text:
\section{Introduction}

Dark matter (DM) remains one of the most important mysteries in modern cosmology. One of the hypotheses for the nature of DM are Primordial black holes (PBHs) made in the early Universe from large amplitude primordial fluctuations \citep{1971MNRAS.152...75H, 1974MNRAS.168..399C, 1975Natur.253..251C, 1975ApJ...201....1C}. The abundance of PBHs is limited by several observational constraints;
reviews over the entire PBH mass range can be found, for example, in \citet{2018CQGra..35f3001S,2021RPPh...84k6902C,2021arXiv211002821C}. While their abundance is severely limited below a mass of $\sim 10^{-16.5}\, {\rm M_{\odot}}$ by the contribution from Hawking evaporation to the $\gamma$-ray background and other impacts on the Cosmic Microwave Background \citep{2020PhRvD.101l3514L}, PBHs might still constitute all of the DM in the Universe in the asteroid-mass range from $10^{-16} \, {\rm M_{\odot}}$ to $10^{-11}\, {\rm M_{\odot}}$.
Despite several early claims for closing this mass window, follow-up studies raised objections on their validity \citep{2019JCAP...08..031M,2020PhRvD.101f3005S}. In addition, PBH might
account for a substantial fraction, but not all, the DM at higher masses while avoiding gravitational microlensing and other constraints \citep{2000ApJ...542..281A,2019NatAs...3..524N, 2020PhRvD.101f3005S}.

Alternative ways to find observational consequences of PBHs in this mass range are therefore of strong interest. A novel approach was proposed in \citet{2009arXiv0901.1093R}, where the idea that these asteroid-mass PBHs might randomly traverse through a star and be captured in the stellar core was presented. If the DM contains PBHs, these would be present in the cool molecular gas clouds where stars form and, if the relative velocities are low enough, they would adiabatically follow the gas contraction during the formation of a protostar, resulting in PBHs bound to the newly born star. A portion of the PBHs in highly eccentric orbits would lose energy via dynamical friction when crossing through the stellar interior and fall to the stellar core. Once settled in the star centre, a PBH would start accreting, growing up to a total mass that can be close to the total mass of the star.

This suggests the possibility to form a black hole with a typical mass of a star but below the Chandrasekhar mass, which cannot be explained in ordinary stellar evolution. Possible origins of these transmuted black holes, as they have been named, include also particle DM \citep{2018PhRvL.121v1102K, 2021PhRvL.126n1105D} and PBH capture by neutron stars \citep{2020PhRvD.102h3004G}.
If a black hole with mass below the Chandrasekhar limit were discovered, it would strongly point to an origin in a PBH or some other process involving DM interaction with stars. However, \citet{2009ApJ...705..659A} questioned that this process could occur in the Milky Way, finding the rate of PBH capture in normal stars to be negligible for the present low DM density and high velocity dispersion.

The capture of PBH was also studied by \citet{2009MNRAS.399.1347B}, focusing on massive stars as the possible origin of supermassive black holes at high redshift. The higher density and lower velocity dispersion of DM halos in the early universe leads to more common PBH capture in the first metal-free stars, although these would not leave unique observational signatures, their mass being similar to other stellar black holes formed at the end of the lives of the same stars.

PBH capture in the present Universe was reexamined by \citet{2013PhRvD..87b3507C}, considering star formation in globular clusters formed in dense DM halos made of PBHs in the asteroid-mass range. The impact of eccentric orbits to the capture rate was included in \citet{2014PhRvD..90h3507C}, who found an enhanced capture rate implying that no neutron stars would form because all their progenitors would have captured a PBH that would accrete the star before or during the formation of the neutron star.
This work, however, considered globular clusters with a very high DM and baryonic density, whereas in fact globular clusters may form from gas cloud fragmentation without involving any DM, so their constraints are not readily applicable \citep{2019JCAP...08..031M}.

In this paper we seek to expand on previous work by studying the effect of PBH capture in the evolution and fate of low-mass stars at high redshift, using precise stellar models of low-mass stars with a range of masses from $0.32\, {\rm M_{\odot}}$ to $1\, {\rm M_{\odot}}$, and with very low metallicity (as expected for the first stars). We consider the first stars formed in the Universe at $z \sim 20$, when both DM density is highest and the velocity dispersion in the star-forming DM halos is lowest, therefore maximising the capture rate of PBHs by stars. We focus on low-mass stars because they are uniquely able to produce black holes of stellar mass below the Chandrasekhar value, giving a clear signature that cannot be explained by standard stellar evolution theory. Although the first stars to form from metal-free gas are expected to have a top heavy initial mass function, stars of lower mass should form almost immediately thereafter from gas polluted by the first few supernovae, so they were probably abundantly made in a way similar to present-day galaxies.

The methods of our calculation are described in section \ref{sec:Methods}. In section \ref{sec:Results} we present our results for the PBH capture rates for stellar models of various stellar masses. Finally our conclusions are discussed in section \ref{sec:Conclusions}. We assume a flat cosmology with $\Omega_b = 0.3$ and $H_0= 70 \, \rm km s^{-1} Mpc^{-1}$

\section{Capture of black holes by a Star: Methods}\label{sec:Methods}

Our aim in this section is to calculate the rate at which PBH accounting for the DM in a halo where a star forms are captured by randomly traversing the star and being slowed down by dynamical friction.

\subsection{Dark Matter density profile around first stars}\label{sec:rhoPBH}

 According to the standard Cold Dark Matter model, the first stars
should form in the first halos where the gravitationally collapsed gas
that is shock-heated to the halo virial temperature is able to
radiatively cool. For the gas of primordial composition, this happens
first when trace amounts of molecular hydrogen formed from the remnant
ionization that is left over from the recombination epoch induce a
cooling rate that is higher than the characteristic inverse time between
successive halo mergers, at $z\simeq 20$ in halos of $M\sim 10^6 \, {\rm M_{\odot}}$
\citep[e.g.][]{1984Natur.311..517B, 1997ApJ...474....1T}

Analytic studies and numerical simulations initially suggested that
stars formed in these halos from primordial gas are all of high mass
\citep[][]{2002Sci...295...93A, 2014ApJ...781...60H}. Newer results however have shown strong fragmentation is possible, resulting in zero metallicity low-mass star formation \citep{2002ApJ...569..549N,2011Sci...331.1040C,2015MNRAS.447.3892H, 2014ApJ...792...32S} and even binary systems \citep{2018MNRAS.479..667R}. Even ignoring this and assuming low-mass stars do not form at zero metallicity, they
should start forming soon thereafter. In this paper we are interested
in low-mass stars, which can survive to the present time either as red
dwarfs in the main-sequence or as white dwarfs, but may become a low-mass black hole if they have captured a PBH from the surrounding DM.

We consider that these low-mass stars acquired a quantity of bound DM at
birth owing to adiabatic contraction during the collapse of the gas
cloud that formed the star \citep{2013PhRvD..87b3507C}, which thereafter remains
bound to the star for an arbitrarily long time. In other words, we
assume that the original DM that is left bound to the star is not
removed by tidal disruption at a later time; we will discuss this further when considering the impact of external tides.

We take as fiducial values of the mass and formation time of the halo
that hosts the first generation of low-mass stars $M=10^7\, {\rm M_{\odot}}$ and
$z=20$, to be conservative on the minimum halo mass where low-mass stars
start forming. The mass of DM left bound to the star
within a given radius depends only on the phase-space density of DM in the formation site. We define the virial halo density
$\rho_v$ and virial radius $R_v$ as
\begin{align}\label{eq:densh}
   \rho_v &= \rho_c \Delta_c = 18\pi^2 \rho_c \simeq 0.22
 \left( \frac{1+z}{21} \right)^3 \, {\rm M_{\odot}}\,{\rm pc}^{-3} ~,  \\
   R_v &= \left( \frac{3M}{4\pi \rho_v} \right)^{1/3} \simeq
 220\, \left( \frac{M}{10^7 \, {\rm M_{\odot}}} \right)^{1/3} \frac{21}{1+z} \,{\rm pc} ~,
   \label{eq:rho}
    \end{align}
where $\rho_c=3H^2/(8\pi G)$ is the critical density of the Universe
and we use the critical overdensity of the top-hat spherical model at
virialization, $\Delta_c=18\pi^2$. The implied halo velocity dispersion
depends of course on the assumed density profile, but is approximately
\begin{equation}\label{eq:sigmah}
 \sigma_v \simeq \left( \frac{GM}{2R_v}\right)^{1/2} \simeq 9.9
 \left( \frac{M}{10^7 \, {\rm M_{\odot}}} \right)^{1/3}\,
 \left( \frac{1+z}{21} \right)^{1/2} \,{\rm km}\,{\rm s}^{-1} ~.
\end{equation}

The phase-space density of DM around the star depends on its distance
from the halo center at formation time, increasingly rapidly towards the
center. We assume the halo has a standard Navarro-Frenk-White (NFW,
\citet{1997ApJ...490..493N}) density profile, as found in numerical
simulations of structure formation. The profile is characterized by a
scale radius $R_s=R_v/c$, where $c$ is the concentration parameter:
\begin{align}
  \rho_h (R) = \frac{\rho_{0}}{R/R_s \left(1+R/R_s \right)^2} ~,
  \label{eq:rho_vir}
\end{align}
where R is the radial distance to the center of the halo and $\rho_0$ is a normalization constant that we determine by
requiring the total mass within $R_v$ to be $M$. We use the value
$c=10$ throughout this paper, in agreement with simulations of the
early halo formation \citep{2001MNRAS.321..155L}.

The detailed phase-space density of the DM at a given radius $R$
depends on the velocity as determined by the condition of dynamical
equilibrium. For simplicity, we assume a Gaussian velocity distribution
to compute the phase-space number density
for small velocities,
\begin{equation}
 Q_g= \frac{\rho_h(R)}{m_b \left[2\pi\sigma_h^2(R)\right]^{3/2} } ~,
\end{equation}
where $\sigma_h(R)$ is the DM one-dimensional velocity dispersion in
the halo as a function of radius, assuming an isotropic model, and $m_b$ is the mass of each PBH. The
DM phase-space density bound to the star is assumed to have a constant
value $Q=f_s Q_g$, where $f_s$ is a dimensionless constant of
order unity that depends on the velocity of the star, and is used also to absorb the difference between the detailed velocity distribution of the NFW profile at a specific radius $R$ and a Gaussian distribution. The maximum amount of bound
DM will be acquired when the star is at rest relative to the mean
surrounding DM, but typically the star is moving at an
rms velocity $\sim \sqrt{3}\sigma_h$ and will be acquiring bound DM at the
phase-space density near the star velocity.

The density profile of bound DM, $\rho_{bd}(r)$, after the star of mass
$M_*$ is formed, at a distance $r$ from the star, is determined by an
approximately constant DM phase-space density up to a maximum of the escape velocity
relative to the star $v=(2GM_*/r)^{1/2}$, and is therefore given by
\begin{equation}
\label{eq:rbd}
 \rho_{bd}= Q m_b \int_0^{\sqrt{2GM_*/r}} 4\pi v^2\, dv =
 \frac{4f_s}{3 \sqrt{\pi} }\, \frac{\rho_h(R)}{\sigma_h(R)^3}\,
 \left(\frac{GM_*}{r} \right)^{3/2} ~.
\end{equation}
Therefore the DM that is left bound around the star has a density
profile proportional to $r^{-3/2}$, with a normalization that is
determined by the density and velocity dispersion of the dark matter
halo at the formation site of the star, and the dimensionless factor of
order unity $f_s$. In this paper we will consider only the capture of
this bound DM, and neglect any unbound DM that may be also be captured
by the star when traversing it, if the incoming velocity is low enough
to allow capture after a single passage through the star. From \citet{2009ApJ...705..659A, 2013PhRvD..87b3507C} we expect the total capture rate should generally be dominated by this
originally bound DM, especially when taking into account the effects of likely
external perturbations on the DM orbits around the star that can
randomly change orbital eccentricities.

