# Permanent Capture into the Solar System

EDWARD BELBRUNO  Yeshiva University, Department of Mathematical Sciences, New
York, NY 10016, USA<EMAIL_ADDRESS>(corresponding author) and  JAMES
GREEN  NASA HQ, Washington, D.C. 20546-0001 USA<EMAIL_ADDRESS>
(Date: March 30, 2024)

###### Abstract.

In this note, an interesting region about the Sun in phase space is described
where the permanent capture of an object, $P$, of small mass from interstellar
space can occur, under the gravitational perturbation of the resultant mass of
the Galaxy. $P$ is never ejected back into interstellar space and won’t
collide with the Sun. It cycles about the Sun for all time, asymptotically
approaching the capture set. In addition to being permanently captured, $P$ is
also weakly captured. A recent result in [5] describes this region in general.
It has a fractal structure. This is applied to the permanent capture of rogue
planets and other objects from interstellar space

###### Key words and phrases:

gravitation, dark matter, local interstellar matter, solar neighbourhood,
celestial mechanics

##  1\. Introduction

The permanent capture of a small body, $P$, about the Sun, $S$, from
interstellar space occurs when $P$ can never escape back into interstellar
space and remains captured within the Solar System for all future time, moving
without collision with the Sun. The perturbing gravitational field is the
resultant gravititational field of the Galaxy due to dark and baryonic matter,
as estimated in [6]. The gravitational field of the Galaxy as a tidal force in
the Solar System, modeling dark matter, is previously studied in [22], [29],
[28], [35], [26], and in other studies.

The capture of comets into our Solar System has been previously studied (see
[28]). However, the capture need not be permanent. When objects are captured
into the Solar System they could eventually be ejected after some time. In
this note, we describe a mechanism where this cannot happen. A simplified
modeling is assumed, and the goal is to show that such a mechanism exists for
permanent capture and how it works. Its extension to a more realistic modeling
is not described, although it is conjectured that permanent capture regions
would still exist.

A dynamical mechanism is described for the permanent capture of $P$ from
interstellar space about $S$, making use of recent results in [5] (since this
reference is cited frequently, it is denoted by $B24$). These results show
that when $P$ is on a permanent capture trajectory from interstellar space, it
also is approaching a special weak capture set about $S$, in phase space, as
time , $t\rightarrow\infty$. This is where the Kepler energy between $P$ and
$S$ becomes negative. Taken together, this is permanent weak capture. The
permanent weak capture set is an interesting fractal set of zero measure in
phase space.

To show how this can occur, a simplified modeling is assumed. It assumes three
point masses: $P$, $S$, and $P_{MW}$, which has the virial mass of the Galaxy
placed at $CG$, the center of the Galaxy. $S$ is assumed to move about
$P_{MW}$ in a circular orbit. $P$ has a negligble mass. This defines the
three-dimensional restricted three-body problem.

The definition of permanent capture used here is more restrictive that what is
used in the classical definiton by J. Chazy for the Newtonian three-
dimensional three-body problem for arbitrary mass points $P_{0},P_{1},P_{2}$
([14]). In that case $P_{0}$ is permanently captured by $P_{1},P_{2}$ if
$P_{0}$ approaches $P_{1},P_{2}$ from infinitely far away, where for all
future time, $P_{0}$ remains a finite distance from $P_{1},P_{2}$, and where
$P_{1},P_{2}$ stay a bounded distance from each other and no bodies collide.
It is shown in [14] that the set of permanent capture trajectories in the
general three-body problem is a small set of measure zero. Showing it actually
existed for $P_{i},1=0,1,2$ was an important conjecture, proven many years
later by V. Alekseev for the Sitnikov problem, using properties of chaotic
dynamics ([2]) .

In this note, $P_{0}=P$, $P_{1}=P_{MW}$, $P_{2}=S$. We consider a special case
of Chazy’s definition where the permanent capture of $P$ is about $S$, and $P$
does not approach $S$ from infinitely far away, but rather from a finite
distance. Also, the requirement of $P$ remaining a bounded distance from $S$
is replaced by $P$ remaining within the Solar System. The approximate extent
of the Solar System is estimated to be $3.81$ LY in radius from $S$ (see
Section 3). It is also shown that if dark matter were not modeled, then the
the Solar System would extend out much further.

In numerous numerical studies for two general mass points, $P_{1}$, $P_{2}$,
moving about their common center of mass in circular orbits, where $P_{1}$ is
much more massive than $P_{2}$, temporary weak capture of $P_{0}$ about
$P_{2}$ is typically considered. Temporary weak capture has been used in
practice to find low energy trajectories from the Earth to the Moon ($P_{1}$ =
Earth, $P_{2}$ = Moon) where spacecraft are automatically captured about the
Moon [8, 9].

To show that permanent capture exists, a planar motion is considered, where
$P$ moves in the same plane as the circular motion of $P_{1},P_{2}$. This is
done since the rigorous results and formulation for weak capture are for the
planar case. This is sufficient for this Note since its goal is to show
dynamically how permanent weak capture can occur. A similar process would be
expected in the three-dimensional case, but this is beyond the scope of this
note. Understanding these results in a more realistic setting is also beyond
the scope of this note.

