# An Earth Encounter As the Cause of Chaotic Dynamics in Binary Asteroid
(35107) 1991VH

Alex J. Meyer Smead Department of Aerospace Engineering Sciences, University
of Colorado Boulder, 3775 Discover Dr, Boulder, CO 80303, USA  Oscar Fuentes-
Muñoz Smead Department of Aerospace Engineering Sciences, University of
Colorado Boulder, 3775 Discover Dr, Boulder, CO 80303, USA  Ioannis Gkolias
Aristotle University of Thessaloniki, Thessaloniki, Greece  Kleomenis Tsiganis
Aristotle University of Thessaloniki, Thessaloniki, Greece  Petr Pravec
Astronomical Institute of the Academy of Sciences of the Czech Republic,
Fričova 298
Ondřejov, CZ 25165 Czech Republic  Shantanu Naidu  Jet Propulsion Laboratory,
California Institute of Technology, Pasadena, CA, USA  Daniel J. Scheeres
Smead Department of Aerospace Engineering Sciences, University of Colorado
Boulder, 3775 Discover Dr, Boulder, CO 80303, USA

###### Abstract

Among binary asteroids, (35107) 1991VH stands out as unique given the likely
chaotic rotation within its secondary component. The source of this excited
dynamical state is unknown. In this work we demonstrate that a past close
encounter with Earth could have provided the necessary perturbation to allow
the natural internal dynamics, characterized by spin-orbit coupling, to evolve
the system into its current dynamical state. In this hypothesis, the secondary
of 1991VH was previously in a classical 1:1 spin-orbit resonance with an orbit
period likely between 28-35 hours before being perturbed by an Earth encounter
within $\sim 80,000$ km. We find if the energy dissipation within the
secondary is relatively inefficient, this excited dynamical state could
persist to today and produce the observed ground-based measurements. Coupled
with the orbital history of 1991VH, we can then place a constraint on the
tidal dissipation parameters of the secondary.

##  1 Introduction

Near-Earth binary asteroids are typically comprised of an oblate primary
rotating rapidly, with a tidally locked, elongated secondary in its orbit.
These systems are generally thought to be formed by mass shedding from a
rapidly rotating primary driven by YORP spin-up (Walsh et al., 2008), and
tidal dissipation tends to relatively quickly synchronize the secondary’s
rotation with the orbit period (Taylor & Margot, 2014). In the synchronous
state, these dissipative forces can eventually be balanced by radiative forces
in the form of the binary YORP (BYORP) effect (Ćuk & Burns, 2005). This BYORP-
tide singly-synchronous equilibrium is a steady state found among binary
asteroids (Jacobson & Scheeres, 2011). The binary asteroid (35107) 1991VH is
an exception to this singly-synchronous paradigm, as its secondary is not only
in an asynchronous rotation state (Pravec et al., 2016), it is most likely
experiencing non-principal axis (NPA) rotation and undergoing an exchange of
angular momentum between the secondary and the orbit (Naidu & Margot, 2015;
Pravec et al., 2021). As a result, the orbit period and secondary rotation
period have evolved over observations. This intriguing dynamical state has
motivated substantial photometry and radar observations of the system. And
with 1991VH’s accessibility from Earth, it was originally selected as a target
by NASA’s SIMPLEx Janus mission (Scheeres et al., 2021). As a result, there is
a wealth of data and analysis surrounding this system. One question that
remains is the source of the excited dynamical state of 1991VH, which we
attempt to answer in this work.

Binary asteroids are characterized by strong spin-orbit coupling (Scheeres,
2006, 2009). These internal dynamics can easily excite the secondary into a
complex rotational state given an external perturbation (Agrusa et al., 2021;
Meyer et al., 2023b). Fluctuations of the spin and orbit period of a system
experiencing only principal axis rotation are generally not large enough to
explain the variations seen in the observations (Meyer et al., 2021). Thus, it
seems the most natural explanation for the current dynamical state of 1991VH
is NPA rotation within the secondary caused by an external perturbation and
driven by the internal dynamics. This dynamical state has been postulated by
Pravec et al. (2016) and Naidu & Margot (2015), and in this work we provide a
more quantitative analysis of this possibility.

The lifecycle of NEAs is chaotic and driven by orbital resonances with the
giant planets and close encounters with the terrestrial planets (Liberato et
al., 2023). Therefore a reasonable explanation for the excited dynamics of
1991VH is a recent close planetary encounter with the Earth. Fuentes-Muñoz et
al. (2022) showed that close Earth encounters with 1991VH are common, and it
has also been shown that such close encounters can excite both the internal
orbital and spin dynamics of binary asteroids (Fang & Margot, 2011; Meyer &
Scheeres, 2021). Thus, it appears all the pieces are present to provide a
satisfactory explanation of the current dynamical state of 1991VH.

We test the hypothesis that 1991VH was previously in a singly-synchronous
state, and possibly in the BYORP-tide equilibrium. This state is long-lived,
but can be broken by resonances with the primary’s spin (Wang & Hou, 2021),
destabilizing impacts on the secondary (Cueva et al., 2024), or planetary
encounters (Merrill et al., 2024). These destabilizing events can lead to non-
principal axis (NPA) rotation in the secondary, which has the effect of
weakening or eliminating the BYORP effect (Quillen et al., 2022; Ćuk et al.,
2021). For the current asynchronous secondary rotation in 1991VH, the BYORP
effect is not present.

In this work we specifically focus on the effects of a single close encounter
with the Earth as a possible explanation for the unstable dynamics in 1991VH.
This dynamical evolution has not been shown in the literature, and in the
current work we are filling this gap. To achieve this, we perform a suite of
Monte Carlo simulations, where a singly-synchronous 1991VH-like system is
perturbed by the Earth in a variety of encounter geometries. We then compare
the orbit period, secondary rotation period, semimajor axis, and eccentricity
of the post-flyby system to check for similarities with the current
observations.

We provide a background discussion of the observations and dynamics of 1991VH
in Section 2. Then, we detail the results of the Monte Carlo flyby simulations
in Section 3. In Section 4 we perform a more detailed analysis of a single
flyby case that produces a system very similar to 1991VH. We then give a
detailed analysis of how long-lived the current dynamical state of 1991VH is
expected to be in Section 5, and summarize and discuss implication in Section
6.

##  2 1991VH Observations and Dynamics

Observations of 1991VH date back to 1997 and have continued on through 2020,
both through photometric lightcurves and radar range-Doppler imaging. This
extensive set of observations has revealed the secondary in 1991VH to be in an
asynchronous, likely chaotic spin state (Pravec et al., 2016, 2021; Naidu &
Margot, 2015). Due to spin-orbit coupling, this spin state has resulted in an
evolving orbit (Pravec et al., 2021; Meyer et al., 2022a, b).

The observations of the secondary rotation period and orbit period are listed
in Table 1. These measurements range between around 11.5 to 14.2 hours for the
secondary spin period. Several estimates of the orbit eccentricity provide
$e=0.05\pm 0.02$ (Pravec et al., 2006; Naidu et al., 2012, 2018). Estimates of
the orbit semimajor axis range from $a=3.24$ km (Pravec et al., 2016) to
$a=3.26$ km (Naidu et al., 2018). Both photometric and radar observations give
a volume-equivalent diameter of the primary of 1.2 km (Pravec et al., 2016;
Naidu et al., 2018), although more recent observations suggest a smaller
diameter of 0.9 km (Nugent et al., 2016). Radar observations reveal the
primary to be top-shaped.

Photometric observations suggest the secondary has a diameter of 450 m with an
elongation of $a_{2}/b_{2}=1.33\pm 0.1$ (Pravec et al., 2016), while radar
observations give a more elongated value of $a_{2}/b_{2}=1.5$ (Naidu & Margot,
2015). Here, $a_{2}>b_{2}>c_{2}$ are the semiaxes of an ellipsoid fit to the
shape of the secondary.