We compute the DM phase-space density in our model using $\sigma_h(R)$ from equation (13) of \citet{2001MNRAS.321..155L}, for the isotropic case. For reference, we show in Figure \ref{fig:sigma_halo} the computed one-dimensional velocity dispersion $\sigma_h(R)$, for our fiducial model of a halo mass $M=10^7\, {\rm M_{\odot}}$ at $z=20$.

\begin{figure}
	\includegraphics[width=\columnwidth]{Pictures/Sigma_halo.pdf}
    \caption{One-dimensional velocity dispersion of an isotropic NFW halo of mass $M=10^7 \, {\rm M_{\odot}}$ at $z=20$,
    used to compute the DM phase-space density around a star formed at a radius $R$ in the halo according to equation (5).}
    \label{fig:sigma_halo}
\end{figure}

\subsection{Black Hole capture: dynamical friction inside the star}\label{sec:PBH_capt}

Once the PBH is in a bound orbit around the star, the mechanism to dissipate its orbital energy and gradually fall to the core of the star is dynamical friction when it traverses the stellar interior. This requires the orbital eccentricity to be close enough to one for the periastron to be in the stellar interior.

To compute the dynamical friction effect accurately, a stellar model for the interior density, temperature, and sound speed profiles must be used, which will be described below in section \ref{sec:Stellar}.
The orbit external to the star is purely Keplerian, characterized by the specific energy $u$ and angular momentum $\ell$,

\begin{align}
    \ell =

    \sqrt{GM_{\ast} a \left(1-e^2\right)} ~,

    \label{eq:mom}
\end{align}
\begin{align}
    u = \frac{-G M_{\ast}}{2a} = \frac{1}{2}v^2 - \frac{GM_*}{r} \;,
    \label{eq:ener}
\end{align}
where $e$ is the eccentricity and $a$ the semimajor axis.

To compute the PBH capture rate, we need to calculate the time required for capture as a function of the semimajor axis and orbital eccentricity. This is determined by the amount of energy the PBH loses per stellar crossing. As long as the PBH periastron is within the stellar radius, the energy dissipation process continues with a decreasing semimajor axis and orbital period, and more frequent crossings that result in faster energy dissipation.

Therefore, to compute the time required for capture we do not need to worry about the detailed evolution of the energy and angular momentum of the orbit as the PBH is gradually slowed down through a large number of stellar crossings, because the semimajor axis remains much larger than the periastron, implying a negligible change in the part of the orbit in the stellar interior,
over most of the capture process time. In fact, we can simply compute the loss of energy over one stellar crossing by integrating over the unperturbed orbit of the PBH through the interior of the star, because the change in orbital energy is very small in a single stellar crossing compared to the kinetic energy of the PBH in the stellar interior.

We also approximate dynamical friction to depend only on the local plasma density and sound speed in the PBH vicinity (this is equivalent to neglecting the contribution to the Coulomb logarithm from the largest distances, comparable to the stellar radius, where these thermodynamic variables have substantial variation). Then, the dynamical friction acceleration $a_{df}$ is always in the direction opposite to the orbital velocity $v$, and the rate of energy loss is $du= - a_{df}\, v\, dt$, so the loss of specific orbital energy per stellar crossing is given by
\begin{equation}
  \Delta u= - \int_{t1}^{t2} a_{df}\, v \, d t = - 2\int_q^{R_{\ast}} a_{df} \, v \frac{d t}{d r}\, dr ~,
\end{equation}
where the first integral is from the time $t_1$ when the PBH enters the star to the time $t_2$ when it exits it. In the second integral, we change variables to the radial coordinate $r$ and use the fact that the integral contains two symmetric parts, from the stellar radius $R_*$ to the periastron $q$ and viceversa.

As explained before, we compute this integral for the unperturbed orbit of the PBH moving in the gravitational field of the star without the dynamical friction, because the modification of the orbit in a single passage is very small. The radial time derivative is related to the conserved specific angular momentum as $dr/dt=v[1-(\ell/rv)^2]^{1/2}$, so the above integral becomes
\begin{equation}\label{eq:Du}
  \Delta u = - 2\int_q^{R_{\ast}} \frac{a_{df}}{\sqrt{1-(\ell/rv)^2}}\,  dr ~.
\end{equation}

The acceleration caused by dynamical friction is the quantity that depends on the stellar interior model. A first approximation one can use is
Chandrasekhar's formula \citep[][]{1949RvMP...21..383C}, but this is valid for collisionless matter only, which does not apply to the interior plasma in stars. Instead, the adequate computation to use is for a collisional fluid, as presented by \citet[][]{1999ApJ...513..252O}. This fluid friction is close to the collisionless formula for a Mach number $M_a=v/c_s > 2$, where $c_s$ is the sound speed, but is substantially larger  for $1<M_a<2$, a common value for the capture process because the free-fall speed of the PBH is larger than the sound speed in the stellar interior by a small factor (as an example, the escape velocity at the surface of the Sun is $v\approx 615\, {\rm km/s}$ compared to a typical sound speed of $c_s \sim 350 \, {\rm km/s}$ \citep{2001ApJ...555..990B}). For the collisional case, we will actually use a formula from a simplified analytic estimate by \citet{2016A&A...589A..10T}, obtained from numerical simulations.

In general, the dynamical friction can be written as
\begin{align}\label{eq:adf}
  a_{df} = \frac{4\pi G^2 \rho_s m_b}{v^2} \, I(v,\Lambda) ~,
  \end{align}
where $\rho_s$ is the density of the star, $m_b$ is the PBH mass, and the dimensionless function $I(v,\Lambda)$ contains the detailed physical dependence on the velocity dispersion and Coulomb logarithm for the collisionless or collisional cases. The Chandrasekhar formula is written as
  \begin{align}
 I_C(v, \Lambda) = \left[ \, {\rm erf} \left( \frac{v}{\sqrt{2}\sigma_s}\right) -\frac{2 v}{\sqrt{2\pi}\sigma_s}\, \exp\left(-\frac{v^2}{2\sigma_s^2} \right) \right]\, \ln\Lambda ~,
    \label{eq:Chandra}
  \end{align}
while for the fluid case, the equation from \citet{2016A&A...589A..10T} is
  \begin{align}
 I_T(v, \Lambda) = {\rm ln} \left[ 2\Lambda \left(1-1/M_a^2 \right) \right] ~,
    \label{eq:Thun}
  \end{align}
where erf is the error function, $\sigma_s = (3kT/\mu)^{1/2}$ is the three-dimensional velocity dispersion for the plasma, and $\Lambda = R_{\rm max}/R_{\rm min}$ is the Coulomb logarithm. The lengths $R_{\rm max}$ and $R_{\rm min}$ are the usual maximum and minimum impact parameters for an effective gravitational interaction to produce dynamical friction, while $T$ is the temperature, $k$ the Boltzmann constant, and $\mu$ the mean particle mass. Equation (\ref{eq:Thun}) is valid for $M_a>1$, which is generally correct for an object moving near the escape speed in the stellar interior. In addition, the factor of $2$ in this equation is a numerical result obtained for an adiabatic index of $5/3$ for the plasma, the value for a monatomic gas that we assume here.
We use the radius of the star for $R_{\rm max}$, and $R_{min}=(2Gm_b)/v^2$. Below $R_{\rm min}$, the deflection of gas by the PBH gravity is much less effective at slowing it down; in particular, any accretion into the PBH is negligible as far as the rate of slowing down its velocity is concerned.

 \subsection{Number of Captured Black Holes}\label{sec:PBH_num}

If a PBH follows the trajectory determined only by the gravity of the star and the dynamical friction when it traverses the stellar interior calculated in the previous subsection, it will slow down gradually and reduce its semimajor axis until the orbit moves entirely to the interior of the star and the PBH settles on the stellar core. For the PBH to complete the process of orbital energy loss during the present age of the Universe, the initial orbital period has to be short enough so that the energy loss at each passage can add up roughly to the initial orbital energy.

As in previous work \citep[e.g.,][]{2013PhRvD..87b3507C},
we assume the orbital energy loss at each crossing of the stellar interior, $| \Delta u|$, is small compared to $| u|$, so we can integrate the evolution of the orbital energy with time $t$ with the simple equation
  \begin{equation}
      du = \Delta u \, \frac{dt}{P} = \Delta u \frac{dt}{P_0} \left( \frac{u}{u_0} \right)^{3/2} ~,
  \end{equation}
where $u_0$ is the initial orbital energy and $P_0=\pi GM_* (|u_0|)^{-1}\, (2|u_0|)^{-1/2}$ is the initial orbital period. Solving this equation, we find the time $t_c$ required to capture the PBH to the stellar interior (i.e., to reduce the semimajor axis to a value much smaller than the initial one) is
\begin{equation}\label{eq:tc}
    t_c = P_0\, \frac{2u_0}{\Delta u} ~.
\end{equation}
Note that as long as the semimajor axis remains much larger than the stellar radius, the trajectory of the PBH through the stellar interior at each crossing remains nearly the same, so $\Delta u$ stays almost constant and this simple solution is a good estimate. The detailed orbital evolution during the late stages of the capture, when the semimajor axis becomes comparable to the stellar radius, do not matter because the orbital period is then very short and capture takes a short time to be completed. Most of the required time for capture is at large semimajor axis, close to its initial value. In our model we still compute the time and energy loss for each orbit allowing for $\Delta u$ to vary until the orbit is wholly within the star, but the analytic estimate gives results very close to the ones we found.

For each initial orbital energy $u_0$, there is a critical value of the eccentricity $e_c(u_0)$ that makes the capture time $t_c$ equal to the present age of the Universe. The condition for the PBH to be captured (in the absence of other orbital perturbations, which we will discuss below) is then that for a fixed energy the orbital eccentricity is larger than this critical value. We assume the distribution of eccentricities follows the thermal distribution that is implied when the phase-space density is constant, so the probability for the eccentricity to be above $e_c$ is $1-e_c^2$.

We can now calculate the total number of PBH that will be captured by the star over a time $t_c$. At a fixed radius $r$ from the star, the number density of PBH with phase-space density $Q$ that will be found with an orbital semimajor axis $a$, and therefore orbital energy $u=-GM_*/(2a)$ and velocity $v=(2GM_*/r-GM_*/a)^{1/2}$, is
\begin{equation}
    n_b(a)\, da= 4\pi Q v^2\, dv = {2\pi Q} \,(GM_*)^{3/2}\,
    \left(\frac{2}{r} - \frac{1}{a} \right)^{1/2}\, \frac{da}{a^2} ~,
\end{equation}
where we have used $v\, dv= GM_*\, da/(2a^2)$ at fixed $r$. Integrating over the volume, from $r=0$ to $r=2a$ and replacing $x=r/a$, the total number of PBH with semimajor axis $a$ is found to be
\begin{equation}
    N_b(a)\, da= {8\pi^2 Q}\, (GM_*)^{3/2}\, a^{1/2}\, da\, \int_0^2 dx\, x^2\,
    \sqrt{\frac{2}{x}-1} ~,
\end{equation}
which yields
\begin{equation}
    N_b(a)\, da= {4\pi^3 Q}\, (GM_*)^{3/2}\, a^{1/2}\, da ~.
\end{equation}
Finally, the total number of captured PBH is expressed as
\begin{equation}\label{eq:ncap}
    N_c = \frac{\sqrt{2\pi^3}\, f_s \rho_h}{m_b\sigma_h^3} (GM_*)^{3/2} \int_0^{a_m} \left[1-e_c^2(a)\right] \sqrt{a}\, da ~,
\end{equation}
where $a_m$ is the maximum semimajor axis allowing capture within the age of the universe, at which $e_c(a_m)=1$, and we have used equation (\ref{eq:rbd}) to express the phase-space density in terms of the DM density and velocity dispersion around the star at its formation time.