A region about $P_{2}$ in the phase space of position and velocity coordinates
is first estimated by monitoring the stability of cycling trajectories of $P$
about $P_{2}$ after $n$ cycles, $n=1,2,\ldots$. It defines points where there
are transitions between stable and unstable cycling motion for trajectories
and starting with negative Kepler energy relative to $P_{2}$ ([8, 18, 7]).
This region is called the $n$th weak stability boundary, labeled $W_{n}$. It
consists of points giving unstable cycling for trajectories about $P_{2}$. By
symmetry, this yields weak capture trajectories to points on $W_{n}$ after $n$
cycles. Although this region is rigorously defined in the planar three-body
problem, it was initially defined in the general three-dimensional three-body
problem in 1987 [8].

The structure of $W_{n}$ about $P_{2}$ has been an open problem. It was
conjectured to be fractal in nature ([18]) (GG07 for brevity) and also to give
rise to chaotic motion for any trajectories taking initial values on it.
Recent results in $B24$ has determined the structure of this region in the
case $n\rightarrow\infty$, that is, for infinitely many cycles. It is defined
by the set $W^{\prime}=\lim{n\rightarrow\infty}W_{n}$. $W^{\prime}$ turns out
to be fractal set with a Cantor set structure. Moreover, $W^{\prime}$ bounds a
region about $P_{2}$ where only infinite stable cycling motion can occur for
all time. This region shares similar properties to the classical Mandelbrot
set for a simple complex quadratic map’

In this note it is shown that permanent weak capture of $P$ occurs on a
special subset of $W^{\prime}$ (Section 4). This is applied to the case where
$P_{2}=S$ and $P_{1}=P_{MW}$. The gravitational forces acting on $P$ is
$F_{S}$, the gravitational force of the Sun, and $F_{G}$, the resultant
gravitational field due to $P_{MW}$ relative to the CG. This is a tidal force
relative to $S$. The permanent capture of $P$ occurs from interstellar space
where $P$ first moves through a channel into the Hill’s region about $P_{2}$
at $3.81$ LY from $S$, and then performs infinitely many cycles about $S$ to
asymptotically approach weak capture about $S$ at one of the special points of
$W^{\prime}$. $P$ can never escape back into interstellar space.

The capture of rogue planets is discussed in Section 5.1. The gravity of such
objects traveling within the inner solar system could perturb the orbits of
the planets. It is possible such a perturbation could be detected.

The permanent capture of Rogue planets is an important application since they
can accumulate over time. This implies that even though the permanent capture
set is small and therfore capture of a rogue planet is unlikely, over time on
the order of hundreds of millions of years, the liklihood of capture
increases.

The analytic model for $F_{G}$ is described in Section 2. This, together with
$F_{S}$ and the centripetal force, models the motion of $P$ with a classical
set of differential equations, the restricted three-body problem (see Section
3). $W^{\prime}$ and permanent weak capture is described in Section 4.

##  2\. Galactic Force Modeled as a Classical Point-Mass Inverse Square
Newtonian Force

In [6] ($BG22$ for brevity) ${\bf F_{G}}$ is determined within our Solar
System, due to dark and baryonic matter of the Milky Way, MW.111By Solar
System we mean the Sun together with all the planets, asteroids, Kuiper belt
objects, etc. It’s magnitude relative to the centre of the Galaxy, CG, at the
distance of the Sun is

(2.1) $F_{G}=1.8\times 10^{-10}m/s^{2}$

per unit mass. In a Sun-centered coordinate system, $F_{G}$ is reduced by a
factor of approximately $\mathcal{O}(10^{-7})$ since it is measured as a tidal
force. $F_{G}$ is dominant over the gravitational force of the Sun on a body
moving sufficiently far from the Sun, say 10s of thousands of AU. If $P$ is
moving in interstellar space near the Solar System at distances sufficiently
far from the Sun, the gravitational forces acting on $P$ are $F_{G}$ and
$F_{S}$.

The derivation of (2.1) is briefly summarized. The gravitational force acting
on $P$, located near the Sun due to both dark and baryonic matter in MW can be
modeled using a Hernquist model, applied in two steps.

The virial mass of MW, $M_{vir}$ as estimated in [44],

(2.2) $M_{vir}=1.54_{-.44}^{+.75}\times 10^{12}M_{\odot},$

for a virial radius $R_{vir}=300$ kpc (see also [12]). The Hernquist model
gives the mass density $\nu(r)=(M_{vir}/2\pi)[d/(r(r+d)^{3}]$ at a distance
$r$ from the Galactic Center, GC (see [23]). The scale length $d=21.5$ kpc is
assumed. The virial mass includes both baryonic and dark matter without regard
to the relative geometry of baryonic matter in the disc, bulge and stellar
halo, and dark matter in the dark matter halo.

The mass distribution is $M(r)=M_{vir}r^{2}/(r+d)^{2}$ for $r\leq R_{vir}$.
$M_{vir}\equiv M(R_{vir})$. This yields the total mass of MW for $r\leq
r_{s}$, $M(r_{s})=.077M_{vir}$, where $r_{s}=8.29$ kpc is the distance of the
Sun to GC. Thus, the mass within the sphere of radius 300 kpc for the galactic
halo of MW is used to estimate the mass within the smaller sphere of radius
$r_{s}$.

