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Investigation of the dynamics of interaction of a cluster of primordial black holes with stellar cluster

Konstantin Belotsky

https://orcid.org/0000-0003-4617-8819

National Research Nuclear University MEPhI, Moscow 115409, Russia

Novosibirsk State University, Novosibirsk 630090, Russia

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Maxim Krasnov

https://orcid.org/0009-0003-1740-5397

National Research Nuclear University MEPhI, Moscow 115409, Russia

E-mail Address: [email protected]

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, and

Stanislav Pugachev

National Research Nuclear University MEPhI, Moscow 115409, Russia

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https://doi.org/10.1142/S0218271823400102Cited by:1 (Source: Crossref)

This article is part of the issue:

SPECIAL ISSUE — New Trends in Cosmology, Particle Physics and Astrophysics: Selected Papers from the 6th International Conference on Particle Physics and Astrophysics (ICPPA2022)
Guest editor: Maxim Yu. Khlopov

Abstract

In this paper, we revise dynamical constraint on primordial black hole (PBH) abundance following the effect of their interaction with stellar cluster in dwarf galaxy Eridanus II in case of PBH clustering. We consider scattering of stars on PBH cluster as a whole, dynamical friction of stars inside cluster and finally we are taking into account tidal forces’ effects due to the finite size of PBH cluster. We obtain the rate of change of kinetic energy of the stars and PBHs to make conclusion about constraints on clusters.

Keywords:

1. Introduction

There is no need to justify the current relevance of the topic of primordial black holes (PBHs). Possibility of such object existence was first discussed in 1966 by Zel’dovich and Novikov¹ and also by Hawking et al.^2,3,4 It is now believed that PBHs are not only candidates for dark matter (DM), requiring minimal new physical assumptions, but probably also explanation for other phenomena.⁵ We omit all the arguments “pro” and “contra” PBH referring the readers to the aforementioned review as the last (at the time of writing this paper) from the many ones. Many observations are analyzed from point of view of possible PBH existence what defines/restricts allowable PBH abundance.

We focus here on one special case which allows to constrain the relative PBH contribution to DM density ( $f_{PBH}$ ) coming from an observation of stellar cluster in dwarf galaxy Eridanus II. Respective constraint on PBH abundance with their mass $M \sim 1 0^{3} M_{⊙}$ – $1 0^{5} M_{⊙}$ was imposed recently⁶ (and updated just before this paper issue⁷) from the fact of (old) stellar cluster existence⁸ in DM (i.e. presumably, PBH) enriched dwarf galaxy. In the process of motion within a star cluster, the PBHs could destroy (“heat up”) it. This is due to the fact that lighter stars will scatter on heavier PBHs and as the result of this interaction, the stars will gain energy.⁹

But in the models where PBHs are clustered (at birth or during evolution) some constraints should be reconsidered,¹⁰ “dynamical constraints” are in particular.

In this paper, we consider the case when compact objects of aforementioned mass are a PBH cluster with mass of each PBH $m_{bh} ≪ M_{⊙}$ . The interaction of such objects with stellar cluster in Eridanus II is the subject of given consideration. Respectively, the following effects are taken into account:

•	Scattering of stars on PBH cluster as a whole (as on point-like one),^a that was the case of previous consideration.⁶
•	Tidal forces’ effects due to finite size of PBH cluster.
•	Dynamical friction of stars inside PBH cluster.

This continues our previous work¹¹ where the third point was only considered to make limiting estimates.

In this work, we consider the effect in terms of energy change (loss and gain) of PBH cluster, which is the same for a stellar cluster with opposite sign. Conclusion of the work is reduced to comparison of energy change rates of the PBH (stellar) cluster in the cases when it is point-like and of finite size.

2. Scattering of Stars on PBH Cluster as a Whole

We assume in this paper the mass of any PBH to be much less than solar mass. If one considers PBHs of solar masses and above, then effect of dynamical heating of the stellar cluster would be stronger, which would lead to stronger constraints. The size and mass of PBH cluster will vary around 1 pc and $1 0^{4} M_{⊙}$ , respectively (which is somehow model-motivated¹⁰).

