Daniel Baguet
email: daniel.baguet@utinam.cnrs.fr

Alessandro Morbidelli

Jean-Marc C Petit

J.-M Petit

Positions of the secular resonances in the primordial Kuiper Belt disk

Keywords: celestial mechanics, Kuiper Belt, secular resonance

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Introduction

The Kuiper Belt is the residual disk of a more massive primordial planetesimal disk. The Kuiper Belt objects, in their orbital structure, keep a trace of the past dynamical evolution of the giant planets. Studying their dynamics is therefore important to acquire a better understanding of the dynamical evolution of the giant planets during the life of the Solar System.

It has been shown that before the dissipation of the protoplanetary gas disk, the giant planets should have followed an inward migration until they got locked in a multiresonant configuration where each planet is in mean motion resonance (MMR) with the others [START_REF] Masset | Reversing type II migration: resonance trapping of a lighter giant protoplanet[END_REF], [START_REF] Morbidelli | Dynamics of the Giant Planets of the Solar System in the Gaseous Protoplanetary Disk and Their Relationship to the Current Orbital Architecture[END_REF]), with a preference for the configuration where Jupiter and Saturn are in a 3:2 MMR because it allows their gaps in the protoplanetary gas disk to overlap. This overlap of the gaps is needed to explain why our solar system does not contain a hot Jupiter, because it refrains Jupiter from migrating close to the Sun.

From this multiresonant configuration, the giant planets, by interacting with the planetesimal disk, were driven in a migration phase until they reached their current positions and during which a dynamical instability among them occurred. This instability, which has been proposed in the framework of the Nice model [START_REF] Tsiganis | Origin of the orbital architecture of the giant planets of the Solar System[END_REF], [START_REF] Morbidelli | Chaotic capture of Jupiter's Trojan asteroids in the early Solar System[END_REF], [START_REF] Gomes | Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets[END_REF] and the planetesimal-driven migration allow to reproduce many of the orbital features of the Kuiper Belt [START_REF] Nesvorný | Dynamical Evolution of the Early Solar System[END_REF].

In order to explain the difference in color, size distribution and binary fraction between the dynamically hot and the dynamically cold populations in the classical Kuiper Belt, it has been suggested that the hot population formed in a region between Neptune and 30 AU, from a primordial massive planetesimal disk of mass in the range ∼10 -60 M ⊕ [START_REF] Gomes | The origin of the Kuiper Belt high-inclination population[END_REF], [START_REF] Levison | Origin of the structure of the Kuiper belt during a dynamical instability in the orbits of Uranus and Neptune[END_REF], [START_REF] Nesvorný | Neptune's Orbital Migration Was Grainy, Not Smooth[END_REF]). The outer edge of this disk is placed at 30 AU because it allows to stop at this location the planetesimal-driven migration of Neptune which corresponds to its current semi-major axis [START_REF] Gomes | Planetary migration in a planetesimal disk: why did Neptune stop at 30 AU?[END_REF]. In other respects, the cold classical population is probably the remnant of a lighter planetesimal disk formed in-situ (Nesvorný, 2015b). Besides, numerical integrations that tried to reproduce the cold Kuiper Belt from a single massive disk extending from the primordial position of Neptune to 50 AU did not succeed to reach a sufficient rate of mass depletion in the region of the current cold Kuiper Belt [START_REF] Gomes | Checking the compatibility of the cold Kuiper belt with a planetary instability migration model[END_REF]. However, it is still unclear where the inner and the outer edge of this last one were located and why there is this sharp transition between the massive planetesimal disk and the light planetesimal disk.

In the rest of this paper, we will call the massive disk (MD) the part of the planetesimal disk between Neptune and 30 AU, and the light disk (LD) the part of the planetesimal disk beyond 30 AU.

The cold classical Kuiper Belt allows us to put strong constraints on the dynamical evolution of the giant planets and in particular Neptune [START_REF] Batygin | Retention of a Primordial Cold Classical Kuiper Belt in an Instability-Driven Model of Solar System Formation[END_REF], [START_REF] Dawson | Neptune's Wild Days: Constraints from the Eccentricity Distribution of the Classical Kuiper Belt[END_REF]), because the latter must migrate in such a way that the hot population forms from the MD while keeping the LD located in the region corresponding to the current classical Kuiper Belt cold enough by not exciting it. Given that the MD has a mass comparable to that of an ice giant planet, it can have a significant influence on the dynamics of an object in the LD and in particular, it will bring a perturbation on the apsidal and nodal precessions of the orbits of the planets and of the small bodies. This can have an important effect in determining the positions of the secular resonances. The latter occur where the frequency of the free apsidal or nodal precessions of a small body, depending on its semi-major axis, equals one of the eigenfrequencies of the apsidal or nodal precessions of the planetary system, respectively. [START_REF] Batygin | Retention of a Primordial Cold Classical Kuiper Belt in an Instability-Driven Model of Solar System Formation[END_REF] did numerical integrations where they follow the evolution of the MD and of the LD. In their paper, it is said that the presence of the massive Kuiper Belt enhanced the free precession of the Kuiper Belt objects, so it suggests that the MD objects act on the LD objects. However, they did their numerical integrations by using the Mercury6 integration software package. Yet, in the standard version of Mercury6 the planetesimals of the disk do not interact with each other. Therefore, it is likely that in their numerical integrations the MD does not directly perturb the LD objects but that the latter see only the indirect effect through the action of the planets. On the other hand, [START_REF] Dawson | Neptune's Wild Days: Constraints from the Eccentricity Distribution of the Classical Kuiper Belt[END_REF] analytically explored the excitation of the eccentricities of the LD objects caused by Neptune alone but where the other giant planets indirectly act on the LD objects through their influence on the apsidal precession of Neptune. However, they did not include the contributions of the other giant planets and of the MD to the apsidal precession of the LD objects. In their numerical study, where only Neptune is present, they included fictitious forces on the ice giant so as to reproduce its radial migration and its eccentricity damping under the effect of the interactions with the planetesimal disk as well as its apsidal precession. The MD itself is represented by a set of massless particles and does not act on the LD objects. Nesvorný (2015b) included fictitious forces on the giant planets so as to reproduce their radial migrations and their eccentricity and inclination damping but the MD is not present so it does not directly affect the apsidal and nodal precessions of the giant planets and of the LD objects Thus, in this work we investigate the positions of the secular resonances by considering both the effects of the MD on the giant planets and on the LD objects. In order to do so, we represent the MD by an axisymmetric continuous thick disk and we take into account its contribution to the apsidal and nodal precession frequencies in the linear secular theory.

Studies of the positions of the secular resonances and of their effects in the

Kuiper Belt under the perturbation of the gravitational potential of a disk have already been made by [START_REF] Nagasawa | Sweeping Secular Resonances in the Kuiper Belt Caused by Depletion of the Solar Nebula[END_REF] and [START_REF] Li | The Effect of Sweeping Secular Resonances on the Classical Kuiper Belt[END_REF]. They showed that the locations of the secular resonances due to the giant planets are shifted when they account for the solar nebula. They modeled the latter based on the minimum mass solar nebula of [START_REF] Hayashi | Structure of the Solar Nebula, Growth and Decay of Magnetic Fields and Effects of Magnetic and Turbulent Viscosities on the Nebula[END_REF]. Then, the depletion of the nebula causes the sweeping of the secular resonances in the Kuiper Belt.

With the current orbital architecture of the giant planets, they showed that if the exponential depletion timescale is 10 7 yr, the sweeping secular resonances are efficient in exciting the eccentricities and inclinations of the Kuiper Belt objects at the observational level, provided that the inclination between the invariant plane of the planets and the midplane of the solar nebula is high enough.

Moreover, with a more compact orbital architecture of the giant planets, they showed that the exponential depletion timescale has to be 10 8 yr. However, the lifetime of gas nebulae around stars is of the order of 10 6 -10 7 yr [START_REF] Haisch | Disk Frequencies and Lifetimes in Young Clusters[END_REF], [START_REF] Mamajek | Initial Conditions of Planet Formation: Lifetimes of Primordial Disks[END_REF]). Thus, with a compact orbital configuration, this mechanism cannot be sufficient to rise the eccentricities and inclinations of the Kuiper Belt objects. Besides, the mass distribution of the primordial planetesimal disk is not the same as the mass distribution of the solar nebula and the former is much less massive than the latter, while still as massive as the ice giant planets. Hence, we think that it is worthwhile to explore the positions of the secular resonances caused by the gravitational effects of the primordial planetesimal disk.

We present the method used in the next section. In section 3, we explore the positions of the secular resonances during the period when the giant planets were placed in a multiresonant configuration and the timescale needed to excite an object located in it. As it will be explained in the concerned section, we can only look at the nodal secular resonances in such multiresonant configurations.

In section 4, we explore the positions of the secular resonances just before the dynamical instability. We conclude in section 5.

