# Gliders in shape-changing active matter

Akash Vardhan  School of Physics, Georgia Institute of Technology, Atlanta,
Georgia 30332, USA  Ram Avinery  School of Physics, Georgia Institute of
Technology, Atlanta, Georgia 30332, USA  Hridesh Kedia  School of Physics,
Georgia Institute of Technology, Atlanta, Georgia 30332, USA  Shengkai Li
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332,
USA  Kurt Wiesenfeld  School of Physics, Georgia Institute of Technology,
Atlanta, Georgia 30332, USA  Daniel I. Goldman  School of Physics, Georgia
Institute of Technology, Atlanta, Georgia 30332, USA  School of Physics,
Georgia Institute of Technology, Atlanta, Georgia 30332, USA  School of
Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA  School
of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332,
USA  School of Physics, Georgia Institute of Technology, Atlanta, Georgia
30332, USA  School of Physics, Georgia Institute of Technology, Atlanta,
Georgia 30332, USA

(August 28, 2024)

###### Abstract

We report in experiment and simulation the spontaneous formation of
dynamically bound pairs of shape changing smarticle robots undergoing locally
repulsive collisions. Borrowing terminology from Conway’s simulated Game of
Life, these physical ‘gliders’ robustly emerge from an ensemble of
individually undulating three-link two-motor smarticles and can remain bound
for hundreds of undulations and travel for multiple robot dimensions. Gliders
occur in two distinct binding symmetries and form over a wide range of angular
flapping extent. This parameter sets the maximal concavity which influences
formation probability and translation characteristics. Analysis of dynamics in
simulation reveals the mechanism of effective dynamical attraction – a result
of the emergent interplay of appropriately oriented and timed repulsive
interactions.

Active matter studies focusing on collisional interactions among entities
reveal fascinating emergent behaviors. Molecular motors [1, 2], self-propelled
rods, [3] and shaken grains [4, 5] demonstrate that patterns can form and
overall order can emerge from purely repulsive interactions. And active
extended objects undergoing shape-changing motions have appeared as a novel
class of systems with their own repertoire of rich behavior,especially in
regard to their locomotion [6, 7, 8, 9, 10]. Novel behaviors such as
mechanical diffraction [11, 12, 13] and emergent locomotion [14, 15, 16] have
been observed in shape-changing active matter systems. Furthermore, emergent
collective behaviors such as collision-mediated gait synchronization in
clusters of nematode worms [17] and in undulatory robots [18], and collective
transport by worm blobs [19], have also been observed in this novel class of
systems.

Figure 1:  Observation of gliders in relaxation experiments. (A)
Configurations of a single smarticle at the corners of a square gait. (B) Each
smarticle performs a continuous square gait in its shape space
($\alpha_{1},\alpha_{2}$) with amplitude $\alpha_{max}$. Arm angles are
defined such that equal values are mirror-symmetric configurations. A
smarticle rests on its middle body with its arms mounted slightly raised above
the ground. The direction of red arrows represent the chirality of the gait
executed. (C) An unconfined collective of 7 smarticles in a compressed
configuration, at the beginning of the experiment ($t=0$). (D) Spontaneous
emergence of a locomoting bound pair (glider) from the collective as seen from
smarticle body center tracks. (E) Snapshots of exemplary glider configurations
C1 and C2. Arrows represent the body normal vectors of individual smarticles;
colorbar is time in seconds.

Smarticles—smart active particles [20, 10, 14, 15, 21, 19], are a convenient
system to study shape-changing active matter. These two motor, 3 link robots
resemble Purcell’s swimmer [22] in their structure and their gait. However,
they differ from swimmers at low Reynolds number [22, 7] or their terrestrial
counterparts [9], since an isolated smarticle has limited motility as its arms
do not interact with flat ground (see Fig. 8) Despite such restrictions,
previous studies on collections of smarticles revealed diverse behaviors
resulting solely from collisions. These included expansion, contraction and
diffusion of unconfined “clouds” of smarticles and directed locomotion of
confined smarticles (a phototaxing “supersmarticle” [15, 10]); further these
robots were used to test a new principle of active collectives, the
spontaneous selection of “low-rattling” states [21].

Here we study in laboratory experiment and numerical simulation unconfined
collections of smarticles and report the discovery of _gliders_ —dynamically
bound pairs of smarticles which form spontaneously from a cloud and travel for
long distances, interacting solely via locally repulsive rigid body
collisions. These gliders are reminiscent of ‘Gliders’ in Conway’s CA Game of
Life [23], which are patterns that move via periodic changes to their
configuration. Although an individual smarticle has limited motility,
smarticle gliders travel superdiffusively as a result of the collisions caused
by the periodic shape-changing motions of the smarticles. Experimentally
validated simulations give insight into glider formation and transport
properties, revealing principles that could potentially guide the development
of dynamically entangling collectives for self assembly and transport in
shape-changing active matter.

_Experimental apparatus._ The smarticles studied have a mass of _m_ = 34.8
$\pm$ 0.6g, and are composed of three links: two side arms of length _L_ = 5.4
cm and thickness of 0.3 cm, and a central link (body) of length _W_ = 5.0 cm
and thickness of _D_ = 2.2 cm. The smarticles were placed on a $60\times
60\,\mathrm{cm}^{2}$ aluminium plate, levelled to $\leq 0.1^{\circ}$. The
gait, i.e., the sequence of shape-changing motions, of each smarticle,
depicted in Fig. 1A,B, was inspired by the dynamics of Purcell’s three-link
swimmer [22, 24]. The periodic shape-changing moves are achieved by actuating
the revolute joints connecting the central link to the side arms with a
programmable servo motor with a gait period of 1.6 sec. The side arms were
rotated at the maximal motor speed until reaching the target arm rotation
amplitude angle. Since the moving arms of the smarticle rest above the central
link, they rotate without interacting with the ground, and the inertial
impulses they impart on the central link typically induce negligible motion.
The friction between the smarticle and the underlying surface limited the
motion induced by the arm rotations to about 0.0015 $W$ (0.0075 cm) per cycle,
where $W$ is the length of the central link. Therefore, individual smarticles
do not move significant distances on their own.

_Observation of Gliders._ A collection of 7 smarticles was initialized in a
densely packed configuration, where all smarticles began their shape-changing
motions at the same time and in phase, as shown in Fig. 1C. We hypothesized
that the smarticles would push each other away via collisions thus expanding
to a relaxed state where no smarticles are in contact with each other.
Although previous studies of cloud dynamics [15] had hinted at geometry-
induced cohesion via fluctuations in the area fraction. It was frequently
observed (77 out of 151 total trials) that a pair of smarticles would move
together, as shown in Fig. 1D (Supplementary Movie 1), for $\geq 100$ gait
periods, traveling an average distance of $2.9\pm 0.2$ $W$. The formation,
stability and locomotion of these _gliders_ was remarkable given that the
smarticles only interacted via collisions which are locally repulsive. The
discrete and highly nonlinear nature of their collisions made it challenging
to analyze them theoretically. In order to systematically study the gliders, a
simulation was developed based on the open source physics engine DART [25]
(see SM for details), and calibrated against experiment (see Fig. 5). In the
following sections we discuss the anatomy, robustness, binding mechanism, and
locomotion of the gliders.

Figure 2: Glider configurations. (A) Glider described by a pair of smarticles
with relative position in polar coordinates ($r,\theta$), and relative body
orientation ($\phi$) calculated by taking the angle between the two normal
vectors. (B) Relative coordinates observed for a glider over 7 gait periods.
(C-D) In both experiment and simulation, distributions of relative coordinates
observed each gait period display two distinct clusters, C1 and C2.