Table 1: The observed secondary rotation and orbit period of 1991VH, calculated from lightcurves. Uncertainties are $\leq 0.02$ hr. The bracketed value is not unique, and solutions exist between 7 and 29 hours. Orbit periods for all 2020 dates were determined from all the January-March data. Data taken from Pravec et al. (2021). Epoch UT | Secondary Period (hr) | Orbit Period (hr)
---|---|---
1997-03-17 | n/a | 32.69
2003-02-25 | 12.84 | 32.63
2008-07-01 | (14.18) | 32.8
2020-01-27 | 11.57 | 32.50
2020-02-18 | 11.78 | 32.50
2020-02-26 | 11.55 | 32.50
2020-03-02 | 11.61 | 32.50

###  2.1 Binary Asteroid Population

The excited state of 1991VH is relatively unique among binary asteroids. Using
data from Pravec et al. (2016) which has been kept up-to-date in an online
repository (Pravec et al., 2023), we see only about $15\%$ of binary asteroids
have asynchronous secondaries. This statistic is calculated using only systems
which have a solution for both the orbit period and secondary rotation period.

We demonstrate the uniqueness of 1991VH in Fig. 1, in which we plot the
secondary spin to orbit period ratio for the population of binary asteroids,
as a function of the separation distance between the two asteroids. Among
close binary asteroids, which we define here as having $a/D_{A}<5$, 1991VH has
the smallest secondary to orbit period ratio (about 0.4). Here $a$ is the
binary asteroid semimajor axis and $D_{A}$ is the diameter of the primary.
1991VH has a separation of about 3 primary diameters. Any of the systems which
have a larger discrepancy between these periods are wide systems with orbit
periods on the order of 100 hours.

Figure 1: The secondary spin to orbit period ratio as a function of the
separation between the two asteroids. 1991VH is plotted as a star, and has a
nominal separation $a/D_{A}\approx 3$ and $T_{B}/T_{orb}\approx 0.4$. Error
bars show the range of possible values.

###  2.2 Dynamical Model

To capture the spin-orbit coupling in binary asteroids one generally uses the
full two-body problem (F2BP). This gives a 9 degree-of-freedom system, as the
attitude of both bodies must be computed in addition to the relative position.
We track the attitude of these bodies using the coordinate transformation
matrices $\mathbf{C_{A}}$ and $\mathbf{C_{B}}$, which map vectors written in
either the primary ($A$) or the secondary ($B$) body-fixed frame,
respectively, to the inertial frame. In this environment, the mutual potential
is

$U=G\int_{A}\int_{B}\frac{dM_{A}dM_{B}}{|\vec{r}-\mathbf{C_{A}}\vec{r}_{A}+\mathbf{C_{B}}\vec{r}_{B}|}$
(1)

where $\vec{r}$ is the relative position of body $B$ with respect to body $A$,
$\vec{r}_{A}$ is the position of an infinitesimal mass element $dM_{A}$ of $A$
with respect to the body’s barycenter, and similarly for $\vec{r}_{B}$ for
$dM_{B}$ in $B$. The equations of motion for the F2BP are derived in
Maciejewski (1995), and are implemented in the numerical tool General Use
Binary Asteroid Simulator (gubas) (Davis & Scheeres, 2020, 2021). gubas
integrates the F2BP using a user-specified expansion order for the mutual
gravitational potential.

While gubas is an efficient implementation of the F2BP, we can simplify the
problem further to allow for even faster calculation when appropriate. By
reducing the primary to be spherical, we can ignore its attitude. This
simplification means we will ignore the oblateness of the primary which can be
important, but in this work we are more focused on the rotation state of the
secondary. Using a second-order gravity expansion, this is equivalent to the
sphere-ellipsoid problem (Scheeres, 2006). In higher-fidelity simulations, we
will remove this simplification and return to the full primary shape. But for
now, we use the sphere-ellipsoid model, where the mutual potential energy can
be approximated to second order using MacCullagh’s formula (Murray & Dermott,
2000):

$U=-\frac{GM_{A}M_{B}}{r}-\frac{GM_{A}(I_{x}+I_{y}+I_{z}-3\Phi)}{2r^{3}}$ (2)

where

$\Phi=\frac{I_{x}x^{2}+I_{y}y^{2}+I_{z}z^{2}}{r^{2}}$ (3)

and $I_{i}$ is the principal inertia about axis $i$, and $I_{x}<I_{y}<I_{z}$.
$x$, $y$, and $z$ define the location of the spherical primary in the body-
fixed frame of the secondary.

The sphere-ellipsoid formulation allows for more efficient computation while
still allowing for three-dimensional rotation of the secondary, which will be
vital for understanding the chaotic dynamics of 1991VH. However, in a binary
asteroid, the widely-separated assumption used in MacCullagh’s formula may not
be sufficiently accurate. Ideally, the separation would be at least an order
of magnitude larger than the primary’s radius, whereas for 1991VH the
separation is only around 6-8 primary radii. For a more accurate calculation
we turn to a more traditional spherical-harmonic approximation. For an
ellipsoid, the spherical harmonics can be simply written up to degree and
order 4 (Scheeres, 2016),

$\displaystyle C_{20}$
$\displaystyle=\frac{1}{5r_{0}^{2}}\left(c_{2}^{2}-\frac{a_{2}^{2}+b_{2}^{2}}{2}\right)$
(4) $\displaystyle C_{22}$
$\displaystyle=\frac{1}{20r_{0}^{2}}(a_{2}^{2}-b_{2}^{2})$ (5) $\displaystyle
C_{40}$ $\displaystyle=\frac{15}{7}(C_{20}^{2}+2C_{22}^{2})$ (6)
$\displaystyle C_{42}$ $\displaystyle=\frac{5}{7}C_{20}C_{22}$ (7)
$\displaystyle C_{44}$ $\displaystyle=\frac{5}{28}C_{22}^{2},$ (8)

where $r_{0}$ is an arbitrary normalization radius. All other spherical
harmonic terms are zero. These can be easily used in secondary-centered
equations of motion using an algorithm such as the one defined in Gottlieb
(1993).

We will use MacCullagh’s approach to obtain Poincaré maps for 1991VH in Sec.
2.3, since the approximation is sufficient to qualitatively capture the
behavior of the system and the computational efficiency allows us to generate
more initial conditions for these maps. For the remainder of this work we will
use the spherical harmonics approach.

Given the mass parameters of the system, the equilibrium spin rate in the
sphere-ellipsoid model at a given separation distance $r$ is given by
(Scheeres, 2009):

$\dot{\theta}^{2}=\frac{GM_{A}M_{B}}{r^{3}}\left(1+\frac{3}{2r^{2}}\left(\bar{I}_{y}+\bar{I}_{z}-2\bar{I}_{x}\right)\right)$
(9)

where $\bar{I}_{i}$ is the mass-normalized inertia. This corresponds to the
spin rate of the system when it is in the perfect 1:1 spin-orbit resonance.

###  2.3 Dynamical Structure

We first explore the dynamical structure of 1991VH using a secondary with
$a_{2}/b_{2}=1.3$, $b_{2}/c_{2}=1.2$, in the sphere-ellipsoid problem. This
shape is the nominal estimate for the elongation of the secondary of 1991VH
from Pravec et al. (2016). Since there is no information on the $b_{2}/c_{2}$
for 1991VH, we choose $b_{2}/c_{2}=1.2$ as this is near the values of other
secondaries of near-Earth binary asteroids (Naidu et al., 2015; Ostro et al.,
2006). We start the system in the synchronous equilibrium near its current
dynamical state, then add eccentricity by perturbing the velocity such that
the eccentricity becomes approximately 0.05.