Method

We base our approach on the linear secular theory (see for example [START_REF] Murray | Solar System Dynamics[END_REF], chapter 7, for details). Given the gravitational potential of a disk, the secular theory allows us to determine its contribution to the apsidal and nodal precessions of bodies. We model the MD as an axisymmetric continuous thick disk with a planar symmetry and centered on the Sun. We arbitrarily choose the reference frame such that its origin is also centered on the Sun and such that the plane (Oxy) corresponds to the plane of symmetry of the MD. Consequently, the MD has a radial density profile symmetric around the z-axis and a vertical density profile symmetric in regards to the plane z = 0. In the following, the inclinations of the bodies are referred with respect to the latter plane. With the help of the numerical method developed by [START_REF] Fukushima | Numerical computation of gravitational field for general axisymmetric objects[END_REF] we calculate the gravitational potential induced by the MD from its density profile. In the linear secular theory, the disturbing function developed to the second order in eccentricities and in inclinations is the gravitational potential averaged on one orbital period along a fixed ellipse. Therefore, in order to include the dynamical effect of the MD in the linear secular theory, we calculate its contribution to the second order disturbing functions of the giant planets and of a LD object, which is considered as a massless particle. For that, in the first place, for each of the giant planets, let us consider its total disturbing function R tot j where the subscript j denotes the jth giant planet, from the closest to the Sun to the outermost. We can write R tot j = R P j +R M D j

, where the superscript P denotes that this is the part of the disturbing function due to the giant planets only and the superscript M D denotes that this is the part of the disturbing function due to the MD. They are expressed as:

R P j =n j a 2 j 1 2 A P jj e 2 j + N k=1,k =j A P jk e j e k cos( j -k ) + 1 2 B P jj I 2 j + N k=1,k =j B P jk I j I k cos(Ω j -Ω k ) , (1)
R M D j = n j a 2 j 1 2 A M D jj e 2 j + 1 2 B M D jj I 2 j , (2)
where N is the number of giant planets, the coefficients precessions of the jth giant planet with the presence of the MD only. We can see that in equation 2, there are no coupling terms between the eccentricities of the giant planets and the "eccentricity" of the MD and between the inclinations of the giant planets and the "inclination" of the MD because of the axisymmetry and the planar symmetry of the MD. We can determine R P j analytically (see appendix A). To determine the MD contribution to the apsidal precession frequency A M D jj , we calculate the gravitational potential numerically, and we averaged it on one period of the orbit of the planet, keeping its inclination fixed to zero, for values of the eccentricity starting from zero until e max = 0.002 and by increasing the eccentricity of 0.0001 at each step. We obtain the averaged gravitational potential as a function of the eccentricity, corresponding to the disk contribution of the apsidal part of the disturbing function which is of the form R M D j = 1 2 n j a 2 j A M D jj e 2 j (see appendix A, figure 6, top panel). Second, we use the same method to determine the MD contribution to the nodal precession frequency B M D jj . We fix the eccentricity equal to zero and starting with an inclination equal to zero, we increase it with a step of 0.0001 rad until it reaches the value I max = 0.002 rad. At each step, we calculate the gravitational potential averaged on one period of its orbit. We obtain the disk contribution to the nodal part of the disturbing function which is of the form ) 3:2,3:2,4:3,4:3;(b) 3:2,3:2,3:2,3:2;(c) 3:2,3:2,2:1,3:2;(d) 3:2,3:2,2:1,2:1

{A P jk , 1 ≤ j ≤ N, 1 ≤ k ≤ N } and {B P jk , 1 ≤ j ≤ N, 1 ≤ k ≤ N }
R M D j = 1 2 n j a 2 j B M D jj I 2 j (see appendix A,

Positions of the nodal secular resonances

In the secular theory, if we do not consider the MD but only the five giant planets, one of the eigenfrequencies f i associated to the inclinations and to the nodal precessions is null. The reason is that the choice of the reference plane being arbitrary, the inclination of a planet in regards to the reference plane evolves as a function of its inclination with respect to the other planets. In other words, there is an invariant plane, corresponding to the plane orthogonal to the total angular momentum vector of the planetary system (hereafter called the planetary plane), that does not precess because of the invariance of the total angular momentum. The frequency in question (usually denoted f 5 for the Solar System) is associated to the precession rate of the planetary plane relative to the arbitrary reference plane, and therefore it is zero. Now, if we add the MD, the total angular momentum of the giant planets is not conserved anymore and the planetary plane precesses with the frequency f 5 which is no longer equal to zero. However, the MD being static in our model, if we put the constraint that the orthogonal direction to the plane of symmetry of the MD is aligned with the total angular momentum of the planets, the amplitude of the oscillation corresponding to the f 5 frequency is null. Nevertheless, if the MD is in such a way that its orthogonal direction is misaligned with the total angular momentum of the planets, the amplitude associated to the f 5 frequency is not null anymore.

Figure 1 shows the positions of the nodal secular resonances for the orbital configuration 3:2, 3:2, 3:2, 3:2. The frequencies are negative because the nodal precessions are retrograde. We arbitrarily choose the subscripts of the eigenfrequencies according to their values: the lower is the absolute value of the frequency the higher is the subscript, except for the f 5 frequency corresponding to the frequency of the precession of the planetary plane of the giant planets.

We the three different masses of the disk. In the configuration 3:2, 3:2, 2:1, 3:2, with M disk = 40M ⊕ and M disk = 60M ⊕ , the resonance is located at the outer border of the current cold Kuiper Belt, near the 2:1 MMR. With M disk = 20M ⊕ , it is located in a region where an extension of the cold population beyond the 2:1 MMR has been discovered [START_REF] Bannister | OSSOS. VII. 800+ Trans-Neptunian Objects-The Complete Data Release[END_REF]. In the configuration 3:2, 3:2, 2:1, 2:1, the resonance is located in a region where cold Kuiper Belt objects have still not been observed so far. Finally, in the configuration 3:2, 3:2, lifetime of an object on an orbit in that region is short, in particular because of the presence of the current g 8 secular resonance. The other secular resonances remain below 33 AU in all the configurations and never reach the region of the current classical Kuiper Belt.

Efficiency of the f 5 secular resonance

We performed simple numerical integrations in order to illustrate the efficiency of the f 5 secular resonance. We cannot include directly the massive disk and compute the forces it exerts on the bodies because it is computationally too heavy. Therefore, we only include the nodal precession it exerts. To do so, we proceed as follow: as a first step, for a given shape and mass of the disk, we use the method described in section 2 in order to sample the values of the nodal precession frequency as a function of the semi-major axis. We sample the frequency every 0.1 AU in a range from 1 AU to 60 AU. It is illustrated in figure 2. Then, we use a linear interpolation in order to have a continuous function of the frequency. In a second step, to allow the variation of the longitude of the node of a body, we use the method of [START_REF] Lee | Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets[END_REF] implemented in the same way as [START_REF] Wolff | Neptune on Tiptoes: Dynamical Histories that Preserve the Cold Classical Kuiper Belt[END_REF] in the Mercury6 N-body integrator package [START_REF] Chambers | A hybrid symplectic integrator that permits close encounters between massive bodies[END_REF], adding acceleration and velocity perturbation terms corresponding to the variation of the longitude of the node, as described in appendix B.

Here, the goal of our numerical integrations is not to include the effect of the f 5 secular resonance in a complete model of dynamical evolution of the giant planets and of the Kuiper Belt and then to compare the Kuiper Belt obtained with the observations. It would demand to construct a strong model where the evolution of the shape and of the mass of the disk is coherent with the migration of the giant planets. The construction of such a model is beyond the scope of this study and is left for future work. The aim of the present study is qualitative and is to illustrate how efficient the f 5 secular resonance can be in a numerical integration. Therefore, in the present model, the giant planets remain unrealistically at their initial semi-major axes during the whole numerical integration and the shape of the MD does not evolve but we allow its mass to decrease exponentially with time, simulating the migration of the f 5 secular resonance. We do not allow the giant planets to migrate because, as we can see in figure 2, there is a strong variation of the nodal precession frequency in the region of the MD. However, as Neptune follows its outward migration, the inner border of the MD moves back and Neptune never reaches the region where this strong variation takes place. So, if we allow Neptune to migrate but we do not adapt the shape of the disk to it, the nodal precession frequency of Neptune would be more enhanced than it is in a more realistic model. We do not explore the effect of the f 8 and f 9 secular resonances either because the decaying disk would cause them to move inward whereas the migration of the giant planets would make the resonances to migrate outward. Therefore, if the effect of the outward migration is stronger than that of the inward migration, a model that accounts for the two effects would see the outward migration of the resonances whereas in our model we only see an inward migration, thus the representation would be wrong.

We illustrate our numerical integrations with the example of the multiresonant configuration 3:2, 3:2, 2:1, 3:2 which is evaluated as the most probable by [START_REF] Deienno | Constraining the Giant Planets' Initial Configuration from Their Evolution: Implications for the Timing of the Planetary Instability[END_REF]. We choose the initial mass of the massive disk to be

M disk = 20M ⊕ .
We use fictitious forces of [START_REF] Lee | Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets[END_REF] to keep the semi-major axes of the giant planets in the multiresonant configuration and to damp their eccentricities. Given that the efficiency of the f 5 secular resonance depends on the inclination of the planetary plane with respect to the mean plane of the disk, we did numerical integrations with different initial inclinations of the planetary plane. We place the giant planets initially on coplanar orbits.

We put massless particles from 45 AU to 55 AU, with one particle every 0.1 AU. We generate the eccentricities and the inclinations of the particles with a Rayleigh distribution with a mode of 0.01 and 0.01 radians respectively. The other angles of the giant planets and of the particles are uniformly distributed over 360 • . Given that in the linear secular theory, the precession frequency is linearly proportional to the mass of the disk, we can account for the depletion of the disk directly by modifying the precession frequency. We represent the depletion of the mass of the disk by an exponential decay with a timescale τ .

In a first set of numerical integrations, we explore the effect of the inclination of the planetary plane. We choose a disk depletion timescale τ = 50 Myr and we run three numerical integrations over 10 Myr with a timestep of 100 days and with three different initial inclinations of the planetary plane: 0 • , 1 • and 2 • . Figure 3 presents the results and shows that the higher the inclination of the planetary plane, the higher the efficiency of the f 5 secular resonance.

In a second set of numerical integrations, we fix the initial inclination of the planetary plane at 1 • but we try three values of the depletion timescale: τ = 50

Myr, τ = 30 Myr and τ = 10 Myr. We run our numerical integrations for 200 Myr. The time step is kept at 100 days. Figure 4 shows the results, with the semi-major axes averaged over the last 50 Myr and the proper inclinations.

In the case τ = 50 Myr the resonance sweeps from ∼48 AU to ∼55 AU in ∼100 Myr. The rising of the proper inclinations after 54 AU is associated to a dephasing effect. In the case τ = 30 Myr it sweeps in ∼60 Myr and for τ = 10

Myr it sweeps in ∼30 Myr.

