Quantitative Criteria for Defining Planets

The current IAU definition of"planet"is problematic because it is vague and excludes exoplanets. Here, we describe aspects of quantitative planetary taxonomy and examine the results of unsupervised clustering of Solar System bodies to guide the development of possible classification frameworks. Two unsurprising conclusions emerged from the clustering analysis: (1) satellites are distinct from planets and (2) dynamical dominance is a natural organizing principle for planetary taxonomy. To generalize an existing dynamical dominance criterion, we adopt a universal clearing timescale applicable to all central bodies (brown dwarfs, stars, and stellar remnants). Then, we propose two quantitative, unified frameworks to define both planets and exoplanets. The first framework is aligned with both the IAU definition of planet in the Solar System and the IAU working definition of an exoplanet. The second framework is a simpler mass-based framework that avoids some of the difficulties ingrained in current IAU recommendations.


INTRODUCTION
In 2006, the International Astronomical Union (IAU) adopted resolution B5, which contains the following definition: A planet is a celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit.
The current IAU definition of "planet" is problematic both because it is not quantitative and because it excludes exoplanets.In a previous paper, one of us proposed a possible solution to remedy both problems (Margot 2015).
In 2018, IAU Commission F2 "Exoplanets and the Solar System" promulgated the following working definition for exoplanets: Objects with true masses below the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 Jupiter masses for objects of solar metallicity) that orbit stars, brown dwarfs or stellar remnants and that have a mass ratio with the central object below the L4 / L5 instability (M/M central < 2/(25 + √ 621) ≃ 1/25) are "planets", no matter how they formed.The minimum mass/size required for an extrasolar object to be considered a planet should be the same as that used in our Solar System, which is a mass sufficient both for self-gravity to overcome rigid body forces and for clearing the neighborhood around the object's orbit.
The rationale for this working definition is explained in detail by Lecavelier des Etangs & Lissauer (2022).Notably, the minimum mass/size required for an extrasolar object to be considered a planet remains undefined because IAU resolution B5 does not define it precisely, except through vague implications about the mass required to clear a zone (how large?how clear?) and to approach hydrostatic equilibrium (how closely?).
In this paper, we present two perspectives to stimulate conversations about improving the current IAU planet definition.Both proposals include a quantitative, unified framework to define both planets and exoplanets.The first proposal is aligned with the spirit of IAU Resolution B5 and incorporates the working definition of IAU Commission F2.The second proposal is a simpler mass-based criterion.In both instances, we follow a common approach to astronomical nomenclature, which is to use physical principles to establish meaningful thresholds and then adhere to the thresholds for definitional purposes.
We outline a metric for dynamical dominance in Section 2, describe desirable features of a planetary taxonomy in Section 3, use unsupervised clustering techniques to guide a proposed planetary taxonomy in Section 4, and extend an existing dynamical dominance criterion in Section 5. We propose an IAU-aligned taxonomic framework in Section 6 and a simpler mass-based framework in Section 7.

DYNAMICAL DOMINANCE
Dynamical dominance looms large in planetary taxonomy because both Ceres and Pluto lost their status as planets once they were found to belong to a belt of small bodies.Margot (2015) developed the planetary discriminant Π to quantify what it takes to clear a well-defined orbital zone around a planetary body, echoing the IAU definition.A logical choice for the extent of the zone to be cleared around a planetary body is the canonical feeding zone (Birn 1973;Artymowicz 1987;Ida & Makino 1993;Gladman 1993), which is 2 √ 3 times the Hill radius of the planetary body: where m is the mass of the planetary body, m central is the mass of the central body or host star, and a is the semimajor axis.
Following Tremaine (1993), one can model the ejection of planetesimals by a dynamically dominant body as a diffusion process and derive the minimum orbit-clearing mass required to clear a zone of extent C times the Hill radius in a given clearing timescale t clear (Margot 2015, Equation 8): where a is expressed in au and m clear , m central , and t clear are expressed in units of earth masses, solar masses, and years, respectively.This expression was derived for definitional purposes in the context of circular orbits and is not meant to capture all astrophysical situations.Clearing in the context of eccentric orbits does not yield a simple expression (e.g., Quillen & Faber 2006;Morrison & Malhotra 2015).Likewise, additional perturbations due to other planets, dynamical friction between the smaller bodies, and the effects of gas drag and disk tides are not included.The parameter Π is simply the mass of a planetary body expressed in units of the corresponding orbit-clearing mass: Values of Π larger than 1 convey an ability to clear the feeding zone, whereas values of Π smaller than 1 convey an inability to do so.In the Solar System with a clearing timescale t clear =10 billion years and C = 2 √ 3, the expressions for the minimum orbit-clearing mass and Π reduce to and Π = 807 ma −9/8 , ( where m is the mass of the planetary body in Earth masses and a is the semimajor axis in au. We emphasize the focus on the ability to clear a zone in a specified timescale as opposed to the state of having cleared a zone.This distinction is important because dynamically dominant bodies retain their ability to clear a zone -and their status as planets -even if small bodies enter the zone or if the zone is still in the process of being cleared.In other words, the ability to clear a zone is relatively impervious to various evolutionary phases in the lives of planets (e.g., early formation epoch, Nice-like migration, gravitational scattering).

