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\section{Introduction}
Star-forming regions in the Galaxy are distributed in a complex web of filaments that resemble a highly hierarchical network \cite[e.g.][]{2018A&A...610A..77H, 2010A&A...518L.100M, 2010A&A...518L.102A}. While open clusters are typically found in the densest parts of the structure, nearly 90\% of newborn stars become gravitationally unbound soon after the birth due to their dynamic interactions. Such loose ensembles of dispersing coeval stars are observed as stellar associations that keep the kinematic imprint of their local birth site up to $\sim$30~Myr before they become a part of the Galactic disk \citep{2019ARA&A..57..227K}. Because such groups of hundreds to thousands of stellar siblings were born from the same molecular cloud, they all have similar surface abundances \citep{2007AJ....133..694D}. These moving groups are thus the fossil records of the Galaxy that have a potential to link together star formation sites with the larger structures of the disk. They resemble an ideal laboratory to study a wide variety of important topics, from star- and planetary formation environments, the initial mass function and sequentially triggered star formation to dynamical processes that lead to the evaporation and finally the dispersal of an association.
A reliable reconstruction of stellar associations is thus of critical importance. While observations from the Hipparcos space astrometry mission allowed a major improvement in the search of overdensities in the kinematic phase space using stellar positions, parallaxes and proper motions \citep{1999AJ....117..354D}, it is high precision measurements from the Gaia space telescope -- including radial velocities for a subset of 7,000,000 stars -- that is revolutionising Galactic astrophysics \citep{2018A&A...616A...1G}. It has facilitated numerous attempts to study young stars above the main sequence and identify new members of the known moving groups in the Solar neighbourhood
\cite[e.g.][]{2018ApJ...862..138G, 2020MNRAS.491..215B}. Additionally
\citet{2018A&A...618A..93C} studied young populations on much larger Galactic scales and reported on the discovery of $\sim$1500 clusters.
Although a selection of the candidate members of a particular moving group is often based on the cuts in the kinematic space \cite[e.g.][]{2020arXiv200204801U},
the true nature of these groups appear to be diffuse due to their gradual dispersal. \citet{2019A&A...621L...3M} recently described extended structures emerging as the tidal streams of the nearby Hyades cluster, while \citet{2019A&A...623A.112D} found 11,000 pre-main sequence members of the Scorpius-Centaurus OB2 association residing in both compact and diffuse populations. Kinematic cuts in such cases are prone to be biased against the low-mass stars that are most likely to evaporate first.
Numerous works on young associations rely on multi-dimensional clustering algorithms. For example, \cite[e.g.][]{2019AJ....158..122K} report on the discovery of 1,900 clusters and comoving groups within 1~kpc with HDBSCAN (Hierarchival Density-Based Spatial Clustering of Applications with Noise described by \citealp{campello_density-based_2013}). However, the arrival of the Gaia's high precision parallaxes and proper motions enables reliable orbital simulations for the first time. For instance, \citet{2019MNRAS.489.3625C} were able to model an association at its birth time using Chronostar, perform its orbital trace-forward and blindly reconstruct the known Beta Pictoris association, reliably determine its members and, importantly, its kinematic age.
Stellar age is, besides the kinematics, one of the decisive parameters in the characterization of the young moving groups. Parallaxes of nearby stars with uncertainties better than 10\% enable the placement of stellar populations on the color-magnitude diagram. However, due to the numerous effects including the evolutionary model uncertainties and inflated radii on low-mass end of the population, the presence of binaries, background contamination and spread due to metallicity effects, and the variability of young stars, isochronal dating techniques remain non-trivial.
While gyrochronology relies on the multiple photometric measurements to determine the rotation period of a star, it is spectroscopic youth indicators that require only one observation for the estimation of stellar age.
Spectroscopic features of solar-like and cooler young stars up to the solar age are straightforward to observe. They emerge from the processes related to the magnetic activity of a star and manifest themselves in the excess emission in calcium H\&K lines (Ca~II~H\&K, 3969 and 3934~\AA; \citealt{2008ApJ...687.1264M}), H$\alpha$ line (6563~\AA; \citealt{2005A&A...431..329L}) and infrared calcium triplet (Ca~II~IRT; 8498, 8542 and 8662~\AA; \citealt{2013ApJ...776..127Z}). \citealp{2008ApJ...687.1264M} describe an age--activity relation that estimates age from the Ca~II~H\&K emission in the range from $\sim$10~Myr up to 10~Gyr, although \citealp{2013A&A...551L...8P} has shown later that there is no decay in chromospheric activity beyond 2~Gyr. The decline of the emission rate is the fastest in the youngest stars. Despite the variable nature of magnetic activity, especially in the pre-main sequence stars, it is easy to differentiate between stars of a few 10 and a few 100~Myr.
On the other hand, the presence of the lithium 6708~\AA~ line in GKM dwarfs directly indicates their youth and is a good age estimator for stars between 10-30~Myr -- which is a typical age of a stellar association.
Follow-up observations with the goal to detect the lithium line in young candidates have been performed by \citet{2019ApJ...877...60B} (who found lithium in 58 stars)
while \citet{2009A&A...508..833D}
report on the lithium measurements for $\sim$400 stars.
Over 3000 young K and M stars with a detectable lithium 6708~\AA~ line have recently been identified in the GALAH dataset \citep{2019MNRAS.484.4591Z}. While the majority of young early K dwarfs in the GALAH sample have practically settled on the main sequence, young late K and M stars with a detectable lithium line still reside 1~magnitude or more above the main sequence.
\citealp{2015MNRAS.448.2737R} have kinematically and photometrically selected candidate members of the Upper-Scorpius association and discovered 237 new members by the presence of lithium absorption.
In the Gaia era, the majority of stars in the Solar neighbourhood have parallaxes and proper motions precisely determined while spectroscopic age indicators and precise radial velocities are missing for a large fraction of low-mass young stars. Large spectroscopic surveys, such as GALAH \citep{2020arXiv201102505B}, typically avoid the crowded Galactic plane where most of the young stars reside.
This work aims to fill the gap and presents spectroscopic observations, their age indicators and radial velocities of 799 young star candidates within 200~pc with no pre-existing lithium measurements.
Section \ref{sec.data} describes the kinematically unbiased selection of all overluminous late K and early M stars within 200~pc.
We measure equivalent widths of the lithium absorption lines and the excess flux in Ca~II~H\&K and H$\alpha$ lines, as described in Section \ref{sec.youth_indicators}. Section \ref{sec.discussion} discusses age estimation and strategy success. The dataset is accompanied with radial velocities. Concluding remarks are given in Section \ref{sec.conclusions}.
\section{Data} \label{sec.data}
\subsection{Selection function}
Candidate young stars with Gaia magnitudes $10<G<14.5$ were selected from the \textit{Gaia}~DR2 catalogue \citep{2018A&A...616A...1G}. We focused only on the low-mass end of the distribution. The selection was based on their overluminosities in the colour-magnitude diagram. The colour index was chosen to be BP-W1 because it gives the narrowest main sequence with overluminous stars clearly standing out. BP is taken from \citet{2018A&A...616A...1G} and is described in more detail by \citet{2018A&A...616A...4E} while W1 is from \citet{2014yCat.2328....0C}. The relation used as a lower main sequence parametrisation $G(c)$
\begin{dmath}
G(c) = 4.717 \times 10^{-3} \; c^5 -0.149 \; c^4 + 1.662 \; c^3 - 8.374 \; c^2 + 20.728 \; c - 14.129
\end{dmath}
where G is absolute Gaia G magnitude and $c$=BP-W1 is described in more detail in \citet{2019MNRAS.484.4591Z} together with the arguments leading to the choice of BP-W1 being the best colour index for this purpose.
The colour-- temperature relation is determined from synthetic spectra while the temperature-spectral type relation is based on \citet{2013ApJS..208....9P}\footnote{In the version from 2018.08.02, available online: \url{http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt}}.
Our criteria further exclude older stars and keep only objects that are found 1~magnitude or more above the main sequence. This approach largely avoids main sequence binaries (at most 0.75~mag above the main sequence).
The sample was color cut to include only stars between 3$<$BP-W1$<$5.6. This limit corresponds to K5-M3 dwarfs with $T_{\rm eff}=3400$--4400\,K and allows the optimisation of the observation strategy and a focus on the cool pre-main sequence objects with the fastest lithium depletion rate. The blue limit is chosen so that it minimises the contamination with subgiants but keeps most of the late K dwarfs in the sample. The red limit is set on the steep region of the lithium isochrones that divides early M dwarfs with the fast depletion processes from those cooler ones that need more than 100~Myr to show a significant change in lithium.
The upper luminosity boundary
\begin{equation}
G > G(c) - (1.33 c -3)
\end{equation}
rejects giants from the sample.
Since all stars disperse with time in the kinematic parameter space, young objects are found only in regions with low velocities. To avoid the kinematic bias towards the pre-selected clumps of young stars in the velocity parameter space that disfavors the low-mass stars, and to remove old stars, we compute the mean $UVW$ value of the sample and keep all objects within ($\pm$15, $\pm$15, $\pm$10)~$\mathrm{km\,s^{-1}}$ of the median $UVW$ = (-11.90, 215.77, 0.19)~$\mathrm{km\,s^{-1}}$.
No kinematic cut was performed on stars that have no radial velocities available in the Gaia catalogue \citep{2018A&A...616A...6S}.
A declination cut with $\delta<30\,\mathrm{deg}$ eliminated objects not visible from the Siding Spring Observatory, Australia, where the observations took place.
Known young stars
from the Simbad database and stars observed with the GALAH survey \citep{2018MNRAS.478.4513B} were removed from the list to maximise survey efficiency at detecting new young stars. This selection results in 799 candidate stars.
Finally, our sample of stars described in this work includes observations of 756 candidate objects from this list.
A color-magnitude diagram with all the candidates is shown in Figure \ref{fig.cmd}. Parallaxes are taken from Gaia~DR2 \citep{2018A&A...616A...1G}.
\begin{figure}
\includegraphics[width=\columnwidth]{cmd.pdf}
\caption{Colour-magnitude diagram with candidate young stars and their reddening estimated in this work.
Details on the reddening estimation are described in Sec. \ref{sec.reddening}. The most crowded region ($\sim$ K5 dwarfs) is contaminated with reddened hotter stars while M dwarfs show less contamination due to their proximity.
Red lines denote the main sequence (dashed line) and the selection function 1 magnitude above (solid line).
Contours show the density of stars in the Gaia catalog.
}
\label{fig.cmd}
\end{figure}
\subsection{Observations}
Observations were carried out between November 2018 and October 2019 over 64 nights with the ANU 2.3m telescope at Siding Spring Observatory.
In order to achieve better radial velocity precision, 349 stars brighter than G=12.5 were observed with the slit-fed Echelle spectrograph in the Nasmyth focus, covering wavelengths between $\sim$3900 and $\sim$6750~\AA~ at R=24,000. Exposure times were between 600~sec for the brightest and 1800~sec for the faintest objects, resulting in a typical S/N of 20 in the order containing the H$\alpha$ line. Blue wavelengths with the calcium H\&K lines have poor S/N but clearly show strong emission above the continuum when present (Fig. \ref{fig.echelle_calcium}). The spectra were reduced as per \citet{2014MNRAS.437.2831Z}. Wavelength calibration was provided by bracketing Thorium-Argon lamp exposures.
Fainter stars (449) between $12.5<G<14.5$ were observed with the Wide Field Spectrograph (WiFeS; \citealt{2007Ap&SS.310..255D}), namely with resolving power of 3000 in the blue and 7000 in the red, covering 3500-7000~\AA.
We typically used a RT480 beam splitter.
Typical exposure times were 5~minutes per star that resulted in the median S/N of 13 and 31 for the blue and the red band, respectively. Thorium-Argon lamp frames were taken every hour to enable wavelength calibration. WiFeS spectra were reduced with a standard PyWiFeS package \citep{2014Ap&SS.349..617C}, updated to be better suited for stellar reductions of a large number of nights.
\subsection{Synthetic Spectra} \label{sec.synthetic}
For computation of radial velocities and parameter estimation, we use a template grid of 1D LTE spectra that was previously described by \citet{2019MNRAS.488L.109N}.
Briefly, spectra were computed using the TURBOSPECTRUM code (v15.1; \citealt{1998A&A...330.1109A, 2012ascl.soft05004P}) and MARCS model atmospheres \citep{2008A&A...486..951G}.
For models with $\log\,g > 3.5$, we use $v_{\rm mic} = 1\,{\rm km\,s^{-1}}$; for models with $\log\,g \le 3.5$, we use $v_{\rm mic} = 2\,{\rm km\,s^{-1}}$ and perform the radiative transfer calculations under spherical symmetry taking into account continuum scattering.
The spectra are computed with a sampling step of $1\,$km$\,$s$^{-1}$, corresponding to a resolving power $R\sim300\,000$. We adopt the solar chemical composition and isotopic ratios from \citet{2009ARA&A..47..481A}, except for an alpha enhancement that varies linearly from $\text[\alpha / \text{Fe}] = 0$ when $\rm [Fe/H] \ge 0$ to $\text[\alpha/\text{Fe}] = +0.4$ when $\rm [Fe/H] \le -1$.
We use a selection of atomic lines from VALD3 \citep{2015PhyS...90e4005R} together with roughly 15 million molecular lines representing 18 different molecules, the most important of which for this work are CaH (Plez, priv. comm.), MgH \citep{2003ApJS..148..599S,1995ASPC...78..205K}, and TiO \citep[with updates via VALD3]{1998A&A...337..495P}.
We use this grid to generate two synthetic libraries for radial velocity determination and parameter estimation.
For the WiFeS spectra, we use a coarsely sampled version of this grid, broadened to $R\sim7000$ with $5400 \leq \lambda \leq 7000$, $3000 \leq T_{\rm eff} \leq 8000\,$K, $3.0 \leq \log g \leq 5.5$, and $-1.0 \leq $[Fe/H]$ \leq 0.5$, in steps of $100\,$K, $0.25\,$dex, and $0.25\,$dex respectively.
For the Echelle spectra, we adopted R=24,000 for $3000 \leq T_{\rm eff} \leq 6000\,$K, $4 \leq \log g \leq 5$, and [Fe/H]=0, in steps of $250\,$K and $0.5\,$dex, respectively. Additionally, $\log g$ for $T_{\rm eff}<4000\,$K was extended to 5.5. Spectra cover wavelengths from 4800 to 6700~\AA.
\subsection{Radial velocities}
Radial velocities for datasets from both instruments were determined with the same algorithms using synthetic spectra described in the previous section.
\subsubsection{WiFeS}
Radial velocities of the WiFeS R7000 spectra were determined from a least squares minimisation of a set of synthetic template spectra varying in temperature (see Section \ref{sec.synthetic} for details of model grid). We use a coarsely sampled version of this grid, computed at R$\sim7000$ over $5400 \leq \lambda \leq 7000$ for $3000 \leq T_{\rm eff} \leq 5500\,$K, $\log g = 4.5$, and [Fe/H]$= 0.0$, with $T_{\rm eff}$ steps of $100\,$K for radial velocity determination.
Prior to computing radial velocities, we normalise both our observed and synthetic template spectra. For warmer stars without the extensive molecular bands and opacities present in cool stars, continuum regions are typically used to continuum normalise the spectrum. For observed cool star spectra however, such regions are unavailable in the optical, so we must opt for another normalisation formalism, which we term here \textit{internally consistent normalisation}:
\begin{equation}
f_{\rm norm} = f_{*} \times e^{\big(a_0 + \frac{a_1}{\lambda}+\frac{a_2}{\lambda^2}\big)}
\end{equation}
where $f_{\rm norm}$ is the internally consistent normalised flux vector, $\lambda$ is the corresponding wavelength vector, and $a_0$, $a_1$, and $a_2$ are coefficients of a second order polynomial fitted to the logarithm of $f_{*}$, which is either an observed flux corrected spectrum, or a synthetic template. This functional form of normalisation has chosen to be largely independent of reddening.
Once generated, a given synthetic template (initially in the rest frame) can be interpolated and shifted to the science velocity frame as follows:
\begin{equation}
f_{\rm temp,~rvs} = f_{\rm t}\big[\lambda \times \big(1-\frac{v_r-v_b}{c}\big)\big]
\end{equation}
where $f_{\rm temp,~rvs}$ is the RV shifted normalised template flux, $f_{\rm temp}$ is the template flux in the rest frame, $v_r$ and $v_b$ the radial and barycentric velocities respectively, and $c$ is the speed of light. $v_b$ is computed using the \texttt{ASTROPY} package \citep{astropy:2018} in \texttt{PYTHON}.
Given a grid of $k$ different synthetic template spectra, the final radial velocity value is found by finding the synthetic template that best minimises:
\begin{equation}
R(v_r) = \displaystyle\sum_{j}^{N}\bigg(\frac{{f_{\rm obs,~j} - f_{\rm temp,~rvs,~k,~j}(v_r)}}{\sigma_{{f_{\rm obs,~j}}}}\bigg)^2 M_j
\end{equation}
where $R$ is the total squared residuals as a function of radial velocity offset, $j$ is the pixel index, $N$ the total number of spectral pixels, $f_{\rm obs,~j}$ is the normalised observed flux at pixel $j$, $\sigma_{{f_{\rm obs,~j}}}$ is the uncertainty on $f_{\rm obs,~j}$, and $M_j$ is a masking term set to either 0 or 1 for each pixel. This step is done twice for each template spectrum, initially masking out only pixels affected by telluric contamination (H$_2$O: 6270-6290$\,$\SI{}{\angstrom}, and O2: 6856-6956$\,$\SI{}{\angstrom}), but then additionally masking out further pixels with high fit residuals. This second mask has the effect of excluding any pixels likely to skew the fit such as science target emission not present in the synthetic template (such as H$\alpha$).
Least squares minimisation was done using the leastsq function from \texttt{PYTHON}'s \texttt{SCIPY} library, implemented in the \texttt{PYTHON} package \texttt{plumage}\footnote{\url{https://github.com/adrains/plumage}}. Statistical uncertainties on this approach are on average 430$\,$m$\,$s$^{-1}$, however per the work of \citet{2018MNRAS.480.5099K} we add this in quadrature with an additional 3$\,$km$\,$s$^{-1}$ uncertainty to account for WiFeS varying on shorter timescales than our hourly arcs can account for, and effects of variable star alignment on the slitlets. Note however that we do not employ corrections based on oxygen B-band absorption, demonstrated by \citealt{2018MNRAS.480.5099K} to improve precision, as such additional precision is unnecessary for this work and is difficult for cooler stars.
Comparison of radial velocities for cool dwarf standard stars (e.g. from \citealp{2015ApJ...804...64M} and \citealp{2012ApJ...748...93R}, observed with the same instrument setup as part of Rains et al. in prep) with the Gaia catalogue \citep{2018A&A...616A...6S} shows an offset of WiFeS values for -1.7~$\mathrm{km\,s^{-1}}$ and a standard deviation of 3.2~$\mathrm{km\,s^{-1}}$ (Figure \ref{fig.rvs}). We suspect that most of the outliers are binary stars. Some of them are confirmed by either visual inspection or significally different radial velocities in case of repeated observations while there is not enough information available to investigate the rest of the interlopers.
\begin{figure}
\includegraphics[width=\linewidth]{rv_wifes_gaia_comparison.pdf}
\caption{A comparison between radial velocities from Gaia and from our pipeline for the WiFeS spectra. Standard stars (blue) have high S/N and small uncertainties. Binary star candidates (stars with repeated observations that show standard deviation of radial velocities greater than $5\,\mathrm{km\,s^{-1}}$ and stars that were classified as binaries by visual inspection) are marked in red.
}
\label{fig.rvs}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{rv_echelle.pdf}
\caption{A comparison between radial velocities from Gaia and our Echelle pipeline. Stars with the biggest disagreement with Gaia appear to be binary star candidates (red circles) or active (measured by calcium~II~H\&K emission $\mathrm{\log{R'_{HK}}}$, see Section \ref{sec.calcium}). The match with best-fitting template has been visually inspected for all stars in the sample.
}
\label{fig.rvs_echelle}
\end{figure}
\subsubsection{Echelle}
The same routine was utilized for the Echelle spectra on wavelengths between 5000 and 6500~\AA~ using their own synthetic library described in Sec. \ref{sec.synthetic}. As the correction for the blaze function and flux calibration were not performed in the data reduction step, each order within the relevant wavelength range was continuum normalized with a low order polynomial. Orders were then combined together into one spectrum in the range between 5000 and 6500~\AA. To match the continua of measured and synthetic libraries, fluxed model spectra were cut into wavelengths corresponding to Echelle orders, normalized and stitched back together with the same procedure. Finally, synthetic spectra were scaled to match 90th percentile of Echelle continua.
All spectra were visually inspected for any major reduction issues or other sources of peculiarity. Obvious double-lined binaries were flagged and their radial velocities are not reported in this work. Binary detection is reported in a separate column in Table \ref{tab.results}.
Median internal uncertainty of derived radial velocity is 0.06~$\mathrm{km\,s^{-1}}$, but a combination of the systematic uncertainty and radial velocity jitter characteristic to young stars account for 1.5~$\mathrm{km\,s^{-1}}$.
Most of the stars have radial velocities consistent with Gaia (Figure \ref{fig.rvs_echelle}). Mean absolute deviation for stars with difference less than 10$\mathrm{km\,s^{-1}}$ is 0.6$\mathrm{km\,s^{-1}}$. There are a handful of outliers, and they all have large uncertainties in Gaia values. Some of those appear to be binary stars discovered either by visual inspection or large radial velocity difference in case of the repeated measurements. At the same time, a lost of such stars show high activity level (depicted by a measure of activity in calcium HK lines) that might dominate Gaia's calcium infrared triplet region used to determine radial velocities and cause systematic offsets. All Echelle stars have been visually inspected for possible peculiarity and their match with the best-fitting template.
\subsection{Reddening} \label{sec.reddening}
The M dwarf candidates are too close to be significantly reddened (<200~pc), but on the other hand they could remain embedded in their birth cocoons. At the same time, the sample is contaminated with hotter stars that lie in regions of more heavy extinction within the Galactic plane.
To derive an estimate for the intrinsic colour index (BP-W1)$_0$, temperatures of the best-matching templates were used as an input in the colour-temperature relation derived from the synthetic spectral library. Although Solar values were used to calibrate the zero point, a degree of uncertainties remains (increasing with colour) and the relation is only approximate.
The resulting E(BP-W1) reveals a number of interlopers with temperatures higher than 4500~K. In particular, 156 WiFeS stars have E(BP-W1)$>$1 (20\% of the entire sample).