_Glider Configurations._ Several (N = 151) trials of experiments and
simulations revealed that the gliders existed in two distinct configurations.
These configurations bear resemblance to the anti-phase and in-phase modes
observed in Huygens’ clocks [26] and oscillons [27] in vibrated granular
media. We refer to them as :(i) “$C1$” (anti-phase), in which the two
smarticles are almost anti-aligned one with the other, their orientations
differing by approximately $180$ degrees and (ii) “$C2$” (in-phase) in which
the two smarticles are oriented in approximately the same direction. Snapshots
of glider configurations are depicted in Fig. 1E. To characterize these
distinct configurations, we measured their inter-smarticle distance
$|\vec{r}|$, the polar angle $\theta$ and their relative orientation $\phi$,
defined in Fig. 2A. The time evolution of the relative coordinates $|r|$,
$\theta$ and $\phi$ over 7 periods for a glider trajectory (Fig. 2B) showed
consistent behavior over many periods, indicative of a limit cycle
[eldering_role_2016]. The distribution of the relative coordinates of gliders
observed in experiments and simulation showed distinct peaks in their
histograms (Fig. 2C-D), corresponding to the distinct glider configurations
$C1$ and $C2$ (Supplementary Movie 2). Further, the two glider configurations
possessed distinct lifetimes (Fig. 6), with $C1$ gliders persisting longer
than $C2$ gliders. Trials were conducted for 300 gait periods, and in some
cases a $C1$ glider was still bound and locomoting when the trial was stopped.
The lack of perfect synchronization in the gaits of the smarticle pair forming
the glider in experiments lead to a higher variance in the distribution of the
relative coordinates $r,\phi$ as compared to simulations (see Fig. 2C-D).

Figure 3: Bound pair displacement dynamics. (A) MSD
$\langle\sigma^{2}(t)\rangle$ vs time-delay for 5 arm amplitude values with
snapshots of the bound pairs shown besides the plot. (B) Probability to bind
_$P_{b}$_ vs. arm amplitude, for random initial configurations; blue curve
represents the bound pairs found active at the end of the trial, red curve
represents pairs that remained active throughout the trial. (C) MSD Exponents
($\beta$) vs. $\alpha_{max}$ for bound pairs, from msd slopes fit at long time
delays. Blue and orange curves show experiment and simulation data,
respectively; Bars represent standard-deviation. The black curve is the
exponent calculated for a time symmetric gait (S1 to S3 to S1, see Fig. 1A) in
simulations. (D) Bound pair center-of-mass speed ($V_{com}$) vs.$\alpha_{max}$
. (E) Steady state separation $|r|$ vs. $\alpha_{max}>45^{\circ}$, from
simulations of smarticles started off in bound states.

_Glider robustness and transport._ To test the robustness of bound-pair states
and their subsequent locomotion, we systematically varied the arm amplitude,
i.e. the maximum rotation angle $\alpha_{max}$ of the smarticle arm, while
keeping the smarticle arm rotation speed fixed and studied the persistence of
bound pairs for shapes with varying degrees of concavity [28]. We performed
experiments and numerical simulations, fixing one smarticle and initializing
the other in a polar grid of constant radius with 1000 points in simulations
and 21 points in experiments (see Fig. 7) for a range of $\alpha_{max}$. We
observed a rich repertoire of bound states i.e., where smarticles made contact
at least once during their gait, for all amplitudes, suggesting that these
bound states robustly form for a variety of $\alpha_{max}$.

The mean-square displacement (MSD) vs time plot of all glider trajectories
(surviving greater than 30 cycles) for 5 different arm amplitudes is shown in
Fig 3A. The bound pairs for $90^{\circ}$ and $10^{\circ}$ move nearly
ballistically with the $\langle{\sigma}^{2}(t)\rangle\propto t^{\beta}$ (where
MSD
$\langle{\sigma}^{2}(t)\rangle=\langle\vec{x}^{2}(t)\rangle-{\langle\vec{x}(t)\rangle}^{2}$
with exponent $\beta\approx 1.8$), while for the intermediate values of arm
amplitude the motion is almost diffusive. We observed that gliders studied in
experiments have a higher $\beta$ and $V_{com}$ gliding speed (speed of the
com of the 2 particle bound excitation) than those studied in simulations as
shown in Fig. 3. We posit that the noise in experiments from the motors, the
collisions and the friction with the floor enhances the transport of the bound
pairs with intermediate arm amplitudes ($40^{\circ}-60^{\circ}$), but the
pairs bind and unbind frequently. This lack of stability for intermediate
$\alpha_{max}$ correlates with a decrease in binding probability (Fig. 3B).
The different bound states in experiments and simulations for the three
different amplitude regimes are shown in Supplementary Movie 3.

Bound pair speed and transport decreased sharply as $\alpha_{max}$ decreased
from $90^{\circ}$, but surprisingly increased again in the low amplitude
regime (see $10^{\circ}-20^{\circ}$ range in Fig. 3C-D). We attribute this
anomaly to inertial effects which enhanced the drift of a single smarticle as
shown in Fig. 8. Since the smarticle arms were driven at a constant motor
speed, for low rotation angles the time of rotation of a smarticle arm became
comparable to the time of overlap between the motions of two arms as shown in
Fig. 9. This led to a significant increase in the motility of an individual
smarticle. Consequently gliders at low $\alpha_{max}$ also showed an
improvement in their gliding speed as shown in Fig. 3(C-D).

To probe how arm amplitude affects glider locomotion, we performed simulations
of pairs of smarticles in a bound formation, performing gaits with varying
$\alpha_{max}$ for values greater than $45^{\circ}$. A clear trend appears,
where the distance monotonically decreases with increased $\alpha_{max}$ Fig.
3E, with a rapid drop in distance around $85^{\circ}$. Coincidentally, this is
also roughly the amplitude at which the $\beta$ and $V_{com}$ shoot up (Fig.
3C-D). Simulations revealed that for arm amplitudes below $85^{\circ}$ the
smarticles only interact with one arm, and above they experience an additional
collision – with the other arm, as depicted in the middle configuration of the
attractive case in Fig. 4C and Supplementary Movie 3. The additional collision
seems to result in closer proximity between the smarticles, which leads to a
third collision between an arm and the middle link. Since super-diffusive
transport becomes effective above arm amplitude of $85^{\circ}$, we attribute
most of the locomotion to these additional collisions.

_Binding mechanism and factors responsible for gliding._ Though two convex-
shaped bodies interacting only via repulsive collisions must repel ( Fig. 4A),
pairs of smarticles undergoing repulsive collisions can remain bound since
they develop concave shapes during their shape-changing motions. This makes it
possible for the net contact impulse during a collision of two smarticles to
have an attractive effect $((\vec{I}_{N}+\vec{I}_{F})\cdot\vec{r}>0)$ for
atleast one collision during the gait, typically this is when the 2 red arms
collide for the C1 glider (Fig. 4A) and results in the two smarticles coming
closer. Friction between the smarticles and the base plate prevents coasting,
and the periodic attractive contact forces during collisions ensure they
remain bound over multiple gait periods. Most of the stable glider
configurations converge to the periodic deformation pattern characteristic of
the attractor within a few periods; moreover, their ultimate fate – binding or
not – could be determined from the configuration following the first gait
period - which orients the robots such that they are either attracted or
repelled by the subsequent collisions. We show the first 5 periods of 2
representative initial conditions stroboscopically at the same phase of the
gait with the corresponding $|r|$ values (Fig. 4B) to illustrate how
collisions in the same phase of the gait cycle could produce binding vs.
repulsion depending on how the smarticles started off relative to each other.

To identify the conditions leading to binding, we performed simulations for
5000 uniformly randomly sampled configurations in the $(|r|,\theta,\phi)$
space (Fig. 4C). Once initialized, we tracked the lifetimes of bound pairs for
75 gait periods and separated the relative configuration space into regions of
attraction vs. repulsion based on the fate of the corresponding smarticle duo.
The glider configurations $C1$ and $C2$ correspond to attractors in these
selected regions of the configuration space. Fig. 4D shows the structure of
the attractor in chosen coordinates, with corresponding projections on the
axes.

Figure 4: Interactions leading to binding and locomotion. (A) Schematic
showing how locally repulsive collisions can produce attraction in extended
rigid bodies undergoing shape change contrasted with point particles. Color
represents current frame while grey indicates past frame. (B) Stroboscopic
representation of 2 initial conditions leading to attraction and repulsion
respectively for the first 5 periods with the corresponding $|r|$. Blue cones
represent the trajectory of the arms 20 frames into the future, while black
cones represent the same 20 frames in the past. (C) Simulations of smarticle
pairs initialized in random configurations (all points), of which many led to
stable gliders (red points). (D) At steady state the configurations
corresponding to the stable gliders show up as distinct attractors for the C1
and C2 type gliders. (E) Role of relative position and orientation E(i,ii) and
gait chirality E(iii,iv) (asymmetries) on glider locomotion.

The basin of attraction for the $C1$ glider was larger than that for the for
the $C2$ type glider. As a result, anti-aligned smarticle pairs were likely to
form a $C1$ type glider if initialized sufficiently close together. Fig. 7 (A)
shows the schematic for the initial condition scan and Supplementary Movie 4
shows the flow of the initial configurations onto the attractors that form the
gliders. Though glider formation is highly likely, the boundary between
attractive and repulsive initial conditions depends on relative angle and
position, at a given phase, in a non-trivial way. See Fig. 7 (B-C) for
details.