PRACTICAL CONSIDERATIONS FOR A TAXONOMY OF PLANETARY BODIES
We would like our taxonomy of planetary bodies to be useful and consistent.A taxonomy is useful if it helps guide, or contributes to, our scientific understanding of the population.Conversely, a taxonomy that obscures fundamental relationships among specimens is not useful.In order to be useful, a taxonomy must also be based on features that are readily observable and measurable.A taxonomy based on unobservable or difficult-to-observe properties has no value.
Shortly after the discovery of a planetary system, the first properties that we can measure are the orbital elements, starting with the orbital periods and semimajor axes.In many situations, we can then obtain mass or minimum-mass estimates, either from radial velocity measurements, transit-timing variations, orbital perturbations, or the presence of satellites.In some situations, size is easier to measure than mass, either from transit depths, secondary eclipses, or optical/IR flux measurements.In these instances, a radius-mass relationship can be used to provide a mass estimate.Popular radius-mass relationships and their domain of applicability include Fang & Margot (2012 Fabrycky et al. (2014), Wolfgang et al. (2016 Chen & Kipping (2017, 9 orders of magnitude), Otegi et al. (2020, M p < 120M ⊕ ).To summarize, we can generally expect measurements of orbital properties and masses shortly after discovery.
In contrast, the shape of planetary bodies in newly discovered planetary systems cannot be determined with technology available now or in the foreseeable future.Lightcurve information can sometimes provide useful limits on an object's axial ratio and convex shape.However, for distant objects where the viewing and illumination geometries change slowly, the spin orientation and 3D shape remain undetermined.For instance, the shape of the ∼360 km diameter Neptunian moon Nereid remains undetermined ("somewhere between roughly spherical and greatly nonspherical"), despite an extensive set of ∼600 lightcurves spanning 20 years (Schaefer et al. 2008).Stellar occultations that involve multiple chords can provide a 2D projection of an object's shape at the occultation epoch.Although quite useful, these data are usually insufficient to make a confident determination about the 3D shape of an object (Ortiz et al. 2020).Because shapes are so difficult to measure, even the shapes of the largest trans-Neptunian bodies at mere distances of ∼40 au remain enigmatic.For instance, the shapes of the ∼920 km diameter Orcus and ∼850 km diameter Salacia remain uncertain, despite discoveries ∼20 years ago (Grundy et al. 2019;Emery et al. 2024).For exoworlds, the problem is much worse because the spatial resolution is degraded by a factor of ∼10,000 compared to the trans-Neptunian region, even for the nearest stars.Therefore, shape information is either difficult or impossible to obtain, so planetary taxonomies that rely on the knowledge of shapes are unworkable.
Taxonomic classification may be aided by the presence of gaps between individual clusters of planetary bodies.As described by Soter (2006), nature does not provide an obvious gap between spheroidal and nonspheroidal shapes (Figure 1).The transition between these two regimes depends on multiple properties, including bulk density and material strength (Tancredi & Favre 2008), as well as collisional, tidal, and thermal evolution histories.Most of these properties are not observable remotely.Transitional objects exhibit a range of mass values spanning ∼2 orders of magnitude, such that mass (or diameter) is not a good proxy for roundness.However, it appears that Solar System objects with masses larger than 10 21 kg have sufficient mass to be approximately in hydrostatic equilibrium and adopt a nearly triaxial figure of equilibrium (Figure 1).This mass threshold, which corresponds approximately to the masses of Ceres and Dione, could perhaps be used if one had to make an informed guess about the state of hydrostatic equilibrium of a planetary body in the absence of detailed shape information.

