Keywords:

In antiquity, all of the enduring celestial bodies that were seen to move relative to the background sky of stars were considered planets. During the Copernican revolution, this definition was altered to objects orbiting around the Sun, removing the Sun and Moon but adding the Earth to the list of known planets. The concept of planet is thus not simply a question of nature, origin, composition, mass or size, but historically a concept related to the motion of one body around the other, in a hierarchical configuration. After discussion within the IAU Commission F2 "Exoplanets and the Solar System", the criterion of the star-planet mass ratio has been introduced in the definition of the term "exoplanet", thereby requiring the hierarchical structure seen in our Solar System for an object to be referred to as an exoplanet. Additionally, the planetary mass objects orbiting brown dwarfs, provided they follow the mass ratio criterion, are now considered as exoplanets. Therefore, the current working definition of an exoplanet, as amended in August 2018 by IAU Commission F2 "Exoplanets and the Solar System", reads as follows: 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 L 4 / L 5 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.

Here we discuss the history and the rationale behind this definition.

Introduction

The importance of defining terms

The definition of terms may appear to be unimportant for the advancement of science. Whether or not the word "planet" or "exoplanet" is used for a newly discovered object is much less important for our knowledge than the understanding of its physical characteristics, its composition or its history and fate. Nonetheless, the primary purpose of language is to convey information in an efficient and effective manner. Usage of terms contributes to the definition in all living languages. This applies to scientific terminology, where many words, even categories of objects like asteroids (see below), are defined more by usage than by a formal statement [START_REF] Wittgenstein | Philosophical Investigations[END_REF].

Scientific terminology is continuously improving because our collective knowledge is advancing. Refining the definition of terms contributes to the crystallization of knowledge. Clear and agreed-upon definitions facilitate communication of information. In that perspective, the role of International Astronomical Union (IAU), which endeavors to promote global collaboration, is crucial. One of the first achievements of the IAU, just after its birth in 1919, was the definition of the names of constellations and their boundaries on the sky [START_REF] Delporte | Delimitation scientifique des constellations (tables et cartes)[END_REF]. Today, the IAU is tasked with adopting definitions of astronomical terms and names of astronomical objects, including those related to planets within our Solar System and beyond.

In 2001, the IAU's Working Group on Extrasolar Planets wrote a working definition for the term "planet" to be applied to extrasolar planets, which it subsequently amended in 2003. Fifteen years later, knowledge of exoplanets had significantly advanced, with discoveries of a large number of planetary mass objects showing an astonishing diversity in physical properties. Therefore during 2018 we, as then President (ALdE) and Vice-President (JJL) of IAU Commission F2 -Exoplanets & The Solar System, decided to lead our commission in a reassessment of the 2003 working definition.

Historical Background -Planets in the Solar System

The wonders of the night sky, the Moon and the Sun have fascinated mankind for many millennia. Ancient civilizations were particularly intrigued by several brilliant "stars" that move among the far more numerous "fixed" (stationary) stars. Aristotle and many other ancient Greek philosophers used and popularized the word πλαν ήτ ης ("planete"), meaning wandering star, to refer to these objects. The list of "planetes" included seven bodies in this category: the Sun, the Moon, Mercury, Venus, Mars, Jupiter and Saturn.

Although Aristarchus of Samos (∼310 -230 BCE) had proposed the first known heliocentric model, the geocentric models of Aristotle and Ptolemy were more popular until the publication of the seminal work of [START_REF] Copernicus | De revolutionibus orbium coelestium[END_REF]. Following the Copernican revolution, there were six bodies, all known to the ancients, that were classified as planets: Mercury, Venus, Earth, Mars, Jupiter and Saturn. This new classification give a similar list of planets with the exclusion of the Moon and the Sun and the addition of the Earth because it orbits the Sun as do the other planets. The Copernican revolution changed the planet term from "moving on the sky", to "moving around the Sun". In both cases, motion was the key concept for the term "planet": the term planet was not related to the nature of the object given by its shape, size or composition, it was related to its dynamics.

The first newly-discovered planet was Uranus, which was found by William Herschel in 1781. The discovery of Uranus was followed in rapid succession by the detection of four smaller objects traveling between the orbits of Mars and Jupiter: Ceres in 1801, then Pallas, Juno, and Vesta, the last of which was discovered in 1807. The total number of known objects classified as planets had reached 11, and remained at this value for almost four decades. A fifth small body between Mars and Jupiter, Astraea, was found in December of 1845, less than one year prior to the September 1846 discovery of Neptune. Three more small "planets" orbiting between Mars and Jupiter, now known as 6 Hebe, 7 Iris, 8 Flora, were detected in the second half of 1847, and several others were found in the next few years. It became clear that many objects orbited between Mars and Jupiter, and that all of them were much smaller than any of the six planets known to the ancients, as well as Uranus and Neptune. These small bodies were therefore reclassified as "minor planets" or "asteroids", and the number of "planets" dropped down to eight.

Pluto was found in 1930. Although it was much fainter than the other planets, no minor planets more distant than the Jupiter Trojans were known at the time, and Pluto was near the location predicted for a ∼ 6 M ⊕ (Earth mass) planet hypothesized to explain unaccounted for deviations of Neptune's orbit, so Pluto was quickly accepted as the ninth planet orbiting our Sun.

However, subsequent data showed that Pluto was much smaller and less massive than initially thought. Pluto's absolute magnitude decreased as it approached perihelion, and further study showed that Neptune's orbit was better explained without a massive perturber at Pluto's location. Analysis of the orbit of Pluto's large moon Charon, which was discovered in 1978, implied that Pluto was only 4% as massive as Mercury, the smallest of the eight other bodies regarded as planets.

The minor planet 1992 QB1, later named 15760 Albion, which like Pluto had a semimajor axis larger than that of Neptune, was discovered in 1992 [START_REF] Jewitt | Discovery of the candidate Kuiper belt object 1992 QB 1[END_REF]. Many other bodies were discovered in this region, referred to as the Kuiper Belt, in subsequent years. Pluto is by far the brightest Solar System body observed beyond Neptune's distance, but several known Kuiper Belt objects (KBOs) are larger than 1 Ceres, which is by far the largest asteroid. In 2005, the discovery of 136199 Eris, a KBO that is ∼ 25% more massive than Pluto [START_REF] Brown | The Mass of Dwarf Planet Eris[END_REF] and that was initially thought to be slightly larger in size as well, brought some urgency to the question of how large a body needed to be in order to be classified a planet, because it didn't make sense to consider Pluto a planet if Eris was not considered a planet as well.

A contentious discussion at the 2006 IAU General Assembly resulted in the adoption of the following definition for planets in the Solar System2 : A planet is a celestial body that:

1. is in orbit around the Sun, 2. has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and 3. has cleared the neighborhood around its orbit.

Pluto was thus reclassified, and the number of Solar System planets dropped back down to eight. We note here that this definition is aimed at addressing the question of the use of the term "planet" for objects in the Solar System. Therefore, the first requirement to "orbit around the Sun" should not be read as an exclusion of exoplanets for the term "planet", but a restriction of the application of this definition to Solar System objects.

Historical Background -Exoplanets

At the beginning of this millennium, the IAU's Working Group on Extrasolar Planets (WGESP) wrote and then amended a "working definition" for the term "planet" to be applied to exoplanets [START_REF] Boss | Nomenclature: Brown Dwarfs, Gas Giant Planets, and ?[END_REF][START_REF] Boss | Working Group on Extrasolar Planets[END_REF]. For more than 15 years, the definition was unchanged. It read as follows: Position statement on the definition of a "planet"

Working group on extrasolar planets (WGESP) of the International Astronomical Union Created: February 28, 2001Last Modified: February 28, 2003 Rather than try to construct a detailed definition of a planet which is designed to cover all future possibilities, the WGESP has agreed to restrict itself to developing a working definition applicable to the cases where there already are claimed detections, e.g., the radial velocity surveys of companions to (mostly) solar-type stars, and the imaging surveys for free-floating objects in young star clusters. As new claims are made in the future, the WGESP will weigh their individual merits and circumstances, and will try to fit the new objects into the WGESP definition of a "planet", revising this definition as necessary. This is a gradualist approach with an evolving definition, guided by the observations that will decide all in the end. Emphasizing again that this is only a working definition, subject to change as we learn more about the census of low-mass companions, the WGESP has agreed to the following statements:

1. 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 or stellar remnants 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. 2. Substellar objects with true masses above the limiting mass for thermonuclear fusion of deuterium are "brown dwarfs", no matter how they formed nor where they are located. 3. Free-floating objects in young star clusters with masses below the limiting mass for thermonuclear fusion of deuterium are not "planets", but are "sub-brown dwarfs" (or whatever name is most appropriate).