The estimated reddening E(BP-W1) is presented in Figure \ref{fig.cmd} together with the reddening vector\footnote{Reddening vector is determined for $A_V = 1$ and $R_V = 3.1$ from the \citet{1989ApJ...345..245C} model - \texttt{ccm89} in \url{https://extinction.readthedocs.io/en/latest/index.html}.}. Most interlopers with high reddening are found in the two regions in the Galactic plane with the highest concentration of stars in our sample: the Hyades and the Scorpius-Centaurus OB2 region (Fig. \ref{fig.galaxy}). Further analysis revealed that these stars do not show signs of youth and are likely located behind the local dust clouds associated with star-forming regions.
\begin{figure}
\includegraphics[width=\columnwidth]{gal.pdf}
\caption{The distribution of young candidates in the Galaxy. The majority of stars is found in clumps suggesting that they still reside close to their birth sites. The biggest group is found in the direction of the Scorpius-Centaurus OB2 region ($l>280$~deg). The second clump is likely the Hyades stars ($l\sim180$~deg). Colours show the interstellar reddening E(BP-W1).}
\label{fig.galaxy}
\end{figure}
\section{Youth indicators} \label{sec.youth_indicators}
The following subsections address the characterization of the lithium absorption line and the excess emission in H$\alpha$ and Ca~II~H\&K lines for stars in our sample. A combination of all three values provides a robust indicator of the stellar youth. Algorithms used to measure the strengths of lithium and H$\alpha$ lines in this work are similar for data from both instruments WiFeS and Echelle. Excess emission in calcium is measured differently for Echelle due to low signal in the blue.
All spectra, except the WiFeS calcium region, were locally normalized so that the youth features are surrounded by the continuum at 1 (and pseudo-continuum in M dwarfs). Binaries were not treated separately in this work and we provide youth indicators regardless of stars' multiplicity. All spectra were visually inspected for multiplicity and high rotation rate. We flag such cases in the final table and emphasize that this is qualitative inspection only and it is not complete.
\subsection{Lithium}
The primary and most reliable spectroscopic feature sensitive to the age of the pre-main sequence dwarfs in the temperature range observed in our sample is the lithium 6708~\AA~ line. This absorption line is observed in low-mass pre-main sequence stars before the ignition of lithium in their interiors. Since these stars appear to be fully convective before their onset on the main sequence, the depletion of lithium throughout the entire star occurs almost instantly. Lithium is observed in F, G and early K dwarfs for up to $\sim$100~Myr (mass dependent), but late K and early M-type dwarfs deplete lithium much faster. For further information see \cite{2014prpl.conf..219S} and references therein. Both data and theoretical predictions show that at the age between $\sim$15-40~Myr there is practically no lithium left in these stars \citep{2015A&A...577A..42B, 2019MNRAS.484.4591Z}.
The strength of the lithium absorption lines in this work was characterized with the equivalent widths measured within 6707.8$\pm$1.4~\AA.
Our spectra were pseudo-continuum normalized with a second order polynomial between 6700 and 6711~\AA. The lithium line was excluded from the continuum fit. The equivalent width was defined to be positive for lines in the absorption and was measured from the continuum level of 1.
In contrast to the emission-related features superimposed on the photospheric spectrum, the lithium absorption line shows a certain degree of correlation with the stellar rotation rate, e.g. \citet{2018A&A...613A..63B}. Fast rotators found by visual inspection are flagged in the table with results. While it appears to be fairly insensitive to the chromospheric activity \citep[e.g.][]{2019MNRAS.490L..86Y}
it might in some cases be affected by strong veiling present in the classical T~Tauri stars \citep{1989AJ.....98.1444S}. Veiling is an extra source of continuum that causes absorption lines to appear weaker \citep{1990ApJ...363..654B}. However, measurements of H$\alpha$ emission described below reveal that no classical T~Tauri stars are present in the sample. Figure \ref{fig.youth_indicators} confirms a robust correlation between all three measures of the youth.
The distribution of EW(Li) shows a concentration of stars below 0.05~\AA, though we only consider positive detections in stars with values above this level.
Repeated observations (45 stars) show 0.02~\AA~ of variation between individual measurements of the same object.
\subsection{Calcium~II~H\&K} \label{sec.calcium}
It has long been known that atmospheric features associated with stellar activity in solar-like dwarfs anticorrelate with their age \citep{1972ApJ...171..565S, 1991ApJ...375..722S}. Empirical relations derived from chromospheric activity proxies enable age estimation of stars between $\sim$0.6-4.5~Gyr to a precision of $\sim$0.2~dex \citep{2008ApJ...687.1264M}. However, a combination of saturation \citep{2010ApJ...709..332B} and high variability \citep{1995ApJ...438..269B} of activity in younger stars prevents this technique yielding reliable results before the age of $\sim$200~Myr.
Nevertheless, a detection of a strong excess emission in the calcium~II~H\&K lines (Ca~II~H\&K; 3968.47 and 3933.66~\AA, respectively) -- a proxy for chromospheric activity -- helps to distinguish between active young stars and older stars with significantly lower emission rates.
A commonly used measure of stellar activity in solar-type stars is S-index introduced by \citealp{1978PASP...90..267V} and derived as
\begin{equation}
S = \alpha \frac{N_H + N_K}{N_V + N_R},
\end{equation}
where $N_H$ and $N_K$ are the count rates in a bandpass with a width of 1.09~\AA~ in the center of the Ca~II~H and K line, respectively. To match the definition of the first measurements obtained by a spectrometer at Mount Wilson Observatory \citep{1978ApJ...226..379W} and make the measurements directly comparable, counts are adjusted to the triangular instrumental profile as described in \citealp{1978PASP...90..267V}.
$N_V$ and $N_R$ are the count rates in 20~\AA-wide continuum bands outside the lines, centered at 3901.07~\AA~ and 4001.07~\AA.
Constant $\alpha$ is a calibration factor that accounts for different instrument sensitivity and is derived by a comparison with literature S values. For WiFeS we provide a linear relation that converts measured S value on a scale directly comparable with the literature. For derivation see Appendix \ref{sec.appendix_s_index}.
Since $N_V$+$N_R$ has a color term due to nearby continuum shape varying with temperature, and because $N_H + N_K$ accounts for both chromospheric and photospheric contribution, it is more convenient to use the $\mathrm{R'_{HK}}$ index (first introduced by \citealp{1979ApJS...41...47L}) that represents a ratio between the chromospheric and bolometric flux and enables a direct comparison of activity in different stellar types. Using the conversion factor $C_{cf}$ that describes the colour-dependent relation between the S-index and the total flux emitted in the calcium lines, and $\mathrm{R_{phot}}$ that removes the photospheric contribution from the total flux in calcium, $\mathrm{R'_{HK}}$ is obtained as
\begin{equation}
\mathrm{R'_{HK} = R_{HK} - R_{phot}}
\end{equation}
where $\mathrm{R_{HK}} = 1.887 \times 10^{-4} \times C_{cf} \times S$. The constant in the equation is taken from \citealp{2017A&A...600A..13A}.
\citealp{1982A&A...107...31M} and \citealp{1984A&A...130..353R} provide the calibration of $C_{cf}$ and \citealp{1984ApJ...279..763N} and \citealp{1984ApJ...276..254H} for $\mathrm{R_{HK}}$ for the main sequence stars, but their relations become increasingly uncertain above B-V$>$1.2. \citealp{2017A&A...600A..13A} have recently extended the relation to M6 dwarfs (B-V$\sim$1.9) using HARPS data and calibrated the relation for colours that are more suitable for cool stars:
\begin{align}
\log_{10}{C_{cf}} & = - 0.005c^3 + 0.071c^2 - 0.713c + 0.973 \\
\log_{10}{R_{phot}} & = - 0.003c^3 + 0.069c^2 - 0.717c - 3.498.
\end{align}
where $c=$V-K was determined from a low-order polynomial fit to the relation between synthetic BP-RP and V-K from \citealp{2018MNRAS.479L.102C}.
There are 26 stars in the sample with repeated observations. In general more active stars show higher variability rates. We provide a median value of 1.1$ \times 10^{-5}$ for the $\mathrm{R'_{HK}}$ variability. Stars with low levels of activity have measured $\log{\mathrm{R'_{HK}}}=$ -5 or lower and we consider them inactive.
Activity in the Echelle spectra was evaluated in the same way as WiFeS stars. The calibration of the S-index was done using 19 stars observed with both instruments. For more details on the calibration see Appendix \ref{sec.appendix_s_index}.
The distribution of $\log{\mathrm{R'_{HK}}}$ is known to be bimodal for the main sequence stars in the Solar neighbourhood (e.g. \citealp{2003AJ....126.2048G}). Figure \ref{fig.rhk} shows two peaks, but they are centered at higher levels of activity due to our focus on the pre-main sequence stars. The more active peak is found at $\sim -4$ where $\log{\mathrm{R'_{HK}}}$ saturates for stars with rotation rates less than 10~days \citep{2017A&A...600A..13A}. According to \citealp{2008ApJ...687.1264M}, such high activity levels occur at ages of $\sim$10~Myr. We also plot $\log{\mathrm{R'_{HK}}}$ versus colour (the same figure) to confirm that the colour term is minimized.
There are two sets of lines that cause strong emission in this wavelength range: calcium~II~H\&K lines and Balmer emission lines in the youngest stars. Calcium H line is in some cases strongly blended by the Balmer emission line in the WiFeS spectra but count rate was measured within 1.09~\AA~ (see Fig. \ref{fig.wifes_calcium_balmer}).
\begin{figure*}
\includegraphics[width=\linewidth]{echelle_calcium.pdf}
\caption{Calcium lines in the Echelle spectra. Strong emission lines are detectable despite a low signal-to-noise ratio. There is an indication of a weak Balmer emission line at 3970~\AA. The red line is an average spectrum with a marginally detectable calcium emission while the blue line represents an average very active spectrum. Thick black line is a median inactive spectrum. Spectra in this plot were convolved with a smoothing kernel with the of width 7 for noise reduction purposes.}
\label{fig.echelle_calcium}
\end{figure*}
\begin{figure*}
\includegraphics[width=\linewidth]{wifes_calcium_and_balmer.pdf}
\caption{Calcium lines in the WiFeS spectra. Ca~II~H line appears to be wider than Ca~II~K due to the presence of the Balmer emission line at 3970~\AA. Red spectrum is a median spectrum with $\mathrm{logR'_{HK}}<$-4.9. Very active spectra with $\mathrm{logR'_{HK}}>$-4.4 (green) are young and show Balmer emission.
}
\label{fig.wifes_calcium_balmer}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{rhk_distribution.pdf}
\caption{\textit{Upper panel:} The introduction of the $\mathrm{logR'_{HK}}$ index minimises the color term and allows for comparison of activity rates among different spectral types. \textit{Lower panel:} Distribution of $\mathrm{logR'_{HK}}$ index for 680 stars. Nearly all stars with a detectable lithium show very strong calcium emission.
}
\label{fig.rhk}
\end{figure}
\subsection{Balmer series}
While weak and moderate excess emission rates in the H$\alpha$ line (6562.8~\AA) are associated with chromospheric activity \cite[e.g.][]{2004AJ....128..426W, 2008AJ....135..785W}, strong emission in the entire Balmer series, with H$\alpha$ being especially prominent ($>$10~\AA), is typically observed in classical T~Tauri stars that are low-mass objects younger than $\sim$10~Myr \citep{1989ARA&A..27..351B, 1989A&ARv...1..291A, 1998AJ....115..351M, 2006MNRAS.370..580K, 2014prpl.conf..219S}. It is widely accepted that there is a tight correlation between the average chromospheric fluxes emitted by the Ca~II~H\&K and H$\alpha$ lines \cite[e.g.][]{1995A&A...294..165M}. Although \citet{2007A&A...469..309C} report that this relation is more complicated, emission in H$\alpha$ represents a robust indicator of stellar youth.
Characterisation of stellar activity from the H$\alpha$ line is especially convenient in late-type dwarfs that only present a weak photosphere in the blue where Ca~II~H\&K are located.
The equivalent width of H$\alpha$ lines was measured between 6555 and 6567~\AA~ relative to the continuum, e.g. (1~-~flux) in the H$\alpha$ region. Negative values thus indicate absorption while positive values denote emission above the continuum. Interpretation of these results is not straightforward due to a wide range of the H$\alpha$ line profiles being strongly affected not only by the temperature but also the surface gravity. However, most of the stars show strong emission that is in any case an indicator for extreme stellar youth. We make a conservative estimate and only treat spectra with EW(Ha)$>$-0.5~\AA~ as active (see Fig. \ref{fig.wifes_calcium_balmer}).
Repeated observations of 45 stars reveal a typical difference between the maximal and minimal EW(Ha) value of 0.2~\AA. This uncertainty might also include a variability component of stellar activity.
Based on equivalent widths of H$\alpha$, most of the stars with excess emission belong to either weak (EW(Ha)$<$5~\AA) or post-T~Tauri stars. One third of the entire sample shows emission in the entire Balmer series. Column \texttt{Balmer} in Table \ref{tab.results} lists objects with clear Balmer emission that was detected by visual inspection.
\section{Discussion} \label{sec.discussion}
A combination of the three complementary youth features -- excess emission in Ca~II~H\&K and H$\alpha$ associated with magnetic fields active but declining for billions of years, and lithium absorption line present for a few 10~Myr in late K and early M dwarfs -- maximises the estimated age range and the robustness of our young star identification.
\begin{figure}
\includegraphics[width=\columnwidth]{activity_correlation.pdf}
\caption{Youth indicators studied in this work show a high degree of correlation. Chromospheric activity in young stars shows a high level of variability over time, but there appears to be a lower limit for H$\alpha$ emission with respect to the strength of the lithium line. Stars with no $\log{\mathrm{R'_{HK}}}$ available are marked with circles.
}
\label{fig.youth_indicators}
\end{figure}
This work uncovered 549 sources with at least one of the three indicators above the detection limit:
EW(Li)$>$0.1~\AA~ or EW(Ha)$>$-0.5~\AA~ or $\log{\mathrm{R'_{HK}}}>-4.75$. The strategy is thus 70\% successful. In particular, there are 281 stars with all three indicators above the detection limit.
There are 346 stars with a detectable lithium line (44\%), 479 with $\mathrm{EW(H}\alpha)>-0.5$ (60\% of the sample) and 464 objects (60\%) with a detectable calcium emission. Not surprisingly, there are 409 stars that show both calcium and H$\alpha$ youth features, as these two indicators are well correlated due to their common origin in chromospheric activity.
The lithium absorption line undergoes a different mechanism (lithium depletion in the pre-main sequence phase) and is much more short-lived. This causes an overdensity of chromospherically active stars with high H$\alpha$ but no lithium left (Fig. \ref{fig.youth_indicators}).
There are 10 stars in the sample that display lithium absorption but show no chromospheric activity.
The figure also shows that all stars with strong lithium emit excess flux in their chromospheres. This explains the void in the bottom right part of this figure.
Note that a small subset of individual stars only has one or two youth indicators measured due to noise in the respective spectral regions.
All youth indicators, radial velocities and flags denoting Balmer emission, binarity and fast rotation are listed in Table \ref{tab.results}, together with their 2MASS identifiers \citep{2003yCat.2246....0C}.
Even though our selection avoided known young stars, we cross-matched our catalogue with the literature. We found 15 stars in common with the list of association members described by \citet{2018ApJ...856...23G} and 6 from \citet{2018ApJ...860...43G}. We found 9 objects from our list in the work by \citet{2009A&A...508..833D} measuring lithium lines of $\sim$400 objects, and 3 overlapping stars with \citealp{2015MNRAS.448.2737R} who targeted stars from Upper Scorpius that were mostly fainter than our magnitude limit.
In total, 33 unique objects out of 766 from our list (4\%) are known association members or have lithium measured in the literature, and the rest are considered new detections.
The occurrence rate for all youth features is color dependent (Fig. \ref{fig.strategy_success}). Cooler stars in general more likely show signs of youth. Due to their slower evolution they spend more time above the main sequence and display signs of their youth much longer. However, we observe a drop in the occurrence rate of the lithium line in M dwarfs. This is because they deplete lithium the fastest and soon fall below the detection limit.
Lithium isochrones enable age estimation for late K and early M dwarfs younger than 15-40~Myr. We follow \citet{2019MNRAS.484.4591Z} and take indicative non-LTE equivalent widths from \citet{1996A&A...311..961P} for Solar metallicity and $\log{g} = 4.5$. We combine them with the \citet{2015A&A...577A..42B} models of lithium depletion (assuming the initial absolute abundance of 3.26 from \citealp{2009ARA&A..47..481A}) to compute lithium isochrones (Fig. \ref{fig.isochrones}).
Lines indicating abundances in the plot show that EW(Li) in our temperature range practically traces any amount of lithium left in the atmosphere.
There appears to be an overdensity of 278 objects above EW(Li)$>$0.3~\AA~ corresponding to the ages of 15~Myr and younger. Moreover, there are 325 stars lying above the 20~Myr isochrone and the 0.1~\AA~ detection limit.
Figure \ref{fig.youth_indicators} confirms that stars with the strongest lithium have the highest $\log{\mathrm{R'_{HK}}}$ values of -4 which corresponds to $\sim$10~Myr according to the \citealp{2008ApJ...687.1264M} activity-age relation.
These objects likely belong to the Scorpius-Centaurus association -- especially because their $(l,b)$ location overlaps with this region in the sky. However, further kinematic analysis is needed to confirm their membership.
Since our selection encompass all stars above the main sequence, the sample is contaminated with stars with bad astrometric solutions. 45\% of our observed objects have \textit{re-normalised unit weight error} (the \texttt{RUWE} parameter from the Gaia~DR2 tables describing the goodness of fit to the astrometric observations for a single star) greater than 1.4. Gaia~DR2 documentation suggests that such stars either have a companion or their astrometric solution is problematic. There is no detectable lithium left in these stars and they appear to be old in our context with low or zero emission in calcium and H$\alpha$. When stars with \texttt{ruwe}$>$1.4 and high reddening are removed from our catalog, 80\% of stars left show at least one spectroscopic sign of stellar youth. This suggests a high efficiency in selection of young stars from the Gaia catalog based on their overluminosity and a reliable astrometric single star solution.
\begin{figure}
\includegraphics[width=\columnwidth]{strategy_success.pdf}
\caption{Strategy success as a fraction of young stars with detectable spectroscopic features of youth versus their color. Detection rate for calcium and H$\alpha$ increase towards redder stars with different slopes. This might be due to a dependence of EW(H$\alpha$) on the temperature. Lithium absorption line is observed only in the youngest stars. Detection rate drops for early M dwarfs because they deplete lithium the fastest.
The number of all candidates in each colour bin is shown in the plot.
}
\label{fig.strategy_success}
\end{figure}
\begin{figure*}
\includegraphics[width=\linewidth]{lithium_color.pdf}
\caption{Lithium isochrones (blue lines) reveal a number of very young stars in the sample ($<$25~Myr). 349 stars have a detectable lithium with EW(Li)$>$0.1~\AA. Black lines show lithium abundances with their uncertainties (dashed). Lithium strength correlates well with the excess emission in the H$\alpha$ line.
}
\label{fig.isochrones}
\end{figure*}
\section{Conclusions} \label{sec.conclusions}
We selected and observed 766 overluminous late K and early M dwarfs with at least 1~magnitude above the main sequence and with Gaia G magnitude between 12.5 and 14.5. The kinematic cut was wide enough to avoid a bias towards higher-mass stars and include low-mass dwarfs. Observations were carried out over 64 nights with the Echelle and Wide Field Spectrographs at the ANU 2.3m telescope in Siding Spring observatory.
The analysis revealed 544 stars with at least one feature of stellar youth, i.e. the lithium absorption line or excess emission in H$\alpha$ or calcium~H\&K lines. The strength of the lithium absorption line indicates that 349 stars are younger than 25~Myr.
This sample significantly expands the census of nearby young stars and adds 512 new young stars to the list. Only 33 out of 544 objects with at least one youth indicator are listed in external catalogs of young stars. For example, \citealp{2018ApJ...856...23G} characterised known nearby associations and provided a list of 1400 young stars from a wide variety of sources. Our catalog has only 15 stars in common with theirs and thus expands the sample by 35\%. Although a further kinematic analysis is needed to confirm their membership, it is likely that a great fraction of stars from our sample belong to the Scorpius-Centaurus association because they are found in that direction in the sky and all have lithium ages $<$20~Myr. However, we only find 3 stars in common with \citealp{2015MNRAS.448.2737R} who kinematically and photometrically selected and observed mostly fainter stars in Upper-Scorpius.
Strong lithium absorption lines and excess emission in calcium in these objects consistently indicate likely stellar ages of roughly 10~Myr, according to the activity--age relation \citep{2008ApJ...687.1264M} and lithium isochrones (see Fig. \ref{fig.isochrones}). The latter reveal 325 stars with $\mathrm{EW(Li)>0.1}$~\AA~ and above the 20~Myr isochrone.
We report on a high success rate in search for young stars by selecting overluminous objects in the Gaia catalog. After stars with unreliable astrometry ($\mathrm{ruwe}>1.4$ that indicates bad astrometry or multiplicity) and high reddening are removed, the success rate is 80\%.
Radial velocities are determined for spectra from both instruments, with average uncertainties of 3.2~$\mathrm{km\,s^{-1}}$ for WiFeS and 1.5~$\mathrm{km\,s^{-1}}$ for Echelle stars.
This catalog of nearby young stars now has all kinematic measurements available to improve the analysis of young associations and help to find their birthplace. For example, \citealp{2020MNRAS.499.5623Q} have recently shown that stellar associations come from different places in the Galaxy. Follow up work may include e.g. using Chronostar \citep{2019MNRAS.489.3625C} to provide kinematic ages, robust membership estimates and orbital models of young associations to infer the origins of this sample, as well as the extraction and analysis of rotational periods using TESS to obtain ages using gyrochronology where possible.
\section*{Acknowledgements}
We acknowledge the traditional owners of the land on which the telescope stands, the Gamilaraay people, and pay our respects to elders past and present. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. M{\v Z} acknowledges funding from the Australian Research Council (grant DP170102233). ADR acknowledges support from the Australian Government Research Training Program, and the Research School of Astronomy \& Astrophysics top up scholarship. This research made use of Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration 2013, 2018).
Parts of this research were supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.
Parts of this research were conducted by the Australian Research Council Centre of Excellence for Gravitational Wave Discovery (OzGrav), through project number CE170100004. S.-W. Chang acknowledges support from the National Research Foundation of Korea (NRF) grant, No. 2020R1A2C3011091, funded by the Korea government (MSIT).
Software: \texttt{numpy} \citep{harris2020array}, \texttt{scipy} \citep{2020SciPy-NMeth}, \texttt{ipython} \citep{doi:10.1109/MCSE.2007.53}, \texttt{pandas} \citep{mckinney-proc-scipy-2010}, \texttt{matplotlib}
\citep{doi:10.1109/MCSE.2007.55} and \texttt{astropy} \citep{astropy:2018}.
\section*{Data Availability}
This work is based on publicly available databases. Gaia data is available on \href{https://gea.esac.esa.int/archive/}{https://gea.esac.esa.int/archive/} together with the crossmatch with 2MASS and WISE catalogs.