UNSUPERVISED CLUSTERING AS A GUIDE TO TAXONOMIC CLASSIFICATION
Taxonomists usually group specimens according to certain features.However, there are multiple ways to delineate groups in a taxonomy of planetary systems, and we would like to establish the groups in a way that is both logical and dispassionate.Unsupervised clustering algorithms are useful in this context, as they can guide our choices impartially on the basis of well-established algorithms (Kaufman & Rousseeuw 1990;Everitt et al. 2011).Because no training is involved in unsupervised algorithms, there is no mechanism for intentionally or unintentionally biasing the results.

K-means clustering
K-means clustering is a popular nonparametric algorithm designed to partition n observations (x 1 , ..., x n ) into k clusters (C 1 , ..., C k ) in a way that minimizes the within-cluster sum of squared distances.Formally, the objective function to minimize is   where the mean or centroid of cluster i is given by and |C i | is the size of cluster i.

Silhouette analysis
The user may specify the number of desired clusters arbitrarily or may conduct a silhouette analysis to guide the choice of the number desired clusters 1 (Rousseeuw 1987).In a silhouette analysis, one calculates the mean distance a j between point j and all other points in its cluster and the mean distance b j between point j and all other points in the nearest cluster.The silhouette value s j = b j − a j max {a j , b j } ranges between −1 and +1, where −1 indicates that point j would be more appropriately placed in the neighboring cluster, and +1 indicates that point j is appropriately clustered.The mean silhouette value over all data points is an indicator of the quality of the clustering and can be used to determine the most appropriate number of clusters to use for a data set.

Clustering of Solar System bodies according to semimajor axis
We explored k-means clustering of Solar System bodies to guide and motivate a more general taxonomic system for planetary bodies.There is an important limitation to this approach.We used the Solar System because it is the only planetary system for which we have a sufficiently complete inventory to conduct this analysis.However, using a single system to derive a taxonomy applicable to many other systems is precarious.It is entirely possible that the classification system will not generalize well and that revisions will be needed once additional information about exoplanetary systems becomes available.In particular, it is possible that the Solar System is atypical, with perhaps a greater degree of order and stability than other systems, a condition that may be required for life to form and thrive.With this caveat in mind, we investigated the taxonomic insights that can be garnered by clustering Solar System data.
We started by evaluating the two features at our disposal when classifying newly discovered planetary systems: orbital elements and masses (Section 3).To approximate the case of a newly discovered planetary system, we considered only the most massive bodies.
We considered the 35 most massive planetary bodies in the Solar System (8 planets, the Moon, Ceres, 4 Jovian satellites, 5 Saturnian satellites, 5 Uranian satellites, Triton, and 10 trans-Neptunian objects, including Pluto and Charon) and recorded the semimajor axes of their orbits.We applied k-means clustering to the logarithm (base 10) of the semimajor axes and conducted a silhouette analysis, which indicated that the most suitable number of clusters in this data set is two.We verified that the optimal number of clusters and cluster membership are robust against inclusion or exclusion of either one or both of the most extreme objects (Charon and Sedna).The unsupervised clustering algorithm grouped all satellites in one cluster and all objects that orbit the Sun in another (Figure 2).Other clustering algorithms (single linkage, Ward linkage, DBSCAN, Gaussian mixture model) yielded the same result.This finding motivates our first unsurprising conclusion with respect to planetary taxonomy: This taxonomic distinction is useful and desirable because it emphasizes a relationship that is fundamental to the evolution of planetary bodies.Io's volcanic activity and Europa's distinctive cycloids, for instance, would not exist if these worlds were in orbit around the Sun.Their orbit around Jupiter is an essential, defining feature of their planetary identities.We argue that such a fundamental feature ought to be recognized in any planetary taxonomy.Consequently, we strongly disfavor "Moons are Planets" taxonomies (Metzger et al. 2022) because they obscure a fundamental defining feature of planetary bodies.
We are not asserting that planets and satellites will always be neatly divided into two clusters according to semimajor axis.For instance, we anticipate that the two groups may overlap when including short-period exoplanets, distant irregular satellites, or quasi satellites.Likewise, the evidence for preferring the Gaussian mixture model with two clusters instead of four clusters is quite weak, with Bayesian information criterion (BIC) values of 133.99 and 134.07, respectively.Nevertheless, the planet-satellite distinction remains an important consideration when developing a planetary taxonomy.
We note that a distinction between satellites and planets is also expected when orbital distances to the primary are expressed in units of Hill radii or mutual Hill radii.Most planets will orbit tens of (mutual) Hill radii away from the central body simply by virtue of the required spacing between neighboring planets.Planets on orbits closer than ∼3.5 mutual Hill radii are not stable (Gladman 1993;Chambers et al. 1996); the oligarchic growth stage during planet formation yields embryos spaced ∼10 mutual Hill radii apart (Kokubo & Ida 2002;Thommes et al. 2003); and the observed distribution of the spacings between neighboring exoplanets peaks near ∼20 mutual Hill radii (Fang & Margot 2013;Lissauer et al. 2014).In contrast, satellites orbit planets at distances smaller than the Hill radius (Hamilton & Burns 1991).