We can expect this definition to evolve as our knowledge improves.

Toward a new definition of an exoplanet

Need for an update

As stated in the Introduction, the definition of a class of objects is a way to formalize our knowledge on these objects. By 2018, knowledge on exoplanets had significantly advanced since 2003. The number of known objects had been multiplied by a factor of 40 with the discovery of numerous exoplanets. These exoplanets have a wide range of masses, radii and orbits, with many being quite different from any planets (in our Solar System or beyond), known in 2003. The diversity of system primaries also expanded, e.g., a 5 Jupiter mass object was discovered in orbit around a 25 Jupiter mass brown dwarf [START_REF] Chauvin | A giant planet candidate near a young brown dwarf. Direct VLT/NACO observations using IR wavefront sensing[END_REF], raising new questions on the planethood of this kind of objects and others (e.g., [START_REF] Basri | Planetesimals to Brown Dwarfs: What is a Planet?[END_REF]. So we considered it time for the 2003 working definition to be reassessed and, if needed, updated.

The process

The Organizing Committee (OC) of IAU Commission F2 "Exoplanets and the Solar System" addressed the subject during the 2015-2018 triennium. After considering various possibilities, it was decided to consult the exoplanet community by questioning the whole membership of the commission. With about 400 members, this commission includes only a small subset of all astronomers working in the field. Nonetheless, it can be considered as a representative sample of experienced exoplanet researchers because of the wide diversity of its membership, both geographically and regarding research activities (theory, observations, dynamics, characterization, etc.).

The Commission F2 Organizing Committee concluded that, while the definition should evolve with time, unless fundamental discoveries dictated otherwise, the evolution in the definition should be in continuity with the previous definition. As a consequence, it was decided that any change in the definition of an exoplanet should be made only when a substantial majority of the community considers it to be an improvement. To translate the principle of a "substantial majority" into numbers, a limit of a qualified majority of 2/3 was agreed upon within the OC.

Also, to make the decision practical and the process efficient, the possibility to ask general question to the community such as "Should the definition (or some specific portion thereof) be changed" has been excluded. Indeed, a large majority can be in favor of some type of change, while no majority agrees on any specific change to be made. Therefore, specific amendments to the definition were prepared by the OC and proposed to a vote by the whole membership of the Commission.

Several issues on the mass of an exoplanet (lower and upper limits) were discussed within the OC, but only those supported by a majority of the OC were proposed to the vote of the entire commission membership. For instance, the OC agreed on introducing the planet-star mass ratio in the definition but did not agree on amending or discarding the 13 Jupiter mass limit, so no amendment on this last issue was included in the plebiscite ( §2.4).

The lower limit

Although the smallest extrasolar planets known to orbit actual stars (objects supported against gravitational collapse by self-sustained fusion, as opposed to stellar remnants like white dwarfs and neutron "stars") are only ∼ 10 -3 times as massive as any of the exoplanets known to orbit stars in 2003, none are in crossing orbits and all are significantly larger than Pluto and Eris. Also, perhaps more surprising, the planetary nature of PSR 1257+12b, with a mass of 1.8 times that of the Moon and orbiting a pulsar, has never been contested in the literature [START_REF] Schneider | Defining and cataloging exoplanets: the exoplanet.eu database[END_REF]. Still, even for this extreme case, given the short orbital distance it can be shown that the criterion on the cleaning of the orbital neighbourhood is well satisfied [START_REF] Margot | A Quantitative Criterion for Defining Planets[END_REF]. The same is the case for the Moon-size transiting exoplanet Kepler-37 b [START_REF] Barclay | A sub-Mercury-sized exoplanet[END_REF].

Thus, there has not been any significant controversy of the lower size/mass limit for any observed object to be considered an exoplanet. Given the lack of clear scientific direction from observations, the OC decided not to amend the wording of the 2003 provisional definition on the lower limit for an exoplanet, which explicitly refers to the criterion "used in our Solar System". Indeed, in 2006 the IAU formally adopted a definition for planets in the Solar System that specifically addresses the issue of the minimum mass (Sect. 1.2), and which can presently be used for extrasolar planetary systems.

The minimum planetary mass is therefore the mass sufficient for self-gravity to overcome rigid body forces and for clearing the neighborhood around the object's orbit. In other words, an object can be considered a planet if it is capable to influence dynamically the evolution of other bodies in its vicinity, sculpting their orbital distribution. This criterion has the advantage that the question of the evolution of other bodies in the neighborhood can be addressed through theoretical calculations or numerical simulations (see, e.g., [START_REF] Margot | A Quantitative Criterion for Defining Planets[END_REF]. Note, however, that this definition of the lower limit relies on a dynamical criterion, which has been written with the architecture of the Solar System in mind. In future, it might need to be adapted to extrasolar systems showing different architectures, e.g.,with very eccentric orbits.

The mass upper limit

The upper mass limit of the 2003 definition has been controversial [START_REF] Chabrier | Giant Planet and Brown Dwarf Formation[END_REF], and various proposals have been made to alter it. Some researchers prefer definitions based on formation mechanism; although this has aesthetic appeal, it is highly impractical given the difficulty in determining the formation process of most bodies that might be reclassified by this change. The deuterium burning criterion should not be understood as a criterion linked to the formation mechanism.

Some researchers have proposed increasing the boundary to ∼ 25 Jupiter masses (M Jup ) based upon the location of the "driest" region of the brown dwarf desert of the population of objects observed to orbit within a few AU of sunlike stars. However, there is no "gap", the location of this minimum appears to be a function of the mass of the stellar host, and it may also vary with orbital period. Furthermore, recent results show that the stellar metallicity -giant planet occurrence rate correlation goes mostly away for companion masses above 4 M Jup (e.g., [START_REF] Schlaufman | Evidence of an Upper Bound on the Masses of Planets and Its Implications for Giant Planet Formation[END_REF], which suggests it might be more appropriate to move the mass boundary downwards rather than upwards.

At present, the majority of known exoplanets have been discovered via transits, which observe size rather than mass. But a boundary based upon size would be inappropriate because the planetary radius function is not a monotonic function of planetary mass, even for objects of a specified composition [START_REF] Stevenson | Interior models of Jupiter[END_REF].

As the Universe is less than 14 billion years old, sizes of cool H/He-dominated objects increase with mass up to a peak at ∼4 M Jup , and then drop (due to self-compression) to a minimum near 70 M Jup , just below the start of the stellar main sequence. The radius of the star TRAPPIST-1 is very close to that of Jupiter, and even the Solar System's nearest stellar neighbor, Proxima Centauri, is smaller than many hot Jupiters.

Finally, the Organizing Committee decided to keep the current mass limit at the thermonuclear fusion of deuterium. This limit corresponds roughly to 13 Jupiter mass, with some dependence on the metallicity [START_REF] Spiegel | The Deuterium-burning Mass Limit for Brown Dwarfs and Giant Planets[END_REF].

In conclusion, no amendment on this point was included in the plebiscite of the commission membership.

The mass ratio

At the time of the previous amendments to the exoplanet definition in 2003, no planetary mass objects had been found in orbit about brown dwarfs, and such objects were not considered in the 2003 definition. Members of this class have subsequently been found, and they have typically been referred to as exoplanets. Most planetary mass objects orbiting brown dwarfs seem to fall cleanly into one of two groups: (1) very massive objects, with masses of the same order as the object that they are bound to and (2) much lower mass objects. The high-mass group of companions appear akin to stellar binaries, whereas the much less massive bodies appear akin to planets orbiting stars. There is a wide separation in mass ratio between these two groupings Figures 1 and2, so any definition with a ratio between ∼1/100 and 1/10 would provide similar results in the classification of known objects.

It is noteworthy that the limiting mass ratio for stability of the triangular Lagrangian points, M/M central < 2/(25 + √ 621) ≈ 1/25, falls in the middle of this range. Moreover, this ratio is a limit based on dynamical grounds, which distinguish between star-planet couples where the star dominates and the planet can "clear the neighborhood around its orbit" (when the mass ratio is below 1/25), and pairs of objects where the more massive body does not dominate the dynamics to the extent that the less massive body can be to a good approximation considered to be orbiting about an immobile primary. Therefore, we proposed using this dynamically-based criterion as the dividing point.

