Thérèse Encrenaz
email: therese.encrenaz@obspm.fr

Several temperate Jupiters have been discovered to date, but most of them remain to be detected. In this note, we analyse the expected infrared transmission spectrum of a temperate Jupiter, with an equilibrium temperature ranging between 350 and 500 K. We estimate its expected amplitude signal through a primary transit, and we analyse the best conditions for the host star to be filled in order to optimize the S/N ratio of its transmission spectrum. Calculations show that temperate Jupiters around M stars could have an amplitude signal higher than 10 -4 in primary transits, with revolution periods of a few tens of days and transit durations of a few hours. In order to enlarge the sampling of exoplanets to be observed with ARIEL (presently focussed on objects warmer than 500 K), such objects could be considered as additional possible targets for the mission.

Introduction

Temperate Jupiters, with equilibrium temperatures ranging between 350 and 500 K, are not expected to exist according to the standard nucleation model, which requires that giant planets are formed at large distances from their host star, at temperatures low enough for water and other hydrogenated molecules to be in the form of ices (see e.g. [START_REF] Pollack | Formation of the giant planets by concurrent accretion of solids and gas[END_REF]. However, several temperate exoplanets with radii larger than half the Jupiter one and with periods ranging between 100 and 300 days have been detected around F, G and K stars (some examples are listed in Table 1, for semi-major axes ranging between 0.5 and 1.0 AU an eccentricities smaller than 0.1). It can be seen that their equilibrium temperatures range between 250 and 500 K. Their presence at relatively close distances to their host star might be the result of migration processes, or other possible formation mechanisms, still to be investigated. Although most of the objects detected so far are not suitable or probably too faint for transit spectroscopy, their detection demonstrates that this category of objects actually exists. If such objects are also transiting around low-mass stars, they would be suitable targets for transit spectroscopy, in particular with the ARIEL space mission. Future surveys with space missions like GAIA (Global Astrometric Interferometer for Astrophysics), but also CHEOPS (CHaracterizing ExOPlanets Satellite), TESS (Transiting Exoplanet Survey Satellite) or PLATO (PLAnetary Transits and Oscillations of stars), are likely to provide new targets for this category of exoplanets.

Table 1

Examples of temperate giant exoplanets detected around F, G and K stars, with a mass larger than 0.5 Jovian mass and an eccentricity smaller than 0.1 (from www.exoplanets.eu). The equilibrium temperature is calculated assuming an albedo a = 0.03 and a fast-rotating planet (Equation 4, see text below, Section 2). In this paper, we investigate what could be the infrared spectrum of a temperate Jupiter and we study its detectability with the ARIEL space mission using transit spectroscopy, both through primary transit and secondary transit. We then analyse the most favorable type of star in order to optimize the primary transit signal.

The objective of ARIEL is the characterization of a large sample of exoplanets'atmospheres (500 -1000 targets) using transit spectroscopy in the midinfrared range (2 -8 m), with a resolving power between 100 and 200. It consists in an off-axis 90-cm telescope, passively cooled at 60-70 K, with a spectrometer cooled at 50 K and MCT detectors also cooled at 40 K. The stellar light is simultaneously recorded by a photometer operating at 1 -2 m. ARIEL is primarily designed for observing various kinds of exoplanets with equilibrium temperatures higher than 500 K. In order to enlarge the diversity of targets observable with ARIEL, we analyze in this paper the feasibility of observing cooler giant exoplanets, with equilibrium temperatures ranging between 350 and 500 K. 

The equilibrium temperature of an exoplanet

The equilibrium temperature Te of a slow-rotating exoplanet is given by this equation: In the case of a solar type star, taking as extreme conditions albedo values of 0.3 and 0.03 (as most commonly observed in the solar system), we find distances to the host star ranging between 0.26 AU (for a fast-rotating planet, Te = 500 K and a = 0.3) and 0.9 AU (for a slow-rotating planet, Te = 350 K and a = 0.03). For a solar-type star, the corresponding revolution periods range between 48 days and 312 days. The periods decrease if low-mass stars are considered. In order to perform transit spectroscopy of temperate Jupiters, several transits have to be performed over the lifetime of the mission, so, in what follows, we will focus on stars less massive than the Sun which allow a larger number of transits.

In order to estimate the equilibrium temperature of an exoplanet, we need to make an assumption on its rotation period. As a first approximation, this can be estimated from a comparison of its distance to the star versus the tidal lock radius, i.e. the distance to the star within which synchronous rotation takes place as a result of tidal forces. According to Murray and Dermott (1999) and [START_REF] Léger | Transiting exoplanets from the CoRoT space mission. VIII CoRoT-7 b: the first super-Earth with measured radius[END_REF], the characteristic time for a planet to reach synchronous rotation is given by:

synch = l(n -P)l x MP x D 6 x I x Q/k2P /[3/2 xG x RP 3 x M* 2 ] ( 5
)
where n is the mean motion of the revolution rate of the planet,  is its rotation rate, I is its moment of inertia, Q its dissipation constant, k2P its Love number of second order. MP is the planetary mass and RP is its radius, D is the distance of the planet to the star, M* is the stellar mass and G is the gravitational constant. It can be seen that for a given planet, synch is proportional to D 6 /M* 2 . Thus, the tidal lock radius Dsynch, corresponding to a given value of synch, is proportional to [M*] 1/3 . This is shown in the study by [START_REF] Kasting | Habitable zones around mainsequence stars[END_REF]Fig. 16) who indicate the tidal lock radius for Earth-like exoplanets; they find a tidal-lock radius Dsynch of 0.4 AU for a solar-mass star and 0.2 AU for a star of 0.1 solar mass. The corresponding revolution period of the planet is about 90 days, independent of the stellar mass.

For the present study, we need to estimate the tidal lock radius for giant exoplanets. In the lack of information about the values of the tidal parameters of these planets, we can compare the physical and tidal parameters of giant solar system planets as compared with terrestrial ones. Planetary moments of inertia range between 0.2 and 0.35. Rotation periods of the Earth and Jupiter are within a factor of 2. The Q/k2P factor is about 1000 for the Earth while this quantity is in the range 10 3 -10 5 for Jupiter and Saturn (Lainey, 2006). This leads to a synch value between 1 and 100 times the terrestrial value for giant exoplanets. Accordingly, Dsynch is expected to range between the terrestrial value (as indicated by [START_REF] Kasting | Habitable zones around mainsequence stars[END_REF]) and half this value, and the corresponding limit for the revolution period is between 90 days and 33 days. As a consequence, the values of Te shown in Table 1 are calculated using Equation 4, corresponding to fast-rotating exoplanets.

The atmosphere of a temperate Jupiter

Let us assume a Jupiter-type planet with an equilibrium temperature ranging between 350 K and 500 K under thermochemical equilibrium. The atmosphere is expected to be characterized by a troposphere where the temperature decreases as the altitude increases, following the adiabatic gradient. The atmospheric composition is expected to be dominated by hydrogen and helium. As for solar system giant planets, we assume for the tropopause a pressure level of 0.1 bar, constrained by the hydrogen and helium opacities. Figure 1 shows a P(T) diagram of a temperate Jupiter. If we assume thermochemical equilibrium, it can be seen that, in the exoplanet's environment, nitrogen and carbon are expected to be in the form of NH3 and CH4 respectively. In the same way, oxygen is expected in the reduced form of water H2O.

Figure 1. The T(P) diagram of a temperate

Jupiter with an equilibrium temperature ranging between 350 K and 500 K. The range corresponding to temperate Jupiters is shown with the black dots. It can be seen that carbon and nitrogen are expected to be in the form of NH3 and CH4 respectively. The figure is adapted from the EChO proposal [START_REF] Tinetti | The EChO Mission Proposal -A candidate for the ESA M3 mission[END_REF]).

In the troposphere, according to thermochemical equilibrium, CH4, NH3 and H2O are expected to be the main minor species. We assume for them, in the troposphere, the cosmic abundances. We know, however, that in the case of the giant planets of the solar system, these molecules are enriched by a factor about 4 as a result of the nucleation formation Owen and Encrenaz 2006). This factor of 4 depends upon the relative mass of the initial icy core with respect to the total mass of the planet (3% in the case of Jupiter); in the case of Saturn, for which the initial icy core is about 15% of the total mass, the expected enrichment is a factor of about 9. When the icy core becomes a major part of the total mass of the giant planet, as for Uranus and Neptune, the enrichment factor is above 30-40. We note that, in any case, in the nucleation model, the enrichments are expected to be the same for O, C, and N; so we can assume that the relative abundances of CH4, NH3 and H2O in the troposphere of the exoplanet are consistent with the cosmic abundances [START_REF] Grevesse | The new solar chemical composition[END_REF]. In our calculations, we assume for CH4 and NH3 the same abundances as measured in Jupiter below the condensation levels: CH4/H2 = 2 x 10 -3 [START_REF] Encrenaz | The atmospheric composition and structure of Jupiter and Saturn from ISO observations: What have we learnt? Plan[END_REF]), NH3/H2 = 2 x 10 -4 [START_REF] Encrenaz | The atmospheric composition and structure of Jupiter and Saturn from ISO observations: What have we learnt? Plan[END_REF][START_REF] Bolton | Jupiter's interior and deep atmosphere: The initial pole-to-pole passes with the Juno spacecraft[END_REF].

For H2O we assume H2O/H2 = 4 x 10 -3 below the water cloud level. We thus adopt H2O:CH4:NH3 = 2:1:0.1. It must be noted that these numbers are no more than examples, chosen to build a typical spectrum of a temperate Jupiter.