The critical eccentricity is related to a critical extrapolated periastron, $q_c(a)=a[1-e_c(a)]$, where the true periastron is larger than $q_c(a)$ because of the reduced gravitational potential in the stellar interior compared to the Kepler one, owing to the extended mass distribution. Typically, $q_c(a)$ is of order the radius of the stellar core, where the stellar density is close to the maximum, at the values of $a$ close to $a_m$ that dominate the contribution to the integral in equation (\ref{eq:ncap}). We define the effective mean capture periastron, $\bar q_c$, as
\begin{equation}\label{eq:barqc}
    \bar q_c\, 2\sqrt{a_m} = \int_0^{a_m} q_c(a)\,
    \frac{da}{\sqrt{a}} ~.
\end{equation}
The captured number of black holes is then, approximating $1-e_c^2\simeq 2(1-e_c)$,
\begin{equation}
    N_c= \frac{2(2\pi GM_*)^{3/2} f_s \rho_h}{m_b \sigma_h^3}\, \bar q_c \sqrt{a_m} ~.
    \label{eq:Capture_rate}
\end{equation}
It is also useful to express this in terms of fiducial values,
\begin{equation}
    N_c= \frac{f_s \rho_h/m_b}{10^{14} {\rm pc}^{-3} }\,
    \left( \frac{M_{\ast}}{\, {\rm M_{\odot}}} \right)^{3/2} \left(\frac{10\, {\rm km/s}}{ \sigma_h}\right)^3 \frac{\bar q_c}{0.05 R_{\odot}} \sqrt{ \frac{a_m}{\rm pc}} ~.
    \label{eq:generalized_capture_rate}
\end{equation}
From equations (\ref{eq:densh}), (\ref{eq:sigmah}) and (\ref{eq:rho_vir}), we find that for a star formed at radius $R\sim 0.1 R_v$, a typical density $\rho_h\sim 10 \, {\rm M_{\odot}}\,{\rm pc}^{-3}$ is expected, so for PBHs of mass $m_b=10^{-12}\, {\rm M_{\odot}}$ that are the DM and follow a thermal eccentricity distribution, a star
of mass close to $\, {\rm M_{\odot}}$ would have a probability of order 0.1 to capture a black hole during the age of the Universe if $a_m$ is as large as a parsec.

It is useful to estimate at this point a rough value for $a_m$ for some fiducial parameters. The change in orbital energy per stellar crossing can be approximated, from equations (\ref{eq:Du}) and (\ref{eq:adf}), as
\begin{equation}
    \Delta u \sim \frac{8\pi G^2\rho_c r_c m_b}{v_e^2} \ln \Lambda ~,
\end{equation}
where the stellar density near the core is $\rho_c\sim 100\, {\rm g\, cm}^{-3}$, the stellar core has size $r_c\sim 10^{10}\rm cm$, $v_e\sim 1000 \rm km/s$ is the escape velocity from the core, and $\Lambda=R_{\rm max}/R_{\rm min}\simeq R_*v_e^2/(2Gm_b) \sim M_*/m_b$, so we find
\begin{align}\label{eq:anest}
    \Delta u\sim ~ 6\times 10^{-5} \frac{\rho_c r_c}{ 10^{12}{\rm g\, cm^{-2}} }\left(\frac{10^3\, {\rm km/s}}{ v_e}\right)^2  \frac{m_b}{10^{-12}\, {\rm M_{\odot}}}\, \frac{{\rm km}^2}{{\rm s}^2 } ~.
\end{align}
Precise calculations of $\Delta u$ will be presented in Section \ref{sec:Results} for specific stellar models. For a capture time $t_c= 10^{10}\, {\rm yr}$ and $M_*=\, {\rm M_{\odot}}$, the maximum semimajor axis at which equation (\ref{eq:tc}) is obeyed is
\begin{align}\label{eq:ams}
    a_m\sim (2\, {\rm pc})\,
    \left( \frac{\rho_c r_c}{10^{12}{\rm g\, cm^{-2}}} \frac{m_b}{10^{-12}\, {\rm M_{\odot}}}
    \right)^2 \left(\frac{10^3\,{\rm km/s}}{v_e}\right)^4 ~.
\end{align}
However, the ideal case of a Keplerian orbit around the single star of mass $M_*$ is not realistic for the large values of $a_m$ implied for the typical parameters in equation (\ref{eq:ams}), because tidal perturbations by the host DM halo and possibly other factors perturb the orbit, as we discuss next.

\subsection{Impact of Tidal Perturbations on the Capture Rate}\label{sec:Perturbations}

We have so far assumed that the PBH moves in a Kepler orbit around the star of mass $M_*$ with no gravitational perturbations. This assumption is clearly unrealistic for a PHB mass as low as $m_b\sim 10^{-12}\, {\rm M_{\odot}}$, because the tidal acceleration caused by the host DM halo, $g_h$, at the maximum semimajor axis $a_m$ is
\begin{equation}
    g_h\sim \frac{GM_h a_m}{R^3} \sim g_* \frac{M_h a_m^3}{M_* R^3}~,
\end{equation}
where $g_*=GM_*/a_m^2$ is the gravitational acceleration from the star on the PBH. For $M_h/M_*\sim 10^6$, the external tidal acceleration is larger than $g_*$ at $a_m> 0.01 R$. Taking as an example a typical halo radius where the star is located as $R\sim 0.1 R_v\sim 10\, {\rm pc}$, we would expect any dark matter further than $\sim 0.1$ pc from the star to actually be tidally disrupted from the host halo tide.

Moreover, the external tide perturbs the orbital eccentricity of any PBH, deviating it from the nearly radial orbit required to cross the stellar interior. The change in orbital eccentricity over one period induced by the external tide is related to the change in specific angular momentum as
\begin{equation}
    \delta e\simeq \frac{\sqrt{1-e^2} \delta\ell}{e \sqrt{GM_* a} } ~.
\end{equation}
We can reasonably assume that the external perturbation causes a change $\delta \ell / \sqrt{GM_* a} \sim g_h/g_* $ over one period, so for nearly radial orbits ($1-e \ll 1$), the eccentricity perturbation in one orbit is
\begin{equation}\label{eq:deg}
    \delta e \sim \sqrt{2(1-e)}\, \frac{g_h}{g_*} ~.
\end{equation}
At the same time, the minimum eccentricity at each semimajor axis $a$ required for the PBH to be effectively slowed down as it crosses the stellar interior is $1-e_c(a)=\bar{q}_c(a)/a$. This small window of eccentricity was denominated the ``loss-cone'' in \citet{1976MNRAS.176..633F} to refer to the region in velocity space where an orbiting object is lost because of the interaction with the central object, but we use the term ``loss-cylinder'' here because of the cylinder shape of this velocity space region. When $\delta e > 1-e_c$, the PBH is removed from the loss-cylinder. This clearly has the effect of decreasing the capture rate at large semimajor axis.

However, at small semimajor axis the PBH capture rate can be increased by the external perturbations. The reason is that in the absence of perturbations, PBHs that are initially at $a\ll a_m$ and within the loss-cylinder are captured over a time $t\ll t_c$, so the loss-cylinder is depleted and further captures can occur only from PBH near the edges of the loss-cylinder that cross the star through the low-density envelope, with reduced friction and energy loss. When perturbations are present, the loss-cylinder is refilled and the capture rate increases back to its most effective rate.

For orbits of semimajor axis $a$, the ideal orbital perturbation rate that leads to the maximum PBH capture rate is that which produces, over a time $t_c$, a change in eccentricity of
\begin{equation}\label{eq:De}
    \Delta e \sim \frac{\bar q_c a_m^{1/2}}{a^{3/2} } ~,
\end{equation}
because the time required for the PBH to lose its orbital energy at $a$ when the periastron is within $\bar q_c$ is only $t_c\, (a/a_m)^{1/2}$, so the PBH will be captured if perturbations induce a random-walk of the eccentricity within the characteristic interval $\Delta e$ over time $t_c$.

In general, orbital perturbations can greatly reduce PBH captures from $a\sim a_m$ because PBH are always removed from the loss-cylinder before they are captured, but as $a$ is decreased, the interval $\delta e$ over which the eccentricity random-walks over time $t_c$ decreases until it equals $\Delta e$ in equation (\ref{eq:De}). At this value of $a$, the PBH capture rate will be roughly the same as it was at the maximum semimajor axis $a_m$ in the absence of perturbations. Therefore, we conclude that despite the presence of orbital perturbations, the PBH capture rate will always be roughly the same as obtained from equation (\ref{eq:ncap}). Perturbations imply that most PBH are actually captured from orbits much closer to the star than $a_m$, but the total capture rate should not be greatly modified.

It is possible that the bound DM around the star has been tidally stripped at some time down to a semimajor axis $a\ll a_m$, and subsequently any external perturbers are removed so that perturbations are absent but the PBH are no longer available at large $a$. In this case  $a_m\sim 0.1 pc$, so the capture rate may obviously be greatly reduced. As we expect the capture rate to have a dependence of $a_m^{1/2}$ at larger $a$, when the change in $\bar{q}_c$ becomes negligible, this should result in a capture rate less than one order of magnitude smaller compared to previous $a_m\sim 1 \, {\rm pc}$. We will assume here that enough DM has been retained to maintain the capture rate close to the value computed with equation (\ref{eq:ncap}) in the presence of tidal perturbations by refilling of the loss-cylinder.

\subsection{Black Hole Growth after Capture}

Once the PBH has settled in the center of the stellar core by the continuous action of dynamical friction, it will start growing in mass by accreting the surrounding stellar plasma. As discussed previously by \citet{
1995MNRAS.277...25M,
2009arXiv0901.1093R,
2009MNRAS.399.1347B,
2019JCAP...08..031M}, the PBH can accrete rapidly at the Bondi accretion rate if photons are trapped with the accreting plasma, and proceeds more slowly if an accretion disk is formed that can radiate efficiently and slow down the accretion by emitting close to the Eddington luminosity. The Bondi accretion rate is
\begin{align}
\frac{\Dot{M}_{\rm B}}{m_b } = \, & \frac{\pi G^2 m_b \rho_s}{c_{s}^3} \simeq
 3.24 \cdot 10^{-6} \, {\rm yr}^{-1} ~ \times \nonumber \\
 & \frac{m_b}{10^{-12}\, {\rm M_{\odot}}} \frac{\rho_s}{100 \, \rm g/cm^3} \left(\frac{c_{s}}{300 \rm km/s}\right)^{-3} ~,
 \end{align}
 where $\rho_s$ and $c_s$ are the plasma density and sound speed in the stellar center. We see that at the Bondi accretion rate and typical values presented above, even an initial PBH mass as low as $10^{-16}\, {\rm M_{\odot}}$ will grow its mass in less than $10^{10}$ years.
 The growth rate becomes faster as the mass increases so the PBH can accrete all the stellar mass if Bondi accretion continues. If accretion becomes Eddington-limited at some stage due to the angular momentum of accreting matter and formation of an accretion disk, the accretion rate becomes
 \begin{align}
\frac{\Dot{M}_{E}}{m_b} = \frac{4\pi G m_{p}}{c\;\eta\;\sigma_{T}} \simeq  2.2\cdot 10^{-8} \frac{0.1}{\eta} \rm yr^{-1} \, ,
 \end{align}
where $\rm m_{p}$ is the proton mass, $\eta$ is the radiative efficiency of the accretion disk matter and $ \sigma_{T}$ the Thompson cross-section. In this case, the constant e-folding time for mass growth is also much less than the present age of first stars $t_c\simeq 10^{10}$ years. Therefore, the PBH will continue to grow until it has accreted a substantial mass of the star.