A compilation of known young stars with S-indices from \citealp{2013A&A...551L...8P} is available on \href{http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/A+A/551/L8&-to=3}{http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/A+A/551/L8\&-to=3}.
All measurements from this work are provided in the appendix with a full table available online.
\bibliographystyle{mnras}
|
\section{Introduction}
\label{sec:introduction}
The cytoplasmic membrane separates the living cell from its extra-cellular surroundings,
while other intra-cellular membranes compartmentalize cellular organelles.
Biomembranes are constructed from two monolayers (leaflets) in a back-to-back arrangement, and are in general asymmetric in their lipids composition~\cite{Devaux,Meer}. For example, in human red blood cells, the inner cytoplasmic leaflet is composed mostly of phosphatidylethanolamine (PE) and phosphatidylserine (PS), while the outer cytoplasmic
leaflet is composed of phosphatidylcholine (PC), sphingomyelin (SM) and a variety of glycolipids~\cite{Opden,Devaux1}.
The asymmetric nature of the cell membrane plays a key role in a variety of cellular processes such as endocytosis~\cite{Pomo}, vesicle budding and trafficking~\cite{Dal1}. Furthermore, in living cells, the composition asymmetry is an active and energy-consuming process. It is maintained by several membrane proteins such as flippase and floppase that allow lipids to exchange between the two leaflets with the aid of adenosine triphosphate (ATP)~\cite{Dal}.
In artificial multi-component lipid bilayers, the two
membrane leaflets can undergo a lateral phase separation. Several authors made
the connection between such a phase separation in artificial membranes
and existence of small dynamic domains (``rafts") in biological
membranes~\cite{KA14}. It should be noted, however, the size of rafts
in biological membranes are expected to be in the range of 10--100~nm~\cite{Simons,Munro}.
Raft are believed to be enriched mixtures of cholesterol and SM in a liquid-ordered phase
(L$_{\rm o}$), embedded in a background of a liquid-disordered phase (L$_{\rm d}$). Despite the lack of an ultimate proof for the existence of rafts, they have been advocated in relation with their potential influence on biological cellular processes.
It has been suggested that rafts act as organizing centers for the
assembly of signaling molecules, influencing membrane fluidity, and
regulating receptor trafficking~\cite{Simons,Munro}.
\begin{figure*}[tbh]
\begin{center}
\includegraphics[scale=0.4]{fig1.eps}
\end{center}
\caption{\textsf{
Schematic vertical cut through a membrane consisting of several domains.
The domains are embedded in a lipid matrix (yellow) and have a line tension acting
along the domain rim.
Three domain types can be seen and are further explained in Fig.~2: Flat (F), Budded (B)
and Dimpled (D).
Each domain is formed by two lipids (red and blue) that partition themselves differently
on the two domain leaflets.
}}
\label{fig1}
\end{figure*}
A theoretical model for domain-induced budding of planar membranes was proposed by
Lipowsky~\cite{Lipowsky92} some years ago, and later was extended for closed
vesicles~\cite{JL93,JL96}.
In the model, the competition between the membrane bending energy and domain line tension leads to a budding transition.
More recently, a model describing domain-induced budding in bilayers composed of a binary mixture of lipids was proposed by us~\cite{Wolff}.
In particular, we have shown that dimpled domains are formed and remain stable due to an
asymmetry between the two compositions of the corresponding domain leaflets,
given that the line tension along the domain rim is not too large.
The calculations in Ref.~\cite{Wolff} were done for a specific case where
the relative concentration between the two lipids on the bilayer domain stays constant, while
the lipids are allowed to freely exchange between the two leaflets.
In the last decade, however, techniques such as Langmuir-Blodgett or
Langmuir-Sch\"{a}efer have enabled a control over the asymmetric composition
of artificial membranes~\cite{Kiessling, Cheng}.
For example, unsupported bilayers via the Montal-M\"uller method have been used to form asymmetric bilayers in which the composition of each leaflet was independently controlled~\cite{Collins}. A key ingredient in understanding those experiments is the fact that
the flip-flop process that exchanges lipids across the leaflets is slower than experimental times.
Given these experimental findings, it is worthwhile to consider bilayers where each of the leaflet domain composition (rather than the overall bilayer composition) can be controlled in a separate and independent way.
In this paper, we generalize our previous budding model~\cite{Wolff} and extend it to asymmetric two-component lipid domains.
Each domain leaflet has a conserved lipid composition that is independent from that of the other leaflet.
We consider the possibility of domains curved in the third dimension, which can produce buds as shown in Fig.~\ref{fig1}.
In our model, the composition-dependent spontaneous curvature leads to coupling between curvature and lipid composition in each of the domain leaflets~\cite{Leibler,SPA,MS}.
Such a coupling leads to rich phase behavior including various phase coexistence regions.
For the sake of clarity, we do not take into account any direct interaction between two domains that face each other, although various possibilities have been previously proposed~\cite{May}. In addition, we would like to mention that in a separate set of studies for completely planar membranes, asymmetric bilayers composed of two modulated monolayers
(leaflets) were considered~\cite{HKA09,HKA12}. In these works, the static and dynamic properties of concentration fluctuations have been presented together with the related micro-phase separation.
Our model may have several experimental implications. We predict that bilayer domains can exist in three states having different equilibrium shapes: fully
budded, dimpled, and flat states. Their relative stability depends on controlled system parameters: temperature, degree of compositional asymmetry
between the two leaflets and domain size.
We find that the dimpled state is the most stable one in some of the parameter range and this is in accord
with recent experiments~\cite{RUPK}.
Based on the calculated phase diagrams, we anticipate that membranes should exhibit in some parameter range two-phase
and three-phase coexistence between different domain states.
The outline of this paper is as follows.
Section~\ref{sec:model} generalizes our previous budding model~\cite{Wolff}, and the free-energy describing asymmetric two-component lipid domains is discussed. In Sec.~\ref{sec:diagram}, we explain the conditions for various phase equilibria and how to calculate the phase diagrams.
We then proceed by presenting the phase diagrams in Sec.~\ref{sec:result}, and discuss the resulting global phase behavior.
Finally, a more qualitative discussion is provided in Sec.~\ref{sec:discussion}.
\section{Model}
\label{sec:model}
\begin{figure}[tbh]
\begin{center}
\includegraphics[scale=0.4]{fig2.eps}
\end{center}
\caption{\textsf{
A schematic vertical cut through three possible domain shapes:
(a) a flat bilayer domain composed of a mixture of A and B lipids (red and blue, respectively), and embedded in an otherwise flat membrane (yellow).
The circular flat domain (F) has a radius $L$ and area $S=\pi L^2$.
(b) A partial bud (dimpled domain) curved in the out-of-plane direction.
The bud (D) of the same area $S$ forms a spherical cap of radius $1/C$, where $C$ is the curvature. (c) A fully-budded domain (B) has a spherical shape of total area $S$, and just touches the flat membrane. The line tension $\gamma$ acts along the line boundary between the domain (red/blue lipids)
and the flat membrane matrix (yellow lipids).
}}
\label{fig2}
\end{figure}
We consider a membrane consisting of two monolayers (the terms ``monolayer" and ``leaflet" will be used interchangeably in this paper), each composed of an A/B mixture of lipids, which partition themselves asymmetrically between the two leaflets.
We assume that the membrane can undergo a lateral phase separation creating domains of different lipid composition. The domains are taken to fully span the two monolayers, but
the leaflet compositions in these domains can be different. Hence, the formed lipid domains are, in general, asymmetric.
We further assume that the two leaflet compositions are conserved and can be taken as independent from each other.
In the following, we discuss the thermodynamic behavior of a system consisting of a large number of such asymmetric domains as shown in Fig.~\ref{fig1}. For this purpose, we explain below the different terms that contribute to the domain free-energy.
In Fig.~\ref{fig2}, we show a vertical cut through three possible domain states: flat (F), dimpled (D) and fully-budded (B).
The flat circular domain (F) in (a) has an area, $S=\pi L^2$, which is assumed to remain constant during the budding process.
For simplicity, we consider in (b) dimpled buds (D) whose shape is a spherical cap of radius $1/C$, and in (c) the extreme case of a completely detached spherical bud (B).
The total bending energy of the budded domain is given by the curvature
contributions from its two monolayers~\cite{Helfrich73,SafranBook}:
\begin{equation}
E_{\rm bend} =2 \pi L^2 \kappa \left[ (C -C_0)^2
+ (C + C_0)^2 \right],
\label{bend}
\end{equation}
where $\kappa$ is the bending rigidity modulus (assumed to be a constant that is independent of lipid composition) and $C_0$ the monolayer spontaneous curvature. As shown in Fig.~\ref{fig2}, the two monolayer deform with curvatures $+C$ and $-C$, respectively.
The second energy contribution is the domain edge energy that is proportional to the perimeter length and to the line tension, $\gamma$~\cite{Lipowsky92}:
\begin{equation}
E_{\rm edge} = 2 \pi L \gamma\sqrt{1-(LC/2)^2}.
\label{edge}
\end{equation}
Note that in the extreme case, when the domain buds into a complete spherical shape (B) as in Fig.~\ref{fig2}(c), $C=\pm 2/L$ and $E_{\rm edge}=0$.
In the above, the strong variation in composition between the domain and its surrounding matrix is effectively taken into account through the line tension, $\gamma$, which is treated as an external control parameter.
This situation can be justified for a strong segregation that results
in a sharp boundary between the domain and its flat matrix surroundings.
As each domain is composed of an A/B mixture, we define
$\phi_{\rm A}$ ($\phi_{\rm B}$) as the area fraction (assumed to be equal to the molar fraction) of the
${\rm A}$ lipid ($\rm B$ lipid) in the upper leaflet domain, and similarly,
$\psi_{\rm A}$ ($\psi_{\rm B})$ for the lower leaflet domain.
We assume that each monolayer is incompressible so that
$\phi_{\rm A}+\phi_{\rm B}=1$ and $\psi_{\rm A}+\psi_{\rm B}=1$.
Hence, the two relevant order parameters are the relative composition
in the upper leaflet:
\begin{equation}
\phi= \phi_{\rm A}-\phi_{\rm B} ,
\end{equation}
and in the lower leaflet,
\begin{equation}
\psi= \psi_{\rm A}-\psi_{\rm B} .
\label{def}
\end{equation}
As in any A/B mixture, the possibility of a phase separation can be described by a phenomenological Landau expansion of the free-energy in powers of $\phi$ and $\psi$. This expansion is done separately for
each monolayer, and the total contribution
to the free-energy is the sum over the two monolayers:
\begin{equation}
E_{\rm phase} = \pi L^2 \frac{U}{\Xi^2}
\left[ \frac{t}{2} (\phi^2 +\psi^2)+ \frac{1}{4} (\phi^4+\psi^4) \right],
\label{phase}
\end{equation}
where $\Xi \equiv \kappa/\gamma$ is the invagination length,
$U$ is a parameter that sets the energy scale, and
$t \sim (T-T_{\rm c})/T_{\rm c}$ is the reduced temperature ($T_{\rm c}$ being the critical temperature).
In Eq.~(\ref{phase}) above, we multiply by the domain area, $\pi L^2$, to obtain the domain free-energy.
Hereafter, we will use several dimensionless variables: a rescaled curvature $c \equiv LC$, rescaled spontaneous curvature $c_0 \equiv LC_0$, and rescaled invagination length $\xi\equiv \Xi/L$.
The coupling between curvature and composition is taken into account by
assuming a linear dependence of the spontaneous curvature $c_0$ on the relative composition in each of the leaflets~\cite{Leibler,SPA,MS}:
\begin{equation}
c_0(\phi) =\bar{c}_0-\beta \phi,
\label{spontaneous1}
\end{equation}
\begin{equation}
c_0(\psi) =\bar{c}_0-\beta \psi,
\label{spontaneous2}
\end{equation}
where $\bar{c}_0$ is the monolayer spontaneous curvature for the symmetric 1:1 composition,
$\phi=\psi=0$, and $\beta$ is a coupling parameter that has the same value for the two monolayers.
Since $\bar{c}_0$ is a constant, it merely shifts the origin of the chemical potential, and will be dropped out without loss of generality.
The total free-energy per domain is then given by the sum of Eqs.~(\ref{bend}), (\ref{edge}) and (\ref{phase}): $E_{\rm tot}=E_{\rm bend}+E_{\rm edge} + E_{\rm phase}$, and
its dimensionless form, $\varepsilon=E_{\rm tot}/2\pi \kappa$, is expressed as
\begin{align}
\varepsilon(\phi, \psi, c)& =(c+\beta\phi)^2+(c-\beta\psi)^2
+\frac{1}{\xi}\sqrt{1-c^2/4} \nonumber \\
&+\frac{1}{\xi^2} \left( \frac{U}{2 \kappa} \right)
\left[\frac{t}{2}(\phi^2+\psi^2)+\frac{1}{4}(\phi^4+\psi^4)
\right].
\label{eq6}
\end{align}
We note that Eq.~(\ref{eq6}) depends on three dimensionless parameters: $\beta$,
$\xi$, and $U/(2 \kappa)$, while the thermodynamic variables are the reduced temperature
$t$ and the three order parameters: $\phi$, $\psi$ and $c$.
In the calculations presented hereafter, we set $U/(2 \kappa)=1$ and vary the
values of $\beta$ and $\xi$.
Within mean-field theory, the equilibrium states and phase transitions
are determined by minimizing $\varepsilon$ with respect to $\phi$, $\psi$ and $c$, under the condition that $\phi$ and $\psi$ are conserved order parameters while $c$ is not. From the minimization of $\varepsilon$ with respect to $c$, we obtain the condition
\begin{equation}
2(c+\beta\phi)+2(c-\beta\psi)-\frac{c}{4\xi\sqrt{1-c^2/4}}=0.
\label{eq7}
\end{equation}
Then, by substituting the curvature $c=c(\phi, \psi)$ into Eq.~(\ref{eq6}), results in a partially minimized free-energy, $\varepsilon^*$
\begin{equation}
\varepsilon^{\ast}(\phi, \psi) = \varepsilon(\phi, \psi, c(\phi, \psi)),
\label{epsast}
\end{equation}
as a function of $\phi$ and $\psi$.
Typical experimental values of domain sizes are in the range of $L\simeq 50$--$500$~nm~\cite{Simons}, the bending rigidity
$\kappa \simeq 10^{-19} {\rm J}$~\cite{SafranBook},
and the line tension
$\gamma \simeq 0.2$--$6.2 \times 10^{-12}$~J/m~\cite{baumgart,tian}.
Hence, the scaled invagination length is estimated to be in the range, $\xi \simeq 0.01$--$10$. These values will be used in the next section where we calculate numerically the phase diagrams.
\section{Phase equilibria conditions}
\label{sec:diagram}
In order to obtain various phase coexistence regions, $\varepsilon^{\ast}$ in Eq.~(\ref{epsast}) should be further minimized with respect to the conserved order parameters, $\phi$ and $\psi$. Hence, we consider the following thermodynamical potential
\begin{equation}
g(\phi, \psi)= \varepsilon^{\ast}(\phi,\psi)-\mu_{\phi}\phi-\mu_{\psi}\psi,
\label{grand}
\end{equation}
where $\mu_{\phi}$ and $\mu_{\psi}$ are the chemical potentials coupled with the A/B
relative compositions in the upper and lower domains, respectively.
They act as Lagrange multipliers that take into account the conserved $\phi$
and $\psi$ compositions.
In general, these two chemical potentials have different values, $\mu_{\phi} \ne \mu_{\psi}$.
The special case of $\mu_{\phi} = \mu_{\psi}$, for which only the total relative composition,
$\phi + \psi$, is conserved was investigated in our previous work~\cite{Wolff}, while here we
deal with a general situation where each of the compositions, $\phi$ and $\psi$,
are conserved independently.
\begin{figure}[tbh]
\begin{center}
\includegraphics[scale=0.4]{fig3.eps}
\end{center}
\caption{\textsf{
(a) Plot of the partially minimized free-energy $\varepsilon^{\ast}(\phi, \psi) $ as a function of $\phi$ and
$\psi$ for $\xi=0.25$, $\beta=1$ and $t=-0.2$ (below $T_{\rm c}$).
(b) A cut through the free-energy landscape $\varepsilon^{\ast}$ in the particular direction,
$\phi+\psi=0$, plotted as a function of $\Phi = (\phi-\psi)/2$.
The two P points are cusps at which the slope of $\varepsilon^*$ changes discontinuously, although the Hessian $H$ remains positive (see Appendix).
The two Q points correspond to the location at which $H$ vanishes.
}}
\label{fig3}
\end{figure}
The thermodynamic equilibrium between the two coexisting phases denoted as `1' and `2' and characterized by $(\phi_1, \psi_1)$ and $(\phi_2, \psi_2)$, satisfies the conditions~\cite{Gibbs}
\begin{eqnarray}
\left. \partial_\phi g(\phi,\psi)\right|_1&=&
\left. \partial_\phi g(\phi,\psi)\right|_2=0,\nonumber\\
\left. \partial_\psi g(\phi, \psi)\right|_1&=&
\left. \partial_\psi g(\phi,\psi)\right|_2=0,\nonumber\\
g(\phi_1, \psi_1)&=& g(\phi_2, \psi_2).
\label{eq123}
\end{eqnarray}
Similarly, for a three-phase coexistence between phases `1', `2' and `3', the following set of conditions should be satisfied:
\begin{eqnarray}
\label{3phase}
\left. \partial_\phi g(\phi,\psi)\right|_1&=&
\left. \partial_\phi g(\phi, \psi)\right|_2=
\left. \partial_\phi g(\phi, \psi)\right|_3=0\nonumber\\
\left. \partial_\psi g(\phi, \psi) \right|_1&=&
\left. \partial_\psi g(\phi, \psi) \right|_2=
\left. \partial_\psi g(\phi, \psi) \right|_3=0\nonumber\\
g(\phi_1, \psi_1)&=& g(\phi_2, \psi_2)=g(\phi_3, \psi_3).
\end{eqnarray}
In Fig.~\ref{fig3}(a), we show an example of the partially minimized free-energy, $\varepsilon^{\ast}(\phi,\psi)$, as a function of $\phi$ and $\psi$ at a fixed temperature $t=-0.2$ (below $T_{\rm c}$), and for given
values of $\xi$ and $\beta$. In order to have a better view of the free-energy surface, we show in Fig.~\ref{fig3}(b) a
cross-section cut of the free-energy surface in the direction of $\Phi = (\phi-\psi)/2$, while keeping $\phi+\psi=0$.
Here, we see two singular cusps (points P) where the determinant of the Hessian matrix, $H$ (see Appendix) does not vanish. At these cusps, $H$ changes discontinuously although it remains positive. At points Q, on the other hand, the Hessian $H$ vanishes.
More details on the Hessian matrix and determinant, and their relation to the phase stability and spinodal lines are presented in the Appendix.
By calculation the Hessian $H$, it is possible to derive the spinodal lines and critical points. Note that in most cases, the critical points and spinodal will not be shown on the phase diagrams, because they are preempted by the first-order phase transition lines and coexistence regions.
The phase diagrams are obtained by further minimizing $\varepsilon^{\ast}$, with respect to the two independent variables, $\phi$ and $\psi$. Convex regions of the free energy
correspond to single thermodynamical phases. Two-phase coexistence regions correspond to non-convex regions, where we can
construct a common tangent plane. The plane touches the free-energy surface at two points that determine the two phases in coexistence.
A more special three-phase coexistence region corresponds to a plane that touches the free-energy surface at three points.
More details on the numerical procedure of finding the phase diagrams are given below.
The numerical computation of the phase diagram is performed using a public-domain software
called ``Qhull"~\cite{qhull}.
The Qhull software generates initially a fine grid of triangulation,
which approximates the free-energy surface,
$\varepsilon^{\ast}$.
The three-phase coexistence regions correspond to facets with all sides being much larger than
the initial discretization.
The two-phase coexistence regions are associated with elongated triangles having one short side
that is much smaller than the other two longer sides that approximate the tie-lines.
Finally, small triangular facets of the free-energy surface are associated with stable one-phase regions.
The projection of the triangulated free-energy surface onto the composition plane provides
a systematic approximation for the phase diagram.
The Qhull results are then used as an initial condition in calculating more precisely
the equilibrium phase diagrams, including the various phase coexistence, Eqs.~(\ref{grand})--(\ref{3phase}).
\begin{figure}[tbh]
\begin{center}
\includegraphics[scale=0.5]{fig4.eps}
\end{center}
\caption{\textsf{
(a) Phase diagram in the ($\phi$, $\psi$) plane for $\beta=1$, $\xi=0.25$ and $t=-0.45$.
The corners of the phase diagram indicates the four one-phases: D$_\pm$ and B$_\pm$, while
the flat F phase (not drawn) strictly lies only on the diagonal $\phi=\psi$.
The black lines represent tie-lines in the two-phase regions, and the two triangles are the
three-phase regions (see text for more details).
(b) The phase diagram is plotted as in (a) but with a superimposed colored plot for the
curvature, $c$.
As one crosses the major diagonal, $\phi=\psi$,
there is a smooth change from D$_+$ ($c>0$) through the flat F ($c=0$)
to the D$_-$ ($c<0$).
Furthermore, the curvature also changes smoothly inside
the $D_\pm$ phases, but
the gradient in orange (D$_-$) and light blue (D$_+$) colors is not shown for clarity. However, the curvature
has a jump between B$_-$ ($c=-2$, red) and D$_-$ ($-2< c <0$, orange) regions, as well as
between B$_+$ ($c=2$, dark blue) and D$_+$ ($0< c<2$, light blue) ones.
}}
\label{fig4}
\end{figure}
\begin{figure}[tbh]
\begin{center}
\includegraphics[scale=0.5]{fig5.eps}
\end{center}
\caption{\textsf{
(a) Phase diagram in the ($\phi$, $\psi$) plane as in Fig.~\ref{fig4} but with
$\beta=3$, $\xi=0.25$ and $t=-0.45$.
(b) The phase diagram is plotted as in (a) but with a superimposed colored plot for the
curvature, $c$.
}}
\label{fig5}
\end{figure}
\section{Results}
\label{sec:result}
\subsection{Phase diagrams}
\label{sec:diagrams}
\begin{figure}[tbh]
\begin{center}
\includegraphics[scale=0.5]{fig6.eps}
\end{center}
\caption{\textsf{
(a) Phase diagram in the ($\phi$, $\psi$) plane as in Fig.~\ref{fig4}
but with $t=-0.2$, $\xi=0.25$ and $\beta=1$.
(b) The phase diagram is plotted as in (a) but with a superimposed colored plot for the
curvature, $c$.
}}
\label{fig6}
\end{figure}
\begin{figure}[tbh]
\begin{center}
\includegraphics[scale=0.5]{fig7.eps}
\end{center}
\caption{\textsf{
(a) Phase diagram in the ($\phi$, $\psi$) plane as in Fig.~\ref{fig4}
but with $t=-0.02$, $\xi=0.25$ and $\beta=1$.