The change of kinetic energy of any system can be written as (accordingly to König’s theorem) :

Δ ℰ_{k} = \sum_{i} \frac{m_{i} Δ v_{i}^{2}}{2} = Δ ℰ_{cm} + Δ ℰ_{in},

where

Δ ℰ_{cm}

and

Δ ℰ_{in}

are the changes of energy of the center of system mass and its inner energy, respectively, the sum is over the system components.

Let us consider the change of the energy of the PBH cluster as a whole — as a point-like — in laboratory reference frame (LRF) $Δ ℰ_{cm}$ , when the PBH cluster moves through medium of (lighter) stars (stellar cluster). This is virtually what was done for point-like MACHO objects in the paper.⁶ At this motion, energy change (losses) is induced by dynamical friction^9,12,13 which is accounted for by the fact of perturbation of trajectories of light particles (stars) by gravitational field of the moving massive particle (PBH cluster). Massive particle transfers to light ones a part of its momentum :

M \frac{d V}{d t} = - \frac{4 π G^{2} M^{2}}{V^{2}} ln (Λ) ρ (< V),

where V is the velocity of the heavy object (PBH cluster

v_{cl}

) of mass M,

ρ (< V)

is the density of particles with velocities less than V, and

Λ = b_{max} / b_{min}

. The values

b_{max}

and

b_{min}

are the impact parameters in interactions of light particles with heavy particles. Usually

b_{max} \approx L

, where L is the size of the system (stellar cluster), and

b_{min} \approx max (G M / V^{2}, l)

with

l \equiv R_{cl}

being the size of the heavy object (PBH cluster).

One can reproduce Chandrasekhar’s formula for dynamical friction as applied to the case of PBH cluster (see Appendix A) :

\begin{matrix} F_{df} = M_{cl} \frac{d v_{cl}}{d t} & = & - \frac{4 π G^{2} M_{cl} (M_{cl} + m_{star}) ρ_{s}}{v_{cl}^{3}} ln (\frac{R_{s} v_{cl}^{2}}{G (M_{cl} + m_{star})}) \\ \cdot [Erf (X) - \frac{2 X}{\sqrt{π}} e^{- X^{2}}] v_{cl} . \end{matrix}

Here,

m_{star}

ρ_{s}

and

R_{s} \equiv L

are the mass and density of the stars and their cluster size,

X = v_{cl} / (\sqrt{2} σ)

with

σ

being the dispersion of star velocities in the cluster. The size

R_{s}

will be taken here from 10 to 25 pc. Since

v_{cl} ≫ σ \sim 2 km/s

(therefore

X ≫ 1

) we will put

[Erf (X) - \frac{2 X}{\sqrt{π}} e^{- X^{2}}] \approx 1

. The rate of the energy change of the PBH cluster as a whole is

\frac{d ℰ_{cm}}{d t} = (F_{df} v_{cl}) = - \frac{4 π G^{2} M_{cl} (M_{cl} + m_{star}) ρ_{s}}{v_{cl}} ln (\frac{R_{s} v_{cl}^{2}}{G (M_{cl} + m_{star})}) .

3. Tidal Forces’ Effects due to Finite Size of PBH Cluster

Let us now estimate contribution to the inner PBH cluster energy gain, $Δ ℰ_{in}^{(1)}$ , from the stars which fly outside it.

For it, let us first calculate the change in the (total) kinetic energy of the PBH cluster, $Δ E_{k}^{'}$ , due to a single star scattering on the PBHs of the cluster in the reference frame where its center of mass is fixed at the origin and initially at the rest (the corresponding values will be marked with a prime). To solve this problem analytically, we use momentum approximation (which was first used by Fermi in calculations for quantum scattering¹⁴). In the momentum approximation, we assume that the PBHs in the cluster are static objects, whose velocities change only during (instant) gravitational interaction with the passing star. This is justified by the condition that the velocity of PBH cluster center mass (in LRF), $v_{cl} \approx V \equiv | v_{cl} - v_{star} |$ , is assumed to be much greater than inner velocities of the stars and PBHs inside their clusters (with respect to their centers of mass). The latter are determined by the clusters’ virial velocities, $v_{vir} \sim \sqrt{G M / R} \sim 2 km/s$ , and assumed here with some tension to be $≪ v_{cl} \sim 10 km/s$ . The change of the cluster energy will be expressed by the same formula as Eq. (1), but in the reference frame where PBH cluster is initially at the rest¹⁵ :