If Bondi accretion continues, the star would only start being dynamically affected by the black hole accretion a few days before being completely accreted \citep{1995MNRAS.277...25M}. On the other hand, if an Eddington luminosity is emitted the total stellar luminosity will obviously be dominated by the accretion already when $m_b \sim 10^{-5}\, {\rm M_{\odot}}$ for $M_*=1\, {\rm M_{\odot}}$. At this late stage, formation of a jet or other mechanical energy release resulting from accretion may result in the ejection of the stellar envelope, and the end of PBH accretion, but the details of this final process are complex and it is not clear what the final mass of the PBH will be. Depending on stellar rotation and perhaps other stellar properties, the final mass may be close to the initial mass of the star, or may be much lower if mechanical energy can disperse the stellar material.

\subsection{Stellar Models} \label{sec:Stellar}

In the present work, we consider low-mass stars formed at high redshift in low-mass DM halos. Low-mass stars live for a long time ($\gtrsim 10^{10}$ yrs) and thus might have a considerably high probability of capturing a PBH. Consequently, they might form low-mass black holes that, if discovered, would point to a formation process beyond the standard stellar formation channels.
Low-mass stars were expected to form soon after the first metal-free stars, once the first supernovae increased the heavy element abundance of the gas enough to lead to fragmentation and collapse of low-mass cloud cores
\citep{2021MNRAS.508.4767S, 2022MNRAS.510.4019P}. The concept of critical metallicity, that is, a minimum metallicity below which low-mass stars could not form \citep{2005IAUS..228..121B} seemed to have observational support \citep{2007MNRAS.380L..40F}. However recent observations of metal-poor stars in the halo keep pushing this low threshold to increasingly low values. The star by \citet{2014Natur.506..463K} holds the current record, with a metallicity [Fe/H]=-7.1 .

Even in the primordial metal-free gas, disks around massive stars might fragment into low-mass objects that form stars in the range $0.1$ to $1 \, {\rm M_{\odot}}$ \citep{2002ApJ...569..549N, 2020ApJ...901...16D, 2011Sci...331.1040C,2015MNRAS.447.3892H, 2014ApJ...792...32S, 2022ApJ...925...28L}, though the Initial Mass Function (IMF) of the most primitive stars is still largely uncertain\citep{2016MNRAS.462.1307S, 2015MNRAS.447.3892H}. Any such stars formed near the center of a low-mass DM halo are the best candidates for PHB capture.

In this study, we use models of very metal poor stars,  that is, of $Z =10^{-4}$. This is the typical metallicity at which we expect the transition from a top-heavy to a bottom-heavy IMF to happen \citep{2013MNRAS.432L..46S,2019ffbh.book...67K, 2021MNRAS.503.2014S}. Six different stellar models with stellar masses in the range $0.32\, {\rm M_{\odot}} < M_* < 1 \, {\rm M_{\odot}}$, were computed for the present study with the open-source software instrument Modules for Experiments in Stellar Astrophysics (MESA) \citep{2011ApJS..192....3P, 2013ApJS..208....4P, 2015ApJS..220...15P, 2018ApJS..234...34P, 2019ApJS..243...10P}.
These models provide the stellar density, temperature, sound speed and mean atomic weight as functions of the radii for models ranging from the zero age main-sequence till times above the age of the universe, from which the loss of energy per orbit depending on orbital energy and angular momentum can be calculated as described in the previous subsections.

A precise calculation of the PBH capture rate for a given stellar mass would involve averaging the capture rate over all ages, from the stellar birth to the present time. Instead of this, we present results for six cases of fixed stellar mass and age, assuming a constant stellar profile over the time $t_c$ at a fixed age. In practice, stars are in the main-sequence most of the time and low-mass stellar evolution is very slow, so this is a good approximation as long as we use an age when the star has already settled on its main-sequence. However, for stars of $M< 0.4 \, {\rm M_{\odot}}$ the time required to stabilize near the main-sequence is as long as $\sim 10^9$ years, which causes a complex dependence of the central stellar density on mass if an early age is used. To illustrate the dependence on both mass and age, we present results for the six stellar models with masses
$1$, $0.79$, $0.63$, $0.5$, $0.4$, and $0.32 \, {\rm M_{\odot}}$, and ages $1.5$, $3.2$, $1.3$, $1.3$, $10$, and $10$ Gyr, respectively. The density and temperature radial profiles of these six models are presented in Figures \ref{fig:Densstellmod} and \ref{fig:Tempstellmod}.

 \begin{figure}
  \includegraphics[width=\columnwidth]{Pictures/Stellar_models_density.pdf}
  \caption{Mass density profile for the six stellar models used in this paper, with stellar masses 1, 0.79, 0.63, 0.5, 0.4, and 0.32 $\, {\rm M_{\odot}}$, and ages 1.5, 3.2, 1.3, 1.3, 10, and 10 Gyr,  respectively.}
  \label{fig:Densstellmod}
\end{figure}
 \begin{figure}
  \includegraphics[width=\columnwidth]{Pictures/Stellar_models_temperature.pdf}
  \caption{Temperature profiles for the same stellar models as in Figure \ref{fig:Densstellmod}.}
  \label{fig:Tempstellmod}
\end{figure}

\section{Results for six stellar models}\label{sec:Results}

We now present specific results for our six stellar models, starting with the energy loss in a single stellar crossing, then the critical eccentricity for a given capture time of $t_c= 10^{10}$ years, and then the total number of captured black holes. Our results are presented for $m_b=10^{-12}\, {\rm M_{\odot}}$, but can be scaled to other PBH masses in the asteroid mass range as good estimates in the way described previously.

\subsection{Energy loss by dynamical friction} \label{sec:Dyn_fric_rlts}
Using equations (\ref{eq:Du}) and (\ref{eq:adf}), we can find the loss of energy over a single passage as a function of the PBH specific energy and angular momentum. As an illustration, we show this energy loss for the highest mass stellar model, with $M=1\, {\rm M_{\odot}}$ and $t=1.5$ Gyr.
Results are plotted as a function of specific angular momentum in
Figure \ref{fig:Eloss} for the collisionless case using equation (\ref{eq:Chandra}), and for the fluid case, using equation
(\ref{eq:Thun}), for four different values of the specific energy: the parabolic case with zero energy, an unbound case, and two bound orbits, with specific energies as indicated in the figures. The assumed PBH mass is $m_b=10^{-12}\, {\rm M_{\odot}}$, but note that the energy loss is proportional to $m_b$, from equation (\ref{eq:adf}).

\begin{figure}
  \includegraphics[width=\columnwidth]{Pictures/PBH_energy_loss_sel_1.pdf}
  \caption{Specific energy loss of a $10^{-12}\, {\rm M_{\odot}}$ PBH when crossing our model star with $M_*=1\, {\rm M_{\odot}}$ and $t=1.5$ Gyr, as a function of the orbital
  angular momentum, using both the Chandrasekhar dynamical friction formula for collisionless matter (Chandra, thin line) and the hydrodynamic formula (Thun, thick line) based on \citet{2016A&A...589A..10T} and \citet{1999ApJ...513..252O}. The four curves of each case are for the indicated specific orbital energies (zero energy is a parabolic orbit, and negative energy is for bound elliptical orbits).}
  \label{fig:Eloss}
\end{figure}

The energy loss $\Delta u$ in Figure \ref{fig:Eloss} agrees with the simple analytic estimate from equation (\ref{eq:anest}). When the PBH is captured from a semimajor axis much larger than the stellar radius, the relevant result for calculating the time it takes for the PBH to finalize the capture process is the one for the parabolic orbit, with zero total energy. The unbound case is never used for our computation (we neglect any contribution to the capture rate from unbound, incoming PBHs that were not placed in bound orbits when the star formed). The bound orbits show appreciable differences in $\Delta u$ in Figure \ref{fig:Eloss} only when the apoastron is already not much larger than the stellar radius; by then, the capture process is almost finished. The critical eccentricity $e_c(a)$ therefore depends, to an excellent approximation, on the zero energy curve for $a\gg R_*$. Note that the characteristic specific angular momentum where $\Delta u$ drops corresponds to $\sim v_e \bar q_c$, as we would expect from equation (\ref{eq:barqc}).

The hydrodynamical expression leads to an energy loss that is a factor $\sim 1.6$ larger than that obtained from the Chandrasekhar formula.

\subsection{Critical eccentricity for capture}

Using our stellar models we compute the critical eccentricity as a function of the initial semimajor axis $a$ required for the PBH to be captured over a time $t_c= 10^{10}$ years, assuming there are no gravitational perturbations to the potential of the spherical star of mass $M_*$. All results shown from now on are for the fluid case, which is the valid one for stellar interiors,
with the use of equation (\ref{eq:Thun}), but use of equation (\ref{eq:Chandra}) results in only minor changes. Equation (\ref{eq:tc}) can be used for this purpose to a very good approximation, except when $a$ is not very large compared to $R_*$, but here we have carried out the exact calculation of the orbital evolution up to the point where the PBH orbit is completely absorbed in the stellar interior. The result is shown in Figure \ref{fig:Critecc_1}, as the critical extrapolated periastron $q_c(a)=a[1-e_c(a)]$ (or the periastron the orbit would have if the star were replaced by a point mass), for our six stellar models. Note that the probability for a random orbit of semimajor axis $a$ to have a periastron below $q_c$ is $2q_c/a$, if $q_c \ll a$.

As expected, $q_c$ has a very weak variation with $a$ when $a\ll a_m$, and then drops sharply when $a$ becomes close to $a_m$.
The friction when the PBH moves through the dense stellar core determines the maximum semimajor axis $a_m$ where the energy loss allows the PBH to be captured in the time $t_c$. When the PBH starts at $a\ll a_m$, it can cross the star many times resulting in $q_c \sim R_{\ast}$, but if the crossing occurs in the outer envelope the friction is greatly reduced; this causes the slow decrease of $q_c$ with $a$.

In the absence of orbital perturbations, PBHs that start on an orbit with eccentricity $e> e_c=1-q_c/a$ are the ones that are inside the loss-cylinder and will therefore be captured by the stars, and those outside the loss-cylinder will not be captured in the time $t_c$.

\begin{figure}
	\includegraphics[width=\columnwidth]{Pictures/q_c_various_radius.pdf}
    \caption{Critical extrapolated periastron $q_c$ of a PBH of $10^{-12} \, {\rm M_{\odot}}$ as defined in section \ref{sec:PBH_num}, as a function of the initial semimajor axis $a$, for capture to occur in a time $t< 10^{10}$ years.

