B Sicardy

R Leiva

S Renner

F Roques

M El Moutamid

P Santos-Sanz

J Desmars

Ring dynamics around non-axisymmetric bodies with application to Chariklo and Haumea

Dense and narrow rings have been discovered recently around the small Centaur object Chariklo 1 and the dwarf planet Haumea 2 , while being suspected around the Centaur Chiron 3 , although this point is debated [START_REF] Ruprecht | stellar occultation by 2060 Chiron: Symmetric jet-like features[END_REF] . They are the first rings observed in the Solar System elsewhere than around giant planets. Contrarily to the latters, gravitational fields of small bodies may exhibit large non-axisymmetric terms that create strong resonances between the spin of the object and the mean motion of rings particles. Here we show that modest topographic features or elongations of Chariklo and Haumea explain why their rings are relatively far away from the central body, when scaled to those of the giant planets [START_REF] Esposito | Planetary rings[END_REF] . Resonances actually clear on decadal time-scales an initial collisional disk that straddles the corotation resonance (where the particles mean motion matches the spin rate of the body). Quite generically, the disk material inside the corotation radius migrates onto the body, while the material outside the corotation radius is pushed outside the 1/2 resonance, where the particles complete one revolution while the body completes two rotations. Consequently, the existence of rings around non-axisymmetric bodies requires that the 1/2 resonance resides inside the Roche limit of the body, favoring faster rotators for being surrounded by rings.

Chariklo and Haumea's non-axisymmetric gravity fields raise new and rich dynamical issues that are not encountered for rings around the giant planets. The strong coupling between the body and the surrounding collisional disk put tight constraints on the ring final location. This is the main topic of this paper, while the possible formation scenarios for those rings, that are discussed elsewhere [START_REF] Braga-Ribas | A ring system detected around the Centaur (10199) Chariklo[END_REF][START_REF] Ortiz | The size, shape, density and ring of the dwarf planet Haumea from a stellar occultation[END_REF][START_REF] Sicardy | Planetary ring systems[END_REF][START_REF] Pan | On the mass and origin of Chariklo's rings[END_REF][START_REF] Hyodo | Formation of centaurs' rings through their partial tidal disruption during planetary encounters[END_REF][START_REF] Araujo | The rings of Chariklo under close encounters with the giant planets[END_REF][START_REF] Wood | The Dynamical History of Chariklo and Its Rings[END_REF][START_REF] Melita | Assessment of different formation scenarios for the ring system of (10199) Chariklo[END_REF] , will not be addressed here.

To be more specific, Haumea is a triaxial ellipsoid with principal semi-axes A>B>C and elongation ∼ 0.61 (see Table 1). Chariklo's shape is less constrained due to scarce observations. Extreme solutions [START_REF] Leiva | Size and Shape of Chariklo from Multi-epoch Stellar Occultations[END_REF] are a spherical Chariklo of radius R sph = 129 km with topographic features of typical heights z ∼ 5 km, or an ellipsoid with elongation ∼ 0.20. The two bulges associated with those elongations contain large masses (of order ) compared to the body itself. Even a 5km topographic feature on Chariklo represents a mass anomaly µ ∼ (z/2R sph ) 3 ∼ 10 -5 relative to the body. This is much larger than the mass of Janus (a small satellite that confines the outer edge of Saturn's main rings) with µ ∼ 3 × 10 -9 , or putative Saturnian mass anomalies [START_REF] Hedman | More Kronoseismology with Saturns rings[END_REF] , with µ < 10 -12 , supporting the strong coupling mentioned earlier.

Fig. 1 outlines two possible configurations of Chariklo's dynamical environment, one with a mass anomaly and one with an elongated body. In both cases, there are four fixed points C 1 , ...C 4 near the corotation radius a cor ∼ (GM/Ω 2 ) 1/3 = R/q 1/3 , where the dimensionless rotation parameter q is defined by

q = Ω 2 R 3 GM , (1)
G being the gravitation constant, M the mass of the body, Ω its spin rate, and R denoting either the radius R sph of a sphere or the reference radius of the ellipsoid (Table 1).

In principle, the region around C 2 or C 4 may host ring arcs, but these points being potential maxima, arcs are unstable against dissipative collisions over time scales of some 10 4 years at most. Moreover, for Chariklo's elongations larger than the critical value crit ∼ 0.16 (which is the case here with the adopted value = 0.20), the points C 2 and C 4 are linearly unstable (see Methods). Consequently, particles moving away from C 2 or C 4 rapidly collide with the body (Fig. 1), This problem is exacerbated in the case Haumea, because of its larger elongation, ∼ 0.61.

Particles with mean motion n and epicyclic frequency κ experience Lindblad Resonances (LRs) for κ = m(n -Ω), m integer.

(

) 2
The resonances occur either inside (m>1) or outside (m<0) the corotation radius (Fig. 1), with only even values of m allowed in the ellipsoid case, due to symmetry reasons (Methods). Only disks revolving in a prograde direction are considered here. Retrograde resonances are weaker 14 and will not be studied here. Since κ ∼ n, the relation above reads n/Ω ∼ m/(m -1), referred to as an m/(m-1) LR. In a disk dense enough to support collective effects (self-gravity, pressure or viscosity), an m/(m -1) LR forces a m-armed spiral wave that receives a torque

Γ m = sign(Ω -n) 4π 2 Σ 0 3n (GM ) 2 ΩR 2 A 2 m . (3)
This formula encapsulates in separate factors the sign of the torque, the physical parameters of the disk (n and its surface density Σ 0 ) and of the perturber (M , R, Ω), and an intrinsic dimensionless strength factor A m , see Methods. Importantly, this is a generic formula that applies in contexts as different as galactic dynamics [START_REF] Goldreich | The excitation of density waves at the Lindblad and corotation resonances by an external potential[END_REF][START_REF] Hopkins | An analytic model of angular momentum transport by gravitational torques: from galaxies to massive black holes[END_REF] , circum-stellar accretion disks 17 , protoplanetary disks [START_REF] Goldreich | Disk-satellite interactions[END_REF] or planetary rings and interiors [START_REF] Goldreich | The dynamics of planetary rings[END_REF][START_REF] Meyer-Vernet | On the physics of resonant disk-satellite interaction[END_REF][START_REF] Marley | Planetary acoustic mode seismology: Saturn's rings[END_REF] . Moreover, both the sign of the torque and its value are largely independent of the physics of the disk 20 , providing a robust estimation of Γ m even ignoring the detailed processes at work.

Eq. (3) shows that the LRs cause the migration of the disk material away from the corotation. An annulus of width W and average radius a has most of its angular momentum H ∼ 2πaW Σ 0 √ GM a = 2πΣ 0 W (ΩR 3 /q) transfered to the body over a migration time scale

t mig ∼ H | Γ m | = 3q 4π 2 W R T rot [(m -1)/m]A 2 m , (4)
where T rot = 2π/Ω is the rotation period of the body. The summation includes the relevant torques Γ m that apply inside the annulus (Fig. 2). Note that the current angular momentum of Chariklo's rings is less than 10 -5 of that of the body [START_REF] Braga-Ribas | A ring system detected around the Centaur (10199) Chariklo[END_REF][START_REF] Sicardy | Planetary ring systems[END_REF] . Even considering an initial disk one hundred times more massive, the reaction torque of the disk on the body has a negligible effect on Chariklo's rotation rate, with similar conclusions for Haumea.

We estimate t mig for two annuli around Chariklo, one initially placed inside the corotation radius, and one placed outside. Fig. 2 shows that (i) a difference A-B as small as a kilometer ( < ∼ 0.01) causes a rapid, decadal scale outward migration of the outer annulus; (ii) the resonances on the inner annulus are weaker, but t mig remains geologically short ( < ∼ Myr) for A -B > ∼ 5 km; (iii) even ∼ 5-km topographic features are sufficient to induce migration time scales of a few Myr.

Numerical simulations can test those mechanisms. Global collisional codes have been run [START_REF] Michikoshi | Simulating the Smallest Ring World of Chariklo[END_REF] , but with no torque appearing as the potentials considered were axisymmetric. Other local simulations do consider elongated bodies [START_REF] Gupta | Rings of non-spherical, axisymmetric bodies[END_REF] , but not rotating, hampering again any torque. Here we performed numerical integrations using a simple Stokes-like friction acting on the particles,

γ Stokes = -ηΩv r , (5)
where v r is the particle radial velocity and η is a dimensionless friction coefficient. This friction dissipates energy while conserving angular momentum, thus being a good proxy for collisions at low computing cost. Fig. 3 shows results using η = 0.01 (see Methods for the choice of η). As mentioned earlier, the specific form of γ Stokes and the value of η have little effects on the resonant torque Γ m , when compared to more realistic situations including collisions and self-gravity.

We have checked numerically the dependence t mig ∝ A -2 m (Eq. ( 4)). This permits to save computing time in the case of a mass anomaly by using µ = 0.005 (instead of ∼ 10 -5 ), hence speeding up migration time scales by a factor 500 2 = 2.5 × 10 5 , an effect accounted for in Fig. 3a,b,c,d. In contrast, the integration shown in Fig. 3e,f,g,h uses a realistic Chariklo's elongation = 0.20, with no further corrections applied. Fig. 3 confirms our calculations, i.e. (i) the rapid infall of particles onto Chariklo's equator inside the corotation radius, exacerbated by the fact that the large orbital eccentricities enhance collisions with the body, and due to the unstable character of the C 2 and C 4 points for = 0.20 (Fig. 1); (ii) the strong torques up to the 1/2 resonance, that pushes the disk material outwards. Our integrations actually show that inward of the 1/2 resonance, and between discrete LRs, the orbital eccentricities remain sufficiently excited to cause a slower but still significant migration away from the corotation radius.