Clustering of Solar System bodies according to mass
Having identified satellites as a distinct group, we continued our exploration and applied k-means clustering to the logarithm (base 10) of the masses of the 18 non-satellite objects (8 planets, Ceres, 9 trans-Neptunian objects, including Pluto).The silhouette analysis revealed that the data are best partitioned in two or three clusters, with silhouette scores that are essentially tied (0.702 and 0.704, respectively).Figures 3 and 4 reveal the groupings, which are also unsurprising.When asked to produce two clusters, the unsupervised clustering algorithm grouped the eight planets in one cluster and all remaining objects in another.When asked to produce three clusters, it grouped the four giant planets in one cluster, the four terrestrial planets in another cluster, and all remaining objects in a third cluster.Another clustering algorithm based on a Gaussian mixture model yielded the same results.Either grouping could be used to further develop a taxonomic system, with no strong preference for either one.However, as motivated by IAU resolution B5, we refined our exploration to include a combination of orbital elements and masses in order to diagnose dynamical dominance.

Clustering of Solar System bodies according to dynamical dominance
The planetary discriminant Π quantifies an object's ability to clear its feeding zone in a given timescale (Section 2), also known as dynamical dominance.It can be evaluated with knowledge of the mass of the planetary body and its semimajor axis (Equation 5).Conveniently, both of these quantities can generally be measured or estimated shortly after the discovery of a planetary body.We applied k-means clustering to the logarithm (base 10) of the Π values for the 18 non-satellite objects.The silhouette analysis showed that there are two clusters in this data set.For additional support of this conclusion, see Appendix A. Notably, all eight planets belong to the same group and there is a gap of three orders of magnitude between the two groups (Figure 5).A similar gap was previously identified and used for taxonomic purposes by Stern & Levison (2002) and Soter (2006).
This finding motivates our second conclusion with respect to planetary taxonomy: Dynamical dominance provides a natural organizing principle for planetary taxonomy.
The taxonomic distinction provided by dynamical dominance is useful and desirable because it identifies planetary bodies that are able to accrete or disperse most of the mass in their feeding zone, which fundamentally governs their subsequent evolution.For instance, Earth's dynamical dominance allowed it to accrete 6×10 24 kg of material, which ultimately enabled our planet to hold onto an atmosphere and oceans, to differentiate, and to generate a magnetic field.Although these features are neither necessary nor sufficient to label Earth a planet, they clearly distinguish Earth from other planetary embryos near 1 au, which did not accrete as much material as Earth and, as a result, were unable to experience a similar evolution.Collisional evolution also yielded different outcomes for Earth and the much less massive embryos nearby.Therefore, dynamical dominance is another essential, defining feature of planetary identity.
We can be relatively confident that dynamical dominance is also a general feature of planetary systems.Planet formation simulations consistently indicate that a small number of dynamically dominant bodies emerge after a chaotic period of accretion (Chambers & Wetherill 1998;Agnor et al. 1999;Raymond et al. 2006).
A dispassionate analysis of the features of Solar System bodies yielded a distinct group of eight bodies that have been historically referred to as planets.Nothing in the clustering method is engineered to exclude any particular object or to keep the number of planets small.The unsupervised clustering could have resulted in a less numerous or more numerous group.
Readers who are chagrined that smaller bodies are not recognized as planets should take comfort in the fact these these bodies are no less worthy of exploration.Indeed, some of the best laboratories for studying planetary processes can be found on smaller bodies.In addition, our understanding of planetary systems would be deficient without a careful study of planetary bodies of all sizes.In other words, a taxonomic classification in one group or another is not an indicator of scientific importance.