Δ E_{k}^{'} = \sum_{i} \frac{m_{i} Δ {v_{i}^{'}}^{2}}{2},

where subscript “i” refers to PBHs inside their cluster (

m_{i} \equiv m_{pbh}

Formula (5) can be generalized in integral form :

Δ E_{k}^{'} = \frac{1}{2} \int d^{3} r ρ_{cl} (r) {(Δ v)}^{2} .

For simplicity, we set the density of PBHs in the cluster

ρ_{cl} (r) = ρ_{cl} = const.

, the justification for it will be commented on at the end.

We assume that initially the stars are not bound to each other and approach along a hyperbolic orbit with initial relative velocity v and impact parameter b. For sufficiently large v, the deviation of the stars from their initial trajectory due to gravitational interaction is small, and it can be treated as a straight line (see Fig. B.1 in Appendix B). One can get (see Appendix B)

Δ E_{k}^{'} = 8 π {(\frac{G m_{star}}{v_{cl}})}^{2} ρ_{cl} [R_{cl} - \sqrt{b^{2} - R_{cl}^{2}} arctan (\frac{R_{cl}}{\sqrt{b^{2} - R_{cl}^{2}}})] .

Equation (7) gives us the total kinetic energy received by the PBH cluster. We are interested in the change in inner energy of the cluster, $Δ E_{in}^{(1)}$ , which is the change of energy relative to the rest system of the PBH cluster center of mass. For this purpose, we must subtract the change in the energy of the center of mass, $Δ E_{cm}^{'}$ .

The latter in the considered reference frame is [see Appendix B for derivation of $Δ V$ being the change of relative star-PBH cluster (center of its mass) velocity]

Δ E_{cm}^{'} = \frac{M_{cl} Δ V^{2}}{2} = {(\frac{G m_{star}}{v_{cl}})}^{2} \frac{2 M_{cl}}{b^{2}} = 8 π {(\frac{G m_{star}}{v_{cl}})}^{2} \frac{ρ_{cl} R_{cl}^{3}}{3 b^{2}} .

The energy change we are looking for will be then

Δ E_{in}^{(1)} = Δ E_{k}^{'} - Δ E_{cm}^{'} = Δ E_{0} \cdot [1 - \sqrt{\frac{b^{2}}{R_{cl}^{2}} - 1} \cdot arctan (\frac{R_{cl}}{\sqrt{b^{2} - R_{cl}^{2}}}) - \frac{R_{cl}^{2}}{3 b^{2}}],

where

Δ E_{0} = 8 π (\frac{G m_{star}}{v_{cl}})^{2} ρ_{cl} R_{cl}

As can be seen, there is a restriction on the impact parameter — it cannot be smaller than the cluster size $R_{cl}$ . The change in the internal energy of the PBH cluster is presented in Fig. 1.

Fig. 1. Variation of the internal energy of the PBH cluster from the impact parameter, $Δ E_{in}^{(1)} / Δ E_{0}$ .

Knowing how much inner energy the cluster receives from a single star, we can sum over all star encounters that occur at a frequency of $2 π b d b n_{star} v_{cl}$ per unit time :

\begin{matrix} \frac{d ℰ_{in}^{(1)}}{d t} & = & \int_{R_{cl}}^{R_{s}} Δ E_{in}^{(1)} 2 π b d b n_{star} v_{cl} \\ = & Ė_{0} [R_{cl} (R_{s}^{2} - R_{cl}^{2}) - {(R_{s}^{2} - R_{cl}^{2})}^{3 / 2} \cdot arctan (\frac{R_{cl}}{\sqrt{R_{s}^{2} - R_{cl}^{2}}})], \end{matrix}

where

Ė_{0} = \frac{16 π^{2} G^{2} m_{star} ρ_{s} ρ_{cl}}{3 v_{cl}}

. This dependence is shown in Fig. 2.