Next, the gravitational force for $r\leq r_{s}$ is computed. This uses a
Hernquist model defined for the less massive stellar component of MW. The
force at a point ${\bf x}$ within this spherical region is

(2.3) ${\bf{F}_{G}}({\bf{x}})=-\frac{GM(r_{s}){\bf{x}}}{r(r+b)^{2}},$

where ${\bf{x}}=(x_{1},x_{2},x_{3})$ is relative to $GC$ at the origin,
$r=|{\bf{x}}|=(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{1/2}\leq r_{s}$, and where $b$
is a new scale length for the stellar halo of MW, $\leq r_{s}$. $G$ is the
Newtonian gravitational constant. $b=1$ kpc. $F_{G}(r)=GM(r_{s})(r+b)^{-2}$ is
the magnitude of $(\ref{eq:GalForce})$. Setting $r=r_{s}$ yields (2.1).

It is noted that in $BG22$, $F_{G}(r_{s})$ is also estimated using both the
Navarro-Frenk-White (NFW) model (see [34]) for dark matter distribution in MW
together with a point-mass Newtonian force model for the baryonic
contribution, whose value agrees with (2.1). The multi-step Hernquist model
yields a formulation more suitable for the purposes of this paper.

This model accurately estimates the velocity of the Sun about GC. The circular
velocity of the Sun about GC is defined to be
$V_{C,s}=\sqrt{GM(r_{s})/r_{s}}$. This yields, $V_{C.s}=248$ km/s, lying
within the acceptable range $220-250$ km/s (see [44]).

Point-Mass Newtonian Approximation

$F_{G}$ can be approximated by a classical point-mass Newtonian force for MW,
$P_{MW}$, with mass $M(r_{s})$ placed at CG. This plays a key role in a
simplified modeling near our Solar System for the dynamics of motion for a
test particle, $P$, due to the gravity of MW and the Sun. It is assumed that
$P$ will move within a few 10s of LY from $S$.

Approximation of $F_{G}(r)$ For $r$ near $r_{s}$, the gravitational force due
to MW is approximated as

(2.4) $F_{G}(r)\approx\frac{GM(r_{s})}{r^{2}}.$

This is a point mass Newtonian approximation for $F_{G}(r)$, where MW is
represented as point mass $P_{MW}$ at CG.

$r$ near $r_{s}$ means that $r$ is within $4$ LY to $r_{s}$, or equivalently,
that $P$ is within approximately $4$ LY so as not to extend to the distance of
the closest star, Proxima Centauri at $4.2$ LY. In the units of kpc which are
used for the length scale in $F_{G}(r)$, where a distance of $1$ corresponds
to $1$ kpc, then $4$ LY represents approximately $.0001$. (2.4) is verified by
noting that

(2.5) $(r+b)^{-2}=r^{-2}(1+x)^{-2},$

where $x=b/r\approx b/r_{s}=1/8.29\approx.12<1$, and $x$ is dimensionaless.
Thus, $(1+x)^{-2}$ can be expanded in a convergent binomial series,
$(1+x)^{-2}=1-2x+\mathcal{O}(x^{2})$ . This implies,

(2.6) $F_{G}(r)=\frac{GM(r_{s})}{r^{2}}(1-2x+\mathcal{O}(x^{2})).$

The leading term yields the value given by (2.1). The second term has fractor
$2x\approx.24$. This has the effect of reducing (2.1), and hence (2.4) for $r$
near $r_{s}$ by an approximate order of magnitude, which given the small size
of (2.1) can be omitted as long as $r(t)\approx r_{s}$. The higher order terms
in the binomial expansion reduce (2.1) to even higher orders of magnitude.
This yields (2.4).

It is noted that ignoring the terms $-2x+\mathcal{O}(x^{2})$ can be done only
because (2.1) is already small. Dropping these terms makes sense for
sufficiently short time spans where their effect on the dynamics of the motion
of $P$ is negligible.

##  3\. Modeling: Restricted Three-Body Problem

A simplified model is made for the dynamics of a test particle, $P$, as it
moves near the Sun due to the gravitational forces of both MW and the Sun.

It is assumed that $S$ movies about $P_{MW}$ at GC in a circular orbit with
constant radius $r_{s}$, constant circular velocity $V_{C,s}$, on a plane
$\Sigma$ that passing through$P_{MW}$ and that $P$ is contrained to move on
$\Sigma$.222It is noted that both $P_{MW}$, $S$ perform mutual circular motion
about their common center of mass, essentially at CG. These assumptions make
sense relative to the galactic scale, locally for the motion of $P$ near $S$
(assumed within 10 LY). It is also assumed that the only forces acting on $P$
are the gravitational forces due to point masses $S$, $P_{MW}$ and the
centripetal force due to the rotation of $S$ about $CG$ .

Inertial coordinates on the plane $\Sigma$ are ${\bf{x}}=(x_{1},x_{2})$,
centered on $P_{MW}$. The mass of $P$ mass is negligible with respect to the
Sun, taken to be 0. The mass of $P_{MW}$ is $M(r_{s})=1.19\times
10^{11}M_{\odot}$. The mass of $S$, $M_{S}=1M_{\odot}$.

These assumptions define the classical planar circular restricted three-body
problem for the motion of $P$, going back to H. Poincaré ([38]). It is
convenient to choose a rotating coordinate system ${\bf y}=(y_{1},y_{2})$ that
rotates about $P_{MW}$ with angular velocity $\omega=V_{C,s}$ in the
$\Sigma$-plane. In this system, $S$ is fixed and lies on the $y_{1}$-axis.