We remind that those numerical integrations are not autocoherent and it would demand to do stronger effort in order to perform numerical integrations including the effect of the massive disk with the latter evolving with respect to the migration of Neptune in a consistent way. Besides, there is an uncertainty of the time of the triggering of the instability. It has a consequence on the timescale during which the f 5 secular resonance is efficient. During the preinstability period, the mass of the disk is slowly depleted and remains high enough, so the resonance slowly moves from its initial location. Then, during the instability, the mass of the disk is quickly depleted and the secular resonance migrates very rapidly toward higher semi-major axes.

-0.01

Positions of the secular resonances just before the planetary instability

During the migration of the giant planets, they interact with the planetesimals of the MD and some of the latter get ejected. Therefore, the MD is gradually depleted. The aim of this section is to check whether the shift of the secular resonances remains important or not and to check the position of the f 5 secular resonance, just before the planetary instability. For that purpose, we use the results of a numerical integration for a Neptune migration scenario developed by Nesvorný (2015a). It also includes the planetary instability, represented via a jumping Neptune, which occurred when the ice giant is near 28 AU in order to explain the kernel in the cold population (Nesvorný, 2015b). The context of this numerical integration is different than that of numerical integrations of [START_REF] Deienno | Constraining the Giant Planets' Initial Configuration from Their Evolution: Implications for the Timing of the Planetary Instability[END_REF]. The aim of the study of Nesvorný (2015a) was to reproduce the inclination distribution of the hot Kuiper Belt population from a set of massless particles initially located between Neptune and 30 AU with a slow radial migration of Neptune. Here, there are only the four giant planets and they are not in a multiresonant configuration anymore. Thus, it is possible to look at the positions of the apsidal secular resonances. The initial positions of Jupiter, Saturn and Uranus correspond to their current ones and Neptune was placed with an initial semi-major axis lower than 25 AU. Its radial migration and its eccentricity and inclination damping were driven by fictitious forces.

Model

From the orbital elements of the particles provided by the numerical integration of Nesvorný (2015a) we estimate the distribution of the MD at t = 3

Myr before the instability between the giant planets. Let (r, θ, z) be the heliocentric coordinates. We assume that the MD distribution is axisymmetric and uncoupled in r and z. We construct histograms of the r and z distributions from which we obtain the following density profiles:

σ r (r) =                  23.5 2π if r ≤ 28 AU, 2961 πr(r -19) if 28 AU < r ≤ 1000 AU, 0 if r > 1000 AU, and
λ z (z) =        exp(- |z| 10 ) if -100 AU ≤ z ≤ 100 AU, 0 otherwise,
where r and z are expressed in AU, σ r (r) is the radial surface density in kg.AU -2 and λ z (z) is the vertical linear density in kg.AU -1 . Here, those two last quan- tities are not normalized. The density profile of the MD is:

ρ(r, z) = M disk σ r (r)λ z (z) r=R r=0 z=zmax z=zmin θ=2π θ=0 σ r (r)λ z (z)r dr dz dθ , (3)
in kg.AU -3 , where the denominator is used to normalize the function. The mass of the MD is chosen with an initial value of 20 M ⊕ , because the migration timescale of Neptune in that numerical integration is typical of that with such a mass. It has the value 5.9604 M ⊕ at the time t = 3 Myr before the planetary instability.

Results

We apply the method described in section 2 to evaluate the apsidal and nodal eigenfrequencies of the planetary system and the free frequencies of the LD object.

Table 2 shows the semi-major axes and eccentricities of the giant planets 3

Myr before the planetary instability. The f 5 secular resonance is located beyond 150 AU, because the mass of the MD is low and the radial distance of the f 5 secular resonance increases as the disk loses mass.

Figure 5 presents the results that we obtained for the positions of the g 8 secular resonance, exhibited by the dashed vertical lines, according to several perspectives. A P (red curve) and g P 8 (red dashed horizontal line) are the frequencies in the case where the apsidal precessions are only driven by the giant planets, whereas A tot (blue curve) and g tot 8 (blue dashed horizontal line) are the frequencies in the case where the apsidal precessions are driven by the giant planets but also by the MD. The red dashed vertical line shows the position of the resonance in the case where the MD does not act neither on the giant planets nor the LD objects and then is located where A P = g P 8 , at 37.21 AU.

The purple dashed vertical line corresponds to the case where the MD acts on the giant planets but not on the LD objects and is located where

A P = g tot 8 ,
at 37.59 AU. Finally, the blue vertical line is the ideal case where the MD acts both on the giant planets and on the LD objects. It is located where A tot = g tot 8 , at 37.44 AU. It makes a shift of 0.6% compared to the case where the MD is neglected. The other resonances are also shifted by less than 1%. We conclude that the shifts imposed by the MD are negligible.

We also check the positions of the secular resonances 50 Myr after the planetary instability. In this case, the mass of the disk is 2.9006 M ⊕ , two times lower than at 3 Myr before the instability, but the peak of the mass distribution is located at ∼30 AU, which is closer from the Kuiper Belt than in the previous case. However, we find that the f 5 secular resonance remains located beyond 150 AU and the shift of the other secular resonances is even smaller than in the previous case.

Discussion and conclusion

Studies of the primordial dynamical evolution of the giant planets and of the primordial planetesimal disk have shown that the latter must have a sharp edge near 30 AU in its mass distribution. The part of the disk beyond this sharp edge is a light extension of the disk so as to make unefficient the planetesimal-driven migration of Neptune beyond this edge and to stop the ice giant at its current position. Besides, current models starting with initial conditions presenting a continuous mass distribution in the planetesimal disk between Neptune and 50 AU do not succeed in sufficiently depleting the mass of the cold Kuiper Belt region. The cold Kuiper Belt population is supposed to be formed from this light extension. That being said, dynamical studies of the objects of the light part of the disk have so far always been restricted to the dynamical evolution of those objects under the effect of the gravitational interactions with the giant planets only. In section 3, we have investigated on the positions of the secular resonances once the solar nebula vanished and the giant planets were still locked in a multiresonant configuration. We could only look at the positions of the nodal secular resonances because the resonant dynamics is dominant in the evolution of the longitudes of the perihelion of the giant planets. We have found that the f 5 secular resonance, driven by the precession of the total angular momentum of the giant planets, is located in the region of the LD in several configurations. It can have an effect in exciting the inclinations of the LD objects. Consequently, it can create a population of objects with high inclinations but low eccentricities.

Its efficiency in rising their inclinations mainly depends on the angle between the planetary and the MD planes. While revising this manuscript we became aware of the work of [START_REF] Toliou | Secular resonance sweeping and orbital excitation in decaying disk[END_REF], performed independently, that find results similar to ours on the existence and the role of a non-zero f 5 frequency during the disk phase.

In planetary formation models, the planets and the planetesimal disk formed from the same protoplanetary disk in the plane of the solar equator. In the current Solar System, we observe an inclination of ∼6 • between the solar equator and the invariant plane. It has been shown that the presence of a ninth planet located in the outer Solar System can explain this tilt [START_REF] Bailey | Solar Obliquity Induced by Planet Nine[END_REF], [START_REF] Gomes | The Inclination of the Planetary System Relative to the Solar Equator May Be Explained by the Presence of Planet 9[END_REF] and [START_REF] Lai | Solar Obliquity Induced by Planet Nine: Simple Calculation[END_REF]). Another possibility, as suggested by [START_REF] Batygin | A primordial origin for misalignments between stellar spin axes and planetary orbits[END_REF], is that the tilt can be produced by the passage of a star in the birth cluster. The passage of a star can warp the protoplanetary disk, making its outer part inclined with respect to the inner part where the planets are formed. Moreover, [START_REF] Izidoro | Accretion of Uranus and Neptune from inward-migrating planetary embryos blocked by Jupiter and Saturn[END_REF] have shown that during the accretion of Uranus and Neptune, ejections of planetary embryos caused by close encounters were common. As ejected objects take away with them a part of the total angular momentum, the ones remaining in the Solar System have an angular momentum different from the total one and that is not conserved. Therefore, the orientation of the plane orthogonal to the angular momentum of the remaining objects varies with time, until the Solar System reaches its current architecture and the ejections of objects stopped, leaving what is currently called the invariant plane.

All of these elements give weight to the conception that the invariant plane of the planets got misaligned with the plane of the disk.

In section 4, we have looked at the positions of the secular resonances during the migration of Neptune just before the instability between the giant planets occurred, when Neptune is near 28 AU, from data provided by the works of Nesvorný (2015a). We have found that during this phase of the planetary evolution, the mass of the MD is not high enough anymore and that consequently the positions of the secular resonances are only shifted by less than 1% with respect to the case where the MD is neglected. In this case, if numerical integrations omit the effects of the MD, the evolution of the LD objects will be qualitatively the same as if they do not.

where m c is the mass of the Sun, m j is the mass of the jth giant planet,

α jk =        a k /a j if a j > a k , a j /a k if a j < a k , ᾱjk =        1 if a j > a k , a j /a k if a j < a k , α j =        a j /a if a > a j , a/a j if a < a j , ᾱj =        1 if a > a j , a/a j if a < a j ,
and where b

(n) 3/2 (α) = 1 π 2π 0 cos nψ dψ (1 -2α cos ψ + α 2 ) 3 2 , (14)
is called the Laplace coefficient.