The triangular Lagrangian points are potential energy maxima, but in the circular restricted three-body problem the Coriolis force stabilizes them for the secondary to primary mass ratio (m 2 /m 1 ) below 1/25, which is the case for all known examples in the Solar System that are more massive than the Pluto-Charon system. The precise ratio required for linear stability of the Lagrangian points L 4 and L 5 is m 2 /m 1 < 2/(25 + √ 621) ∼ 1/25 (see [START_REF] Danby | Fundamentals of celestial mechanics[END_REF]. If a particle at L 4 or L 5 is perturbed slightly, it will start to librate about these points (i.e., oscillate back and forth, without circulating past the secondary).

From an observational point of view, mass ratios are commonly used in statistical studies of exoplanet discoveries by microlensing (e.g., [START_REF] Suzuki | The Exoplanet Mass-ratio Function from the MOA-II Survey: Discovery of a Break and Likely Peak at a Neptune Mass[END_REF][START_REF] Suzuki | Microlensing Results Challenge the Core Accretion Runaway Growth Scenario for Gas Giants[END_REF] or by transit photometry (e.g., [START_REF] Pascucci | A Universal Break in the Planet-to-star Mass-ratio Function of Kepler MKG Stars[END_REF]). It appears that the planet-to-star mass ratio is not only the quantity that is best measured in microlensing light curves analysis, but also it may be a more fundamental quantity in some aspects of planet formation than planet mass [START_REF] Pascucci | A Universal Break in the Planet-to-star Mass-ratio Function of Kepler MKG Stars[END_REF]. It can be considered as a natural criterion to be used in the definition of the term exoplanet.

The question of unbound planetary mass objects

Motion relative to the "fixed" stars was the defining aspect of the ancient definition of the term "planet". Motion about the Sun became a requirement for planethood as a result of the adoption of the heliocentric model of the Solar System. Orbiting a star (or a similar object) is a natural extension of this requirement for exoplanets. We therefore agreed to keep that requirement for an object to be considered as an exoplanet to be "orbiting" around a more massive object. The need to orbit around a star or an analogous massive object to be considered as a planet is effectively an extension of the Copernicus revolution.

For planetary mass objects that do not orbit around a more massive central object, the term "sub-brown dwarf" has not been adopted in the usage by the community; rather, these objects are often referred to as "free floating planetary mass objects". These two terms are nowadays considered as synonymous. An alternative to the rather ambiguous term "free floating", to specifically underline the presence or absence of a central object, would be to use the term "unbound". Note that neither of these terms accounts for the situation when the planetary mass object is orbiting a companion whose mass is below the deuterium-burning limit and/or the minimum mass ratio for stability of the triangular Lagrangian points, but the advantages of brevity in terminology may well dictate that one of these terms is nonetheless optimal.

The C.F2 consultation of July 2018

Following the process of a qualified majority of 2/3 within the OC as defined in §2.2, four questions were asked the whole membership of Commission F2 of the IAU. Questions #3 and #4 were about some recommendations for exoplanet nomenclature and the approval of the criterion of a qualified majority of 2/3 within the whole commission for future changes in the definition of an exoplanet, planetary nomenclature, and discovery criteria, respectively.

The first two questions in the plebescite asked to the commission were directly related to the new definition of an exoplanet:

1. Question 1: Should the term "planets" only apply to objects that have a mass ratio to the central object below the L4/L5 instability limit:

(M/M central < 2/(25 + √ 621) ≈ 1/25)? 2. Question 2: In the WGESP definition, to be considered to be a "planet", the object needs to orbit a star or a stellar remnant. It is proposed to extend the criterion for the central object to include also the "brown dwarfs".

Questions #1 and #2 got approved by more than 2/3 of the expressed opinions, with 71% and 90% of the vote, respectively. Thus, both amendments were approved by large majorities and have been incorporated into the working definition given in Section 3.1. Question #4 also passed, so future amendments to the definition will also require a two-thirds majority for passage.

The new definition

The current IAU working definition of an exoplanet

The current working definition of an exoplanet is as follows:

1. 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 L 4 /L 5 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.

2. Substellar objects with true masses above the limiting mass for thermonuclear fusion of deuterium are "brown dwarfs", no matter how they formed nor where they are located. 3. Free-floating objects in young star clusters with masses below the limiting mass for thermonuclear fusion of deuterium are not "planets", but are "sub-brown dwarfs" (or whatever name is most appropriate).

What's new? A few examples

The two changes between the new and the previous working definition of an exoplanet are the addition of the ∼1/25 mass ratio criterion and the possibility to orbit a brown dwarf. These changes remove from the exoplanet list some massive objects, whose masses place them above the mass ratio criterion, and add new objects that orbit brown dwarfs.

As a ten known systems with a mass ratio above 1/25 do not meet the criteria be considered as exoplanets, but as gravitationally linked binary systems composed of a sub-brown dwarf and a slightly more massive object. In one case, the host is a star, so the companion was classified as a planet under the previous definition, whereas in nine cases the host is a brown dwarf, so the secondary does not fulfil either the old or the new requirements for planethood. The characteristics of these binary systems are summarized in Table 1.

In the other direction, five objects less massive than 13 Jupiter mass and orbiting around a brown dwarf are now considered to be exoplanets. The characteristics of these exoplanetary systems are summarized in Table 2.

Future

There are some minor items that we think should be changed in the future. Near the beginning of item (3), the text "in young star clusters" is outdated, since similar, older objects have been seen outside of clusters, so it should be deleted. At the end of item (3), the text: "sub-brown dwarfs" (or whatever name is most appropriate) should be updated to the currently accepted term "free-floating planetary mass objects".

The exoplanet definition discussed herein is a working definition. The question of if/when it should go through a full IAU process to become the official IAU definition remains open. Indeed, in astronomy all definitions are to some extent "working", and even official ones are subject to amendments, because our knowledge and our understanding of the true nature of objects changes with time.

Note that the deuterium-burning limit is for solar metallicity. Strange objects like planetary mass bodies orbiting black widow pulsars (e.g., [START_REF] Bailes | Transformation of a Star into a Planet in a Millisecond Pulsar Binary[END_REF], which probably lack hydrogen and are primarily composed of carbon and oxygen, are called planets because they are below 13 Jupiter masses and orbit a stellar remnant.

Both the WGESP and IAU Commission F2 have avoided discussing the low mass limit because of the lack of data on exoplanets in the corresponding mass domain that precludes from addressing this issue. Very low mass "exocomets" have been observed to transit the star β Pictoris since the mid-1980's, with detection of both the gas and the dust cometary tails [START_REF] Kiefer | Two families of exocomets in the β Pictoris system[END_REF][START_REF] Zieba | Transiting exocomets detected in broadband light by TESS in the β Pictoris system[END_REF]. Small "asteroids" have been detected in very shortperiod orbits about white dwarfs [START_REF] Vanderburg | A disintegrating minor planet transiting a white dwarf[END_REF], and "disintegrating planets" (e.g., [START_REF] Croll | Multiwavelength Observations of the Candidate Disintegrating Sub-Mercury KIC 12557548b[END_REF] have been seen. The situation may change in the future when very low mass extrasolar orbiting objects will be discovered. It appears that the current definition used for the Solar System, which defines the limit between planets and dwarf planets, can be used for extrasolar objects, at least for those that clearly fall into one category or the other. When a significant number of intermediate extrasolar objects have been discovered, there will be more information to address the possible need for new criteria in the definition.

In conclusion, it must be emphasized that the current definition remains a working definition. Current space missions like Gaia, TESS and CHEOPS, the missions to come like PLATO and Nancy Grace Roman Space Telescope, as well as ground-based observations, will increase the number of known objects significantly. As our knowledge of the exoplanetary zoo improves, the working definition might need to be amended or rewritten. measured, and the red ones are for the cases where only the secondary M sin i has been measured using radial velocimetry. The green line corresponds to the 13 Jupiter mass limit. The blue line corresponds to the Secondary/Primary mass ratio limit of 1/25. The data are from The Extrasolar Planets Encyclopaedia accessed on September 16, 2021 (exoplanet.eu). For OGLE-2016-BLG-1195L the tabulated values are from [START_REF] Shvartzvald | An Earth-mass Planet in a 1 au Orbit around an Ultracool Dwarf[END_REF] ; with a mass of 0.2 M estimated by [START_REF] Bond | The lowest mass ratio planetary microlens: OGLE 2016-BLG-1195Lb[END_REF], the primary may not be a brown dwarf but a low mass star.