The following nomalizations can be made without loss of generality:
$\omega=1$, mass of $S=\mu$, where $\mu=M_{\odot}/(M(r_{s})+M_{\odot})\approx
8.4\times 10^{-12}$, $G=1$, the distance from $P_{MW}$ to $S$ is $1$, where
$S$ is located on the $y_{1}$-axis at $y_{1}=1$ and $P_{MW}$ is at the origin.
The center of mass between $P_{MW}$ and $S$ is at $(\mu,0)$. Under this
scaling, the distance of the center of mass to $P_{MW}$ is just a few thousand
km.

The differential equations for the motion of $P$ are given in the more general
case for any two bodies $P_{1},P_{2}$, where $P_{1}$ is the more massive
primary and $P_{2}$ is less massive secondary ([39]). The motion of $P$ for
${\bf y}(t)$ as a function of time, $t$, is given by the system of
differential equations,

(3.1) $\ddot{y}_{1}-2\dot{y}_{2}=\Omega_{y_{1}},\hskip
7.22743pt\ddot{y}_{2}+2\dot{y}_{1}=\Omega_{y_{2}},\hskip 7.22743pt,$

where
$\Omega=(1/2)[(y_{1}-\mu)^{2}+y_{2}^{2}]+(1-\mu){r^{-1}_{1}}+\mu{r^{-1}_{2}}+(1/2)\mu(1-\mu)$,
and $r_{1}^{2}=y_{1}^{2}+y_{2}^{2}$, $r_{2}^{2}=(y_{1}-1)^{2}+y_{2}^{2}$,
$\Omega_{x}\equiv\partial\Omega/\partial x$, ${}^{.}\equiv$ d/dt.

The Jacobi energy integral is

(3.2) $J=J({\bm{y,\dot{y}}})=2\Omega-|{\bm{\dot{y}}}|^{2},$

$|{\bm{\dot{y}}}|^{2}=(\dot{y}_{1}^{2}+\dot{y}_{2}^{2})$ . Along a trajectory,
${\bm{\psi}}(t)=({\bm{y}}(t),{\bm{\dot{y}}}(t))$, $t\in\mathbb{R}$,
$J({\bm{\psi}}(t))=C$, the Jacobi constant. The set
$\\{({\bm{y,\dot{y}}})|J=C\\}$ defines a three-dimensional Jacobi surface.

The motion of $P$ is dynamically understood about $P_{1}$, starting with
elliptical initial conditions for $\mu$ small, by the Kolmogorov-Arnold-Moser
(KAM) Theorem ([39]), provided the ratio of the two frequencies, $\omega$ and
$\omega^{*}$ (the frequency of $P$ for $\mu=0$) are sufficiently irrational.
Then the trajectory of motion for $P$, ${\bm{\psi}}(t)$, $t\in\mathbb{R}$, in
four-dimensional position-velocity space, $({\bm{y,\dot{y}}})$,
${\bf{y}}=(y_{1},y_{2}),{\bm{\dot{}}{y}}=(\dot{y}_{1},\dot{y}_{2})$, on fixed
three-dimensional Jacobi energy surfaces lie on two-dimensional invariant tori
for $\mu>0$ sufficiently small. It is quaisperiodic. The motion between the
tori is chaotic, but also stable since the tori separate the phase space on
each Jacobi energy surface.

However, for motion about $P_{2}$, the general motion for $P$ starting with
Keplerian elliptical initial conditions is not dynamically understood, unless
the motion is arbitrarily close to $P_{2}$ for $C$ sufficiently large where
the KAM theorem is valid ([27]). This situation isn’t realistic and not in the
range of $C$ required in this study.

New results in $B24$ on the weak stability boundary and its analogy with the
boundary of a Mandelbrot set give an understanding of some of the dynamics
about $P_{2}$, including permanent capture $P$ and the nonexistence of KAM
tori for a suitable range of $C$.

Remark (Dark Matter)  It is noted that (3.1) is incorporating dark matter into
its definition by virtue of $M(r_{s})$ obtained from $BG22$, where it is shown
that near the Solar System, the ratio of dark matter to baryonic matter is
roughly 45$\%$ to 55$\%$, respectively. Thus, the analysis made on the
properties of the motion of $P$ depends significantly on dark matter.

###  3.1. Assumptions, Solar System Extent, Dark Matter, Lagrange Points

The value of $C$ determines the geometry of the Hill’s regions in
$y_{1},y_{2}$ coordinates were $P$ is able to move [41], [15]. They are
defined by ${H}(C)=\\{{\bf y}|2\Omega\geq C\\}$. Let $C_{i},\ i=1,2,3,4,5$ be
the values of $C$ corresponding to the Lagrange-Euler points $L_{i}$,
respectively, $3=C_{5}=C_{4}<C_{3}<C_{2}<C_{1}$. For $C\geq C_{1}$ $P$ cannot
pass between $P_{2}$ to $P_{1}$ since the Hill’s regions about $P_{1}$ and
$P_{2}$, $H_{1},H_{2}$, repectively, are not connected. For $C$ slightly less
than, but not equal to $C_{1}$, $C\lessapprox C_{1}$, the Hill’s regions about
$P_{1},P_{2}$ are connected near $L_{1}$ in a channel $\mathcal{C}_{1}$ about
a Lyapunov periodic orbit, $\gamma_{1}$. When $C$ decreases to $C\lessapprox
C_{2}$ there is another channel $\mathcal{C}_{2}$ about $L_{2}$ connecting
$H_{2}$ to the outer Hill’s region, $H_{O}$ about both $P_{1},P_{2}$. Another
Lyapunov orbit, $\gamma_{2}$, appears in $\mathcal{C}_{2}$.