4. Dynamical Friction of Stars Inside PBH Cluster

Now, it is necessary to find the energy that the cluster receives when stars pass through it; we denote this energy as $Δ ℰ_{in}^{(2)}$ . This simple estimate was done in the previous work¹¹ as limiting one. For it, one finds the energy that a star loses as it passes through the PBH cluster per unit time. For this purpose, Chandrasekhar’s formula used in the previous section can be applied again :

\frac{d E}{d t} = - \frac{4 π G^{2} (m_{bh} + m_{star}) m_{star} ρ_{cl}}{v_{cl}} ln (\frac{R_{cl} v_{cl}^{2}}{G (m_{star} + m_{bh})}),

where

m_{bh} ≪ M_{⊙}

is the PBH mass. One summarizes this effect from all the stars that pass through the cluster :

\begin{matrix} \frac{d ℰ_{in}^{(2)}}{d t} & = & \frac{d E}{d t} \frac{4 π R_{cl}^{3}}{3} n_{star} \\ = & - \frac{4 π G^{2} (m_{star} + m_{bh}) M_{cl} ρ_{s}}{v_{cl}} ln (\frac{R_{cl} v_{cl}^{2}}{G (m_{star} + m_{bh})}) . \end{matrix}

Further, one neglects

m_{bh}

with respect to

m_{star}

. The change in the internal energy of the PBH cluster is, including two aforementioned contributions :

\begin{matrix} \frac{d ℰ_{in}^{Σ}}{d t} = \frac{d ℰ_{in}^{(1)}}{d t} + \frac{d ℰ_{in}^{(2)}}{d t} & = & Ė_{0} [R_{cl}^{3} ln (\frac{R_{cl} v_{cl}^{2}}{G m_{star}}) + R_{cl} (R_{s}^{2} - R_{cl}^{2}) \\ - {(R_{s}^{2} - R_{cl}^{2})}^{3 / 2} arctan (\frac{R_{cl}}{\sqrt{R_{s}^{2} - R_{cl}^{2}}})] . \end{matrix}

We illustrate the obtained rate in Fig. 3.

5. The Rate of Change of the Cluster Energy

Taking into account all the contributions in energy change of the PBH cluster considered, total energy change effect can be estimated on the base of it which is believed to be sign-symmetrically the same for the stellar cluster.

For our problem, we set the PBH cluster mass $M_{cl} = 10^{4} M_{⊙}$ , the typical star mass $m_{star} = 1 M_{⊙}$ , the velocity dispersion in the globular cluster $σ \approx 2 km/s$ , and the velocity of the PBH cluster center of mass $v_{cl} \approx 10 km/s$ , what justifies (though with some tension) the use of the momentum approximation. It is also possible to simplify Eq. (4) :

\frac{d ℰ_{cm}}{d t} = - \frac{4 π^{} G^{2} M_{cl}^{2} ρ_{s}}{v_{cl}} ln (\frac{R_{s} v_{cl}^{2}}{G M_{cl}}) .

Change in the total energy of the PBH cluster is finally :

\begin{matrix} \frac{d ℰ_{k}}{d t} & = & \frac{d ℰ_{in}^{Σ}}{d t} + \frac{d ℰ_{cm}}{d t} = \frac{4 π G^{2} M_{cl} m_{star} ρ_{s}}{v_{cl}} [\underset{passage through PBH cluster}{\underset{︸}{ln (\frac{R_{cl} v_{cl}^{2}}{G m_{star}})}} + \frac{R_{s}^{2}}{R_{cl}^{2}} - 1 \\ - {(\frac{R_{s}^{2}}{R_{cl}^{2}} - 1)}^{3 / 2} arctan (\frac{R_{cl}}{\sqrt{R_{s}^{2} - R_{cl}^{2}}}) - \underset{passage through globular cluster}{\underset{︸}{\frac{M_{cl}}{m_{star}} ln (\frac{R_{s} v_{cl}^{2}}{G M_{cl}})}}] . \end{matrix}

We subscribed the physical sense of some of the terms for clarity. Since $M_{cl} = 1 0^{4} M_{⊙}$ , the last term is dominant within the brackets of (15). Thus, Eq. (15) gives nearly constant value while varying the radii of the clusters.