We test different values of the secondary spin rate, keeping the velocity
perturbation constant so the translational kinetic energy is the same for each
initial condition. Following Wisdom et al. (1984), we plot maps of the
normalized secondary spin rate $\omega_{B}/\dot{\theta}$ as a function of the
secondary’s libration angle $\phi$ at each periapsis crossing. In binary
asteroid dynamics, the periapsis is not necessarily well defined, as the
Keplerian orbit precesses at a rate comparable to the orbit rate (Scheeres,
2009) In these plots, we define a periapsis crossing as a local minimum in the
orbit separation. Thus, we adopt a purely numerical definition for the
periapsis crossing. We define the libration angle as the angle between the
secondary’s longest axis and the vector from the secondary to the primary. The
results, restricted to planar motion, are shown in Fig. 2.

These Poincare maps show spin-orbit resonances in blue, quasiperiodic dynamics
in green (in which the Poincare map will trace out a curve but never exactly
repeat), and chaotic dynamics in black. This demonstrates how different
initial conditions result in varying dynamical behavior.

These results are similar to the results obtained by Naidu & Margot (2015),
which is expected as our approach is nearly the same. We see stable equilibria
around the 1:1, 2:1, and 5:2 spin-orbit resonances. However, the perturbation
is strong enough to force overlapping of the 2:1 and 3:2 resonances, leading
to a very large chaotic sea. Several distinct initial conditions result in the
chaotic sea shown in Fig. 2. Above this chaotic sea, the system follows
predictable quasi-periodic curves. We also note the resonances are shifted
away from exactly 1, 2, or 2.5 due to the effect of libration in the system,
which shifts the secondary’s spin rate at periapsis (Naidu & Margot, 2015).

Figure 2: A map of the sphere-ellipsoid problem limited to planar motion,
plotting the normalized secondary spin rate as a function of the libration
angle at periapsis. Periodic resonances are plotted in blue, quasi-periodic
curves in green, and chaotic motion in black.

Around the 1:1 spin-orbit resonance of the full 3 dimensional problem, Agrusa
et al. (2021) showed that resonances among the natural frequencies of the
system can excite out-of-plane rotation. Thus, certain shapes of the secondary
are inherently unstable. The large uncertainty on the secondary’s shape in
1991VH allows for instabilities in the system driven by the internal spin-
orbit coupling in binary asteroids. Notably, the nominal shape of
$a_{2}/b_{2}=1.3$, $b_{2}/c_{2}=1.2$ is near a 2:1 resonance between the
nutation and libration of the secondary, and as we will show next, this system
is susceptible to chaotic motion.

While Fig. 2 presents a classical result in the spin-orbit problem, it is
limited to planar motion. However, it is well known that out-of-plane rotation
in the secondary is important in binary asteroids. Furthermore, observations
suggest the secondary of 1991VH could be in a state of non-principal axis
(NPA) rotation. So we expand our analysis to include out-of-plane rotation.
Taking the same initial conditions as those in Fig. 2, we add now a small
perturbation to the secondary’s spin axis of $(1\times 10^{-6})^{\circ}$ along
its minimum principal inertia direction. The results of these simulations are
shown in Fig. 3.

We see a significant change to the structure of the dynamics. Most
importantly, because we have increased the dimensionality by 2, the resonant
and quasi-periodic curves of the two dimensional problem are no longer
barriers to chaotic motion in three dimensions. We see some solutions around
the 1:1 resonance become chaotic, even though the spin perturbation is very
small. The chaotic sea now crosses the 1:1, 2:1, and 5:2 resonances, along
with the quasiperiodic curves. However, these quasiperiodic curves and the 2:1
and 5:2 resonances generally keep the same structure as in the planar case.
These solutions have sufficient angular momentum that the small spin
perturbation is not large enough to cause these solutions to become unstable.
The instabilities we see in Fig. 3 are not unexpected, as a linear stability
analysis predicts this behavior for a wide range of secondary shapes (Wisdom
et al., 1984).

Figure 3: A map of the sphere-ellipsoid problem perturbed into three-
dimensional rotation, plotting the normalized secondary spin rate as a
function of the libration angle at periapsis. The colors match the same
initial conditions as in Fig. 2.

This analysis demonstrates the importance of considering full three-
dimensional rotation in dynamical analyses of 1991VH, and binary asteroids in
general. Even very small perturbations can destroy the structure that exists
when the system is restricted to planar motion and rotation.

Applied specifically to 1991VH, this shows that a very small perturbation to
the orbit, sufficient to produce the observed eccentricity, could kick-start a
period of NPA rotation, even if the system was previously in the 1:1 spin-
orbit resonance. This provides one possible evolutionary pathway to place
1991VH in the dynamical state observed today.

###  2.4 External Perturbation

The most likely source of external perturbation to 1991VH is a close
gravitational encounter with a terrestrial planet. Over the last 100,000
years, there is around a 25% chance that 1991VH had a close encounter with
either Mars or Earth sufficient to increase the eccentricity to 0.1. (Fuentes-
Muñoz et al., 2022).

On average, the change in eccentricity to a binary asteroid can be calculated
by (Fang & Margot, 2011):

$|\Delta e|\approx
1.89\sqrt{\frac{G}{M_{A}+M_{B}}}\frac{M_{S}a^{3/2}}{v_{\infty}q^{2}}$ (10)

where $M_{S}$ is the mass of the perturbing planet, $a$ is the semimajor axis
of the binary, $v_{\infty}$ is the hyperbolic excess speed of the flyby, and
$q$ is the closest approach distance between the planet and the binary’s
barycenter. Using a semimajor axis of 3.25 km and a hyperbolic excess velocity
of 10 km/s, 1991VH would gain an eccentricity of 0.05 from a close Earth
encounter with a periapsis of approximately $160,000$ km. From the semi-
analytical propagation carried out by Fuentes-Muñoz et al. (2022), such an
encounter was possible no sooner than $12,000$ years ago.

The average change in semimajor axis is similarly written as (Fang & Margot,
2011):

$|\Delta a|\approx
1.48\sqrt{\frac{G}{M_{A}+M_{B}}}\frac{M_{S}a^{5/2}}{v_{\infty}q^{2}}.$ (11)

Using the same numbers for a $160,000$ km Earth flyby, this corresponds to a
change in the semimajor axis of about 140 m. Thus, on average, we would expect
the pre-encounter 1991VH to have a semimajor axis around 3.22-3.28 km, with
the flyby either increasing or decreasing the semimajor axis to its currently
estimated value.

In their high-fidelity simulations of a 1991VH-like binary asteroid undergoing
an Earth encounter, Meyer & Scheeres (2021) find a flyby distance of $100,000$
km results in an eccentricity of around $0.05\pm 0.01$. This also corresponds
to a change in the binary’s mutual semimajor axis of about $100\pm 100$ m. The
binary asteroid parameters used in that work are slightly different from those
of 1991VH, but the results are still generally consistent with the analytic
predictions.

##  3 Earth Encounter Simulations

Our hypothesis is that 1991VH was previously in a singly-synchronous state,
similar to the majority of observed binary asteroids (Pravec et al., 2016),
before a close Earth encounter added eccentricity to the system and excited
the system into its currently-observed chaotic state. To test this hypothesis,
we vary possible pre-encounter orbit periods for the singly-synchronous state,
then simulate a variety of Earth encounter conditions.

###  3.1 Simulation Inputs

In these simulations, we fix the bulk density of the system to be
$\rho=1.6$g/cm3 (Naidu & Margot, 2015), and set the primary and secondary
densities equal to one another. Thus, when we choose the pre-encounter orbit
period, this fixes the pre-encounter semimajor axis. Instead of using the
Keplerian relationship to determine the semimajor axis, we adopt the approach
used by Meyer et al. (2023b), who demonstrated the inaccuracy of the Keplerian
elements in binary asteroid dynamics due to relaxing the point-mass
assumption. Thus, we use the stroboscopic orbit period, also called the plane-
crossing orbit period, which is determined by the differences in timings of
the secondary crossing of an arbitrary inertial plane. Equivalently, this is
the time required for the secondary to complete one full revolution around the
primary in inertial space. Following Meyer et al. (2023b), we iteratively
calculate the separation distance required to achieve a desired stroboscopic
orbit period using a numerical secant search algorithm.