    }
    \label{fig:Critecc_1}
\end{figure}

Finally, we list the results of our calculation for the mean extrapolated pericenter $\bar q_c$ in Table \ref{tab:XiforR}, for our six stellar models and several values of $a$ within which the average is made. As previously seen in figure \ref{fig:Critecc_1}, the average $q_c$ will be of the order of the stellar radius but as we get closer to $a_{m}$,  $\bar q_c$ falls as PBHs at such distances need to cross the core to get captured. If trying to reproduce the capture rate for a lower $a$ it is enough to use the corresponding value of $\bar q_c$ in the table on equation (\ref{eq:generalized_capture_rate}) with the new value for $a_m$.

\begin{table}
	\centering
	\setlength\tabcolsep{4pt}
	\begin{tabular}{lcccccr}

    \multicolumn{6}{c}{}\\
    	\hline
		 $M_{\ast}\,  [\, {\rm M_{\odot}}]$ & 1 & 0.79 & 0.63 & 0.50 & 0.40 & 0.32 \\

		 $R_{\ast}\,  [\, {\rm R_{\odot}}]$ & 0.91 & 0.76 & 0.57 & 0.43 & 0.33 & 0.28\\

        $t\,  \rm [Gyr]  $ &  1.5 &  3.2  &  1.3 & 1.3  & 10 & 10 \\
    	\hline
    	 $a_{m}\,  [pc]$ &  0.98 &  0.93  &  0.88 &  0.83 & 0.78 & 0.72 \\
		 \hline
		 $a\,  [km]$ &   &  \  &  $\bar q_c\, (a) \,[\, {\rm R_{\odot}}]$ &   &  &  \\
		 \hline
        $5\cdot 10^8$& 0.64 &	0.63 &	0.52 &	0.40 &	0.31 &	0.27 \\
		\hline
		 $10^9$& 0.64 &	0.62 &	0.52 &	0.40 & 0.31	 &	0.27 \\
		\hline
		 $3\cdot 10^9$& 0.63 &	0.62 &	0.52 &	0.40 &	0.31 &	0.27 \\
		\hline
		 $10^{10}$& 0.59 &	0.58 &	0.51 &	0.40 &	0.31 &	0.27\\
		\hline
		$3\cdot 10^{10}$&0.58 &	0.56 &	0.51 &	0.40 &	0.31 &	0.27 \\
		\hline
		 $10^{11}$& 0.53 &	0.51 &	0.48 &	0.39 &	0.31 &	0.27 \\
		\hline
		 $3\cdot 10^{11}$& 0.50 &	0.48 &	0.45 &	0.38 &	0.31 &	0.27 \\
		\hline
		 $10^{12}$& 0.44 &	0.43 &	0.41 &	0.36 &	0.30 &	0.26 \\
		\hline
		$3\cdot 10^{12}$& 0.38 & 0.38 &	0.36 &	0.33 &	0.28 &	0.25\\
		\hline
		$10^{13}$& 0.32 & 0.34 & 0.31 &	0.30 &	0.27 &	0.23 \\
		\hline
		$a_{m}$& 0.25 & 0.27 & 0.25 &	0.25 &	0.25 &	0.21 \\
		\hline
	\end{tabular}
	\caption{Values of  $\bar q_c$ for $10^{-12} \, {\rm M_{\odot}}$ PBHs for various upper limits of $a$ and stellar models. $a_{m}$ is defined as the maximum a in which $e_c = 1$ results in capture within $t_c$, as discussed in section \ref{sec:Perturbations}, but we give $\bar q_c$ for various additional upper limits of $a$.}
	\label{tab:XiforR}
\end{table}

\subsection{Results for captured PBH}

\begin{figure}
	\includegraphics[width=\columnwidth]{Pictures/PBH_Capture_rate_all_Ostriker_mod.pdf}
    \caption{Mean number of PBHs captured by the star with initial semimajor axis within $a$, for the phase-space density of PBH at the halo radius $R=0.1 R_v$. The calculation is done for no external perturbations which we could expect to happen above the vertical red line but do not greatly alter the total capture rate, see main text for details.}
    \label{fig:PBH_capture_rate_all_stars}
\end{figure}

Figure \ref{fig:PBH_capture_rate_all_stars} shows the mean number of PBHs captured from orbits within an initial semimajor axis $a$. As expected from equation (\ref{eq:ncap}), this total number increases as $a^{1/2}$, except when $a$ is already close to $a_m$ and $q_c$ starts declining rapidly with $a$.

The assumption that there are no external tidal perturbations used for this calculation is not realistic, because even the tidal perturbation of the host DM halo becomes comparable to the acceleration by the star at $a\simeq 0.1$ pc, as discussed in section \ref{sec:Perturbations}. Even at the smaller radius $a\simeq 0.01$ pc (indicated by the vertical red line in the figure), where the tidal perturbation from the minihalo is $10^3$ times smaller than the stellar acceleration, the critical eccentricity is only $1-e_c=q_c/a\simeq 10^{-6}$, so the tidal perturbation can move the periastron outside the loss-cylinder in just one period, according to equation (\ref{eq:deg}).

However, while the perturbations eliminate any PBH captures from semimajor axes as large as $0.01$ pc, they should correspondingly increase captures from smaller $a$, as discussed in section \ref{sec:Perturbations}. As $a$ decreases, the loss-cylinder from which the PBH can be captured increases in width as $\delta e \propto a^{-3/2}$ if we assume that the PBH random-walks through the interval $\delta e$ owing to the tidal perturbations, spending a fraction of the total time $t_c$ proportional to $a^{1/2}$ in the region of width $\delta e_c\propto a^{-1}$ where the stellar core is crossed. This fraction of time is enough to capture a PBH that starts on an orbit at semimajor axis $a$. The eccentricity interval swept by the random-walk caused by an external tide narrows down as $a$ decreases, and when it coincides with the capture region with $\delta e\propto a^{-3/2}$, a total dominant capture rate is produced that is roughly independent of $a$ if the total number of PBH within $a$ increases as $a^{3/2}$.

Based on this argument, our calculation of the total capture rate shown in Figure \ref{fig:PBH_capture_rate_all_stars} should have a wide range of validity, even in cases where tidal perturbations are added from a variety of causes such as passing stars or crossing of galactic disks. The predicted total capture rate for a star that acquired its bound DM at birth at $R=0.1 R_v$, in our standard halo of $M_h=10^7\, {\rm M_{\odot}}$ at $z=20$, is $\sim 0.3$ PBH for the $1\, {\rm M_{\odot}}$ star, and 10 times lower for stars of $0.3\, {\rm M_{\odot}}$. This is a substantial probability for low-mass stars born at this high-redshift to have formed low-mass stellar black holes by the present time. Tidal perturbations may still reduce this probability, for example if the bound DM is first tidally disrupted and then perturbations cease when the star is ejected to a region of very low density. But, as we have argued, our calculation should be realistic for many of the low-mass stars formed at high-redshift.

Furthermore, many stars may form much closer to the halo center, increasing the phase-space density of the bound PBH acquired at birth. Figure \ref{fig:PBH_capture_rate_Halo} shows the result for the total capture rate, up to $a=a_m$, as a function of the initial halo radius $R$ where the star is formed, assuming the validity of our argument that tidal perturbations never reduce this rate. We assume the NFW profile with isotropic velocity dispersion, with the velocity dispersion profile shown in Figure \ref{fig:sigma_halo}, and the normalizing factor $Q$ for the bound DM density in equation (\ref{eq:rbd}), $Q \propto \rho_h/\sigma_h^3$. We see that the mean number of PBH captured actually reaches unity for stars born at $R\sim 0.03 R_v$, which is a typical formation radius for stars in present-day galaxies in galactic DM halos.

Considering PBH of different masses, we note that the specific energy loss in a stellar crossing is proportional to $m_b$, implying that lighter PBH are more difficult to capture, and that the maximum semimajor axis for capture $a_m$ increases as $m_b^2$ in equation (\ref{eq:ams}), so lighter PBH are captured only from smaller radii. However, for fixed $Q$ and therefore a fixed {\it mass} density of PBH, the number density is proportional to $m_b^{-1}$ and the total PBH capture rate is independent of $m_b$. We therefore conclude with the robust conclusion that if PBH over the broad asteroid-mass range account for most of the DM, a substantial fraction of low-mass stars formed at high redshift will inevitably form low-mass black holes after capturing a PBH and accreting onto them, and these low-mass stellar black holes should be present in the Universe today.

\begin{figure}
	\includegraphics[width=\columnwidth]{Pictures/PBH_Capture_rate_all_sigma_iso.pdf}
    \caption{Mean number of PBHs captured by a star as a function of halo radius of the stellar birth site, for a NFW halo with DM halo with isotropic velocity dispersion.}
    \label{fig:PBH_capture_rate_Halo}
\end{figure}

\section{Discussion and Conclusions} \label{sec:Conclusions}

Observational constraints for the abundance of PBHs have left only the asteroid-mass window, $10^{-16} \lesssim m_b/M_\odot \lesssim 10^{-11}$, as the one where all the DM may be composed of PBH \citep{2021RPPh...84k6902C}. This manuscript shows that if the DM is indeed made of these PBH, the first generation of low-mass stars formed at $z\sim 20$ in low-mass halos would have a high probability to make stellar black holes with masses less than a Chandrasekhar mass, through the process of capture of a PBH by the main-sequence star and the subsequent accretion and growth of the PBH.

The final black hole may reach a mass comparable to the initial stellar mass, with uncertainties related to the possible formation of an accretion disk around the growing black hole. The reason is that the radiative efficiency of such a disc might be able to hamper accretion by dispersing the stellar material when only a small fraction of the stellar mass has been accreted. Other uncertainties include the possible role of external tidal perturbations or internal ones due to, for example, a planetary system around the star. The PBH capture rate would also be heavily modified in binary stars \citep{2012PhRvL.109f1301B}. Further work will be needed to clarify some of these uncertainties; nevertheless, the calculations presented in this paper suggest as a likely outcome of this asteroid-mass PBH scenario that many low-mass stellar black holes formed in this process may exist today in the Milky Way, after having originated in early dwarf galaxies that later merged into our Galaxy.

This paper extends earlier works \citep{2009ApJ...705..659A,2009MNRAS.399.1347B,2013PhRvD..87b3507C} with
calculations that use models of very metal-poor main-sequence stars of low mass, an analysis of the impact of external tidal perturbations on the PBH capture rate (with an improved treatment compared to previous work, e.g. \citet{2013PhRvD..87l3524C, 2014PhRvD..90h3507C}), and its dependence on the PBH mass. We reach the remarkable conclusion that the capture of PBH by these low-mass stars formed in the early DM halos with highest phase-space densities should occur for most stars formed within a halo radius $R\sim 0.03 R_v$ (a typical location for star formation in galactic halos), over this whole asteroid-mass range for PBH.

If many of these early low-mass stars have indeed collapsed to low-mass stellar black holes, their remnants should be found today among the Milky Way stellar populations, because the early dwarf galaxies where they formed may merge into more massive halos and end up tidally disrupted in the Milky Way halo over a wide range of radii. Their spatial distribution in the Milky Way is not easy to predict: early simulations proposed that
the remnants of the most ancient stellar populations should be found near the centre of the Galaxy, owing to the high bias factor of DM halos formed at high redshift \citep{2000fist.conf..327W,2006ApJ...653..285S}, but more recent work has found that baryonic effects may imply a broader radial distribution over the outer halo \citep{2018MNRAS.480..652E}.