With respect to the true atmosphere of Jupiter, the main difference of a temperate Jupiter is that there is no cold trap at the tropopause: at a temperature above 350 K, all molecules, including H2O and NH3, keep a constant mixing ratio with the altitude, up to a level where they become photo-dissociated. As a result, the infrared spectrum of a temperate Jupiter is expected to be fully dominated by CH4, NH3 and H2O.

Our transmission spectra are generated using a line-by-line calculation of the absorption coefficient, assuming a single pressure layer (1 bar) and a single temperature (300 K).

We use for NH3 a column density of 3 m-amagat, as measured in Jupiter in the 1.56-m spectral window which probes down to a pressure of a few bars (Owen et al. 1977); we then derive column densities of 30 m-amagat for CH4 and 60 m-amagat for H2O, respectively. Our transmission spectrum is convolved with a Gaussian function with a resolving power of 100 at 2 m, 200 at 4 m and 400 at 8 m. The resolution is comparable to the ARIEL one in the 2 -4-m range and higher at 4 -8 m range, which allows a better description of the band shapes.  Figure 2 shows the synthetic transmission spectrum of a temperate Jupiter. For comparison, Figure 3 shows the synthetic spectrum of a cold Jupiter, where most of the water is trapped in the form of ice in the lower troposphere (in the gaseous phase, H2O:CH4:NH3 = 0.07:1:0.1). This corresponds to the spectrum of the actual Jupiter if it were seen in transmission in front of the Sun.

Figure 2. A typical transmission spectrum for a temperate Jupiter (T = 350 K). Jupiter-like relative abundances are assumed for the molecules in the troposphere below the clouds, and no condensation is assumed for H2O: H2O:CH4:NH3 = 2.0:1.0:0.1. The full scale is equal to the amplitude of the transmission signal calculated in Table 2.

Total& CH 4& NH 3& H 2 O&
Figure 3. A typical transmission spectrum for a cold Jupiter (T = 150 K). Jupiter-like relative abundances are assumed for the molecules in the troposphere, and condensation is assumed for H2O: H2O:CH4:NH3 = 0.07:1.0:0.1. The full scale is equal to the amplitude of the transmission signal calculated in Table 2.

Above the tropopause, other minor molecules are expected as products of photodissociation: C2H2 and C2H6 from the methane photo-dissociation, CO and CO2 from the water photo-dissociation. By analogy with the case of Jupiter, we estimate the possible contribution due to photochemistry in the transmission spectrum by assuming the following abundances:

(1) at P = 100 mb (tropopause), H2O:CH4:NH3 = 2:1:0.1 (as in the troposphere); (2) in the stratosphere, at P = 1 mb, CH4:C2H2:C2H6 = 1:10 -4 :2 10 -3 , and H2O:CO:CO2 = 1:1:0.2. Calculations show that CO and CO2 contribute as minor absorbers in the infrared spectrum, while the contribution of hydrocarbons is negligible (Figure 4). including the effect of photochemistry. Jupiter-like relative abundances are assumed for the molecules in the troposphere and no condensation is assumed (P = 100 mb, H2O:CH4:NH3 = 2.0:1.0:0.1). In the stratosphere (P = 1 mb), the relative abundances are H2O:CO:CO2 = 1:1:0.2. The contribution of hydrocarbons is negligible. The full scale is equal to the amplitude of the transmission signal calculated in Table 2.

!!!!CH 4! !!!!NH 3! !!!!H 2 O! !!!!CO 2 !!CO! !!!!!!!l!!!!!!!l!
What could be the condensates present in a temperate Jupiter? We know that hazes may be important contributors that flatten the infrared transmission spectrum of hot or warm exoplanets, as illustrated by the flat spectrum of GJ1214 b. For temperate objects colder than 500 K, all ices are in gaseous form. We can thus expect temperate Jupiters to be relatively free of condensates, as discussed by [START_REF] Sudarsky | Albedos and reflection spectra of extrasolar giant planets[END_REF]. [START_REF] Morley | Neglected clouds in T and Y dwarf atmospheres[END_REF], however, have shown that some condensates (including Na2S, ZnS and KCl) might be present in the atmospheres of T-dwarfs, so the possible presence of clouds in temperate exoplanets is still an open question. In the lack of more information, we adopt in our study a low value of the albedo (0.03), as has been actually observed on hot Jupiters (Rowe et al. 2008).

Our analysis is consistent with the study of [START_REF] Sudarsky | Albedos and reflection spectra of extrasolar giant planets[END_REF] who have characterized the various types of giant exoplanets as a function of their equilibrium temperature. Giant exoplanets in the 350 -500 K range, as considered in the present study, are intermediate between Class II and Class III and show no NH3 nor H2O clouds; they are supposed to have a very low albedo. The situation would be different for objects with an equilibrium temperature of 250 K (Class II), as the presence of a water cloud would lead to a high albedo. However, Class II objects are not considered in the present study because their expected transmission signal would be below the sensitivity limit of ARIEL (see Section 7).

Primary transit spectroscopy of a temperate Jupiter

Let us assume a Jupiter-size exoplanet transiting in front of its host star. During a primary transit, its atmosphere is observed in transmission at terminator in front of the stellar light. The area of planetary atmosphere observed in transmission is an annulus around the planet with a radial height of about 5 x H, where H is the scale height. H is equal to RT/μg, where R is the perfect gaz constant, T the temperature, μ the mean molecular weight of the atmosphere and g the planet's gravity. The amplitude of the absorption within the exoplanet's atmosphere can be approximated as follows [START_REF] Tinetti | The EChO Mission Proposal -A candidate for the ESA M3 mission[END_REF])

A = 5 x [2RpH/R* 2 ] ( 6
)
where Rp and R* are the radii of the planet and the star respectively. For a hydrogen-helium rich planet with μ = 2.4, the scale height (in km) can be expressed as

H = 3.46 x TP/g (7)
where TP is the equilibrium temperature of the exoplanet in Kelvins and g is the surface gravity in m/s 2 (at a pressure level of 1 bar). The surface gravity g can be expressed as g = 25 x MP/RP 2 (8)

where MP and RP are the mass and the radius of the planet, expressed in Jovian masses and radii, respectively. As a result, the amplitude of the absorption can be written (with R* expressed in solar radii):

A = 1.4 10 -6 x RP x H/R* 2 (9)
For a hydrogen-rich exoplanet, we have A = 1 .94 10 -7 x T x RP 3 /MP/R* 2 = 1.94 10 -7 x T /R* 2 (10)

where  is the planet's density, expressed in Jovian units. For hot Jupiters, typical values of A are a few 10 -4 . The amplitude of the signal is especially strong for planets having a high temperature (and thus being close to their star), a large radius and a low molecular weight.

For a temperate Jupiter with an equilibrium temperature of 400 K around a solar-type star, we find A = 7.75 10 -5 . As compared with a hot Jupiter (T = 1600 K), the amplitude is lower by a factor of 4 due to the low temperature. As compared with a super-Earth with a density equal of 4 times the Jovian value, with an effective temperature of 1600 K, the amplitude of a temperate giant exoplanet is expected to be of the same order. Temperate Jupiters could thus be considered to exist in the detectability range of ARIEL.

Secondary transit spectroscopy of a temperate Jupiter

Secondary transits provide a direct measurement of the dayside emission of the exoplanet. The ratio of the planet/star flux can be approximated as follows [START_REF] Tinetti | The EChO Mission Proposal -A candidate for the ESA M3 mission[END_REF]:

-in the visible and near-infrared range:

1 = [RP/R*] 2 x [TP/T*] 4 (11)
-in the mid and far-infrared (Rayleigh-Jeans approximation):

2 = [RP/R*] 2 x [TP/T*] ( 12
)
For a temperate Jupiter (T = 400 K), as compared to a hot Jupiter (T = 1600 K), 1 is lower by a factor 4 4 = 256, and 2 by a factor of 4. As compared with a hot super-Earth, 1 is lower by a factor of 10 and 2 larger by a factor of 7.5. These numbers show that, in the near-infrared range, temperate Jupiters are not suitable targets for secondary transit spectroscopy. The comparison would be more favorable for 2; however the Rayleigh-Jeans approximation holds only at long wavelengths (above about 20 m), and is not valid in the spectral range of ARIEL (2-8 m).

It can be noted that the secondary transit spectrum of a temperate Jupiter could be significantly more complex than the primary transit spectrum, for two reasons: (1) if a temperature inversion takes place above the tropopause, the bands of CH4, NH3 and H2O could be present in emission;

(2) photo-dissociation products of methane (i.e. C2H2, C2H6) could have strong signatures in the mid-infrared, as observed in the case of solar system giant planets.

In summary, we consider that, within the spectral range of ARIEL, temperate Jupiters are suitable targets for primary transit spectroscopy only.

How to optimize the primary transit spectroscopy of temperate Jupiters

We now consider temperate giant exoplanets orbiting low-mass stars. The advantage is twofold: (1) the amplitude of the primary transit signal is larger by a factor [1/R*] 2 where R* is the radius of the star;

(2) the revolution period is shorter than a year, which allows repeated transits of the object during the lifetime of the ARIEL mission.

For a star of mass M* and radius R* (in solar units) and effective temperature T* (in K), the star-planet distance D (in AU) of a fast-rotating exoplanet with an equilibrium temperature TP is given by the following equation: where a is the albedo of the planet and TP its equilibrium temperature; M*, R* and T* are the mass, radius and effective temperature of the host star, expressed in solar units.