The range of $C$ is assumed to be $C_{a}<C<C_{1}$, where $C_{a}<C_{2}$ is
defined in Section 4. Thus, there are channels $\mathcal{C}_{i},1=1,2$ on both
sides of $P_{2}$ near $L_{1},L_{2}$, respectively, that $P$ can pass through.

It is assumed $P_{1}=P_{MW},\ P_{2}=S$, and that $P$ moves from $H_{1}$ or
$H_{O}$, defined to be interstellar space, into $H_{2}$ about $S$. $H_{2}$
defines the extent of the Solar System for this study. This implies that the
distance of $L_{1},L_{2}$ to $S$, $\rho_{1},\rho_{2}$, respectively,
approximates the extent of the solar system. For the small value of $\mu$,
$\rho_{1},\rho_{2}$ are approximately the same.

The distance of $L_{1},L_{2}$ to $S$ can be estimated. If this distance is
labeled $\rho_{1},\rho_{2}$, respectively, then for any $P_{1},P_{2}$ with
$\mu$ sufficiently small,

(3.3) $\rho_{1}\approx\gamma(1-(1/3)\gamma),\hskip
7.22743pt\rho_{2}\approx\gamma(1+(1/3)\gamma),$

within $\mathcal{O}(\gamma^{3})$, where $\gamma=(\mu/3)^{1/3}$ ([8], page 130)
(see also [41]). $\mathcal{O}(\gamma^{3})$ is negligibly small given the
magnitude of $\mu$. From the value of $\mu$, $\rho_{i}$ can be computed, where
$\rho_{i}\approx\mathcal{O}(10^{-4})$, and converting to LY,

(3.4) $\rho_{1}\approx 3.81088\hskip 7.22743pt\rho_{2}\approx 3.81092,$

$\rho_{1}\lessapprox\rho_{2}$. These distances correspond to approximately
$241,005.5$ AU. In scaled units, the distance $d$ from $S$ to either $L_{1}$
or $L_{2}$ is given by $d\approx 1.41\times 10^{-4}$.

Thus, the Solar System extends to approximately $3.81$ LY from $S$.

It is noted that if dark matter were not included for $M_{vir}$ then the value
of $3.81$ LY would be substantially larger: The baryonic mass of the Galaxy is
estimated as $M_{B}=5\times 10^{10}M_{\odot}$ ([30]). Using this for $M{vir}$,
then $M(r_{s})\equiv M(r_{s},B)=M_{B}r_{s}^{2}/(r_{s}+d)^{2}=.077M_{B}$ and
$\mu\equiv\mu_{B}=M_{\odot}/(M(r_{s})+M_{\odot})\approx 2.60\times 10^{-10}$.
This yields, $\rho_{i}\equiv\rho_{i,B}\approx 11.97$ LY.

Conclusion The effect of the resultant dark matter of the Galaxy is to reduce
the distance of $L_{1,2}$ to $S$ from $11.87$ LY to $3.81$ LY by a factor of
3.14.

The openings about $\gamma_{i},\ i=1,2$ connecting $H_{2}$ with $H_{1},H_{O}$,
respectively, are small channels that $P$ can move through to pass from
$H_{1}$ or $H_{O}$ into $H_{2}$. They are labeled $\mathcal{C}_{i},\ i=1,2$,
respectively.

It is assumed that the motion of $P$ starts in interstellar space in $H_{1}$
or $H_{O}$ outside $\mathcal{C}_{1}$ or $\mathcal{C}_{2}$, respectively. The
initial distance of $P$ from $S$ is assumed to be approximately $4$ LY. This
distance is chosen to be less than the distance to Proxima Centauri.

##  4\. Permanent Weak Capture Mechanism

A $P_{2}$-centered coordinate system ${\bf Y}=(Y_{1},Y_{2})$ is adopted, by
the translation $Y_{1}=y_{1}-1,Y_{2}=y_{2}$. The two-body Kepler energy of $P$
with respect to $P_{2}$, $E_{2}$, is given in Section A.1. Although this is
stated for general $P_{1},P_{2}$, we are assuming $P_{1}=P_{MW},P_{2}=S$.

Let ${\bm{\psi}}(t)=({\bm{Y}}(t),{\bm{\dot{Y}}}(t))$ be a solution to the
restricted problem for $P$ defining a trajectory as a function of time, $t$,
in the $P_{2}$-centered rotating system ${\bf{Y}}=(Y_{1},Y_{2})$. The Kepler
energy along the trajectory is defined as $E_{2}(t)=E_{2}({\bm{\psi}}(t))$.
Let $\tilde{J}(\bm{Y},\bm{\dot{Y}})$ be the Jacobi integral in these
coordinates.

Weak capture is defined along ${\bm{\psi}}(t)$ at a time $t=t_{1}$: If
$E_{2}(t_{0})\geq 0$ and at a later time, $t=t_{1}>t_{0}$, $E_{2}(t_{1})<0$,
then $P$ is weakly captured by $P_{2}$ at $t=t_{1}$.