Glider trajectories are governed by gait chirality and configurational
asymmetry. The chirality of the performed gait results in a consistent pattern
of arm to arm collisions where, starting from a non-overlapping configuration
(“facing away”, as in Fig. 4E(i)), the first arm to collide with the opposing
arm is always the same one (shown in red). When smarticles are programmed to
perform a gait with the opposite chirality, they form glider pairs that engage
first with the opposite arm (Fig. 4E(ii)). Gliders of a given chirality are a
mirror image of gliders of the opposite chirality. It is therefore expected
that the resulting trajectories will diverge in opposing directions, and as
seen in multiple trials, they do (Fig. 4E(i, ii)). In addition, the chirality
of each gait appears to manifest in the chirality of the resulting
trajectories. Imposing a time reversible gait makes the system lose persistent
locomotion and the impending motion of the bound state turns diffusive. We
performed experiments and simulations with time symmetric gaits (diagonal in
the configuration space, S1 to S3 to S1 in Fig. 1) for varying arm amplitude,
as mentioned above, and observed that the value of $\beta$ was close to one
(black curve in Fig. 3 C). See Supplementary Movie 5 for an example of a time-
symmetric gait.

Although glider pairs appear symmetric under $180^{\circ}$ rotation (and half
gait period shift for C2). Such symmetry should not result in a directional
bias, yet gliders exhibit persistent, nearly ballistic, locomotion, in which
one smarticle consistently appears somewhat ahead of the other. Upon
observation it is revealed that relative orientation between the smarticles
deviates from $180^{\circ}$ (Fig. 4E(iii)), in a consistent manner. The
direction of this deviation from (anti-)alignment controls the direction of
the trajectory, as evident from multiple trials (Fig. 4E(iii, iv)). This is
expected, since the configuration in Fig. 4E(iv) can be obtained by an almost
$180^{\circ}$ rotation of the configuration in Fig. 4E(iii), this effect is
also evident in the structure of the C1 attractor as the two clusters which
have an apparent reflection symmetry (Fig. 4D).

_Conclusion._ We studied ensembles of shape-changing robots (smarticles) in
experiments and simulations, and reported the novel phenomenon of _gliders_
—spontaneously formed dynamically bound pairs of smarticles that travel
together with almost ballistic locomotion, over multiple body lengths,
enduring hundreds of rigid-body collisions between them. We characterized
gliders by the relative spacing and orientation between their smarticles and
showed that there is a vast number of initial configurations that bind and
locomote. We attribute the nearly ballistic nature of glider motion to the
breaking of time-reversal symmetry by their gait, as well as the spontaneously
broken rotational symmetry of their configuration. Further, we showed that
gliders exist and travel superdiffusively for a wide range of arm amplitudes,
a parameter which essentially controls the degree of concavity manifest by
smarticles, as they changed their shape. Surprisingly these physical gliders
also display some other unexpected similarities with their CA counterparts,
like assembling into long structures upon collision with other smarticles in
their vicinity (Supplementary Movie 6). Our study suggests that gliders are
likely to arise in a wide variety of active matter systems where shape-
changing particles take concave shapes with time irreversible gaits in a
sufficiently damped environment and adds further evidence of rich dynamics in
active collisions.

_Acknowledgments:_ We thank Will Savoie for discovery of gliders, Pavel
Chvykov and Zachary Jackson for stimulating discussions on smarticles, Yunbo
Zhang for help with learning DART, and all members of the lab for support on
miscellaneous fronts. Support provided by ARO MURI grant W911NF-13-1-034,
School of Physics Georgia Institute of Technology, QBIOS PhD program and a
Dunn Family Professorship (D.I.G).

Gliders in shape-changing active matter
Supplementary Information

Akash Vardhan Ram AvineryHridesh KediaShengkai Li

Kurt Wiesenfeld

Daniel I. Goldman

August 28, 2024

##  I SUPPLEMENTARY INFORMATION

_Comparison between simulation and experimental tracks._ A simulation of the
smarticles was created using the open source physics engine Dart to seek
understanding of the phenomena. The rigid body equations of motion are
formulated as an implicit time-stepping, velocity based LCP (Linear
complimentarity problem) which are then solved using standard constraint
solvers like the Lemke and Dantzig methods, after [29] . The time step used in
the simulations was 1e-4 seconds which guaranteed the best convergence of
trajectories between simulations initialized from a set of initial conditions.
Fig. 5 shows the tracks of the body centers of the 2 smarticles for 90 periods
in experiment and simulation. Experimental tracks tend to curve more which can
be attributed to the various sources of noise in the real world, in our
simulations we did not add any external noise (over the accumulated numerical
error), since we wanted to understand the phenomena in a cleaner more pristine
setting, the tracks in the simulation have much lower direction modulation
consequently, which also shows up as tighter peaks in the relative coordinate
PDF’s as shown in Fig. 2 C.

Figure 5:  Representative trajectories from an experimental and simulated
trials over 90 gait periods.

_Lifetimes and contour lengths of emergent bound pairs._ It usually took the
closed packed configurations about 30 periods to reach steady state. So, to
characterize the lifetime of our bound pairs we decided to plot the lifetimes
vs. displacements (contour length) of all 2 smarticle bound pairs Fig. 6 A and
observed a linear correlation between the two suggesting that the longest
surviving bound pairs would also travel the largest distance. We then
characterized the lifetime of individual glider configurations and observed
that the C1 type gliders were more stable and lived longer than the C2 counter
parts Fig. 6 B. To measure the displacements of individual glider types we
initialized the smarticles from known configurations, and observed that for
the C1 glider the 2 smarticles would travel roughly the same distance Fig. 6
C, while in the C2 type glider there would constantly be one smarticle which
would end up travelling more than the other as they would curve as seen in
Fig. 6 D and Supplementary movie 2. The C2 configuration was susceptible to
tiny phase differences in the arms of the 2 smarticles, and required near
perfect synchronization, as a result the 2 smarticles would break apart from
the C2 configuration causing them to have a lower lifetime than their C1
counterparts, as shown in Fig. 6B and supplementary movie 2.

Figure 6:  Trajectory contour length and lifetime for emergent gliders. (A)
Contour length vs. contour time for all bound pairs emerging from the
relaxation experiments (see Fig. 1), for experimental trials recorded over 300
gait periods. (B) Lifetimes of the glider configurations in the relaxation
experiments. (C-D) Contour lengths of C1 (C) and C2 (D) type gliders after
initializing them from known initial configurations, over 21 trials.

_Pictorial representation of basin scan._ To probe the binding mechanism and
the robustness of gliders we initialized smarticles, by fixing one and varying
the position of the second one around it in polar coordinates and also the
absolute orientation and separated the configuration space into regions of
attraction vs. repulsion. This provided us with the control and resolution to
zoom into the events within a gait period and understand how the same
collisions bring about the 2 effects due to the geometrical configurations
they start off with. Fig 7 (A-B) shows how we set up the scan, and the
corresponding configurations which got attracted vs. repelled. To demonstrate
positional angle and orientation dependence, we show scan results in Fig 7
(B-C), for a fixed radius of 1.3 $W$. Fig 7 (C) shows a phase diagram in the
initial angular coordinate space, with cyan corresponding to the repelled
configurations, red corresponding to configurations that ended up forming
bound pairs and white being the invalid configurations, of the physical
configurations sampled in Fig 7 B. The attracted configurations seem to be
almost symmetrically distributed about the normal vector of the fixed
smarticle, and this showed up as the symmetry in the attracted configurations
about ($\theta\approx 270^{\circ})$ .

Figure 7:  Configurations leading to attraction. (A) Smarticle pair
configurations were sampled on a polar grid of positions on the the upper half
circle around a fixed smarticle. At each position, the orientation of the
placed smarticle was uniformly sampled. The polar plot at each position
represents valid (nonintersecting, filled - gray or blue) or invalid (white)
configurations, where blue (gray) indicates attraction (repulsion). (B)
Smarticle pair configurations were also sampled at a constant radius, in all
positional angles and uniformly over orientation. Polar plots are same as in
A. (C)._Angular coordinate phase diagram._ Initial configurations in B that
led to attraction are depicted as red, while those that led to repulsion are
depicted as cyan points, with white being the physically invalid
configurations over the positional ($\theta$) and orientation ($\phi$) angles.
A dashed line of $\phi=(360-\theta)^{\circ}$ was added as a visual aid.