In these simulations, we will use the F2BP dynamics, and thus allow for the
primary to be aspherical. We set the primary $J_{2}=0.02$ (S. Naidu, personal
communication, April 2024) and assume it is axisymmetric, and use a second
degree-and-order expansion of the mutual potential. We fix the primary volume-
equivalent diameter to 1.2 km.

We generate realistic flyby geometries of 1991VH relative to the Earth by
propagating its orbit backward in time, taking into account the uncertainties
in its ephemeris. We do this using the semi-analytical NEO propagation tool
developed by Fuentes-Muñoz et al. (2022). We use the heliocentric orbit
nominal solution and full covariance from JPL’s SSD/CNEOS Small-Body
Database111Orbit solution retrieved from JPL’s SSD/CNEOS SBDB API available at
https://ssd-api.jpl.nasa.gov/doc/sbdb.html. Data accessed 03-08-2024.. This
solution uses a 29 year data arc, including 91 high-precision observations
from the Gaia Focused Product Release (Gaia Collaboration et al., 2023),
incorporated in the solution following the data treatment of Fuentes-Muñoz et
al. (2024). Even though the latter observations further constrain 1991VH’s
orbit, the long-term propagation leads to a statistical distribution of the
Earth flyby parameters. The results of this propagation for the past $100,000$
years are shown in Fig. 4, showing every close approach within $150,000$ km of
Earth. In the propagation we occasionally find close encounters with Mars and
Venus, but based in the estimated perturbation (Fang & Margot, 2011) and the
lower frequency of those encounters (Fuentes-Muñoz et al., 2022) we exclude
them from this analysis.

There are a cluster of possible flybys around $25,000$ years ago, followed by
another cluster around $40,000$ years ago. After these periods, uncertainties
grow large enough that we have a more uniform distribution in flybys. The
distribution of the longitude of the ascending note is generally uniform for
all flybys, but the hyperbolic excess velocity has a clear evolution with
time, generally decreasing into the past until around $80,000$ years ago. The
inclination of the flybys appears roughly Gaussian, and centered around
$90^{\circ}$. The argument of periapsis is centered around either
$0/360^{\circ}$ or $180^{\circ}$. We generate a uniform distribution of the
Earth-secondary phase angles at closest approach, indicating we are testing a
wide range of possible geometries Since there is no data on the system’s phase
angle, we randomly draw from all possible values.

Figure 4: The close approach distance and hyperbolic excess speed, along with
the Keplerian angles of the hyperbolic flyby trajectories as a function of
time in the past. We also plot their overall distributions as histograms.
These flybys are calculated by propagating the heliocentric orbit of 1991VH
into the past and finding close approaches with Earth. We also plot the
distribution of Earth-secondary phase angle at closest approach used in our
Monte Carlo analysis.

For the pre-encounter binary asteroid, we use the range of possible values
outline in Table 2. This gives the pre-encounter orbit period, as well as the
size and shape of the secondary. The separation distance is calculated from
the orbit period such that the secondary is in a physically circular orbit,
using the F2BP dynamics to eliminate pre-encounter libration from the
secondary.

Table 2: The physical parameters used in the Monte Carlo simulations of Earth close encounters. Parameter | Symbol | Range
---|---|---
Pre-encounter Binary Orbit Period | $T_{pre}$ | 27-37 hr
Secondary Diameter | $D_{B}$ | 400-500 m
Secondary Prolateness | $a_{2}/b_{2}$ | 1.1-1.5
Secondary Oblateness | $b_{2}/c_{2}$ | 1.1-1.5

We use the Spherical-Restricted Full 3-Body dynamical model defined and
implemented by Meyer & Scheeres (2021). This model calculates the full spin-
orbit coupling between the primary and secondary while also influenced by a
distant and large spherical perturber, in our case the Earth. We simulate the
flyby for 20 days with closest approach centered at 10 days. After this, we
hand off the results to the pure F2BP to speed up computational time without
needing to track the effect of the Earth, and simulate the system for a full
year. Meyer & Scheeres (2021) showed that plus or minus 2 days from closest
approach, the effect of the perturbing planet was negligible, so we are
conservative in our use of plus and minus 10 days from closest approach.

###  3.2 Results

As already discussed, Meyer et al. (2023b) showed the inaccuracy of Keplerian
elements in representing binary asteroid orbits. As such, we will also adopt
the use of the ‘observable’ semimajor axis and eccentricity in our results.
These quantities are defined using only the maximum and minimum separation
distances within some time interval; in this work we will use a sliding time
window of 5 days.

This still leaves the question of how best to calculate the secondary’s spin
period, which is a key piece of information provided by observations. We are
concerned with the dynamics of 1991VH that will reproduce these observations,
so our approach should mimic how the spin period is calculated in these
observations. Generally this is done using lightcurves of the secondary
(Pravec et al., 2016). We define synthetic lightcurves, which we will
calculate using the visible cross section of the secondary’s cross-sectional
area:

$A=\pi\sqrt{\hat{x}^{2}b_{2}^{2}c_{2}^{2}+\hat{y}^{2}a_{2}^{2}c_{2}^{2}+\hat{z}^{2}a_{2}^{2}b_{2}^{2}}$
(12)

where $\hat{x}$, $\hat{y}$, and $\hat{z}$ is the direction of the observer
defined in the body-fixed coordinates of the secondary.

After calculating the cross-section area, we perform a 1-degree Fourier fit
over the same 5-day sliding observation window. If the r-squared value of the
fit is at least 0.5, we keep the fit as a measurement of half the secondary
spin period. Due to the symmetry of the perfect ellipsoid, the Fourier fit
will only measure half the spin period, so we correct by a factor of 2 to
obtain the true spin period.

Our observer is fixed at an arbitrary position in inertial space, so these
synthetic lightcurves calculate the sidereal period. The real period
measurements are reported as synodic values, so we do not have an exact
comparison between the synthetic and true data. However, the difference
between the sidereal and synodic periods are less than 0.1 hour for all
observations (Pravec et al., 2006, 2021), so this difference is small. Due to
the uncertainties and chaotic dynamics in the system, we will not be able to
exactly reproduce the true measurements, so this difference is acceptable.

We perform $4,600$ Monte Carlo simulations of Earth close encounters, randomly
sampling from the distribution in Fig. 4. In each of these simulations we
calculate the stroboscopic orbit period of the post-encounter system. To match
the observations, we remove any results that have an average orbit period less
than 30 hours or greater than 35 hours. After this removal, we obtain $1,866$
results, which are plotted in Fig. 5. This plots the secondary spin period
measurements as a function of closest approach distance, with dashed lines
bracketing the observed values.

Figure 5: The secondary spin periods as a function of Earth closest approach.
The dashed lines bracket the range of observed values, and the color gradient
shows eccentricity. The observed eccentricity of 1991VH is between 0.03 and
0.07.

From Fig. 5, we see a large grouping of secondary spin periods in the 30-35
hour range. These are the systems which remain synchronous with the orbit
period after the flyby. We see most simulations fall in this category, but a
significant number of simulations do have faster or shorter secondary spin
periods, indicating these systems have become asynchronous. The number of
results falling within the observed window, bracketed by dashed lines, is
small but significant, indicating it is possible to reproduce the observed
systems after a single Earth close encounter.