Name

Name

Figure 1 :

 1 Figure1: Histogram of the Secondary/Primary q mass ratios for the known objects with a mass of the secondary below 13 Jupiter mass. The black histogram is for the measured mass, and the red histogram is for the M sin i measured using radial velocimetry. The blue vertical line corresponds to the limit ratio of 1/25. The data are from The Extrasolar Planets Encyclopaedia accessed on September 16, 2021 (exoplanet.eu).

Figure 2 :

 2 Figure 2: Primary-Secondary mass diagram. The black dots are for the cases where the secondary mass hasmeasured, and the red ones are for the cases where only the secondary M sin i has been measured using radial velocimetry. The green line corresponds to the 13 Jupiter mass limit. The blue line corresponds to the Secondary/Primary mass ratio limit of 1/25. The data are from The Extrasolar Planets Encyclopaedia accessed on September 16, 2021 (exoplanet.eu).

Table 1 :

 1 Table of known binary systems including an object of less than 13 Jupiter mass and with a Secondary/Primary mass ratio above 1/25. The data are from the Extrasolar Planets Encyclopaedia[START_REF] Schneider | Defining and cataloging exoplanets: the exoplanet.eu database[END_REF]) accessed on September 16, 2021 (exoplanet.eu).

		Primary Secondary	M1/M2 q=M2/M1
		mass	mass Mass ratio Mass ratio
		(M Jup )	(M Jup )
	MOA-2010-BLG-073L	160.	11.00	15.3	0.065
	KMT-2016-BLG-1820L	41.	4.57	8.8	0.113
	OGLE-2011-BLG-0420L	26.2	9.8	2.65	0.377
	OGLE-2012-BLG-0358L	23	1.85	8.0	0.125
	MOA-2016-BLG-231L	21.0	9.00	2.33	0.430
	2M 0441+23	20.	7.50	2.67	0.370
	OGLE-2009-BLG-151L	18.9	7.9	2.39	0.419
	OGLE-2016-BLG-1266L	15.7	12.0	1.31	0.762
	Oph 98	15.4	7.8	1.97	0.510
	WISE J1355-8258	11.	9.00	1.22	0.820

Table 2 :

 2 Table of known exoplanetary systems where an exoplanet of less than 13 Jupiter mass orbit a brown dwarf.

Neither of the authors of this article was present for the 2006 planet definition and vote.

Acknowledgement

The two authors contributed equally to this manuscript and are listed in alphabetical order.

We thank Antonella Barucci, Julio A. Fernandez, René Heller, Antoniadou Kyriaki, Eric Mamajek, Rosemary Mardling, and Alessandro Morbidelli for constructive comments on preliminary drafts that enabled us to improve the manuscript. We thank Clément Ranc for very useful discussions about the microlensing exoplanets.

This research has made use of data obtained from the portal exoplanet.eu of The Extrasolar Planets Encyclopaedia.


Siti Mutmainah
email: siti.mutmainah@uin-suka.ac.id

Samir Hachour

Frédéric Pichon

David Mercier

Improving an evidential source of information using contextual corrections depending on partial decisions

Keywords: Belief functions, Contextual corrections, Partial decisions, Interval dominance

In this paper, an improvement of the quality of an evidential source of information is proposed using contextual corrections depending on partial decisions obtained from an interval dominance relation on the source outputs. Numerical experiments with the EkNN classifier and synthetic and real data allows us to illustrate the performances and the interest of this method.

Introduction

In pattern recognition [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF][START_REF] Duda | Pattern Classification[END_REF], the quality of the information provided by a source (e.g. a sensor, a classifier, . . .) plays an important role in the success of the pattern recognition task as the information may be false, biased or irrelevant.

The belief function framework (or Dempster-Shafer theory [START_REF] Shafer | A mathematical theory of evidence[END_REF]) provides a flexible mathematical framework for dealing with imperfect information. In this theory, the quality of a source of information is classically managed by means of the discounting operation introduced by G. Shafer in his seminal book [18, chapter 11, page 251]. This method has since been refined using so-called contextual correction mechanisms [START_REF] Mercier | Refined Modeling of Sensor Reliability in the Belief Function Framework Using Contextual Discounting[END_REF][START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF] taking into account more refined knowledge about the quality of a source: its reliability (in the sense of its relevance, meaning the capacity of the source to answer the question of interest) and its truthfulness (meaning its capacity to tell what it knows; this capacity being possibly conscious -as a lie for example -or unconscious -as a bias for example) [START_REF] Pichon | Relevance and truthfulness in information correction and fusion[END_REF]. More specifically, three mechanisms have been introduced. They are respectively called Contextual Discounting (CD), Contextual Reinforcement (CR) and Contextual Negating (CN). They all can be mathematically derived [START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF] from these notions of reliability and truthfulness: CD, which generalizes the discounting operation, Mrs. Mutmainahs research is supported by the overseas 5000 Doctors program of Indonesian Religious Affairs Ministry (MORA French Scholarship) can adjust the output of a source in accordance with information about its reliability, while CN, which generalizes the negation of a source [START_REF] Dubois | A set-theoretic view of belief functions: logical operations and approximations by fuzzy sets[END_REF], can adjust the source according to its truthfulness, and at last, CR is the dual operation of CD, it may reinforce a too cautious source [START_REF] Mercier | Belief functions contextual discounting and canonical decompositions[END_REF][START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF].

In this paper, we propose to improve the quality of a source of information outputting belief functions regarding a question of interest using contextual corrections CD, CR or CN, and partial decisions computed from the outputs using the relation of interval dominance [START_REF] Troffaes | Decision making under uncertainty using imprecise probabilities[END_REF][START_REF] Yang | Nested Dichotomies with probability sets for multi-class classification[END_REF], also called strong dominance in [6,[START_REF] Ma | Partial classification in the belief function framework[END_REF]. More specifically, the source is considered as a black box meaning we have no access to the manner it works to make its evidential outputs. This situation occurs for example when a company buys a sensor from another one to perform a given task, and the decision making process (or the algorithm) used by this sensor is protected [START_REF] Mercier | Decision fusion for postal address recognition using belief functions[END_REF]. A learning set is available. It is composed of outputs of the source (as mass functions for example), regarding data the ground truth of which is known. As a simple example [START_REF] Elouedi | Assessing sensor reliability for multisensor data fusion within the transferable belief model[END_REF], we may have in the learning set the following information m S output by the source regarding the true class of an object o, which belongs to a universe

Ω = {ω 1 , ω 2 , ω 3 } ω 1 ω 2 ω 3 {ω 1 , ω 2 } {ω 1 , ω 3 } {ω 2 , ω 3 } Ω Ground truth of object o m S {o} 0 0 0.5 0 0 0.3 0.2 ω 1
Now, instead of learning a best possible correction among CD, CR and CN from the whole learning set as proposed in [START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF], we would like to propose a method that takes advantage of all the corrections, and so propose to regroup the outputs leading to the same partial decisions to learn the best possible correction between CD, CR and CN in each of these groups of outputs to reach better global performances (certainly at the cost of learning more models). With this strategy, an output is thus adjusted differently depending on the partial decision it leads to. This paper is organized as follows. In Section 2, the basic concepts and notations on belief functions used in this paper are presented, as well as a reminder on decision making with interval dominance with belief functions. Reminders on contextual corrections are given in Section 3. Thereafter, in Section 4, the proposed method to learn contextual corrections depending on partial decisions is exposed. It is tested with synthetic and real data in Section 5. A discussion is also added in this last Section to conclude the paper.

Belief functions: necessary concepts and notations

In this Section, necessary concepts and notations used in this paper are quickly reminded. Further details can be found for example in [START_REF] Shafer | A mathematical theory of evidence[END_REF][START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem[END_REF][START_REF] Denoeux | Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence[END_REF].

With Ω = {ω 1 , ..., ω K } the universe (or the frame of discernment) representing the finite set of answers to a given question of interest, a piece of evidence regarding the answer to this question of interest induces a mass function (MF) m Ω (or m if no ambiguity) defined from 2 Ω to [0, 1], verifying A⊆Ω m Ω (A) = 1.

The focal elements of a MF m are the subsets A ⊆ Ω s.t. m(A) > 0. A MF having only one focal element A is called a categorical MF and can be simply denoted by m A .