_Single Smarticle Locomotion vs Amplitude._ The anomalous transport of the
bound pairs that were formed at lower arm opening angles was attributed mostly
due to the enhanced drift of a single smarticle at lower amplitudes. We wanted
to characterize the behavior of a single robot as a function of the amplitude.
Fig 8 (A-B) depict the body center tracks and the angular drift of a single
robot for the different gait amplitudes. The motion of a single smarticle is
super-diffusive at higher and intermediate amplitudes and approaches the
ballistic regime for lower values of the amplitude and it caused an increased
drift speed, as shown in Fig 8 (C-D).

Figure 8: Motion of individual smarticles as a function of gait amplitude. (A)
Tracks of the body center of the smarticle for different gait amplitudes, over
75 gait periods. (B) Time series for the angular coordinate of the smarticle.
(C) $\beta_{single}$ (MSD exponent) of a single smarticle vs. arm amplitude.
(D) Speed of the body center vs. arm amplitude.

_Experimental track of arm angles of single smarticle._ We hypothesized that
this anomaly was due to inertial effects from the overlap in the motions of
the 2 arms, since the gait was executed at the maximal motor speed for all arm
amplitudes and we adjusted the dwelling time at the corners of the square to
ensure the time period scaled commensurately with the amplitude. At lower
amplitudes, the value of this delay was so small that the arms wouldn’t get to
the prescribed angle, and the other arm would move before the first arm
completed its motion. This caused a drift in the center of the middle link.
Fig. 9 (A,C) shows experimentally tracked arm angles of a single smarticle for
90 and 10 degrees, and when plotted in the shape configuration space 9 (B,D),
it becomes clear that in the low amplitude regime the arms don’t finish the
whole prescribed range of motion.

Figure 9:  Experimentally tracked arm angles for 90 and 10 degree amplitude.
(A,C) Time-series of the first arm angle ($\alpha_{1}$) over 3 gait periods
for 90∘ (A) and a single period for 10∘ (C) degree arm amplitudes. Blue and
orange curves represent arms 1 and 2 respectively. (B,D) Gait represented in
arm-angle configuration space ($\alpha_{1},\alpha_{2}$), averaged over 10
periods, for 90 (B) and 10 (D) degree arm amplitudes.

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# Coronal Line Emitters are Tidal Disruption Events in Gas-Rich Environments

Jason T. Hinkle1, Benjamin J. Shappee1, and Thomas W.-S. Holoien2
1Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Dr., Honolulu,
HI 96822, USA
2The Observatories of the Carnegie Institution for Science, 813 Santa Barbara
St., Pasadena, CA 91101, USA
<EMAIL_ADDRESS>Einstein Fellow

###### Abstract

Some galaxies show little to no sign of active galactic nucleus (AGN)
activity, yet exhibit strong coronal emission lines (CLs) relative to common
narrow emission lines. Many of these coronal lines have ionization potentials
of $\geq 100$ eV, thus requiring strong extreme UV and/or soft X-ray flux. It
has long been thought that such events are powered by tidal disruption events
(TDEs), but owing to a lack of detailed multi-wavelength follow-up, such a
connection has not been firmly made. Here we compare coronal line emitters
(CLEs) and TDEs in terms of their host-galaxy and transient properties. We
find that the mid-infrared (MIR) colors of CLE hosts in quiescence are similar
to TDE hosts. Additionally, many CLEs show evidence of a large dust
reprocessing echo in their mid-infrared colors, a sign of significant dust in
the nucleus. The stellar masses and star formation rates of the CLE hosts are
consistent with TDE hosts, and both populations reside within the green
valley. The blackbody properties of CLEs and TDEs are similar, with some CLEs
showing hot (T $\geq 40,000$ K) blackbody temperatures. Finally, the location
of CLEs on the peak-luminosity/decline-rate parameter space is much closer to
TDEs than many other major classes of nuclear transients. Combined, these
provide strong evidence to confirm the previous claims that CLEs are indeed
TDEs in gas-rich environments. We additionally propose a stricter threshold of
CL flux $\geq 1/3$ $\times$ [O iii] flux to better exclude AGNs from the
sample of CLEs.

###### keywords:

accretion, accretion discs — ISM: dust — galaxies: active — galaxies: emission
lines — galaxies: nuclei

††pubyear: 2023††pagerange: Coronal Line Emitters are Tidal Disruption Events
in Gas-Rich Environments–Coronal Line Emitters are Tidal Disruption Events in
Gas-Rich Environments

##  1 Introduction

Supermassive black holes (SMBHs) reside in the centers of nearly all massive
galaxies (e.g. Kormendy & Richstone, 1995; Magorrian et al., 1998; Ho, 2008;
Kormendy & Ho, 2013). While direct detections of quiescent SMBHs are difficult
to obtain, their presence can be illuminated when a star passes within the
tidal radius of a SMBH and is torn apart, powering a luminous accretion flare
(e.g., Rees, 1988; Phinney, 1989; Evans & Kochanek, 1989; Ulmer, 1999). Such
events are known as tidal disruption events (TDEs; e.g., van Velzen et al.,
2011; Holoien et al., 2014; Gezari, 2021).

Many optically-selected TDEs seem to prefer host galaxies where a recent burst
of star formation has occurred (e.g., Stone & van Velzen, 2016; French et al.,
2016). These include quiescent Balmer strong and post-starburst galaxies, both
likely post-merger systems (French, 2021). The enhanced TDE rate is likely due
to the high stellar densities in the cores of these post-merger galaxies
(French et al., 2020). However, some studies suggest that the TDE rate can be
even higher for systems with higher current star-formation rates, especially
luminous infrared galaxies (Tadhunter et al., 2017; Mattila et al., 2018; Kool
et al., 2020), where high nuclear stellar densities may also be present due to
the ongoing or very recent merger funneling gas to the nucleus.

TDEs exhibit a wide range of emission across the electromagnetic spectrum, but
nevertheless have clear defining features as a class. The majority of TDEs
display luminous UV/optical emission which is well-fit by a blackbody with
$T\sim 15,000-50,000$ K (e.g., Gezari et al., 2012; Holoien et al., 2014,
2016a). Many TDEs also exhibit X-ray emission (e.g., Holoien et al., 2016a;
Wevers et al., 2019; Hinkle et al., 2021a) well-modeled by a hotter blackbody,
with kT $\sim 30-60$ eV (Auchettl et al., 2017). TDE light curves often
exhibit a smooth rise and monotonic decline (e.g., Auchettl et al., 2018;
Holoien et al., 2020; Hinkle et al., 2021a; van Velzen et al., 2021b),
generally without stochastic variability. Additionally there appears to be a
correlation between the peak luminosity of a TDE and its decline rate after
peak (Hinkle et al., 2020; Hammerstein et al., 2023).

One class of transients that has been linked to TDEs are the so-called coronal
line emitters (CLEs; Komossa et al., 2008; Wang et al., 2011; Wang et al.,
2012; van Velzen et al., 2021a). Observationally, these events are
characterized by strong high ionization coronal lines (CLs) such as [Fe x]
$\lambda$6376, [Fe xi] $\lambda$7894, [Fe xiv] $\lambda$5304, and [S xii]
$\lambda$7612, with ionization potentials of up to several hundred eV. In
particular CLEs are characterized by CLs with strong emission relative to
common AGN lines such as [O iii] $\lambda$5007 (e.g., Wang et al., 2012).
While AGNs are known to sometimes power strong high ionization CL emission
(e.g., Negus et al., 2021), the spectral shapes and emission line ratios of
these CLEs are generally not consistent with strong AGNs.

The most likely explanation for these events are TDEs occurring in gas-rich
environments, where the strong extreme UV and soft X-ray flux from TDEs can
photoionize gas present in the host galaxy. To date, the connection of CLEs to
TDEs has been largely circumstantial, relying on the transient nature of the
CLs (Komossa et al., 2009; Yang et al., 2013), a TDE-like decline slope for a
small number of cases (Palaversa et al., 2016), and similar rate estimates
(Wang et al., 2012).