A LR opens a cavity in the disk if Γ m exceeds the viscous torque 19 Γ ν = 3πna 2 νΣ 0 , where the kinematic viscosity ν = h 2 n is related to the ring vertical thickness h, see Methods. From Eq. (3), we obtain

Γ m Γ ν ∼ 4π 9q 4/3 m -1 m 5/3 R h 2 A 2 m . (6)
Using h ∼ 10 m (see Methods) and z ∼ 5 km we get |Γ -2 /Γ ν | ∼ 3 × 10 -2 for m = -2 (2/3 outer LR). Thus, a 5-km feature is too weak to open a cavity, but not by much owing to the steep dependence of A 2 m ∝ µ 2 ∝ z 6 . In contrast, the torque exerted by an ellipsoid with = 0.20 is overwhelming (by six orders of magnitude) at the 2/3 LR compared to Γ ν . Since A -2 ∝ (see Methods), ellipsoids with A -B as small as 0.1 km are actually able to carve a cavity inside the 2/3 LR.

Chariklo and Haumea's elongations considered here are large enough to strongly perturb a ring near the 1/2 resonance, although no torque formula is available at that resonance in the ellipsoid case, because it is of second order nature (it must actually be noted 2/4, and is not a LR, see Methods). This said, the final radius of the cavity depends on processes that are not considered here, since our friction law is an oversimplification of actual collisions. More importantly, accretion into satellites takes over as the Roche limit is approached, leading to complex ring-satellites interaction like shepherding. Nevertheless, our results show that either due to mass anomalies or body elongation, rings should not exist inward of the 1/2 resonance at a 1/2 = 2 2/3 a cor . This is consistent with what is observed for Chariklo and Haumea.

In fact, the ring existence requires that a space exists between a 1/2 and the Roche limit a Roche , to prevent the ring accretion into satellites. From a Roche ∼ (3/γ) 1/3 (M/ρ ) 1/3 , where ρ is the density of the ring particles, and γ is a factor describing the particle shape [START_REF] Tiscareno | Compositions and Origins of Outer Planet Systems: Insights from the Roche Critical Density[END_REF] , the condition a 1/2 < a Roche reads

γρ < ∼ 3 4 Ω 2 G . (7)
Thus, a non-axisymmetric body must rotate fast enough and/or the particles be underdense enough for a ring to exist. Although γ and ρ are poorly known, we can consider the preferred value γ = 1.6 that describes particles filling their lemon-shaped Roche lobes [START_REF] Porco | Saturns small inner satellites: clues to their origins[END_REF] , and ρ ∼ 450 kg m -3 , typical of the small moons orbiting near Saturn's rings [START_REF] Thomas | shapes, and derived properties of the saturnian satellites after the Cassini nominal mission[END_REF] , and a good proxy of ring particle densities. Eq. ( 7) then requires rotation periods shorter than about 7 h, a condition met by both Chariklo and Haumea.

Our model predicts that the inner part of the disk may be deposited on the equator of the body, forming a ridge akin to that of the Saturnian satellite Iapetus. It has been proposed that this ridge is due to the presence of a transient ring that rained down onto Iapetus' equator due to the torque from of a former subsatellite [START_REF] Ip | On a ring origin of the equatorial ridge of Iapetus[END_REF][START_REF] Levison | Ridge formation and de-spinning of Iapetus via an impact-generated satellite[END_REF][START_REF] Dombard | Delayed formation of the equatorial ridge on Iapetus from a subsatellite created in a giant impact[END_REF] . This may happen on short time scales. For instance Eq. ( 4) shows that a 10 km-radius satellite orbiting at 500 km pushes an initial disk extending from the surface to 400 km in less than 1 Myr. In our case, the disk decay is caused by the body itself. The infall time scales being of many years (Fig. 2), impact angles on the surface are very shallow, with velocities less than a km s -1 , ensuring that the material piles up as a ridge instead of forming craters. Future stellar occultations might detect such ridges on Chariklo or Haumea.

In a broader and more speculative perspective, it is interesting to consider the orbital distribution of satellites of asteroids and Trans-Neptunian Objects. Supplementary Fig. 1 displays the histogram of the satellite orbital periods, expressed in units of the rotation periods of the primaries. Apart from a conspicuous peak corresponding to synchronous, tidally evolved orbits, this histogram indicates a clearing between the corotation radius and the outer 1/2 resonance, followed by a steady increase beyond this resonance. This distribution might be the signature of satellite formation proceeding from an initial collisional disk that has been pushed away by the resonant mechanism described here. (a) No error bars are considered here, the adopted parameters being representative of typical cases examined in this work.

(b) See Eq. ( 1).

(c) Defined as R = √ 3(1/A 2 + 1/B 2 + 1/C 2 ) -1/2
, see Methods Eq. ( 16). (d) See Methods Eq. ( 18). (e) Assuming a spherical body of radius R sph = 129 km (ref. 12). This corresponds to a mass anomaly µ ∼ (z/2R sph ) 3 ∼ 10 -5 . (f ) Using a cor = Rq -1/3 , from Eq. ( 1) and Kepler's third law.

(g) Using a 1/2 = 2 2/3 a cor , from Kepler's third law. (h) Using the classical expression a Roche ∼ (3/γ) 1/3 (M/ρ ), with γ = 0.85 and icy ring particles with density ρ = 1000 kg m -3 . More realistic values of γ and ρ are discussed in the text. In both panels (topographic feature on the left, elongated body on the right), the red (resp. blue) circles correspond to inner (resp. outer) m/(m -1) Lindblad resonance (LR) radii with m>1 (resp. m<0), see Eq. ( 2). The black lines show isopotential curves in a frame corotating with Chariklo, and the gray lines outline the limb of the body. The dots C 1 , ...C 4 mark the corotation fixed points. The points C 2 and C 4 are local potential maxima and are linearly stable if the mass anomaly or the elongation of the body are not too large, see Methods. Left -A topographic feature of height z = 5 km (gray half dome, not on scale) is sitting at the surface a body of radius R sph = 129 km, and corresponds to a mass anomaly µ ∼ 10 -5 . For better viewing, the isopotential black lines have been radially stretched by a factor of 50 with respect to the corotation radius. A few inner (m = 2, 3, 4, 5) and outer (m = -1, -2, -3, -4) LR radii are shown. Right -The same for a Chariklo shape solution with elongation = 0.20. The limb of the body and the isopotential lines are plotted on scale. Only LRs with m even are now allowed, the inner one corresponding to m = 6, and the two outer ones corresponding to m = -2, -4. The green curve is an orbit that starts at C 2 without applying any friction (conservative case). It is unstable because the elongation is larger than the critical value crit = 0.16 (see Methods), leading to a collision with the body after three months in the example shown here. 3)) vs. the resonant radii on each side of the corotation radius (dotted line). The values of A m are evaluated from Supplementary Table 1, using a Chariklo equatorial topographic feature of height z = 5 km, corresponding to a mass anomaly µ = 10 -5 (blue squares), or a difference of semi-axes A -B = 18 km (Table 1), corresponding to an elongation parameter = 0.20 (red squares). Note the steep decrease of the torques as the corotation radius is approached, due to the exponential decrease of A 2 m as |m| increases, see Methods. The light gray region at left encloses Chariklo's largest semi-axis A = 157 km, inside which particles collide with the body in the ellipsoidal case, while the dark gray region encloses Chariklo's radius R sph = 129 km in the spherical case (Table 1). b, Solid lines: migration times (Eq. ( 4)) of an outer annulus of width 100 km that extends outside the corotation (see a), either due to the topographic features (blue) of heights z or ellipsoids with various A -B (red). Dotted lines: the same for an inner annulus of width 20 km in the ellipsoidal case, and 60 km in the spherical case, see a. 5)). The radii of the corotation point C 2 (a cor ), the 2/3 and 1/2 outer Lindblad Resonances (LRs) between the particle mean motions and Chariklo's rotation period are marked at the bottom, together with the location of Chariklo's main ring 1 C1R. a, b, c, d, The effect of an equatorial topographic feature (black dot) with mass µ = 5 × 10 -3 relative to Chariklo. Initially, 701 particles are regularly placed between 0.7a cor and 2.2a cor . In all panels, each particle is plotted over twenty regular time steps spanning 40,000 years. a, After 40,000 years, the clearing of the corotation region is ongoing; b, After 2.5 × 10 5 years, some particles remain near C 2 , while others are pushed outside the 2/3 LR; c, After 2.5 × 10 6 years, all the particles inside the corotation radius and near C 2 have collided with Chariklo; d, After 6.3 × 10 6 years, all the remaining particles are now outside the 1/2 LR. e, f, g, h, Effect of an ellipsoid with elongation = 0.20, displayed with its longest axis face on. The particles now start between 1.1a cor and 2.2a cor (particles inside 1.1a cor collide with Chariklo after a few days). e, After three months, most of the particles have been pushed outside the 2/3 LR; f, g and h, After one, five and twelve years, respectively, all the particles have either collapsed onto Chariklo, or continue their outward migration at decreasing pace outside of the 2/4 resonance. Note that time scales of same order (but shorter) would be obtained for particles orbiting around Haumea, which has a larger elongation = 0.61.

Supplementary Figure 1 | Distribution of orbital periods of satellites around asteroids and Trans-Neptunians Objects. The orbital period P s of 179 satellites known around binary or multiple asteroids and Trans-Neptunians Objects (taken from http://www.johnstonsarchive.net/ astro/astmoontable.html as of April 2018) are plotted in units of the rotation period P p of their primaries. The resulting histogram of P s /P p shows a peak near unity, corresponding to tidally evolved systems, in which the primary rotates synchronously with the satellite orbital period. The vertical dotted line correspond to the outer 1/2 resonance, where the satellite completes one revolution while the primary completes two rotations. The steady increase of satellite presence beyond that resonance is in line with the model presented in the text, i.e. satellite formation in a primordial collisional disk that has been pushed outwards by the resonant torque of the 1/2 resonance.