In the case of a slow-rotating (tidally locked) planet, equations ( 13 (17) Assuming for our exoplanet two possible values of TP (350 K and 500 K), we determine in each case its distance to its host star D. For each spectral type, we consider the two cases (fast rotating or tidally locked object) and we compare the two values of D with Dsynch . It can be seen that the tidally locked regime is found for the M5 and M8 stars, while fast-rotating objects are expected in all other cases. Then we calculate the revolution period P of the exoplanet, using the equation:

D 3 /P 2 = M* ( 18
)
where D is in AU, P in year and M* in solar mass.

We then calculate the amplitude of the transmission spectrum using Equation ( 9) for the two values of the exoplanet's equilibrium temperature. The transit time t is estimated using the following equation:

t = [2R*/D] x P/2 (19)
with t expressed in hours, P in days, R* in solar radii, and D in AU, this equation becomes:

t = 0.0359 x R* x P /D (20)
Table 2 lists the result for a variety of stars, ranging from solar-type stars to M8 dwarfs.

Table 2

Estimated semi-major axis, rotational period, amplitude of primary transit signal and transit time for a Jovian-like exoplanet transiting around a star of spectral type between G2 and M8, using Equations ( 9), ( 13), ( 15), ( 16), ( 17), ( 18) and ( 20), with an albedo a = 0.03, assuming either a fast rotator (columns 6 and 8) or a tidally locked object (columns 7 and 9). Two cases are considered: TP = 350 K and TP = 500 K.

Sensitivity estimate

In order to estimate the observing time needed to obtain a transmission spectrum of a temperate Jupiter, we use as a calibrator the exoplanet WASP-76 b, for which a synthetic transmission spectrum is shown in the ARIEL proposal. This object has a mass of 0.92 MJ, a radius of 1.83 RJ, a temperature of 2200 K. The star WASP-76, of V magnitude 9.5, located at 120 pc from the Sun, has a mass of 1.46 Ms, a radius of 1.73 Rs, and a temperature of 6250 K. The star-planet distance is 0.033 AU and the rotation period of the exoplanet is 1.81 days. The amplitude of primary transit is about 1.0 10 -3 . The time transit is 3.4 hours. A summation of 25 transits (corresponding to 85 hours of total observing time) is needed to achieve a S/N of about 10.

We now estimate the number of transits needed for a temperate Jupiter around a Mdwarf to be observed by transmission spectroscopy with a total observing time of 100 hours. This corresponds to a S/N of about 12 for WASP-76 b, and about 2.5, 9.5 and 57 for temperate Jupiters around M0, M5 and M8 stars respectively. Table 3 summarizes the results. We also indicate the total time in orbit needed to accumulate the required number of transits; this time must be shorter than the lifetime of the ARIEL mission of 4 years. The case of WASP-76 is added for comparison. It can be seen that all three classes of objects can be considered within a lifetime of 4 years; however, temperate Jupiters around M5 to M8 dwarfs should be favored for a higher S/N ratio and a shorter required time of observation. As shown in Fig. 4, a signalto-noise of 10 or higher with the resolving power of ARIEL (100 -200) should be sufficient for separating the signatures of the main molecular species (H2O, CH4, NH3, CO, CO2).

How to find exoplanets around M-dwarfs?

So far we know of no example of temperate Jupiters around M stars. Can we estimate the probability of finding such target in the coming decade? We know that M-dwarfs are dominant in stellar populations, however they are difficult to identify because of their low luminosity. According to some authors [START_REF] Henry | [END_REF][START_REF] Reylé | Habilitation à Diriger des Recherches[END_REF]), 70% of these systems (corresponding to about 3000 samples) might exist remaining to be detected within 25 pc from the Sun. Several hundred of objects have already been found using the ground-based infrared surveys DENIS and 2MASS [START_REF] Reylé | Habilitation à Diriger des Recherches[END_REF], Rajpurohit et al. 2014).

In addition, the CHEOPS, TESS and PLATO space missions are expected to provide new targets for transit spectroscopy. However, they will be oriented rather toward bright stars and small exoplanets. More importantly, the GAIA space mission is expected to bring an enormous database for this study. According to a simulation of stars observable with GAIA performed by [START_REF] Robin | [END_REF], a total of 1.3 10 7 stars with a visual magnitude less than 12 (corresponding to the sensitivity limit of ARIEL) are expected to be observed by GAIA, including 11% of M-type stars. Knowing that about each star is expected to have a planet, the number of potential candidates is expected to be huge. In addition, GAIA will be able to perform photometry transits of Jupiter-like exoplanets around stars brighter than the 14 th magnitude, including M-type stars.

It should be noted that transit spectroscopy around a M-dwarf star might be more difficult than around a G or K star, because of the intermittent flares emitted by these stars. However, the continuous near-infrared monitoring of the stellar flux by ARIEL, in parallel with the transit measurements, should make possible the identification and the removal of these anomalies.

Assuming M-stars can be found in sufficient numbers, one may question the probability of finding giant planets around such low-mass stars. We have no answer to this question yet, but some lessons can be drawn on the basis of the present exoplanet catalogs (Schneider, 2010). We know of about 150 exoplanets with masses greater than 10 jovian masses orbiting F, G or K stars, which seems to indicate that the mass ratio between a giant planet and its host star may actually be greater than the Jupiter/Sun mass ratio; a few examples of such objects can be found in Table 1. In addition, the recent detection of 7 planets in orbit around the ultra-cool M8 star TRAPPIST-1 [START_REF] Gillon | Seven temperate terrestrial planets around the nearby utra-cool dwarf star TRAPPIST-1[END_REF], possibly the result of a migration from beyond the ice line, may suggest that the presence of larger objects is also possible. In addition, the exoplanet demographics from microlensing suggest that giant planets around M-stars are more frequent than expected from radio-velocimetry studies, with a frequency of 0.11 for planets more massive than 50 terrestrial masses [START_REF] Clanton | Synthetizing exoplanet demographics from radial velocity and microlensing surveys. II. The frequency of planets orbiting M-dwarfs[END_REF].

Conclusion

In summary, it appears that temperate Jupiters orbiting M stars at semi-major axes between about 0.1 and 0.5 AU could be suitable targets for ARIEL. If such targets are identified in future surveys, their study with ARIEL would allow us to enlarge the range of physical atmospheric parameters to be covered, and thus to extend the exoplanets' exploration by the mission.

  Te = (1 -a) 0.25 x 331.0 x (T*/5770) x R* 0.5 )/ D 0the albedo, T* the effective temperature of the star (in K), R* its radius and M* its mass (in solar units), and D the distance of the exoplanet to the star (in AU). For a fast-rotating exoplanet, these equations become: Te = (1 -a) 0.25 x 279.0 x (T*/5770) x R* 0.5 )/ D 0

Figure 4 .

 4 Figure4. A typical transmission spectrum for a temperate giant exoplanet (T = 350 K), including the effect of photochemistry. Jupiter-like relative abundances are assumed for the molecules in the troposphere and no condensation is assumed (P = 100 mb, H2O:CH4:NH3 = 2.0:1.0:0.1). In the stratosphere (P = 1 mb), the relative abundances are H2O:CO:CO2 = 1:1:0.2. The contribution of hydrocarbons is negligible. The full scale is equal to the amplitude of the transmission signal calculated in Table2.

  D = [(1-a) 0.25 x 279.0 (T*/5770)x R* 0.5 ]/TP] 2

  ) an (15) become: D = [(1-a) 0.25 x 331.0 (T*/5770)x R* 0.5 ]/TP]

Table 3

 3 Number of transits required to obtain 100 hours of integration time on a temperate Jupiter around a M star and total time required to accumulate these transits.

	Spectral type R*(Rs) A	S/N in	Period	t(h)	Number	Total time
				100 hours	(d)		of	needed for
							transits	ttot = 100 h (d)
	WASP-76	1.73	1.0 10 -3	12	1.81	3.4	30	54
	M0(TP=350K)	0.63	1.71 10 -4	2	48	5.4	19	912
	M0(TP=500K)		2.44 10 -4	3	17	3.9	26	442
	M5(TP=350K)	0.32	6.64 10 -4	8	18	2.6	38	684
	M5(TP=500K)		9.50 10 -4	11	6	1.7	59	354
	M8(TP=350K)	0.13	3.98 10 -3	47	6	1.0	100	600
	M8(TP=500K)		5.70 10 -3	67	2	0.6	167	334


C Krafft

A Volokitin

Whistler envelope solitons. II. Interaction with non relativistic electron beams in plasmas with density inhomogeneities

Russia

The paper studies the self-consistent interactions between whistler envelope solitons and electron beams in inhomogeneous plasmas, using a Hamiltonian model of waveparticle interaction where nonlinear equations describing the dynamics of whistler and ion acoustic waves and including a beam current term are coupled with Newton equations. It allows to describe the parallel propagation of narrowband whistlers interacting with arbitrary particle distributions in irregular plasmas. It is shown that the whistler envelope soliton does not exchange energy with all the resonant electrons as in the case of whistler turbulence but mostly with those moving in its close vicinity (locality condition), even if the downstream particle distribution is perturbed. During these interactions the soliton can either damp and accelerate particles, either absorb beam energy and cause electron deceleration. If the energy exchanges are significative, the envelope is deformed; its upstream front can steepen whereas oscillations can appear on its downstream side. Weak density inhomogeneities as the random fluctuations of the solar wind plasma have no strong impact on the interactions of the whistler soliton with the resonant particles.