We see a narrow range of flyby conditions that are able to reproduce both the
observed secondary rotation period and the orbital eccentricity. These flybys
have a close approach distance within the range of $50,000$ \- $80,000$ km.
Closer than this produces an eccentricity that is too large, while further
than this does not provide a sufficient perturbation to induce asynchronous
rotation in the secondary. This range is smaller than the analytic prediction
in Section 2.4. This suggests reproducing the observed secondary spin period
is more difficult than simply achieving the observed eccentricity.

In Fig. 6, we plot the post-encounter orbit period as a function of pre-
encounter orbit period. Again, the dashed lines bracket the observed values.
In order to achieve these observed values, the pre-encounter orbit period of
the system was likely between 28 and 35 hours, but to match the observed
eccentricity, the pre-encounter orbit period was likely not within the 30-33
hour range. Thus, the most likely pre-encounter orbit period was either 28-30
or 33-35 hours. This indicates the orbit period and semimajor axis of 1991VH
have likely not changed significantly as a result of a possible Earth
encounter in its past, and the main change is the eccentricity and rotation of
the secondary.

Figure 6: The post-encoutner orbit period as a function of pre-encounter orbit
period. The dashed lines bracket the range of observed values, and the color
gradient shows eccentricity. The observed eccentricity of 1991VH is between
0.03 and 0.07.

In our simulations, we do not find a single case in which a system remains in
planar rotation and has a secondary rotation period near the observed values.
This is another indication of chaotic rotation in 1991VH.

##  4 Example History

Here we present an example simulation providing a close match with
observations. This system was initially in a circular orbit with a separation
distance of 3.11 km and an orbit period of 29.9 hours. The secondary ellipsoid
has a volume-equivalent diameter of 458 m, and axis ratios $a_{2}/b_{2}=1.39$
and $b_{2}/c_{2}=1.17$. This system undergoes an Earth encounter with a close
approach distance of $46,580$ km with a hyperbolic excess speed of 7.6 km/s.
The hyperbolic elements are $i=78^{\circ}$, $\Omega=217^{\circ}$, and
$\omega=193^{\circ}$.

To study this history, we integrate the system forward for 10 years after the
flyby and plot the post-encounter secondary spin period and orbit period in
Fig. 7. The results are not a perfect match, but lie very close to the
observed values. The secondary spin period alternates between epochs of more
chaotic rotation and quasi-constant spin. The periods of quasi-constant spin
last for several years at around a 10 hour spin period, generally consistent
with observations. During the more chaotic epochs the spin period varies
between 10 and 40 hours, and larger values are also possible. We note the
actual observations of the secondary’s rotation period in 1991VH do not see
these longer rotation periods, although there is a possible estimate of a spin
period up to 29 hours (see Table 1).

Likewise, the orbit period fluctuates between around 31.5 and 33 hours. The
observed orbit period ranges between 32.5 and 32.8 hours. Thus, our simulated
orbit period is not an exact match, but very close. There is a clear
correlation with the secondary spin period. During the quasi-constant
secondary spin period, the orbit period is slightly less than 32 hours, but
increases during the periods of chaotic rotation.

Figure 7: The fitted secondary spin periods (top) and stroboscopic orbit
period (bottom) for one example simulation over 10 years after an Earth
encounter.

In Fig. 8 we plot the observable semimajor axis and eccentricity. The
semimajor axis ranges between 3.23 and 3.32 km. Similar to the orbit period,
these values are a close match to the true observations of 32.4-32.6 km. Again
there is a correlation with the secondary spin, with a lower semimajor axis
during quasi-constant spin and a larger value during the chaotic rotation.

The eccentricity ranges between 0.02 and 0.15. The true observed eccentricity
values range from 0.03 to 0.07. The periods of higher eccentricity correlate
the the chaotic rotation and lower eccentricity during quasi-constant spin. So
while our simulated eccentricity values can exceed these observations, they
are overall generally consistent with observations.

Figure 8: The observable semimajor axis (top) and eccentricity (bottom) for
one example simulation over 10 years after an Earth encounter.

We track the secondary’s orientation with a classical set of 1-2-3 roll,
pitch, yaw Euler angles relative to the rotating Hill frame with $z$-axis
aligned with the orbit’s angular momentum vector. In this construction, a
tidally-locked secondary would have zero rotation in all three axes. Fig. 9
shows these Euler angles over time for this example. During the period of
quasi-constant spin, the secondary is generally only librating in its roll
angle, either around $0^{\circ}$ or $180^{\circ}$. However, the other angles
are circulating, indicating even in this period of quasi-constant spin, the
secondary is still fully tumbling. During the periods of chaotic rotation, all
three angles are circulating.

Figure 9: The roll (top), pitch (middle), and yaw (bottom) angles tracking the
secondary’s orientation relative to the singly-synchronous equilibrium.

In this simulation, the secondary enters a state of fully chaotic rotation,
and in particular the barrel instability is prominent in the roll angle (Ćuk
et al., 2021). As this dynamical state closely reproduces the observations,
this is another indicator that 1991VH is indeed experiencing chaotic rotation
in the secondary.

Overall, this simulation demonstrates that due to the chaotic nature of binary
asteroids, it is possible for a single Earth encounter to transform an
equilibrated, singly-synchronous binary asteroid into a system that looks
similar to the current 1991VH.

##  5 Secular Energy Dissipation

While we have demonstrated a close Earth encounter can create a system similar
to 1991VH from a previously singly-synchronous binary asteroid, it is unclear
how long-lived this excited state will last. The earliest Earth encounter
sufficiently close to produce these dynamics was around 12,000 years ago
(Fuentes-Muñoz et al., 2022).

In the singly-synchronous configuration, for a given spin rate of the primary
the system’s energy is minimized for the amount of angular momentum in the
system. A perturbation such as a close planetary flyby will change the energy
and angular momentum such that NPA rotation is allowable, precipitating an
exchange of angular momentum between the secondary and the orbit. Since the
secondary’s angular momentum is much smaller than that of the orbit and
primary, a small excess in energy at a given angular momentum level allows for
NPA rotation.

In the literature, it has been argued that NPA rotation can be long-lived
(Wisdom et al., 1984), including specifically for binary asteroids (Quillen et
al., 2022). In their analysis, Naidu & Margot (2015) make arguments that the
asynchronous state of 1991VH can also be persistent over long times. However,
recently Meyer et al. (2023a) demonstrated that eccentricity damping in binary
asteroids can be much faster than classical analytic predictions due to the
spin-orbit coupling and the relationship between libration and eccentricity.
In this section, we provide a more concrete analysis to investigate the
question of the persistence of the NPA rotational state of 1991VH.

In the minimum-energy configuration, the equilibrium spin rate is defined by
Eq. 9. Because the secondary is in the 1:1 spin-orbit resonance in this
configuration, its spin rate relative to an inertial frame is also defined by
9. Thus, the angular momentum for this configuration is simply written as:

$H=(I_{Bz}+Mr^{2})\sqrt{\frac{G(M_{A}+M_{B})}{r^{3}}\left(1+\frac{3}{2r^{2}}\mathcal{I}\right)}+I_{Az}\omega_{A}$
(13)

where

$\mathcal{I}=\bar{I}_{By}+\bar{I}_{Bz}-2\bar{I}_{Bx}$ (14)

and

$M=\frac{M_{A}M_{B}}{M_{A}+M_{B}}.$ (15)

Here we are explicitly assuming the primary is limited to principal-axis
rotation about its major axis.