Supplementary Figure 2 | Phase portrait of the 2/4 outer spin-orbit resonance. The phase portrait of the 2/4 resonance is shown for an ellipsoidal Chariklo with elongation = 0.20 (Table 1), with X = e cos(φ 2/4 ) and Y = e cos(φ 2/4 ), where e is the particle eccentricity,

φ 2/4 = 2λ -λ A -
is the resonant angle, and the various other angles are defined in the Methods. All the trajectories share the same Jacobi constant, see Murray & Dermott, Solar system dynamics, Cambridge University Press (1999) for details. This constant has been chosen so that the particle that starts at the origin (X, Y ) = (0, 0) is at exact resonance, i.e. with semi-major axis a 2/4 = a 1/2 , see Main Text. The origin is then an unstable hyperbolic point that forces particles initially on a circular orbit to acquire high eccentricities of the order of e ∼ 0.2, see Methods. This kind of topology occurs for a narrow semi-major axis range of a 1/2 (1 -0.25 ) < ∼ a < ∼ a 1/2 (1 + 0.25 ) around the resonance.

Methods

We calculate the potential outside a body in two simple cases: a topographic feature located at the equator of a spherical object and a homogeneous triaxial ellipsoid. Calculations are restricted to the equatorial plane of the body, where a collisonal disk is expected to settle.

Topographic feature

We consider a spherical body of mass M , radius R sph and center C with an equatorial topographic feature of mass µ relative to the mass of the body, that rotates with period T rot and angular velocity Ω = 2π/T rot . We denote O the center of mass of the body plus the topographic feature, r the position vector of the particle, measured from the center of the body, r = ||r||, θ = L -L A , where L is the true longitude of the particle, and L A = Ωt is the orientation angle of the topographic feature, counted from an arbitrary origin. Finally, R is the vector that connects the center of mass O to the topographic feature and ∆ = r-R The potential acting on the particle at position r in a frame fixed at C is:

U (r) = - GM r -µ GM ∆ + Ω 2 (CO • r), (8)
where the last term is the indirect part stemming from the motion of C around O. Using CO = µR and the definition of the rotational parameter q (Main Text Eq. ( 1)), we obtain

U (r) = - GM r -GM µ 1 ∆ -q R • r R 3 sph = - GM r - GM 2R sph µ +∞ m=-∞ b (m) 1/2 (r/R sph ) cos(mθ) -2q r R sph cos(θ) , (9)
where the b (m) 1/2 's are the classical Laplace coefficients.

Homogeneous triaxial ellipsoid

We now consider a homogeneous triaxial ellipsoid of mass M and semi-axes A>B>C. The potential U (r) can again be expanded in a series in cos(mθ), where θ is now the difference between the true longitude of the particle and the orientation of the largest axis of the ellipsoid. Only even values of m must appear to ensure the invariance of the potential under a rotation of π radians. Thus, posing m = 2p,

U (r) = +∞ p=-∞ U 2p (r) • cos (2pθ) . ( 10
)
In the 3-D case, we use the Eq. (64) of ref. 33, where the m index used by this author is replaced by p in our version. Correcting for the typo of ref. 33, in which the index m should vary from 0 to l (and not from 1 to l):

U (r) = - GM r    +∞ p=0 cos (2pθ)   +∞ l=p R r 2l C 2l,2p P 2l,2p (sin ϕ)      , ( 11
)
where ϕ is the latitude. The coefficient P l,p is the associated Legendre polynomial, whose classical expression is

P l,p (u) = (-1) p (1 -u 2 ) p/2 2 l l! d l+p du l+p (u 2 -1) l . ( 12
)
Note that the factor (-1) p is absent in Eq. ( 2) of ref. 33, but is transfered to the coefficient K lp used by this author (his Eq. ( 3)).

Here we consider particles moving in the equatorial plane of the body, i.e. ϕ = 0. Using the binomial expansion of (u 2 -1) l , we obtain:

P 2l,2p (0) = (-1) (l-p) (2l + 2p)! 2 2l (l + p)!(l -p)! (13)
Moreover, from the last equation of ref. 34, we have:

C 2l,2p = 3 R 2l l! (2l -2p)! 2 2p (2l + 3) (2l + 1)! 2 -δ (0,p) × int( l-p 2 ) i=0 (A 2 -B 2 ) p+2i C 2 -1 2 (A 2 + B 2 ) l-p-2i 16 i (l -p -2i)! (p + i)!i! , (14)
where δ (0,p) is the Kronecker delta function. Because we want both negative and positive values of p in Eq. ( 10), it is convenient to modify Eq. ( 11) so that p varies from -∞ and +∞. This can be accommodated by replacing the factor (2 -δ (0,p) ) by 1 in Eqs. 14, and replacing p by |p| when calculating P 2l,2p and C 2l,2p . Thus,

U 2p (r) = - GM r +∞ l=|p| R r 2l Q 2l,2|p| , (15)
where Q 2l,2|p| = C 2l,2|p| P 2l,2|p| (0). Moreover, the reference radius R is defined by

3 R 2 = 1 A 2 + 1 B 2 + 1 C 2 . (16)
Finally, a closed form of U (r) outside the body that depends on A, B and C can be derived, with

Q 2l,2|p| = 3 2 l+2|p| (2l + 3) (2l + 2|p|)!(2l -2|p|)!l! (l + |p|)!(l -|p|)!(2l + 1)! × int( l-|p| 2 ) i=0 1 16 i |p|+2i (|p| + i)!i! f l-|p|-2i (l -|p| -2i)! . ( 17
)
The dimensionless parameters and f measure the elongation and oblateness of the body, respectively:

= A 2 -B 2 2R 2 and f = A 2 + B 2 -2C 2 4R 2 . ( 18
)
Note that in the limiting case where tends to zero, ∼ (A -B)/A. In the limiting case where f tends to zero and the body is axisymmetric ( = 0), f ∼ (A -C)/A, the usual definition of oblateness.

The term Q 2l,2|p| is of order l in ( f ). For evaluating the effect of Lindblad resonances, it is enough to consider the term of lowest order in R/r in Eq. ( 15), corresponding to l = |p|. Defining the sequence S |p| = Q 2|p|,2|p| / |p| and from m = 2p, we obtain

U (r) = - GM r +∞ m=-∞ R r |m| S |m/2| |m/2| cos (mθ) (m even), (19)
where S |p| is recursively given by

S |p|+1 = 2 (|p| + 1/4)(|p| + 3/4) (|p| + 1)(|p| + 5/2) × S |p| with S 0 = 1. ( 20
)
The potential ( 19) has been implemented in numerical schemes to integrate the motion of particles around an elongated body (adding the Stokes-like friction of Main Text Eq. ( 5)). We have truncated the expansion of the potential above |m| > 10, which is justified by the fact that the resonance strength rapidly decreases as m increases (Fig. 2). In the numerical integrations, the initial velocities assigned to the particles is simply the local circular keplerian velocity around a body of mass M . The Stokes-like friction rapidly damps any radial motions caused by this choice, over a transient time scale 1/ηΩ. Taking η = 0.01 (see below), this corresponds to about 15 revolutions around the body, a short time compared to the integrations performed here. Moreover, Eq. ( 15) yields the axisymmetric part of the potential by making p = 0,

U 0 (r) = - GM r +∞ l=0 Q 2l,0 R r 2l , (21)
which in turns provides the particle mean motion and horizontal epicyclic frequency:

n 2 (r) = 1 r dU 0 (r) dr and κ 2 (r) = 1 r 3 d(r 4 n 2 ) dr . ( 22
)
In our numerical integrations, we have kept the lowest order term in R/r, corresponding to l = 1, i.e. U 0 (r) = -(GM/r) [1 + (f /5)(R/r) 2 ]. Identification with classical formulae shows that this corresponds to a body with dynamical oblateness J 2 = (2/5)f . This approximation provides:

n 2 (r) ∼ GM r 3 1 + 3f 5 R r 2 and κ 2 (r) ∼ GM r 3 1 - 3f 5 R r 2 (23)

Expansion of the disturbing function

The potential in Eq. ( 9) can be expressed as U (r) = -GM/r -R, where R is the classical disturbing function:

R(r) = GM 2R sph µ +∞ m=-∞ b (m) 1/2 (r/R sph ) cos(mθ) -2q r R sph cos(θ) ( 24
)
It can be expressed in terms of the orbital elements of the particle, a, e, λ and (semimajor axis, eccentricity, mean longitude and longitude of periapse, respectively), using e.g. the formalism presented in Chapter 6 and Appendix B of ref. 37 to describe perturbations by a satellite. This transformation uses operators f n that contain multiplications by powers of the ratio β = a/R sph and the differential operator D = d/dβ. The only difference with the case of satellite perturbations is that the ratio a/R sph replaces the ratio a/a s , where a s is the satellite semi-major axis.

In the case of an ellipsoid, R is derived from Eq. ( 19),

R(r) = GM r +∞ m=-∞(m =0) R r |m| S |m/2| |m/2| cos (mθ) ( 25
)
The same formalism as in ref. 37 can be applied, i.e. using exactly the same operators f n . Identifications term by term in the summations above show that it is sufficient for that to replace R sph by R and µb

(m) 1/2 (r/R sph ) by 2(R/r) |m|+1 S |m/2| |m/2| .

Corotation resonance

The potential near the corotation radius a cor , as observed in a frame corotation with the body, is

V (r) = U (r)- Ω 2 r 2 2 ∼ - 3 2 Ω 2 a 2 cor ∆r a cor 2 - GM a cor f (θ) = - 3 2 Ω 2 a 2 cor ∆r a cor 2 - Ω 2 R 2 q 2/3 f (θ), ( 26
)
where the azimuthal function f (θ) is given in Supplementary Table 1, and ∆r = r -a cor a cor . Examples of isopotential levels for the two cases examined here are displayed in Fig. 1. Note that the corotation points associated with the mass anomaly mimic the Lagrange points L 1 , ...L 5 , except that L 1 and L 2 have merged into a single saddle point C 1 where the potential remains finite. Also, the points C 2 and C 4 are close to but not at 60 degrees from C 1 . That angle actually depends on q (see Supplementary Table 1) and is close to 70 deg in the particular example displayed in Fig. 1.