I. INTRODUCTION

Interactions between whistler waves and electron fluxes have been extensively studied owing to space measurements 2-7 and laboratory experiments [START_REF] Krafft | Whistler wave emission by a modulated electron beam[END_REF][START_REF] Starodubtsev | Resonant Cherenkov emission of whistlers by a modulated electron beam[END_REF][START_REF] Starodubtsev | Resonant cyclotron emission of whistler waves by a modulated electron beam[END_REF][START_REF] Starodubtsev | Whistler wave emission by a modulated electron beam through transition radiation[END_REF][12] , as well as in analytical works and numerical simulations [START_REF] Lee | Stability analysis of whistler amplification[END_REF][START_REF] Hashimoto | Temperature anisotropy and beam type whistler instabilities[END_REF][START_REF] Omura | Computer experiment on whistler and plasma wave emissions for spacelab-2 electron beam[END_REF][START_REF] Zhang | Linear and nonlinear interactions of an electron beam with oblique whistler and electrostatic waves in the magnitosphere[END_REF][START_REF] Nishikawa | New aspects of whistler waves driven by an electron beam studied by a 3D electromagnetic code[END_REF][START_REF] Volokitin | Whistler waves emission by a modulated electron beam : nonlinear theory[END_REF][START_REF] Krafft | Nonlinear interaction of whistler waves with a modulated thin electron beam[END_REF][START_REF] Krafft | Nonlinear electron beam interaction with a whistler wave packet[END_REF][START_REF] Krafft | Interaction of suprathermal solar wind electron fluxes with sheared whistler waves : Fan instability[END_REF][START_REF] Trakhtengerts | Whistler-electron interactions in the magnetosphere: new results and novel approaches[END_REF][START_REF] Eliasson | Nonlinear evolution of the electromagnetic electron-cyclotron instability in bi-kappa distributed plasma[END_REF] . Moreover, the interactions of whistlers with particles in inhomogeneous plasmas have been investigated theoretically and numerically. For example, such interactions were studied [START_REF] Hikishima | Full particle simulation of whistlermode rising chorus emissions in the magnetosphere[END_REF] in a plasma with a non uniform magnetic field, considering parallel propagating coherent whistler waves interacting with energetic resonant electrons. A lot of studies were focused on the problem of whistler chorus emissions where the inhomogeneity of the ambient magnetic field plays an important role. However, the influence of other types of inhomogeneities, as random plasma density irregularities, on whistlers' interactions with electron fluxes were only rarely investigated, in particular during the occurrence of nonlinear effects; such plasma irregularities are present in the solar wind or in other space plasmas. They are particularly important as they can strongly reduce the efficiency of wave-particle interactions, as it was evidenced recently for the case of Langmuir turbulence in the solar wind [START_REF] Krafft | Interaction of energetic particles with waves in strongly inhomogeneous solar wind plasmas[END_REF][START_REF] Krafft | Langmuir wave decay in inhomogeneous solar wind plasmas: Simulation results[END_REF] .

Measurements by the satellite Helios have shown that whistler turbulence with wave frequencies up to the electron gyrofrequency is present in the solar wind, together with the occurrence of ion acoustic-like type oscillations [START_REF] Beinroth | Properties of whistler mode waves between 0.3 and 1.0 au from Helios observations[END_REF] . This observation was also confirmed by the spacecraft Ulysses [START_REF] Lengyel-Frey | Ulysses observations of whistler waves at interplanetary shocks and in the solar wind[END_REF] which detected an ubiquitous whistler wave background in the solar wind, that can be the source of many wave-particle phenomena. When the whistler waves' intensities become sufficiently large, nonlinear effects can occur, and in particular modulational instabilities which can give rise to soliton formation. Large amplitude whistlers have been measured by the Wind and Stereo satellites [START_REF] Cattell | Discovery of very large amplitude whistlermode waves in Earth's radiation belts[END_REF] and a statistical study of such waves and their association with energetic electron distributions was reported [START_REF] Wilson | The properties of large amplitude whistler mode waves in the magnetosphere: Propagation and relationship with geomagnetic activity[END_REF] . Localized whistler wave packets were observed in the solar wind [START_REF] Russell | Ogo 5 observations of upstream waves in the interplanetary medium: Discrete wave packets[END_REF] and the authors suggested they could be soliton structures. Such packets were also detected by the spacecraft Freja and Cluster in connection with density cavities, and were reported to be whistler envelope solitons [START_REF] Huang | Whistler waves in freja observations[END_REF][START_REF] Moullard | Density modulated whistler mode emissions observed near the plasmapause[END_REF] . At the same time, satellites detected different distributions of energetic particles in the plasma regions where such observations were performed. Consequently the questions arise under what conditions these whistler envelope solitons are able to interact with particles, whether or not they can exchange a noticeable amount of energy and momentum with them and in what extent they can loose their stability.

The interactions of Langmuir and ion acoustic solitons with electron fluxes have been studied theoretically and numerically in non magnetized plasmas by different authors [START_REF] Kingsep | Spectra of strong Langmuir turbulence[END_REF][33][START_REF] Valeo | Solitons and resonant absorption[END_REF][START_REF] Gorev | Interaction of Langmuir solitons with plasma particles[END_REF][START_REF] Karpman | The effects of the interaction between ion-sound solitons and resonance particles in a plasma[END_REF][START_REF] Rypdal | Interaction of Langmuir solitons with resonant particles[END_REF] .

They have shown that the solitons can damp and that, as a consequence, the kinetic energy of the resonant particles' population can increase. For electromagnetic waves in a magnetized plasma, the situation is obviously more complex and only very rare works have been undergone on this subject. To our knowledge, only in Ref. 37 the case of whistler solitons interacting with an electron beam is investigated; however the authors limited their study to the case of whistlers with frequencies very small compared to the electron gyrofrequency and a parallel beam velocity distribution modeled by a Dirac function. They only calculated the rate of energy loss of the solitons to the beam in conditions typical for the solar wind, using an approach involving a gas of solitons; they did not consider the possibility for the soliton to gain energy from the particles and did not perform numerical simulations to study the dynamics of the whistler envelope solitons interacting with the electrons.

In view of the above, we have built a Hamiltonian model describing the interaction of whistler waves with electron fluxes in a inhomogeneous magnetized plasma, in order to perform numerical simulations aimed to study various nonlinear phenomena. The model involves in particular a parabolic-type equation including self-consistently a beam current term, which is coupled to Newton equations describing the individual motion of the beam electrons as well as to lower frequency equations taking into account ponderomotive effects [START_REF] Krafft | Dynamics of whistler enevelope solitons in inhomogeneous plasmas[END_REF] . This paper is aimed to present this model in detail and to apply it to study the interactions of whistler envelope solitons with electron fluxes (in homogeneous or inhomogeneous plasmas), i.e. to understand the dynamics, the stability and the energy exchanges of the soliton during its interactions with the resonant particles. Such model can also be used to understand whistler chorus modulation by density variations. As a first approach, we consider hereafter coherent narrowband whistler waves propagating parallel to the ambient magnetic field and interacting with arbitrary electron distributions.

II. LINEAR EXCITATION OF COHERENT WHISTLERS BY BEAMS

Most of whistler waves' instabilities are driven by the free energy contained in anisotropic electron velocity distributions as loss cones, rings, horseshoes or beams, which is released via normal cyclotron resonant interactions, as well as in distributions presenting perpendicular temperatures larger than parallel ones. For example, in the solar wind, the temperature anisotropy and the electron heat flux instabilities of whistlers can likely occur.

Our study concerns narrowband whistler waves excited by beams and propagating parallel to the ambient magnetic field. In this view, studying the wave emission of a hot anisotropic electron beam drifting along a constant magnetic field in a homogeneous cold plasma for a wide range of parameters, some authors 16 calculated the variation of the maximum growth rates γ max of all types of waves emitted as a function of their propagation angle θ, showing that for the electromagnetic whistler mode, γ max reaches its highest value at parallel propagation θ = 0 • and remains almost unchanged when θ increases until 30 • . However, the electrostatic mode excited (with phase velocity in the direction of the drifting electron beam) presents a larger growth rate for θ = 0 • , even if it decreases strongly with θ to reach, around θ 30 • , roughly the same value as that of the electromagnetic mode.

So, for quasi-parallel propagation (θ 30 • ), both the whistler and the electrostatic modes can be excited simultaneously. On the contrary, for strict parallel propagation, electrostatic waves grow more fast, so that the beam electrons are trapped and diffused, reducing by the way significantly the whistler growth rates. However, the whistler mode is excited via the normal cyclotron resonances and the electrostatic one by the Landau resonances; therefore, if the beam is not monoenergetic but warm, both modes, excited on different time scales, are also characterized by different ranges of wavenumbers and frequencies. Indeed, at parallel propagation, the electrostatic and electromagnetic modes are uncoupled as they act on different time scales, having different growth rates. But at oblique propagation their growth rates can become comparable and they can act simultaneously.

In Ref. [START_REF] Omura | Competing processes of whistler and electrostatic instabilities in the magnetosphere[END_REF] it was shown that a cold beam should not be able to excite efficiently whistler waves, due to the fact that electrostatic waves grow more fast than whistlers. However, a sufficiently warm beam is able to radiate whistlers in spite of the diminution of their growth rates during the quasi-linear electrostatic diffusion process. Moreover they estimated that for a dense plasma the whistler instability can dominate if the beam pitch angle

θ p = tan -1 ( v 2 ⊥ /v 2 b )
1/2 is large enough, i.e. if the perpendicular thermal velocity is large enough (v b is the beam parallel velocity and v 2 ⊥ is the square perpendicular velocity proportional to the perpendicular thermal energy). For warm beams, the wavenumber width ∆k wherein the whistler growth rates are positive can be rather narrow (i.e. ∆k/k 1), as shown below (see also Refs. [START_REF] Hashimoto | Temperature anisotropy and beam type whistler instabilities[END_REF][START_REF] Omura | Computer experiment on whistler and plasma wave emissions for spacelab-2 electron beam[END_REF][START_REF] Zhang | Linear and nonlinear interactions of an electron beam with oblique whistler and electrostatic waves in the magnitosphere[END_REF] ). In this case the growth of narrowband whistlers occurs for a limited range of frequencies and wavenumbers; note that, for such type of emissions, the ranges of k where normal cyclotron and Landau resonant velocities are lying are narrow and, for adequate conditions, they can be well separated one from the other. This circumstance justifies the possibility to consider intense parallel propagating whistler emissions and assume hereafter that ∆k/k 0.1.