A MF m is in one-to-one correspondence with belief and plausibility functions Bel and P l respectively defined for all A ⊆ Ω by Bel(A) = B⊆A m(B), and P l(A) = B∩A =∅ m(B).

The contour function pl corresponds to the restriction of the plausibility function to the singletons of Ω, it is defined for all ω ∈ Ω by pl(ω) = P l({ω}).

If a source S provides an output m S , and if it is also known that this source is reliable with a degree of belief

β = 1 -α ∈ [0, 1], then this original MF m S can be discounted into a MF m s.t. m = A → βm S (A) ∀A ⊂ Ω Ω → βm S (Ω) + α (1)
Equation ( 1) can also be simply rewritten as m = β m S + α m Ω , with m Ω the categorical MF defined by m Ω (Ω) = 1, and it can also be shown (see for example [START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF]Prop. 11]) that the contour function associated with the discounted MF is defined for all ω ∈ Ω by pl(ω) = 1 -(1 -pl S (ω))β, with pl S the contour function of m S . Derivations of this operation can be found in [START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem[END_REF][START_REF] Mercier | Refined Modeling of Sensor Reliability in the Belief Function Framework Using Contextual Discounting[END_REF][START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF]. At last, when a decision has to be made [START_REF] Denoeux | Analysis of evidence-theoretic decision rules for pattern classification[END_REF][START_REF] Ma | Partial classification in the belief function framework[END_REF], if we consider that the set of possible decisions (or acts) is equal to Ω, we can use the following relation of dominance between the singletons of Ω:

ω ω ⇐⇒ Bel({ω}) ≥ P l({ω }) , (2)
and make a partial decision composed of the non dominated singletons according to relation [START_REF] Denoeux | A k-nearest neighbor classification rule based on Dempster-Shafer theory[END_REF]. Due to lack of space, details cannot be written, but Equation (2) comes from for example [6][Equation 43] or [START_REF] Ma | Partial classification in the belief function framework[END_REF][Page 6, Strong dominance criterion] with 0 -1 utilities and pieces of information represented by belief functions.

Contextual corrections and learning

In this section, the definitions of CD, CR and CN are recalled as well as the possibilities to learn them from labelled data [START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF] composed of the outputs of a source regarding objects whose true class is known.

The contour functions resulting from CD, CR and CN are recalled with K parameters, K being the number of elements in Ω. These mechanisms can indeed be used with more parameters [START_REF] Mercier | Refined Modeling of Sensor Reliability in the Belief Function Framework Using Contextual Discounting[END_REF][START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF] but as shown in [START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF], these configurations with K parameters for each corrections are expressive enough to reach the lowest possible values (with these mechanisms) of the following measure of discrepancy [START_REF] Mercier | Refined Modeling of Sensor Reliability in the Belief Function Framework Using Contextual Discounting[END_REF][START_REF] Pichon | Proposition and learning of some belief function contextual correction mechanisms[END_REF] between the source outputs adjusted with these corrections and the ground truth:

E pl (β) = n i=1 K k=1 (pl i (ω k ) -δ i,k ) 2 , ( 3
)
where n is the number of objects in the learning set, β = (β ω , ω ∈ Ω) is the vector composed of the K parameters of each correction, pl i is the contour function regarding the class of the object i resulting from a contextual correction (CD, CR or CN) of the MF provided by the source for this object, and δ i,k is the indicator function of the truth of all the instances i ∈ {1, . . . , n}, meaning

δ i,k = 1 if the class of the instance i is ω k , otherwise δ i,k = 0.
The discrepancy measure E pl yields a linear least-squares optimization problem, which can be then efficiently solved using standard algorithms.

As for the discounting (1), we consider a source of information outputting a MF m S regarding a question of interest. The corresponding contour functions of each contextual correction of m S are summed up in Table 1.

(ω) = 1 -(1 -plS(ω))βω CR pl(ω) = plS(ω)βω CN pl(ω) = 0.5 + (plS(ω) -0.5)(2βω -1)

Contextual corrections depending on partial decisions

In this paper, we propose to improve the previous learning method exposed in Section 3, meaning we would like to reach better performances. For this, we will consider, in the learning set, groups of distinct partial decisions the outputs of the source lead to.

The idea is to consider that the quality of the source, and then the way we have to adjust it, may depend on the outputs it gives regarding the objects whose true classes are to be found. For example, the source may be quite right when it decides a certain type of class, while having some bias when it decides another class or a group of classes, and another bias for another class or group of classes, etc.

Specifically we investigate the quality of the source depending on groups of partial decisions according to relation [START_REF] Denoeux | A k-nearest neighbor classification rule based on Dempster-Shafer theory[END_REF]. CD, CR and CN best parameters β according to E pl (3) are then computed in each group. The correction with the lowest value of E pl is kept in each group.

This learning procedure is summarized by Algorithm 1.

Numerical Experiments and Discussion

In this Section we present experiments made with the EkNN classifier [START_REF] Denoeux | A k-nearest neighbor classification rule based on Dempster-Shafer theory[END_REF][START_REF] Denoeux | Evclass: Evidential distance-based classification[END_REF] with k = 5 as the source of information on synthetic and real data.

Algorithm 1 Learning procedure

Input: A set I of instances (mi{oi}, ωi), i ∈ {1, . . . , n}, mi{oi} being the output of the source regarding object oi whose true class is ωi.

Outputs: Groups G of partial decisions and best corrections for each group. 1: procedure LearningProcedure 2: G initially empty 3:

for each instance i in I do 4:

Compute the partial decision coming from mi using relation (2) 5:

Add i to the group g in G associated with this partial decision. 6:

for each group of partial decisions g in G do 7:

Compute CD, CR and CN best parameters β according to E pl (3) restricted to the instances in this group 8:

Keep for this group the correction reaching the lowest value of E pl .

Synthetic data are composed of 5000 instances, 5 classes and 2 features, which were generated from a multivariate normal distribution with means µ 1 = (0, 0), µ 2 = (2, 0), µ 3 = (0.2), µ 4 = (2, 2), µ 5 = (1, 1) for respectively class 1, 2, 3, 4 and 5, and the same covariance matrix Σ for each class: Σ = 1 0.9 0.9

An illustration of a generated synthetic data set with these parameters is given in Figure 1.

The real data sets used from UCI [START_REF] Dua | UCI Machine Learning Repository[END_REF] are described in Table 2. For each data set, the following experiment is repeated 10 times: one half of the data (L 1 ) is used to learn the EkNN classifier; then a 10-fold cross validation is performed on the second half of the data with 9 folds (L 2 ) to learn the best correction in each group of partial decisions using Algorithm 1, and 1 fold for testing (meaning for the test phase).

Some possible partial decisions obtained from the outputs of the source (the EkNN classifier) with the generated data set illustrated in Figure 1 are given in Figure 2 for some points in the feature space. Note that these groups of partial decisions are not necessary computed in the training phase L 2 . They are just given here as an illustration. As exposed in Algorithm 1, only the points (and their associated partial decisions) in the fold L 2 are considered to learn the corrections. That is why it may happen that during the test phase, a partial decision did not happen in the learning phase. In this situation, we propose to use the best correction for the whole set of L 2 . Note that in our experiments presented here, this case very rarely occurred.

To measure the performances, we used the measure E pl (3) with which corrections are learnt, and we also wanted to use another measure with which corrections are not learnt. For this second choice, we opted for the u 65 utility measure, introduced by Zaffalon et al. [START_REF] Zafallon | Evaluating credal classifiers by utilitydiscounted predictive accuracy[END_REF], which allows one to take into account the advantages of partial decisions concerning the fact of preferring imprecision to being randomly correct. The u 65 utility measure gives indeed a greater utility to imprecise but correct partial decisions of size n (meaning decisions equal to a set of n singletons one of them being the true class) than precise decisions (in favor of one singleton) only randomly correct with probability 1 n . Formally, the U 65 value of a partial decision d, possibly in favor a set of singletons, is defined by

U 65 (x) = 1.6x -0.6x 2 (5)
with x the so called discounted accuracy of d defined by I(ω∈d) |d| , with I the indicator function, ω the true class of the instance, and |d| the number of elements in d.

Performances according to E pl (3) and the averages of the u 65 utility measures [START_REF] Zafallon | Evaluating credal classifiers by utilitydiscounted predictive accuracy[END_REF] of the partial decisions obtained from the corrected outputs are regrouped in Table 3. As it can be seen in Table 3, the proposed method using the groups of partial decisions obtains almost in each situation better results than the previous learning using only CD or CR or CN.