How could these low-mass stellar black holes be detected at present? The most obvious possibility is via gravitational wave emission, if the black hole is in a binary system that leads to a merger. Some binary systems might contain two stars that both collapsed to low-mass black holes below the Chandrasekhar mass; alternatively, one of these low-mass black holes might be left in orbit around a more massive star that collapses to a black hole or neutron star following the standard evolution. In both cases, these binaries could spiral down into a merger at the present time that is detectable by the LIGO-Virgo collaboration \citep{2015CQGra..32g4001L,2021arXiv211103634T}. The detection of a merging black hole with a mass below the Chandrasekhar value would naturally indicate new physics beyond standard stellar evolution theory, and has already been searched in recent gravitational wave searches \citep{2021arXiv210912197T}.

Another way of detecting these low-mass black holes is when they are observed in a binary system where the other object is a luminous star. For a main-sequence companion, a normal star would be seen orbiting around a dark object. The difficulty in this case would be to rule out an old white dwarf as the unseen companion, because white dwarfs are very faint and difficult to see when they are unresolved from the main-sequence star. If the companion is a white dwarf then the system is much fainter and difficult to discover, but it is then easier to rule out another white dwarf companion for an unseen compact object below the Chandrasekhar mass. The additional difficulty to accomplish this type of detection is that the low-mass black holes formed from PBH capture would be rare and present only among very low-metallicity stars, so a lot of these possible binaries would have to be examined, most of which would contain regular old white dwarf companions.

If the low-mass black hole that is formed by a PBH captured by a star is isolated, then it is extremely difficult to discover. Microlensing seems the only possibility \citep{1986ApJ...304....1P, 2000ApJ...542..281A}, but these black holes would actually pass for M-dwarfs, white dwarfs or brown dwarfs, depending on their mass. These other objects are all very faint and therefore usually not possible to distinguish from low-mass black holes, with a much lower expected abundance in our PBH scenario.

In summary, the open asteroid-mass window for PBH as DM is a possibility in which we expect that low-mass black holes of stellar mass, but below the Chandrasekhar limit, exist today. These can be found if they are in binaries that are either tight enough to lead to mergers with other compact objects detectable via gravitational waves at present, or through the direct detection of the binary companion in the Milky Way and identification of the unseen object as a low-mass black hole. These binaries are expected to be among the first stellar systems to have formed, and therefore of very low-metallicity, which may help in their identification among halo stars in our vicinity or closer to the Milky Way centre.

\section*{Acknowledgements}

We would like to acknowledge helpful discussions and advice from  N. Bellomo, J. L. Bernal, A. Escriv\`a, C. Germani, and J. Salvad{\'o}. This work was supported in part by Spanish grants CEX-2019-000918-M funded by MCIN/AEI/10.13039/501100011033, AYA2015-71091-P, and PID2019-108122GB-C32.

\bibliographystyle{mnras}
\typeout{}

redpajama_set_name: RedPajamaArXiv


Clays or charcoal? Which team will you join?
There are only two types of people – those who prefer taking a shower and those who'd rather go for a nice relaxing bath. Those who don't mind squeezing a tube of toothpaste wherever it suits them, and those who always, always squeeze from the end. Whichever one you are, one thing is clear: Garnier will help you find just the right products for your skin type!
Do you struggle with various imperfections such as spots, blackheads or oily skin? Garnier designed two unique skin care products specifically for your needs: Pure 3in1 with clays and eucalyptus and Garnier PureActive 3in1 with active charcoal. They differ in colour, ingredients and the skin problem they target, but their ultimate goal is the same: to give you clear, healthy looking skin without imperfections. They are suitable for both women and men and you can use them in three different ways – as a cleansing gel, exfoliator or a mask.
So now to the important decision. Clays or charcoal? White Pure 3in1 is formulated to eliminate imperfections in your skin, mainly spots. It leaves your skin perfectly clean and prevents it from producing an excess sebum thanks to natural clays. This product for daily skin care is also enriched with eucalyptus extract to soothe your skin.
On the other hand, black Pure Active 3in1 is an immediate solution for blackheads. Its formula contains active charcoal which binds impurities like a magnet and deeply cleanses problematic skin.
Share your experience with either of these little beauty helpers and follow us on Instagram on @notino_co_uk @garnieruk.


text:
Crimes Against People, Crimes in Concert, Federal, State
What is Aiding & Abetting?
Aiding & Abetting is in the same family as Conspiracy. Think of this as being charged for helping someone else commit a crime, even just a little bit. They may not have had the original idea to do the crime, but at some point found out about it, and helped it get accomplished. In order to convict someone for aiding and abetting the commission of a crime, the Government must prove that the defendant, before or at the time the crime was committed
Knew that the crime was being committed or going to be committed;
Had enough advance knowledge to the extent and character of the crime that they were able to make the relevant choice to walk away from the crime before all of the elements of it were complete; and
Knowingly acted in some way for the purpose of causing, encouraging, helping, or aiding the commission of the crime.
Furthermore, the Government also has to prove beyond a reasonable doubt that all of the elements of the crime were committed by other person or people, and that the defendant affirmatively aided and abetted the commission of that crime. Just merely being associated with people committing crimes or being present at a scene where a crime was committed doesn't count.
An example of when it did work out for the defendant in State Court:
Where it didn't work out so well for the defendant in Federal Court:
A woman named Helen went on trial for aiding and abetting first degree murder. She had moved in with her son David, and his wife Carol, and their children. But it was not going well. Carol and David's marriage was failing and the mother-in-law and daughter-in-law did not like or trust each other. It was a toxic situation. David told Helen, along with his friends, that he would kill Carol and make it look like an accident. Eventually he choked Carol to death and later admitted it. The state charged Helen with helping with the murder because David told investigators that he told Helen that day that he was going to kill Carol and when he finally did it, Helen was home when he got rid of the body. Ultimately on appeal, the Minnesota Supreme Court reversed the conviction, finding that her actions did not constitute conduct that encouraged another to act; while she disliked Carol, she did not advise David to murder her; and her presence in the house wasn't enough to satisfy the elements of aiding and abetting. That is the case of State v. Ulvinen 313 N.W.2d 425 (Minn. 1981).
Two men were pulled over in a car for a traffic offense. A trooper ended up searching the car and finding a large bag of methamphetamines. The driver swore up and down that he had no knowledge of the drugs in the car, but was charged with aiding and abetting drug distribution and ultimately convicted. The court found that because the driver knew that the passenger was a drug dealer, the purpose of the trip was to collect drug money, and that he had a small amount of meth in his sock, he was guilty of aiding and abetting drug distribution. That is the case of United States v. Santana, 524 F.3d 851 (8th Cir. 2008).
Catherine is an experienced criminal defense lawyer that defends against charges involving Aiding & Abetting. If you or a loved one have been accused of Aiding and Abetting, she can help. Call today for a free consultation.
abetting, aiding, Aiding & Abetting, Crimes Against People, defense, drug dealer, drugs, enhanceable offenses, Federal, female criminal, first degree murder, lawyer, methamphetamines, minnesota, State, traffic offense

redpajama_set_name: RedPajamaCommonCrawl


Both products in issue are iMessage and VPN On-Demand. The obvious infringements identified in VPN protection came to about $ 335 million, as well as in addition, the many infractions identified in face-time led to yet another $ 290 million. Read more about VPN in this vpncomparison.
Named as a "patent troll," VirnetX does not have have any commodity, plus it generates money through licensing patents and following-up with law suits. Instances in this way are not totally fresh, as they still happen every so often, particularly in Eastern Texas.
In accordance with the judgement form (PDF), the court ruled that Apple really infringed on the patent statements offered, showing the amounts of cash as well as the general agreement of the jurors.
The Cupertino manufacturer stated that it is going to challenge the consensus when it had been requested to get a remark.
"We're amazed and disappointed from the judgement and we will charm. Our workers alone developed this technology over a long time, and we obtained patents to protect this intellectual property. All of VirnetX's patents are found unacceptable from the patent office. Instances in this way just augment the urgent dependence on patent reform," an Apple representative claims.
That isn't the first moment that Apple and VirnetX have confronted each other in court, and this newest situation is really a rehashing of a dispute that times right back to 2012. The court in the moment dominated that Apple ought to pay about $ 368 thousand in damages. Nevertheless, a U.S. appeals court determined that the jurors didn't compute the amount right, organizing however yet another demo.
To be particular, the patents included comprise the 1998 U.S. obvious 7921211, where the the master title is recorded as "Research Programs Global Company." In 2006, the stated evident was paid to VirtnetX.
Meanwhile, the U.S. evident 6502135 that contains the name "Nimble community proto-col for secure communications with guaranteed program accessibility" is the obvious that was at issue in the 2012 test. In spite of the purchase of the U.S. appeals courtroom for a re-scheduled day, the recent advancement suggests that this was re-used, where VirtnetX won the same amount of funds with the first demo.
What this requires is the judgement will impact the obvious reforms scenario, where the federal courtroom in Eastern Texas could possibly function as topic of examination for arriving obvious tests.
In accordance with experts, the regulators in the condition have assembled a sanctuary for evident-keeping businesses like VirtnetX, where the tribunals in the condition may find considerate juries that may let such plaintiffs to acquire a money settlement. That mentioned, many businesses would rather to stay rather of taking expensive steps to win an out of state test.
The Apple v. VirtnetX case isnot establishing a precedent at whichat which complainant wins a "obvious trolling" demo, as Microsoft also needed to cope with the obvious-keeping organization, spending $ 22 million including the movie technologies located in Skype.com.
Anyway, circumstances similar to this will popup sometimes if points tend not to change, and large technology firms might have at least something to keep an eye away for a side from your warmed rivalry in the marketplace.


Whether you're a Star Wars fan or not, the series is making a comeback like none other. Not only are more movies on the way, but a virtual reality game is also in the pipeline to be released. Today, we got our first look at Lucasfilm's ILMxLAB Vader Immortal: A Star Wars VR Series — Episode I, a new virtual reality game set to be released on Oculus. This is just the first of three upcoming installments, but what all does this game have in store?
This is the first in three titles that will tell a new story written by David S Goyer. All of the stories will center around the iconic villain, Darth Vader, hence the name. You'll become a smuggler who was pulled out of hyperspace near the planet Mustafar. From the trailer, it looks like you start out life trapped on a ship, attempting to escape. Once down on the planet's surface, you're united with a lightsaber.
After coming into contact with multiple hostiles while wielding a lightsaber, dodging blaster fire, and running through corridor, you'll finally come up against Darth Vader. He looks right at you, through the magic of virtual reality and said "You are the one I've been searching for. Do as I command." This cryptic message is the only thing we hear from the Dark Lord in the entire trailer, begging to know what he will be telling you to do, and if you'll choose to follow orders.
"Is there any version of this plan that doesn't end up with us being dead?" your droid co-pilot ZOE3 says as you hurtle through space toward a desolate planet. The trailer ends with the simple tagline "Fate has chosen you," leaving you to wonder whether or not you'll join the Sith mastermind or fight against him to stop his advances.
It looks like the main storyline itself will take you around 45-60 minutes to complete, and looks to be filled with fun missions where you'll use tools and more to complete your goals.
Vader Immortal: A Star Wars VR Series — Episode 1 will hit Oculus Quest and Oculus Rift devices later this year. There's no exact price information or release date yet, though we can't wait to get our hands on it.
I love seeing a series that I grew up watching come to life. This game looks like it could create an entirely new storyline, one that could eventually end up as a movie now that Disney is producing new titles again.