In this work we strengthen the proposed link between CLEs and TDEs by
leveraging nearly a decade of fruitful searches for TDEs. In Section 2 we
discuss our CLE sample selection in addition to our comparison sample of TDEs.
We compare the host galaxy properties of CLEs to those of TDEs in Section 3
and compare the spectral energy distributions of the transients in Section 4.
Then, in Section 5 we discuss the selection methods for CLEs. Finally, in
Section 6, we provide a discussion and our conclusions. Throughout the paper
we assume a cosmology of $H_{0}$ = 69.6 km s-1 Mpc-1, $\Omega_{M}=0.29$, and
$\Omega_{\Lambda}=0.71$ (Wright, 2006; Bennett et al., 2014).

Figure 1: Optical spectra of our sample of CLEs. The vertical lines mark
spectral features common in AGNs and TDEs, with hydrogen lines in red, helium
lines in blue, and oxygen lines in green. Selected coronal lines of iron and
sulfur are shown in purple, which can be relatively weak in sources without
significant [O iii] emission. The source names are given at the right of each
spectrum.

##  2 Sample Selection

In this section, we discuss our sample selection of CLEs from spectroscopic
and photometric surveys. We also introduce and our comparison sample of TDEs.

###  2.1 Coronal Line Emitters

Our sample of CLEs come from two distinct selection methods which we will call
photometrically-selected and spectroscopically-selected CLEs throughout this
work.

The photometrically-selected CLEs are sources that were initially discovered
based on a large-scale rise in the continuum flux and then classified through
spectroscopic follow-up. In our search for such objects, we include recent
discoveries of singular nuclear flares from all-sky surveys for which
classification or follow-up spectra reveal coronal line emission that is
strong relative to narrow AGN lines. We find 4 events, which were initially
discovered based on large-scale continuum flux changes and were selected in a
similar manner to other optically-selected TDEs, resulting in initial
spectroscopic classifications as TDE candidates. In this manuscript, we will
denote the CLEs discovered in sky surveys as photometrically-selected CLEs.
These are ATLAS17jrp (Onori et al., 2022; Wang et al., 2022), ASASSN-18jd
(Neustadt et al., 2020), AT2021dms, and AT2022upj (Newsome et al., 2022;
Fulton et al., 2022).

The spectroscopically-selected CLEs are events that were initially discovered
based on emission line properties in a spectrum, often without accompanying
photometry. In particular, this class includes the spectroscopically-selected
CLEs from SDSS (e.g., Komossa et al., 2008; Wang et al., 2011; Wang et al.,
2012). These were selected by systematically searching the archive of SDSS
spectra for sources with coronal lines strong relative to typical narrow AGN
emission lines, such as [O iii] $\lambda$5007, and removing sources classified
as AGNs through typical line diagnostic diagrams, yielding 7 CLEs. We will
refer to these CLEs as spectroscopically-selected in the remainder of this
manuscript. Throughout the manuscript, we will shorten the names of the SDSS
sample of CLEs to improve readability (e.g. SDSSJ074820.66+471214.6 becomes
0748+4712). The full sample of 11 events are listed in Table 1.

The optical spectra for our sample of CLEs are shown in Figure 1. For the SDSS
sample, we use the SDSS spectra (Aguado et al., 2019), the same as shown in
Wang et al. (2012). For ATLAS17jrp, we show a publicly available ePESSTO
(Smartt et al., 2015; Barbarino et al., 2019) spectrum taken with the ESO
Faint Object Spectrograph and Camera (EFOSC2) from the European Southern
Observatory archive. For ASASSN-18jd we show a Southern African Large
Telescope (SALT) spectrum from Neustadt et al. (2020). We obtained a spectrum
of AT2021dms using the Low Dispersion Survey Spectrograph (LDSS-3) on the
Magellan Clay telescope. This new spectrum was reduced using standard IRAF
(Tody, 1986, 1993) procedures, such as bias subtraction, flat-fielding,
wavelength calibration with arc lamps and flux calibration with
spectrophotometric standard stars. Finally, for AT2022upj, we use a
classification spectrum on the Transient Name Server obtained with the
FLOYDS-S spectrograph on Faulkes Telescope South (Newsome et al., 2022). The
CLE nature of these transients is confirmed by spectroscopic similarity to the
spectroscopically-selected CLEs.

Table 1: Sample of Coronal Line Emitters

Object | TNS ID | $z$ | Right Ascension | Declination | References
---|---|---|---|---|---
SDSSJ074820.66+471214.6 | — | 0.0615 | 07:48:20.674 |  $+$47:12:14.30 | Wang et al. (2011); Wang et al. (2012)
SDSS J093801.64+135317.0 | — | 0.1006 | 09:38:01.636 |  $+$13:53:17.07 | Wang et al. (2012)
SDSS J095209.56+214313.3 | — | 0.07949 | 09:52:09.563 |  $+$21:43:13.31 | Komossa et al. (2008); Wang et al. (2012)
SDSS J105526.43+563713.3 | — | 0.07396 | 10:55:26.411 |  $+$56:37:13.04 | Wang et al. (2012)
SDSS J124134.26+442639.2 | — | 0.0419 | 12:41:34.253 |  $+$44:26:39.25 | Wang et al. (2012)
SDSS J134244.42+053056.1 | — | 0.03655 | 13:42:44.416 |  $+$05:30:56.13 | Wang et al. (2012)
SDSS J135001.49+291609.7 | — | 0.07768 | 13:50:01.500 |  $+$29:16:09.70 | Wang et al. (2012)
ATLAS17jrp | AT2017gge | 0.066 | 16:20:35.004 |  $+$24:07:26.57 | Onori et al. (2022); Wang et al. (2022)
ASASSN-18jd | AT2018bcb | 0.1192 | 22:43:42.871 |  $-$16:59:08.49 | Neustadt et al. (2020)
ZTF20acsacog | AT2021dms | 0.031102 | 03:21:24.069 |  $-$11:08:45.71 | Forster et al. (2021)
ZTF22abegjtx | AT2022upj | 0.054 | 00:23:56.846 |  $-$14:25:23.22 | Newsome et al. (2022); Fulton et al. (2022)

The 11 CLEs analyzed in this manuscript. Those with full SDSS names are
spectroscopically-selected and those with survey names are photometrically-
selected. The TNS ID is the identification given for objects reported on the
Transient Name Server, and is only given for the photometrically-selected
CLEs. The references are the discovery paper and key follow-up efforts where
available or the transient announcement/classification as needed.

###  2.2 Tidal Disruption Event Comparison Sample

Within the past several years the sample of well-studied TDEs has grown. In
particular, the number of TDEs discovered prior to their peak and with
significant multi-wavelength follow-up has increased dramatically. In this
manuscript we compare our CLEs to the sample of Hinkle et al. (2021b), as
these sources each have significant multi-wavelength data and have been
reduced in a uniform manner. That work compiled observations of 27 well-
studied TDEs (Holoien et al., 2014, 2016a, 2016b; Hung et al., 2017;
Wyrzykowski et al., 2017; Blagorodnova et al., 2017, 2018; Holoien et al.,
2019a, b; Leloudas et al., 2019; Nicholl et al., 2020; Wevers et al., 2019;
van Velzen et al., 2021b; Hinkle et al., 2021a).

##  3 Comparison of Host-Galaxy Properties

We first compare the host galaxies of our sample of CLEs and TDEs. In Section
3.1 we examine archival WISE photometry and in Section 3.2 we examine key host
properties such as stellar mass and star-formation rate (SFR).

###  3.1 WISE

Mid-infrared (MIR) photometry has long been used to distinguish AGNs from
normal galaxies based on color cuts (e.g., Stern et al., 2004; Assef et al.,
2010, 2013). The all-sky Wide-field Infrared Survey Explorer (WISE; Wright et
al., 2010) has been especially useful in selecting AGNs, as it mapped the
entire sky in four mid-infrared (MIR) bands: $W1$ (3.4 $\mu m$), $W2$ (4.6
$\mu m$), $W3$ (12 $\mu m$) and $W4$ (22 $\mu m$). As compared to inactive
galaxies, AGNs show redder WISE colors due to the strength of the non-thermal
power-law component of the AGN SED as compared to the stellar population of
the galaxy.

Figure 2: WISE color-color diagram used to discriminate strong AGNs from star-
forming galaxies. The gray box is the AGN region determined by Assef et al.
(2013). The orange points are the spectroscopically-selected CLEs, the dark
red points are the photometrically-selected CLEs, and the blue points are the
comparison sample of TDEs. The orange points without black borders use the
AllWISE catalog photometry and are connected by a dashed line to points with
$W1-W2$ colors taken from the most recent epoch of NEOWISE photometry. The
background points are a sample of HyperLeda (Makarov et al., 2014) galaxies
shown to illustrate the distribution of normal galaxies.