The red circles correspond to the critical points.
(b) The phase diagram is plotted as in (a) but with a superimposed colored plot for the
curvature, $c$.
}}
\label{fig7}
\end{figure}
Four representative types of phase diagrams are shown in Figs.~\ref{fig4}--\ref{fig7}.
In the first two figures, we show the calculated phase diagram for $\beta=1$ (Fig.~\ref{fig4})
and $\beta=3$ (Fig.~\ref{fig5}), while the other parameters are fixed to
$\xi=0.25$ and $t=-0.45$.
The latter two phase diagrams are for $t=-0.2$ (Fig.~\ref{fig6}) and
$t=-0.02$ (Fig.~\ref{fig7}), while keeping $\beta=1$ and $\xi=0.25$.
For presentation purposes, in (b) of each figure, we have superimposed the color plot for
the curvature $c$ on the phase diagram presented in (a).
As denoted above, the various stable phases are the fully-budded phase (B) with
curvature $\vert c\vert=2$, the dimpled or partial budded phase (D) with curvature
$0<\vert c \vert< 2$, and the flat membrane (F) with zero curvature $c=0$.
The subscripts $\pm$ denote whether the bud is curved positively or negatively with
respect to the positive normal direction of the planar membrane.
Figure~\ref{fig4} includes five homogeneous phases: B$_\pm$, D$_\pm$ and F.
The B$_+$ and B$_-$ occupy the regions around the $\phi=-\psi=1$ and $\phi=-\psi=-1$ corners, respectively.
On the other hand, the D$_\pm$ occupy the two remaining corners: $\phi=\psi=\pm 1$. The difference between D$_+$ and D$_-$ is only associated with their curvature ($c\lessgtr 0$), which changes continuously.
This is further clarified in part (b) of the figure.
For all $t$-values, the flat F phase (not seen on the figure) strictly exists only on the $\phi=\psi$ diagonal, but is important for the multi-phase coexistence regions (as discussed below). As one approaches this line from above or below, the D$_\pm$ phases change-over smoothly into the F phase (with $c=0$) on the diagonal line.
Moreover, five two-phase coexistence regions are shown in Fig.~\ref{fig4} together with their calculated tie-lines: two
B$_+$/D$_+$ and two B$_-$/D$_-$ along the boundaries of the phase diagram, and one B$_+$/B$_-$ along the major diagonal, $\phi=-\psi$.
In addition, two three-phase coexistence regions: B$_+$/B$_-$/F can be seen. These are the two triangular regions lying above and below the $\phi=-\psi$ diagonal. Note that the F corner of the three-phase region lies close to the D$_\pm$ corners, but since it lies on the $\phi=\psi$ diagonal, it is identified as the F phase with $c=0$.
The three-phase coexistence region between F and B$_\pm$ phases means that
each point inside the triangular region is composed of three relative area fractions of the three coexisting phases: the flat (F) and budded (D$_\pm$) phases.
The nearly horizontal (D$_-/$B$_-$) or vertical (D$_+/$B$_+$) tie-lines on the boundaries of phase diagram indicate that the two $\phi$ and $\psi$ monolayers are almost decoupled, because either $\phi$ or $\psi$ do not vary along the tie-line. On the other hand, tie-lines that lie along the major diagonal, $\phi=-\psi$, indicate a strong coupling between the $\phi$ and $\psi$ monolayers in the B$_+/$B$_-$ coexistence region.
In Fig.~\ref{fig5} with $\beta=3$ and the same temperature as in Fig.~\ref{fig4}, we see that the central binary coexistence region (B$_+$/B$_-$) becomes much larger. On the other hand, the four two-phase coexistence regions of Fig.~\ref{fig4} have
shrunken because the extent of the dimpled phase becomes smaller for larger coupling parameter, $\beta$. When $\beta$ gets large values in Eqs.~(\ref{spontaneous1}) and (\ref{spontaneous2}),
the composition of each monolayer induces higher curvature that promotes budding. In this case, the central two-phase region, B$_+$/B$_-$, occupies a large fraction of the phase diagram, and most of its tie-lines are parallel to the diagonal $\phi = -\psi$, suggesting a strong coupling between the two monolayers. On the other hand, the two three-phase coexistence regions, B$_+$/B$_-$/F, become smaller in Fig.~\ref{fig5}. Furthermore, we note that in Fig.~\ref{fig5}(b), the regions of the dimpled phases,
D$_\pm$, represented by the light blue and orange regions are narrower as compared with those of Fig.~\ref{fig4}(b).
It is of interest to explain how one of the monolayers induces a phase
transition in the second monolayer.
As an illustrative example, Let us consider a point in Figs.~\ref{fig4} and
\ref{fig5} with average leaflet composition, $(\phi,\psi)=(0,-0.9)$.
At these compositions, the bilayer separates in Fig.~\ref{fig4} into a D$_-$ phase with compositions
$(-0.75,-0.9)$ and a B$_-$ phase with $(0.75, -0.9)$.
In this weak coupling case, $\beta=1$, the phase separation in one monolayer
does not induce any instability leading towards phase separation in the second monolayer, because the tie-line is nearly
parallel to the horizontal $\phi$-axis.
On the other hand, in Fig.~\ref{fig5} with a larger value of $\beta=3$, the bilayer separates into a B$_-$ phase with $(0.1, -0.9)$ and a B$_+$ phase with $(-0.9, 0.1)$.
As the tie-line in this case lies along the major diagonal,
$\phi=-\psi$, the two monolayers are influencing each other.
Such a situation results from a strong composition-curvature coupling
when the parameter $\beta$ is large enough.
A more general dependence of the phase diagram on the parameter
$\beta$ will be further discussed in the next subsection.
Figure~\ref{fig6} is plotted for a higher temperature $t=-0.2$ than in Figs.~\ref{fig4} and {\ref{fig5}, while we fix $\beta=1$ as in Fig.~\ref{fig4}. The dimpled region expands both toward the corners and the middle of the phase diagram.
Moreover, there are two new one-phase regions of the dimpled phase (D$_\pm$)
resulting in four additional two-phase coexistence regions:
two D$_+$/D$_+$ and two D$_-$/D$_-$.
The system exhibits a first-order phase transition in composition,
while the transition is second-order in curvature.
In Fig.~\ref{fig7}, the temperature is increased to $t=-0.02$, while $\beta$ and $\xi$ stay as in Fig.~\ref{fig4}. The chosen $t$-value is higher than in the previous figures, and approaches the critical temperature, $t_{\rm c}=0.04$~\cite{Wolff}.
The central region of the phase diagram is dominated by the dimpled phase (D$_\pm$), while the budded regions (B$_\pm$) exist only close to the two corners, with two-phase coexistence regions, B$_+$/D$_+$ and B$_-$/D$_-$.
As seen in (b), the regions of the dimpled phases (D$_\pm$) are determined by $\beta$, and do not depend on the temperature $t$ as long as $\xi$ is fixed.
Furthermore, there are four critical points appearing in the central region (marked by red circles), for which the system exhibits a second-order phase transition both in curvature and composition.
\subsection{Global phase behavior}
\label{sec:global}
\begin{figure}[tbh]
\begin{center}
\includegraphics[scale=0.6]{fig8.eps}
\end{center}
\caption{\textsf{
Schematic phase-behavior plot in the ($1/\xi, \beta$) plane for temperatures
(a) $t=-0.25$ and (b) $t=-0.02$.
The diagram types, I, II, III and IV, correspond to representative examples
as in Figs.~\ref{fig4}, \ref{fig5}, \ref{fig6}, and \ref{fig7}, respectively.
}}
\label{fig8}
\end{figure}
Next, we investigate how the different phase-diagram types appear and change as we adjust the system parameters in a global way.
In Fig.~\ref{fig8}, we present the global phase behavior in the ($1/\xi, \beta$) plane for $t=-0.25$ in (a) and $t=-0.02$ in (b).
We show how the four different types of phase diagrams of Figs.~\ref{fig4}, \ref{fig5}, \ref{fig6}, \ref{fig7}, labelled as I, II, III and IV, respectively, evolve as function of $\xi$ and
$\beta$.
Figure~\ref{fig8}(a) summarizes the result for $t=-0.25$ and is valid even for lower $t$-values.
For $\beta>2$ (strong coupling), the diagram is that of type II in which the dimpled phase (D) region shrinks while the budded phase (B) expands.
For $\beta<2$ (weak coupling), on the other hand, the diagram is mostly of type I for which the wide three-phase coexistence occurs.
For $\beta<1$ and $1/\xi<4$, type III diagram is found, and contains
coexistence regions of the dimpled phases (D$_+$/D$_+$ and D$_-$/D$_-$).
In Fig.~\ref{fig8}(b), for $t=-0.02$ that is closer to the critical temperature, type III is replaced by type IV for $\beta<1$ and $1/\xi<5$, while type II also extends to smaller $\beta$ when $1/\xi$ is large enough.
When the temperature is higher, the phase transition in curvature
from the flat (F) phase to the dimpled (D)
one, becomes continuous~\cite{Wolff}.
\section{Discussion}
\label{sec:discussion}
In this paper, we have discussed a phenomenological model that accounts for the budding transition of asymmetric two-component lipid domains, where the two domains have in general different average compositions, represented by $\phi$ and $\psi$. Assuming a linear composition dependence of the spontaneous curvature, we have taken into account a coupling between the local curvature and local lipid composition in each of the two leaflets. We then explored the morphological changes between the flat and budded domains by using a thermodynamic argument. Our free-energy model contains three contributions: bending energy, accounting for domain deformation in the normal direction; line tension along the rim of the budded or flat domain;
and a Landau free-energy expansion, which accounts for a phase separation of the two-component lipid domains. We have assumed, in addition, that the domain area remains constant during the budding process.
Our model predicts three different states for the domains: fully budded (B), dimpled (D), and flat (F) states. In particular, in some ranges of parameters, the D state is found to be the most stable one, as observed in the experiment~\cite{RUPK}.
Within mean-field theory, we have calculated various phase diagrams in terms of two compositions for different temperatures $t$, domain size $1/\xi$, and coupling parameter, $\beta$.
The resulting phase behavior is very rich.
The calculated phase diagrams include various one-phase, two-phase (e.g., B$_+$/D$_+$, B$_-$/D$_-$ or B$_+$/B$_-$) and three-phase coexistence regions (e.g., B$_+$/B$_-$/F), depending on the curvature-composition parameter $\beta$ as well as the temperature. Finally, four different types of phase diagram morphologies are found, and we have analyzed the global phase behavior in terms of the coupling parameter $\beta$, and
the domain size $\xi$. The model analysis suggests that the asymmetry in the lipid composition between the two leaflets can lead to complex morphological and thermodynamic behavior of lipid domains.
The most important mechanism that leads to the inter-leaflet correlated phase separation, such as the two-phase coexistence seen in Fig.~\ref{fig5} between two budded phase (B$_+$/B$_-$), is the composition-dependent spontaneous curvature, introduced in Eqs.~(\ref{spontaneous1}) and (\ref{spontaneous2}).
With this mechanism, the coupling between domain curvature and its composition is controlled by the parameter $\beta$. Although we did not include any direct interaction between the domains occurring on the two leaflets, the fact that the two domains are in full registry to each other and have opposite curvatures, $\pm C$, results in a strong correlation between the opposing domains. We consider that such an effect generally exists in bio-membranes, even in the absence of specific proteins that maintain the compositional asymmetry between the two leaflets.
When the coupling parameter $\beta$ is made larger at a fixed temperature
(compare Figs.~\ref{fig4} and \ref{fig5}), the two-phase coexistence region,
B$_+$/B$_-$, dramatically expands due to the coupling between the two leaflets. Such a strongly correlated phase separation can be in general observed for lower temperatures. We also note that the stable budded (B) phase occupies the two asymmetric corners of the phase diagram, i.e., $\phi=-\psi=\pm 1$ . Moreover, the region of the dimpled (D) phase increases as the temperature is raised towards the critical temperature from below (compare Figs.~\ref{fig4}, \ref{fig6} and \ref{fig7}).
Especially, in Fig.~\ref{fig7}, the dimpled one-phase region dominates the central region of the phase diagram and four different critical points are expected to appear. In the intermediate temperature, as in Fig.~\ref{fig6}, the phase diagram contains several types of two-phase coexistence regions.
The present work for asymmetric domains is a generalization of our previous model~\cite{Wolff}, where only the average composition of the domains in the two leaflets is controlled by a single chemical potential.
Here, we have considered a more general situation, where each of the two domain composition is controlled in a separate and independent way. This is done by introducing two independent chemical potentials coupled to the two domain compositions, $\phi$ and $\psi$, as in Eq.~(\ref{grand}).
Hence, each domain has a conserved lipid composition that is independent from the other one. The resulting phase diagrams show a much richer phase behavior, while the previous results~\cite{Wolff} can be recovered by considering the special case of $\phi+\psi={\rm const.}$
Our assumption that the two opposing domains in different leaflets are correlated to each other is in accord with several experimental observations. For studies of planar but asymmetric composition in the two leaflets, Collins \textit{et al.}~\cite{Collins} reported that in some cases, one leaflet can induce a phase separation in the other leaflet, depending on local lipid composition. Whereas in other cases, the two leaflets do not interact. A similar experimental phenomenon was observed for membranes composed of lipids extracted from biological cells~\cite{Kiessling}.
Such situations for asymmetric membranes can be partially described by the different phase behaviors of asymmetric membrane depending on the coupling parameter $\beta$ as shown in Fig.~\ref{fig8}.
Moreover, both in our model and in experiments~\cite{Collins}, domain-induced processes take place in lipid membrane without any proteins.
These results may suggest a cellular mechanism for regulating protein function by modulating the local lipid composition or inter-leaflet interactions.
Although we have mainly discussed the domain-induced budding, our model
can be applied to describe the formation of vesicles in mixed amphiphilic systems~\cite{SH}. It was observed in experiment that mixtures of anionic and cationic surfactants in solution form disk-like bilayers in some range of the relative amphiphilic composition. As these disk-shaped bilayers grow in size, they transform into spherical caps and eventually become spherically closed vesicles. In such cases, the spontaneous curvature of bilayer membranes may be induced due to the compositional asymmetry between the two monolayers.
Finally, our model suggests that the asymmetry in the lipid composition between the two leaflets
leads to a complex behavior of lipid
domains even in the absence of any specific enzymes or proteins, which can induce additional coupling between
the two leaflets~\cite{Gri04}. The importance of such a pure physical mechanism can be verified in experiments on asymmetric model membranes involving only a lipid mixture. They also
can be of relevance to signal transduction~\cite{Simons00},
membrane fusion~\cite{Puri06}, or penetration of viruses into cells~\cite{Chazal03}. We hope that
additional experiments will address these issues in the future.
\bigskip
{\em Acknowledgments.~}
J.W.\ acknowledges support from the Service de Coop\'eration Scientifique et Universitaire de
l'Ambassade de France en Isra\"el, the French ORT association, and ORT school of Strasbourg.
S.K.\ acknowledges support from the Grant-in-Aid for Scientific Research on
Innovative Areas ``\textit{Fluctuation and Structure}" (Grant No.\ 25103010) from the Ministry
of Education, Culture, Sports, Science, and Technology of Japan, and
the Grant-in-Aid for Scientific Research (C) (Grant No.\ 15K05250)
from the Japan Society for the Promotion of Science (JSPS).
D.A. acknowledges the hospitality of the KITPC and ITP, Beijing, China, and partial support from
the Israel Science Foundation (ISF) under Grant No.\ 438/12, the United States--Israel Binational Science Foundation (BSF) under Grant No.\ 2012/060, the ISF-NSFC joint research program under Grant No.\ 885/15, and a CAS President's International Fellowship
Initiative (PIFI, China).
|
\section{Introduction}
Deep visuomotor policies that map from pixels to actions end-to-end can represent complex manipulation skills \cite{levine_finn_2016}, but have shown to be sensitive to the choice of action space \cite{martin2019iros} -- e.g., Cartesian end effector actions (i.e. task space \cite{berenson2011task}) perform favorably when learning policies for tabletop manipulation, while joint actions have shown to fare better for whole-body motion control \cite{peng2017learning, Tan-RSS-18}.
In particular, policies modeled by deep networks are subject to spectral biases \cite{ronen2019convergence,battaglia2018relational, bietti2019inductive} that make them more likely to learn and generalize the low-frequency patterns that exist in the control trajectories.
Hence, choosing the right action space in which to define these trajectories remains a widely recognized problem in both reinforcement learning \cite{peng2017learning} as well as imitation learning, where demonstrations can be provided in a wide range of formats -- e.g., continuous teleoperation in Cartesian or joint space, kinesthetic teaching, etc. -- each changing the underlying characteristics of the training data.
\begin{figure}[t!]
\centering
\vspace{0.5em}
\noindent\includegraphics[width=1.0\linewidth]{images/splash-v4.png}
\caption{Enabled by an implicit policy formulation, Implicit Kinematic Policies (IKP) provide both Joint and Cartesian action representations (linked via forward kinematics) to a model that can pick up on inductive patterns in both spaces. Subsequently, this allows our end-to-end policies to extract the best action space for a wide variety of manipulation tasks ranging from tabletop block sorting to whole-arm sweeping.}
\label{fig:teaser}
\vspace{-1.5em}
\end{figure}
Although considerable research has been devoted to finding the right action space for a given application \cite{peng2017learning}, much less attention has been paid to figuring out how our models could instead discover for themselves which optimal combination of action spaces to use.
This goal remains particularly unclear for most \textit{explicit} policies $f_\theta(\textbf{o})=\hat{\textbf{a}}$ with feed-forward models that take as input observations $\textbf{o}$ and output actions $\textbf{a}\in\mathcal{A}$. Explicit formulations must either specify a single action space to output from the model and potentially convert predictions into another desired space (e.g. with inverse kinematics), or use multiple action spaces with multiple outputs and losses which can be subject to conflicting gradients \cite{yu2020gradient} from inconsistent predictions \cite{martin2019iros}.
In this work, we demonstrate that we can train deep policies to learn which action space to use for imitation, made possible by implicit behavior cloning (BC) \cite{florence2021implicit}. In contrast to its explicit counterpart, \textit{implicit} BC formulates imitation as the minimization of an energy-based model (EBM) $\hat{\textbf{a}}=\arg\min_{\textbf{a} \in \mathcal{A}} E_{\theta}(\textbf{o},\textbf{a})$ \cite{lecun2006tutorial}, which regresses the optimal action via sampling or gradient descent \cite{welling2011bayesian, du2019implicit}. Since actions are now inputs to the model, we propose presenting the model with the same action represented in multiple spaces while remaining consistent between each other, allowing the model to pick up on inductive patterns \cite{battaglia2018relational, bietti2019inductive} from all action representations.
We study the benefits of this multi-action-space formulation with Cartesian task and joint action spaces in the context of learning manipulation skills. Since both spaces are linked by the kinematic chain, we can integrate a differentiable kinematic module within the deep network -- a formulation which we refer to as Implicit Kinematic Policies (IKP). IKP can be weaved into an implicit autoregressive control strategy introduced in \cite{florence2021implicit}, where each action dimension is successively and uniformly sampled at a time and passed as input to the model. This exposes the model to not only the joint configuration and end-effector pose, but also the Cartesian action representations of {\em{every rigid body link}} of the arm as input. This can provide downstream benefits when learning whole-body manipulation tasks that may have emergent patterns in the trajectory of any given link of the robot. Furthermore, a key aspect of Implicit Kinematic Policies is that in addition to exposing the implicit policy to both Cartesian and joint action spaces, it also enables incorporating learnable layers throughout the kinematic chain, which can be used to optimize for residuals in either space
Our main contribution is Implicit Kinematic Policies, a new formulation that integrates forward kinematics as a differentiable module within the deep network of an autoregressive implicit policy, exposing both joint and Cartesian action spaces in a kinematically consistent manner. Behavior cloning experiments across several complex vision-based manipulation tasks suggest that IKP, without any modifications, is broadly capable of efficiently learning both prehensile and non-prehensile manipulation tasks, even in the presence of miscalibrated joint encoders -- results that may pave the way for more data efficient learning on low-cost or cable-driven robots, where low-level joint encoder errors (due to drift, miscalibration, or gear backlash) can propagate to large non-linear artifacts in the Cartesian end effector trajectories. IKP achieves 85.9\%, 97.5\% and 92.4\% on a sweeping, non-prehensile box lifting and precise insertion task respectively -- performance that is on par with or exceeds that of standard implicit or explicit baselines (using the best empirically chosen action space per individual task). We also provide qualitative experiments on a real UR5e robot suggesting IKP's ability to learn from noisy human demonstrations in a data-efficient manner.
Our experiments provide an initial case study towards more general action-space-agnostic architectures for end-to-end robot learning.
\section{Related Work}
\subsection{Learning Task-Specific Action Representations}
While much of the early work in end-to-end robot learning has focused on learning explicit policies that map pixels to low-level joint torques \cite{ddpg2016, levine_finn_2016}, more recent work has found that the choice of action space can have a significant impact on downstream performance for a wide range of robotic tasks ranging from manipulation to locomotion \cite{martin2019variable, duan2021learning, peng2017learning, Viereck_2018, varin2019comparison}. In the context of deep reinforcement learning for locomotion, \cite{peng2017learning} find that using PD joint controllers achieves higher sample efficiency and reward compared to torque controllers on bipedal and quadrupedal walking tasks in simulation, whereas \cite{Viereck_2018} interestingly find that torque controllers outperform PD joint controllers for a real robot hopping task. In contrast, \cite{duan2021learning} shows that using task space actions are more effective for learning higher level goals of legged locomotion compared to joint space controllers and demonstrate results on a real bipedal robot.
Careful design of the action space is equally important when learning manipulation skills with high DoF robots \cite{martin2019variable, varin2019comparison}. Prior approaches successfully use direct torque control or PD control with joint targets to learn a variety of tasks such as pick-and-place, hammering, door opening \cite{rajeswaran2018learning} and stacking \cite{popov2017dataefficient}, but these methods often require lots of data and interaction with the environment to generalize well and are also sensitive to feedback gains in the case of PD control.
Other works default to low-level residual cartesian PD controllers in order to learn prehensile tasks such as block stacking \cite{johannink2018residual} and insertion \cite{silver2019residual}, but \cite{martin2019variable, varin2019comparison} instead vouch for a task-space impedance controller to support more compliant robot control. However, this approach is agnostic to the full joint configuration of the robot and focuses on tabletop manipulation which is generally better defined in task space. Fundamentally, these works share the common theme of choosing the single best action space for a particular task. In this work, we are more concerned with the problem of extracting the best combination of action spaces.