Furthermore, the source was supposed to be a black box, but we were also curious to see the performances of this source if we were able to improve its performances by using the data we use to learn the corrections. These performances are given in the column EkN N + in Table 3. We can see that the new learning method succeeds to even improve these results for some data sets.

As summarized in [START_REF] Ma | Partial classification in the belief function framework[END_REF] by Ma and Denoeux, relation ( 2) is only one among others. First future works will then consist in testing these other possible relations (s.t. weak dominance, maximality, . . .) to compute the groups of partial decisions and see if better performances can be reached.

Fig. 1 :

 1 Fig. 1: Example of a generated data set.

Fig. 2 :

 2 Fig. 2: Points in the feature space belonging to different groups of partial decisions obtained from the generated synthetic data set composed of 5 classes and 2 attributes/features illustrated in Figure 1. The legend for the partial decisions obtained from the EkNN classifier and relation (2) are given above the figure. We can see that for these points, 10 possible partial decisions have been obtained: 1 (meaning a decision for class 1), 2, 3, 4, 5, 14 (meaning a partial decision in favor of class 1 or class 4), 15, 45, 145 (meaning a partial decision in favor of class 1 or class 4 or class 5), 12345 (meaning a partial decision in favor of all the classes -total uncertainty -each class can be the good one).

Table 1 :

 1 Contour functions of each contextual correction of a MF mS given for any ω ∈ Ω. Each parameter βω may vary in [0, 1].

	Corrections Contour functions
	CD	pl

Table 2 :

 2 Descriptions of used UCI data sets.

	Data sets	# instances # features # classes
	Breast cancer	569	31	2
	Glass	214	10	6
	Haberman	306	3	2
	Ionosphere	350	34	2
	Iris	150	4	3
	Liver	345	6	2
	Lymphography	146	18	3
	Pima	768	8	2
	Red wine	1599	12	6
	Sonar	208	60	2
	Transfusion	748	3	2
	Vehicles	846	19	4
	Vertebral	310	6	3

Table 3 :

 3 Performances (Average E pl values, the lower the better, and average U65 utilities, the greater the better) obtained for EkNN learnt with L1 (denoted by EkNN), EkNN learnt with groups L1 and L2 (denoted by EkNN+), EkNN learnt with L1 followed by corrections (CD, CR, CN) learnt with L2, and EkNN learnt with L1 followed by the new method using groups of partial decisions learnt with L2. Standard deviations are indicated in parenthesis. Vertebral 80.40 (10.14) 82.09 (8.85) 80.29 (10.04) 79.95 (10.27) 80.36 (10.19) 81.78 (9.40)

	Data	EkNN	EkNN+	CD	CR	CN	Method
	Average Epl values (the lower the better)
	Synthetic	188.27 (26.48) 181.27 (16.23) 187.91 (26.56) 134.57 (9.83) 174.06 (21.38) 106.76 (7.29)
	Breast cancer 3.74 (2.50)	3.59 (2.43)	3.65 (2.35)	3.64 (2.34) 3.56 (2.22)	3.56 2.21)
	Glass	8.16 (3.61)	8.97 (3.19)	8.13 (3.57)	5.11 (1.40) 6.93 (2.04)	4.81 (1.49)
	Haberman	8.59 (2.40)	8.79 (2.34)	8.57 (2.32)	5.97 (1.84) 6.61 (1.07)	5.87 (1.83)
	Ionosphere	2.31 (1.66)	1.64 (1.18) 2.31 (1.62)	2.19 (1.49) 2.25 (1.52)	1.81 (1.49)
	Iris	0.56 (0.87)	0.59 (1.02)	0.56 (0.85)	0.55 (0.82) 0.56 (0.84)	0.56 (0.86)
	Lymphography 3.07 (1.44)	2.32 (1.27) 3.08 (1.44)	2.74 (1.24) 3.02 (1.18)	2.74 (1.40)
	Liver	12.66 (2.08) 12.21 (1.56)	12.66 (2.02) 8.17 (1.08) 8.31 (0.64)	8.01 (1.12)
	Pima	20.01 (3.32) 18.12 (3.21)	20.01 (3.32) 15.67 (2.20) 16.76 (1.57) 14.93 (2.40)
	Red wine	162.09 (21.75) 140.60 (13.98) 162.09 (21.75) 48.62 (2.96) 116.83 (4.28) 45.92 (3.44)
	Sonar	4.04 (1.43)	3.06 (1.33) 4.05 (1.43)	3.50 (1.08) 3.80 (1.05)	3.30 (1.37)
	Transfusion	20.54 (4.91) 19.77 (4.37)	19.43 (4.19) 15.21 (3.45) 15.44 (2.21) 13.56 (2.85)
	Vehicles	34.26 (5.66) 28.14 (4.74)	34.26 (5.66) 24.07 (2.35) 32.40 (4.20) 20.69 (2.96)
	Vertebral	5.08 (2.29)	4.68 (2.05)	5.04 (2.19)	4.64 (1.89) 4.85 (1.89)	4.31 (1.81)
	Average U65 values (the greater the better)
	Synthetic	66.52 (2.68) 66.40 (2.63)	66.49 (2.73) 66.10 (2.69) 65.46 (2.89) 68.36 (2.59)
	Breast cancer 91.77 (4.65) 93.74 (3.90) 92.76 (4.60) 92.75 (4.64) 92.80 (4.62) 92.76 (4.60)
	Glass	61.51 (14.47) 63.38 (13.44) 61.56 (14.25) 64.53 (13.87) 45.93 (13.22) 66.24 (12.35)
	Haberman	74.28 (8.12) 74.45 (8.57)	74.61 (7.97) 74.84 (9.80) 71.69 (8.04) 75.13 (9.26)
	Ionosphere	93.02 (4.67) 94.71 (4.16) 93.02 (4.67) 93.12 (4.62) 92.98 (4.64) 93.86 (4.99)
	Iris	96.78 (6.78) 96.51 (6.07)	96.78 (6.78) 96.84 (6.72) 96.78 (6.78) 97.06 (6.20)
	Liver	66.74 (5.86) 67.14 (5.53) 66.73 (5.86) 66.64 (6.82) 64.70 (2.25) 65.45 (7.42)
	Lymphography 80.23 (13.13) 83.84 (12.65) 80.21 (13.20) 79.42 (13.65) 79.26 (12.26) 79.11 (14.49)
	Pima	72.64 (5.06) 73.00 (5.56)	72.64 (5.06) 73.34 (5.50) 71.97 (3.91) 73.38 (5.02)
	Red wine	45.39 (4.17) 53.29 (4.23)	45.39 (4.17) 57.60 (4.14) 25.00 (0)	59.12 (4.12)
	Sonar	78.34 (8.53) 84.50 (7.69) 78.27 (8.60) 78.96 (8.95) 78.00 (8.03) 79.32 (10.49)
	Transfusion	72.76 (5.68) 72.87 (5.41)	74.20 (5.29) 74.39 (6.38) 72.63 (5.42) 75.91 (6.03)
	Vehicles	61.20 (5.77) 63.46 (5.62)	61.20 (5.77) 60.56 (5.65) 58.11 (5.60) 63.61 (6.20)

Acknowledgements The authors would like to thank the anonymous reviewers for their constructive comments and suggestions that helped them to clarify parts of this paper and will help them for future researches.


Geir Ringen
email: geir.ringen@ntnu.no

Halvor Holtskog

Torgeir Welo

A novel value driven co-creation framework

Keywords: Innovation model, co-creation, aluminium product case study

The Environmental awareness is growing throughout the society and the pressure on manufacturing companies, and their extended value chain, to operate in a more sustainable way. Designing products that can be controlled throughout their value chain and life cycle is one option to secure reuse of the material stream. We hypothesize that increased collaboration and common understanding among industrial stakeholders will promote improved products with regards to sustainability criteria. To test our hypothesizes a model for cross-industry and cross industry and academia collaboration is outlined. This model emphasizes three distinct levels of collaboration, team, company and cluster, and the life cycle stages of a product. Understanding these levels and stages are important not only for assuring correct and rational behavior among actors but also for facilitating a long-term sustainable business model development. This article presents the theories and arguments behind the model and initial results from an aluminum product case. Further verification and calibration of the model will be done by adding case studies as input to the collaboration model.