Approximate expressions for the linear growth rate of whistler waves excited in a cold plasma by a warm anisotropic non relativistic beam of velocity distribution function F b (v z , v ⊥ ) were determined [START_REF] Bell | Plasma instability in the whistler mode caused by a gyrating electron stream[END_REF] , neglecting the parallel temperature effects compared to the perpendicular ones. The parallel propagating whistler dispersion relation can be written as follows, when the beam parallel thermal velocity v T bz is very small compared to the

resonance velocity v R = (ω -ω c ) /k (i.e. kv T bz / (ω -ω c ) 1) c 2 k 2 -ω 2 + ω 2 p ω ω -ω c + n b n 0 ω 2 p ω -kv b ω -kv b -ω c + k 2 v 2 ⊥ 2 (ω -kv b -ω c ) 2 + k 2 v 2 z ω c 2 (ω -kv b -ω c ) 3 0, (1)
where n b and n 0 are the beam and the plasma densities; ω and k are the frequency and the wavenumber of the whistler; ω p and ω c are the electron plasma and cyclotron frequencies;

v 2 z = ∞ -∞ F z (v z )v 2 z dv z and v 2 ⊥ = 2π ∞ 0 F ⊥ (v ⊥ )v 3
⊥ dv ⊥ are the mean square velocities along and perpendicular to the magnetic field, respectively, where F z and F ⊥ are the parallel and the perpendicular beam velocity distributions. This equation has been solved numerically

(see Fig. 1) when F b (v z , v ⊥ ) is a DGH (Dory-Guest-Harris) function 41 of index ν F b (v z , v ⊥ ) = 1 π 3/2 Γ (ν/2 + 1) v ν ⊥ v T bz v ν+2 T b⊥ exp - v 2 ⊥ v 2 T b⊥ exp - (v z -v b ) 2 v 2 T bz = F z (v z )F ⊥ (v ⊥ ), (2)
where v T b⊥ is the perpendicular beam thermal velocity, with 2π

∞ 0 F ⊥ (v ⊥ )v ⊥ dv ⊥ = 1 and ∞ -∞ F z (v z )dv z = 1;
ν is an integer and Γ is the Gamma function.

The beam drifts in the direction opposite to the ambient magnetic field B 0 = B 0 z. The interactions between the beam and the waves take place at normal cyclotron resonance conditions kv z = ω -ω c . Using (2) one can calculate that v 2 ⊥ = v 2 T b⊥ (ν/2 + 1) . We define in this case the anisotropy factor as A = (ν + 1)T b⊥ /T bz , which reduces to A = T b⊥ /T bz for a bi-Maxwellian, where T bz and T b⊥ are the paralllel and the perpendicular beam temperatures.

Figure 1 shows the variations of the growth rate γ/ω c of whistler waves as a function of their normalized wavenumber ck/ω c , for three different anisotropy factors A and typical conditions of the Earth magnetosphere (upper panel) and the solar wind (lower panel). The maximum growth rate increases with A. One can see in both cases that the wavenumber region ∆k where γ/ω c is maximum is narrow, as ∆k/k 0.1. Therefore coherent narrowband whistlers propagating parallel to the ambient magnetic field can be efficiently excited by a warm anisotropic beam. Note that the maximum growth rates can reach values of the order of γ/ω c 0.01 even for weak beams (n b /n 0 10 -4 ) and rather small anisotropies. Then it is of interest to develop a nonlinear model of coherent whistlers which takes into account the interactions of electromagnetic waves with particle fluxes together with different nonlinear effects : waves' interactions with plasma inhomogeneities, soliton propagation, wave decay, etc. Such model is developed in the next Section.

III. NONLINEAR HAMILTONIAN MODEL

Let us build a model describing the self-consistent interactions of coherent electromagnetic whistlers with particles in a plasma presenting inhomogeneities, by following the approach developed in our companion paper [START_REF] Krafft | Dynamics of whistler enevelope solitons in inhomogeneous plasmas[END_REF] . The electric field of the whistler wave is given by

E = Re E (z, t) e -iω 0 t+ik 0 z (x+iy) , (3)
where

E = E x -iE
D (ω 0 , k 0 ) = k 2 0 c 2 -ω 2 0 + ω 2 p ω 0 ω 0 -ω c = 0. (4)
Then the equation of evolution of the wave electric field envelope is obtained in the form

42 i ∂E ∂t + iv g0 ∂E ∂z + v g0 2 ∂ 2 E ∂z 2 = - ω 0 ω 2 p (ω 0 -ω c ) D 0 ρ + ω c ω 0 -ω c k 0 V ω 0 E, (5)
where ρ = δn e /n 0 and V = δv e are the perturbations of the slowly varying electron density n e and fluid velocity v e , respectively; n 0 is the unperturbed plasma density at equilibrium, and D 0 = (∂D/∂ω) 0 = (∂D/∂ω) k=k 0 ,ω=ω 0 (see also below a similar notation for other variables). The group velocity and its derivative are given by

v g0 = dω dk 0 = - (∂D/∂k) 0 (∂D/∂ω) 0 = - 2c 2 k 0 D 0 , v g0 = d 2 ω dk 2 0 = v g0 k 0 1 - v 2 g0 c 2 1 + ω 2 p ω c (ω c -ω 0 ) 3 . (6)
In the case of quasineutral slow oscillations (ρ δn e /n 0 δn i /n 0 , where δn i is the ion density perturbation) and, consequently, due to the charge conservation at lowest order (V δv e δv i , where δv i is the ion population's velocity perturbation), one can obtain the ion acoustic dynamics in the form 42

∂ ∂t V - k 0 ω c ω 2 p ω 2 0 (ω 0 -ω c ) 2 |E| 2 16πn 0 m i + ∂ ∂z c 2 s ρ + ω 2 p ω 0 (ω 0 -ω c ) |E| 2 16πn 0 m i 0 (7) ∂ρ ∂t + ∂V ∂z 0, (8)
where the hydrodynamic nonlinear terms have been neglected compared to the ponderomotive terms, which include a stationary and a nonstationary part; c s is the ion acoustic velocity and m i is the proton mass. Defining the hydrodynamic flux Ψ in the form

∂Ψ ∂z = V - k 0 ω c ω 2 p ω 2 0 (ω c -ω 0 ) 2 |E| 2 16πn 0 m i , (9)
and identifying the couples of canonical variables as (Ψ, -k 2 0 ρ/D 0 ) and (C, C * ), with C = σ 0 Ee -iω 0 t , we get the Hamiltonian of the system without resonant particles as

H w = L H w dz/L with H w = iv g0 2 C ∂C * ∂z -C * ∂C ∂z + v g0 2 ∂C ∂z 2 + ω 0 ω 2 p ρ |C| 2 (ω c -ω 0 ) D 0 + Ln 0 m i 2   c 2 s ρ 2 + ∂Ψ ∂z + k 0 ω c ω 2 p ω 2 0 (ω c -ω 0 ) 2 |E| 2 16πn 0 m i 2   , (10)
where we will show below that

σ 2 0 = Lε 0 16π . (11)
ε 0 = -D 0 /ω 2 0 is the derivative of the dielectric constant with respect to frequency, taken at k = k 0 and ω = ω 0 ; L is the size of the system along z. The Hamilton equations are then given by

- k 2 0 D 0 ∂ρ ∂t = δH w δΨ , k 2 0 D 0 ∂Ψ ∂t = δH w δρ , ∂C ∂t = -i δH w δC * , (12)
where δ refers to the functional derivative. Note that the choice of the canonical variables (C, C * ) in the form C = σ 0 Ee -iω 0 t is due to the fact that they have also to be canonical variables for the full system, i.e. the plasma where a flux of resonant electrons is now introduced. This flux has a density n b which is small compared to that of the background plasma, i.e. n b n 0 . Let us determine the energy conservation law starting from the electrons' motion

dv p dt = - e m e E - e m e c v p × B, (13)
where v p is the velocity of the particle p; m e and -e < 0 are the electron mass and charge.