\subsection{Difficulties in Learning High-Frequency Functions}
The proposed benefits of implicit kinematic policies rely on the hypothesis that simpler patterns are easier for deep networks to learn. Although theoretical analyses of two-layer networks and empirical studies of random training labels have shown that deep networks are prone to memorization \cite{Zhang2017UnderstandingDL}, further studies on network convergence behavior and realistic dataset noise have found that this result does not necessarily explain performance in practice. Instead, deep networks first learn patterns across samples before memorizing data points \cite{Arpit2017ACL}. \cite{Rahaman2019Spectral} formalize this finding and show that deep networks have a learning bias towards low-frequency functions by using tools from Fourier analysis and \cite{Basri2019TheCR} extend this result by showing that networks fit increasingly higher frequency functions over the course of training. Within learning-based robotics, multiple works have found the related result that higher-level action spaces lead to improved learnability. Specifically, the frequency of the action space can have an outsized effect on the robustness and quality of learned policies \cite{peng2017learning}. This could explain why learned, latent actions improve performance and generalizability for policies trained either with reinforcement learning \cite{Haarnoja2018LatentSP,Hausman2018LearningAE,Nair2020HierarchicalFS,Pflueger2020PlanSpaceSE} or imitation learning \cite{fox2017multi}.
\begin{figure}[t]
\centering
\noindent\includegraphics[width=\linewidth]{images/implicit_vs_explicit.png}
\caption{Integrating forward kinematics (FK), in particular, synergizes with implicit policies (a, b), where actions $\textbf{a}$ are sampled in joint space, and both joints and corresponding Cartesian actions along with observations $\textbf{o}$ can be fed as inputs to the EBM $f_\theta$. For explicit policies (c), it becomes less clear how to expose both action spaces in a kinematically consistent way to the deep network, except via predicting joints from observations, then using forward kinematics to backpropagate gradients from losses imposed on the final output Cartesian actions (in red). In this work, we investigate both formulations and find that implicit (b) yields better performance.
}
\label{fig:key-idea}
\vspace{-2em}
\end{figure}
\section{Background}
\subsection{Problem Formulation}
We consider the imitation learning setting with access to a fixed dataset of observation-action pairs. We frame this problem as supervised learning of a policy $\pi: \mathcal{O} \rightarrow \mathcal{A}$ where $\mathbf{o} \in \mathcal{O} = \mathbb{R}^m$ represents an observational input to the policy and $\mathbf{a} \in \mathcal{A} = \mathbb{R}^d$ represents the action output. The policy $\pi$, with parameters $\theta$, can either be formulated as an {\em{explicit}} function $f_{\theta}(\mathbf{o}) \rightarrow \hat{\mathbf{a}}$ or as an {\em{implicit}} function $\arg\min_{\mathbf{a}\in\mathcal{A}} E(\mathbf{o}, \mathbf{a}) \rightarrow \hat{\mathbf{a}}$. In this work, we adopt the implicit formulation and build upon recent work which we describe in more detail in Section \ref{implicitbcsec}. We operate in the continuous control setting, where our policy infers a desired target configuration at 10 Hz, and a low-level controller asynchronously reaches desired configurations with a joint-level PD controller. Our $\mathcal{O}$ contains both image observations and proprioceptive robot state (i.e. from joint encoders).
\subsection{Implicit BC}
\label{implicitbcsec}
We use an energy-based, contrastive loss to train our implicit policy, modeled with the same late-fusion deep network architecture as in \cite{florence2021implicit} with 26 convolutional ResNet layers \cite{resnets_2016} for the image encoder and 20 dense ResNet layers for the EBM. Specifically, to optimize $E_{\theta}(\cdot)$, we train an InfoNCE style loss \cite{Oord2018RepresentationLW} on observation-action samples \cite{florence2021implicit}. Our dataset consists of $\{\mathbf{o}_i, \mathbf{a}^*_i\}$ for $\mathbf{o}_i \in \mathbb{R}^m$ and $\mathbf{a}_i \in \mathbb{R}^d$ and from regression bounds $\mathbf{a}_{\min} , \mathbf{a}_{\max} \in \mathbb{R}^d$ we generate negative samples $\{\tilde{\mathbf{a}}_i^j\}_{j=1}^{N_{\text{neg.}}}$. The loss equates to the negative log likelihood of $p(\mathbf{a}|\mathbf{o})$ where we use negative samples to approximate the normalizing constant:
\begin{align*}
\mathcal{L}_{\text{InfoNCE}} &= \sum_{i=1}^N -\log(\tilde{p}_\theta(\mathbf{a}^*_i|\mathbf{o}_i,\{\tilde{\mathbf{a}}_i^j\}_{j=1}^{N_{\text{neg.}}})) \\
\tilde{p}_\theta(\mathbf{a}^*_i|\mathbf{o}_i,\{\tilde{\mathbf{a}}_i^j\}_{j=1}^{N_{\text{neg.}}}) &= \frac{e^{-E_\theta(\mathbf{o}_i, \mathbf{a}^*_i)}}{e^{-E_\theta(\mathbf{o}_i, \mathbf{a}^*_i)} + \sum_{j=1}^{N_{\text{neg.}}}e^{-E_\theta(\mathbf{o}_i, \tilde{\mathbf{a}}_i^j)}}
\end{align*}
To perform inference, given some observation $\mathbf{o}$, we minimize our learned energy function, $E_{\theta}(\mathbf{o}, \mathbf{a})$, over actions in $\mathcal{A}$ using a sampling-based optimization procedure.
\label{implicitbc}
\label{background}
\section{Method}
Here we present the details of our proposed Implicit Kinematic Policies (IKP) method. First, we will present our extension to the implicit policy formulation which provides multiple action space representations as input. (Sec.~\ref{subsec:multi-action-space}). Sec.~\ref{subsec:multi-action-space-explicit} also compares the implicit multi-action-space formulation to its explicit counterpart and discusses the relevant tradeoffs. We then describe how to perform autoregressive training and inference with implicit policies in a way that exposes the action trajectories of every joint and link in the robot to the model (Sec.~\ref{subsec:autoregressive-formulation}). Finally, we discuss a motivating application of controlling miscalibrated robots for precise tasks, and how IKP is uniquely suited to automatically compensate for this noise through the use of strategically placed residual blocks in the model (Sec.~\ref{subsec:residual-formulation}).
\subsection{Multi-Action-Spaces With Implicit Policies}\label{subsec:multi-action-space}
Our formulation enables the implicit policy $\hat{\mathbf{a}} = \arg\min_{\mathbf{a}} E_{\theta}(\mathbf{o}, \mathbf{a})$ to have access to multiple action spaces, which are constrained to be consistent. Specifically, our implicit multi-action-space formulation is of the form:
\vspace{-.85em}
\begin{equation}
\begin{aligned}
\underset{\mathbf{a}, \mathbf{a}', ...}{\arg\min} \quad & E_{\theta}(\mathbf{o}, \mathbf{a}, \mathbf{a}', ...)\\
\textrm{s.t.} \quad & \mathbf{a} = \mathcal{T}(\mathbf{a}'), ...
\end{aligned}
\label{eq:general-multi-action-space}
\end{equation}
where $\mathbf{a} \in \mathcal{A}$ is one parameterization of the action space, $\mathbf{a}' \in \mathcal{A}'$ is a different parameterization of the action space, and $\mathcal{T}(\cdot): \mathcal{A}' \rightarrow \mathcal{A}$ is a transformation between the two action spaces. Of course, $N$ different consistent parameterizations of the action space could be represented, which is depicted in Eq.~\ref{eq:general-multi-action-space} by the ellipses (...).
In particular for robots, we are interested in representing both {\em{joint-space}} and {\em{cartesian-space}} actions, which is a case of Eq.~\ref{eq:general-multi-action-space} in which the transformation between the two action spaces is forward kinematics (FK):
\vspace{-0.75em}
\begin{equation}
\begin{aligned}
\underset{\mathbf{a}_{\text{joints}}, \ \mathbf{a}_{\text{cartesian}}}{\arg\min} \quad & E_{\theta}(\mathbf{o}, \mathbf{a}_{\text{joints}}, \mathbf{a}_{\text{cartesian}})\\
\textrm{s.t.} \quad & \mathbf{a}_{\text{cartesian}} = FK(\mathbf{a}_{\text{joints}})
\end{aligned}
\label{eq:joints-cartesian-implicit}
\end{equation}
Specifically, we accomplish this by first sampling input actions in joint space $\mathbf{a}_{\text{joints}} \in \mathcal{A}_{\text{joints}}$, then computing the corresponding Cartesian (task) space actions via forward kinematics ($FK$), $\mathbf{a}_{\text{cartesian}} = FK(\mathbf{a}_{\text{joints}})$, and finally concatenating and passing the combined representation into the model $E_{\theta}(\mathbf{o}, \mathbf{a}_{\text{joints}}, \mathbf{a}_{\text{cartesian}})$. A visualization of this model is shown in the middle portion of Fig.~\ref{fig:key-idea}. Our hypothesis is that the model can use this redundant action representation to exploit patterns in both spaces, akin to {\em{automatically}} discovering the best combination of action spaces -- this hypothesis will be tested in our Experiments section.
As we will show in Sec.~\ref{subsec:autoregressive-formulation}, we can perform training and inference for $E_{\theta}$ through using autoregressive derivative-free optimization.
\begin{figure*}[t!]
\label{fkdiagram}
\centering
\noindent\includegraphics[width=\textwidth]{images/overview.png}
\caption{ \textbf{Method overview.} Given an image captured from an RGB camera overlooking the robot workspace (a), we feed it as input to a deep convolutional network (b) to get a latent state representation. We then predict desired robot joint actions by leveraging implicit autoregression \cite{florence2021implicit, nash2019autoregressive} (c) with state-conditioned EBMs to progressively sample each action dimension (joint angle) at a time: i.e., we uniform sample $\text{j}_n$, feed it to $\text{FK}_n$ (forward kinematics to link $n$, prepended with deep layers) to get the Cartesian pose $\text{C}_{n}$, which is then concatenated with the latent state and all previously sampled argmin joint dimensions $[\text{j}_n, \text{j}_{n-1}, ..., \text{j}_0]$ and Cartesian representations $[\text{C}_{n-1}, ..., \text{C}_0]$ and fed to an 8-layer EBM $\text{E}_{n}$ to compute the argmin over $\text{j}_n$.
}
\label{fig:method-overview}
\vspace{-1.5em}
\end{figure*}
\subsubsection{Multi-Action-Spaces With Explicit Policies}
\label{subsec:multi-action-space-explicit}
Consider the above formulation in contrast to an explicit policy, $\hat{\mathbf{a}}=f_{\theta}(\mathbf{o})$, where the mapping $f_{\theta}(\cdot)$ maps to one specific action space. It is possible to provide different losses on different transformations of the explicit policy's action space, for example $\mathcal{L}_{\text{joints}}(\hat{\mathbf{a}}, \mathbf{a}^*)$ and $\mathcal{L}_{\text{cartesian}}(FK(\hat{\mathbf{a}}), FK(\mathbf{a}^*))$, where $FK$ represents differentiable forward kinematics. However, this can lead to conflicting gradients and requires choosing relative weightings, $\lambda$, between these losses: $\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{joints}} + \lambda \mathcal{L}_{\text{cartesian}}$. This issue of loss balancing is exacerbated if the explicit policy were to regress the Cartesian representation of every link in the robot in addition to the joint configuration and end-effector pose. Additionally, the explicit policy predicts a single joint configuration even though there may be multiple correct configurations when using kinematically redundant robots. It is possible to first regress Cartesian actions and subsequently recover the corresponding joint configuration via inverse kinematics, however, IK solvers are only approximately differentiable which poses challenges for end-to-end training. A visualization of this explicit policy is shown in the rightmost column of Fig.~\ref{fig:key-idea}.
\subsection{Autoregressive Implicit Training and Inference}
\label{subsec:autoregressive-formulation}
As in \cite{florence2021implicit}, we employ an autoregressive derivative-free optimization method for both training and inference. Sampling-based, derivative-free optimization synergizes with the multi-action-space constrained energy model (Eq.~\ref{eq:general-multi-action-space}), since gradient-based optimization (i.e. through Langevin dynamics \cite{florence2021implicit}), while possible, would require a solution to synchronize between the different action spaces.\footnote{Specifically, when using the implicit model gradients, with learning rate $\lambda$, the actions are updated as $\mathbf{a}_{\text{joints}}^{k+1} = \mathbf{a}_{\text{joints}}^{k} - \lambda \nabla_{\mathbf{a}_{\text{joints}}} E_{\theta}(\cdot)$ and $\mathbf{a}_{\text{cartesian}}^{k+1} = \mathbf{a}_{\text{cartesian}}^{k} - \lambda \nabla_{\mathbf{a}_{\text{cartesian}}} E_{\theta}(\cdot)$. The updated actions
actions may not be consistent with each other -- even for an analytical model, there will be differences in the first-order derivatives of the different spaces.}
The autoregressive procedure uses $m$ models, one model $E_{\theta}^{j}(\mathbf{o}, \mathbf{a}^{0:j})$ for each dimension $j = 1, 2, ...,m$. In contrast with prior work \cite{florence2021implicit}, instead of $\mathbf{a}$ only representing a single action space, our proposed method represents, for each $j$th model, both the 6DoF Cartesian pose of the $j$th link {\em{and}} the joint of that link into $\mathbf{a}$. Similarly in $\mathbf{o}$ we also represent the dual Cartesian-and-joint state of each link. Since our method strictly samples actions in joint space and the state is represented by the joint configuration, the Cartesian representations for both the state $\mathbf{o}$ and actions $\mathbf{a}$ are acquired through $FK(\mathbf{o})$ and $FK(\mathbf{a})$ for every dimension $j = 1, 2, ...,m$. We visualize this procedure in Fig. \ref{fig:method-overview}
\subsection{Residual Forward Kinematics}
\label{subsec:residual-formulation}
In robotics, we often assume access to accurate kinematic descriptions and joint encoders, but this assumption can be a significant source of error in the context of low-cost or cable-driven robots. In these settings, the errors often take the form of fixed, unknown linear offsets in each joint motor encoder or link representation. If calibration isn't performed, both cases can cause heavy non-linear offsets in end-effector space which can drastically affect the performance on high-precision tasks. By adding dense residual blocks both before and after joint actions are sampled in the autoregressive procedure, we can encourage our model to learn these linear offsets at each joint and link rather than solving the more difficult problem of learning a highly non-linear offset in task space. This extension is only possible due to the full differentiability of the forward kinematics layer, which allows gradients to flow through the residual blocks. Concretely, we redefine our multi-action-space formulation with residuals as follows:
\vspace{-1.5em}
\begin{equation}
\begin{aligned}
\underset{\mathbf{a}_{\text{joints}}, \ \mathbf{a}_{\text{cartesian}}}{\arg\min} \quad & E_{\theta}(\mathbf{o}, \mathbf{a}_{\text{joints}}, \mathbf{a}_{\text{cartesian}})\\
\textrm{s.t.} \quad & \mathbf{a}_{\text{cartesian}} = FK\big(\mathbf{a}_{\text{joints}} + \Delta_{\theta}(\mathbf{a}_{\text{joints}})\big)
\end{aligned}
\label{eq:joints-cartesian-residuals-implicit}
\end{equation}
where $\Delta_{\theta}(\mathbf{a}_{\text{joints}}) \in \mathcal{A}_{\text{joints}}$ is a new learnable module in the implicit model which gives the model the inductive bias that there may be imperfect calibration of the joint measurements (i.e., biased joint encoder measurements). The module $\Delta_{\theta}(\cdot)$ can be learned during training and we hypothesize that the EBM may be able to automatically learn the $\Delta_{\theta}(\cdot)$ which provides the lowest-energy fit of the data. This can be interpreted as automatically calibrating the biased joint encoders. Experiments testing this hypothesis are in Section~\ref{subsec:residual-experiments}.
\begin{figure}[t]
\centering
\noindent\includegraphics[width=\linewidth]{images/Sweeping+Flipping.png}
\caption{Expert trajectories for bimanual sweeping are characterized by distinct patterns in Cartesian space (b), e.g., y-values experience a mode-change in the latter half of the episode when the policy switches bowls in which to drop particles. Such patterns are less salient in joint space (a). The opposite is true for bimanual flipping, where linear patterns emerge in joint space (c), whereas they are less salient in Cartesian space (d) especially with randomized end effector poses (semi-transparent plots).}
\label{fig:tasks}
\vspace{-1.5em}
\end{figure}
\section{Experiments}
We evaluate IKP across several vision-based continuous control manipulation tasks with quantitative experiments in simulation, as well as qualitative results on a real robot. All tasks require generalization to unseen object configurations at test time. The goals of our experiments are two-fold: (i) across tasks where one action space substantially outperforms the other, we investigate whether IKP can achieve the best of both and perform consistently well across all tasks, and (ii) given a miscalibrated robot with unknown offsets in the joint encoders (representing miscalibration or low-cost encoders), we study whether IKP can autonomously learn to compensate for these offsets while still succeeding at the task. Across all experiments, a dataset of expert demonstrations is provided by a scripted oracle policy (in simulation), or by human teleoperation (in real).
\subsection{\textbf{Simulated Bimanual Sweeping and Flipping}}
In a simulated environment, we evaluate on two bimanual tasks -- sweeping and flipping -- both of which involve two 7DoF KUKA IIWA robot arms equipped with spatula-like end-effectors positioned over a $0.4m^2$ workspace. The setup for sweeping is identical to that presented in Florence et al. \cite{florence2021implicit}, where given a pile of 50-60 particles on the workspace, the task is to scoop and evenly distribute all particles into two bowls located next to the workspace. For flipping, given a bowl of 20-30 particles attached to the top of a $0.2m^3$ box, the task is to flip the box to pour the particles into a larger bowl positioned near the base of the arms. Both tasks are designed to involve significant coordination between the two arms. During sweeping, for example, scooping up particles and transporting them to the bowls requires carefully maintaining alignment between the tips of the spatulas to avoid dropping particles. Flipping, on the other hand, requires aligning the elbow surfaces against the sides of the box, and using friction to carry the box as it pours the particles into the bowl -- any subtle misalignment of the elbow against the sides can lead to the box slipping away or tipping over. For both tasks, we generate fixed datasets of 1,000 demonstration episodes using a scripted oracle with access to privileged state information, including object poses and contact points, which are not accessible to policies.
We train two Implicit BC \cite{florence2021implicit} baselines \textit{without} Residual Kinematics: one using a 12DoF Cartesian action space (6DoF end-effector pose for each arm), and another using a 14DoF joint action space (7DoF for each arm). From the quantitative results presented in Tab. \ref{table:bimanual-experiments}, we observe that the performance of Implicit BC on bimanual sweeping degrades substantially when using a joint action space. This is likely that learning the task (\i.e. distribution of oracle demonstrations) involves recognizing a number of Cartesian space constraints that are less salient in joint space: e.g., keeping the Cartesian poses of the spatulas aligned with each other while scooping and transporting particles, and maintaining z-height with the tabletop.
For bimanual flipping, on the other hand, we observe the opposite: where the performance of Implicit BC performs well with joint actions, but poorly with Cartesian actions. We conjecture that the task involves more whole-body manipulation, where fitting policies in joint space are more likely to result in motion trajectories that accurately conform to the desired elbow contact with the box.
\vspace{-.7em}
\begin{table}[h]
\centering
\caption{\scriptsize Performance Measured in Task Success (Avg $\pm$ Std \% Over 3 Seeds)}
\begin{tabular}[b]{@{}lcccccccc@{}}
\toprule
Task & Sweeping & Flipping \\
{\em{Oracle's action space}} & \em{cartesian} & \em{joints} \\
Policy Action Space & & \\
\midrule
Cartesian & 79.4 $\pm$ 2.1 & 38.6 $\pm$ 5.2\\
Joints & 44.3 $\pm$ 3.2 & \textbf{98.4 $\pm$ 1.4} \\
Joints + Cartesian (Ours) & \textbf{85.9 $\pm$ 1.5} & \textbf{97.5 $\pm$ 1.2} \\
\bottomrule
\end{tabular}
\vspace{-1.0em}
\label{table:bimanual-experiments}
\end{table}
Our proposed method, Implicit Kinematic Policies (labeled as Joints + Cartesian in Tab. \ref{table:bimanual-experiments}) leverages both action spaces. Results suggest its performance is not only on par with the best bespoke action space for the task, but also surprisingly exceeds the performance of Cartesian actions for bimanual sweeping. Upon further inspection of the action trajectories for bimanual sweeping, it is evident that the Cartesian space action trajectories (Fig. \ref{fig:tasks}b) are lower frequency and contain more piecewise linear structure in comparison to the joint space action trajectories (Fig.~\ref{fig:tasks}a). Implicit policies have shown to thrive in conditions where this structure is present \cite{florence2021implicit}. In contrast, for bimanual flipping, the joint space action trajectories (Fig.~\ref{fig:tasks}c) appear much more linear and low frequency than the corresponding arc-like Cartesian trajectories (Fig.~\ref{fig:tasks}d). We also add small perturbations to the end-effector pose highlighted by the semi-transparent lines in plot (d) of Fig.~\ref{fig:tasks}. Note that the Implicit (Cartesian) policy uses a generic IK solver which can return multiple distinct joint target commands, causing less stable trajectories overall.
\begin{figure}[t]
\centering
\noindent\includegraphics[scale=0.2]{images/Insertion+Sweeping.png}
\vspace{-0.5em}
\caption{Simple constant joint encoder errors (which may appear from drift in low-cost or cable-driven robots) can propagate to non-linear offsets in Cartesian space. By learning joint space residuals to compensate for these offsets, IKP is better equipped to generalize over such errors than standard end-to-end policies trained in Cartesian space for tabletop manipulation.}
\label{fig:residual-deltas}
\vspace{-1.5em}
\end{figure}
\subsection{\textbf{Simulated Miscalibrated Sweeping and Insertion}}
\label{subsec:residual-experiments}
We design two additional tasks to evaluate IKP's ability to learn high precision tasks in the presence of inaccurate joint encoders, described in Sec.~\ref{subsec:residual-formulation}. To replicate such a scenario, we simulate these errors by adding constant $<$ 2 degree offsets to each of the six revolute joints on a UR5e robot. These offsets are visualized and described in more detail in Fig.~\ref{fig:residual-deltas}. We test the effect of these offsets on the bimanual sweeping task and a new insertion task where the goal is to insert an L-shaped block into a tight fixture. Demonstrations for both tasks are provided in the form of a Cartesian scripted oracle with privileged access to the underlying joint offsets and can compensate for them. This is akin to a human using visual servoing to compensate for inaccurate encoders when teleoperating a real robot to collect expert demonstrations. Although both tasks are generally well-defined in Cartesian space as a series of linear step functions, the joint offsets induce high-frequency non-linear artifacts in the end-effector trajectory (Fig.~\ref{fig:residual-deltas}b and \ref{fig:residual-deltas}d) causing poor performance with the Implicit (Cartesian) policy (results in Tab.~\ref{table:noisy-experiments}). Poor performance persists for miscalibrated insertion despite using 10x the data. Alternatively, even though the encoder offsets only cause linear shifts in the joint trajectories (Fig.~\ref{fig:residual-deltas}a and \ref{fig:residual-deltas}c), both tasks are less structured in joint space and as a result Implicit (Joints) performs significantly worse on miscalibrated bimanual sweeping and comparatively worse on miscalibrated insertion. Implicit (Joints) performs relatively well on the miscalibrated insertion task due to the simplicity of the pick and place motion, but struggles to generalize when the block nears the edge of workspace as the joint trajectories become increasingly non-linear in those regions. The performance only slightly improves when using 1000 demonstrations. IKP provides a best-of-all-worlds solution by learning the linear offsets in joint space through the residual blocks while simultaneously exploiting the unperturbed Cartesian trajectories through forward kinematics on the shifted joint actions to generalize. IKP achieves the highest performance on miscalibrated insertion with both 100 and 1000 demonstrations, and significantly higher performance on miscalibrated bimanual sweeping. Interestingly, not only do Explicit and ExplicitFK perform significantly worse than both Implicit (Joints) and IKP for both tasks, but ExplicitFK also provides little to no additional benefit over Explicit, even in the presence of more data.