We write A P and B P the matrices of dimension N × N containing the

elements {A P jk , 1 ≤ j ≤ N, 1 ≤ k ≤ N } and {B P jk , 1 ≤ j ≤ N, 1 ≤ k ≤ N }
respectively. By introducing h P ≡ e sin , k P ≡ e cos , p P ≡ I sin Ω and q P ≡ I cos Ω, then by using the Lagrange planetary equations with the disturbing functions R P j and R P , we obtain the solutions:

h P = e P f ree sin(A P t + β P ) + h P 0 (t),

k P = e P f ree cos(A P t + β P ) + k P 0 (t), p P = I P f ree sin(B P t + γ P ) + p P 0 (t),

q P = I P f ree cos(B P t + γ P ) + q P 0 (t), (15)
(16) (17) (18)
where e P f ree , I P f ree , β P and γ P are constants determined by the initial conditions. The first terms of the right members of the equations 15, 16, 17 and 18 correspond to the free oscillations while the terms h P 0 (t), k P 0 (t), p P 0 (t) and q P 0 (t) correspond to the forced oscillations: where the frequencies g P i and f P i are the eigenvalues (which we will call eigenfrequencies in the following) of the matrices A P and B P respectively. We also

h P 0 = - N i=1
R M D = na 2 1 2 A M D e 2 + 1 2 B M D I 2 , ( (23)
) 24
where the terms with the superscript M D denotes that they are due to the MD.

As described in section 2, we can determine R M D j and R M D computationnally.

For example, figure 6 shows the apsidal part (top panel) and the nodal part x =r cos Ω cos(ω + f ) -r cos i sin Ω sin(ω + f ) y =r sin Ω cos(ω + f ) + r cos i cos Ω sin(ω + f ) z =r sin i sin(ω + f ) ẋ = cos Ω ṙ cos(ω + f ) -r ḟ sin(ω + f ) -sin Ω ṙ cos i sin(ω + f ) + r ḟ cos i cos(ω + f ) ẏ = sin Ω ṙ cos(ω + f ) -r ḟ sin(ω + f ) + cos Ω ṙ cos i sin(ω + f ) + r ḟ cos i cos(ω + f ) ż = ṙ sin i sin(ω + f ) + r ḟ sin i cos(ω + f ), (25) where r is the heliocentric distance, f the true anomaly, i the inclination, ω the argument of the perihelion and Ω the longitude of the ascending node. Using the notations and the method of [START_REF] Lee | Dynamics and Origin of the 2:1 Orbital Resonances of the GJ 876 Planets[END_REF], we can obtain the terms needed to update the acceleration and the veolocity:

  are frequencies associated to the apsidal and nodal precessions, respectively, of the giant planets without the MD and from which the eigenfrequencies g P i and f P i , respectively describing the apsidal and nodal evolutions of the planetary system, are determined. The coefficients A M D jj and B M D jj are frequencies associated to the apsidal and nodal

  figure 6, bottom panel). Then, using a linear regression we obtain the values of the apsidal precession frequency A M D jj and of the nodal precession frequency B M D jj . By using the frequencies A tot jj = A P jj + nodal and apsidal evolutions of the giant planets which account for the effect of the MD. To calculate the MD contribution to the free frequencies A tot = A P + A M D and B tot = B P + B M D of the apsidal and nodal free oscillations of a LD object, we can operate in the same way as previously for the giant planets. The apsidal and nodal secular resonances are localized at semi-major axes where A tot = g tot i and B tot = f tot i respectively.3. Positions of the nodal secular resonances in the primordial orbital configuration of the giant planets3.1. ModelIn the first part, we look at the positions of the secular resonances in the initial conditions of the models proposed by[START_REF] Deienno | Constraining the Giant Planets' Initial Configuration from Their Evolution: Implications for the Timing of the Planetary Instability[END_REF]. They studied the dynamical evolution of the giant planets, interacting with the MD, from several multiresonant configurations. In their models, there are initially Jupiter, Saturn, Uranus and Neptune as well as an extra ice giant. This additional planet is ejected during the dynamical instability. They found that only four multiresonant configurations can lead to the current orbital architecture of the outer solar system. These configurations are the following: (a

  , with the semi-major axis of Jupiter initially at ∼5.4 AU. We use those four multiresonant configurations. We place the semi-major axis of Jupiter at 5.4 AU, the inner edge of the MD at 1 AU from Neptune and its outer edge at 30 AU from the Sun.The radial density profile of the MD is of the form Σ(r) = Σ 0 r0 r . Its vertical density profile follows a Gaussian law with σ z = r tan(σ i ) where σ i = 2 • . We choose to fix the mass of the extra ice giant to 15 M ⊕ and we place it between Saturn and Uranus. The giant planets being in a chain of first order resonances, in their disturbing functions the apsidal and nodal parts of the resonant part contain terms of orders O(e) and O(eI 2 ) respectively whereas in the secular part, the apsidal and nodal terms are of orders O(e 2 ) and O(I 2 ). Thus, for the apsidal part, the resonant effects are stronger than the secular ones so the computation of the apsidal eigenfrequencies of the planetary system by using the linear secular theory would be wrong and it would not give the correct positions of the secular resonances. Hence, we cannot use the secular theory to look at the positions of the apsidal secular resonances. For the nodal part, the secular effects are more important than the resonant ones, so we are allowed to use the secular theory. However, for the configurations (c) and (d), some giant planets have a ratio between their semi-major axes corresponding to a 3:1 MMR. In such a resonance, the nodal part of the resonant part of the disturbing function is of order O(I 2 ) and it is as high as in the secular part. Depending on the libration or circulation of the resonant angles involved in this resonance, it can have the effect of slightly moving the positions of the secular resonances from the original ones.

Figure 1 :

 1 Figure1: Nodal free frequency of a LD object as a function of its semi-major axis in the configuration 3:2, 3:2, 3:2, 3:2, in a model without the MD (top panel) and in a model with a MD of mass M disk = 40M ⊕ (bottom panel). The horizontal lines are the nodal eigenfrequencies. In the bottom panel, the dashed curve is the nodal free frequency in the case where the MD does not act on the LD object and the full black curve is the nodal free frequency but in the case where the MD acts on the LD object. The nodal secular resonances occur at the semi-major axes where the nodal free frequency crosses one of the nodal eigenfrequencies.

  4:3, 4:3, with M disk = 40M ⊕ and M disk = 60M ⊕ the resonance is located in a region where the current cold Kuiper Belt does not exist anymore because the Orbital configuration M disk (M ⊕ ) Position of f 5 (AU)

Figure 2 :

 2 Figure 2: The frequency of the nodal precession caused by the MD as a function of the semimajor axis. The mass of the MD is M disk = 20M ⊕ , its inner edge is at 21 AU and its outer edge at 30 AU.

Figure 3 :

 3 Figure3: Inclinations of the particles after 10 Myr. The multiresonant configuration of the giant planets is 3:2, 3:2, 2:1, 3:2. The initial mass of the MD is M disk = 20M ⊕ , its inner edge is at 21 AU, its outer edge at 30 AU and its depletion timescale is 50 Myr. The three colors of the particles represent the results of three different numerical integrations, for which the initial inclination of the planetary plane with respect to the mean plane of the disk is i = 0 • (blue dots), i = 1 • (green dots) and i = 2 • (red dots).

Figure 4 :

 4 Figure 4: Proper inclinations of the particles after 200 Myr numerical integrations. The multiresonant configuration of the giant planets is 3:2, 3:2, 2:1, 3:2 and the initial inclination of the planetary plane with respect to the mean plane of the disk is i = 1 • . The initial mass of the MD is M disk = 20M ⊕ , its inner edge is at 21 AU and its outer edge at 30 AU. The blue dots represent the results of the particles for a numerical integration where the nodal precession due to the MD is not included. The other colors represent the results for numerical integrations whith different depletion timescales τ of the mass of the MD. The green dots are for τ = 10 Myr, the orange dots for τ = 30 Myr and the red dots for τ = 50 Myr. The semi-major axes are averaged over the last 50 Myr of the numerical integrations.

Figure 5 :

 5 Figure 5: Position of the g 8 secular resonance in the case without the MD (red curve) and with it (blue curve), 3 Myr before the instability between the giant planets. Horizontal dashed lines represent the value of g 8 . We exhibit the intersection of the curves with the horizontal dashed lines by dashed vertical lines which is the position of the g 8 secular resonance. The purple dashed vertical line shows the position of g 8 in the case where the MD acts on the giant planets but not on the LD object.

  However, given the important mass of the disk between Neptune and 30 AU (∼10 -60 M ⊕ ), it can have a non-negligible influence. This is what we wanted to check in this paper. We have called the massive disk (MD) the part of the planetesimal disk between Neptune and 30 AU, and the light disk (LD) the part of the planetesimal disk beyond 30 AU. We have studied the different positions that the secular resonances can have in three different situations: (a) a case where the MD is not included, (b) a case where the MD modifies the apsidal and nodal precessions of the giant planets but not those of the LD objects and (c) a case where the MD also acts on the apsidal and nodal precessions of the LD objects.

  ji where e P ji , I P ji , β P i and γ P i are constants depending on the eigenvectors associated to the eigenfrequencies and on the initial conditions of the giant planets. These equations bring out the existence of resonances when the secular frequency A P and B P of the LD object, which depends on its semi-major axis, are equal to one of the eigenfrequencies g P i and f P i respectively. The resonances involving the frequencies A P and g P i are called apsidal secular resonances whereas those involving the frequencies B P and f P i are called nodal secular resonances. Now, if we consider the disturbing function R M D j of the jth giant planet due to the MD and the disturbing function R M D of the LD object due to the MD,

(

  bottom panel) of the disturbing function of Jupiter due to the MD as a function of the eccentricity and the inclination of the planet respectively.