In Figure 2 we show CLE and TDE hosts in a common WISE color-color selection
diagram (Assef et al., 2013). In general, AGNs lie above a $W1-W2$ color of
$\sim$0.8 Vega mag. Here we see that, like the other TDEs, the
photometrically-selected CLEs have WISE colors typical of non-active galaxies.
Simply using the AllWISE catalog (Wright et al., 2010) photometry (data points
without black borders in Fig. 2), it appears that nearly all the
spectroscopically-selected CLEs would be selected as AGNs. However, it is
likely that many hosted transient events (e.g., Yang et al., 2013; Palaversa
et al., 2016) temporally overlapping with the AllWISE mapping.

This raises the possibility that these AGN-like WISE colors were the result of
nuclear dust heated by a transient event (van Velzen et al., 2016; Jiang et
al., 2021a; Hinkle, 2022). Fortunately, NEOWISE (Mainzer et al., 2014), the
successor mission to WISE, was activated in October 2013. This allows for
continued MIR mapping of the sky, albeit at a low cadence and only in the W1
and W2 filters. If we use the most recent NEOWISE epoch to compute the $W1-W2$
color for these spectroscopically-selected CLEs instead of AllWISE, we find
the majority of the spectroscopically-selected CLEs appear to be normal
galaxies in the WISE color-color space. Nevertheless, two sources, 0938+1353
and 1055+5637, do not show significant changes in their $W1-W2$ color.

To further test the possibility of transient-heated dust, we made WISE light
curves of these events, combining the individual AllWISE epochs with NEOWISE
survey photometry. To create our NEOWISE light curves, we stacked the intra-
visit exposures to create deeper W1 and W2 light curves with $\sim$6 month
cadence. We show long-term WISE color curves in Figure 3. The WISE color
curves for most of the spectroscopically-selected CLEs become bluer over time,
consistent with transient heating of nuclear dust. The $W1$ and $W2$ light
curves for these sources also show declines. The CLEs 0938+1353 and 1055+5637
show relatively flat WISE colors, with small-scale variability consistent with
AGN activity (Sheng et al., 2017). Like their WISE colors, the $W1$ and $W2$
light curves for 0938+1353 and 1055+5637 are flat, without any consistent
decline. Yang et al. (2013) also find that both of these sources do not
exhibit significant changes in their CL emission over long baselines, likely
suggesting persistent AGN activity. We will discuss these sources in more
detail in Section 5. We will treat 0938+1353 and 1055+5637 differently from
the other spectroscopically-selected CLEs as they are likely related to AGN
activity rather than a transient event.

The early evolution of the photometrically-selected CLEs all show WISE colors
typical of normal galaxies and are consistent with the sample of TDE hosts.
ASASSN-18jd is slightly redder than the typical TDE host, but not to a level
where strong AGN emission must be present. Some photometrically-selected CLEs
show signs of large-scale dust echoes at late times; these are particularly
notable for ATLAS17jrp and ASASSN-18jd, whose WISE colors grow significantly
redder after their peak emission. These dust echoes indicate significant dust
in the nucleus (Jiang et al., 2021a; Hinkle, 2022). Given a typical gas-to-
dust ratio of $\sim$100 in the interstellar medium (e.g., Hildebrand, 1983;
Draine & Lee, 1984), this implies significant amounts of gas present in the
nuclei of these galaxies. As AT2021dms and AT2022upj are relatively recent
events, their dust echoes are likely still rising and future NEOWISE releases
will likely reveal dust reprocessing echoes, as suggested in Figure 3.

Figure 3: WISE $W1-W2$ color curves for our sample of CLEs. AllWISE photometry
is shown as squares and NEOWISE photometry is shown as circles. The orange
lines are spectroscopically-selected CLEs with time relative to the SDSS
spectrum and the dark red lines indicate photometrically-selected CLEs with
time relative to the peak/discovery time. The individual point colors denote
separate events as shown in the legend. The dashed black line is the typical
AGN color cut. The dashed blue line and shaded region are the median and
1$\sigma$ range of the WISE $W1-W2$ colors of the TDE comparison sample.

###  3.2 Stellar Mass and Star Formation Rate

Figure 4: Host-galaxy star formation rate as compared to stellar mass. The
orange points are the spectroscopically-selected CLEs, the dark red points are
the photometrically-selected CLEs, and the blue points are the comparison
sample of TDEs. Upper-limits on SFRs are shown as downward-facing triangles.
The two points with golden borders are 0938+1353 and 1055+5637, which each
show evidence for AGN activity. In the two outer panels, we show normalized
KDEs of the distributions for the mass and SFRs of the two sub-samples of CLEs
and the TDE comparison sample. We have excluded 0938+1353 and 1055+5637 from
the spectroscopically-selected CLE KDEs. The background points are a sample of
SDSS DR8 (Eisenstein et al., 2011) galaxies with properties taken from the
MPA-JHU catalog (Brinchmann et al., 2004), with corresponding KDEs as the
dotted lines.

While the WISE colors provide insight into the possible presence of AGN
activity and the heating of nuclear dust, we are also interested in the bulk
properties of the galaxies in which these CLEs reside. As such, we next fit
archival photometry of the host galaxies with the Fitting and Assessment of
Synthetic Templates (Fast; Kriek et al., 2009) as we have done for previous
transient events (e.g., Holoien et al., 2016a; Hinkle et al., 2021a; Hinkle et
al., 2021b). When available, we obtained photometry in the UV from the Galaxy
Evolution Explorer (GALEX; Martin et al., 2005), optical photometry from SDSS
(Aguado et al., 2019) or Pan-STARRS (Chambers et al., 2016; Flewelling et al.,
2016), and infrared photometry from the Two Micron All-Sky Survey (2MASS;
Skrutskie et al., 2006) and AllWISE catalogs.

However, as the spectroscopically-selected CLEs clearly have contamination in
their AllWISE data, we have excluded it from their host-galaxy fits. We also
consider the survey timescales for the other sources of our galaxy
photometery. GALEX mapped the sky from 2003 to 2013, SDSS conducted its
imaging survey between 2000 and 2005, and 2MASS took images between 1997 and
2001. Therefore, while there is mild overlap between some GALEX and SDSS
images and the times at which the SDSS spectra of our CLEs were taken, the
likelihood of significant transient flux in the final photometry is low. This
is especially true as the SDSS spectra do not show significant optical
continuum excesses that indicate strong transient flux contributions. The
GALEX and 2MASS data are most important for constraining SFR and stellar mass
from our SED fits respectively, and each has a low probability of transient
flux significantly affecting the photometry. We also compared our Fast results
with estimates for the spectroscopically-selected CLEs available in the MPA-
JHU catalog (Brinchmann et al., 2004) and find excellent agreement in mass and
good agreement in SFR, although the emission-line based SFR estimates are
occasionally higher than our estimates.

In Figure 4 we compare the CLE hosts to a sample of TDEs with host-galaxy
properties computed in a similar manner by Hinkle et al. (2021b). We also
compare our CLE hosts to a broader galaxy sample from the MPA-JHU catalog
(Brinchmann et al., 2004). While these stellar mass and SFR estimates are
computed from spectra, they provide a large sample of uniformly-computed
galaxy properties. In addition to the individual source host properties, we
show kernel density estimates (KDE) computed using scipy.stats.gaussian_kde
and Scott’s Rule to estimate the underlying distribution of these properties
for each sub-sample, excluding the sources 0938+1353 and 1055+5637.

In terms of stellar mass, we find that the CLE hosts are generally similar to
TDE hosts, but prefer the more massive end of the TDE distribution.
Interestingly, of the CLE sample, the photometrically-selected sub-sample
resides at the very highest end of the TDE stellar mass distribution, with
ASASSN-18jd lying well above the stellar mass of any other TDE or CLE in our
samples. Conversely, for star-formation rate, the CLE hosts from both
selection methods prefer relatively high star-formation rates as compared to
TDE hosts, although they are consistent in most cases with the TDE hosts with
the highest SFRs. This is not a large surprise as gas is needed for CL
emission to be seen and many TDE hosts are known to be relatively gas-poor
(e.g., French et al., 2017, 2020; Jiang et al., 2021b).