A REVISED CRITERION FOR DYNAMICAL DOMINANCE
The unsupervised clustering of Solar System bodies suggests that a reasonable approach to planetary taxonomy is to recognize satellites and planets as distinct and to recognize planets as capable of local dynamical dominance.Dynamical dominance can be ascertained easily because it is a simple function of mass and semimajor axis.
The choice of the clearing timescale in Equation ( 2) is essentially the only arbitrary choice.Margot (2015) proposed to use the main-sequence lifetime of the host star, which can be expressed as a simple function of stellar mass.Objects in orbit around brown dwarfs were explicitly excluded from this initial formulation.However, IAU Commission F2 included brown dwarfs in its working definition, and we are interested in developing a quantitative criterion that is fully consistent with this working definition.A timescale based on the main-sequence lifetime of hydrogen-fusing stars must therefore be abandoned.
Here, we propose to adopt either the approximate main-sequence lifetime of the Sun (10 billion years) or the current age of the universe (13.8 billion years) as a universal clearing timescale applicable to all central bodies (brown dwarfs, stars, and stellar remnants).With these choices for the clearing timescale t clear , the orbit-clearing mass is  1).For t clear = 13.8 by, they are augmented by ∼27%, a change that does not affect the classification of any Solar System body.
We verified that all confirmed exoplanets known to date easily satisfy the criterion for dynamical dominance.We downloaded the Planetary Systems Composite Data from the NASA Exoplanet Archive (2024, March 13 version), which contains information about 5595 confirmed exoplanets.We eliminated objects with masses (or M sin i) values larger than 13 Jupiter masses and computed values of Π for the remaining 5443 objects (Figure 6).The smallest value is Π ≃ 85 for the pulsar planet PSR B1257+12 b with a = 0.190 au and m = 0.02 Earth masses.

PROPOSED IMPROVEMENTS TO IAU RESOLUTION B5 (2006)
Because it is relatively straightforward to apply a quantitative orbit-clearing criterion to exoworlds, it is possible to extend the 2006 IAU planet definition to brown dwarfs and stars other than the Sun and to remove ambiguity about what it means to clear an orbital zone.Likewise, it is possible to use a specific mass threshold to replace a vague and impractical prescription regarding roundness.We hope that these considerations will help start the conversation about making planetary taxonomy both quantitative and useful.One possible IAU-aligned formulation is as follows: A planet is a celestial body that (a) orbits one or more stars, brown dwarfs, or stellar remnants, and (b) has sufficient mass to dynamically dominate the neighborhood around its orbit, i.e., m > 0.0012 m 5/8 central a 9/8 , where m is the mass of the planetary body expressed in Earth masses, m central is the mass of the central body expressed is solar masses, and a is the semimajor axis expressed in astronomical units, and (c) has sufficient mass for its self-gravity to overcome rigid body forces so that it is approximately in hydrostatic equilibrium and assumes a nearly triaxial shape, i.e., m > 10 21 kg, and (d) has a true mass below the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 Jupiter masses for objects of solar metallicity), and (e) has a mass ratio with the central object below the L4/L5 instability, i.e., m/m central < 2/(25 + √ 621) ≃ 1/25.
A satellite is a celestial body that orbits a planet.
In this formulation, clauses (a), (d), and (e) follow the recommendations from the working definition of IAU Commission F2 (Lecavelier des Etangs & Lissauer 2022).Clause (c) is a remnant of IAU Resolution B5 (2006) and "overcoming rigid body forces" is also part of the IAU working definition of an exoplanet.However, over a wide range of conditions, bodies that satisfy the dynamical dominance criterion also satisfy the hydrostatic equilibrium criterion (Margot 2015), so clause (c) may be superfluous.There may be exceptions, especially for bodies in close-in orbits around brown dwarfs.For instance, a Mimas-mass object (3.8 × 10 19 kg) around a 13 Jupiter-mass brown dwarf is dynamically dominant at 0.1 au but may not be in hydrostatic equilibrium (Figure 7).We encourage community conversations about the value of clause (c) and whether this clause could be eliminated.Two of us (J.L.M. and B.G.) submitted a proposed IAU resolution (Appendix B) in accordance with IAU procedures to IAU Commission F2 (Dec. 22, 2023 andFeb. 13, 2024) and IAU Division F (Jan. 17, 2024 and Feb. 13, 2024) with the goal of clarifying the definition of planet and extending it to exoplanets.As of March 2024, it appeared that the IAU had chosen not to publish the proposed resolution to allow for community feedback and voting at its 2024 General Assembly.Meaningful action does not appear likely until the next General Assembly in 2027 at the earliest.