The components of the electric and magnetic fields E and B along the axes x and y are

given by

E x = Re k E k e -iω 0 t+i(k 0 +k)z , E y = Re k iE k e -iω 0 t+i(k 0 +k)z , (14)
where k k 0 , and

B x = - ck 0 ω 0 E y , B y = ck 0 ω 0 E x , B z = B 0 = ω c m e c e . (15)
Then we can calculate that 1 2

dv 2 p dt = - e m e (v px E x + v py E y ) = - e m e Re k (v px + iv py ) E k e -iω 0 t+i(k 0 +k)z , (16)
which allows to write that

1 N d dt p m e v 2 p 2 + e Re k E * k J k = 0, (17)
with

J k = 1 N p (v px -iv py ) e iω 0 t-i(k 0 +k)zp , (18)
where z p is the coordinate of the particle p along z and N = Ln b is the number of macroparticles representing the resonant electrons; v px and v py are the components of the velocity of the particle p along x and y. As in a homogeneous plasma the equation of wave evolution

is of the form 43 ∂E k ∂t = αJ k = 8πen b ω 0 ε 0 J k , (19)
we get the following energy conservation (17) in the absence of inhomogeneities

d dt p m e v 2 p 2 + L k ω 0 ε 0 |E k | 2 16π = 0. ( 20
)
The wave energy density is given by

E * i E j 16π ∂ ω 2 εh ij ω∂ω 0 = ω 0 ε 0 |E| 2 16π , (21)
where εh ij is the Hermitian part of the dielectric tensor εij with elements

ε xx = ε yy = 1 - ω 2 p ω 2 -ω 2 c , ε xy = -ε yx = iω c ω 2 p ω (ω 2 -ω 2 c ) . ( 22
)
Therefore, the total Hamiltonian H t corresponding to the whistlers propagating in the inhomogeneous plasma and interacting with the flux of resonant particles can be written in the form

H t = H w + H p = H w + N p=1 h p + k ω 0 |C k | 2 , ( 23
)
where the kinetic energy of a particle p is defined by (with the gauge ϕ = 0)

h p = m e v 2 p 2 = 1 2m e P p + e c A 0 (z p ) + e c Re A (z p , t) e -iω 0 t+ik 0 zp (x+iy) 2 . ( 24
)
C k is the Fourier component of the canonical variable C = σ 0 Ee -iω 0 t ; P p is the generalized momentum of the particle p; A 0 is the stationary part of the vector potential A; A is the envelope of A. The full set of Hamilton equations is then provided by (12) applied to the

Hamiltonian H t (23) instead of H w (10) - k 2 0 D 0 ∂ρ ∂t = δH t δΨ , k 2 0 D 0 ∂Ψ ∂t = δH t δρ , ∂C ∂t = -i δH t δC * , (25)
together with the two additional relations for each particle p

dP p dt = - ∂H t ∂z p , dz p dt = ∂H t ∂P p , (26)
which provide the Newton equations. Using (25) in the Fourier space, the following relation can be derived from the term H p describing the wave-particle interaction in ( 23)

i ∂C k ∂t = ∂ δC * k k ω 0 |C k | 2 = ω 0 C k . ( 27
)
For the first term h p of H p (23) we can calculate, using the relation

C = σ 0 Ee -iω 0 t = iω 0 σ 0 Ae -iω 0 t /c, that δ δC * k p h p = p m e v p • δv p δC * k = p v p • δ δC * k P p + e c A 0 (z p ) + e c Re c iω 0 σ 0 C (z p , t) e ik 0 zp (x+iy) , (28)
which provides the following expression

i ∂C k ∂t = ie 2ω 0 σ 0 p v p • (x-iy) e -ikzp e -ik 0 zp . (29)
Taking into account ( 27) and ( 29), we obtain the evolution of the envelope C in the form

i ∂C ∂t -ω 0 C = ie -iω 0 t ∂ ∂t Ce iω 0 t = ie 2ω 0 σ 0 k p v p • (x-iy) e -ik 0 zp e -ikzp e ikz , (30)
where summations on all waves and resonant particles are performed in the right hand side term. The parameter σ 0 can be determined by writing that (see (20))

L k ω 0 ε 0 |E k | 2 16π = k ω 0 |C k | 2 = k σ 2 0 ω 0 |E k | 2 , (31)
which leads to (11), so that (30) can be written as

i ∂E ∂t = 8πie ε 0 ω 0 n b N k p v p • (x-iy) e iω 0 t e -ik 0 zp e -ikzp e ikz . ( 32
)
Adding the terms provided by the Hamilton equations for H w (10)-( 12), we finally obtain (32) in the form

i ∂E ∂t +iv g0 ∂E ∂z + v g0 2 ∂ 2 E ∂z 2 + ω 2 p ε 0 ω 0 (ω c -ω 0 ) ρE = - 8πieω 0 D 0 n b N k p (v px -iv py ) e -i(k+k 0 )zp e iω 0 t e ikz , (33)
which describes the interaction of the whistlers' electric field envelope with resonant electrons in a plasma with density and velocity inhomogeneities. The right hand side term represents the current contribution of the beam electrons.

The full system to be solved self-consistently consists in the set of equations ( 7), ( 8), (13) and (33). A numerical code has been written to solve these equations by using discretiza-

IV. INTERACTION OF WHISTLER ENVELOPE SOLITONS WITH

PARTICLES

In this Section we present results of numerical simulations of whistler envelope solitons interacting with electron fluxes at various conditions typical of the heliospheric plasmas.

Note that we present hereafter examples only for a few sets of parameters, but the same qualitative conclusions can be stated for quite different plasma conditions, as shown by our simulations' results. This paper is not devoted to fulfill a parametric analysis but to evidence and explain physical mechanisms. We consider weak and warm electron beams with initial velocity distributions F b (v z , v ⊥ ) , that propagate in the direction opposite to the ambient magnetic field with a velocity v b and a density n b n 0 , and interact with the whistler solitons at normal cyclotron resonance velocities v zk = (ω k 0 +k -ω c ) / (k 0 + k) distributed within a narrow velocity width ∆v around the central velocity v z0 = (ω k 0 -ω c ) /k 0 . Initially the wave packet is chosen in the form of a whistler envelope soliton 42 moving with the group

velocity v g0 B(z, t = 0) = B s sec z -z 0 l s . ( 34
)
The soliton is located at the initial position z 0 and propagates along a simulation box of length L; its amplitude B s and width l s are related as

l s B s = (1/β) (ck 0 /ω 0 ) with 42 β = 1 16πn 0 m i c 2 ω 4 p (ω 0 -ω c ) 2 v g0 2k 0 v g0 1 + ω c ω 0 -ω c k 0 v g0 ω 0 2 1 c 2 s -v 2 g0 1/2 . ( 35
)
The corresponding density and fluid velocity perturbations [START_REF] Krafft | Dynamics of whistler enevelope solitons in inhomogeneous plasmas[END_REF] , i.e. ρ(z, t = 0) = ρ 0 sec 2 ((z -z 0 ) /l s ) and V (z, t = 0) = V 0 sec 2 ((z -z 0 ) /l s ), are moving with the same velocity as the field envelope. Note that in (35) the ratio v g0 /k 0 is always positive, as well as the term v g0 c 2 s -v 2 g0 ; the soliton is either supersonic (v g0 > c s ) when v g0 < 0, either subsonic (v g0 < c s ) when v g0 > 0.

Figure 2 shows the damping of a high intensity whistler soliton when it interacts with the tail of a Maxwellian electron velocity distribution centered at v z = 0. The slow decrease with time of the energy density variation of the soliton, i.e. ∆W = W (t) -W (0) (Fig. 2a), shows that it keeps most of its energy during several tens of thousands of cyclotron periods ω -1 c ; eventually the soliton has lost only a few percents of its energy (∆W/W (0) 2% at ω c t 35000). During its propagation the magnetic field envelope (34) is accompanied by the slow fluid velocity and density perturbations V and ρ which move with the same velocity (see Fig. 3). The profile of the magnetic field envelope is only very slightly deformed (at the bottom edge of the upstream front) after its travel along the whole box. Meanwhile resonant particles are accelerated, as shown by the time evolution of the variations of their parallel, perpendicular and total kinetic energy densities, i.e. ∆K z = K z (t) -K z (0) < 0, ∆K ⊥ = K ⊥ (t) -K ⊥ (0) > 0 and ∆K t = K t (t) -K t (0) > 0, respectively (Fig. 2b). The parallel kinetic energy of the particles is decreased whereas the perpendicular and the total ones are increased. Moreover, ∆K t is shown on Fig. 2a together with ∆W as well as the sum ∆K t + ∆W 0 (dashed line) demonstrating the total energy's conservation.

As the size of the simulation box is limited and the boundary conditions are periodic, one has to take care in the frame of this study that all electrons should only travel one time across the soliton. Therefore the parameters have been chosen so that the solitons have the time to travel along roughly the full length of the box before they cross the counterstreaming particles for a second time. When this moment is reached the simulation has to be stopped.

In order to understand how the soliton interacts with the particles, we examined the time evolution of the resonant electrons' velocity distributions in the plane v z -v ⊥ for three different populations : (i) the first one corresponds to the particles located inside the soliton region, (ii) the second one to those moving upstream the soliton (right part of the simulation box) and (iii) the third one to those moving downstream it (left part of the box, where the soliton is initially located). Note that the particles move with a negative parallel velocity and the soliton with a positive one so that, at a given time, those located in the downstream (resp. upstream) region have already interacted (resp. did not already interact) with the soliton; moreover, the upstream population can contain some electrons that have been reinjected at the right boundary of the box (due to the periodic boundary conditions) and are originating from the downstream region. The beam velocity distributions F b (v z , v ⊥ ) are represented in Fig. 4 for each population (columns) and for 3 time moments (rows). The first statement provided by this analysis is that the soliton interacts locally with the particles; indeed, one observes that the distributions upstream and downstream the soliton are only weakly affected by their interactions with the nonlinear waves, contrary to those of the electron population located in the soliton region, i.e. of particles with positions z p (t) satisfying the condition |z p (t) -z 0s (t)| l s , where z 0s (t) is the coordinate of the center of the soliton at time t. One If the interactions of the resonant electrons with the solitary structure occur mainly locally, the soliton is however responsible for the perturbation of the particles' distribution in its downstream region which is modified by its passage, what is not expectable a priori. Note that the position of the soliton is not connected with the resonance conditions (depending on the velocities) that govern its interactions with the beam. The condition on locality is not a sufficient condition for efficient interactions between the solitary structure and the particles.

A second example shows the interaction of a whistler soliton of weak intensity with an electron velocity distribution in conditions when the soliton can absorb energy from the beam particles, as shown by the evolution of the energy density variation ∆W (Fig. 6a).