Introduction

Increasing pressure to develop products of higher quality, with added functionality, at a lower cost, and in shorter time frames unquestionably brings about some dichotomies. The examples of high quality vs. low cost, less resources and time vs. higher performance, and increased robustness vs. lower weight all illustrate well known contradictions which become more and more important to optimize. Only companies that can manage such conflicting objectives and in an adaptive manner consistently and timely bring new and innovative products to market will be regarded as long-term partners. In this regard, Utterback [START_REF] Utterback | Mastering the Dynamics of Innovation[END_REF] claimed that the main challenge is to develop the ability to innovate products, processes, and the organization, seeing them dependent of each other as a whole. To this overall picture, companies must comply to requirements and expectations regarding sustainability criteria as well as increase their understanding and ability to utilize enabling technologies for improving product lifecycle performance. Product development capability [START_REF] Stalk | Competing on Capabilities: The New Rules of Corporate Strategy[END_REF], the ability to use and integrate existing organizational and inter-organizational competences, is seen as fundamental to introducing a successful new product. It is argued that success is especially challenged by technology changes and global competition, meaning that product development capability must contain features beyond competence utilization. For instance, Barney [START_REF] Barney | Firm Resources and Sustained Competitive Advantage[END_REF] emphasized that such capability must be valuable, rare, and imperfectly imitable. Others start from the firm level [START_REF] Prahalad | The Core Competence of the Corporation[END_REF] and see resource utilization as a driving force behind successful product development and competitive advantage. A third stream of literature, with contributors like Nelson and Winter [START_REF] Nelson | An Evolutionary Theory of Economic Change[END_REF], concluded that technological and organizational progress is driven by mechanisms of variation, selection, and retention. Hence, this evolutionary perspective focuses heavily on continuous innovation and mastering of change.

The coming decade is expected to become a crucial period for industrial companies as they more frequently have to respond to global challenges. Environmental awareness is growing throughout the society and the pressure on manufacturing companies to operate in a more sustainable way is increasing. Having control of, and insight into, the entire value chain is becoming a significant part of running a business [START_REF] Hollander | Product Design in a Circular Economy: Development of a Typology of Key Concepts and Terms[END_REF]. However, for many manufacturing companies, that are not distributing their products themselves, knowledge of the complete life cycle of the products is limited. Furthermore, insights into users and end of life stages are particularly restricted. Understanding these stages are important not only for assuring correct behavior but also for facilitating a long-term sustainable business development. On the other side, the amount of available technologies for enabling information tracking throughout the product life cycle is increasing.

Many manufacturing companies do not have control over the distribution chain, and find it difficult to obtain complete information on use and reuse [START_REF] Gmelin | Determinants of a sustainable new product development[END_REF]. In many value chains, information exchange is very limited due to lack of interest of sharing knowledge or understanding of what information that is important. The point of departure for industrial design is the end user, on a general level or through deep insights. Insights can be achieved through interviews, observations, experiences or co-design where design for sustainable behaviour aims at supporting greener choices by buyers and users of products.

This paper seeks to initiate a framework for cross-collaboration co-creation strategies towards improved product development processes for a more circular economy.

Theory

Innovation today is about bringing together different actors at different levels, ranging from the individual, to the group, organization and inter-organizational levels. The concept of organizational learning is therefore dependent on those groups and teams that are able to learn; if they cannot, the organization cannot learn [START_REF] Pawlowsky | The Treatment of Organizational Learning in Management Science[END_REF]. This view is supported by Senge [START_REF] Senge | The Fifth Discipline: The Art & Practice of The Learning Organization[END_REF], who stated, "team learning is vital because teams, not individuals, are the fundamental learning units in modern organizations." Groups are social systems in which sharing, learning, and organizational behavior take place, and they can play a central role in both supporting individual learning and opening new opportunities for interventions. Individual, group and departmental learning are thoroughly discussed in the previous chapters, so the main body of this chapter is concerned with the organizational and the inter-organizational level. In a complex world system are developed to reduce complexity, and trust may serve as a building block in creating these systems. Luhmann [START_REF] Luhman | Trust and Power: Two Works by Niklas Luhmann[END_REF] regarded trust as reduction of complexity; he saw trust as a function for establishing and maintaining a complex world. He also argued for trust as an input condition to abstract systems, stating that "without trust it cannot stimulate supportive activities in situations of uncertainty or risk." Giddens [START_REF] Giddens | The Consequences of Modernity[END_REF] defined trust as "confidence in the reliability of a person or system, regarding a given set of outcomes or events, where that confidence expresses a faith in the probity or love of another, or in the correctness of abstract principles and technical knowledge." This means he believed that the result of actions is not random, but is directly linked to the characteristics of the system. As discussed above, the automotive industry has clear characteristics described in the transaction cost economic. By showing trust Giddens [START_REF] Giddens | The Consequences of Modernity[END_REF] argued for a mild form of hostage taking or being called upon. The trusted party will then feel an obligation to act in a certain way, and the will be revised during the time of the relationship. In such relations based on trust lies an idea that humans create risk and that this risk cannot be eliminated, only reduced. This is done in a rational and conscious way when the risk can be calculated.

Bowersox et al. [START_REF] Bowersox | 21st Century Logistics: Making Supply Chain Integration a Reality[END_REF] defined supply chains as value-adding relations of partially discrete yet interdependent units that cooperatively transform raw materials into finished products through sequential, parallel, and/or network structures. The supply chain management (SCM) literature analyses how firms are affected by their participation in global production networks [START_REF] Dicken | Global Shift. Mapping the Changing Contours of the World Economy[END_REF], and it views supply chains as networks of organisations that are involved in different ways to produce value in the form of products and services in the hands of the ultimate consumer [START_REF] Christopher | Logistics and Supply Chain Management. Creating Value-Adding Networks[END_REF].

The key technologies developed in the early and mid-20 th century were developed primarily by large diversified enterprises. This diversification, along with vertical integration, provided these firms with a competitive advantage over smaller competitors. Realizing the increased knowledge mobility and availability of venture capital to capitalize on knowledge, Chesbrough and Vanhaverbeke concluded that the traditional model had reached its limits [START_REF] Chesbrough | Open Innovation: Researching a New Paradigm[END_REF], and suggested open innovation means a break with the traditional paradigm.

From this Chesbrough and Vanhaverbeke [START_REF] Chesbrough | Open Innovation: Researching a New Paradigm[END_REF] derived the following definition of open innovation: "Open innovation is the use of purposive inflows and outflows of knowledge to accelerate internal innovation, and expand the markets for external use of innovation." The increased global flow of goods, services, knowledge and capital implies that firms increasingly obtain ideas and knowledge from distributed knowledge networks and open innovation [START_REF] Chesbrough | Open Innovation: The New Imperative for Creating and Profiting from Technology[END_REF] in addition to their internal knowledge base. This transition from stable conditions, often in local and national markets, to global competition and more tailor made products has also increased the interest in innovation and organizational learning [START_REF]Hvordan Organisasjonen Fungerer[END_REF]. Tidd et al. [START_REF] Tidd | Managing Innovation: Integrating Technological, Market and Organizational Change[END_REF] asserted that innovation networks have become popular as a means of innovation across organizations. Such networks may be viewed as a new hybrid form of organization that has the potential to replace both firms and markets and create a kind of a "virtual corporation". The model of open innovation by Chesbrough [START_REF] Chesbrough | Open Innovation: The New Imperative for Creating and Profiting from Technology[END_REF] is based on the same idea: to have an open flow of ideas and technology between the firm and its environment. He focused on the notion that good ideas are worthless without capital and a superior business model, saying that firms can and should use both external and internal ideas to create value and internal and external paths to market [START_REF] Chesbrough | Open Innovation: Researching a New Paradigm[END_REF]. This concept draws on for instance the theories of absorptive capacity and spillovers of industrial R&D.

Network ties often reflect formal collaboration, but the knowledge flow, which is the main focus, may be separated into formal and informal ties. The former reflects planned channels for knowledge exchange, whereas the informal ties are viewed as more interesting and valuable due to unforeseen knowledge opportunities [START_REF] Chesbrough | Open Innovation: Researching a New Paradigm[END_REF]. In this perspective the open innovation theory turns into more of a networking perspective, where the competitive advantage can be found in the community of companies rather than in the individual company. On the other hand Lorenz and Valeyre [START_REF] Lorenz | Organizational Forms and Innovative Performance: A Comparison of the EU-15[END_REF] asserted that innovation activities still critically depend on factors internal to the firm, in particular on the job training related to solving technical and production related problems. Another perspective of network innovation models is co-opetition, where collaboration and competition are not mutually exclusive, but can actually coexist and create benefits from joint dynamics among competitors [START_REF] Ritala | Introduction to coopetition and innovation: Contemporary topics and future research opportunities[END_REF]. Evans et al. discuss the advances of sustainable business models, which take into account the extended responsibility companies has for their products and services in value chain and life cycle perspective [START_REF] Evans | Business Model Innovation for Sustainability: Towards a Unified Perspective for Creation of Sustainable Business Models[END_REF].