$n$-stable cycling motion is defined assuming $P$ starts on a reference line
$L(\theta)$ from $P_{2}$, at $t=t_{0}$, making an angle $\theta\in[0,2\pi]$
with the $Y_{1}$-axis. The initial conditions of ${\bm{\psi}}(t_{0})$ on
$L(\theta)$ with resepct to $P_{2}$ it is assumed: $E_{2}(t_{0})<0$, the
eccentricity $e\in[0,1)$ for $P$ and the initial velocity vector is
perpendicular to $L(\theta)$, assuming counterclockwise motion. $P$ cycles
about $P_{2}$ $n$ times, $n\geq 1$, without cycling around $P_{1}$. Each of
the $n$ crossings with $L(\theta)$ is transversal with $E_{2}(t)<0$. Otherwise
it is $n$-unstable (see $GG07$, [42], [7]($BGT10$ for brevity) ). $n$-unstable
cycling can happen several diferent ways: (i.) $P$ cycles about $P_{1}$ before
making $n$ cycles about $P_{2}$, (ii.) A crossing with $L(\theta)$ is
tangential, (iii.) $E_{2}\geq 0$ at a crossing, (iv.) $P$ does not return to
$L(\theta)$ prior to $n$ cycles.

The points along $L(\theta)$ that are initial conditions for trajectories at
the boundary between $n$-stable and $n$-unstable motion about $P_{2}$ in
$Y_{1},Y_{1}$-space is parameterized by $e,\ \theta,\ n$. It is labeled
$W_{n}(\theta,e)$ and called the $n$th weak stability boundary. Taking the
union over all $e\in[0,1)$ yields $W_{n}(\theta)$, of particular interest for
this paper. The Jacobi integral value $C$ varies over the points of
$W_{n}(\theta,e)$. The points of $W_{n}(\theta,e)$ are themeselves
$n$-unstable. (The case of $n=1$ was first defined in [8])

In $BGT10$, it is shown through numerical and analytic analysis (semi-
analytic) that $W_{n}(\theta)$ is equivalent to the points on the stable
manifolds associated to $\gamma_{i}$, $i=1,2$ in the Hill’s region $H_{2}$
about $P_{2}$ satisfying $\dot{r}=0,E_{2}(\bm{Y},\bm{\dot{Y}})<0$, where
$r=|\bf{Y}|$. These manifolds are labeled $W^{s}(\gamma_{i})$. Set
$\Lambda=\\{{\bm{Y}},{\bm{\dot{Y}}}\ |\ \dot{r}=0,E_{2}<0\\}$. Thus, the
points of $W_{n}(\theta)$ belong to $\Lambda$. The points of $W_{n}(\theta)$
are obtained as the intersections of $W^{s}(\gamma_{i})$ on a two-dimensional
surface of section $S_{\theta}$, with coordinates $r,\dot{r}$, for each fixed
$\theta$ and a fixed admissible energy value within
$C=\in\\{C_{a}<C<C_{1}\\}$. $C_{a}$ is the largest value of $C$ where
$W^{s}(\gamma_{i}$ collides with $P_{2}$.

The appearance of $W_{n}(e)$ (= union of $W_{n}(\theta,e)$ over
$\theta\in[0,2\pi]$) was noted in $GG07$, $BGT10$ to appear fractal for a
small set of $e$, but wasn’t justified.

The fractal structure is proven in $B24$ for the limiting set
$W^{\prime}(\theta)=\lim_{n\rightarrow\infty}W_{n}(\theta)$ over infinitely
many cycles about $P_{2}$. More precisely, it is shown that
$W^{\prime}(\theta)$ on $S_{\theta}$ is a Cantor set on each Jacobi energy
surface $\\{\tilde{J}(\bm{Y},\bm{\dot{Y}})=C\\}$ for each $C$ in a special
subset of $C_{a}<C<C_{1}$. Besides the Cantor set being everywhere
disconnected, it is also self-similar, so that when any part of
$W^{\prime}(\theta)$, no matter how small, is magnified, it equals all of
$W^{\prime}(\theta)$. Each of the points of $W^{\prime}(\theta)$ have one
dimensional stable and unstable manifolds. Let $W^{\prime}$ be the union of
$W^{\prime}(\theta)$ over all $\theta\in[0,2\pi]$. $W^{\prime}$ is invariant
under the flow of the restricted problem. The points of $W^{\prime}$ belong to
$\Lambda$. Trajectories ooving within $W^{\prime}$ are chaotic, and
trajectories moving near $W^{\prime}$ are unstable.

Permanent Capture Set

A trajectory ${\bm{\psi}}(t)$ starting on $W^{\prime}$, starts with $E_{2}<0$
and generally on this trajectory $E_{2}(t)\geq 0$ for some $t$ since
$W^{\prime}$ consists of unstable cycling points. This occurs for the unstable
cycling points satisfying $n$-unstable conditions (i.), (iii.) as
$n\rightarrow\infty$ which we restrict to. By the symmetry condition,
$(Y_{1}(t),Y_{2}(t),\dot{Y}_{1}(t),\dot{Y}_{2}(t))\rightarrow(Y_{1}(-t),Y_{2}(-t),\dot{Y}_{1}(-t),\dot{Y}_{2}(-t))$
this yields a trajectory, ${\bm{\bar{\psi}}}(t)$ that starts with $E_{2}\geq
0$ and movies to weak capture at a point $p$ on $W^{\prime}$. This occurs with
${\bm{\bar{\psi}}}(t)$ asymptotically approaching $p$ as $t\rightarrow\infty$
by cycling about $P_{2}$ infinitely many times.