Near a cor , the particles follow the trajectory (3/8)(∆r/a cor ) 2 + f (θ) ∼ constant. They are nothing else than the level curves of the potential in a frame corotating with the body (Fig. 1), except for a dilation by a factor two with respect to a cor (ref. 35). The full radial width of the trajectory is then W cor = 2 8∆f /3, where ∆f = f max -f min is the total variation of f (θ) over [0,2π[. For order of magnitude considerations, we note that ∆f ∼ µ for the case of the mass anomaly. In the case of the ellipsoid, and for sake of estimation, we can simplify the expression (19) further by taking the lowest orders p = 0 and |p| = 1, i.e. (noting that S 1 = 0.15)

V (r) ∼ - 3 2 Ω 2 a 2 cor ∆r a cor 2 - 3 10 R 2 Ω 2 cos (2θ) , (27)
so that f (θ) ∼ (3/10)q 2/3 cos(2θ) and thus ∆f ∼ (3/5)q 2/3 , from which we derive

W cor,µ ∼ 4Rq -1/3 2 3 µ and W cor, ∼ 4R 2 5 . ( 28
)
in each of the two cases examined here. For a typical Chariklo topographic feature (µ ∼ 10 -5 ), we obtain a narrow corotation region with W cor,µ ∼ 2 km only, while for ∼ 0.20, W cor, ∼ 130 km, meaning that the corotation region fills in all the space between a cor and Chariklo's surface (Fig. 1).

If ring arcs are present near C 2 and C 4 , they should be destroyed by viscous spreading time scales t spread ∼ W 2 cor /ν, where ν is the kinematic viscosity. This quantity can be parametrized as ν = h 2 n, where h typically represents, for a dense disk, the size of the largest particles, or equivalently, the vertical thickness of the ring [START_REF] Schmidt | Dynamics of Saturn's dense rings[END_REF] . The local velocity field in Chariklo or Haumea's rings are comparable to those of Saturn 1 . Consequently, the collisional physics in those systems is expected to be similar 6 , i.e. h ∼ 10 meters (ref. 36). From the expressions of W cor derived above, we obtain t spread,µ ∼ 2µ(R/h) 2 T rot for a mass anomaly µ, and t spread, ∼ (R/h) 2 T rot for an ellipsoid. With µ ∼ 10 -5 , we obtain very short escape times (a few years) of the arc material from the corotation region, if caused by a mass anomaly. The spreading time is longer, some 10 4 years, but still geologically short if the corotation is controled by an ellipsoid with elongation ∼ 0.20.

The corotation points C 2 and C 4 are linearly unstable if the potential V (r) meets the condition

4Ω 2 + V xx + V yy 2 ≤ V xx V yy -V 2 xy , (29)
where the indices x and y are short-hand notations for partial derivatives [START_REF] Murray | Solar system dynamics[END_REF] .

For the classical L 4 and L 5 points (corresponding to q = 1), this condition leads to the Gascheau-Routh criterium µ > 0.0385.... For the cases examined here, q is smaller than but of order unity, so that the critical value of µ remains close to 0.04. This value is safely avoided for Chariklo, as it would correspond to an unrealistic feature with z = 80 km.

In the case of the ellipsoid, it is found from Eqs. ( 27) and ( 29) that C 2 and C 4 are unstable for:

> crit ∼ 0.06 q 2/3 . (30)
Using q = 0.226 for Chariklo implies crit ∼ 0.16, which is a bit smaller than to Chariklo's adopted elongation (Table 1), making the points C 2 and C 4 unstable, see Main Text. Haumea's elongation = 0.61 is well beyond the critical value, making C 2 and C 4 highly unstable.

Lindblad resonances

A particle revolving around the central body is submitted to a potential of generic form

U (r) = +∞ -∞ U m (r) • cos (mθ) = +∞ -∞ U m (r) • cos [m(L -λ A )] . (31)
The quantities r and L can be expressed in terms of the keplerian elements of the particle, a, e, λ and . In doing so, terms with frequency jκ -m(n -Ω) appear in the expansion of U (r), where κ is the particle horizontal epicyclic frequency and j is a non negative integer. Each term for which jκ ∼ m(n -Ω) (j integer > 0) (32) describes a resonance between the mean motion of the particle and the spin rate of the body.

For bodies close to spherical, we have κ ∼ n, and the condition above reads

n Ω ∼ m m -j , (33)
referred to as an orbit-spin m/(m -j) resonance. Its associated resonant critical angle is

φ m,j = [mλ A -(m -j)λ -j ]/j. ( 34
)
In the case of an ellipsoid, the potential (19) contains only even terms of the form 2pθ, so that the only resonances encountered have the form

n Ω ∼ 2p 2p -j . ( 35
)
The classical d'Alembert's rule implies that the term responsible for the m/(m -j) resonance is of order e j . Consequently, the strongest resonances are those with j = 1, and are classically referred to as Lindblad Eccentric Resonances, or simply Lindblad Resonances (LRs).

Note that we classically restrict the qualifier "Lindblad" to resonances of first order in e, i.e. of the form κ = m(n -Ω). Lindblad resonances of higher orders k > 1 are possible when they are forced by a satellite with eccentric orbit e s . Then, the resonant term in the forcing potential is proportional to ee k-1 s , i.e. still of first order in e, but of total order k in ee s . In those cases, Ω must be replaced by a pattern speed Ω pat that accounts for higer-order harmonics in the satellite motion [START_REF] Murray | Solar system dynamics[END_REF] . Those complications do not arise here because all the mass elements of the body execute circular motions around the center of mass.

The corresponding terms in the expansion of U (r) are easily obtained by using the first order expansions, r ≈ a -ae cos(λ -) and L ≈ λ + 2e sin(λ -). Introducing them into

+∞ -∞ U m (r) • cos [m(L -λ A )],
we obtain to first order in eccentricity

U (r) = +∞ k=-∞ U k (a) • cos [k(λ -λ A )] -e +∞ m=-∞ A m (a) • cos(φ m ), (36)
where

φ m = mλ A -(m -1)λ - (37)
is the resonant angle associated with the m/(m -1) LR and

A m (a) = [2m + a(d/da)] U m (a). ( 38
)
In order to separate the effects of the physical parameters of the body (Ω, R, q) and the effects of the resonances per se, we define a new dimensionless coefficient A m = -(q/Ω 2 R 2 )A m , so that the potential can eventually be split into a corotation and a Lindblad resonance part,

U (r) = +∞ k=-∞ U k (a) • cos [k(λ -λ A )] + e Ω 2 R 2 q +∞ m=-∞ A m (a) • cos(φ m ). ( 39
)
The coefficients A m are obtained from Eqs. 9, 19 and 38, and are listed in Supplementary Table 1. For large values of |m|, A m has the exponential behaviour A m ∝ K |m| (K being a constant depending on the problem considered). Using Chariklo's parameters (q = 0.226), we obtain asymptotically that A m ∝ 0.54 |m| µ ∝ 0.54 |m| z 3 in the mass anomaly case, and A m ∝ (1.93 q 2/3 ) |m/2| = (0.72 ) |m/2| in the ellipsoid case, using q = 0.226 (Chariklo case). Note that m is even in the latter case and that the 2/3 LR is the strongest of all (Fig. 2), with A -2 ∝ .

Choice of the friction coefficient η

Main Text Eq. ( 5) introduces a dimensionless friction coefficient η that quantifies the drag applied to the ring particles in our numerical integrations. Note that we do not consider any other forces acting on the particles, such as radiation pressure or Poynting-Roberston (PR) drag. This is justified by the fact that both Chariklo and Haumea's rings probably contain mainly cmto m-sized particles, which are stable against PR drag over hundreds of millions years [START_REF] Braga-Ribas | A ring system detected around the Centaur (10199) Chariklo[END_REF][START_REF] Sicardy | Planetary ring systems[END_REF] . This said, the choice of η is rather arbitrary as it does not enter in the expression of the torque Γ m (Eq. ( 3)). However, it does define the typical width ∆a m of the m/(m -1) LR, defined as the region over which most of the torque Γ m is deposited around the resonance radius a m . In order to be as realistic as possible about ∆a m , we choose the value of η to match the expected disk properties.

Following the formalism of ref. 20, the dimensionless width α = ∆a m /a m of the resonance is determined by the dominant physical process at work in the disk, which can be self-gravity, viscosity or pressure. In the self gravity case, α is given by

α G = 2π|m -1|GΣ 0 3m 2 Ω 2 a m , ( 40
)
where G is the gravitational constant and Σ 0 is the disk surface density. Using Σ 0 = 500-1000 kg m -2 (ref. 6), a rotation period of 7 h (Table 1), we obtain a typical value α G ∼ 2×10 -3 .

If viscosity prevails, then α takes the form α ν = [7ν/(9|m|Ωa 2 m )] 1/3 = [7/(9|m-1|)] 1/3 (h/a m ) 2/3 . Taking h ∼ 10 m and a typical a m ∼ 250 km, we obtain α ν < ∼ 10 -3 , with similar values if the disk is pressure-dominated. This shows that Chariklo's rings are likely to be dominated by self-gravity near LRs. Finally, the coefficient α associated with a Stokes-like force as in Main Text Eq. ( 5) is α η = 2η/3|m|, so that η ∼ 0.01 provides a realistic estimation of the LR widths in Chariklo's rings, i.e. α η ∼ α G . The same exercise can be performed for Haumea's rings, yielding smaller values of η, since both the spin rate Ω and the radii a m are larger in this case. However, the orders of magnitude remain the same and the main conclusions of this work are not altered.