text:
Japan's New Gender Equality Policy Takes a Step Back
Tokyo Report | Politics | Society | East Asia
Japan's law requiring married couples to use the same surname has come under the spotlight as a significant obstacle to women's empowerment.
By Thisanka Siripala for The Diplomat
Credit: Pixabay
On December 25, a fifth basic policy draft for gender equality, which made concrete recommendations regarding allowing different surnames after marriage, was dropped after heated pushback from conservative lawmakers.
Japan's new five-year gender equality promotion policy fell short of embracing the use of premarital surnames despite growing calls within the ruling Liberal Democratic Party (LDP) to introduce a dual-surname policy. The original, more positive phrasing was slimmed down from three pages into just 10 lines which consisted of ambiguous wording: "many Japanese women continue working after marriage and the current system presents an obstacle for their daily lives."
The LDP is generally known for its conservative stance on gender roles, but the resignation of Prime Minister Abe Shinzo has unlocked more support for women's social advancement. The shift to a dual-surname system, however, continues to divide the ruling party. An earlier meeting to discuss the basic gender equality promotion policy on December 8 ended in a near stalemate after 19 lawmakers expressed caution and 18 lawmakers voted in support of separate surnames.
On December 25 at the Council for Gender Equality meeting, Prime Minister Suga Yoshihide refrained from directly addressing the controversy and stated, "We will form a policy which reflects women's voices and aim for a society without gender bias for people in leadership positions."
Japan is the only country in the world that does not officially allow married couples to have different surnames. The law requiring married couples to have the same surname – in practice requiring the woman to change her name to match her husband's – was first introduced in 1898. But as women continue to work after marriage, maintaining their identity in the academic and business spheres is a practical step that can prevent career disruptions. The basic plan for gender equality draft acknowledged that mandating the same surnames for couples was hindering women's success as their achievements are not carried over under their married name.
The United Nations Committee on the Elimination of Discrimination against Women expressed concern over the current family system in Japan and requested international perspectives be taken into consideration when reviewing the civil code and the Family Register Act.
Meanwhile, backlash from conservative LDP lawmakers has been driven by ideals around traditional family values. In 2015 the Supreme Court upheld the same surname system as constitutional, citing entrenched Japanese views on family and child rearing. The court judgement rejected the notion that the current system and changing surnames after marriage promotes gender discrimination or damage to individual identities. The court argued that "the family system is rooted in the history and culture of each country and international comparisons which ignore it are meaningless."
Although 96 percent of women in Japan adopt their husband's surname after marriage, a growing number of women are choosing to maintain their maiden name in social and professional settings, especially if they remain at the same company. A 2018 survey by the National Institute of Population and Social Security Research found that 50.5 percent of married women believe couples should be allowed to retain different surnames. Public opinion appears to have grown increasingly accepting of a dual-surname system, with approval rising 9 points compared to 2013.
Minister of Gender Equality Hashimoto Seiko expressed support for introducing the option of different surnames and highlighted that the same surname law is a factor in Japan's rapidly declining birth rate. It's believed that some resistance to marriage can stem from wanting to continue one's own family generational register, which under law certifies family relationships, birth, deaths, marriages, divorce, and designates a head of the household. While separate surnames are permitted for marriages between Japanese and foreign nationals, marriage between Japanese nationals means leaving one's birth family registry for another.
There have been incremental inroads to permit the use of premarital names in social settings. In 2016 the Ministry of Foreign Affairs approved the use of premarital names in passports, albeit in parenthesis. That was expanded to resident cards and social security cards, without the need for official legal backing, in 2017. Judges and prosecutors are also able to work under their premarital name and arrests can be made in accordance with a premarital name on a driver's license. Some conservative lawmakers have proposed expanding the option of using premarital names unofficially, in parentheses, in order to minimize the inconvenience women face after marriage.
Ultimately the revised basic plan for gender equality highlighted the need for further discussion on the issue, underscoring Japan's difficult progress toward gender equality.
Thisanka Siripala
Thisanka Siripala is an Australian-Sri Lankan cross platform journalist living in Tokyo.
Japan gender equality
Japan married names
Japan women's rights
In Japan's Post-Abe Era, Addressing Political Gender Inequality is Essential
By Nick Stores
A more uncertain and competitive political climate could lead to urgently needed quota legislation for female politicians.
After Abe, Will Press Freedom Improve in Japan?
By Henry Laurence
New Prime Minister Suga Yoshihide is widely seen as a continuity choice. Will he change course on media freedoms?
Japan Domestic Violence Fears Rise Amid Coronavirus Pandemic
By Thisanka Siripala
As stay-at-home appeals start to take effect in Japan, alarms have been raised over a growing worldwide trend in domestic abuse cases.
Japan's Leader Slammed Over 'Stay Home' Shutdown Tweet
"An elegant tweet despite a national crisis," blared a headline in the Nikkan Sports tabloid. "Abe the aristocrat!!"