A SIMPLER PROPOSAL
We recognize potential limitations and difficulties related to clauses (b), (c), and (e) of the framework described in Section 6.
Although dynamical dominance is the cleanest criterion we have identified to classify planetary bodies (Figure 5), the semimajor axis dependence of the clearing mass (clause (b)) leads to the unfortunate consequence that the planetary status of a body depends on its distance from the central object.Some of our colleagues regard this characteristic as problematic, although there is ample precedent for assigning location-dependent taxonomic labels to celestial objects (e.g., trans-Neptunian object vs. centaur vs. Jupiter-family comet, Amor vs. Apollo vs. Aten asteroid, and planet vs. free-floating planetary-mass object).
The shape-based aspects of the framework (clause (c)) are generally recognized as problematic.For almost all practical purposes in the foreseeable future, we will not be able to obtain sufficient information about the shape of distant objects, and the criterion, if upheld, will likely reduce to a mass threshold.However, a mass estimate is a poor predictor of an object's shape (Figure 1).
The current framework includes a mass ratio (clause (e)) proposed by the IAU Working Group on Extrasolar Planets.This ratio is also regarded as problematic because it would exclude Jovian planets around certain hosts from the planet category, e.g., a >4 Jupiter-mass exoplanet orbiting an M9V star.The necessity of this ratio, which was apparently recommended to distinguish objects with different formation mechanisms, is debatable.Our planetary taxonomy might be more robust if we agreed to deemphasize formation hypotheses and instead agreed that a celestial object orbiting a brown dwarf is a planet, whether its mass is 1 Jupiter mass or 12 Jupiter masses.
As a result of these three legitimate concerns, we also propose a much leaner definition with simple mass limits.With such a definition, the connection to physical principles is less apparent, although we suggest a minimum mass that recognizes the gap between planets and non-planets in the Solar System (Figure 3) and a maximum mass that recognizes the nominal deuterium burning limit.
A planet is a celestial body that (a) orbits one or more stars, brown dwarfs, or stellar remnants, and (b) is more massive than 10 23 kg, and (c) is less massive than 13 Jupiter masses (2.5 × 10 28 kg).
A satellite is a celestial body that orbits a planet.
The IAU WG on Exoplanetary System Nomenclature has stated that freely floating bodies ("rogue planets") are not planets (Lecavelier des Etangs & Lissauer 2022).We propose that rogue planets -whatever they are called -ought to satisfy criteria (b) and (c).

CONCLUSIONS
Precise definitions are needed to communicate and organize thoughts.The IAU definition of "planet" has been criticized with good reason since 2006.Our community and the public deserve better definitions for such important astrophysical terms as "planet" and "satellite".
In this work, we examined the results of unsupervised clustering of Solar System bodies to propose guiding principles for planetary taxonomy.This analysis revealed the presence of groupings among solar system bodies.These groupings are not accidental and instead reveal profound differences in formation and evolution circumstances.We suggest that our community would be better served with a quantitative taxonomic framework that recognizes such groupings and that is applicable to both Solar System bodies and exoworlds.To facilitate further reflection on these topics, we have provided an IAU-aligned proposal and a simpler proposal.These proposals are meant as a starting point for community conversations, and we welcome feedback on all aspects of both proposals.

ACKNOWLEDGMENTS
We thank Jack Lissauer and Eiichiro Kokubo for insightful suggestions.We thank two reviewers, John Chambers and Jason W. Barnes, and the AAS Statistics Editor for judicious comments that improved the manuscript.

A. ADDITIONAL SUPPORT FOR TWO-CLUSTER SOLUTION
We applied a variety of unsupervised, nonparametric clustering algorithms to the logarithm (base 10) of the Π values for the 18 non-satellite objects: k-means, single linkage, Ward linkage, and DBSCAN (eps=2).All algorithms yielded the same solution, suggesting that the two-cluster solution for Π values is robust.
Furthermore, we applied a Gaussian mixture model, which is a parametric clustering algorithm, to the same data and computed the Bayesian information criterion (BIC) values to assist in model selection.We found BIC values of 77.2, 84.3, 87.8, and 94.2 for 2, 3, 4, and 5 clusters, respectively.The difference in BIC between the two-cluster and three-cluster solutions is 7.1, which is considered "strong evidence" to prefer the former over the latter (Kass & Raftery 1995).In this case, the posterior odds are greater than 35 to 1.