Higher order resonances

Besides the first order LRs considered in the Main Text (Eq. ( 2)), higher order n/Ω = m/(m -j) resonances appear, corresponding to j > 1 (as explained above, they are not Lindblad resonances). Being of order e j , they are weaker than the LRs. Nevertheless, they may have significant effects in the ellipsoidal case, owing to the large values of Chariklo and Haumea's elongation parameters .

In that case, combining d'Alembert's rule and Eq. ( 19), we see that a m/(m -j) resonance is of global order e j |m/2| (m even). For instance, while the outer 1/2 (first order) LR appears in the case of a mass anomaly, it only exists in its second order version 2/4 (m = -2, j = 2) in the case of the ellipsoid. Similarly, the outer 1/3 LR appears in its second order version with a mass anomaly, but only in its fourth order version 2/6 (m = -2, j = 4) when caused by an ellipsoid.

To our knowledge, no evaluation of the torque exerted at a m/(m-j) resonance with j > 1 has been published. There are two reasons for that. First, the hydrodynamical equations describing the disk must be expanded to j th order in the perturbations, a challenging task. Second, such resonances cause streamline self-crossings. It can be shown (Sicardy et al. 2018, in preparation) that near a m/(m -j) resonance, where m and j are relatively prime, a periodic resonant streamline has j braids with |m|(j -1) self-crossing points. This creates singularities in the hydrodynamical equations (shocks), even for vanishingly small perturbations, thus requiring new kinds of treatments. This said, we see that although the 2/4 resonance (ellipsoid case) is of second order in the particle eccentricity, it does not induce self-crossing streamlines since the ratio 2/4 can be reduced to 1/2, resulting in | -1|(1 -1) = 0 self-crossing points. Still, as mentioned above, no expression of the resonance torque exists because of the second-order nature of that resonance. A general behaviour can nevertheless be sketched. At second order in eccentricity, Eq. ( 39) is replaced by

U (r) = +∞ k=-∞ U k (a) • cos [k(λ -λ A )] + e 2 Ω 2 R 2 q +∞ m=-∞ B m (a) • cos(2φ m ), ( 41
)
where now φ m = [mλ A -(m -2)λ -2 ]/2 (with m even). The expression of B m can be retrieved from ref. ) and Y = e cos(φ 2/4 ) (φ 2/4 = 2λ -λ A -), it can be shown that the origin (X, Y ) = (0, 0) is always a fixed point. It is stable, except for a narrow interval of initial semi-major axes a 1/2 (1 -0.25 ) < ∼ a < ∼ a 1/2 (1 + 0.25 ), the coefficient ∼0.25 stemming from the particular values of R and q used here. In that interval (of width ∼25 km for Chariklo and ∼375 km for Haumea, from Main Text Table 1), the origin (X, Y ) = (0, 0) is an unstable hyperbolic point, so that ring particles initially orbiting on those circular orbits periodically acquire orbital eccentricities of order e ∼ √ 0.25 , This shows that a Chariklo with elongation ∼ 0.20 forces large excentricities (e ∼ 0.2) at the second-order 2/4 resonance (see an example in Supplementary Fig. 2), while Haumea ∼ 0.61 forces even larger values (e ∼ 0.35) that lead to collisions with the body. The 2/4 resonant zone is thus a highly perturbed region where no ring is expected to survive.

Turning to the second order 1/3 (mass anomaly) and fourth order 2/6 (ellipsoid) resonances, we see that it is the unique prograde resonant orbit with only one self-crossing point (corresponding to m = -1 and j = 2, so that |m|(j -1) = 1). Our numerical integrations show no significant effect of the 2/6 resonance on the particle motion, even with an elongation as high as = 0.61 (Haumea's case). This stems from the fourth-order nature of that resonance. It is noteworthy that both Chariklo and Haumea's rings are close to the 1/3 resonance configuration [START_REF] Ortiz | The size, shape, density and ring of the dwarf planet Haumea from a stellar occultation[END_REF][START_REF] Leiva | Size and Shape of Chariklo from Multi-epoch Stellar Occultations[END_REF] , possibly leading to yet-to-be explicited more subtle confining effects of a narrow ring at that location. This makes further investigations (in particular using collisional codes) highly desirable.

Figure 1 |

 1 Figure 1 | Corotation and Lindblad resonances around Chariklo.In both panels (topographic feature on the left, elongated body on the right), the red (resp. blue) circles correspond to inner (resp. outer) m/(m -1) Lindblad resonance (LR) radii with m>1 (resp. m<0), see Eq. (2). The black lines show isopotential curves in a frame corotating with Chariklo, and the gray lines outline the limb of the body. The dots C 1 , ...C 4 mark the corotation fixed points. The points C 2 and C 4 are local potential maxima and are linearly stable if the mass anomaly or the elongation of the body are not too large, see Methods. Left -A topographic feature of height z = 5 km (gray half dome, not on scale) is sitting at the surface a body of radius R sph = 129 km, and corresponds to a mass anomaly µ ∼ 10 -5 . For better viewing, the isopotential black lines have been radially stretched by a factor of 50 with respect to the corotation radius. A few inner (m = 2, 3, 4, 5) and outer (m = -1, -2, -3, -4) LR radii are shown. Right -The same for a Chariklo shape solution with elongation = 0.20. The limb of the body and the isopotential lines are plotted on scale. Only LRs with m even are now allowed, the inner one corresponding to m = 6, and the two outer ones corresponding to m = -2, -4. The green curve is an orbit that starts at C 2 without applying any friction (conservative case). It is unstable because the elongation is larger than the critical value crit = 0.16 (see Methods), leading to a collision with the body after three months in the example shown here.

Figure 2 |

 2 Figure 2 | Torques strengths at Lindblad Resonances around Chariklo and migration time scales. a, The dimensionless coefficients A 2 m providing the torque value at m/(m -1) Lindblad resonances (Eq. (3)) vs. the resonant radii on each side of the corotation radius (dotted line). The values of A m are evaluated from Supplementary Table1, using a Chariklo equatorial topographic feature of height z = 5 km, corresponding to a mass anomaly µ = 10 -5 (blue squares), or a difference of semi-axes A -B = 18 km (Table1), corresponding to an elongation parameter = 0.20 (red squares). Note the steep decrease of the torques as the corotation radius is approached, due to the exponential decrease of A 2 m as |m| increases, see Methods. The light gray region at left encloses Chariklo's largest semi-axis A = 157 km, inside which particles collide with the body in the ellipsoidal case, while the dark gray region encloses Chariklo's radius R sph = 129 km in the spherical case (Table1). b, Solid lines: migration times (Eq. (4)) of an outer annulus of width 100 km that extends outside the corotation (see a), either due to the topographic features (blue) of heights z or ellipsoids with various A -B (red). Dotted lines: the same for an inner annulus of width 20 km in the ellipsoidal case, and 60 km in the spherical case, see a.

Figure 3 |

 3 Figure 3 | Migration of ring particles around Chariklo. The particles are submitted to Chariklo's gravitational field (topographic feature on the left, elongated body on the right), plus a radial Stokes-like friction with η = 0.01 (Eq. (5)). The radii of the corotation point C 2 (a cor ), the 2/3 and 1/2 outer Lindblad Resonances (LRs) between the particle mean motions and Chariklo's rotation period are marked at the bottom, together with the location of Chariklo's main ring 1 C1R. a, b, c, d, The effect of an equatorial topographic feature (black dot) with mass µ = 5 × 10 -3 relative to Chariklo. Initially, 701 particles are regularly placed between 0.7a cor and 2.2a cor . In all panels, each particle is plotted over twenty regular time steps spanning 40,000 years. a, After 40,000 years, the clearing of the corotation region is ongoing; b, After 2.5 × 10 5 years, some particles remain near C 2 , while others are pushed outside the 2/3 LR; c, After 2.5 × 10 6 years, all the particles inside the corotation radius and near C 2 have collided with Chariklo; d, After 6.3 × 10 6 years, all the remaining particles are now outside the 1/2 LR. e,

  Figure 1

Figure 3

 3 Figure 3

Table 1 -

 1 Chariklo and Haumea's adopted parameters(a)

	Parameters

  37. It involves the operator which reduces to B -2 = -2.55(R/a 1/2 ) 3 , where a 1/2 is the radius of exact 2/4 resonance. The phase portrait of this resonance is found in various works (e.g. ref.37). Posing X = e cos(φ 2/4

	f 45 =	1 8	(4m 2 -5m) + 2(2m -1)a	d da	+ a 2 d 2 da 2	(42)

that must be applied to each term of the expansion given in Eq. (

19

). This provides

B m (a) = -1 4 S |m/2| |m/2| (4m 2 -5m) -2(2m -1)(|m| + 1) + (|m| + 1)(|m| + 2) R a |m|+1 ,

(43)

Acknowledgements

The work leading to this results has received funding from the European Research Council under the European Community's H2020 2014-2020 ERC Grant Agreement No. 669416 "Lucky Star". P.S.S. acknowledges financial support by the European Union's Horizon 2020 Research and Innovation Programme, under Grant Agreement No. 687378. We thank Françoise Combes for discussions on corotation and Lindblad resonances in the context of galactic dynamics, and Timothée Vaillant for comments on satellite formations and migrations. We thank the reviewers for comments that improved the content and presentation of this work.

Author contributions B.S., R.L. and M.E.M. contributed to the analytical calculations that describe the resonance dynamics around a non-axisymmetric body. B.S. wrote the paper and made the figures, with contributions from R.L., S.R., F.R. and P.S.S. F.R. provided insights for the application of this work to the formation of satellites around small bodies. Numerical integrations were independently performed by B.S, S.R. and F.R.