Figure 6 :

 6 Figure 6: The disturbing function (or equivalently, the gravitational potential averaged on one orbital period along a fixed ellipse) of Jupiter (subscript j = 1) due to the MD as a function of the eccentricity (top panel) and of the inclination (bottom panel). The dots are the values for which we have computed the averaged gravitational potential. The curves are the best fit functions of the form R 1 M D = αe 2 (top panel) and R 1 M D = βI 2 (bottom panel), where α and β are coefficients.

  r sin Ω cos(ω + f ) -r cos i cos Ω sin(ω + f )cos Ω cos(ω + f ) -r cos i sin Ω sin(ω + f )Ω ṙ cos(ω + f ) -r ḟ sin(ω + f ) -cos Ω ṙ cos i sin(ω + f ) + r ḟ cos i cos(ω + f ) ṙ cos(ω + f ) -r ḟ sin(ω + f ) -sin Ω ṙ cos i sin(ω + f ) + r ḟ cos i cos(ω + f )

Table 1 :

 1 Position of the f 5 secular resonance for four orbital configurations and for several masses of the MD.

Table 2 :

 2 Semi-major axes and eccentricities of the giant planets 3 Myr before the planetary instability.

			Orbital elements
		a (AU)	e
	Jupiter	5.201731	0.01824838
	Saturn	9.578610	0.05086522
	Uranus	18.16524	0.03557539
	Neptune	27.80457	0.01387243

We thank the reviewers for helpful comments that improved the manuscript.

We acknowledge support by the French ANR, project number ANR-13-13-BS05-0003-01 project MOJO (Modeling the Origin of JOvian planets). We acknowledge David Nesvorný for providing us with data of a numerical integration.

Appendix A

In a planetary system with N giant planets, the secular part of the disturbing functions R P j of the jth giant planet and R P of a LD object are [START_REF] Murray | Solar System Dynamics[END_REF]:

where the terms with the superscript P denotes that they are due to the planetary system only and the variables with the subscript j and without it respectively relate to the jth giant planet and to the LD object. n is the mean motion, a the semi-major axis, e the eccentricity, I the inclination, the longitude of the perihelion and Ω the longitude of the node. The coefficients A P jj , A P jk , B P jj , B P jk , A P , A P j , B P and B P j are frequencies, which depend on the masses and semi-major axes of the bodies such that:

3/2 (α jk ),

3/2 (α jk ),

(8) (9)

3/2 (α j ),

3/2 (α j ),

3/2 (α j ),

Appendix B

Here we describe the terms that we need to add to the user-defined acceleration and to the user-defined velocity in order to allow the longitudes of the node of the bodies to evolve as defined by the user in addition of the mutual interactions between the bodies. The components of the position vector (x, y, z) and the components of the velocity vector ( ẋ, ẏ, ż) can be expressed as a function of the osculating elements:


Maxime Parisotto

Elise Grelet

Rana El Bizri

Daniel Metzger
email: metzger@igbmc.fr

Senescence controls prostatic neoplasia driven by Pten loss

Keywords: prostate cancer, mouse model, senescence, replicative stress, DNA damage

We report that Pten (phosphatase and tensin homologue) ablation in prostatic epithelial cells of adult mice promotes cell proliferation to generate prostatic intraepithelial neoplasia. Moreover, our results demonstrate that proliferating Pten-deficient cells undergo replication stress and exhibit a DNA damage 10 response, leading to cell senescence, as seen in oncogene-induced senescence.

Prostate cancer, the most common male visceral cancer, is a multistage disease that progresses over decades. About 40% of men over 50 have prostatic intraepithelial neoplasia (PIN), and even though most of PINs will remain indolent, about 1% 15 of the patients will develop clinically significant tumors. If detected when locally confined, prostate cancer is eradicated in 70 -80% of cases by radical prostatectomy, radiation therapy or cryotherapy. In contrast, no efficient treatments are available for metastatic prostate cancer, which remains the 20 second leading cause of male cancer-related death. [START_REF] Hodson | Small organ, big reach[END_REF] The tumor suppressor gene PTEN (phosphatase and tensin homologue) is frequently mutated or deleted in prostate cancer. PTEN is a lipid phosphatase that antagonizes the production of Phosphatidylinositol (3,4,5)-trisphosphate by 25 phosphoinositide 3-kinase. Its loss leads to a constitutive hyperactivation of the Protein kinase B (AKT)-mammalian target of rapamycin (mTOR) pathway. Genetic ablation of Pten in prostatic epithelial cells of mice induces PINs. [START_REF] Chen | Crucial role of p53-dependent cellular senescence in suppression of Pten-deficient tumorigenesis[END_REF][START_REF] Ratnacaram | Temporally controlled ablation of PTEN in adult 145 mouse prostate epithelium generates a model of invasive prostatic adenocarcinoma[END_REF] Progression toward adenocarcinoma has been shown to be 30 restrained by a cell senescence barrier, 2 but as no hyperproliferation phase, replication stress or DNA damage response were observed in Pten-deficient cells, in contrast to oncogeneinduced senescence (OIS), it was concluded that Pten lossinduced senescence (PICS) is a new type of senescence. [START_REF] Alimonti | 155 Tumour-infiltrating Gr-1+ myeloid cells antagonize senescence in cancer[END_REF] It was 35 proposed that interleukin-1 receptor antagonist (Il-1ra) secreted by infiltrated Gr-1 positive myeloid cells promotes the proliferation of some Pten-deficient prostatic epithelial cells, leading to PIN formation. 5 To further characterize the impact of Pten loss in prostatic 40 epithelial cells of the mature prostate, we analyzed PTEN (i)pe-/- mice, in which Pten is selectively ablated in prostatic luminal epithelial cells of mice after puberty. [START_REF] Parisotto | PTEN deletion in luminal cells of mature prostate induces replication stress and senescence in 160 vivo[END_REF] Our results show that the proliferation of Pten-null prostatic epithelial cells is stimulated within 1 month, reaches a maximum at 3 months, and is very low after 6 months. Between 5 and 12 months, most of the prostatic epithelial cells in PINs are senescent, produce a number of chemokines and cytokines, and are infiltrated by Gr-1 positive myeloid cells. As senescence markers and Gr-1 positive cells were however only detected only after 2 months, our data demonstrate that Pten loss in prostatic epithelial cells initiates a phase of enhanced proliferation in the absence of GR-1 positive myeloid cells and before senescence entry. In addition, many Pten-null prostatic luminal epithelial cells exhibited phosphohistone H2AX (γH2AX) nuclear foci, reflecting an activation of a DNA damage response, [START_REF] Mah | gammaH2AX: a sensitive molecular marker of DNA damage and repair[END_REF] and those proliferating presented characteristics of replication stress [e.g. nuclear foci of Rpa32 (Replication Protein A 32), Rpa32 phosphorylated at Serine 4 and Serine 8 and Atr (Ataxia telangiectasia and Rad3-related)]. Thus, Pten 60 loss-induced senescence resembles OIS, and strategies for cancer prevention or therapies based on PICS induction are at risk, as they will promote replication stress that might lead to additional mutations, allowing senescence escape and cancer progression, and induce the secretion of cytokines and chemokines that might contribute to tumor development.

Interestingly, our results show that transformation related protein 53 (Trp53, best known as p53) is stabilized in Ptendeficient prostatic epithelial cells by replication stressmediated DNA damage response activation. However, at early time after Pten ablation, p53 is down-regulated by mouse double-minute 2 (Mdm2), most probably through Akt-mediated Mdm2 phosphorylation, which increases its E3 ligase activity towards p53. The progressive p53 stabilization likely involves Casein Kinase I (CkI). Indeed, CkI transcript levels are induced 3 and 5 months after Pten ablation, and CkI has been shown to be induced by a DNA damage response and to decrease the interaction between Mdm2 and p53 and/or promote Mdm2 degradation. [START_REF] Knippschild | The CK1 family: contribution to cellular stress response and its role in carcinogenesis[END_REF][START_REF] Inuzuka | Phosphorylation by casein kinase I promotes the turnover of the Mdm2 oncoprotein via the SCF (beta-TRCP) ubiquitin ligase[END_REF] Ablation of p53 in Pten-deficient prostatic epithelial cells allows to bypass/avoid 80 the senescence barrier, accelerates the development of adenocarcinoma and promotes the emergence of metastatic sarcomatoid tumors [START_REF] Parisotto | PTEN deletion in luminal cells of mature prostate induces replication stress and senescence in 160 vivo[END_REF] (and unpublished data).

Our data highlight the importance of analyzing relevant mouse models. Obviously, the possibility to control the time 85 of gene ablation by induction of the CreER T2 recombinase activity in luminal epithelial cells of the mature prostate, is a major advantage compared to other models, where target genes are ablated in prostatic epithelial cells before puberty. Moreover, as cell culture conditions imply high levels of growth 90 factors and high partial O 2 pressure, stressors that might facilitate cellular senescence induction, [START_REF] Serrano | Putting the stress on senescence[END_REF] in vitro data on senescence should be extrapolated to in vivo contexts with care.

Taken together, our study provides important insights into the latency phase of prostate cancer. Indeed, our results show 95 that Pten loss in prostatic epithelial cells induces replication stress-mediated activation of a DNA damage response and progressive cell senescence. Akt activation after Pten ablation is likely to promote Mdm2-mediated p53 degradation and delays senescence until CkI levels are sufficient to allow p53 stabiliza-100 tion (Figure 1). Even though selective inhibitors of Akt or of Mdm2-p53 interaction might enhance senescence and thus increase the latency phase, therapeutic approaches allowing to eradicate PTEN-deficient senescent prostatic epithelial cells will probably be required to prevent prostate cancer progression.

Figure 1 .

 1 Figure 1. Mechanisms whereby Pten loss drives prostatic intraepithelial neoplasia under the control of cellular senescence. Ablation of Pten (phosphatase and tensin homologue) in prostatic luminal epithelial cells of adult mice induces the phosphorylation of Protein Kinase B (P-Akt), epithelial cell proliferation and prostatic intraepithelial neoplasia (PIN) formation. Enhanced proliferation promotes replicative stress, a DNA damage response, transformation related protein 53 (p53) stabilisation and epithelial cell senescence. At early time after Pten ablation, Mdm2 (mouse double-minute 2) is phosphorylated (P-Mdm2), most probably by P-Akt, and p53 is downregulated, until DNA damage response-induced Casein Kinase I (CkI) levels are sufficient to promote Mdm2 degradation and/or impair the interaction between Mdm2 and p53.