In order to make a conclusion about constraints on abundance of the PBHs in form of clusters, we consider the ratio between energy change per unit time of a nonpoint-like cluster to a point-like one. The ratio is shown in Fig. 4. One can see that effect of clustering at the considered cluster parameters is vanishing with respect to point-like object of the same mass. This seems to make the use of more accurate calculation approximations (as to density profile of PBH cluster, velocity distributions, adiabatic approximation of description of star-PBH cluster interaction, etc.) rather unjustified, but nonetheless some specific issues of the problem will still be worth considering.

Fig. 4. The ratio of the energy change per unit time of a nonpoint-like cluster to a point-like cluster as a function of PBH cluster’ size.

6. Conclusion

In this paper, we studied the interaction of a PBH cluster with the stars of a globular cluster. The purpose of this work is to revise the constraint on the PBHs as a DM obtained for single PBHs coming from their possible interaction with a cluster of stars. In the course of this work, an expression for the PBH cluster energy change per unit time (15) was derived. The following dynamical effects were taken into account: tidal forces when a star scatters on a PBH cluster, the scattering of a star on the cluster as a whole, the dynamical friction when a PBH cluster passes through a globular cluster of stars, and the dynamical friction when a star passes through a PBH cluster. Possible ranges of different parameters of the PBH cluster were considered.

Based on it, the ratio of the nonpoint-like PBH cluster’ energy change to the point-like PBH cluster’ energy change was obtained. As can be seen from Fig. 4, in the framework of the used approximation, the effects associated with a finite size of the PBH cluster can be considered small. Thus, we conclude that the clusters of PBHs should approximately obey the same constraints as single PBHs.

Acknowledgment

The authors would like to thank Viktor Stasenko for fruitful discussions. The work was supported by RSF Grant 23-42-00066, https://rscf.ru/project/23-42-00066.

Appendix A

Here, we will derive dynamical friction force, experienced by a cluster of PBHs of mass $M_{cl}$ moving in a homogeneous field of noninteracting stars of mass $m_{star}$ .

Let us denote the position, velocity of the center of mass, and mass of the PBH cluster by $R_{cl}, v_{cl}, M_{cl}$ , and the corresponding parameters of the stars by $r_{star}, v_{star}, m_{star}$ .

Our goal is to reproduce the famous Chandrasekhar formula, for this purpose, let us consider the two-body problem. Let us move to a frame of reference where the star moves relative to the cluster, the motion of the star relative to the point cluster of PBHs is shown in Fig. A.1. The position of the star relative to the cluster is $r = r_{star} - R_{cl}$ , the relative velocity of the star is $V = v_{star} - v_{cl}$ , the change in the position of $r$ is determined by

μ \ddot{r} = - \frac{κ}{r^{2}} r,

where

μ = M_{cl} m_{star} / (M_{cl} + m_{star})

κ = G M_{cl} m_{star}

. Change of relative velocity is given by

Δ V = Δ v_{star} - Δ v_{cl} .

Let’s also use the law of conservation of momentum :

M_{cl} Δ v_{cl} + m_{star} Δ v_{star} = 0 .

From the previous two equations, we obtain the change in cluster’s velocity :

Δ v_{cl} = - \frac{m_{star}}{M_{cl} + m_{star}} Δ V .

Fig. A.1. Motion of a star relative to a point cluster of PBHs.