Overall, accounting for the need for increased nuclear gas, the host
properties of CLEs are generally similar to TDE hosts in both mass and SFR.
Perhaps most interesting when comparing the CLE hosts to the distribution of
SDSS galaxies in mass and SFR is that they seem to reside either in the green
valley or the lower edge of the blue sequence, much like TDE hosts (e.g..,
French et al., 2020; Hammerstein et al., 2021). Even with slightly higher SFRs
than TDE hosts, the non-AGN CLE hosts do not belong in the population of
heavily star-forming galaxies.

Figure 5: Temporal evolution of the UV/optical blackbody luminosity (top
panel), radius (middle panel), and temperature (bottom panel) for the
photometrically-selected CLEs (red shades) and the comparsion sample of TDEs
(blue). The solid lines are the median values and the semi-transparent shading
corresponds to the 1$\sigma$ uncertainty. Time is in observer-frame days
relative to the earliest Swift epoch. We have not shown the very-late time
blackbody properties for several comparison TDEs.

##  4 Comparison of SED Properties

In addition to the properties of the host galaxies, we can compare the
properties of the transient events themselves. Unfortunately, this is only
doable for the photometrically-selected sub-sample, for which real-time
follow-up was conducted and multi-wavelength coverage exists. In this section,
we compare the evolution of blackbody parameters and decline rates of these
CLEs to well-studied TDEs.

###  4.1 Blackbody Parameters

Each of our photometrically-selected CLEs has follow-up imaging from the Neil
Gehrels Swift Observatory (Swift ; Gehrels et al., 2004) UltraViolet and
Optical Telescope (UVOT; Roming et al., 2005). Many of these epochs use the
six typical filters (Poole et al., 2008): $V$ (5425.3 Å), $B$ (4349.6 Å), $U$
(3467.1 Å), $UVW1$ (2580.8 Å), $UVM2$ (2246.4 Å), and $UVW2$ (2054.6 Å), where
the wavelengths quoted here are the pivot wavelengths calculated by the SVO
Filter Profile Service (Rodrigo et al., 2012).

Two sources have published Swift UVOT photometry and blackbody fits. These are
ASASSN-18jd (Hinkle et al., 2021b) and ATLAS17jrp (Hinkle, 2022), which we
adopt in this manuscript. For the other sources, we reduced the UVOT data
following the procedures of Hinkle et al. (2021b), first summing the
individual exposures per epoch in each filter. For AT2021dms and AT2022upj, we
used our FAST SED fits and computed synthetic photometry using the
corresponding filter response functions to estimate synthetic host fluxes
which we then subtracted from our reduced Swift . For AT2021dms (PI: Hinkle)
and AT2022upj (PIs: Newsome & Wang), we computed UVOT magnitudes using a
radius that best matched the radius used for the archival host photometry.
While we do not include the spectroscopically-selected CLEs here, as their
time of peak is highly uncertain and each has little UV follow-up data
available, the SDSS CLE 0952+2143 was shown to have modest UV emission even at
late times (Palaversa et al., 2016), consistent with a TDE.

We then fit each of these host-subtracted and Galactic extinction-corrected
UVOT epochs using a blackbody model, similar to many previous TDEs (Holoien et
al., 2014, 2016a; van Velzen et al., 2021b; Hinkle et al., 2021b). For each
epoch of host-subtracted UV photometry, we used Markov Chain Monte Carlo
(MCMC) methods and a forward-modeling approach to fit a blackbody model.
Figure 5 shows these fits in comparison to fits of well-observed TDEs from
Hinkle et al. (2021b).

Similar to previous TDEs (Holoien et al., 2020; Hinkle et al., 2021a; Hinkle
et al., 2021b), we bolometrically corrected optical survey light curves using
the fitted blackbody luminosity. This was done by scaling either the ATLAS $o$
(for ATLAS17jrp, ASASSN-18jd, and AT2021dms) or ZTF $g$ (for AT2022upj) light
curves to match the bolometric luminosity estimated from the blackbody fits.
In between Swift UVOT epochs, the scaling is interpolated. Outside of this
range, we assume the correction factor of the nearest Swift point, effectively
assuming a constant blackbody temperature. This is reasonable for most TDEs
(e.g., Hinkle et al., 2021b; van Velzen et al., 2021b), but introduces a
possible source of systematic uncertainty. We show these bolometric luminosity
curves in the top panel of Fig. 5.

Unfortunately, AT2021dms and AT2022upj do not have particularly long Swift
UVOT coverage. Nevertheless, after bolometrically correcting the optical light
curves it is clear that bolometric luminosities of the CLEs and TDEs are
generally similar. Of the CLE photometric sample, only ASASSN-18jd is
especially luminous, but even so it is consistent with the most luminous TDEs
in our comparison sample. It is also well within the luminosity range seen for
the recent class of featureless TDEs (Hammerstein et al., 2023). While the
temporal coverage is much shorter on average than the TDEs, the CLE blackbody
radii and temperatures are similar to TDEs as well. ASASSN-18jd has a larger
radius than most TDEs, but its temperature is fully consistent with the cooler
end of the TDE temperature distribution. We additionally note that 2 of the
CLEs have blackbody temperatures among the highest seen for any TDE. As the
production of CL emission requires significant extreme UV flux, this may not
be surprising. However, as other CLEs have relatively modest blackbody
temperatures, CL emission cannot rely solely on extremely hot thermal
emission.

###  4.2 Decline Rates

Figure 6: The peak luminosity versus the decline rate of the photometrically-
selected CLEs (red shades) and TDEs (blue). The uncertainties on the CLEs are
likely underestimated as for most objects the temperature constraint comes
after the time of peak luminosity. The decline rate $\Delta$L40 is the
difference between the log of the peak luminosity and the log of the
luminosity at 40 days after peak. The black solid line is the best-fit line
for the TDEs (Hinkle et al., 2021b) and the dashed black lines are the allowed
range of uncertainty from the best-fit line.

It has been noted previously (Hinkle et al., 2021a; Hinkle et al., 2021b;
Hammerstein et al., 2023) that there is a relationship between the peak
luminosity of a TDE and its bolometric decline rate, namely that more luminous
TDEs decline more slowly after peak. As such, we test if the photometrially-
selected CLEs in our sample follow a similar trend to the TDEs. Following the
procedures of Hinkle et al. (2020) we measure the peak bolometric luminosity
and decline rate at 40 days after peak. Figure 6 shows this measurement for
our CLEs as compared to the comparison sample of TDEs. For these CLEs, the
uncertainties on the peak luminosity and decline rate are likely
underestimated as the temperature estimate comes from Swift photometry taken
after peak emission. We add a 15% error in quadrature to the peak luminosity
values for ATLAS17jrp, AT2021dms, and AT2022upj following the estimates of
Hinkle et al. (2020) for objects without Swift data at peak.

The CLEs straddle the peak-luminosity/decline-rate relationship of TDEs.
Previous studies have shown the positions of various classes of supernovae
(Hinkle et al., 2020) and ambiguous nuclear transients (ANTS; Hinkle et al.,
2022) in this space. Both ANTs and supernovae reside off the TDE relationship,
unlike the CLEs in our sample. For ANTs, the average distance is 4.3$\sigma$
and for SNe the median distance is 1.8$\sigma$. When including all the CLEs,
the median offset is 1.2$\sigma$, substantially less than both ANTs and SNe.
Nevertheless 2 of our 4 photometrically-selected CLEs lie below the best-fit
line, at a level of $2.1\sigma$ for ATLAS17jrp and $2.6\sigma$ for AT2021dms,
solely accounting for the intrinsic scatter in the relationship computed by
Hinkle et al. (2021b) and excluding the uncertainties on the points
themselves. This may be a natural consequence of the somewhat more massive
hosts in which they reside which presumably also host more massive SMBHs for
which the fallback timescale is increased.

##  5 Coronal Line Selection

In Wang et al. (2012), one of the key selection criteria for CLEs was at least
one CL with a flux of more than 20% that of the [O iii] emission line. This
selected sources with strong coronal lines, but did seem to allow a fraction
of AGNs into the sample, as evidenced by the flat CL emission over a several-
year baseline (e.g. Yang et al., 2013). The CLEs 0938+1353 and 1055+5637,
without fading CLs, are also shown in Fig. 3 to have flat and red WISE light
curves consistent with AGNs. This suggests that even when attempting to
restrict the sample of CLEs to exclude strong AGNs, which can power CLs in
their own right, that a population of [O iii]-weak sources may exist that are
not related to transient events, but rather unusual AGNs.