In the equilibrium configuration, the orbital velocity is $v=r\dot{\theta}$.
Using this, the corresponding minimum energy can be written as (Scheeres,
2009):

$E^{*}=\frac{1}{2}(I_{Bz}+Mr^{2})\frac{G(M_{A}+M_{B})}{r^{3}}\left(1+\frac{1}{3r^{2}}\mathcal{I}\right)-\frac{GM_{A}M_{B}}{r}\left(1+\frac{1}{2r^{2}}\mathcal{I}\right)+\frac{1}{2}I_{Az}\omega_{A}^{2}.$
(16)

When the system is perturbed away from this equilibrium, it will dissipate
energy at a constant angular momentum. As the orbit expands and the primary’s
rotation rate slows, it will eventually be constrained to synchronous rotation
in the secondary. This happens at the combination of separation and primary
spin rate defined in Eq. 16. At this point, energy dissipation will continue
as long as the primary rotates asynchronously. However, there will not be
enough energy to allow for a complex spin state in the secondary. As
dissipation continues within the primary, the separation distance $r$ will
increase while the primary’s spin rate $\omega_{A}$ will decrease.

The primary methods of energy dissipation considered are tidal torques.
Starting from the classical tidal torque equation in Murray & Dermott (2000),
Meyer et al. (2023a) generalized this to three dimensions as:

$\vec{\Gamma}_{B}=\frac{3GM_{A}^{2}M_{B}^{2}}{2r^{6}R_{B}}\left(\frac{3}{4\pi\rho}\right)^{2}\frac{k_{B}}{Q_{B}}\left(\frac{-(\dot{\vec{\phi}}-(\dot{\vec{\phi}}\cdot\hat{r})\hat{r})}{|\dot{\vec{\phi}}-(\dot{\vec{\phi}}\cdot\hat{r})\hat{r}|}\right)$
(17)

where

$\dot{\vec{\phi}}=\vec{\omega}_{B}-\vec{\omega}_{orbit}.$ (18)

The same equation can be applied to the primary, and the torque on the orbit
is equal and opposite to the sum of the torques on the primary and secondary.
As derived in Meyer et al. (2023a), this adds an acceleration to the orbit:

$\ddot{\vec{r}}=\frac{\vec{\Gamma}_{orbit}\times\vec{r}}{Mr^{2}}.$ (19)

where the torque on the orbit is simply
$\vec{\Gamma}_{orbit}=-(\vec{\Gamma}_{A}+\vec{\Gamma}_{B})$ to conserve
angular momentum in the system.

For the ratio of tidal quality factor to Love number, we initially use the
radius-dependency derived by Jacobson & Scheeres (2011). This was shown to be
a good approximation for rubble-pile asteroids (Nimmo & Matsuyama, 2019; Pou &
Nimmo, 2024), and is given as

$\frac{Q_{i}}{k_{i}}\approx 300R_{i}$ (20)

for body $i$, where $R_{i}$ is in meters. However, there is significant
uncertainty around these values, and in reality could be larger or smaller
than this value. Nominally, this gives us $Q_{A}/k_{A}=180,000$ for the
primary and $Q_{B}/k_{B}=67500$ for the secondary.

Using the results from Section 4, we integrate the equations of motion forward
for another 100 years using the sphere-ellipsoid equations of motion. Because
of the asynchronous rotation of the secondary, we do not need to include the
BYORP effect in these simulations. The percent change in total energy, free
energy, and primary spin rate in these units are plotted in Fig. 10,
demonstrating the energy dissipation. Here we are defining free energy as the
total energy minus the contribution from the primary.

In Fig. 10, we also plot the total and free angular momentum. This ensures our
system is conserving total angular momentum accurate to within $10^{-6}$
percent. However, due to the secular evolution, the free angular momentum (the
contribution of the secondary and the orbit) is increasing.

Figure 10: The percent change in total energy, free energy, primary spin rate,
total angular momentum, and free angular momentum. We split between increasing
(to plot) and decreasing (bottom plot) quantities on a logarithmic scale.

To predict the system’s future behavior, we perform a linear fit to both the
total energy and the primary spin rate. The linear fit is a conservative
estimate, as dissipation will in reality slow as the separation distance
increases. Using this fit, we can extrapolate the total energy and the
primary’s spin rate at a future time. Using the primary’s spin rate,
numerically solving Eq. 13 provides the separation distance required to
conserve angular momentum, then Eq. 16 at these values gives the energy for
the singly-synchronous configuration at this angular momentum level, what we
call the minimum energy. The difference between the total energy and this
minimum energy is the excess energy, which is plotted in Fig. 11.

As we see, the excess energy is greater than 0 for around $5,000$ years in
this case, suggesting NPA rotation would in fact be relatively short lived for
these tidal parameters. Indeed, a close Earth encounter for 1991VH did not
occur within the past $12,000$ years (Fuentes-Muñoz et al., 2022).

Figure 11: The excess energy of the system at this level of angular momentum.
When the curve reaches zero, there is no longer any excess energy and the
system must be in the singly-synchronous configuration.

To validate this predicted time for NPA rotation, we integrate the system for
$10,000$ years. The results of this integration are shown in Fig. 12, plotting
the libration angle, semimajor axis, and eccentricity. In this plot, the
libration angle is the angle between the secondary’s body-fixed $x$-axis and
the position vector of the secondary relative to the primary.

From Fig. 12, we see very good agreement with the predicted duration of NPA
rotation and these fully numerical results. The libration angle stops
circulation a little before $5,000$ years, very similar to the prediction by
our linear extrapolation. In this example, the secondary settles into the
anti-synchronous case where it has flipped $180^{\circ}$ from its initial
orientation. This plot also shows the reduction of eccentricity, which happens
much faster than predicted in analytical models (Taylor & Margot, 2011; Meyer
et al., 2023a). During this initial energy dissipation, the eccentricity is
dissipating quicker than the semimajor axis is expanding, although we do see a
secular increasing trend in the semimajor axis envelope toward the end of the
simulation.

Figure 12: The libration angle (top), semimajor axis (middle), and
eccentricity (bottom) of our example system undergoing tidal dissipation.

With the linear dissipation model validated by this detailed numerical
integration, we now extend the analysis to different values of secondary tidal
parameters. The secondary’s libration dissipation is driven primarily by the
parameters of the secondary (Murray & Dermott, 2000; Meyer et al., 2023a), so
we can limit our analysis to only these parameters while keeping the primary’s
parameters constant.

Fig. 13 plots the total time for which NPA rotation is permissible as a
function of secondary tidal parameters, keeping the tidal parameters of the
primary constant. The latest a flyby could have occurred was around $12,000$
years ago, which is plotted as a dashed black line. Assuming it was an Earth
encounter that provided the excitation to place 1991VH in an excited dynamical
state, this places a lower bound on $Q_{B}/k_{B}$ of around $2\times 10^{5}$,
although in reality it would likely be higher. This is relatively large for a
rubble pile secondary (Pou & Nimmo, 2024), and using the model of Nimmo &
Matsuyama (2019) would require a 15 m or smaller dissipative regolith layer or
less around the secondary. However, these values are still within the realm of
possibility.

Figure 13: The permissible time of NPA rotation as a function of secondary
tidal parameters. Close Earth encounters were possible starting around 12,000
years ago, which is marked as a dashed black line. This places a minimum
boundary on secondary tidal parameters assuming 1991VH was excited by a past
Earth encounter.

##  6 Discussion

In this work we have investigated the feasibility that the current excited and
chaotic dynamical state of 1991VH is the result of a previous close encounter
with Earth. We find that it is relatively easy to excite NPA rotation of the
secondary of 1991VH, and this NPA rotation is generally chaotic and can cross
between planar spin-orbit resonances thanks to the addition of two out-of-
plane rotational degrees of freedom. We find it is most likely that 1991VH was
in a singly-synchronous configuration with an orbit period between 28 and 35
hours, although probably not within the 30-33 hour range. A single close Earth
encounter around between $50,000$ km and $80,000$ km could have then placed
the system into its current dynamical state, slightly closer but still in line
with our initial analytic prediction. Because the current eccentricity of
1991VH is relatively small, this indicates the semimajor axis and orbit period
of the system did not significantly change as a result of the flyby. However,
the spin-orbit coupled dynamics are sufficient to place the secondary into its
currently excited rotation state as a result of this flyby. A flyby closer
than $50,000$ km is possible if secular dissipation has reduced the
eccentricity into the range observed today.

