}^{2}}}{ \sigma \ \text{in barn}}, $$
(A.20)

where \(\sigma \) is the absorbtion cross-section in barns. Given that most interactions are scattering, effective absorbtion cross-section has to be calculated.

We calculate the attenuation coefficient of photons. In Table 2, the first column is the photon energy. The second column is the total photon absorbtion and scattering cross-section per amu (Hubbell 1969, pp. 41–42). The third column is the mass attenuation coefficient. It is calculated in (A.20). For low energy photons, all of the events are scattering. For high energy photons, there is another algorithm for interaction with matter—production of \(\{ e^{-}, e^{+} \} \) pairs. The forth column denoted PPF is the pair production fraction of all events.

Table 2 Photon mass attenuation coefficient

The minimal hot tail radius can be obtained from (A.9):

$$ r_{{mh}}=\sqrt{\frac{3 P_{{T}}}{5 \pi v_{0} \rho T C_{v}}}, $$
(A.21)

where \(P_{{T}}\) is the part of MBH power used to produce heat rather than the sound wave.

Appendix B: Estimation of a MBH kinetic energy loss on passage through a sun-like star

This subsection consists of mostly numerical calculations. In our calculations we use the force given in (3.1), which is given in Abramowicz and Becker (2009, p. 8) and also derived in Appendix C. We use \(r_{{ \text{min}}}=0.1\mbox{ m}\) and \(r_{{\text{max}}}=5 \times 10^{7}\mbox{ m}\). Expressing (3.1) in numerical terms we obtain:

$$\begin{aligned} F_{t}&=\frac{4 \pi ( M G )^{2} \rho }{v_{0}^{2}} \ln \biggl( \frac{r _{{\text{max}}}}{r_{{\text{min}}}} \biggr) \\ &= \bigl( 1.12 \times 10^{9} \mbox{ N} \bigr) \frac{M_{18}^{2} \rho _{3}}{v_{6}^{2}}. \end{aligned}$$
(B.1)

Above we tabulate (see Table 3) several parameters for a MBH passing through a sun-like star. We use the density data from Solar interior given in Guenther et al. (1992). The first column stands for the fraction of Solar radius denoted by \(R_{\text{Sun}}\). The second column is the density in \(\mbox{g}/\mbox{cm}^{3}\). The third column is an estimated speed of a MBH which arrives at the sun from a large distance. The forth column is \(F_{t}\) for \(M_{18}=1\).

Table 3 Parameters for MBH passing through a Sun-like star

The Solar radius is \(R_{\odot }=6.96 \times 10^{8}\mbox{ m}\). Thus we estimate the energy loss of a MBH passing through the center of a Sun-like star:

$$ \triangle E = \int _{-R_{\odot }}^{R_{\odot }} F_{t}\ dx = \bigl( 2.0 \times 10^{19}\mbox{ J} \bigr) M_{18}^{2}. $$
(B.2)

Appendix C: Derivation of (3.1)

In this section we show that a small black hole passing through gas at supersonic speed experiences a drag force given in (3.1).

The black hole passes along the \(\mathbf{x}\)—axis in \(\hat{x}\) direction. The black hole \(x\)—coordinate is the function of time:

$$ x(t)=v_{0} t. $$
(C.1)

During the passage, it changes the speed of gas by providing a push into \(-\hat{r}\) direction. For any volume \(dV\) of gas located at the distance \(r\) from x-axis, the speed supplied by the gravitational push is (see Fig. 4)

$$\begin{aligned} \triangle v & = \int _{-\infty }^{\infty } ( \text{Acceleration in $-\hat{r}$ direction at time $t$} ) dt \\ & = \int _{-\infty }^{\infty } MG \frac{r}{ (r^{2}+x^{2} )^{1.5}} \biggl( \frac{dx}{v_{0}} \biggr) \\ &=\frac{MG}{v_{0}} \int _{-\infty }^{\infty } \frac{r dx}{ (r ^{2}+x^{2} )^{1.5}} = \frac{MG}{v_{0} r} \int _{-\infty }^{ \infty } \frac{d ( x/r )}{ (1+ (x/r )^{2} ) ^{1.5}} \\ &= \frac{2MG}{v_{0} r}. \end{aligned}$$
(C.2)
Fig. 4
figure4

Effects of black hole passage through gas

The energy supplied to the gas within the volume \(dV\) is

$$ \triangle E = ( \rho \triangle V ) \frac{ (\triangle v )^{2}}{2} = \frac{2 (M G )^{2}}{v_{0}^{2} r^{2}} \rho \triangle V. $$
(C.3)

The energy supplied to a cylindrical shell of length \(dx\) and thickness \(dr\) can be expressed by

$$\begin{aligned} \frac{dE}{dr dx} & =\triangle E \cdot ( \text{Area} ) = \biggl( \frac{2 (M G )^{2}}{v_{0}^{2} r^{2}} \rho \biggr) \cdot ( 2 \pi r ) \\ &= \frac{4 \pi ( M G )^{2} \rho }{v_{0}^{2} r}. \end{aligned}$$
(C.4)

Integrating the above we obtain the resistive drag force:

$$\begin{aligned} F_{t} & =\frac{dE}{dx}= \int _{-\infty }^{\infty } \frac{dE}{dr dx} dr = \int _{-\infty }^{\infty } \frac{4 \pi ( M G )^{2} \rho }{v _{0}^{2} r} dr \\ & = \frac{4 \pi ( M G )^{2} \rho }{v_{0}^{2}} \ln \biggl( \frac{r_{{\text{max}}}}{r_{{\text{min}}}} \biggr). \end{aligned}$$
(C.5)

The power is

$$ P=v_{0} \frac{dE}{dx} = \frac{4 \pi ( M G )^{2} \rho }{v_{0}} \ln \biggl( \frac{r_{{\text{max}}}}{r_{{\text{min}}}} \biggr). $$
(C.6)

Notice, that the above result is valid if and only if the flow is supersonic. For transonic and subsonic flows, the gas is stopped by pressure before it is accelerated to speed \(\triangle v\). For transonic and subsonic flows, the tidal drag is much less.

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Shubov, M.V. Ramjet acceleration of microscopic black holes within stellar material. Astrophys Space Sci 364, 220 (2019). https://doi.org/10.1007/s10509-019-3707-9

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Keywords

  • Microscopic black hole
  • Stellar material
  • Gravitational drag
  • Accelerating forces
  • Mach number