B. PROPOSED RESOLUTION
The text of the proposed resolution submitted to IAU Commission F2 (Dec. 22, 2023 andFeb. 13, 2024) and IAU Division F (Jan. 17, 2024 andFeb. 13, 2024) is reproduced below, in part because it includes arguments in favor or revising the current definition.There are minor differences in notation with Section 6 of this manuscript (Mar. 27, 2024) but the fundamental concepts are the same.Our manuscript also contains a simplified proposal in Section 7.
Proposed Follow-up to IAU Resolution B5 ( 2006) "Definition of a Planet in the Solar System" Proposers: Jean-Luc Margot (UCLA) and Brett Gladman (UBC)

Rationale
The current IAU definition of planet is inadequate due to vague (non-quantified) terms and it does not include exoplanets in a cohesive framework.The purpose of this document is to propose a clarification of the IAU definition of "planet" and an extension to exoplanets.A reasonable approach to astronomical nomenclature is to use physical principles to establish a meaningful threshold then adhere to the threshold for definitional purposes, as in the case of near-Earth asteroid dynamical classes.We propose both quantitative and non-quantitative versions of the resolution.In the quantitative version, clauses (d) and (e) are reproduced exactly from the Working Definition of the IAU WG on Exoplanetary System Nomenclature.

Resolution A (Quantitative)
The IAU resolves that a planet is a celestial body that (a) orbits one or more stars, brown dwarfs, or stellar remnants, and (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it is approximately in hydrostatic equilibrium and assumes a nearly triaxial shape, and (c) has sufficient mass to dynamically dominate the neighborhood around its orbit, i.e., M p > 0.0012 M 5/8 ⋆ a 9/8 p , where M p is the mass of the planetary body expressed in Earth masses, M ⋆ is the mass of the central body expressed is solar masses, and a p is the semi-major axis expressed in astronomical units, and (d) has a true mass below the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 Jupiter masses for objects of solar metallicity), and (e) has a mass ratio with the central object below the L4/L5 instability, i.e., M p /M ⋆ < 2/(25 + (621) 1/2 ) ≃ 1/25.

Resolution B (Non-quantitative)
The IAU resolves that a planet is a celestial body that (a) orbits one or more stars, brown dwarfs, or stellar remnants, and (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it is approximately in hydrostatic equilibrium and assumes a nearly triaxial shape, and (c) has sufficient mass to dynamically dominate the neighborhood around its orbit, and (d) has a true mass below the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 Jupiter masses for objects of solar metallicity), and (e) has a mass ratio with the central object below the L4/L5 instability (approximately 1/25).
The 2024 GA Opportunity IAU Resolution B5 is problematic and the problems will not go away on their own.We have had 18 years to identify the problems and consider possible ways forward.There are good reasons to believe that we are better equipped in 2024 than in 2006 to produce a good outcome.
(1) Increased transparency and community participation The 2006 IAU Planet Definition Committee did not allow for community review of its work and did not communicate its proposed planet definition to IAU members until after the start of the 2006 GA (via an embargoed press release released first to reporters).The proposal was flawed and had to be revised in relative haste during the short two-week span of the 2006 GA.This was ultimately the cause of the bad publicity and ensuing trauma.We have an opportunity to publicize a proposal prior to the GA and consider community feedback in a transparent and measured manner, without rushing the review process during the short time span of a GA.Both the IAU Resolutions Committee and the IAU Executive Committee have the prerogative to recommend approval or rejection of the proposed resolution (IAU Working Rules IV.18).
(2) Better voting representation IAU procedures in 2006 required in-person voting and only a small fraction of IAU members (approximately 400 out of 9000 at the time) had the opportunity to vote on IAU Resolution B5.The IAU has since implemented an electronic voting system that allows broad participation and voting by IAU members.
(3) Improved clarity IAU Resolution B5 is not quantitative and has generated profound disagreements about its interpretation.For instance, prominent planetary scientists have stated that Mercury and Venus would not be planets under IAU Resolution B5 because they may be slightly out of hydrostatic equilibrium, or that Earth and Jupiter may not be planets in that they have not have "cleared" the neighborhood around their orbits (because the term "cleared" is not quantitatively defined).We have an opportunity to clarify the language of the resolution, address these criticisms, and possibly quantify what we mean.
(4) Link to exoplanets The IAU Working Group on Exoplanetary System Nomenclature has promulgated a Working Definition of an Exoplanet that specifically refers to the Solar System: "The minimum mass/size required for an extrasolar object to be considered a planet should be the same as that used in our Solar System."However, IAU Resolution B5 does not define what the minimum mass is.We have an opportunity to establish a unified definition for planets and exoplanets.
(5) Elimination of an impossible standard IAU Resolution B5 requires that a body has "cleared an orbit".Complete clearing is an impossible standard to meet and a planet could lose its planetary status if a new small body began to cross the planet's orbit.Evaluation of this criterion also requires difficult-to-attain knowledge about the existence or properties of other bodies in the vicinity (requiring a complete inventory of the entire system down to very small sizes).We have an opportunity to frame a definition that is aligned with IAU resolution B5's orbit-clearing idea and to eliminate this impossible standard.We are proposing to use the concept of "dynamical dominance", which is a natural organizing principle for classifying planets that relies on the ability to clear a zone and does not require instantaneous or complete clearing.A young body that is dynamically dominant may not yet have cleared its orbit but would still qualify as a planet because it is expected to clear its orbit over time (e.g., our Jupiter in the early planetesimal-clearing epoch).
(6) Congruence with the observed dichotomy of Solar System objects In the Solar System, planets and dwarf planets are cleanly separated by orders of magnitude by dynamical dominance criteria (e.g., Stern and Levison 2002, Soter 2006, Margot 2015).The figure below illustrates the latter criterion, which is used in clause (c).material in the canonical feeding zone of extent 2 √ 3 times the Hill radius, as opposed to ejection of the material to infinity.
-Timescale for clearing: The proposed characteristic timescale for the ability to clear a zone in the dynamical dominance criterion is set to 10 billion years, which is comparable to the main-sequence lifetime of a solar-mass star.There is no profound meaning attached to this timescale and it could be equally set to, say, the current age of the universe.The coefficient in clause (c) comes from equation (8) in Margot, A Quantitative Criterion for Defining Planets, Astronomical Journal 150, 2015 with C = 2 √ 3 and t ⋆ =10 by.With t ⋆ =13.7 by, the coefficient would be 0.0010.
-Mass ratio: The mass ratio of 1/25 proposed by the IAU WG on Exoplanetary System Nomenclature may exclude certain planets (e.g., a >4 Jupiter-mass exoplanet orbiting a M9V star).The motivation for this threshold is explained in detail in Lecavelier des Etangs and Lissauer, The IAU working definition of an exoplanet, New Astronomy Reviews 94, 2022.
-Complexity of derivations: The derivation of the mass required to clear the canonical feeding zone and the derivation of the stability conditions for Lagrangian points are both within reach of professional astronomers.
-Inability to clear at large distances from the central body: With the proposed criterion, an Earth-mass body around a solar mass star is dynamically dominant out to 383 au.When orbital periods are long and the volume is enormous (at large distances), even an Earth-mass body does not dominate its neighborhood.
-In the quantitative proposal, clause (b) is currently non-quantitative.One could in principle use physical principles to establish a meaningful mass threshold for clause (b) as well.