text:
Who Was? A Series of Biographies for Kids
Description: This series tells the incredible stories of trailblazers, legends, innovators, and creators. Explore this non-fiction collection of biographies written especially for children. #kids #series
Showing 76 through 100 of 138 results
Sort: By titleBy date added to Reading ListBy author
Who Is Wayne Gretzky?
by Gail Herman and Nancy Harrison and Ted Hammond
After breaking or tying more than sixty records in hockey, it's no wonder that Wayne Gretzky is known as "The Great One." Born in Brantford, Ontario, on January 26, 1961, in a nation obsessed with the sport, he threw himself into the game practically from the time he first laced up a pair of skates. When he retired from the NHL in 1999, he had led several teams to Stanley Cup victories, competed in the Olympics, and changed the way hockey was played forever. Known for his love for family and as a truly decent human being, Wayne Gretzky is revealed as more than a sports legend in this easy-to-read biography.
Who Was Robert Ripley?
by Nancy Harrison and Kirsten Anderson and Tim Foley
Enter a world of shrunken heads, mystic holy men, shriveled aliens, and bizarre relics in the delightfully odd tale of Robert Ripley. Born in California, Ripley began his career as a sports cartoonist. He went on to chronicle global records and oddities in his weekly column, Believe It Or Not! After publishing mogul William Randolph Hearst took an interest in the column, it became a syndicated global success. Ripley spent his life traveling to more than 200 countries in search of strange objects and interesting facts. His penchant for the peculiar launched an entertainment empire, and his collection of artifacts can be seen worldwide at his famous Odditoriums. Believe It Or Not!
by Nancy Harrison and Stephen Marchesi and Ellen Morgan
This fascinating addition to the best-selling Who Was. . . ' series does not settle questions of theology. Instead, it presents young readers with a biography that covers what is known historically about Jesus and places in his life in the context of his world when Jerusalem was part of the Roman Empire. In an even-handed and easy-to-read narrative, this title—illustrated with eighty black-and-white drawings—also explains the early origins of Christianity and how it became a major religion.
Who Was Jacques Cousteau?
by Nico Medina and Nancy Harrison and Dede Putra
Jacques Cousteau is the most famous and beloved name in the world of deep-sea exploration. Cousteau discovered his passion in 1938, when he first used a pair of goggles to dive off the coast of France. During his time as a French naval officer, he carried out many deep-sea experiments and improved upon early diving equipment. Soon, Cousteau began filming his underwater excursions and offering the world a glimpse below the surface. The documentary television series The Undersea World of Jacques Cousteau made the man, his work, and his red cap famous throughout the world.
Who Was Winston Churchill?
by Nancy Harrison and Jerry Hoare and Ellen Labrecque
Born into aristocracy, Churchill cut his teeth as a young army officer in British India, the Sudan, and the Second Boer War. He rose in the ranks to First Lord of the Admiralty and was a staunch opponent of the encroaching German Nazis. Churchill served as Prime Minister of the United Kingdom from 1940 to 1945 and again from 1951 to 1955. Widely regarded as one of the greatest wartime leaders of the 20th century, Churchill was also a historian, a writer, and an artist. He is the only British Prime Minister to have won the Nobel Prize in Literature, and was the first person to be made an honorary citizen of the United States.
Who Was Harriet Beecher Stowe?
by Nancy Harrison and Dana Meachen Rau and Gregory Copeland
Born in Connecticut in 1811, Harriet Beecher Stowe was an abolitionist, author, and playwright. Slavery was a major industry in the American South, and Stowe worked with the Underground Railroad to help escaped slaves head north towards freedom. The publication of her book, Uncle Tom's Cabin, a scathing anti-slavery novel, fanned the flames that started the Civil War. The book's emotional portrayal of the impact of slavery captured the nation's attention. A best-seller in its time, Uncle Tom's Cabin sealed Harriet Beecher Stowe's reputations as one of the most influential anti-slavery voices in US history.
Who Was Mother Teresa?
by Nancy Harrison and Jim Gigliotti
Born a humble girl in what is now Albania, Agnes Bojaxhiu lived a charitable life. She pledged herself to a religious order at the age of 18 and chose the name Sister Teresa, after the patron saint of missionaries. While teaching in India, where famine and violence had devastated the poor, Teresa shed her habit and walked the streets of Calcutta tending to the needs of the destitute. Her charity work soon expanded internationally, and her name remains synonymous with compassion and devotion to the poor.
Who Was Steve Irwin?
by Dina Anastasio and Jim Eldridge
By popular demand, the 100th Who Was...? subject is Steve Irwin!Steve Irwin did not have a typical childhood. Born in Melbourne, Australia, on February 22, 1962, he was raised on the wildlife park his parents owned. He cared for the animals and especially loved reptiles--he got a python for his sixth birthday! At nine years old, Steve was already helping his father wrestle small crocodiles. He became a crocodile trapper after graduating high school, catching the creatures before dangerous poachers could. Steve met his wife, Terri, at his family's park, and instead of a honeymoon, the pair filmed a wildlife documentary that led to the creation of the well-known series The Crocodile Hunter. Tragically, Steve was killed on September 4, 2006 while filming a documentary when a stingray attacked, piercing his heart. He was forty-four years old. However, his life's work obviously still strikes a chord with kids who voted him the winner of the 100th Who Was...? title.
Who Was Woodrow Wilson?
by Andrew Thomson and Margaret Frith and Nancy Harrison
First he was known as Tommy, then Woodrow, and eventually, Mr. President. Born on December 28, 1856, in Staunton, Virginia, Thomas Woodrow Wilson was a born leader. He was the president of Princeton University, served as governor of New Jersey after that, and was then elected president of the United States. But not everything was so easy for Wilson. He was ahead of his time in wanting a League of Nations after World War I to help prevent another war like it, but his hopes were dashed when the United States refused to join. Margaret Frith offers a fascinating look at how this magnificent and tragic figure handled debilitating illness, heartbreak, and "the war to end all wars."
Who Was Abigail Adams?
by True Kelley and John O'Brien
Abigail Adams was a strong woman far ahead of her time. She urged her husband, President John Adams, to "remember the ladies" and despite having no formal education herself, she later advocated for equal education in public schools for both boys and girls. She was also the first First Lady to live in the White House! This biography tells the story of Abigail Adams and her role in America's Revolutionary War period.
Who Was Bruce Lee?
by Nancy Harrison and John Hinderliter and Jim Gigliotti
Bruce Lee was a Chinese American action film star, martial arts instructor, filmmaker, and philosopher. His Hong Kong and Hollywood-produced films elevated the traditional martial arts film to a new level of popularity and acclaim. Through such films as Way of the Dragon and Enter the Dragon, Lee helped to change the way Asians were presented in American films and, in the process, he became an iconic figure known throughout the world. Although he died at the young age of 32, Bruce Lee is widely considered to be the one of the most influential martial artists of all time.
Who Was Clara Barton?
by Stephanie Spinner and David Groff
Clarissa "Clara" Barton was a shy girl who grew up to become a teacher, nurse, and humanitarian. At a time when few women worked outside the home, she became the first woman to hold a government job, as a patent clerk in Washington, DC. In 1864, she was appointed "lady in charge" of the hospitals at the front lines of the Union Army, where she became known as the "Angel of the Battlefield." Clara Barton built a career helping others. She went on to found the American Red Cross, one of her greatest accomplishments, and one of the most recognized organizations in the world.
Who Was Theodore Roosevelt?
by Michael Burgan and Nancy Harrison and Jerry Hoare
He was only 42 years old when he was sworn in as President of the United States in 1901, making TR the youngest president ever. But did you know that he was also the first sitting president to win the Nobel Peace Prize? The first to ride in a car? The first to fly in an airplane? Theodore Roosevelt's achievements as a naturalist, hunter, explorer, author, and soldier are as much a part of his fame as any office he held as a politician. Find out more about The Bull Moose, the Progressive, the Rough Rider, the Trust Buster, and the Great Hunter who was our larger-than-life 26th president in Who Was Theodore Roosevelt?
Who Was J. R. R. Tolkien?
by Meg Belviso and Nancy Harrison and Pamela D. Pollack and Jonathan Moore
Best known for his epic Lord of the Rings trilogy and The Hobbit, J.R.R. Tolkien was born in British-occupied South Africa. His early life was full of action and adventure.
Tolkien spent his childhood roaming the British countryside with his family and could read and write by age four. He was naturally gifted with languages and used this skill as a signals officers in World War II as well as in his fantasy writing. By creating alternate universes and inventing languages in his work he demonstrated that imaginary realms were not just for children. Fondly remembered as the "Father of High Fantasy," Tolkien's books have inspired blockbuster movies and legions of fans.
Who Was Beatrix Potter?
by Nancy Harrison and Sarah Fabiny and Mike Lacey
Born into wealth in 1860's London, Beatrix Potter always had a vivid imagination. Her early interests included natural history and archaeology, and Potter delighted in sketching fossils and fungi. After briefly illustrating Christmas cards with her brother, Bertram, Potter wrote and illustrated her well-known book, The Tale of Peter Rabbit. The book was rejected by several publishes until Frederick Warne eventually took a risk and published the story in 1902 - a risk that paid off. Peter Rabbit was a huge success and readers loved hearing about Peter's mischevious adventures in the lush English countryside. As she got older, Beatrix Potter became a proud conservationist, working hard to defend the landscape she loved so well against industrialization and logging. Now over one hundred years old, Peter Rabbit and his animal friends have become cultural touchstones and continue to delight readers of all ages.
Who Is Muhammad Ali?
by Nancy Harrison and Stephen Marchesi and James Buckley
Cassius Marcellus Clay, Jr. won the world heavyweight championship at the age of 22, the same year he joined the Nation of Islam and changed his name to Muhammad Ali. He would go on to become the first and only three-time (in succession) World Heavyweight Champion. Nicknamed "The Greatest," Ali was as well known for his unique boxing style, consisting of the Ali Shuffle and the rope-a-dope, as he was for the catchphrase "float like a butterfly, sting like a bee." He was an uncompromising athlete who brought beauty and grace to a very rough sport and became one of the world's most famous cultural icons. Read Who Is Muhammad Ali? and discover "The Greatest."
Who Is Malala Yousafzai?
by Andrew Thomson and Dinah Brown
Malala Yousafzai was a girl who loved to learn but was told that girls would no longer be allowed to go to school. She wrote a blog that called attention to what was happening in her beautiful corner of Pakistan and realized that words can bring about change. She has continued to speak out for the right of all children to have an education. In 2014 she won the Nobel Peace Prize.
Who Was Jesse Owens?
by James Buckley and Gregory Copeland
At the 1936 Berlin Summer Olympics, track and field star Jesse Owens ran himself straight into international glory by winning four gold medals. But the life of Jesse Owens is much more than a sports story. Born in rural Alabama under the oppressive Jim Crow laws, Owens's family suffered many hardships. As a boy he worked several jobs like delivering groceries and working in a shoe repair shop to make ends meet. But Owens defied the odds to become a sensational student athlete, eventually running track for Ohio State. He was chosen to compete in the Summer Olympics in Nazi Germany where Adolf Hitler was promoting the idea of "Aryan superiority." Owens's winning streak at the games humiliated Hitler and crushed the myth of racial supremacy once and for all.
Who Was Edgar Allan Poe?
by Nancy Harrison and Jim Gigliotti and Tim Foley
Filled with broken hearts and black ravens, Edgar Allan Poe's ghastly tales have delighted readers for centuries. Born in Boston in 1809, Poe was orphaned at age two. He was soon adopted by a Virginia family who worked as tombstone merchants. In 1827 he enlisted in the Army and subsequently failed out of West Point. His first published story, The Raven, was a huge success, but his joy was overshadowed by the death of his wife. Poe devoted his life to writing and his tragic life often inspired his work. He is considered to be the inventor of detective fiction and the father of American mystery writers. His work continues to influence popular culture through films, music, literature, and television.
Who Is Richard Branson?
by Michael Burgan and Ted Hammond
What would you do with a billion dollars? This question gets a definitive answer from billionaire Richard Branson: do everything! Born into a wealthy family in London, Branson suffered from dyslexia and was a poor student. Still, his knack for business started early with a successful parakeet-breeding enterprise at age 11.The charismatic entrepreneur launched his first major business, Virgin Records, at age 22 and spent the next few decades building the Virgin group that now includes more than 400 companies. Known for his eccentric lifestyle and trillion-watt smile, Branson's hot air balloon flights, innovative leadership, and world record attempts have made him an instantly-recognizable global icon.
Who Was Maria Tallchief?
by Catherine Gourley and Nancy Harrison and Val Paul Taylor
Born in 1925, Maria Tallchief spent part of her childhood on an Osage reservation in Oklahoma. With the support of her family and world-renowned choreographer George Balanchine, she rose to the top of her art form to become America's first prima ballerina. Black-and-white illustrations provide visual sidebars to the history of ballet while taking readers through the life of this amazing dancer.
Who Was Julia Child?
by Geoff Edgers and Nancy Harrison and Dede Putra and Carlene Hempel
Born in California in 1912, Julia Child enlisted in the Army and met her future husband, Paul, during World War II. She discovered her love of French food while stationed in Paris and enrolled in Le Cordon Bleu cooking school after her service. Child knew that Americans would love French food as much as she did, so she wrote Mastering the Art of French Cooking in 1961. The book was a success and the public wanted more. America fell in love with Julia Child. Her TV show, The French Chef, premiered in 1963 and brought the bubbling and lovable chef into millions of homes. Find out more about this beloved chef, author, and TV personality in Who Was Julia Child?
Who Was Marie Antoinette?
by John O'Brien and Dana Meachen Rau
From the palaces of Austria to the mirrored halls of Versailles, Marie Antoinette led a charmed life. She was born into royalty in 1755 and married the future king of France at age 15. By 21 she ascended to the throne and enjoyed a lavish lifestyle of masquerade balls, sky-high wigs, and extravagant food. But her taste for excess ruffled many feathers. The poor people of France blamed Marie Antoinette for their poverty. Her spending helped incite the French Revolution. And after much public outcry, in 1793 she quite literally lost her head because of it. Whether she was blameless or guilty is debatable, but Marie Antoinette remains woven into the fabric of history and popular culture.
Who Was Seabiscuit?
In the middle of the worst depression in U.S. history, one young racehorse lifted a nation's spirits. Seabiscuit was born in 1933 on a farm in Kentucky. Though bred for racing, he was weak and undersized. He slept too long and ate too much. Against the odds, he began to win local races. He was given a new coach who trained him to race in larger circuits. Soon enough, this scrappy horse began beating the best racehorses in the country. He became a media darling and won national competitions. In 1938 he was voted U.S. Horse of the Year. Seabiscuit's undying spirit and come-from-behind story made him a celebrity and hero for millions.
Who Was Blackbeard?
by Nancy Harrison and James Buckley
Though much of his early life remains a mystery, Blackbeard most likely began his life as Edward Teach in the sailing port of Bristol, England. He began his career as a hired British sailor during Queen Anne's War. He eventually settled in the Bahamas under Captain Benjamin Hornigold who taught the young sailor to go "a-pirating." Soon enough, Blackbeard was commanding his own fleet and stealing ships around the Caribbean and up and down the Eastern seaboard. Known for his thick, black beard and fearsome reputation, the legend of Blackbeard has influenced pirate legend and lore for over 300 years.
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Ed Harris is heading to TV in a villainous role.
The "Apollo 13" and "The Abyss" actor has signed on to join HBO's "Westworld" in key evil role, Variety has confirmed. He joins an already-announced cast of Evan Rachel Wood, Jeffrey Wright and Shannon Woodward.
J.J. Abrams, Jerry Weintraub and Bad Robot's Bryan Burk are producing the adaptation of Michael Crichton's 1973 film, which followed a robot malfunction that led to terror in a theme park. "Person of Interest's" Jonathan Nolan will produce and direct the Warner Bros. Television one-hour drama, and co-wrote the pilot with exec producer Lisa Joy. Kathy Lingg serves as co-executive producer, with Athena Wickham as a producer, David Coatsworth as co-executive producer and line producer and Susie Ekins as co-producer.
Harris is repped by CAA and Ziffren Brittenham.


Here I am with a baby card.
It's for the Christening of a baby girl named Tiril.
I have used some beautiful stamps from the collection Vintage Baby.
These baby images are so beautiful, and I think both fit both for boys and girls.
The image is stamped on glossy paper.
The whole card is cut out with a heart die from Magnolia. I have craft card stock as a base, and Maja Design papers on top of it.
The ribbons are sprayed with water, curled up and let to dry. All the flowers, lace and ribbons are from Wild Orchid Crafts.
The butterflies are stamps from Stempelglede which I have painted with Distress Inks.
I have added White Gesso here and there, also on the ribbon.The text on the flag is also a Stempelglede stamp.


Chances are if you're shopping here in the 100 foot x 250 foot canopy section that you have a pretty serious event, party, or outdoor rental business that's in need of some suitable equipment. The odds are pretty good, as well, that you know that such occasions can present some unique logistical challenges. Well, this section is here to help you manage to challenges quickly, conveniently, and at an affordable price. No occasion is too large and no budget too small for the outstanding shelters located here in the 100 foot x 250 foot canopy section.


text:
.PHONY: default dev dist packaging test testv deps updatedeps

BOLT_HASH = 2f846c3551b76d7710f159be840d66c3d064abbe

default: dev

# dev creates your platform binary.
dev:
	@cd shell && $(MAKE) $@
	@echo "--> Building demo todo app."
	@go build -o todo demo/todo/todo.go
	@echo "--> Done."

# dist creates all platform binaries.
dist:
	@cd shell && $(MAKE) $@

# packaging creates all platform binaries and rpm packages.
packaging:
	@cd shell && $(MAKE) $@

# destroy remove all vagrant vm used to create packages.
destroy:
	@cd shell && $(MAKE) $@

test:
	cd shell && go get && cd -
	go test . ./shell/... -cover

testv:
	cd shell && go get && cd -
	go test . ./shell/... -v

deps:
	rm -rf vendor
	git clone https://github.com/boltdb/bolt vendor/bolt
	cd vendor/bolt && git checkout $(BOLT_HASH) && rm -rf .git

redpajama_set_name: RedPajamaGithub