Using only the internal spin-orbit coupled dynamics characteristic of binary
asteroids, we find a possible range of close encounter geometries can excite
the system to the level observed today. We emphasize this requires only a
single external perturbation, and the internal system dynamics naturally
evolve the system into this state. However, there is a large range of possible
post-encounter dynamical states, and the number of systems that reproduce the
observed secondary rotation period is the minority. One point of consideration
is that 1991VH itself is a minority among the binary asteroid population, in
that it exists in this excited, asynchronous state (Pravec et al., 2016). In
our simulations, we see most flybys result in a system remaining within the
synchronous configuration, consistent with the observed binary asteroid
population. The second point of consideration is that our simulations are not
calculating the probability that 1991VH experienced this Earth encounter. This
probability was calculated in Fuentes-Muñoz et al. (2022), who find a $50\%$
chance 1991VH was significantly perturbed by a close encounter within the past
$250,000$ years. Instead, our simulations simply show that if 1991VH
experienced such an Earth encounter, this could explain its current dynamical
state.

Finally, in reality, a wider range of flyby geometries is possible, as we have
considered only a single flyby whereas in reality there the system would
experience successive close approaches over time. These successive close
encounters, along with energy dissipation, would further evolve the system and
could be responsible for the observed state of the system today.

One interesting point of discussion is how long-lived we would expect this
state of NPA rotation to last. In our analysis, we only include dissipation
caused by tidal torques, while in reality there are many mechanisms of energy
dissipation, including NPA rotation (Vokrouhlickỳ et al., 2007), tidal
saltation (Harris et al., 2009), YORP (Rubincam, 2000), and surface motion
(Agrusa et al., 2022). While these could increase the rate of tidal
dissipation, tidal torque is the strongest among these (Meyer et al., 2023a),
and these other dissipation mechanisms could in effect be wrapped within the
uncertainty on the tidal $Q/k$ parameters. These additional dissipation
mechanisms could be thought of as simply increasing the rate of dissipation
used in this work. This simplification is adopted to increase numerical
stability for our long-term simulations. Nominally, the proximity and rubble-
pile nature of binary asteroids suggests these systems can re-synchronize
quite quickly (Meyer et al., 2022b). This is contrary to the observed state of
1991VH, as the system has not undergone a close planetary encounter within the
past $12,000$ years (Fuentes-Muñoz et al., 2022). This constraint places a
lower bound on the secondary’s $Q_{B}/k_{B}$ tidal parameter:
$Q_{B}/k_{B}\gtrsim 2\times 10^{5}$. This is consistent with recent results
from Pou & Nimmo (2024), who place a general upper bound for $Q_{B}/k_{B}$
roughly around $\lesssim 1\times 10^{6}$, moderately higher than our minimum
constraint. However, their work also uses the analytical approach to
eccentricity damping (Taylor & Margot, 2011; Boué & Efroimsky, 2019; Murray &
Dermott, 2000), which was shown in Meyer et al. (2023a) to generally over-
estimate the eccentricity damping time in binary asteroids. Thus, while our
resulting $Q_{B}/k_{B}$ is slightly larger than typically found, the required
values of $Q_{B}/k_{B}$ in 1991VH are in general consistent with the current
literature. Another consideration is successive close encounters would
counteract energy dissipation by providing additional perturbations over time.
This means our calculated lower bound of $Q_{B}/k_{B}$ is a conservative
estimate.

When considering only binary asteroids that have estimates for both the orbit
period and secondary rotation period, we find around $15\%$ of these systems
have asynchronous secondaries. If we assume these systems became asynchronous
as a result of close planetary encounters, this potentially provides
information about energy dissipation rates within these systems. In their
work, Fuentes-Muñoz et al. (2022) calculate the probability of a close
planetary encounter for two binary asteroids: (35107) 1991VH and (175706)
1996FG3. For 1991VH, they found a likelihood of $15\%$ that this system
experienced a close encounter within the past $70,000$ years. For 1996FG3,
they found the same likelihood occurs around $30,000$ years ago. These are of
the same order of magnitude, so if we assume these numbers can be generalized
to the population of NEA binary asteroids at large, then $15\%$ of NEA binary
asteroids would be perturbed by these close flybys on the order of $10,000$ to
$100,000$ years ago. If we take 1991VH to be a characteristic binary binary
asteroid, this would place the tidal parameters of secondaries $Q_{B}/k_{B}$
roughly around $1\times 10^{5}$ to $1\times 10^{6}$ in order to maintain
around $15\%$ of binary asteroids to be asynchronous. However, far more data
and analysis is needed to substantiate this argument.

Due to the orbital evolution of NEAs, we argue a planetary encounter is the
most likely source for the excited dynamics in 1991VH. However, many other
possible explanations exist. For example, one or several impacts onto the
secondary could induce these unstable dynamics. This was demonstrated by the
DART impact, which induced NPA rotation in the secondary of (65803) Didymos
(Pravec et al., 2024). However, an NEA the size of the secondary of 1991VH has
an extremely small probability of experiencing a significant collision in the
past million years (on the order of $10^{-13}$) (Bottke et al., 1994). As
discussed, it is unlikely that NPA rotation persists longer than this time for
1991VH. Another possible explanation is if 1991VH was previously in a BYORP-
tide equilibrium, and a resonance between the orbit and primary spin
destabilized the rotation of the secondary (Wang & Hou, 2021). If the
secondary shape is near a resonance between the natural frequencies within the
system, this destabilization would result in NPA rotation. However, the rapid
rotation of the primary in 1991VH means the system is far from any low-order
resonance between its spin and the mean motion. Alternatively, BYORP and tidal
evolution could have migrated the system into a wider configuration where the
secondary’s rotation was no longer synchronized by the primary (Jacobson et
al., 2013). However, this scenario would then require a mechanism to reduce
the semimajor axis. Another cause of the excited dynamics could be if 1991VH
was previously a triple system whose secondaries merged, resulting in an
unstable secondary rotational state. The very elongated shape of the secondary
in 1991VH predicted by radar data could be explained by such a contact binary
(Naidu et al., 2015), although this evidence is circumstantial at best, and
there is no concrete evidence of a contact binary satellite. The contact
binary secondary of (152830) Dinkinesh could be an example of such a merger,
although its satellite is much more elongated than even the upper estimates
for 1991VH (Levison et al., 2024). Furthermore, the merger would have had to
occur relatively recently for the excited rotational state to persist to
today, and the probability of such an event is unknown. All of these scenarios
could be examined in further detail. However, the high probability of recent
and close encounters with the terrestrial planets suggests this is the most
likely cause for the unstable dynamics (Fuentes-Muñoz et al., 2022). While we
have exclusively focused on Earth encounters in this work, encounters with
Mars are also possible.

In this work we have demonstrated one possible cause of the excited dynamical
state of 1991VH. In this approach, a single Earth flyby within the past $\sim
100,000$ years is capable of producing the currently observed state in the
system. This state could then persist to the current day without energy
dissipation returning it to the archetypal singly-synchronous state, provided
the secondary is not very efficient at energy dissipation or repeated close
encounters counteract energy dissipation.

## Acknowledgements

A.J.M. acknowledges support from the Planetary Defense Conference Student
Grant. A portion of this work was conducted at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the National
Aeronautics and Space Administration (80NM0018D0004). The work by P.P. was
supported by the Grant Agency of the Czech Republic, Grant 23-04946S.