One decomposes $Δ V$ into components :

Δ V = Δ V_{⊥} + Δ V_{| |},

where

| Δ V_{| |} | = V_{0} cos θ

| Δ V_{| |} | = V_{0} sin θ

, where

θ

is the scattering angle and

V_{0}

is the velocity of the star at infinity. From geometry, the angle

α

is related to the orbital eccentricity e by the relation¹⁶ :

cos α = \frac{1}{e}, cot \frac{θ}{2} = \sqrt{e^{2} - 1},

where

θ + 2 α = π

. The eccentricity for a hyperbolic orbit is¹⁷

e = \sqrt{1 + \frac{2 E L^{2}}{μ k^{2}}},

where

E = μ V_{0}^{2} / 2

is the kinetic energy,

L = μ b V_{0}

is the momentum :

sin θ = \frac{2 tan \frac{θ}{2}}{1 + {tan}^{2} \frac{θ}{2}}, cos θ = \frac{1 - {tan}^{2} \frac{θ}{2}}{1 + {tan}^{2} \frac{θ}{2}} .

After some transformations, we obtain

| Δ V_{⊥} | = \frac{2 b V_{0}^{3}}{G (M_{cl} + m_{star})} {[1 + \frac{b^{2} V_{0}^{4}}{G^{2} {(m_{star} + M_{cl})}^{2}}]}^{- 1},

| Δ V_{| |} | = 2 V_{0} {[1 + \frac{b^{2} V_{0}^{4}}{G^{2} {(m_{star} + M_{cl})}^{2}}]}^{- 1},

| Δ v_{cl ⊥} | = \frac{2 b V_{0}^{3} m_{star}}{{G (M_{cl} + m_{star})}^{2}} {[1 + \frac{b^{2} V_{0}^{4}}{G^{2} {(M_{cl} + m_{star})}^{2}}]}^{- 1},

| Δ v_{cl | |} | = \frac{2 V_{0} m_{star}}{M_{cl} + m_{star}} {[1 + \frac{b^{2} V_{0}^{4}}{G^{2} {(M_{cl} + m_{star})}^{2}}]}^{- 1} .

Moving in a homogeneous field of noninteracting stars of mass $m_{star}$ all perpendicular deviations of the cluster are compensated by symmetry. However, parallel velocity changes are added and the cluster will experience deceleration. The calculation of the total drag force is as follows. Let $f (v_{star})$ be the number density of the stars. The rate at which the cluster collides with stars with an impact parameter between b and $b + d b$ and velocities between $v_{star}$ and $v_{star} + d v_{star}$ , is equal to

2 π b d b V_{0} f (v_{star}) d^{3} v_{star},

\frac{d v_{cl}}{d t} = 2 π V_{0} f (v_{star}) d^{3} v_{star} \int_{0}^{R_{s}} | Δ v_{cl | |} | b d b .

Integration with respect to the impact parameter, where

R_{s}

is the size of the globular star cluster :

\begin{matrix} \int_{0}^{R_{s}} | Δ v_{cl | |} | b d b & = & \int_{0}^{R_{s}} \frac{2 m_{star} V_{0}}{M_{cl} + m_{star}} {[1 + \frac{b^{2} V_{0}^{4}}{G^{2} {(M_{cl} + m_{star})}^{2}}]}^{- 1} b d b \\ = & \frac{2 G^{2} m_{star} (M_{cl} + m_{star})}{V_{0}^{3}} ln (\frac{R_{s} V_{0}^{2}}{G (m_{star} + M_{cl})}) . \end{matrix}

Integration of the equation over the velocity space of the stars is now required :

\begin{matrix} \frac{d v_{cl}}{d t} & = & 4 π G^{2} m_{star} (M_{cl} + m_{star}) ln (\frac{R_{s} V_{0}^{2}}{G (m_{star} + M_{cl})}) \\ \cdot \int_{0}^{\infty} \frac{v_{star} - v_{cl}}{{| v_{star} - v_{cl} |}^{3}} f (v_{star}) d^{3} v_{star} . \end{matrix}

Now, it remains to impose the condition that

v_{star} < v_{cl}

and put

V_{0} \approx v_{cl}

, then we obtain

\begin{matrix} \frac{d v_{cl}}{d t} & = & - \frac{4 π G^{2} m_{star} (M_{cl} + m_{star})}{v_{cl}^{3}} ln (\frac{R_{s} v_{cl}^{2}}{G (m_{star} + M_{cl})}) \\ \cdot \int_{0}^{v_{cl}} f (v_{star}) d^{3} v_{star} v_{cl} . \end{matrix}