\begin{table}[h]
\centering
\caption{\scriptsize Miscalibrated Joint Encoder Experiments, Performance Measured in Task Success (Avg $\pm$ Std. \% Over 3 Seeds). (``J+C'') is short for Joints+Cartesian.}
\begin{tabular}[b]{@{}lcccccccc@{}}
\toprule
Task & Sweeping & \multicolumn{2}{c}{Block Insertion}\\
{\em{Oracle's action space}} & \em{cartesian} & \multicolumn{2}{c}{\em{cartesian}}\\
Method (Action Space) & 1000 & 100 & 1000\\
\midrule
Explicit (Joints) & 38.2 $\pm$ 3.4 & 73.8 $\pm$ 4.3 & 74.2 $\pm$ 2.5\\
ExplicitFK (J + C) (Ours) & 40.3 $\pm$ 4.6 & 72.4 $\pm$ 3.8 & 75.4 $\pm$ 2.8\\
Implicit (Cartesian) & 3.1 $\pm$ 2.2 & 0.0 $\pm$ 0.0 & 0.0 $\pm$ 0.0\\
Implicit (Joints) & 46.3 $\pm$ 1.8 & 82.3 $\pm$ 2.9 & 85.2 $\pm$ 3.1 \\
IKP (J + C) (Ours) & \textbf{84.5 $\pm$ 1.2} & \textbf{88.8 $\pm$ 3.4} & \textbf{92.4 $\pm$ 2.6}\\
\bottomrule
\end{tabular}
\vspace{-1.0em}
\label{table:noisy-experiments}
\end{table}
\subsection{\textbf{Real Robot Sorting, Sweeping, and Alignment}}
We conduct qualitative experiments with a real UR5e robot on two tasks: 1) sorting two blocks into bowls and subsequently sweeping the bowls, and 2) aligning a red block with a blue block, where both tasks have initial locations randomized). Our goal with these experiments is two-fold: i) to demonstrate that IKP can run on real robots with noisy human continuous teleop demonstrations with as few as 100 demonstrations, and ii) to show that Implicit Kinematic Policies can perform both whole-body and prehensile tasks. Demonstrations for the block sorting portion of the first task are provided using continuous Cartesian position control based teleoperation at 100 Hz with a 3D mouse. Once the blocks are sorted, the teleoperator switches to a joint space PD controller with the same 3D mouse to guide the arm into an extended pose in order to sweep the bowls. For the alignment task, demonstrations are also provided using x, y and z Cartesian PD control at 100 Hz. Both tasks take 480${\times}$640 RGB images (downsampled to 96${\times}$96) from an Intel RealSense D435 camera at a semi-overhead view as input to the policy, and no extrinsic camera calibration is used. We show successful rollouts for both tasks in Fig.~\ref{fig:real-tasks}.
\begin{figure}[t]
\centering
\noindent\includegraphics[scale=0.23]{images/Real_Robot_Figure.png}
\caption{Real robot IKP rollouts on a sweeping and sorting task (top two rows) and a block alignment task (bottom row).}
\label{fig:real-tasks}
\vspace{-2em}
\end{figure}
\section{Discussion}
In future work, we will investigate using the multi-action-space formulation to extend beyond joint and cartesian PD control by incorporating the forward and/or inverse robot dynamics into the network which will allow us to expose joint torques and velocities to our model in a fully differentiable way. We hypothesize that this additional information will be particularly helpful in dynamic manipulation tasks and visual locomotion where torque and velocity action trajectories may appear more structured. We would also like to further test our residual framework within IKP on robots that have non-linear drift in joint space which may represent a larger set of low-cost robots. Finally, we would like to utilize our fully differentiable residual forward kinematics module to learn the link parameters themselves, which can be a promising direction for controlling soft and/or continuum robots.
\section*{Acknowledgments}
\small The authors would like to thank Vincent Vanhoucke, Vikas Sindhwani, Johnny Lee, Adrian Wong, Daniel Seita and Ryan Hoque for helpful discussions and valuable feedback on the manuscript.
\bibliographystyle{IEEEtranS}
|
\section{Introduction}
\label{sec:introduction}
Recent large surveys have revealed dozens of stellar streams in the halo of the Milky Way (MW),
which are one-dimensional substructures of halo stars
resulting from disruption of merging stellar systems, such as dwarf galaxies or globular clusters (GCs) \citep{Malhan2018a}.
These stellar streams not only
confirm the prediction of the standard $\Lambda$CDM cosmology
that the galaxies form through hierarchical mergers,
but also
allow measurement of the large- and small-scale mass distribution in the Milky Way \citep{Koposov2010,Malhan2019,Bonaca2019,BanikBovy2021}.
The GD-1 stream \citep{Grillmair2006} is one of the most studied stellar streams.
This stream is characterized by its thinness ($\sim 70 \ensuremath{\,\mathrm{pc}}$)
and its length (more than several tens of degrees) \citep{Carlberg2013}.
The integrated light from
\new{the visible part of}
the GD-1 stream suggest that
the progenitor system of this stream initially had a mass of $(1.58 \pm 0.07) \times 10^4 M_\odot$ \citep{deBoer2020},
\new{
although this mass estimate might be a lower limit if the stream is much more extended than currently recognized.}\footnote{
\new{
Depending on the accretion history of the Milky Way,
the GD-1 stream may have been affected by long-term gravitational perturbations \citep{Carlberg2020ApJ...889..107C}.
In such a case, the visible part of the GD-1 stream
might be a segment of a longer stream;
and thus the GD-1's progenitor mass might be larger than the estimate by \cite{deBoer2020}.
}}
These structural properties suggest that its progenitor system
(which has been completely disrupted)
was a GC-like system which began disruption at least $\sim 3 \ensuremath{\,\mathrm{Gyr}}$ ago \citep{Bowden2015MNRAS.449.1391B,Erkal2016}.
Recently, there have been discoveries of multiple under-dense regions or `gaps' in the GD-1 stream
\citep{Carlberg2013, deBoer2018, deBoer2020, PriceWhelan2018, Bonaca2019}.
Although there is some argument that these gaps might be due to the formation process of the stream (such as the non-uniform stripping rate; \new{
epicyclic overdensities, \citealt{Kupper2010MNRAS.401..105K,Kupper2015ApJ...803...80K};
} or the disruption of the progenitor star cluster within a host dwarf galaxy, \citealt{Malhan2019ApJ...881..106M,Malhan2021MNRAS.501..179M,Qian2022MNRAS.511.2339Q}),
currently the most widely accepted hypothesis to explain these gaps
is that these gaps were generated by perturbations from massive compact objects, including the dark matter subhaloes in the MW \citep{Carlberg2009,Carlberg2016,Yoon2011ApJ...731...58Y,Erkal2015a,Erkal2015b,Erkal2016}.
In general, when a stream experiences a close encounter with a dark matter subhalo with a mass of $\sim 10^5$-$10^7 M_\odot$, the impulsive force from the subhalo results in a differential velocity kick along the stream and forms a gap in the stream.
Since the gap forming mechanism is well understood, the morphology of the gap and the velocity structure near the gap enable us to investigate the mass and size of the perturber, relative velocity of the perturber with respect to the stream, and the time of the encounter \citep{Erkal2015a}.
Also, the number of gaps in a stream can put a constraint on the abundance of the dark matter subhalos if all of the gaps are generated by the dark matter subhalos \citep{Erkal2015b}.
To use these gaps as a probe of the dark matter, we need to disprove that these gaps are not generated by baryonic effects,
such as the perturbation from
the Galactic bar \citep{Hattori2016,PriceWhelan2016MNRAS.455.1079P},
spiral arms of the MW \citep{BanikBovy2021},
giant molecular clouds in the stellar disk \citep{Amorisco2016},
dwarf galaxies \citep{Bonaca2019},
and
GCs \citep{Bonaca2019,BanikBovy2021}.
In disproving these effects,
it is informative to note that the gaps are more difficult to generate
if a perturber moves at a larger velocity relative to the stream;
A higher-speed encounter results in a shorter interaction time with the stream and, therefore, a smaller effect.
Given that the GD-1 stream is orbiting around the MW in a retrograde fashion,
the perturbations from the bar, spiral arms, and giant molecular clouds -- all of which show a prograde motion -- have negligible effects in generating gaps in the GD-1 stream.\footnote{
Another well-studied stream, the Palomar 5 stream, also has some gaps \citep{Erkal2017MNRAS.470...60E}, but it has a prograde orbit.
}
Since the dwarf galaxies are far away from the GD-1 stream at the current epoch and in the past,
dwarf galaxies also have negligible effect in generating gaps in the GD-1 stream \citep{Bonaca2019}.
In this regard,
it is crucial to investigate whether GCs can explain the gaps in the GD-1 stream.
Previously, \cite{Bonaca2019} tried to pursue this strategy by using the catalog of $\sim 150$ GCs equipped with the astrometric data from Gaia DR2 \citep{Gaia2016A&A...595A...1G, Gaia2018A&A...616A...1G}.
They integrated the orbit backward in time for 1 Gyr to conclude
that the impacts from these GCs are negligible in the last 1 Gyr.
In this paper, we updated their results by using a larger number of GCs (158 GCs) from \cite{Vasiliev2021} equipped with the astrometric data from Gaia EDR3 \citep{Gaia2021A&A...649A...1G},
and, more importantly, by adopting a long enough integration time (6 Gyr in our fiducial model).
We note that \cite{BanikBovy2021} investigated how the baryonic perturbers (including GCs, giant molecular clouds, and spiral arms) affect the density power spectrum along the GD-1 stream by running simulations of the GD-1 stream in the last 3-7 Gyr.
Our work is complementary to \cite{BanikBovy2021}, because we focus on the {\it probability} that all of the three gaps in the GD-1 stream were formed by GCs.
This paper is organized as follows.
In Section 2, we present the observed data of the GD-1 stream and the GCs.
In Section 3, we describe how we set up our test-particle simulations.
In Section 4, we show the results of our analysis, details of the gap-forming GCs, and the show-case stream models.
In Section 5, we present some discussion of our results, including the estimated probability that all three gaps in the GD-1 stream were formed by GCs.
In Section 6, we summarize our paper.
\section{Data}
Here we describe the data of the GD-1 stream and GCs.
The coordinate system is defined in Appendix \ref{appendix:coordinate}.
\subsection{Globular clusters}
Using the astrometric data from Gaia EDR3 \citep{Gaia2021A&A...649A...1G},
\cite{Vasiliev2021} derived the position and velocity of 170 GCs in the Milky Way.
We selected 158 GCs with the full 6D position-velocity data by omitting 12 GCs with incomplete data.
The data include Right Ascension and Declination $(\alpha, \delta)$, distance $d$, line-of-sight velocity $v_{\ensuremath{\mathrm{los}}}$, and proper motion $(\mu_{\alpha*}, \mu_\delta)$, and their associated uncertainties including the correlation between the two proper motion components.
We note that the final sample of GCs contains some of the newly discovered GCs for which previous studies (e.g., \citealt{Bonaca2019}) have not checked whether they have experienced a close encounter with the GD-1 stream.
\subsection{Candidate stars of the GD-1 stream}\label{sec:Data_Malhan}
By using STREAMFINDER algorithm \citep{Malhan2018a, Malhan2018b, Malhan2018c},
\cite{Malhan2019} compiled a catalog of 97 candidate stars of the GD-1 stream
(see Table 2 of \citealt{Malhan2019})
for which line-of-sight velocity $v_{\ensuremath{\mathrm{los}}}$ is taken from
spectroscopic surveys such as SEGUE \citep{Yanny2009} and LAMOST \citep{Zhao2012}.
Since their original catalog is based on Gaia DR2 \citep{Gaia2018A&A...616A...1G},
we crossmatched these 97 stars with Gaia EDR3 \citep{Gaia2021A&A...649A...1G}
to obtain more accurate astrometric data.
Judging from $v_{\ensuremath{\mathrm{los}}}$,
some of their sample stars are probably not a member of the GD-1 stream
(see Fig. 4d of \citealt{Malhan2019}).
However, we did not make any manual selection to discard these outliers,
because such a hard cut might bias our inference on the properties of the GD-1 stream.
Instead, we modeled the position and velocity of
the GD-1 candidate stars
with a mixture model of stream stars and background stars
(cf., \citealt{Hogg2010arXiv}).
\subsection{Density along the GD-1 stream}
By using the photometric data from Pan-STARRS DR1 and astrometric data from Gaia DR2,
\cite{deBoer2020} derived the global properties of the GD-1 stream,
such as the density, distance, and proper motions along the stream.
They found three gaps along the GD-1 stream
at $\phi_1 = -36, -20$, and $-3 \deg$ in the
GD-1 stream coordinate system $(\phi_1, \phi_2)$
(see Appendix \ref{appendix:coordinate}; \citealt{Koposov2010}).
In this paper,
we interpret that all of the three gaps are due to some sort of perturbations,
although some authors interpret that the gap at $\phi_1 = -36 \deg$
corresponds to the location of the already-disrupted progenitor (e.g., \citealt{BanikBovy2021}).
The main goal in this paper is to evaluate
the probability that three gaps are formed
within $-40 \deg \leq \phi_1 \leq 0 \deg$
by close encounters with known GCs.
We note that we do {\it not} aim to re-create these three gaps {\it rigorously} at the observed locations.
\begin{figure}
\centering
\includegraphics[width=3.2in]{fig1__fig_GD1_schematic_ver5.png}
\caption{Schematic diagram for our simulations in this paper.
}
\label{fig:schematic}
\end{figure}
\section{Simulations} \label{sec:simulations}
We analyzed the close encounter with the GD-1 stream and GCs in three steps,
as described in Fig.~\ref{fig:schematic}.
\begin{itemize}
\item Step 1:
By using a static ({\it unperturbed}) Galactic potential,
we generated
(i) an {\it unperturbed} model of the GD-1 stream represented by test particles;
and
(ii) $N_\mathrm{MC}=1000$ Monte Carlo orbit models for each GC
reflecting the observed uncertainties.
We treat each GC as a test particle moving in the Galactic potential in this step.
\item Step 2:
By using the {\it unperturbed} models (i) and (ii) in Step 1,
we select GC orbits that could have encountered the GD-1 stream
with a small impact parameter and a small relative velocity.
\item Step 3:
For each of the selected orbits in Step 2,
we construct a time-dependent, {\it perturbed} potential,
consisting of the static MW potential and the time-dependent potential
from the GC that moves around the MW potential.
Under this composite MW+GC potential model,
we ran a test-particle simulation of the GD-1 stream
{\it perturbed} by a GC due to a close encounter.
\end{itemize}
Throughout this paper,
we used the AGAMA package \citep{Vasiliev2019_AGAMA} to run simulations.
In the following, we describe details of these steps.
\subsection{Model potential of the Milky Way} \label{sec:MWpotential}
In this paper, we used the MW model potential in \cite{McMillan2017}.
This model is static and axisymmetric,
and it consists of atomic and molecular gas disks, thin and thick stellar disks,
bulge, and a dark matter halo.
In Step 3 of our experiments, we add a time-dependent potential
caused by the gravitational force from a GC
to the above-mentioned static MW potential.
In such a case, we assume
that the MW potential is rigid and
that the MW does not wobble,
because the mass of a GC is much smaller than that of the MW.
\subsection{Step 1 (i): Unperturbed model of the GD-1 stream}
To generate an {\it unperturbed} model of the GD-1 stream,
we need to assume the orbit of its progenitor system.
We assume that the GD-1's progenitor
would be at $\alpha=148.91 \deg$ at the current epoch
if it had not been completely disrupted.
This location is chosen following \cite{BanikBovy2021}.
By assuming that the GD-1 stream approximately delineates the orbit of its progenitor,\footnote{
\new{
In general, the orbit of the progenitor is misaligned with the stream \citep{Sanders2013MNRAS.433.1813S,Sanders2013MNRAS.433.1826S}.
Although we start from this crude approximation, the final outcome of our simulation more or less reproduces the global properties of the GD-1 stream (see Fig.~\ref{fig:unperturbedGD1}).
}
}
we derived the remaining 5D information of the GD-1's progenitor at the current epoch.
Specifically,
we fit the 6D phase-space data of the candidates of the GD-1 stream members in \cite{Malhan2019}
with an orbit by adequately taking into account that some fraction of the stars is non-member stars.
Our best-fit orbit of the GD-1's progenitor is characterized by
\eq{
(\alpha^\mathrm{prog}, \delta^\mathrm{prog})|_{t=0}&=(148.91, 36.094) \deg,\\%(148.91, 36.0944389) \deg,\\
d^\mathrm{prog}|_{t=0}&= 7.550 \ensuremath{\,\mathrm{kpc}},\\%7.55009368 \ensuremath{\,\mathrm{kpc}},\\
(\mu_{\alpha*}^\mathrm{prog}, \mu_\delta^\mathrm{prog})|_{t=0}&=(-5.533,-12.600) \ensuremath{\,\mathrm{mas\ yr}^{-1}},\\%=(-5.53293288,-12.6006812) \ensuremath{\,\mathrm{mas\ yr}^{-1}},\\
v_{\ensuremath{\mathrm{los}}}^\mathrm{prog}|_{t=0}&=-14.576 \ensuremath{\,\mathrm{km\ s}^{-1}}.
}
These quantities are broadly consistent with the result in \cite{BanikBovy2021}.
\new{
In general, the orbit of the GD-1's progenitor is quite uncertain, not only because the current location $(\alpha^\mathrm{prog}, \delta^\mathrm{prog})|_{t=0}$
of the invisible (disrupted) progenitor is arbitrary, but also because the MW potential has evolved over the last several Gyr.
However, we use our best-fit orbit throughout this paper,
because our aim is to quantify the encounter rate of the GD-1 stream with GCs and not to reproduce the entire properties of the GD-1 stream.
}
Given this phase-space information, we integrated the orbit of the GD-1's progenitor backward in time from $t=0$ (current epoch) to $t=-T$.
We adopt $T=6 \ensuremath{\,\mathrm{Gyr}}$ as the fiducial value,
and thus all the figures in this paper assumes
this fiducial value.
We comment on how the choice of $T$ affects our results in Section \ref{sec:dynamical_age}.
Under the model potential we adopted \citep{McMillan2017},
the position and velocity of the GD-1's progenitor at $t=-T=-6\ensuremath{\,\mathrm{Gyr}}$ are given by\footnote{
See Appendix \ref{appendix:variousT}
for the corresponding quantites for different choices of $T$.
}
\eq{
&(x^\mathrm{prog}, y^\mathrm{prog}, z^\mathrm{prog})|_{t=-T=-6\ensuremath{\,\mathrm{Gyr}}} \nonumber\\
&= (12.079089, 12.387276, -8.601204) \ensuremath{\,\mathrm{kpc}}, \\
&(v_x^\mathrm{prog}, v_y^\mathrm{prog}, v_z^\mathrm{prog})|_{t=-T=-6\ensuremath{\,\mathrm{Gyr}}} \nonumber\\
&= (-172.22552, 54.99903, -72.57744) \ensuremath{\,\mathrm{km\ s}^{-1}}.
}
The position and velocity of the GD-1's progenitor at $t=-T$
was used to create the initial condition of the GD-1 stream particles.
As a simple prescription to mimic the generation of the GD-1 stream,
we release $10^5$ test particles at $t=-T$
from the same position as the progenitor,
$(x^\mathrm{prog}, y^\mathrm{prog}, z^\mathrm{prog})|_{t=-T}$,
with a relative velocity with respect to the progenitor following an isotropic Gaussian distribution:
\eq{
(v_x - v_x^\mathrm{prog})|_{t=-T} &\sim N(0, \sigma_v^2) , \\
(v_y - v_y^\mathrm{prog})|_{t=-T} &\sim N(0, \sigma_v^2) , \\
(v_z - v_z^\mathrm{prog})|_{t=-T} &\sim N(0, \sigma_v^2) .
}
Here, $N(0, \sigma^2)$ represents a Gaussian distribution
with mean $0$ and dispersion $\sigma^2$.
We regard these $10^5$ particles as the unperturbed GD-1 stream,
and we integrate the orbit of these $10^5$ particles
forward in time from $t=-T$ to $t=0$ (current epoch)
under the unperturbed Galactic potential.
After some experiments,
we chose $\sigma_v = (0.5 \ensuremath{\,\mathrm{km\ s}^{-1}}) (T / 6 \ensuremath{\,\mathrm{Gyr}})^{-1}$
so that the length of the GD-1 stream model at the current epoch is
comparable to the observed extent of the GD-1 stream.
We recorded the snapshot of these particles
every 1 Myr and used this information in Step 2.
Fig.~\ref{fig:unperturbedGD1} shows the stellar distribution of
the unperturbed model at the current epoch.
We see that the unperturbed model reproduces the observed phase-space
distribution of the stars except for the apparent outlier stars.
This unperturbed model is used as the benchmark model
with which we quantify the strength of perturbation from GCs
(see Section \ref{sec:defineGD1} and Appendix \ref{appendix:conditionA}).
We note that randomly chosen 100 stars in the unperturbed model
is used in Step 2 to find GCs that may have experienced
a close encounter with the GD-1 stream.
\begin{figure*}
\centering
\includegraphics[width=5.0in]{fig2__No_perturbation_vs_mal.png}
\caption{
Comparison of the unperturbed GD-1 stream model (blue dots) and
the observed GD-1 candidate stars in \cite{Malhan2019} (red data points with error bar).
On each panel, the the horizontal axis is $\phi_1$,
while the vertical axis is either $(\phi_2, \varpi, v_{\ensuremath{\mathrm{los}}}, \mu_{\alpha*}, \mu_\delta)$.
As seen in the top right plot showing $(\phi_1, v_{\ensuremath{\mathrm{los}}})$ distribution,
some of the GD-1 candidate stars
are outlier (non-member) stars with very different $v_{\ensuremath{\mathrm{los}}}$.