## References

  * Agrusa et al. (2022) Agrusa, H., Ballouz, R., Meyer, A. J., et al. 2022, Astronomy & Astrophysics, 664, L3
  * Agrusa et al. (2021) Agrusa, H. F., Gkolias, I., Tsiganis, K., et al. 2021, Icarus, 370, 114624
  * Bottke et al. (1994) Bottke, W. F., Nolan, M. C., Greenberg, R., & Kolvoord, R. A. 1994, Hazards due to comets and asteroids, 337
  * Boué & Efroimsky (2019) Boué, G., & Efroimsky, M. 2019, Celestial Mechanics and Dynamical Astronomy, 131, 1
  * Cueva et al. (2024) Cueva, R. H., McMahon, J. W., Meyer, A. J., et al. 2024, The Planetary Science Journal, 5, 48
  * Ćuk & Burns (2005) Ćuk, M., & Burns, J. A. 2005, Icarus, 176, 418
  * Ćuk et al. (2021) Ćuk, M., Jacobson, S. A., & Walsh, K. J. 2021, The Planetary Science Journal, 2, 231
  * Davis & Scheeres (2020) Davis, A. B., & Scheeres, D. J. 2020, Icarus, 341, 113439
  * Davis & Scheeres (2021) —. 2021, Astrophysics Source Code Library, ascl
  * Fang & Margot (2011) Fang, J., & Margot, J.-L. 2011, The Astronomical Journal, 143, 25
  * Fuentes-Muñoz et al. (2022) Fuentes-Muñoz, O., Meyer, A. J., & Scheeres, D. J. 2022, The Planetary Science Journal, 3, 257
  * Fuentes-Muñoz et al. (2024) Fuentes-Muñoz, O., Farnocchia, D., Naidu, S. P., & Park, R. S. 2024, The Astronomical Journal, 167, 290, doi: 10.3847/1538-3881/ad4291
  * Gaia Collaboration et al. (2023) Gaia Collaboration, David, P., Mignard, F., et al. 2023, A&A, 680, A37, doi: 10.1051/0004-6361/202347270
  * Gottlieb (1993) Gottlieb, R. G. 1993, Fast gravity, gravity partials, normalized gravity, gravity gradient torque and magnetic field: derivation, code and data, Tech. rep.
  * Harris et al. (2009) Harris, A. W., Fahnestock, E. G., & Pravec, P. 2009, Icarus, 199, 310
  * Jacobson & Scheeres (2011) Jacobson, S. A., & Scheeres, D. J. 2011, The Astrophysical Journal Letters, 736, L19
  * Jacobson et al. (2013) Jacobson, S. A., Scheeres, D. J., & McMahon, J. 2013, The Astrophysical Journal, 780, 60
  * Levison et al. (2024) Levison, H. F., Marchi, S., Noll, K. S., et al. 2024, Nature, 629, 1015
  * Liberato et al. (2023) Liberato, L., Araújo, R., & Winter, O. 2023, The European Physical Journal Special Topics, 232, 3007
  * Maciejewski (1995) Maciejewski, A. J. 1995, Celestial Mechanics and Dynamical Astronomy, 63, 1
  * Merrill et al. (2024) Merrill, C., Kubas, A., Meyer, A., & Raducan, S. 2024, Astronomy & Astrophysics, 684, L20
  * Meyer et al. (2022a) Meyer, A., Gkolias, I., Tsiganis, K., et al. 2022a, in AAS/Division for Planetary Sciences Meeting Abstracts, Vol. 54, 202–05
  * Meyer et al. (2022b) Meyer, A., Scheeres, D., & Benson, C. 2022b, in AAS/Division of Dynamical Astronomy Meeting, Vol. 54, 200–03
  * Meyer & Scheeres (2021) Meyer, A. J., & Scheeres, D. J. 2021, Icarus, 367, 114554
  * Meyer et al. (2023a) Meyer, A. J., Scheeres, D. J., Agrusa, H. F., et al. 2023a, Icarus, 391, 115323
  * Meyer et al. (2021) Meyer, A. J., Gkolias, I., Gaitanas, M., et al. 2021, The planetary science journal, 2, 242
  * Meyer et al. (2023b) Meyer, A. J., Agrusa, H. F., Richardson, D. C., et al. 2023b, The Planetary Science Journal, 4, 141
  * Murray & Dermott (2000) Murray, C. D., & Dermott, S. F. 2000, Solar system dynamics (Cambridge university press)
  * Naidu et al. (2018) Naidu, S., Margot, J.-L., Benner, L., et al. 2018, in AAS/Division for Planetary Sciences Meeting Abstracts# 50, Vol. 50, 312–09
  * Naidu & Margot (2015) Naidu, S. P., & Margot, J.-L. 2015, The Astronomical Journal, 149, 80
  * Naidu et al. (2012) Naidu, S. P., Margot, J., Busch, M., et al. 2012, in AAS/Division of Dynamical Astronomy Meeting# 43, Vol. 43, 7–07
  * Naidu et al. (2015) Naidu, S. P., Margot, J.-L., Taylor, P. A., et al. 2015, The Astronomical Journal, 150, 54
  * Nimmo & Matsuyama (2019) Nimmo, F., & Matsuyama, I. 2019, Icarus, 321, 715
  * Nugent et al. (2016) Nugent, C., Mainzer, A., Bauer, J., et al. 2016, The Astronomical Journal, 152, 63
  * Ostro et al. (2006) Ostro, S. J., Margot, J.-L., Benner, L. A., et al. 2006, Science, 314, 1276
  * Pou & Nimmo (2024) Pou, L., & Nimmo, F. 2024, Icarus, 411, 115919
  * Pravec et al. (2024) Pravec, P., Meyer, A., Scheirich, P., et al. 2024, Icarus, 116138
  * Pravec et al. (2023) Pravec, P., Scheirich, P., Kusnirak, P., et al. 2023, https://www.asu.cas.cz/ ppravec/neo.htm, Retrieved December 11, 2023
  * Pravec et al. (2006) Pravec, P., Scheirich, P., Kušnirák, P., et al. 2006, Icarus, 181, 63
  * Pravec et al. (2016) —. 2016, Icarus, 267, 267
  * Pravec et al. (2021) Pravec, P., Scheirich, P., Scheeres, D., et al. 2021, in 7th IAA Planetary Defense Conference, 24
  * Quillen et al. (2022) Quillen, A. C., LaBarca, A., & Chen, Y. 2022, Icarus, 374, 114826
  * Rubincam (2000) Rubincam, D. P. 2000, Icarus, 148, 2
  * Scheeres (2006) Scheeres, D. J. 2006, Celestial Mechanics and Dynamical Astronomy, 94, 317
  * Scheeres (2009) —. 2009, Celestial Mechanics and Dynamical Astronomy, 104, 103
  * Scheeres (2016) —. 2016, Orbital motion in strongly perturbed environments: applications to asteroid, comet and planetary satellite orbiters (Springer)
  * Scheeres et al. (2021) Scheeres, D. J., McMahon, J., Bierhaus, E. B., et al. 2021, in 7th IAA Planetary Defense Conference, 55
  * Taylor & Margot (2011) Taylor, P. A., & Margot, J.-L. 2011, Icarus, 212, 661
  * Taylor & Margot (2014) —. 2014, Icarus, 229, 418
  * Vokrouhlickỳ et al. (2007) Vokrouhlickỳ, D., Breiter, S., Nesvornỳ, D., & Bottke, W. 2007, Icarus, 191, 636
  * Walsh et al. (2008) Walsh, K. J., Richardson, D. C., & Michel, P. 2008, Nature, 454, 188
  * Wang & Hou (2021) Wang, H.-S., & Hou, X.-Y. 2021, Monthly Notices of the Royal Astronomical Society, 505, 6037
  * Wisdom et al. (1984) Wisdom, J., Peale, S. J., & Mignard, F. 1984, Icarus, 58, 137