Contre le Cancer, The Association pour la Recherche sur les Tumeurs de la Prostate, The Centre d'Ingénierie Moléculaire Européen and by French state funds through the Agence Nationale de la Recherche ANR-10-LABX-0030-INRT under the frame programme Investissements d'Avenir labelled ANR-10-IDEX-0002-02. M.P. was supported by the

Association pour la Recherche à l'IGBMC (ARI), E.G. by the Ministère de l'enseignement supérieur et de la recherche and R.E.B. by the Al Bizri foundation. The authors declare no competing financial interests.

Tumeurs de la Prostate;Al Bizri foundation;Alsace Contre le Cancer; Université de Strasbourg;

MOLECULAR & CELLULAR ONCOLOGY e1511205-3

Acknowledgments

The work was supported by funds from the Centre National de la Recherche Scientifique, the Institut National de la Santé et de la Recherche Médicale, the Université de Strasbourg, the Fondation ARC pour la Recherche sur le Cancer, the Ligue Contre le Cancer, Alsace

Funding

This work was supported by the Centre National de la Recherche Scientifique;Fondation ARC pour la Recherche sur le Cancer;Institut National de la Santé et de la Recherche Médicale;Ligue Contre le Cancer;Ministère de l'Enseignement supérieur, de la Recherche et de

Disclosure of Potential Conflicts of Interest

No potential conflicts of interest were disclosed.

Q8


T Lokotko
email: tlokotko@hku.hk

S Leblond

J Lee

P Doornenbal

A Obertelli

‡ A Poves

F Nowacki

K Ogata

K Yoshida

G Authelet

H Baba

D Calvet

F Château

S Chen

A Corsi

A Delbart

J.-M Gheller

A Gillibert

T Isobe

V Lapoux

M Matsushita

S Momiyama

T Motobayashi

M Niikura

H Otsu

C Péron

A Peyaud

E C Pollacco

J.-Y Roussé

H Sakurai

C Santamaria

Z Y Xu

M Sasano

Y Shiga

S Takeuchi

R Taniuchi

T Uesaka

H Wang

V Werner

F Browne

L X Chung

Zs Dombradi

S Franchoo

F Giacoppo

A Gottardo

K Hadynska-Klek

Z Korkulu

S Koyama

Y Kubota

M Lettmann

C Louchart

R Lozeva

K Matsui

T Miyazaki

S Nishimura

L Olivier

S Ota

Z Patel

E Sahin

C Shand

P.-A Söderström

I Stefan

D Steppenbeck

T Sumikama

D Suzuki

Zs Vajta

J Wu

Shell structure of the neutron-rich isotopes 69,71,73 Co

The structures of the neutron-rich 69,71,73 Co isotopes were investigated via (p, 2p) knock-out reactions at the Radioactive Isotope Beam Factory, RIKEN. Isotopes of interest were studied using the DALI2 γ-ray detector array combined with the MINOS target and tracker system. Level schemes were reconstructed using the γ-γ coincidence technique, with tentative spin-parity assignments based on the measured inclusive and exclusive cross sections. Comparison with shell model calculations using the Lenzi-Nowacki-Poves-Sieja LNPS and PFSDG-U interactions suggests coexistence of spherical and deformed shapes at low excitation energies in the 69,71,73 Co isotopes. The distorted-wave impulse approximation (DWIA) framework was used to calculate the single-particle cross sections. These values were compared with the experimental findings.

I. Introduction

Shape coexistence is one of the central topics in nuclear physics [1]. Investigations have shown that the shape of nuclei, described in terms of multipole moments (quadrupole, octupole, ...), can vary greatly with a change of few protons or neutrons [2,3]. Nuclei located in the valley of stability and close to proton and neutron magic numbers appear to be spherical, while isotopes in the mid-shell region are found to have deformed shapes [START_REF] Casten | Nuclear Structure from a Simple Perspective[END_REF][START_REF] Vingerhoets | [END_REF][6]. Deformation in nuclei in general is caused by the quadrupole interaction [START_REF] Bohr | Nuclear Structure[END_REF][START_REF] Hamamoto | [END_REF]. Examination of shell evolution in regions far from stability is important for assessing nuclear models, as new features arise, for instance new magic numbers [9][10][11][12] and rapid phase transitions [13]. For instance, 68 Ni has attracted much attention due to its neutron N = 40 harmonic oscillator magic number and proton magic number Z = 28 [14]. The observation of a high 2 + 1 excitation energy in 68 Ni has provided a clear signature for its magicity (spherical shape) in the ground state [15,16]. On the other hand, the 2 + 1 excitation energy of 66 Fe, which has only two protons removed, drops significantly, indicating a deformed shape [17,18]. The isotope 67 Co is located between 68 Ni and 66 Fe along the N = 40 isotonic chain, and, interestingly, experiment indicates that it exhibits shape coexistence [19][20][21][START_REF] Walters | AIP Conference Proceedings[END_REF].

Energy systematics of the odd-mass 55-67 Co isotopes, presented in Ref. [19], show that the 9/2 - 1 and 11/2 - 1 excited states closely follow the corresponding 2 + 1 excitation energies in the neighboring Ni isotopes. Excited 9/2 - 1 and 11/2 - 1 states of 55 Co (N = 28) lie at approximately 3 MeV, very close to the 2 + 1 excitation energy of 56 Ni. With increasing neutron number, excited 9/2 - 1 and 11/2 - 1 states of 57 Co drop together with the 2 + 1 excitation energy of 58 Ni. This trend continues with slight variations along the even-odd Co isotopes, suggesting that these 9/2 - 1 and 11/2 - 1 levels can be approximated as πf -1 7/2 ⊗ 2 + 1 (Ni) configurations. For 67 Co, the heaviest isotope with tentatively assigned 9/2 - 1 and 11/2 - 1 levels, their rise in excitation energy is analogous to the rise in 2 + 1 excitation energy of 68 Ni and indicates spherical features.

Conversely, the observation of a low energy isomeric level at 492 keV (T 1/2 = 496 (33) ms) with spin 1/2 -in the isotope 67 Co revealed the existence of a decay band with a different structure from a proton hole configuration, as this state was greatly decreased in energy compared to the excitations in 68 Ni [20]. This low-lying isomer configuration in 67 Co has been interpreted as a prolate proton-intruder state coupled to the ground state of the isotope 66 Fe [21]. The low-energy 1/2 --3/2 -configuration is explained by strong proton-neutron correlations producing a deformed shape in 67 Co [20,21].

So far, among all neutron-rich Co nuclei only 67 Co has shown signs of shape coexistence. Together with indications of shape coexistence reported in 78 Ni [START_REF] Taniuchi | [END_REF], this raises the question regarding the shell evolution towards N = 50. The region beyond N = 40 is experimentally challenging, in particular β-decay studies of 71,73 Co offer only limited yield due to the very low production rates of 71,73 Fe secondary beams and large β-delayed neutron emission probabilities. In this work, the in-beam γ ray spectroscopy technique combined with the quasi-free (p, 2p) reaction was chosen to establish level schemes of 69,71,73 Co. While γ-ray transitions have been measured for 69 Co, but not placed in the level scheme [24], hitherto no data were available on 71,73 Co.

II. Experiment

The experiment was carried out at the Radioactive Isotope Beam Factory, operated by the RIKEN Nishina Center and the Center for Nuclear Study, University of Tokyo, where radioactive isotopes of 70,72,74 Ni were produced by impinging a 345 MeV/u 238 U primary beam (with an average intensity of 12 pnA) onto a 3-mmthick 9 Be target. Fragmentation products, which had an energy of ∼ 270 MeV/u, were separated and identified using the BigRIPS separator [25,26]. Two settings were applied to BigRIPS to produce the isotopes of interest. Particle identification on an event-by-event basis was performed via time of flight (ToF), magnetic rigidities (Bρ), and energy loss (∆E) measurements. The ToF was measured between two plastic scintillators separated by 46.6 m, Bρ was measured by trajectory reconstruction using position-sensitive parallel plate avalanche counters (PPAC) [27], and the energy loss was measured using a Multiple Sampling Ionization Chamber (MUSIC) [28]. Particle identification of the fragments of both BigRIPS settings is shown in Fig. 1.

Excited states of 69,71,73 Co were populated via (p, 2p) reactions of 70,72,74 Ni secondary beams that had intensities of 90 pps, 9 pps, and 8 pps, respectively. The secondary beams were delivered to MINOS [29], a time projection chamber (TPC) surrounding a secondary target of 102(1) mm length filled with liquid hydrogen with an effective thickness of 735(8) mg/cm 2 [30]. Average centerof-target energies were 252, 241, and 234 MeV/u for the three secondary beams. The TPC was used to track protons from (p, 2p) reactions in order to reconstruct the vertex of the reaction, thus improving resolution for Doppler corrections. The reaction vertex was reconstructed with a 95% efficiency for at least one proton tracked from the (p, 2p) channel and with 5 mm resolution (FWHM) along the beam axis [START_REF] Santamaria | Tracking with the MINOS Time Projection Chamber[END_REF]. Reaction products exiting MINOS were identified with the ZeroDegree spectrometer [26] by the ToF-Bρ-∆E technique. The particle identification of the reaction products is shown in Fig. 2.