The energy growth occurs up to ω c t s 15000 where stabilization occurs. The electrons are accelerated along the magnetic field (∆K z > 0) whereas the perpendicular ∆K ⊥ and the total ∆K t kinetic energy density's variations decrease with time (Fig. 6b). The balance of energy ∆K t + ∆W = 0 is represented by the dashed curve.

The dynamics of the interaction of a whistler soliton with a particle distribution in a inhomogeneous plasma characterized by a density well of the order of 1% of the average plasma density and a scale significantly larger than the soliton width has been studied.

One can observe in Fig. 7a that the soliton damps much more quickly than in the case of Fig. 2a; it has lost almost half of its energy at time ω c t 35000 and, consequently, a significative increase of kinetic energy (acceleration of particles) has occurred (∆K t > 0 in Fig. 7b, with ∆K ⊥ > 0, ∆K z < 0). Meanwhile the shape of the soliton experiences a strong modification (Fig. 8) : the upstream front has steepened whereas oscillations have appeared in the downstream region. The dissipation encountered by the soliton is strong enough to lead to such structure. Moreover, we present in Fig. 9 the same kind of picture as in Fig. 4, i.e. the time evolution in the plane v z -v ⊥ of the velocity distributions of the three electron populations (downstream, local and upstream). The conclusion is similar to that stated previously : the soliton interacts mainly with the particles located in its vicinity at a given time, the others participating only weakly. The same simulation has been performed without the particles, but keeping the plasma inhomogeneity. All effects observed above are no more visible : they are actually generated by the interactions with electrons and not by the density inhomogeneity.

For comparison we present the case of a soliton traveling in an inhomogeneous plasma with a density well but interacting with another distribution of particles. The velocity distributions at three different times for the three electron populations mentioned above (see Fig. 10) show that significative particle deceleration and perpendicular heating affect the local population and perturb slightly the downstream one (see the middle and left columns of Fig. 10). Meanwhile the soliton damps and has lost around 5% of its energy at time ω c t 40000, i.e. much less than in the previous case. In spite of this small loss of energy, the shape of the soliton is modified as the front has slightly steepened. Note that a simulation performed with the same parameters but with a homogeneous plasma provides the same results, showing that, as expected, the large scale inhomogeneity does not play a significant role, even if the soliton experiences very small acceleration and deceleration when it travels through it in absence of particles [START_REF] Krafft | Dynamics of whistler enevelope solitons in inhomogeneous plasmas[END_REF] .

On another hand, the rate of exchange of energy (loss or gain) between a whistler soliton and an electron beam can also be estimated analytically in the frame of the electromagnetic quasilinear theory of the weak turbulence. However this approach has some limitations, due partly to the fact that it can not take into account the local character of the interaction evidenced above. Nevertheless we can determine by such calculation under what physical conditions an envelope soliton of small or moderate intensity can loose or gain energy during its interactions with the particles and quantify these exchanges. Moreover the estimation given below on the basis of such analytical study shows a rather good agreement with the numerical simulations' results based on the Hamiltonian model. Another more complex approach, which consists in calculating the rate of energy exchange of a moving envelope soliton only with those particles located in its vicinity at a give time, is currently under investigation and will be the subject of a forthcoming paper.

In the frame of electromagnetic quasilinear theory, the slow time variation of the spaceaveraged electron distribution function F b (v z , v ⊥ , t) can be written in the form 44

∂F b (v z , v ⊥ , t) ∂t = πe 2 m 2 e k |E k | 2 × × kv ⊥ ω k ∂ ∂v z + 1 - kv z ω k 1 v ⊥ ∂ ∂v ⊥ v ⊥ δ(ω k -kv z -ω c ) kv ⊥ ω k ∂ ∂v z + 1 - kv z ω k ∂ ∂v ⊥ F b (v z , v ⊥ , t), ( 36
)
where δ is the Dirac function. Note that |E k | 2 in Eqs. (36) and ( 14) have different dimensions; here |E k | 2 is an energy density per surface whereas in Eq. ( 14) it is an energy density per volume.Thus the time variation of the average kinetic energy density W K of the resonant electrons (of density n b ) is given by

∂W K ∂t = ∞ 0 ∞ -∞ m e n b v 2 2 ∂F b ∂t dv = πe 2 n b 2m e k |E k | 2 × × ∞ 0 ∞ -∞ kv ⊥ ω k ∂ ∂v z + 1 - kv z ω k 1 v ⊥ ∂ ∂v ⊥ v ⊥ (δ(ω k -kv z -ω c )G (v z , v ⊥ , t)) (v 2 ⊥ + v 2 z )dv (37)
where dv = 2πv ⊥ dv z dv ⊥ and

G (v z , v ⊥ , t) = kv ⊥ ω k ∂ ∂v z + 1 - kv z ω k ∂ ∂v ⊥ F b (v z , v ⊥ , t) . (38)
By integrating (37) on v z we get

∂W K ∂t = π 4 ω 2 p n b n 0 k |E k | 2 k ∞ 0 v 2 z0 kv 2 ⊥ ω k ∂G ∂v z v z0 + v 2 ⊥ + v 2 z0 1 - kv z0 ω k ∂ (v ⊥ G) ∂v ⊥ v z0 - kv 2 ⊥ ω k ∂ ∂v z Gv 2 z v z0 dv ⊥ , (39)
where v z0 = (ω k -ω c ) /k < 0 is the normal cyclotron resonant velocity of the wave (ω k , k).

Let us suppose that initially the resonant particles' velocity distribution F b (v z , v ⊥ , t = 0) is a beam propagating along z in the direction opposite to the ambient magnetic field with an arbitrary perpendicular distribution

F ⊥ (v ⊥ ) F b (v z , v ⊥ , t = 0) = F z (v z -v b )F ⊥ (v ⊥ ), (40)
where v b < 0 is the beam velocity. The normalization is done according to 2π v ⊥ F ⊥ (v ⊥ )dv ⊥ = 1 and the mean perpendicular velocity square is given by

v 2 ⊥ = 2π ∞ 0 v 3 ⊥ F ⊥ (v ⊥ )dv ⊥ . (41)
Performing further integrations on v ⊥ in (39), we get the time variation of the density of kinetic energy in the form

∂W K ∂t ω 2 p 8 n b n 0 k |E k | 2 F k k , (42)
F k = k 2 v 2 ⊥ ω 2 k v 2 z0 F z (v z0 ) -2v z0 F z (v z0 ) + 4 1 - kv z0 ω k F z (v z0 ). (43)
F z (v z0 ) and F z (v z0 ) are the first and the second order derivatives of F z (v z ) with respect to v z , at v z = v z0 . As for the system of waves interacting with the resonant particles the total energy is conserved in the considered plasma of size L, we can write for the wave (ω 0 , k 0 ) that

-L ∂ ∂t W K = ∂W p ∂t = 1 + c 2 k 2 0 ω 2 0 + ω 2 p (ω 0 -ω c ) 2 ∂ ∂t |E| 2 4π dξ, (44)
where the energy W p carried by the waves has been derived from ( 21) and ξ = z -v g0 t. Thus the rate of change of the energy W sol of a single soliton inside the box of size L is ∂ ∂t

|E| 2 4π dξ = ∂ ∂t W sol = - ω 2 p 8 n b n 0 1 + c 2 k 2 0 ω 2 0 + ω 2 p (ω 0 -ω c ) 2 -1 L k |E k | 2 F k k . (45)
In the frame moving with the whistler group velocity v g0 , the electric field of the whistler envelope soliton satisfies E 2 (ξ, t) = E 2 s sec 2 (ξ/l s ). The energy carried by a soliton of size l s L is

W sol = |E| 2 4π dξ |E s | 2 l s 2π , (46)
where β -1 = E s l s (35). Then for the single soliton we can write that

E k = 1 L L 0 e -ikξ E dξ = E s πl s L sec π (k -k 0 ) l s 2 ,
where we took into account the form of the electric field packet (3). Considering the quasimonochromatic character of the wave packet, we get

|E k | 2 = E 2 s π 2 l 2 s L 2 sec 2 π (k -k 0 ) l s 2 π 2 E 2 s l 2 s L 2 δ kk 0 , (47)
where the discrete Dirac function δ kk 0 satisfies δ kk 0 = 1 if k = k 0 and δ kk 0 = 0 otherwise. Thus, after summation on the wavenumber k in (45), the rate of energy variation for one soliton inside the box is

∂W sol ω c ∂t = γ ω c W sol , (48)
with the normalized growth/damping rate

γ ω c - π 3 4 ω 2 p ω c n b n 0 1 + c 2 k 2 0 ω 2 0 + ω 2 p (ω 0 -ω c ) 2 -1 F k 0 k 0 l s L . (49)
The sign of the growth rate γ depends only on the sign of F k 0 (43). In the limit when the whistler frequency ω 0 is very low, i.e. when ω 0 ω c < ω p and the dispersion relation is

thus ω 0 c 2 k 2 0 ω c /ω 2 p , we obtain γ ω c - π 3 4 ω c k 0 n b n 0 ω c ω 0 v 2 ⊥ F z (v z0 ) + 2k 0 ω c F z (v z0 ) + 4F z (v z0 ) l s L . (50)
Let us now consider N solitons distributed within the whole box of length L. Their resultant energy is W s = N (l s , k 0 ) W sol (l s , k 0 ) /N , where N (l s , k 0 ) is the number of solitons of width l s and central wavenumber k 0 , and W sol (l s , k 0 ) is the energy of such a soliton (48). If the solitons' positions and phases are not correlated, the rate of the soliton energy density variation is given by

∂W s ω c ∂t = ls,k 0 N (l s , k 0 ) N γ (l s , k 0 ) ω c W sol (l s , k 0 ) . (51)
To use this formula one needs to know the distribution of the solitons' characteristics as a function of the scales l s (or amplitudes E s ) and the wavenumbers k 0 . Unfortunately it is not the case. Nevertheless, let us compare the rate of energy variation of one soliton within the box of length L, obtained owing to the numerical simulations, with the value of γ/ω c calculated according to (49). Figure 11 shows the variation of γ/ω c (49) with the resonant velocity v z0 = (ω -ω c ) /k for the parameters of Fig. 2 (upper panel) and Fig. 6 (lower panel). For the former (latter) case the soliton has lost around 4% (has gained around 8%) of its energy at the time ω c t 1 = 35000. The initial resonant velocity domain of the soliton is located, for the parallel velocity distribution of Fig. 2 (Fig. 6), in the region -12 v z0 -9 where γ/ω c < 0 (-22.5 v z0 -19.5 where γ/ω c > 0 for -21.5 v z0 ).