Method

This study is part of the Norwegian Research Council funded project VALUE, reporting upon 6 in-depth interviews done in one of the application cases. The interviews were recorded and transcribed, and the informants read and approved the transcriptions. This formed the backbone of the case study.

The innovation model, hypothesized to support rapid innovation in the context of developing more complex products with increasing number of interfaces both physically and virtual, is illustrated in Figure 1. There are identified three important, and connected, intra-and inter-organizational levels where learning takes place [START_REF] Rothaermel | Building Dynamic Capabilities: Innovation Driven by Individual-, Firm-, and Network-Level Effects[END_REF]. The first one is at the team level where a group of individuals are dedicated towards closing knowledge gaps related to product features. The next level emphasizes how knowledge at group level relates to the whole organization. This can be horizontally and vertically integrated knowledge, where the particular product innovation is put into a business context and where knowledge accumulates through tacit and explicit channels. The third level involves inter-organizational learning, and how the network of the firm can be involved to speed up open innovation processes. We will in the following project this model through a real innovation case, and discuss how feedback-loops from the different learning levels can improve the next innovation process according to product lifecycle performance.

Results of application case

The case company develops and manufactures critical light-weight safety parts for the automotive industry and has been doing that for more than 50 years. More than 10 million components are supplied to demanding customers annually. However, the electrification trend in the transport sector spurs the need for increasing light-weight solutions to compensate for added total vehicle weight due to batteries. They see that the automotive industry is changing. As a result of the United Nations goal of stopping global warming to a maximum of 2°C increase, heavy regulations have been, and will continue to be implemented to reduce emissions in the transportation industry. This has led to a dramatic and rapid change in the automotive industry towards low and zero emission vehicles. The electrification of passenger vehicles is arguably the biggest shift in the automotive industry since Henry Ford introduced the assembly line in the early 20 th century. The push from the governments are coupled with a few pioneering manufacturers which has demonstrated that the technology is ready to be used and it is possible to produce desirable vehicles which is in demand from customers. Over the past years, especially over the last year, there has been a continuous spate of announcements from car-manufacturers declaring their intensions and plans to electrify their product portfolio. Some have even pledged to only produce electric vehicles, whether they are pure electric or electrified, within the next decade. The change to electric drivetrains comprises new vehicle architecture as traditional requirements for engine, gearbox, driveshafts and fuel tanks are redundant in electric vehicles. These would be replaced by smaller, more compact electric motors coupled with a large battery. As these types of vehicles are weight-critical, the current and future market pull creates new opportunities for suppliers of light-weight materials such as aluminium to be put to use in new areas and products. In this connection, one of the product categories with the larger potential is aluminium battery trays. For the case company battery trays are seen as an interesting and complementary product to the existing product portfolio, where deep knowledge about utilizing and forming aluminium may be a steppingstone towards further growth. However, the case company realises that material and process knowledge derived from the existing markets and product portfolios are not sufficient to take on the assumed complexity of making battery trays. Some of the identified requirements affecting the battery tray are given below:

• The primary task of the battery housing is to protect and enclose the energy required to propel the car. • The battery cells are temperature sensitive, meaning the efficiency of the battery cells are dependent on operating in a temperature range. If the temperature is too high, the capacity will decrease along aging of the battery. If the temperature is too low, the internal resistance in the cells increase and performance decrease. • Protection from moisture, rain, dust and particles is important for the operation and lifetime for the batteries. Therefore, all battery housings are enclosed in one way or another using sealants, high internal air pressure or membranes. • Safety requirements: As most battery packs are substantial in size and weight, they have significant effects on the crash structure of the car. There are also official requirements for transporting lithium-ion batteries. • It is desirable to have a lightweight battery housing, as extra weight decreases the efficiency of the car and shortens the driving range. • The battery housing is not to be a source of vibration, which will affect the safety of the batteries and the comfort of the passengers. • The battery housing is a highly customized component since it has to incorporate and satisfy designs that comply with the defined size and shape of space in the vehicle, as well as the mounting points to the vehicle, and it must incorporate all the internal components to fit inside the housing. At the project team level the new product concept, battery tray, was seen as innovative and interesting, although the potential manufacturing processes to be selected for realizing the product were known from before. The potential risks and rewards were more related to the system complexity, where number of sub-components must be assembled in a cost-effective way. Historically the company was used to do welding, a process now done in other parts of the global production network. The team was also discussing the possibility of just developing a product to be produced elsewhere due to assumed high value chain costs of transporting systems to the customer from remote production sites, and the proposed process of do a lot of welding for final assembly. An interesting finding was that this realization of just being part of the initial development phase, spur some ideation among the local team. A novel tooling approach for extruding aluminium profiles was developed to make innovative profile designs, aiming at reducing volume to be transported (50%), reduce welding (50-60%), and increase the panel width (250%). This is one of many initiatives taken at team level, impacting the maturity of a future product concept.

At the level of organizational learning the company has a long history of producing high volume components to demanding customers, controlling the aluminium value chain from casting, extrusion, forming and to final assembly. The combination of long history, value chain control, and a stable workforce at local and regional level over time, lays a good foundation for identifying, transferring, and utilizing knowledge. However, some dynamics in ownership since mid-1990s have influenced this stock of knowledge, especially in terms of customer proximity and how the is organized towards customer satisfaction. Currently, the company is part of large multinational company with a strong foothold in global value chainssuch also for the automotive segment. This new way of organizing the local manufacturer gives better access, and a stronger tie, to key customers, but also a potential longer way for information and knowledge to travel from the source to those who are in charge of developing new solutions. However, the accumulated knowledge in terms of an innovative battery tray was presented at the Shanghai Motor Show in 2017.

At the inter-organizational learning added complexity in the product interfaces required a more open innovation process, including a network of academia and industrial partners to create feasible solutions. From a business perspective the case company early identified sets of risk related to be in the value chain of supplying battery trays. First, immature industry standards and fast developing technology are key factors to be carefully evaluated before going into a new product/system segment. Second, technology risk was noted as high due to consequences of not being able to safely protect the batteries under all possible conditions. Third; competence capability is seen as crucial to develop a sustainable solution for an emerging and demanding market. Actions taken to reduce risk were comprehensive. But, being in the business of development and manufacturing of critical aluminium safety parts for decades, one of the most trusted customers was accompanied to form a joint development project. Here risks and opportunities were identified and broken down to small tasks, where also external potential knowledge actors were mapped according to capability and capacity. Examples of subtasks are; selection of semi-finished goods, welding techniques, calibration for dimensional accuracy, crash impact, weight optimization etc. Academic partners were selected as contributors to these sub-tasks, but the co-creation intensity and level differentiates depending on actor strength within the field of expertise. On the business side, a broad network of strategic partners was selected to develop a modular and scalable solution for light-weight e-mobility. This solution was ready for launch in 2019.

Conclusion

This paper seeks to illustrate the complexity of bringing new innovations to the market, where all levels of the organization, and it's extended network, have to be mobilized in order to close the knowledge gaps. Innovation theory has in recent years developed towards emphasizing cross-boundary and inter-organizational innovation, seeing the value of bringing a broad array of views into consideration when the landscape of developing new products and services become more complex. However, this paper supports the importance of being open, but without forgetting the core of the company. The latter is exemplified by the innovation capability of local engineering teams, combining extrusion and forming skills to prototype entire new solutions. At the organizational level, stating time dependent objectives for closing critical knowledge gaps turned important for urging the organization. Involving academic and strategic business partners is also viewed as important when newness is defined along all the dimensions: market, process and product. We propose an innovation model that integrates this local team innovation capability with the organizational learning and inter-organizational dimensions. Playing out these three capabilities simultaneously when heading for step-change innovation in in-mature markets is valuable when doing it carefully and with awareness. This is the key finding from this paper. We will continuously develop this innovation model based on additional cases.

Figure 1 .

 1 Figure 1. Innovation model.