Apart from these outlier stars,
our unperturbed GD-1 stream model reproduces
the global properties of the observed GD-1 stream.
We note that this unperturbed GD-1 stream model
is used as a reference throughout this paper.
}
\label{fig:unperturbedGD1}
\end{figure*}
\subsection{Step 1 (ii): Unperturbed orbits of the globular clusters with observational uncertainty}
The GCs in the catalog of \cite{Vasiliev2021} typically have
$\sim 3$ percent error in distance.
This slight difference makes a noticeable difference
in evaluating whether a given GC could have experienced a close encounter with the GD-1 stream in the last few Gyr,
because a slight difference in the initial condition can cause a $\ensuremath{\,\mathrm{kpc}}$ scale difference
in predicting the location of the GC in a few Gyr.
To account for the observational uncertainty,
we sampled the current position and velocity of each GC for $N_\mathrm{MC}=1000$ times
from the error distribution.
Namely,
we randomly sampled $(d, \mu_{\alpha*}, \mu_\delta, v_{\ensuremath{\mathrm{los}}})$ for each GC,
by fully taking into account the correlation between the error in $(\mu_{\alpha*}, \mu_\delta)$.
We neglected the small uncertainty in $(\alpha, \delta)$.
For each GC, we integrated the orbit backward in time from $t=0$ to $t=-T$
under the unperturbed Galactic potential.
We recorded the position and velocity of each GC
every 1 Myr (and used this information in Step 2).
For convenience,
we enumerated each of these Monte Carlo orbits
by two integers $(j,k)$,
where $j \; (0 \leq j \leq 169)$ denotes $j$th GC in the catalog of \cite{Vasiliev2021}
and $k \; (0 \leq k \leq 999)$ denotes $k$th Monte Carlo orbit.
For example, $(j,k)=(4,557)$
corresponds to $k=557$th Monte Carlo orbit of NGC1261,
since NGC1261 is the 4th GC in the catalog.
\subsection{Step 2: Globular clusters with close encounters}
To find GCs that could have experienced a close encounter with the GD-1 stream,
we used the results in Step 1 (i) and (ii).
First, from the unperturbed model of the GD-1 stream computed in Step 1 (i),
we randomly chose 100 particles.
(We note that we sample 100 particles only once;
and we use the identical 100 particles throughout Step 2.)
We checked the relative distance and velocity of these 100 GD-1 particles
with respect to each Monte Carlo orbit of a given GC (computed in Step 1 (ii))
as a function of time.
For each Monte Carlo orbit $(j,k)$ of the GC,
we searched for a close encounter with the GD-1 stream,
which we defined as a moment when
at least one particle in the stream is located within $d_\mathrm{min} < 0.5 \ensuremath{\,\mathrm{kpc}}$ from the GC
having a relative velocity smaller than $v_\mathrm{rel} < 300 \ensuremath{\,\mathrm{km\ s}^{-1}}$.
As a result, we found that 1383 Monte Carlo orbits of 28 GCs
have experienced a close encounter
with the GD-1 stream in the last $T=6 \ensuremath{\,\mathrm{Gyr}}$.
We note that, up to this step,
it was unclear whether each of these GCs could
form a gap,\footnote{
For example, some GCs are not massive enough to form a gap.
}
which was investigated in Step 3.
\input{table1.tex}
\subsection{Step3: The GD-1 stream models perturbed by globular clusters}
For the 1383 Monte Carlo orbits $(j,k)$ found in Step 2,
we ran more detailed simulations.
To efficiently run perturbed model of the GD-1 stream,
we make some simplifying assumptions.
In each simulation specified by $(j,k)$,
we only consider one perturber,
namely the $j$th GC with $k$th Monte Carlo orbit.
In other words, we do not consider a situation
where multiple perturbers exist.
We treat the GD-1 stream particles as test particles
that feel forces from the MW and $j$th GC (with $k$th Monte Carlo orbit).
We do not take into account the self-gravity of the GD-1 stream.
We assume that $j$th GC has a Plummer density profile
with the total mass $M_{\mathrm{GC},j}$ adopted from
the compilation by Holger Baumgardt.\footnote{
\url{https://people.smp.uq.edu.au/HolgerBaumgardt/globular/parameter.html}
}
We assume that the mass of GCs does not change as a function of time.
We assume that the scale radius of the Plummer profile is $10 \ensuremath{\,\mathrm{pc}}$.
We note that changing the scale radius of GCs in the range of 1-20 $\ensuremath{\,\mathrm{pc}}$
does not significantly affect our results.
We assume that the GC only feels the force from the MW,
and therefore
the orbit of the GC in Step 2 is identical to that in Step 3.
We also assume that the MW potential is rigid and does not move
due to the motion of the GC,
which is a natural assumption if we consider the perturbation
from the GC only.\footnote{
In reality, however,
the MW is not an isolated system
due to the strong perturbation from the Large Magellanic Cloud
\citep{Besla2010,Erkal2019,GaravitoCamargo2019,Koposov2019},
so our assumption is simplistic in this regard.
}
Under the above-mentioned assumptions,
the gravitational potential that
a GD-1 stream particle feels
at location ${\bf x}$ and at time $t$ is given by
\eq{
&\Phi_{\mathrm{total},jk}({\bf x}, t) \nonumber \\
&= \Phi_\mathrm{MW}({\bf x}) + \Phi_{\mathrm{Plummer}} (|{\bf x} - {\bf x}_{\mathrm{GC},jk}(t)|; M_{\mathrm{GC},j}) . \label{eq:Phi_t}
}
Here,
$\Phi_\mathrm{MW}({\bf x})$ is the static MW potential
(Section \ref{sec:MWpotential}).
The $k$th orbit of the $j$th GC is denoted as
${\bf x}_{\mathrm{GC},jk}(t)$.
The GC's potential is described by a Pummer potential
\eq{
\Phi_{\mathrm{Plummer}}(r; M_{\mathrm{GC},j}) = - \frac{GM_{\mathrm{GC},j}}{\sqrt{r^2 + (10 \ensuremath{\,\mathrm{pc}})^2}}.
}
We represent the GD-1 stream with $10^5$ test particles,
which is large enough to statistically robustly detect a gap in the GD-1 stream model.
The initial conditions of the GD-1 stream particles
are chosen in the same manner as in Step 1 (i).
Importantly, we use the same random seed to generate the initial condition of the GD-1 stream model for all the simulations in Step 1 (i) and Step 3.
We integrate the orbits of these $10^5$ test particles
from $t=-T$ to $t=0$ under the perturbed, time-dependent potential
(equation (\ref{eq:Phi_t})).
After running the simulation,
we judge whether each stream model contains a GD-1-like gap or not by a simple method in Section \ref{sec:defineGD1}.
As a result, 57 Monte Carlo orbits of 16 GCs
result in a GD-1-like gap,
which will be discussed in Section \ref{sec:result}.
\subsection{Definition of a GD-1-like gap in our simulation} \label{sec:defineGD1}
To assess if a given model in our simulation contains a GD-1-like gap,
we compare the {\it perturbed} model in Step 3
with the {\it unperturbed} model in Step 1 (i).
Because we use the same random seed to create these models,
whenever we detect a notable difference between these models,
the difference can be attributed to the effect of the GC perturbation.
To make a fair comparison,
we first compute the linear density of the unperturbed and perturbed models of the GD-1 stream at the current epoch ($t=0$)
along $\phi_1$-coordinate,
$\rho_\mathrm{unperturbed}(\phi_1)$ and
$\rho_\mathrm{perturbed}(\phi_1)$, respectively,
by using the histogram of $\phi_1$ with a bin size of $\Delta\phi_1 = 2 \deg$.
We define that a GD-1-like gap is seen in the model at the location $\phi_1$
when the perturbed model stream satisfies the following two conditions:
\begin{itemize}
\item condition (A) $\frac{\rho_\mathrm{perturbed}(\phi_1)}{ \rho_\mathrm{unperturbed}(\phi_1)} < 0.8$; and
\item condition (B) $-40 \deg \leq \phi_1 \leq 0 \deg$.
\end{itemize}
The condition (A) is motivated by the fact that
a clear gap in our simulations typically satisfy (A).
The condition (B) is motivated by the fact that
\cite{deBoer2020} identified three gaps in the GD-1 stream
at $\phi_1 = -36 \deg, -20 \deg$, and $-3 \deg$.
We illustrate our procedure in Appendix \ref{appendix:conditionA} and Fig.~\ref{fig:ConditionA}.
\section{Result} \label{sec:result}
As described in Section \ref{sec:simulations},
we checked in total 158,000 Monte Carlo orbits (1000 orbits for each of the 158 GCs).
As a result, we found that 57 Monte Carlo orbits of 16 GCs
resulted in a gap
in the GD-1 stream model at $-40 \deg \leq \phi_1 \leq 0 \deg$.
In Sections \ref{sec:GClist}--\ref{sec:GCorbit},
we describe some details of these GCs.
In Section \ref{sec:showcase},
we present some show-case examples of the gap-forming models.
\subsection{GC candidates that might have formed a gap in the GD-1 stream} \label{sec:GClist}
Among the 158 GCs that we explored,
we identified 16 GCs (57 Monte Carlo orbits)
that formed a GD-1-like gap in the simulation.
These 16 GCs are good candidates of GCs
that might have formed the observed gaps in the GD-1 stream.
These 16 GCs are listed in Table~\ref{table:probability},
along with the gap-forming probability $P_{\text{gap-forming}} = N_\text{gap-forming-orbit}/1000$,
where $N_\text{gap-forming-orbit}$ denotes the number of gap-forming Monte Carlo orbits.
As we see from Table~\ref{table:probability}, $P_{\text{gap-forming}}$ is generally low.
However, there are 6 GCs that can form a GD-1-like gap with $P_{\text{gap-forming}} \geq \frac{3}{1000}$.
NGC5272 (M3) and IC4499 are especially interesting GCs,
which have the highest and second highest probability
(0.022 and 0.009, respectively)
of forming a GD-1-like gap.
Most of the 57 Monte Carlo orbits listed in Table~\ref{table:probability}
have only one close encounter with the GD-1 stream, forming a single gap in the GD-1 stream.
Intriguingly,
a few Monte Carlo orbits of NGC5272 (M3) have two close encounters with the GD-1 stream,
forming one visible gap and another mild under-density region.
\subsection{Gap-forming probability and the GC mass} \label{sec:GCmass}
In Fig.~\ref{fig:mass_prob}(a),
we show the relationship between the mass and the gap-forming probability
for the 16 GCs listed in Table \ref{table:probability}.
The 6 GCs with $P_{\text{gap-forming}} \geq \frac{3}{1000}$ are shown by red filled circle,
while the 10 GCs with $0<P_{\text{gap-forming}} \leq \frac{2}{1000}$ are shown by black open circle.
Except for NGC5272 (M3) and IC4499,
there is a mild trend
that more massive GCs have a higher gap-forming probability.
This trend is understandable in the following manner:
Given that
the maximum impact parameter to form a GD-1-like gap is larger for more massive GCs, more massive GCs have a larger chance of forming a gap.
In contrast, low-mass GCs need to pass very close to the GD-1 stream to form a gap, and thus lower-mass GCs have a smaller chance of forming a gap.
The 10 GCs with $0<P_{\text{gap-forming}}\leq \frac{2}{1000}$ have only one or two Monte Carlo orbits among 1000 trials
that result in a GD-1-like gap.
The mass of these GCs is
$10^{4.6}M_\odot \lesssim M_\mathrm{GC} \lesssim 10^{5.6}M_\odot$,
which roughly covers around 20-80 percentiles of the mass of known GCs
(see gray histogram in Fig.~\ref{fig:mass_prob}(b)).
Thus, these 10 GCs have a typical mass of the MW GCs.
Intriguingly,
the two highest-$P_{\text{gap-forming}}$ GCs,
NGC5272 (M3) and IC4499,
are not very massive
($10^{5.6}M_\odot$ and $10^{5.2}M_\odot$, respectively).
Therefore, their high value of $P_{\text{gap-forming}}$
is not because they are very massive but because their orbits are favorable to form a gap in the GD-1 stream.
In Fig.~\ref{fig:mass_prob}(b),
we show the normalized cumulative distribution of the GC mass
for GCs with different ranges of $P_{\text{gap-forming}}$.
As we can see from this figure,
the median GC mass is approximately
$10^{5.0} M_\odot$, $10^{5.25} M_\odot$, and $10^{5.7} M_\odot$,
for GCs with
$P_{\text{gap-forming}}=0$,
$0<P_{\text{gap-forming}} \leq \frac{2}{1000}$, and
$\frac{3}{1000} \leq P_{\text{gap-forming}}$, respectively.
This result suggests that
more massive GCs tend to have a larger probability of forming a gap in the GD-1 stream, supporting the mild trend seen in Fig.~\ref{fig:mass_prob}(a).
Fig.~\ref{fig:mass_prob}(b) also shows that
there is no GCs with $M_\mathrm{GC} \lesssim 10^{4.5} M_\odot$
that form a gap in the GD-1 stream in our simulation.
This result suggests that it is extremely difficult ($P_{\text{gap-forming}}<\frac{1}{1000}$)
to form a gap in the GD-1 stream by these low-mass GCs.
\begin{figure}
\centering
\includegraphics[width=3.2in]{fig3a__summary_log10Mass_Prob_annotate_GCname_ver2.png}\\
\includegraphics[width=3.2in]{fig3b__summary_hist_Mass_prob.png}
\caption{
(a) The relationship between the mass and the gap-forming probability
for the 16 GCs listed in Table \ref{table:probability}.
The 6 red dots are the GCs with $P_{\text{gap-forming}} \geq \frac{3}{1000}$.
(b) The normalized cumulative distribution for the mass of GCs with
different ranges of $P_{\text{gap-forming}}$.
Those GCs with higher $P_{\text{gap-forming}}$ are typically more massive.
Also, low-mass GCs ($M_\mathrm{GC} \lesssim 10^{4.5} M_\odot$)
can hardly form a gap.
}
\label{fig:mass_prob}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.2in]{fig4__fig_phi1_t__6GCs_ver3.png}
\caption{
The epoch of the close encounter with a GC ($t_\mathrm{encounter}$)
and the location of the GD-1-like gap ($\phi_\mathrm{1,gap}$).
We show all of the 57 simulations listed in Table \ref{table:probability}.
The 6 GCs with $P_\text{gap-forming} \geq \frac{3}{1000}$
are marked by various symbols.
The other 10 GCs are shown by gray, filled circles.
The size of the symbol represents the strength of the gap.
We note that the encounter with NGC5272 (M3) at $t_\mathrm{encounter} \simeq -3.5 \ensuremath{\,\mathrm{Gyr}}$ and that with IC4499 at $t_\mathrm{encounter} \simeq -1.7 \ensuremath{\,\mathrm{Gyr}}$ can form gaps at $-40 \deg < \phi_1 < 0 \deg $ and $-20 \deg < \phi_1 < 0 \deg$, respectively.
}
\label{fig:phi1_t}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=3.2in]{fig5a__fig_my_Lz_E__6GCs_ver2.png}\\
\includegraphics[width=3.2in]{fig5b__fig_my_peri_apo__6GCs.png}
\caption{
The orbital properties of the GD-1 stream and GCs.
(a) The azimuthal angular momentum and energy.
(b) The apocentric and pericentric radii.
The blue cross ($+$) corresponds to the GD-1 stream.
The red dots correspond to the 6 GCs with $P_\text{gap-forming} \geq \frac{3}{1000}$;
among which two GCs (NGC5272 (M3) and IC4499) with the highest-$P_\text{gap-forming}$ are highlighted with red $\odot$.
The black open circles correspond to the 10 GCs with $0<P_\text{gap-forming} \leq \frac{2}{1000}$.
The other GCs (namely, those GCs with $P_\text{gap-forming}=0$) are marked with gray dots.
}
\label{fig:GCorbits}
\end{figure}
\subsection{Epoch of the gap-forming encounters} \label{sec:Encounter_epoch}
Fig.~\ref{fig:phi1_t} shows the relationship between
the final gap location $\phi_{1,\mathrm{gap}}$
and the epoch of the encounter $t_\mathrm{encounter}$
for the 6 GCs with $P_\text{gap-forming} \geq \frac{3}{1000}$.
We see that all of the gap-forming encounters shown here
happen at $t<-1.5 \ensuremath{\,\mathrm{Gyr}}$.
This result is consistent with the previous work by \cite{Bonaca2019} which claims that none of the GCs seems to have experienced
a close encounter with the GD-1 stream in the last $1 \ensuremath{\,\mathrm{Gyr}}$.
This result also suggests that
tracing the past orbits of GCs up to a sufficiently long time ago
(up to $t=-6 \ensuremath{\,\mathrm{Gyr}}$ in our simulation)
is important to claim
whether or not GCs can form a visible gap in the GD-1 stream.
Another intriguing result seen in
Fig.~\ref{fig:phi1_t} is that
some GCs have their preferable epoch to interact with the GD-1 stream.
This result is most prominently seen for NGC5272 (M3) and IC4499.
For example,
for all of the 22 orbits of NGC5272 (M3) that
formed a GD-1-like gap,
the GD-1 stream encounters this GC at $t_\mathrm{encounter} \simeq -3.5 \ensuremath{\,\mathrm{Gyr}}$.
For these 22 orbits, the location of the GD-1 gap in the $\phi_1$ coordinate is distributed at $-40 \deg < \phi_1 < 0 \deg$.
Thus, an encounter with NGC5272 (M3) at $t_\mathrm{encounter} \simeq -3.5 \ensuremath{\,\mathrm{Gyr}}$ can explain any of the observed gaps at $-40 \deg < \phi_1 < 0 \deg$.
As another example,
for all of the 9 orbits of IC4499 that
formed a GD-1-like gap,
the GD-1 stream encounters this GC at $t_\mathrm{encounter} \simeq -1.7 \ensuremath{\,\mathrm{Gyr}}$.
For these 9 orbits, the location of the GD-1 gap in the $\phi_1$ coordinate is distributed at $-20 \deg < \phi_1 < 0 \deg$, and none of them are distributed at $\phi_1<-20 \deg$.
Thus, an encounter with IC4499 at $t_\mathrm{encounter} \simeq -1.7 \ensuremath{\,\mathrm{Gyr}}$ can explain either of the gaps at $\phi_1=-3 \deg$ or $-20 \deg$
but can hardly explain the gap at $\phi_1 = -36 \deg$.
\new{
To understand these preferable epochs to form a gap,
we check the orbital phase of the GD-1 stream
at $t\simeq-3.5 \ensuremath{\,\mathrm{Gyr}}$ when it encounters NGC5272 (M3)
and
at $t\simeq-1.7\ensuremath{\,\mathrm{Gyr}}$ when it encounters IC4499.
Intriguingly, all of these encounters take place
when the GD-1 stream is close to its pericenter
($R\simeq 14 \ensuremath{\,\mathrm{kpc}}$).
The GD-1 stream becomes longest near the pericentric passage,
and therefore the probability of a close encounter is increased
due to the enlarged `cross section' of the GD-1 stream.
As mentioned in Section \ref{sec:GClist},
these two GCs (NGC5272 (M3) and IC4499) have the highest probability
of forming a gap in our fiducial simulations.
Our finding hints that, in general, GCs that can encounter the GD-1 stream
near the GD-1's pericenter are a promising perturber to form a gap.
}
\subsection{Gap-forming probability and the orbit of GCs} \label{sec:GCorbit}
Fig.~\ref{fig:GCorbits} shows the orbital properties of
the GD-1 stream and GCs investigated in this study.
In both panels,
the GD-1 stream is marked by blue cross and
the 6 GCs with $P_\text{gap-forming} \geq \frac{3}{1000}$ are marked with red dots.
As we see in Fig.~\ref{fig:GCorbits}(a),
all of the GCs with $P_\text{gap-forming} \geq \frac{3}{1000}$ have retrograde or mildly prograde orbits
($L_z > -1000 \ensuremath{\,\mathrm{kpc}} \ensuremath{\,\mathrm{km\ s}^{-1}}$).
This tendency is understandable because
the GD-1 stream has a highly retrograde orbit
with $L_{z,\text{GD-1}} \simeq 2800 \ensuremath{\,\mathrm{kpc}} \ensuremath{\,\mathrm{km\ s}^{-1}}$.
When a GC with a certain azimuthal angular momentum $L_{z,\text{GC}}$ has a close encounter with the GD-1 stream at a Galactocentric cylindrical radius $R$,
their relative velocity is at least
$v_\mathrm{rel} \geq |v_{{\phi},\text{GD-1}} - v_{{\phi},\text{GC}}| = |L_{z,\text{GD-1}}-L_{z,\text{GC}}|/R$.
Given that the radial excursion of the GD-1 stream
is $14 \ensuremath{\,\mathrm{kpc}} \lesssim R \lesssim 20 \ensuremath{\,\mathrm{kpc}}$
(see Fig.~\ref{fig:GCorbits}(b)),
those GCs with highly prograde orbits
$L_{z,\text{GC}} < -2000 \ensuremath{\,\mathrm{kpc}} \ensuremath{\,\mathrm{km\ s}^{-1}}$
have at least $v_\mathrm{rel} \gtrsim 200 \ensuremath{\,\mathrm{km\ s}^{-1}}$.
Thus, GCs with highly prograde orbits
are hard form a gap in the GD-1 stream
unless they are very massive.
We note that
the GD-1 stream and NGC3201 share similar orbital properties.
Indeed, \cite{Malhan2022ApJ...926..107M}
recently claimed that these systems are
part of the same merging event dubbed `The Arjuna/Sequoia/I’itoi merger'
(see also \citealt{Bonaca2021ApJ...909L..26B}).
Although their orbital similarity is intriguing,
the fact that most of the high-$P_\text{gap-forming}$ GCs
have very different orbital properties means that
even if a GC has an orbital property similar to that of the GD-1 stream,
such a GC is not necessarily a good candidate for forming a gap.
Rather, as we mentioned in the previous paragraph,
the relative distance and velocity
at their closest approach are more important factors to form a gap.
\input{table2.tex}
\subsection{Details on the show-case models} \label{sec:showcase}
For an illustration purpose, for each of the 6 GCs
with $P_\text{gap-forming} \geq \frac{3}{1000}$,
we selected one show-case model.
The details of the selected show-case models
are summarized in Table~\ref{table:model}.
The models are named
{NGC2808\_698},
{NGC3201\_202}, $\;$
{NGC5272\_M3\_261},
{IC4499\_311},
{FSR1758\_731}, and
{NGC7089\_M2\_138}.
Here,
the last three digits of these names correspond to the value of $k$.
Figs.~\ref{fig:show_case_observables_three_rows_1} and \ref{fig:show_case_observables_three_rows_2}
show the current-day properties of the GD-1 stream models
corresponding to these show-case models.
In Figs.~\ref{fig:show_case_observables_three_rows_1} and \ref{fig:show_case_observables_three_rows_2},
each show-case model is displayed with three rows.