)Figure 1 .
Figure 1.Likelihood of adopting a triaxial figure of equilibrium for 62 Solar System bodies as a function of mass.Objects listed as "close" have been recognized as approaching but clearly deviating from an hydrostatic equilibrium shape.Objects in the 10 19 − 2 × 10 22 kg range are labeled.The vertical dotted line represents a possible mass threshold at 10 21 kg.
Figure 2. K-means clustering of the semimajor axes of the 35 most massive planetary bodies in the Solar System.The unsupervised clustering algorithm grouped satellites (S) in one cluster and all objects that orbit the Sun -planets (P), Ceres (C), and trans-Neptunian primaries (T) -in another.

Figure 3 .Figure 4 .
Figure 3. Two-cluster k-means clustering of the masses of the 18 most massive planetary bodies that orbit the Sun.The unsupervised clustering algorithm grouped the eight planets in one cluster and all remaining objects in another.

Figure 5 .
Figure5.K-means clustering of the planetary discriminant (Π) values of the 18 most massive planetary bodies that orbit the Sun.The unsupervised clustering algorithm grouped the eight planets in one cluster and all remaining objects in another, with a three-order-of-magnitude gap between the groups.The vertical dotted line represents the suggested threshold at Π = 1.

Figure 6 .
Figure 6.Planet mass of confirmed exoplanets in units of the corresponding orbit-clearing mass for a clearing timescale t clear = 10 by.The solid black line shows the proposed boundary between planets and non-planets at Π = 1.The dotted lines represent a Jupiter-mass planet in orbit around a 0.1 (top) and 1 (bottom) solar-mass star.Colors encode the discovery technique.
Figure 7. Orbit-clearing mass as a function of semimajor axis for select values of the mass of the central body.The masses of Iapetus, Vesta, Mimas, and Hyperion are shown with dotted lines.