However, the points of $W^{\prime}$ satisfying (i.) are of particular interest
since in this case the symmetric trajectory ${\bm{\bar{\psi}}}(t)$ enters the
Hill’s region $H_{2}$ through channels $\gamma_{i}$, $i=1$ or $2$ at $180$
degrees on either side of $P_{2}$ and goes to weak capture to a point of
$W^{\prime}$.

Let $\bar{W}^{\prime}$ be the subset of points of $W^{\prime}$ containing only
those points satisfying condition (i.). Therefore, this comprises the subset
of permanent capture points of $W^{\prime}$. Trajectories going to these
points start from outside the Hill’s region $H_{2}$, enter $H_{2}$ through
channels $\mathcal{C}_{i}$ and go to permanent capture at points on the set
$\bar{W}^{\prime}$ after infinitely many cycles about $P_{2}$.

Applications to $S,P_{MW}$

Choosing $P_{1}=P_{MW},\ P_{2}=S$, then the set $\bar{W}^{\prime}$ about $S$
respresents points that are permanently captured from interstellar space. $P$
will cycle about $S$ for all future time approaching a capture point
$p\in\bar{W}^{\prime}$. As $P$ asymptotically approaches $p$,
$\dot{r}(t)\rightarrow 0$, and $E_{2}(t)<0$. The size of the set
$\bar{W}^{\prime}$ is small, which must be a set of measure zero as follows
from $B24$. This is also seen since for each $\theta\in[0,2\pi]$,
$\bar{W}^{\prime}(\theta)\subset W^{\prime}(\theta)$, and $W^{\prime}$ is a
Cantor set of zero topological dimension, also for general $P_{1},P_{2}$
provided $\mu$ is sufficiently small. $\bar{W}^{\prime}$ need not be a Cantor
set unless it were closed.

The distribution of the points of $\bar{W}^{\prime}$ about $P_{2}$ in general
is not known. This represents a substantial numerical study beyond the scope
of this note. Computing $W_{n}$ even for smaller values of $n=8$ (the highest
computed) takes a significant effort. The Cantor points of $W^{\prime}$ for
high values of $n$ would be difficult to discern. The points of
$\bar{W}^{\prime}$ would be among the points of $W^{\prime}$, and even harder
to discern.

It is shown in $B24$ that $W^{\prime}$ projected into $(Y_{1},Y_{2})$-space is
the boundary of a region, $\hat{S}$, about $P_{2}$ where the motion cycles
about $P_{2}$ infinitely many times in a stable manner. That is, it consists
of points as limits of $S_{n}$ as $n\rightarrow\infty$, labeled $\hat{S}$. On
$S_{\theta}$, the points of $W^{\prime}(\theta)$ are themselves unstable
points for infinitely many cycles and lie at the boundary between stable and
unstable cycling motion, whereas the points of $\hat{S}(\theta)$ are stable
for infinitely many cycles about $P_{2}$. The points of $\hat{S}$ also belong
to $\Lambda$. This shows that $W^{\prime}$, and hence $\hat{W}^{\prime}$,
represent boundary points for stable cycling motion about $S$. $W^{\prime}$ is
where stable cycling motion can no longer occur.

It is noted that if a trajectory ${\bm{\psi}}(t)$ starts on a point of
$W^{\prime}(\theta)$ for $t=0$, then for $t>0$, ${\bm{\psi}}(t)$ cycles about
$P_{2}$ in $(Y_{1},Y_{2})$-space remaining on $W^{\prime}$. It repeatably
passes though points of $W^{\prime}(\theta)$ on $S_{\theta}$. This yields a
map $\Phi$ on $S_{\theta}$, mapping $W^{\prime}(\theta)$ into itself.
Interestingly, $\hat{S}$ is shown to share similar properties to a Mandelbrot
set for the much simpler complex map $Q_{c}(z)=z\rightarrow z^{2}+c$ starting
at $z=0$ in the complex plane, where $z,c$ are both complex numbers. The
Mandelbrot set, $M$, consists of those points in the complex $c$-plane that
yield bounded iterates of $Q_{c}(0)$ over infinitely many iterates. The
boundary of the Mandelbrot set is a connnected fractal curve whereas the
boundary of $\hat{S}$, which is $W^{\prime}$, is totally disconnected for each
$\theta$.

##  5\. Capture of Interstellar Objects into the Solar System

Objects in interstellar space outside $H_{2}$ beyond $3.81$ LY can be
permanently captured about the Sun in the manner described in the previous
section by passing through $\gamma_{i},i=1,2$, and cycling to capture at a
point of $\hat{W}^{\prime}$ as $t\rightarrow\infty$. Such objects could be
comets or rogue planets.