The top panel of each show-case model
shows the one-dimensional density $\rho(\phi_1)$ with an arbitrary unit.
The blue solid-line histogram shows the histogram of stars in each model.
The gray dashed line shows the estimated linear density
(arbitrarily scaled by a constant factor) derived in \cite{deBoer2020}.
The contrast between the gap and its surrounding over-dense regions in our models is similar to that in the observed GD-1 stream.
Also, the widths of the gaps in our models are comparable to the observed ones.
The middle and bottom panels in each model in Figs.~\ref{fig:show_case_observables_three_rows_1} and \ref{fig:show_case_observables_three_rows_2}
shows the morphology of the model in $(\phi_1, X)$ space,
where $X$ corresponds to various observables.
The show-case model NGC2808\_698 (Fig.~\ref{fig:show_case_observables_three_rows_1})
shows a hole-like structure in $(\phi_1, \phi_2)$ space,
caused by the perturber (in this case NGC2808)
that penetrated the stream.
Interestingly, due to the hole-like structure,
we see two parallel sequences of the stream
at $-40 \deg < \phi_1 < -20 \deg$,
which is reminiscent of the observed spur-like feature in the GD-1 stream \citep{PriceWhelan2018, Bonaca2019}.
A similar hole-like structure in $(\phi_1, \phi_2)$ space
is also seen in the show-case model NGC3201\_202.
In this case, a hole-like feature is also seen
in $(\phi_1, d)$ space,
indicating that this feature is a three-dimensional structure.
Given that we see a hole-like structure in multiple models,
it may be one of the generic features that GCs can form.
Among the 6 GCs, NGC2808, FSR1758, and NGC7089 (M2) are the most massive GCs
with $M>6\times10^5 M_\odot$.
Figs.~\ref{fig:show_case_observables_three_rows_1} and \ref{fig:show_case_observables_three_rows_2}
show that these massive GCs can form a prominent gap,
even if the relative velocity of the encounter is as large as $\simeq 300 \ensuremath{\,\mathrm{km\ s}^{-1}}$ (see also Table.~\ref{table:model}).
In contrast,
the show-case model IC4499\_311
results in a clear but narrow gap,
due to (i) the relatively small mass of IC4499 ($1.55\times 10^5 M_\odot$);
and (ii) the relatively recent encounter ($\sim 1.7 \ensuremath{\,\mathrm{Gyr}}$ ago).
\begin{figure*}
\centering
\includegraphics[width=6.in] {fig6a__fig_showcase_three_rows_15_698.png} \\
\includegraphics[width=6.in] {fig6b__fig_showcase_three_rows_18_202.png} \\
\includegraphics[width=6.in] {fig6c__fig_showcase_three_rows_34_261.png}
\caption{
Morphology of the show-case GD-1 stream models
NGC2808\_698, NGC3201\_202, and NGC5272\_M3\_261.
Each model is displayed with three rows,
and all the horizontal axes are $\phi_1$.
The upper-most wide panel shows
the one-dimensional density $\rho(\phi_1)$ in our simulation (blue histogram)
and in the observed data \citep{deBoer2020} (gray dashed line).
The middle wide panel shows the full extent of the model stream in $(\phi_1, \phi_2)$ space.
The lower 5 panels show the model stream near the gap region in $(\phi_1, X)$ space,
where $X=\phi_2, d, \mu_{\alpha*}, \mu_\delta$, and $v_{\ensuremath{\mathrm{los}}}$, from left to right.
The red vertical lines surrounding the gap are the same for the upper and lower panels.
}
\label{fig:show_case_observables_three_rows_1}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=6.in] {fig7a__fig_showcase_three_rows_40_311.png} \\
\includegraphics[width=6.in] {fig7b__fig_showcase_three_rows_89_731.png} \\
\includegraphics[width=6.in] {fig7c__fig_showcase_three_rows_165_138.png}
\caption{
The same as Fig.~\ref{fig:show_case_observables_three_rows_1},
but for the show-case GD-1 stream models
IC4499\_311, FSR1758\_731, and NGC7089\_M2\_138.
}
\label{fig:show_case_observables_three_rows_2}
\end{figure*}
\section{Discussion}
\subsection{Are globular clusters responsible for the GD-1 gaps?} \label{sec:discussion_probability}
The 16 GCs with $P_\text{gap-forming} \geq \frac{1}{1000}$ can possibly form a gap in the GD-1 stream.
If we assume, as a working hypothesis, that all three gaps in the GD-1 stream were created by the perturbation from these GCs, we can estimate its probability by using $P_\text{gap-forming}$ listed in Table \ref{table:probability}:
\eq{
&P(\text{GCs formed 3 gaps in the GD-1 stream}) \nonumber \\
=& \sum_{l \in \{4,9, \dots, 165\}} \;\;
\sum_{m\,>\,l} \;\;
\sum_{n\,>\,m}
[
P_\text{gap-forming}(l) \nonumber \\
&
\times P_\text{gap-forming}(m)
\times P_\text{gap-forming}(n) ] \nonumber \\
=& \new{1.7}\times 10^{-5}. \label{eq:probability}
}
Here, $P_\text{gap-forming}(j)$ corresponds to the
gap-forming probability of $j$th GC,
and $\{4,9, \dots, 165\}$ corresponds to the set of $j$ listed in Table~\ref{table:probability}.
Given this tiny probability, our results suggest that perturbations from GCs are difficult to explain all three gaps in the GD-1 stream.
Because other baryonic effects
(e.g., from spiral arms, the Galactic bar, giant molecular clouds, or dwarf galaxies) are even more unlikely to form a gap (see Section \ref{sec:introduction}),
our results favor a scenario in which at least one of the gaps in the GD-1 stream
were formed by dark matter subhalos \citep{Carlberg2009,Carlberg2016,Erkal2015a,Erkal2015b,Erkal2016,Bonaca2019,BanikBovy2021}.
Our results can also be used to estimate
the expected number of gaps formed by GCs in the GD-1 stream, $N_\text{gap}(\text{GCs})$.
By assuming that a GC can form at most one gap (and no GCs can form multiple gaps), we have
\eq{
N_\text{gap}(\text{GCs}) = \sum_{l \in \{4,9, \dots, 165\}} P_\text{gap-forming}(l) = 0.057. \label{eq:N_gap}
}
We note that the gaps considered in this paper satisfy condition (A) in Section \ref{sec:defineGD1}, namely
$\frac{\rho_\mathrm{perturbed}(\phi_1)}{ \rho_\mathrm{unperturbed}(\phi_1)} < 0.8$.
As a reference,
\cite{Erkal2016} estimated the
number of gaps in the GD-1 stream
formed by dark matter subhalos, $N_\text{gap}(\text{subhalos})$.
According to their Table~2,
the expected number of gaps with $\frac{\rho_\mathrm{perturbed}(\phi_1)}{ \rho_\mathrm{unperturbed}(\phi_1)} < 0.75$
formed by dark matter subhalos with $(10^5$-$10^9) M_\odot$
is $N_\text{gap}(\text{subhalos})=0.6$,
which is $\sim10$ times larger than our estimate of $N_\text{gap}(\text{GCs})$.
Although their value of $N_\text{gap}(\text{subhalos})$ is still smaller than $3$ (the observed number of gaps),
this comparison also favors dark matter subhalos as the cause of the GD-1's gaps.
\subsection{Dynamical age of the GD-1 stream} \label{sec:dynamical_age}
In this paper, we assume that all the stars in the GD-1 stream
escaped from the progenitor system at $t=-T$.
The dynamical age of the stream, $T$,
is set to be $T=6 \ensuremath{\,\mathrm{Gyr}}$ in the main analysis of this paper.
To check how our choice of $T$ affects our result,
we ran additional simulations with $T= 1, 2, 3$, and $4 \ensuremath{\,\mathrm{Gyr}}$
(see Appendix \ref{appendix:variousT}).
(We note that we kept our description in Section \ref{sec:simulations} as general as possible so that the readers can see how the change in $T$ affects the numerical setup of the simulations.)
For simulations with $T=1 \ensuremath{\,\mathrm{Gyr}}$ and $T=2 \ensuremath{\,\mathrm{Gyr}}$,
we found that all the GCs have $P_\text{gap-forming} =0$.
For simulations with $T=3 \ensuremath{\,\mathrm{Gyr}}$,
we found only two GCs have $P_\text{gap-forming} \geq \frac{1}{1000}$
(see Table.~\ref{table:probability_3Gyr_4Gyr}),
which means that GCs can form at most two gaps.
For simulations with $T=4 \ensuremath{\,\mathrm{Gyr}}$,
we found 5 GCs with $P_\text{gap-forming} \geq \frac{1}{1000}$
(see Table.~\ref{table:probability_3Gyr_4Gyr}).
By combining the result from our fiducial models with $T=6 \ensuremath{\,\mathrm{Gyr}}$,
we obtain
\eq{
&P(\text{GCs formed 3 gaps in GD-1}) \nonumber \\
&=
\begin{cases}
0 \phantom{.0\times10^{-8}} \;\; \text{(if $T=1 \ensuremath{\,\mathrm{Gyr}}$)}\\
0 \phantom{.0\times10^{-8}} \;\; \text{(if $T=2 \ensuremath{\,\mathrm{Gyr}}$)}\\
0 \phantom{.0\times10^{-8}} \;\; \text{(if $T=3 \ensuremath{\,\mathrm{Gyr}}$)}\\
6.2\times10^{-8} \;\; \text{(if $T=4 \ensuremath{\,\mathrm{Gyr}}$)} \\
\new{1.7}\times10^{-5} \;\; \text{(if $T=6 \ensuremath{\,\mathrm{Gyr}}$)}. \label{eq:probability_variousT}
\end{cases}
}
and
\eq{
N_\text{gap}(\text{GCs})
=
\begin{cases}
0\phantom{.000} \;\; \text{(if $T=1 \ensuremath{\,\mathrm{Gyr}}$)}\\
0\phantom{.000} \;\; \text{(if $T=2 \ensuremath{\,\mathrm{Gyr}}$)}\\
0.003 \;\; \text{(if $T=3 \ensuremath{\,\mathrm{Gyr}}$)}\\
0.010 \;\; \text{(if $T=4 \ensuremath{\,\mathrm{Gyr}}$)} \\
0.057 \;\; \text{(if $T=6 \ensuremath{\,\mathrm{Gyr}}$)}. \label{eq:N_gap_variousT}
\end{cases}
}
We see that both
the total probability and the expected number of gaps
become smaller if we adopt a smaller value of $T$.
We can understand this tendency in two ways.
First,
younger streams have fewer opportunities to interact with GCs.
Second,
it takes some time for a gap to grow and become visible.
We note that the previous work by \cite{Bonaca2019} found that
no GCs experienced a close encounter with the GD-1 stream in the last $1 \ensuremath{\,\mathrm{Gyr}}$.
Their result is consistent with our result with $T=1 \ensuremath{\,\mathrm{Gyr}}$.
Our results suggest that adopting a longer integration time increases
the chance that GCs can form a gap in the GD-1 stream,
but it is extremely hard to explain three gaps
only by the GCs, even if we adopt a long integration time of $T=6\ensuremath{\,\mathrm{Gyr}}$.
We note that \cite{BanikBovy2021} investigated the past orbit of the GD-1 stream
up to $t=-7 \ensuremath{\,\mathrm{Gyr}}$. However, they aimed to assess the power spectrum of the density along the GD-1 steam; and not to focus on the individual gaps.
Thus, our work is complementary to their work.
\subsection{Choice of the model Galactic potential} \label{sec:different_potentials}
\new{
In this paper, we used a model MW potential in \cite{McMillan2017} as the fiducial model.
In order to check how our results are affected by
the chosen MW potential,
we did the same simulations with $T=6 \ensuremath{\,\mathrm{Gyr}}$
but with potential models in
\cite{Bovy2015ApJS..216...29B} and \cite{Piffl2014MNRAS.445.3133P}.
As a result, we found no dramatic changes from our fiducial simulations in the gap-forming probability
\eq{
&P(\text{GCs formed 3 gaps in GD-1}) \nonumber \\
&=
\begin{cases}
4.8\times10^{-5} \;\; \text{(if $T=6 \ensuremath{\,\mathrm{Gyr}}$, \citealt{Bovy2015ApJS..216...29B})} \\
9.7\times10^{-5} \;\; \text{(if $T=6 \ensuremath{\,\mathrm{Gyr}}$, \citealt{Piffl2014MNRAS.445.3133P})}
\label{eq:probability_6Gyr_different_potentials}
\end{cases}
}
and the expected number of gaps
\eq{
&N_\text{gap}(\text{GCs}) \nonumber \\
&=
\begin{cases}
0.076 \;\; \text{(if $T=6 \ensuremath{\,\mathrm{Gyr}}$, \citealt{Bovy2015ApJS..216...29B})} \\
0.105 \;\; \text{(if $T=6 \ensuremath{\,\mathrm{Gyr}}$, \citealt{Piffl2014MNRAS.445.3133P})}. \label{eq:N_gap_6Gyr_different_potentials}
\end{cases}
}
We note that the list of GCs that form the gaps
(i.e., the list of GCs in Table \ref{table:probability} in our fiducial model)
slightly changes if we adopt different MW potentials.
However, some GCs seem to be more likely to form gaps.
For example, if we adopt the Galactic potential model in \cite{Bovy2015ApJS..216...29B}, the three important GCs with the highest $P_\text{gap-forming}$ are IC4499, NGC7089 (M2), and NGC3201, all of which appear in Table \ref{table:probability}.
Also, if we adopt the Galactic potential model in \cite{Piffl2014MNRAS.445.3133P}, the two important GCs with the highest $P_\text{gap-forming}$ are IC4499 and NGC6101, both of which appear in Table \ref{table:probability}.
Intriguingly, the gap-forming probability of IC4499 is always high $P_\text{gap-forming} \gtrsim 0.01$,
independent of the adopted potential.
}
\subsection{Caveats in our analysis} \label{sec:caveats}
As discussed in Section \ref{sec:discussion_probability},
we found that it is extremely unlikely that all three gaps in the GD-1 stream are formed by known GCs.
Here we discuss the limitation of our analysis and possible future directions.
In our simulation, we treated the stars in the GD-1 stream
as test particles that feel the gravitational force from the MW and a perturbing GC.
We assume that all the stars were stripped from the
\new{center of the}
GC-like progenitor system at $t=-T$.
In reality,
the stripping process may be continuous,
\new{
and the stars escape from the progenitor
from the inner and outer Lagrange points
\citep{Kupper2010MNRAS.401..105K, Kupper2012MNRAS.420.2700K, Kupper2015ApJ...803...80K, Mastrobuono-Battisti2013MmSAI..84..240M, Sanders2013MNRAS.433.1813S, Sanders2013MNRAS.433.1826S, Bovy2014ApJ...795...95B, Fardal2015MNRAS.452..301F, Guillaume2016MNRAS.460.2711T, Ibata2020ApJ...891..161I}.
}
Also, we do not take into account the host system of the progenitor system that could affect the morphology of the GD-1 stream
\citep{Malhan2022ApJ...926..107M,Qian2022MNRAS.511.2339Q}.
If we were to reproduce the density profile of the GD-1 stream as a function of $\phi_1$,
the effects mentioned above are important
and the only way to faithfully take these effects into account is
to run $N$-body simulations
\new{
or particle spray simulations (see Appendix \ref{appendix:Lagrange}).
}
However, because we are interested in the probability that the gaps were formed by the GCs, our approach is good enough for our purpose.
In this paper, we assumed that
the MW potential is static and axisymmetric.
These assumptions are simplistic,
given that the MW is growing in time due to mass accretion \citep{Buist2015},
that the MW has a rotating bar \citep{Hattori2016,PriceWhelan2016MNRAS.455.1079P},
and that the Large Magellanic Cloud has been perturbing the MW
\citep{Besla2010,Erkal2019,GaravitoCamargo2019,Koposov2019,Conroy2021Natur.592..534C,Petersen2021NatAs...5..251P,Shipp2021ApJ...923..149S}.
These effects can alter the values of $P_\text{gap-forming}$ for each GC.
However, since we did not tune the MW potential to maximize or minimize $P_\text{gap-forming}$ (instead, we just adopted one of the widely-used MW model potentials
\new{in our main analysis}),
our estimate of the probability that the GCs formed all three gaps in the GD-1 stream,
\new{$1.7\times10^{-5}$}
(see equation (\ref{eq:probability})),
is probably not too far from reality.
\new{
Indeed, our additional analysis in Section \ref{sec:different_potentials}, in which we varied the potential, supports this view.}
Thus, even though our simulation neglects some important physics,
we believe our main conclusion is robust:
the probability that GCs are responsible for all three gaps in the GD-1 stream is extremely low.
\section{Conclusion}
In this paper,
we estimated the probability that Galactic GCs can form a gap in the GD-1 stream
by using test-particle simulations.
The summary of this paper is as follows.
\begin{itemize}
\item
In our fiducial simulations (in which the GD-1 stream is $T=6 \ensuremath{\,\mathrm{Gyr}}$ old),
16 GCs formed a gap in the GD-1 stream at $-40 \deg < \phi_1 < 0 \deg$ (Table \ref{table:probability}).
Among them, 6 GCs can form a GD-1-like gap with
a gap-forming probability
$P_\text{gap-forming} \geq 0.003$.
NGC5272 (M3) ($P_\text{gap-forming}=0.022$) and IC4499 ($P_\text{gap-forming}=0.009$) have much higher $P_\text{gap-forming}$ than other GCs.
\item
There is a moderate trend that more massive GCs tend to have a larger $P_\text{gap-forming}$ (Fig.~\ref{fig:mass_prob}).
However, the relative distance and velocity
\new{
at their closest approach to the GD-1 stream
}
are much more critical factors than the mass of the GCs.
\item
As shown in Figs.~\ref{fig:show_case_observables_three_rows_1} and ~\ref{fig:show_case_observables_three_rows_2}, our perturbed models can capture some of the morphological properties of the observed GD-1 stream, such as the length, widths, and strength of the gaps.
\item
The probability that all three gaps in the GD-1 stream
were formed by the GCs is extremely low.
In our fiducial model with $T=6 \ensuremath{\,\mathrm{Gyr}}$,
this probability is $P=\new{1.7}\times10^{-5}$
(Section \ref{sec:discussion_probability}).
This probability decreases if we adopt smaller $T$
(see equation (\ref{eq:probability_variousT})).
Assuming $T \leq 3 \ensuremath{\,\mathrm{Gyr}}$ results in $P=0$,
which explains the result of \cite{Bonaca2019} who assumed $T=1 \ensuremath{\,\mathrm{Gyr}}$.
\item
The expected number of gaps in the GD-1 stream due to the flyby of GCs
is $N_\mathrm{gap}(\text{GCs})=0.057$ in our fiducial model (equation (\ref{eq:N_gap})).
This number is smaller than that due to the flyby of subhalos ($N_\mathrm{gap}(\text{subhalos})=0.6$) by a factor of 10 \citep{Erkal2016}.
\item
Given (i) that the probability that all three gaps in the GD-1 stream are formed by the GCs is extremely low,
and (ii) that the retrograde orbit of the GD-1 stream makes other baryonic perturbers (e.g., spiral arms, the Galactic bar, or giant molecular clouds) even less likely to form the gaps,
at least one of the gaps in the GD-1 stream is probably formed by the
dark matter subhalos \citep{Carlberg2009,Carlberg2016,Yoon2011ApJ...731...58Y, Erkal2015a,Erkal2015b,Erkal2016,Bonaca2019,BanikBovy2021}.
\item
To sophisticate our analysis,
we need to run $N$-body simulations of the encounters of the GD-1 stream and GCs
by also including the effect from the Large Magellanic Cloud
\citep{Erkal2019,Koposov2019,Shipp2021ApJ...923..149S},
or from the host halo of the GD-1's progenitor
\citep{Malhan2021MNRAS.501..179M, Qian2022MNRAS.511.2339Q}.
\end{itemize}
\acknowledgments
The authors thank the referee for thorough reading and constructive comments that improved the original manuscript.
DY and KH thank NAOJ for financial aid during the 2021 summer student program.
We thank Junichi Baba for sharing his $N$-body code that gave insights into our work.
KH thanks lecturers of $N$-body winter school 2021 held by NAOJ for stimulating lectures.
KH is supported by JSPS KAKENHI Grant Numbers JP21K13965 and JP21H00053.
Numerical computations were in part carried out on GRAPE system at Center for Computational Astrophysics, National Astronomical Observatory of Japan.
This research was supported in part through computational resources and services provided by Advanced Research Computing (ARC), a division of Information and Technology Services (ITS) at the University of Michigan, Ann Arbor.
This work has made use of data from the European Space Agency (ESA) mission
{\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by the {\it Gaia}
Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the {\it Gaia} Multilateral Agreement.
Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.
Funding for the Sloan Digital Sky
Survey IV has been provided by the
Alfred P. Sloan Foundation, the U.S.
Department of Energy Office of
Science, and the Participating
Institutions.
SDSS-IV acknowledges support and
resources from the Center for High
Performance Computing at the
University of Utah. The SDSS
website is www.sdss.org.
SDSS-IV is managed by the
Astrophysical Research Consortium
for the Participating Institutions
of the SDSS Collaboration.
\begin{comment}
including
the Brazilian Participation Group,
the Carnegie Institution for Science,
Carnegie Mellon University, Center for
Astrophysics | Harvard \&
Smithsonian, the Chilean Participation
Group, the French Participation Group,
Instituto de Astrof\'isica de
Canarias, The Johns Hopkins
University, Kavli Institute for the
Physics and Mathematics of the
Universe (IPMU) / University of
Tokyo, the Korean Participation Group,
Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik
Potsdam (AIP), Max-Planck-Institut
f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur
Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur
Extraterrestrische Physik (MPE),
National Astronomical Observatories of
China, New Mexico State University,
New York University, University of
Notre Dame, Observat\'ario
Nacional / MCTI, The Ohio State
University, Pennsylvania State
University, Shanghai
Astronomical Observatory, United
Kingdom Participation Group,
Universidad Nacional Aut\'onoma
de M\'exico, University of Arizona,
University of Colorado Boulder,
University of Oxford, University of
Portsmouth, University of Utah,
University of Virginia, University
of Washington, University of
Wisconsin, Vanderbilt University,
and Yale University.
\end{comment}
\facility{Gaia, LAMOST, SDSS/SEGUE}
\software{
Agama \citep{Vasiliev2019_AGAMA},\;
corner.py \citep{ForemanMackey2016},
matplotlib \citep{Hunter2007},
numpy \citep{vanderWalt2011},
scipy \citep{Jones2001}}
\bibliographystyle{aasjournal}
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"\\section{Introduction}\nThe \\ac{iot} provides a number of benefits,\nto consumers as well as busi(...TRUNCATED) |
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"\\section{Introduction}\nTopological data analysis (TDA) has been used in music analysis recently b(...TRUNCATED) |
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