We assume stars have a Maxwellian velocity distribution :

f (v_{star}) = \frac{n_{star}}{{(2 π σ^{2})}^{3 / 2}} e^{- v_{star}^{2} / (2 σ^{2})},

where

σ

is the dispersion of stellar velocities within the star cluster,

n_{star}

is the number density of stars. The integral depending on

f (v_{star})

can be calculated analytically :

\int_{0}^{v_{cl}} \frac{n_{star}}{{(2 π σ^{2})}^{3 / 2}} e^{- v_{star}^{2} / (2 σ^{2})} d^{3} v_{star} = n_{star} [Erf (X) - \frac{2 X}{\sqrt{π}} e^{- X^{2}}],

where

X = v_{cl} / \sqrt{2} σ

Appendix B

We place the center of mass of the PBH cluster at the origin, as it was said above the center of mass is at rest, the velocity of the star in this coordinate system $V = | v_{star} - v_{cl} |$ , where $v_{star}$ is the velocity of the star and $v_{cl}$ is the velocity of the center of mass of the PBH cluster in the LRF. The coordinates of the PBHs in the cluster are $r = (x, y, z)$ , the coordinates of the passing star are $R = (0, b, V t)$ , $R - r = - x e_{x} + (b - y) e_{y} + (z - V t) e_{z}$ . The motion of the star relative to the PBH cluster is shown in Fig. B.1. Let us find the change of the PBH velocity in the cluster :

Δ v = \int_{- \infty}^{+ \infty} a d t = G m_{star} \int_{- \infty}^{+ \infty} \frac{R - r}{{| R - r |}^{3}} d t = \frac{2 G m_{star}}{v_{cl}} \frac{- x e_{x} + (b - y) e_{y}}{x^{2} + {(b - y)}^{2}} .

Fig. B.1. Illustration of notation introduced for description of the motion of the star relative to the PBH cluster.

Total change in PBH cluster energy as a result of the encounter :

Δ E_{k}^{'} = \frac{1}{2} \int d^{3} r ρ_{cl} (r) {(Δ v)}^{2} .

For simplicity, we set the density of PBHs in the cluster

ρ_{cl} (r) = ρ_{cl} = const.

In spherical coordinates

x = r cos ϕ sin θ

y = r sin ϕ sin θ

, then

\begin{matrix} Δ E_{k}^{'} & = & 2 {(\frac{G m_{star}}{v_{cl}})}^{2} ρ_{cl} \int d^{3} r \frac{1}{r^{2} {sin}^{2} θ + b^{2} - 2 b r sin ϕ sin θ} \\ = & 8 π {(\frac{G m_{star}}{v_{cl}})}^{2} ρ_{cl} [R_{cl} - \sqrt{b^{2} - R_{cl}^{2}} arctan (\frac{R_{cl}}{\sqrt{b^{2} - R_{cl}^{2}}})] . \end{matrix}

Using Eq. (B.1), we find the change in velocity of the center of mass $Δ v_{cl} = Δ V$ :

\begin{matrix} Δ V & = & \frac{1}{M_{cl}} \int_{0}^{R_{cl}} ρ_{cl} r^{2} d r \int_{0}^{π} sin θ d θ \int_{0}^{2 π} Δ v d ϕ = \frac{2 G m_{star}}{v_{cl} M_{cl}} ρ_{cl} \\ \cdot \int_{0}^{R_{cl}} r^{2} d r \int_{0}^{π} sin θ d θ \int_{0}^{2 π} d ϕ \frac{- r sin θ cos ϕ e_{x} + (b - r sin θ sin ϕ) e_{y}}{r^{2} {sin}^{2} θ + b^{2} - 2 b r sin ϕ sin θ} \\ = & \frac{2 G m_{star}}{v_{cl} b} e_{y} . \end{matrix}

ORCID

Konstantin Belotsky https://orcid.org/0000-0003-4617-8819

Maxim Krasnov https://orcid.org/0009-0003-1740-5397

Notes

^a In other words, dynamical friction of PBH cluster inside stellar cluster.

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