###  5.1. Rogue Planets

All stars are born from collapsing interstellar clouds forming widely
dispersed stars in open clusters or close stars in globular clusters. Our Sun
was born in an initial open cluster of about 2,000 to 20,000 interacting stars
before dispersing over time ([3]) and these stars must contain rocky planets,
giant planets, and small bodies. The final architecture of our solar system
will be shaped by the stellar flybys of the adjacent forming star systems
since close encounters can pull planets and small bodies out of the system
creating what are called rogue planets ([11]). It has also been predicted that
our Sun captured comets from other stars in the birth cluster by this close
encounter process ([28]). The region of the Milky Way where our Sun resides
must also be a productive region for planet creation since even close by
stars, like Trappist-1, only 40 LY away, has 7 terrestrial planets ([20]). By
the very nature of the creation of solar systems in clusters of stars many
rogue planets must be created.

In addition to close stellar encounters, planetary scientists have come to
realize that our own Solar System has gone through the process of giant planet
migration after an initial period (believed to be 850 My) of stable planetary
resonance has been broken. An early solar system containing five giant planets
was proposed by [36] after numerical models indicated that this is more likely
to reproduce the architecture of the current solar system after ejecting a
Neptune-sized planet. Currently, an observation campaign is underway to find
the fifth giant planet, which is referred to as Planet 9 ([4]), assuming the
ejected planet is still bound to our sun. The other possibility is that the
ejected planet is a rogue.

When taken together, planet ejection both from early stellar encounters and in
the subsequent evolution of a multi-planet solar system should be common and
supports the evidence for a very large number of rogue planets that are free
floating in interstellar space that perhaps exceed the number of stars ([40])
but this assertion has been controversial. It is also noted that many planets
in a solar system maybe liberated during a supernova explosion of the host
star ([10]). There is very limited observational knowledge that leads to a
better understanding of the number of rogue planets in our galaxy. Therefore,
simulations of stars forming in clusters have been the primary tool used to
obtain this number (e.g. [43], [37]) which is still expected to be a
significant component of the substellar population. With the advent of the
James Webb Space Telescope (JWST) a robust census of rogue planets is possible
since it is the first telescope with the capability to directly image rogue
planets down to a Jupiter mass or even below.

The passage of a rogue planet through our Solar System could cause
gravitational disturbances to the planets. This could affect Earth’s orbit
about the Sun especially if the rogue planet were sufficiently massive.

###  5.2. Close Approach of Other Solar Systems

Today the solar neighborhood, a volume with a radius of 6 pc around the Sun,
contains about 131 stars and brown dwarfs, several of which have known planets
([17]). All these stars move relative to the Sun and over time it is estimated
that on average 2 stars per million years come within a few light years of the
Earth, however, six stars are expected to closely pass by in the next 50,000
years ([31]). Since the outer boundary of the Oort Cloud is about 1.5 light
years distance, stellar encounters must eject Oort Cloud comets inward toward,
or away from, the Sun ([28]). It is important to note that permanent capture,
as described in this paper must also be occurring depending on the location of
the passing star, along with it’s small body debris such as asteroids in
addition to comets, as it passes by the Sun. It may turn out that, over time,
the permanent capture of interstellar objects associated with passing stars
near $H_{2}$ have left our solar system with many such objects. Once
identified, these objects could be important enough that space agency’s send
dedicated space missions to study them up close.

##  6\. Conclusions

Based on gravitational considerations of the Sun and the resulting fields of
the Galaxy due to dark and baryonic matter, for the first time regions in
space are determined where a weak capture process leading to permanent capture
would be occurring, in a simplified setting. Small openings into the solar
Hill’s sphere has been determined to exist at about $3.81$ LY from the Sun in
the direction of the galactic center or opposite to it. Permanent weak capture
of interstellar objects into the Solar System is possible through these
openings. They would move chaotically within the Hill’s sphere to permanent
capture about the Sun taking an arbitrarliy long time by infinitiely many
cycles. They would not collide with the Sun. The permanent capture of
interstellar comets and rogue planets could occur. A rogue planet could
perturb the orbits of the planets that may be possible to detect.

##  Appendix A Supporting Calculations

###  A.1. Kepler Energy, $E_{2}$

In a $P_{2}$-centered inertial coordinate system, $\bf{X}=(X_{1},X_{2})$, the
Kepler energy of $P$ relative to $P_{2}$ is

(A.1) $E_{2}=(1/2)|{\bm{\dot{X}}}|^{2}-\mu|{\bf X}|^{-1},$

where $X_{1}=x_{1}-1,X_{2}=x_{2}$. ${\bf{x}}=(x_{1},x_{2})$ are
$P_{1}$-centered inertial coordinates.

In $P_{2}$-centered rotating coordinates, ${\bf{Y}}=(Y_{1},Y_{2})$, obtained
by setting $Y_{1}=y_{1}-1,Y_{2}=y_{2}$, where ${\bf{y}}=(y_{1},y_{2})$ are
$P_{1}$-centered rotating coordinates defined in (3.1);

(A.2) $E_{2}({\bf Y},{\bm{\dot{}}{Y}})=(1/2)\dot{Y}^{2}-\frac{\mu}{Y}-L({\bf
Y},{\bm{\dot{}}{Y}})+(1/2)Y^{2},$

where $L({\bf Y},{\bm{\dot{}}{Y}})=\dot{Y}_{1}Y_{2}-\dot{Y}_{2}Y_{1}$,
$Y=|\bm{Y}|,\dot{Y}=|\bm{\dot{Y}}|$.

Acknowledgements

Many thanks to Princeton University, Department of Astrophysical Sciences.
E.B. was partially funded by NSF grant DMS-1814543.

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