The DALI2 NaI(Tl) detector array [32] measured prompt γ rays emitted following the reactions. DALI2 consisted of 186 NaI(Tl) crystals, covering angles from 12 • to 96 • degrees in the laboratory frame (between target center and detector center) along the beam axis. Energy calibrations of the detectors were made using stationary 88 Y, 137 Cs, and 60 Co γ-ray sources. The energy resolution of the array was measured to be 9% (6%) (FWHM) for 662 keV (1333 keV) γ rays. Doppler corrections, using the reconstructed vertex position in MI-NOS, were made to obtain the excitation energies in the center-of-mass frame. An add-back procedure, with a maximum distance between two γ rays of 15 cm, was performed to improve the peak-to-background ratio and photo-peak efficiency. The detection threshold was set to approximately 170 keV. From GEANT4 simulations [33], a full-energy photo-peak efficiency of 32% (20%) for 0.5 MeV (1 MeV) γ rays emitted at β = 0.6 was obtained prior to applying add-back procedures.

III. Results

Transition energies for 69,71,73 Co measured in this work are listed in Tab. I together with energy uncertainties and relative γ ray intensities corrected for detection efficiency. Spin-parities of the excited states could not be determined by means of momentum distribution and angular correlations of γ rays due to resolution as the most limiting factor. Experimental data were fitted with generated detector response functions via GEANT4 simulations together with two exponential functions for the background. Intrinsic resolutions of individual DALI2 crystals with the 5 mm FWHM resolution in the vertex reconstruction were taken into account for the simulations. The energy calibration uncertainty (5-keV), statistical uncertainty, and systematic uncertainty were added in quadrature to deduce the final transition energy uncertainties. Lifetime estimates of low energy transitions in 69 Co were obtained from the minimization of χ 2 -distributions as a function of transition energy and lifetime. The χ 2 values were obtained from the fitting of the experimental spectrum with simulated response functions in steps of 25 ps and 5 keV.

Figure 3 presents Doppler-corrected spectra obtained from the 70 Ni(p, 2p) 69 Co reaction. Two γ-γ coincidence spectra for the reaction are shown in the insets. For this isotope, transitions were observed at 240 (11), 287 (11), 427 (11), 488 (11), 662 (20), 1102 (25), 1285 [START_REF] Hamamoto | [END_REF], and 1591(8) keV of which the latter one was the strongest (I 1591 = 100(8)%). For the low energy transitions of 240 (11), 287 (11) and 427 (11) keV, lifetimes were obtained to be 144 (40), 271(100) and 78(60)-ps respectively. Subsequent γ-γ analysis revealed coincidences between the 1102 and 488-keV transitions. As the sum of these two energies matches well with the 1591-keV transition, they are believed to originate from the same level. Intensity arguments put the 1102-keV transition (I 1102 = 54(8)% with respect to the highest intensity γ) below the 488-keV transition (I 488 = 25(6)%). Relative intensities of other transitions were measured to be the following: I 240 = 48(6)%, I 287 = 31(6)%, I 427 = 29(6)%, I 662 = 15(6)%, I 1285 = 22(8)%.

The Doppler-corrected γ-ray energy spectra in coincidence with the 72 Ni(p, 2p) 71 Co reaction are presented (6) 291 [24] 427 (11) 78(60) 29( 6) 446 [24] 488 (11) 25( 6) 1102 662 (20) 15 (6) 648, 663 680 [24] 1102 (25) 54(8) 488 1105 [24] 1285 [START_REF] Hamamoto | [END_REF] 22( 8) 1591 [START_REF] Hamamoto | [END_REF] 100( 8)

71 Co 246( 16) 100( 34) 892 (20) 93(54) 925 925 (27) 88( 56) 73 Co 234( 11) 20( 10) 403( 15) 20( 16) 752 (33) 100( 24) 1044, 403 1044 (37) 48( 24) 752 1790 (20) 82( 30 in Fig. 4. Spectra were fitted in the same way as for 69 Co. Transition energies were observed at 246 (16), 892 (20), and 925 (27) keV, with relative intensities of I 246 = 100(34)%, I 892 = 93(54)% and I 925 = 88(56)% with respect to the highest intensity γ. The latter two transitions were obtained from the self coincidence of the 800-1050 keV region, which suggests the existence of two coincident γ-ray transitions merging into one structure in the spectrum. No γ-ray transitions were observed in coincidence with the 246-keV transition.

The Doppler-corrected γ-ray spectrum following the 74 Ni(p, 2p) 73 Co reaction is shown in Fig. 5. Transi-tion energies were observed at 234 (11), 403 (15), 752 (33), 1044 (37), and 1790(20)-keV. The measured relative intensities were I 234 = 20(10)%, I 403 = 20(16)%, I 752 = 100(24)%, I 1044 = 48(24)% and I 1790 = 82(30)% with respect to the highest intensity γ. The γ-γ coincidence analysis revealed that the 752-keV transition is in coincidence with the 1044-keV transition. Furthermore, the sum of the 752-keV and 1044-keV transitions corresponds, within error bars, to the 1790-keV transition. Therefore, it is assumed that they belong to the same decay branch, with intensity arguments putting the 752-keV transition of the cascade below the 1044-keV transition. Also the 752-keV transition was found to be in coincidence with the 403-keV transition. Possibly due to low statistics and low peak-to-background ratio, the 234-keV transition was not clearly observed in γ-γ coincidence. Proposed level schemes for 69,71,73 Co nuclei are presented in Fig. 6.

Inclusive and exclusive cross sections of the (p, 2p) reaction channels for 69,71,73 Co are presented in Table II. Transmission coefficients of ZeroDegree and reaction losses in the secondary target were taken into account in the extraction of the cross sections. Cross sections to the ground state, σ g.s. , were estimated by subtracting the sum of exclusive cross sections, σ ex , to excited states from the inclusive cross section, σ inc . Due to pandemonium effects and the unplaced transitions they represent upper limits. The ratio σ g.s. /σ inc was then estimated to be less or equal to 73(10)%, 68(18)% and 73(20)% for 69,71,73 Co nuclei, respectively. Quoted uncertainties of exclusive cross sections also include statistical uncertainties arising from the of γ-ray intensities.

IV. Discussion

Level schemes for 69,71,73 Co nuclei were reconstructed based on γ-ray transitions showing clear coincidences with each other. They are presented in Fig. 6. The γ-ray coincidence analysis for 69 Co has shown clear existence of one independent decay path, consisting of the 1102 and 1591-keV levels. Energies of the γ rays for 69 Co, measured in the current experiment, are in good agreement with data from literature [24]. Transition energies of 1102( 25)-keV and 662(20)-keV are in good agreement with the 1105-keV transition and the 648, 663, 680-keV triplet observed following β-decay [24]. Despite similarity between the 1591(8)-keV transition and reported 1582-keV transition [24], it is suggested that two transitions originate from different energy levels. This is supported by the absence of the 488( 11)-keV transition in the previous experiment. The observed γ rays with energies of 240 (11) and 287(11)-keV, which were not placed in the level scheme, agree well with the 250 and 291-keV transitions observed in β-decay [24]. The observed 427(11)-keV transition may correspond to the 446-keV transition from Ref. [24]. Shell model calculations for 69,71 Co were carried out in the pf 5/2 g 9/2 d 5/2 valence space [34] using the LNPS interaction [34] and for 73 Co in the pf-sdg valence space using the PFSDG-U interaction [35]. Calculated level schemes for 69,71,73 Co are shown in Fig. 6 next to their experimental counterparts. Two structures are predicted for all three Co isotopes: A "spherical" band, which corresponds to πf -1 7/2 proton hole states, and a low-lying "deformed" K=1/2 -band that corresponds to πf -2 7/2 pf +1 states. Deformation parameters of the K=1/2 -band for each isotope can be found in Table III. The energy levels of 69 Co measured in the current experiment at 1102 and 1591 keV are in good agreement with the predicted 9/2 - and 7/2 -spherical states at 1130 and 1420 keV, which can be interpreted as members of the πf -1 7/2 ⊗ 2 + 1 ( 70 Ni) multiplet in a weak coupling scheme.

In addition, 69 Co may feature the existence of a lowlying deformed band. These states may be populated in the (p, 2p) reactions by removal of a proton from the p 3/2 or f 5/2 orbital when two f 7/2 protons are elevated above the Z = 28 shell to the 0 + g.s. in the correspond-ing nickel isotope. For 62,64 Ni it was estimated that the p 3/2 orbital is filled up to 12% [37,38], and a similar shell structure is expected in heavier Ni nuclei. "Deformed" states are interpreted in the shell-model as members of a strongly coupled K = 1/2 -band, which has a (1p -2h + 2p -3h) structure across Z = 28 for protons and (4p -4h) across N = 40 for neutrons. The relative position of the 1/2 - def and 7/2 - sph bands can be identified by measuring an M 3, 1/2 -→ 7/2 -transition, as already discussed in Ref. [24] for 69 Co. It was impossible to measure transitions with long lifetimes with the current experimental setup. We further note the existence of a low-lying, prolate deformed 0 + 2 1567-keV level was reported in Ref. [36] and well described theoretically in Ref. [16]. A proton hole 7/2 -state coupled to this deformed 0 + 2 state would produce a deformed, excited state in 69 Co at similar excitation energy. It is stressed, however, that the present knockout experiment was only sensitive to the overlaps between the states in the Co isotopes and the ground state of the corresponding Ni isotopes.

Also the level schemes of 71,73 Co may have low-lying bands, as evinced by the low-energy γ-ray transitions, in addition to the spherical states. Indeed, such bands are predicted with the the LNPS and PFSDG-U interactions, as shown in Fig. 6. In 71 Co, the observed excited state at 1818-keV likely corresponds to the predicted 7/2 -level at 1600-keV, while the experimental level at 892-keV corresponds to the 9/2 -level, in agreement with expected decay patterns. Since the 246-keV transition was not found in coincidence with the transitions from spherical states, it is suggested that this transition belongs to the deformed band. The 246-keV transition likely corresponds to either the predicted 5/2 -→ 1/2 -transition, or the 3/2 -→ 5/2 -transition. The isotope 73 Co manifests a similar level scheme. The experimental level at 1790-keV is well matched by the predicted 7/2 -level at 1560-keV. Likewise, the experimental level at 752-keV corresponds to the predicted 9/2 -counterpart at 710-keV. Analogously, the experimental transition at 234-keV may correspond to a transition between the predicted deformed 3/2 -or 5/2 -levels at 870 and 850-keV decaying into the 1/2 -level at 670-keV. As for 69 Co, no relative position of 1/2 - def and 7/2 - sph states in 71,73 Co was determined with the current experimental set-up.