Competing interests

The authors declare no competing interests.

Author information Correspondence and requests for materials should be addressed to B.S. at bruno.sicardy@obspm.fr.

Author Data and code availability The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. We have opted not to make our codes available as we cannot guarantee their correct performance on different computing platforms.

Supplementary Table 1:

Azimuthal variation f (θ) of the corotation potential (Methods Eq. ( 26))

Mass anomaly

q -1/6   1

Coefficients A m (a) of the m/(m -1) Lindblad resonances (Main Text Eq. ( 3) and Methods Eq. ( 39))

Mass anomaly

The sequence S p is defined by Eq. ( 20). Supplementary Figure 1: Supplementary Figure 2:


M Balat-Pichelin

L Charpentier

F Panerai

O Chazot

K G Nickel

OXIDATION AT HIGH TEMPERATURE IN AIR PLASMA FOR THE TPS OF IXV -PASSIVE/ACTIVE TRANSITION AND TEMPERATURE JUMP

This paper reports the experimental results of oxidation studies for the determination of the transition between passive and active oxidation of two silicon carbidebased materials produced by Herakles-Safran (France) and MT Aerospace (Germany). The transition between active and passive oxidation is determined for both materials using two different plasma facilities -a low enthalpy one and a high enthalpy one -and the experimental results lead to the same transition law. Characterization by SEM of the oxidized samples (surface and cross-section) is shown. Mass loss rates in active oxidation conditions are reported for both Herakles and MT Aerospace materials. Moreover, samples exposed to heat fluxes exceeding 1.2 MW/m 2 and low pressures show a spontaneous jump at surface temperature near 2100 K, rapidly leading to temperatures above 2400 K. Under these conditions, the SiC coating at the sample surface is entirely eroded and the bare carbon fibers of the substrate are left exposed to the incoming flow. The temperature jump phenomenon is described by means of infrared temperature measurements and real time visible recordings of the surface. Space-and time-resolved optical emission spectroscopy and SEM analysis of the samples are used to discuss and explain the chemical processes occurring at the surface.

INTRODUCTION

The material used for the hottest parts of the Intermediate eXperimental Vehicle (IXV) is based on silicon carbide. The IXV Thermal Protection System (TPS) is composed of a C/SiC nose, C/SiC shingles on the windward area and ablative materials on the leeward. The control of the vehicle during re-entry is made using C/SiC flaps. Taking into account the heat flux, wall temperature and total air pressure data calculated by the European Space Agency (ESA) for the IXV trajectory, several points were identified on the surface of the vehicle as critical because of a period of a few seconds in the active oxidation zone. The hottest parts are localized on the nose, front part of the windward and on the flaps. The temperature values are derived from fully catalytic radiative equilibrium wall calculations for a total hemispherical emissivity of 0.85 for the surface material. SiC is oxidized according to two regimes, mainly depending on the oxygen partial pressure and temperature. During passive oxidation (Eqs. 1 and 2)often at low temperature and high partial pressure of oxygen -a silica layer is formed on the surface of SiC and this layer is protective slowing down the oxidation process (passivation):

SiC (s) + 2 O 2(g) = SiO 2(s) + CO 2(g) (1) SiC (s) + 3/2 O 2(g) = SiO 2(s) + CO (g) (2) During active oxidation (Eq. 3) -often at high temperature and low partial pressure of oxygengaseous SiO is produced and consequently, no passive layer is formed and significant mass loss can be observed:

SiC (s) + O 2(g) = SiO (g) + CO (g) (3) Previous studies [START_REF] Balat | Determination of the active-to-passive transition in the oxidation of silicon carbide in standard and microwave-excited air[END_REF][START_REF] Rosner | High temperature kinetics of the oxidation and nitridation of pyrolytic silicon carbide in dissociated gases[END_REF][START_REF] Vaughn | Active-to-passive transition in the oxidation of silicon carbide and silicon nitride in air[END_REF][START_REF] Sekigawa | Endurance test of oxidation-resistant CVD-SiC coating on C/C composites for space vehicle[END_REF][START_REF] Ogasawara | Thermal response and oxidation of Tyranno-fiberreinforced Si-Ti-C-O matrix composites for a TPS in high-enthalpy dissociated air[END_REF][START_REF] Jacobson | Oxidation transitions for SiC. Part 1: active-to-passive transition[END_REF] gave theoretical and/or experimental transitions between passive and active oxidation, depending on total pressure, oxygen partial pressure and temperature. The huge discrepancy between the results can be explained by the difference in the parameters taken for the study (total and partial pressures, gas flow rates…) and also by the nature of the gaseous atmosphere surrounding the sample: standard, dissociated in low or high enthalpy facilities.

FACILITIES USED: MESOX AND PLASMATRON

The MESOX -Moyen d'Essai Solaire d'OXydationfacility used in this study is described in details elsewhere [START_REF] Balat | Determination of the active-to-passive transition in the oxidation of silicon carbide in standard and microwave-excited air[END_REF], only its main features are outlined here. The sample (25 mm diameter and 2 mm thickness) is placed in a plasma reactor consisting of a quartz tube, 50 cm in length and 5 cm in diameter, equipped with a CaF 2 viewport at its top. A regulator, a gauge and a vacuum pump allow the precise control of the total pressure during experiments. The airflow is fixed at 4 l/h (1.4 mg/s) and is coming from to the top of the reactor and then pumped down. The air plasma is generated by a 300 W microwave discharge surrounding the sample leading to a degree of dissociation of nearly 70% for oxygen, nitrogen being not dissociated [START_REF] Balat-Pichelin | Neutral oxygen atom density in the MESOX air plasma solar furnace facility[END_REF]. The sample is placed on a sample-holder at the centre of the plasma discharge. The reactor is positioned in way to allow a sample positioning at 25 mm above the theoretical focus of the solar furnace equipped with a variable opening shutter in order to have a homogeneous temperature on 10-12 mm in the center of the sample. The available incident concentrated solar flux can reach 5 MW/m 2 . The temperature measurement is performed using a monochromatic optical pyrometer and to convert measured radiation into temperature, for the C/SiC material, we chose a normal spectral emissivity of 0.90. The testing conditions offered by this reactor allow reproducing a chemical boundary layer that is the main focus of the gas-surface interaction phenomena. Moreover, this set-up allows the independence of the chemical and thermal conditions, i.e. the plasma can be kept constant (300 W, 4 l/h air flow) and the temperature of the sample can be increased independently using the concentrated solar radiation by the progressive opening of the shutter. Finally, it is equipped with detailed measurements (video, optical emission spectroscopy) that give access to a quite complete characterization for TPS testing. The present research was also carried out at the 1.2 MW Plasmatron facility at VKI. It is a high enthalpy, subsonic wind tunnel that generates plasma by electromagnetic induction, at sub-atmospheric pressure [START_REF] Chazot | Simulation methodology in Plasmatron facility and hypersonic wind tunnels[END_REF][START_REF] Chazot | TPS ground testing in plasma wind tunnel for catalytic properties determination[END_REF]. The Plasmatron allows the in-ground simulation of the chemically reacting boundary layer around a reentry body [START_REF] Kolesnikov | Condition of simulation of stagnation point heat transfer from high enthalpy flow[END_REF][START_REF] Barbante | Flight extrapolation of plasma wind tunnel of stagnation region flowfield[END_REF]. Its long test time (up to hours uninterruptedly) and high purity flow make it an ideal set-up for material thermochemistry testing and studies of gas/surface interactions. Displacement mechanisms mounted on a 1.4 m diameter water-cooled test chamber allow moving the test models into or out of the flow. Optical accesses equip the system for non-invasive measurements. A 50 mm diameter copper water-cooled probe, mounting a copper calorimeter in the center of the front face, is used for the calibration of plasma flow conditions. Both the probe and calorimeter have copper surfaces which are highly catalytic to nitrogen and oxygen atoms recombination. A water-cooled Pitot probe is used to perform stagnation pressure measurements. A pyrometer records the front surface temperature of the sample. It measures over two overlapping wavelengths around 1 µm. Measurements of erosion products from the material surface are performed by means of temporally-and spatiallyresolved optical emission spectroscopy in front of the test model. Volatilization of SiO during active oxidation tests is expected to occur according to the reaction mechanism [START_REF] Vaughn | Active-to-passive transition in the oxidation of silicon carbide and silicon nitride in air[END_REF]. We take into account only the formation of CO instead of CO 2 as the reaction occurs at high temperature where the CO formation is favored. Excited Si atoms, likely to be formed as SiO is transported to the high temperature boundary layer, radiate strongly in the UV-range and can be detected using the available three Ocean Optics HR4000 spectrometers.

TPS MATERIALS STUDIED

Two Ceramic Matrix Composites (CMC) based on a carbon preform and a matrix of silicon carbide have been investigated in this study. Both have a β-SiC coating to increase the protection from high temperature oxidation (Fig. 1). In passive conditions, there is a thin silica layer on the oxidized samples (Fig. 2a) compared to the reference sample, also proved by the light blue color on the edges of the samples. For sample HER I (Fig. 2b), the silicon carbide layer is broken and in the hole, the carbon fibers are clearly visible. Atomic oxygen is highly reactive with carbon at these temperature levels and the mass loss rate increases a lot under these conditions. The obtained transition, taking into account that the temperature measurement carried out on the MESOX facility using the optical pyrometer working at 5 µm was done with a value of 0.9 for the spectral normal emissivity, led to the following expression: p O2 (HER) = 10 15 exp (-55850/T) with a correlation factor of 0.99994 with the oxygen partial pressure p O2 expressed in Pa and T in K. This transition is very close to the one previously obtained by Balat [START_REF] Balat | Determination of the active-to-passive transition in the oxidation of silicon carbide in standard and microwave-excited air[END_REF] for nearly the same material (reported in blue in Fig. 3); the difference between the two curves of around 20 to 30 K is due to a different value chosen for the spectral normal emissivity (0.87) in the past study. However, the activation energy of the process is the same and is equal to almost 465 kJ/mol.