To understand this selection better, we used the published emission line
measurements of Wang et al. (2012) for the SDSS spectroscopically-selected
sample to investigate the ratio of CL flux to [O iii] flux. These results are
shown for this sub-sample in Figure 7. For simplicity, we only looked at the
[Fe x], [Fe xi] and [Fe xiv], although a similar analysis can be done for any
CL as was done in Wang et al. (2012). We find that while the initial 20%
threshold was reasonable, this does allow the non-transient sources to be
selected as CLEs. A threshold of a CL with a flux greater than or equal to 1/3
of the [O iii] flux successfully excludes 0938+1353 and 1055+5637, at least
for this set of CLs. We, therefore, propose a stricter flux-ratio cutoff of
1/3 rather than 1/5 in selecting CLEs for future spectroscopic studies. We
note that ratios of other CLs may require different ratios to exclude AGNs
robustly, as 0938+1353 would not have been selected based on these 3 Fe lines
alone.

Interestingly, 1241+4426 is also included in the sample of sources without
fading CLs in Yang et al. (2013), although the WISE colors indicate a
transient event. In Figure 7, 1241+4426 has [Fe x] and [Fe xi] emission above
this 1/3 threshold, but not [Fe xiv] and some others have fluxes with
uncertainties allowing ratios below 1/5. Given the fluxes of two CLs above a
ratio of 1/3 and the declining WISE light curve, it is likely that 1241+4426
is powered by a TDE rather than an AGN.

When applying such a CL flux ratio threshold to the photometrically-selected
sample, we can see from Fig. 1 that ASASSN-18jd, AT2021dms, and AT2022upj each
surpass this ratio as several CLs are visually stronger than [O iii].
ATLAS17jrp is close to this ratio, with spectra shown in Onori et al. (2022)
supporting the CLE classification. With a larger sample of CLEs, it will be
possible to better probe the temporal evolution of CL properties and their
ratios with respect to other strong narrow emission lines.

Figure 7: Ratio of the flux of [Fe x], [Fe xi] and [Fe xiv] coronal lines to
[O iii] flux shown for each spectroscopically-selected CLE. The horizontal
dotted line is the 20% cutoff originally imposed by Wang et al. (2012), above
which sources are CLEs and below which sources are most likely AGNs. The
horizontal solid line is our proposed threshold of 1/3 that removes the two
SDSS sources convincingly shown to be AGNs.

##  6 Discussion and Conclusions

We have presented comparisons between CLEs and TDEs host-galaxy properties and
temporal evolution. In general, we find that the CLEs are consistent with TDEs
occurring gas-rich environments. This is supported by several key pieces of
evidence.

All but two of the spectroscopically-selected CLEs show declining WISE $W1-W2$
color curves, providing strong evidence for the transient nature of these
events. Even 1241+4426, for which spectroscopic follow-up suggested the
possibility of an AGN (Yang et al., 2013), shows a transient WISE color
evolution. Additionally, all of the photometrically-selected and all but two
of the spectroscopically-selected CLEs show WISE colors consistent with normal
galaxies, after accounting for transient-heated nuclear dust. The only two
sources which consistently show AGN-like WISE colors are 0938+1353 and
1055+5637, which are likely AGNs based on spectroscopic follow-up (Yang et
al., 2013). We also examined the ratio of CL flux to [O iii] flux for the
spectroscopically-selected sources. The AGN-like sources 0938+1353 and
1055+5637 have CL-to-[O iii] flux ratios below 1/3 in all of the Fe CLs we
test in Fig. 7, whereas the transient generally CLEs do not. For that reason,
we propose a stricter CL-to-[O iii] flux ratio of 1/3 as opposed to the ratio
of 1/5 used in Wang et al. (2012) for selecting CLEs in the future.

In terms of host properties, the SFRs of CLEs are generally consistent with
TDEs, but lie on the higher end of the TDE distribution, likely as a result of
the gas-rich environments needed for CL production. The stellar masses of the
spectroscopically-selected CLEs are consistent with TDEs. For the
photometrically-selected CLEs, the stellar masses lie at the very highest end
of the TDE distribution, with ASASSN-18jd having a substantially higher mass.

Our sample of CLEs and TDEs show similar blackbody luminosity, radius, and
temperature evolution. Only in the case of ASASSN-18jd is the luminosity
somewhat higher than seen for TDEs. As compared to other classes of
transients, the CLEs lie remarkably close to TDEs in the peak-
luminosity/decline-rate parameter space. Nevertheless, 2 of the 4 CLEs in our
sample lie slightly below the relationship, but within $\sim 2-3\sigma$ given
the intrinsic scatter in the relationship.

The similarities of CLEs and TDEs both in terms of the galaxies in which they
occur and the evolution of the transient events themselves suggests that CLEs
are driven by TDEs. This supports earlier assertions of this nature (e.g.,
Wang et al., 2012; Palaversa et al., 2016) made with less information on the
transient properties. With improved follow-up enabled by early detections with
transient surveys, we can now firmly connect CLEs with TDEs in gas-rich
environments.

Ongoing and future spectroscopic surveys such as SDSS-V (Kollmeier et al.,
2017), the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration et
al., 2016), and the Large Sky Area Multi-Object Fiber Spectroscopic Telescope
(LAMOST; Cui et al., 2012) will discover many more CLEs. Assuming the initial
rate estimates of $\sim 1\times 10^{-5}$ gal-1 yr-1 (Wang et al., 2012), such
surveys will find between a few CLEs (for LAMOST) and up to hundreds of CLEs
(for DESI) per year. As compared to the original SDSS spectroscopic survey
from which the first sample of CLES was selected, significant advancements can
be made if new CLE spectra can be identified in real-time so that multi-
wavelength follow-up efforts can be initiated to better investigate this
currently-understudied class of nuclear transients.

## Data availability

The data underlying this article are publicly available on the NASA High
Energy Astrophysics Science Archive Research Center (HEASARC), Sloan Digital
Sky Survey (SDSS) Science Archive Server, and the NASA/IPAC Infrared Science
Archive (IRSA).

## Acknowledgements

We thank Nidia Morrell for conducting the Magellan-LDSS3 observations of
AT2021dms, and Jack Neustadt for sharing reduced spectra of ASASSN-18jd. We
also thank Alexa Anderson for helpful comments on the manuscript.

JTH is supported by NASA grant 80NSSC22K0127. BJS is supported by NSF grants
AST-1920392 and AST-1911074. Support for T.W.-S.H. was provided by NASA
through the NASA Hubble Fellowship grant HST-HF2-51458.001-A awarded by the
Space Telescope Science Institute (STScI), which is operated by the
Association of Universities for Research in Astronomy, Inc., for NASA, under
contract NAS5-26555.

Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan
Foundation, the Participating Institutions, the National Science Foundation,
the U.S. Department of Energy, the National Aeronautics and Space
Administration, the Japanese Monbukagakusho, the Max Planck Society, and the
Higher Education Funding Council for England. The SDSS Web Site is
http://www.sdss.org/.

The SDSS is managed by the Astrophysical Research Consortium for the
Participating Institutions. The Participating Institutions are the American
Museum of Natural History, Astrophysical Institute Potsdam, University of
Basel, University of Cambridge, Case Western Reserve University, University of
Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, Johns Hopkins University, the Joint Institute for
Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and
Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences
(LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for
Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico
State University, Ohio State University, University of Pittsburgh, University
of Portsmouth, Princeton University, the United States Naval Observatory, and
the University of Washington.

SDSS-III is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS-III Collaboration including the
University of Arizona, the Brazilian Participation Group, Brookhaven National
Laboratory, Carnegie Mellon University, University of Florida, the French
Participation Group, the German Participation Group, Harvard University, the
Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA
Participation Group, Johns Hopkins University, Lawrence Berkeley National
Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for
Extraterrestrial Physics, New Mexico State University, New York University,
Ohio State University, Pennsylvania State University, University of
Portsmouth, Princeton University, the Spanish Participation Group, University
of Tokyo, University of Utah, Vanderbilt University, University of Virginia,
University of Washington, and Yale University.

This research has made use of the NASA/IPAC Infrared Science Archive, which is
funded by the National Aeronautics and Space Administration and operated by
the California Institute of Technology. We acknowledge the usage of the
HyperLeda database (http://leda.univ-lyon1.fr). This paper includes data
gathered with the 6.5 meter Magellan Telescopes located at Las Campanas
Observatory, Chile.

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