Inclusive and exclusive cross sections presented in Table II provide more insights on the shell structure of 69,71,73 Co isotopes. Cross sections to the ground state were evaluated to be less than or equal to 70% with respect to the inclusive cross sections for 69,71,73 Co isotopes. The tentative spin-parity assignments of excited states of 69,71,73 Co isotopes can be corroborated by these cross sections. In a single particle model, the knock-out of an f 7/2 proton from Ni isotopes by the (p, 2p) reaction simply results in the 7/2 -ground state of the respective Co isotope. This argument is supported by the large spectroscopic factors to the 7/2 -ground states extracted for 61,63 Co nuclei [38][39][40][41]. These studies also found spec-FIG. 6. Proposed level schemes and tentative spin-parity assignments for 69,71,73 Co and according shell model calculations using the LNPS and PFSDG-U interactions. The relative position of spherical and deformed states was not measured during the experiment. The relative position of decay bands is defined with respect to "x" and "y" shifts, which can take on different values for each level scheme. Values of 0 + and 2 + Ni states are taken from Ref. [36,47] See text for details. troscopic factors of C 2 S ≈ 0.6 for the first excited 7/2 - state.

Theoretical single-particle cross sections, σ sp , presented in Tab. II, were calculated in the Distorted Wave Impulse Approximation (DWIA) framework [42] and averaged along the target length. The Bohr-Mottelson single-particle potential [START_REF] Bohr | Nuclear Structure[END_REF] was used to obtain the single-particle wave function and the nuclear density. The depth of the potential was adjusted to reproduce the single-particle energy. The microscopic folding model [START_REF] Toyokawa | [END_REF] with the Melbourne g-matrix interaction [45] and the calculated nuclear density was used to construct the optical potentials for the distorted waves in the initial and final channels. The spin-orbit part of distorting potentials was disregarded. The pp interaction was adopted from the Franey-Love effective interaction [46]. This approach is estimated to provide theoretical single-particle cross sections with a systematic uncertainty of 15% [42]. It resulted in ground state ratios (σ ex g.s /σ sp ) of 5.6(9), 4.3(13) and 4.3(12) for 69,71,73 Co isotopes, respectively. These values are similar to the ground state spectroscopic factors obtained for 61,63 Co nuclei [38][39][40][41], but it must be stressed that due to unobserved feeding and the unplaced transitions the obtained values in the present work are only upper limits. Similarly, large values of σ ex suggest an assignment of the second-excited state of the spherical band in the 69,71,73 Co nuclei to be J π = 7/2 -. Moreover, shell model calculations and decay patterns then suggest the first-excited state of the spherical band in 69,71,73 Co to have a spin-parity of J π = 9/2 -. This state cannot be directly populated by the (p, 2p) reaction, but through feeding from higher-lying excited states. Indeed, low cross sections were observed for the tentatively assigned 9/2 -states.

Finally, it is interesting to compare the energy systematics between the 9/2 - 1 states relative to the 7/2 - 1 states -for most isotopes it is the g.s. -in the 69,71,73 Co isotopic chain and the corresponding 2 + 1 excitation energies for the Ni isotopes. The comparison displayed in Fig. 6 reveal a clear correlation between the 9/2 - 1 states and respective 2 + 1 excitations in the Ni isotopes. This tendency suggests that the 9/2 - 1 states of Co isotopes are part of the πf -1 7/2 ⊗ 2 + 1 (Ni) multiplet. The significant drop of the 9/2 - 1 energies in 69,71,73 Co isotopes, comparing to 9/2 - 1 of 67 Co reported in [19][20][21][START_REF] Walters | AIP Conference Proceedings[END_REF], may be understood as an increase of the paring correlations between protons and neutrons. Shell model calculations presented in Table III also support the changes in shell structure of the Co isotopes, as the intrinsic (mass) quadrupole moment Q 0 , increases by ∼ 120 e•fm 2 from Q 0 = 180 e•fm 2 (calculated in Ref. [19]) for 67 Co to Q 0 ∼ 300 e•fm 2 for 69,71,73 Co.

V. Summary and Conclusions

In the present work, excited states in neutron-rich oddeven 69,71,73 Co nuclei have been populated via quasifree (p, 2p) knockout reactions. BigRIPS and ZeroDegree spectrometers were used together with the DALI2 array and the MINOS hydrogen target and TPC to extract their excitation energies. Level schemes were reconstructed via γ -γ analysis for the first time. Large-scale shell-model calculations using LNPS and PFSDG-U effective interactions were confronted with our experimental findings. Spin-parity of reconstructed levels were proposed based on the measured inclusive and exclusive cross sections. The single-particle cross sections, calculated in DWIA, were used to evaluate the ratio between σ ex and σ sp . Tentatively assigned 9/2 - 1 states of 69,71,73 Co isotopes follow the trend of the 2 + 1 level energies of the nickel isotopes. Taking into account the systematic behavior and the good agreement with shell-model calculations, the excited 7/2 -and 9/2 -states can be interpreted as πf -1 7/2 ⊗ 2 + (Ni) configurations.

FIG. 1 .

 1 FIG. 1. (Color online) Particle identification plot for radioactive ions passing BigRIPS for the first (left) and second setting (right).

FIG. 2 .

 2 FIG. 2. (Color online) Particle identification plot for radioactive ions arriving at ZeroDegree for the first (left) and second setting (right).

FIG. 3 .

 3 FIG. 3. (Color online) (a) Doppler-corrected γ ray spectrum for 69 Co populated via the 70 Ni(p, 2p) 69 Co reaction. γ-γ coincidence spectra for the (p, 2p) reaction are shown in the insets with a gate on the 1102-keV transition, (b), and a gate on the 488-keV transition, (c). The analysis was made for γ ray multiplicity, Mγ = 2 for (b) and Mγ ≤ 3 for (c). Individual response functions for each transition are shown as long-dashed black curves, background with short-dashed red curves, and data fits with solid red curves.

  FIG. 5. (Color online) (a) Doppler-corrected γ-ray spectrum for 73 Co populated via the (p, 2p) reaction. γ-γ coincidence spectra are shown as as insets without background subtraction and all Mγ, a gate on 752-keV, (b), and a gate on 1044-keV (c). Individual response functions for each transition are shown as long-dashed black curves, background as short-dashed red curves, and data fits as solid red curves.

TABLE I .

 I Summary of γ-ray transitions for69,71,73 Co nuclei measured in the current experiment. Relative intensities for 69 Co were measured for Mγ ≤ 3, for 71,73 Co -for full gamma multiplicity Mγ.

	Isotope Eγ (keV) τ (ps)	Iγ(%) γ-γ (keV) Literature
	240(11) 144(40) 48(6)	250 [24]
	287(11) 271(100) 31
	69 Co

TABLE II .

 II Inclusive and exclusive cross sections of the (p, 2p) reaction channel for 69,71,73 Co nuclei. Exclusive cross sections are presented together with proposed values for J π . The single-particle cross sections (σsp) were calculated from the DWIA framework. See text for details.

	Isotope	J π	Eex	σex	σsp	σex/σsp
			(keV)	(mb)	(mb)
	69 Co	(7/2 -1 ) (9/2 -1 ) 1102 sph < 0.7(1) 0 < 9.6(6) (7/2 -2 ) 1591 sph 2.8(2)	1.70 1.66	<5.6(9) 1.7(3)
		inclusive		13.1(6)
	71 Co	(7/2 -1 ) (9/2 -1 ) (7/2 -2 ) 1817 sph 0 892 sph	< 6.6(17) 1.54 <4.3(13) 0.0(10) 1.6(8) 1.44 1.1(6)
			246 def	1.5(6)
		inclusive		9.7(6)
	73 Co	(7/2 -1 ) (9/2 -1 ) (7/2 -2 ) 1790 sph 0 752 sph	< 6.0(14) 1.40 <4.3(12) 0.0(10) 1.5(6) 1.36 1.1(5)
		inclusive		8.0(10)

TABLE III .

 III Properties of the deformed K = 1/2 -bands in69,71,73 Co obtained from shell-model calculations. The LNPS interaction was used for69,71 Co and the PFSDG-U interaction for 73 Co.

		69 Co	71 Co	73 Co
	Quadrupole moment Q0 (e•fm 2 )	300	290	320
	Deformation parameter (β)	0.3	0.3	0.3
	B(E2,5/2 -→1/2 -) (e 2 fm 4 )	500	440	420
	E(1/2 -) def -E(7/2 -) sph (keV)	-240	+60	+670

VI. Acknowledgments

The authors are thankful to the RIBF and RIKEN accelerator teams for stable operation of the uranium primary beam during the experiment and the production of the secondary beams. We acknowledge the support by the University of Hong Kong within RGC Early Career Scheme (ECS) (Contract 27303915). The development of MINOS and the core MINOS team have been supported by the European Research Council through the ERC Grant No. MINOS-258567. A.O. has been supported by the JSPS long-term fellowship L-13520 at the RIKEN Nishina Center. C.S. has been supported by the IPA program at the RIKEN Nishina Center. A.P. has been supported by the Ministerio de Ciencia, Innovacion y Universidades (Spain), Severo Ochoa Prog. SEV-2016-0597 and grant PGC-2018-94583. L.X.C would like to thank MOST for its support through the Physics Development Program Grant No. ÐTÐLCN.25/18. R.T. has been supported by the JSPS Grant-in-Aid for JSPS Research Fellows JP14J08718. K.O. acknowledges the support by Grant-in-Aid for Scientific Research JP16K05352.