Bubbles formation in active oxidation conditions

Figure 6 shows the photos of some samples after oxidation and the still images from video recordings of the in situ oxidation process acquired through the quartz tube and the hole performed in the side face of the wave guide, in the MESOX set-up. Around the melting point of silica (2000 K), the presence of silica bubbles is clearly visible on the video captures in Fig. 6 (white bubbles on the upper surfaces).

The CO partial pressure at the silica/SiC interface being higher than the surface pressure, bubbles are formed and some of them exploded leading to some traces visible on the photos of the samples after oxidation. The samples that present bubbles on their surfaces are classified in active oxidation conditions as the remaining oxide layer is no more protective and passivating as some holes are present on the surface leading to an access for atomic oxygen for further oxidation.

Mass loss rates in active oxidation conditions

In order to have a better control on the mass loss rate in active oxidation, complementary experiments were performed in the MESOX facility at a total pressure of 5000 Pa on smaller samples of 15 mm x 15 mm size that allowed an homogeneous temperature on their whole surface. Figure 7 reports the results obtained for HER (black diamonds) and MTA (open dots) samples. About the mass loss rate results, one can see, except for some points, that the trend is the same for HER and MTA samples with a significant increase of the mass loss rate above 2000 K (melting of SiO 2 as confirmed by the images of the video captures in Fig. 6) followed by a plateau up to almost 2150 K (main vaporization of SiO and CO) and then a huge increase again from 2150 K (beginning of SiC sublimation in gaseous Si and Si 2 C) with maximum mass loss rates of nearly 20 mg cm -2 min -1 at around 2300 K. This trend is in agreement with the thermodynamic calculations where the onset of SiC sublimation is calculated to be around 2100 K. In Fig. 8, the β-SiC layer on the HER reference sample before test is clearly visible with a mean thickness of around 80 µm (Fig. 8a). Cracks are present but their widths are very small. In passive oxidation conditions such as for sample HER G (1922 K, 2000 Pa, Fig. 8b), the SiC coating layer is not affected and presents nearly the same thickness as for the reference sample. The presence of a very thin silica layer is confirmed by XPS analyses. For samples submitted to active oxidation conditions but at temperature lower than 2100 K, the formation of silicon monoxide and carbon monoxide are the main reactions leading to a rapid damage of the SiC coating with large holes showing the substrate carbon fiber structure like for HER I (2006 K, 2000 Pa, Fig. 8c). Around 2100 K, even at 5000 and 2000 Pa, the main reaction is the sublimation of SiC leading to the formation of Si and in lesser extent Si 2 C as predicted by thermodynamics calculations. These reactions can explain why the SiC coating layer is still present, but it is affected by large cracks that allow oxygen to pass through and to react with carbon fibers leading to a more porous substructure and an important mass loss rate (HER 6, 2096 K, 2000 Pa, Fig. 8d). Same analyses and conclusions can be done for the MTA samples.

CHARACTERIZATION OF THE TEMPERATURE JUMP

Studies on the oxidation transition of silicon carbidebased materials, conducted in high enthalpy plasma facilities, led many authors to observe spontaneous jumps in surface temperature, of the order of hundreds of degrees, during exposure of the samples to high heat flux, low pressure, dissociated flows. The phenomenon was firstly observed during arc-jet experiments of C/SiC samples at temperatures up to 2120 K [START_REF] Lacombe | Investigation of C/SiC breaking point under arc-jet environment at NASA-JSC[END_REF]. Hald discussed temperature jumps on similar CMC materials and suggested an explanation based on exothermic catalytic reactions of gaseous silicon with atomic nitrogen, which would raise the temperature of the boundary layer and hence the convective component of heating to the surface [START_REF] Hald | Operational limits for reusable space transportation systems due to physical boundaries of C/SiC materials[END_REF]. The phenomenon was observed by Herdrich et al. [START_REF] Herdrich | Oxidation behavior of silicon carbide-based materials by using new probe techniques[END_REF] and Laux et al. [START_REF] Laux | Oxidation behavior of sintered silicon carbide in various plasma wind tunnels[END_REF] both in inductively heated and arc-jet facilities, during The same protocol as the one followed before for the determination of the transition was followed. The Plasmatron tests are performed under standard air plasma (79% N 2 and 21% O 2 ) at a mass flow of 16 g/s.

The plasma conditions are adjusted based on the twocolor pyrometer reading. Once the specified static pressure values are reached by air by-pass and vacuum pumps regulation, the sample is swung into the plasma stream at low heat flux and the power is progressively increased to obtain the target temperature value. A constant temperature is then maintained for a total steady-state time of 600 s. For some tests, this is shortened in order to avoid damages to the set-up or to prevent the complete destruction of the sample.

Description of the temperature jump phenomenon

Figure 9 shows the temperature measured during the cases that present the temperature jump phenomenon. For both materials, the temperature history measured during a jump test is compared with that in passive and active regime, at the same static pressure, tests of HER samples at 2000 Pa and test of MTA samples at 5000 Pa. Test HER 17 represents a well suited case to illustrate the evolution of the surface. A sintered SiC cover is used to hold the sample. The insertion of the test model into the flow at 1.3 MW/m 2 heat flux leads to a natural rise of the surface temperature, stabilizing to a plateau of 2095 K after approximately 40 s. Differently from tests in passive and active oxidation, where a steady state is reached and maintained constant throughout the test, a temperature plateau is measured only until 120 s, when a sudden jump of ~400 K leads to a surface temperature near 2500 K. Power, pressure, and mass flow (i.e. total enthalpy and total pressure) are maintained constant throughout the test, hence the phenomenon results as spontaneous process at constant freestream conditions.

Microscale morphology and surface chemistry

The information obtained by visual imaging of the surface is confirmed at microscopic scale by SEM analysis, shown in Figs. 10 and11. Micro-cracks are present on the pristine β-SiC layer (Figs. 10a and11a). Microscopic analyses of the material cross-section show that the cracks can extend through the whole coating thickness, nevertheless their width never exceeds 5 µm. The morphology of the cracks and the coating itself are not seen to be affected by exposure to passive oxidation conditions (Figs. 10b and11b). Within the investigated pressure range, passive oxidation occurs at surface temperatures below 2000 K. Under passive oxidation condition, a uniform silica scale is formed constituting a protective barrier for the surface from the incoming reactants. After the passive/active oxidation transition, increasing temperatures promotes the decomposition and volatilization of SiO 2 into SiO and Si. High evaporation rates would eventually lead to its depletion, hence to the onset of active oxidation conditions, where the β-SiC coating is progressively consumed by oxygen reactants. This condition is revealed by Figs. 10c and11c where, despite some traces of SiO 2 being still present, several damages and holes in the SiC layer are observed. During the temperature jump experiments, the samples are injected into the flow at very high enthalpies and a very steep temperature rise is experienced upon insertion. The surface temperature reaches values above 2000 K in less than 30-40 s. The progression of silica formation/evaporation interests only the first 20 s and represents therefore a minor contribution to the chemical history of the surface, while active oxidation starts soon after the injection into the flow. Under this condition, the silicon carbide reacts with oxygen atoms, producing mainly SiO and CO. The higher the incoming flux the higher the rate of consumption of the β-SiC layer. Thermodynamic calculations have shown that, if temperatures exceed 2100 K (jump temperature), sublimation of SiC into Si and in lesser extent into Si 2 C largely contributes to the chemistry of the surface. Active oxidation processes continue at the interstices between the fibers, occupied by the SiC matrix filler used to densify the carbon preform.

It is also remarked that the jump process starts at the edges of the probe, where indeed the heat flux is higher due to the turning of the streamlines and the thinning of the boundary layer and where the SiC coating is thinner, hence more quickly consumed.

Optical emission spectroscopy results

Temporally-resolved spectral profiles within the 240-290 nm band are compared in Fig. 12 where oxidation and nitridation reactions occur. These further increase the temperature accelerating the rate of volatilization of SiC. Once the coating is completely eroded, that is once the observed temperature front has covered the whole surface, the substrate preform is directly exposed to the incoming plasma and carbon oxidation and nitridation become the leading reaction mechanisms. Beside those primary effects, further contributions to the convective heating might come for exothermic processes in gas phase in the vicinity of the surface, like reactions of the sublimated silicon with nitrogen radicals, or O and N atom recombination.

CONCLUSION

Air plasma experiments of C/SiC heat shield materials from Herakles and MT Aerospace were performed in the MESOX facility of PROMES-CNRS laboratory (France) and in the high enthalpy Plasmatron facility at VKI (Belgium) to study their response to high temperatures experienced during IXV re-entry trajectory conditions. The transition between passive and active oxidation regimes was determined on both facilities and materials, and an agreement was found for the results obtained on both facilities and on both materials. SEM micrographs of the surface and of the cross-section of the samples were carried out in order to plot and confirm the transition limit that was finally defined by the following law: p O2 (Pa) = 10 15 exp (-56000/T) with the oxygen partial pressure p O2 expressed in Pa and T in K.

The transition obtained experimentally at PROMES-CNRS and at VKI is the same for the CMC samples from Herakles and MT Aerospace, which is not surprising as the composition is very similar and both have a β-SiC coating on the top surface. About the mass loss rates in active oxidation conditions, the trend is the same for Herakles and MT Aerospace samples with a huge increase of the mass loss rate above 2000 K (melting point of SiO 2 ) followed by a plateau up to 2150 K (main vaporization of SiO and CO) and then a huge increase again (SiC sublimation in Si and Si 2 C) with maximum mass loss rates of nearly 20 mg cm -2 min -1 at around 2300 K. This trend is in agreement with the thermodynamic calculation. Air experiments were performed in low pressure plasma flows at temperatures above 1900 K. These tests aimed to simulate enthalpy levels experienced by the IXV TPS material during hypersonic reentry, and to exceed nominal heat flux conditions. A sudden and spontaneous surface temperature increase of several hundred degrees, at constant freestream condition, was observed under different operating conditions. The phenomenon occurred at wall temperatures near 2100 K. A mass loss exceeding 30% of the initial values was found, one order of magnitude higher than that experienced by the material under active oxidation regime. A strong reduction in thickness and a deep erosion of the surface were also observed, consistent with a strong increase in Si radiation detected by optical emission spectroscopy close to the surface. The jump is explained with augmented surface chemistry characterized by sublimation of SiC around 2100 K and onset of O and N reactions with the carbon substrate, leading to a quick and complete erosion of the superficial β-SiC coating and to the exposure of the carbon fibers to the incoming plasma.

Figure 1 . 4 . 1 .

 141 Figure 1. SEM images of the surface of the materials before test (reference): (a) HER and (b) MTA4. EXPERIMENTAL RESULTS4.1. Results for the CMC from Herakles (HER)Oxidation tests on HER material were carried out in the MESOX facility on 24 samples and in the Plasmatron on 16 samples. The sample is put in the air plasma then the temperature increases up to a plateau where a steady-state is maintained during 600 s then the heat

Figure 2

 2 shows clearly the difference between passive and active oxidation for (a) HER G at 1922 K and (b) HER I at 2006 K for 2000 Pa.

Figure 2 .

 2 Figure 2. SEM images of HER samples:(a) G passive, 1922 K and (b) I active, 2006 K at 2000 Pa The transition is reported in Fig. 3 together with the points representing the several experiments, with green diamonds for samples under passive oxidation and red diamonds for the active ones (plain dots for MESOX experiments and open dots for Plasmatron ones).

Figure 3 .

 3 Figure 3. Experimental points (filled diamonds for MESOX and empty diamonds for Plasmatron) and transition line determined for HER samples on both facilities. Green dots are for passive oxidation and red dots for active oxidation.

4. 2

 2 Results for the CMC from MT Aerospace (MTA)Oxidation tests on MTA material were carried out on 27 samples in the MESOX facility and on 11 samples in the Plasmatron with the same protocol as for HER samples. The classification in passive conditions is determined by a relative mass loss lower than ≈ 0.3 % and by a mass loss rate lower than 0.3 mg cm -2 min -1 . As before, the confirmation is obtained through the SEM images performed on the surface of the samples. The transition was found for 200 Pa air total pressure between 1776 and 1822 K, for 500 Pa between 1841 and 1906 K, for 2000 Pa between 1927 and 1957 K and for 5000 Pa between 2008 and 2052 K.

  (a) MTA 7 (b) MTA 20

Figure 4 .

 4 Figure 4. SEM images of MTA samples:(a) 7 passive, 1927 K and (b) 20 active, 1957 K at 2000 Pa Figure 4 shows the difference between passive and active oxidation for (a) MTA 7 at 1927 K and (b) MTA 20 at 1957 K for 2000 Pa. In passive conditions, there is a thin silica layer on the oxidized sample (Fig. 4a) compared to the reference one. For sample MTA 20 (Fig. 4b), the silicon carbide layer is broken and absent in several parts, the carbon fibers being clearly visible. As for the Herakles ones, the samples present very colored surfaces for the ones under passive oxidation from blue to light brown according to the thickness of the silica layer and light grey ones for the samples under active oxidation conditions. The determined transition is reported in Fig. 5 together with the points representing the several experiments with light green dots for samples under passive oxidation and orange dots for the active ones (plain dots for MESOX experiments and open dots for Plasmatron

Figure 5 .

 5 Figure 5. Experimental points (filled dots for MESOX and empty dots for Plasmatron) and transition line determined for HER samples on both facilities. Light green dots are for passive oxidation and orange dots for active oxidation.

HER 9 (Figure 6 .

 96 Figure 6. Photos of the samples after oxidation and captures from video during oxidation in air plasma at 2000 Pa. On the upper surface of the samples (side view) bubbles are clearly visible (in white).

Figure 7 .

 7 Figure 7. Mass loss rate versus temperature for HER and MTA samples (15 mm x15 mm) at 5000 Pa 4.5. SEM micrographs of the cross-section of oxidized samples and oxidation mechanisms Figure 8 presents the SEM micrographs of the crosssections of the HER samples oxidized in the MESOX facility at PROMES-CNRS. These images are in good agreement with the conclusions obtained for the determination of the active-passive transition and confirm the position of the transition line. These images can also be useful to explain the oxidation mechanisms that seem different according to the temperature levels.In Fig.8, the β-SiC layer on the HER reference sample before test is clearly visible with a mean thickness of around 80 µm (Fig.8a). Cracks are present but their widths are very small. In passive oxidation conditions such as for sample HER G (1922 K, 2000 Pa, Fig.8b), the SiC coating layer is not affected and presents nearly

Figure 8 .

 8 Figure 8. Comparison of SEM cross-sections for HER samples: (a) reference and oxidized at 2000 Pa, (b) G at 1922 K, (c) I at 2006 K and (d) 6 at 2096 K.

Figure 9 .

 9 Figure 9. Temperature jump time-histories during standard model testing: (a) HER, (b) MTA materials

Figure 10 .Figure 11 .

 1011 Figure 10. Micrographs of HER samples at 5000 Pa under different regimes

  for tests, MTA 5, 6 and 12 respectively under passive oxidation (1956 K), active oxidation (2119 K) and jump regime (2064 → 2542 K), at similar oxygen partial pressure (~1000 Pa). No Si emission is recorded for sample MTA 5, consistent with a stable silica scale that protects the SiC, formed under passive oxidation. For MTA 6, as temperature increases to steady state, we have a rise in Si emission, which then stabilizes to constant values throughout the whole length of the test. This case is representative of active oxidation of SiC at low rate producing Si gaseous products in the boundary layer. For the temperature jump case (MTA 12) an initial sharp increase in Si emission is observed, leading quickly to values well above active oxidation. An intensity peak is observed around 40 s after sample injection, followed by a successive attenuation and final steady emission from 55 s till the end of the run. This behavior is consistent to what is described in the previous section about the jump. Strong active oxidation starts soon after the sample injection into higher heat flux flow, with a lot of SiO produced into the gas phase. After ~2100 K the onset of SiC sublimation, producing Si and Si 2 C, would lead to a peak intensity in Si emission, which would then decrease as the β-SiC layer is completely eroded. The final constant phase in Si intensity represents constant rate active oxidation reaction at the SiC filler matrix that starts after the substrate is left exposed and continues to occur until the end of the run.

Figure 12 .

 12 Figure 12. Temporal emission spectra of Si erosion products for experiments on MTA samples at 5000 Pa, under different regimes, measured with the spectrometer at 2 mm ahead of the surface: (a) MTA 5, passive oxidation, (b) MTA 6, active oxidation and (c) MTA 12, temperature jump

  These two materials are implemented on the IXV vehicle for the nose, leading edges and flaps. The composite material used for the IXV nose is a CMC which commercial name is Sepcarbinox ® L6 (called HER after) provided by Herakles (Safran group, France). The L6 CMC material is made from a Guipex ® carbon fibre preform. SiC matrix densification is performed by Chemical Vapor Infiltration (CVI). The part is machined to its final geometry at an intermediate state of densification. A final CVI cycle is performed to achieve the requested density while at the same time ensuring that the SiC matrix is deposited on the entire surface, fibers included.

	The manufacture of MT Aerospace Keraman ® C/SiC
	ceramic composites (called MTA after) is realized by an
	improved near-net shaped lay-up technique of carbon
	fabric layers combined with an optimized SiC matrix
	infiltration via gradient CVI process. The process allows
	the fabrication of complex and integrally shaped CMC
	parts. A process is disclosed for producing a high
	temperature stable fiber composite ceramic by CVI with
	a silicon carbide precursor in a suitable carrier gas.
	The samples used in this study are cylinders with a
	diameter of 30 mm for Plasmatron experiments and 25
	mm in the MESOX facility. The thickness of both
	samples is 2 mm.

  testing of sintered and CVD-β SiC coated C/C. Herdrich used increased N-atom recombination on SiC versus SiO 2 to explain part of the increased heat flux to the surface under temperature jumps, observed during plasma testing of passive/active oxidation transition.

	Silicon emissions, detected by optical emission
	spectroscopy, were used to distinguish between
	passively and actively oxidized samples. Investigations
	were performed in the VKI Plasmatron facility on β-SiC
	specimens at surface temperatures of 2000-2800 K and
	various pressures. Sudden temperature jumps for
	constant freestream conditions were observed,
	corresponding to high mass loss and strong increase in
	silicon volatilization, detected by optical emission
	spectroscopy near the surface.
	This section focuses on the temperature jump
	phenomenon observed on the IXV C/SiC materials.
	High heat flux experiments are conducted in the
	Plasmatron facility. Samples are tested at pressures
	between 2000 and 6000 Pa and temperature above 1900
	K. The practical challenges arising when performing
	temperature jump experiments are discussed. The jump
	phenomenon is illustrated by means of IR pyrometry
	measurements and real-time visible emission recording
	of the sample surface. A discussion on the phenomenon
	is provided, based on SEM analysis, spatially-and
	temporally-resolved optical emission spectroscopy and
	comparison with passive and active oxidation regimes.

ACKNOWLEDGMENTS

This work was funded by the European Space Agency through the TRP on Passive/Active Oxidation of CMC Structural Materials (ESTEC contract n° 4000103145/11/NL/CP) led by PROMES-CNRS. The research of H. B. is supported by a fellowship of the Agency for Innovation by Science and Technology in Flanders (IWT, dossier #111529). Herakles-Safran and MT Aerospace are acknowledged for the procurement of material samples used for experiments.