Estimates calculated using (48)-( 49) are in good agreement with the numerical simulations' results as one obtains in the first case, for N = 1 and l s /L 1/30, a relative loss of energy of 4% and, in the second case, for N = 1 and l s /L 1/40, a relative gain of energy of 19%.

Finally, let us mention that numerical simulations performed in the frame of this study (but not presented here because the interactions between the solitons and the particles are not efficient) have shown that whistler solitons can keep their shape and their stability during time periods reaching up to 10 6 -10 8 ω -1 c and even more, depending on the physical parameters, what corresponds to growth/damping rates γ/ω c significantly smaller than those presented in Figs. 11a-b. Moreover, the occurrence during their travel of additional physical processes able to reduce the efficiency of the wave-particles interactions can increase this range of time by one order of magnitude at least.

V. CONCLUSION

The paper studies the self-consistent interactions of whistler envelope solitons with electron beams in inhomogeneous plasmas, using a Hamiltonian model of wave-particle interactions where a nonlinear whistler equation involving the beam current is coupled with lower frequency fluid and Newton equations. It allows to describe the parallel propagation of narrowband whistlers interacting with arbitrary particle distributions in magnetized plasmas.

It is shown that the whistler envelope soliton interacts mainly locally with the beam particles, i.e. it does not exchange energy with all the resonant electrons but only with those moving in its close vicinity, contrary to the case of particles' interaction with whistler turbulence. Interactions are efficient not only if the wave-particle resonance conditions on the velocities are fulfilled, but also if the resonant particles satisfy a locality condition. Thus, as the number of resonant particles involved in the interactions with the solitary structure is smaller than in the case of whistler turbulence, the energy exchanged with them by the whistler soliton is also smaller. This double condition has a significative impact on the stability of the soliton.

Indeed, estimates based on the numerical simulations show that whistler solitons can remain stable during 10 6 -10 8 ω -1 c , depending on the physical parameters. Moreover, if the particles moving in the vicinity of the soliton (local particles) are only intermittently in resonance with it or if the free energy of their distribution is reduced due to any other physical effect, it can propagate in the plasma during a range of time increased by one order of magnitude at least, i.e. 10 7 -10 9 ω -1 c , travelling up to a distance of 10 4 -10 5 times its own width, and even more. On the contrary, the soliton can be totally damped within a time ranging from several thousands of ω -1 c to a few millions, depending on the physical conditions.

During its interactions with the beam, the soliton can either damp and accelerate particles, either absorb beam energy and cause electron deceleration. If the energy exchanges are significative, the soliton envelope is deformed; its upstream front steepens whereas oscillations appear on its downstream side. Weak density inhomogeneities as the random fluctuations of the solar wind plasma have no strong impact on the interactions of the whistler soliton with the resonant particles.

Electromagnetic quasilinear theory has been used in order to provide an estimate of the loss of energy of a soliton interacting with resonant particles. A rather good agreement is found between these analytical calculations and the numerical simulations based on the Hamiltonian model.

VI. ACKNOWLEDGEMENTS

FIG. 1 .

 1 FIG. 1. Linear growth rates γ/ω c of whistlers as a function of their wavenumbers ck/ω c , for different temperature anisotropy factors A, in the case of a DGH function with ν = 1. (Upper panel) Magnetospheric conditions with ω p /ω c = 5, n b /n 0 = 0.0001, v b /v T = -18, ω 0 /ω c 0.32, T e = 50eV , A = 50, 100, 200. (Lower panel) Solar wind conditions with ω p /ω c = 100, n b /n 0 = 0.00005, v b /v T = -8, ω 0 /ω c 0.03, T e = 20eV, A = 3, 30, 80. For both cases, the maximum growth rate γ max /ω c increases with A. T e and v T are the electron plasma temperature and the corresponding thermal velocity, respectively.

  y is the slowly varying (with space and time) envelope of the right circularly polarized wave; x and y are the unitary vectors along the axes perpendicular to the direction z of the wave propagation and of the ambient magnetic field B 0 = B 0 z; E x and E y are the coordinates of E along x and y; ω 0 and k 0 = k 0 z are the central frequency and wavenumber of the wave packet. The whistler dispersion relation is given by

  tion schemes, pseudo-spectral methods and Fast Fourier Transforms. The three-dimensional motion of the individual particles is computed using a leapfrog-type integrator. Owing to the approach used, a limited number N p of resonant particles (N p = 100000 -500000) is sufficient to provide clearly interpretable simulation results. Packets of 1024 -4096 waves of wavenumbers k 0 + k (|k| k 0 ) and frequencies ω k 0 +k are used, which present initially narrowband spectra peaked at the central wavenumber k 0 at frequency ω 0 . At the initial state, particles are distributed uniformly in space and described by arbitrary velocity distributions. The one-dimensional simulation box extends along z over a normalized distance Lω c /c 3000 -10000. More details concerning the numerical scheme are given in the companion paper 42 .

FIG. 2 .

 2 FIG. 2. Soliton interacting with a Maxwellian tail. (a) Time evolution of the variations of the soliton energy density ∆W = W (t) -W (0) (lower curve) and of the total kinetic energy of the resonant electrons, ∆K t = K t (t) -K t (0) (upper curve), in arbitrary units; the dashed curve represents the vanishing total energy variation ∆W + ∆K t = 0. (b) Time evolution of the variations of the parallel (lower curve), the perpendicular (upper curve) and the total (middle curve) kinetic energy densities ∆K z , ∆K ⊥ and ∆K t of the resonant electrons, in arbitrary units. Main parameters are : Lc/ω c = 3500, v T /c = 0.005, ω p /ω c = 17, ω 0 /ω c 0.35, ck 0 /ω c = 12.6, n b /n 0 = 0.00002, -12.5v T v zk -9.5v T .

FIG. 3 .

 3 FIG. 3. Whistler envelope soliton along the simulation box at different times. (Upper panel) : Magnetic field envelope B/B 0 . (Lower panel) : Fluid velocity perturbation V /c s . Parameters are the same as in Fig. 2.

FIG. 4 .

 4 FIG. 4. (Color online) Velocity distributions in the plane v z -v ⊥ at the times ω c t = 213 (upper row), 1021 (middle row), and 3016 (bottom row), for the three electron populations described in the text. (Left column) : downstream electrons; (Middle column) : local electrons; (Right column): upstream electrons. Velocities are normalized by the electron plasma thermal velocity v T . Parameters are the same as in Fig. 2.

FIG. 5 .FIG. 6 .

 56 FIG. 5. (Color online) Parallel velocity distributions F z (v z ) corresponding to the three electron populations of Fig. 4, at the same three time moments ω c t = 213, 1021, and 3016. For each panel : local electrons (thick black lines), downstream electrons (dashed lines -red dashed online), upstream electrons (thin black lines -blue lines online). Parameters are the same as in Fig. 2.

FIG. 7 .

 7 FIG. 7. Soliton interaction with an electron beam in a inhomogeneous plasma. (a) Time evolution of the variations of the soliton energy density ∆W (lower curve) and of the total kinetic energy of the resonant electrons, ∆K t (upper curve), in arbitrary units; the dashed curve represents the vanishing total energy variation ∆W + ∆K t = 0. (b) Time evolution of the variations of the parallel (lower curve), the perpendicular (upper curve) and the total (middle curve) kinetic energy densities ∆K z , ∆K ⊥ and ∆K t of the resonant electrons, in arbitrary units. Main parameters are : Lc/ω c = 7000, v T /c = 0.006, ω p /ω c = 10, ω 0 /ω c 0.25, ck 0 /ω c = 5.8, n b /n 0 = 0.00005, v b /v T = -25.

FIG. 8 . 3 FIG. 9 .

 839 FIG.8. Profile of the magnetic field envelope soliton B/B 0 as a function of the coordinate ξ = z -v g0 t in the frame moving with the soliton velocity v g0 , at times ω c t = 10500 (thin line) and ω c t = 0 (thick line). Parameters are the same as in Fig.7.

FIG. 10 .

 10 FIG. 10. (Color online) Velocity distributions in the plane v z -v ⊥ at the times ω c t = 119 (upper row), 1021 (middle row), and 5011 (bottom row), for the three electron populations described in the text. (Left column) : downstream electrons; (Middle column) : local electrons; (Right column): upstream electrons. Main parameters are : Lc/ω c = 3500, v T /c = 0.005, ω p /ω c = 17, ω 0 /ω c 0.4, ck 0 /ω c = 14.4, n b /n 0 = 0.00002, v b /v T = -9.

FIG. 11 .

 11 FIG. 11. Variation of γ/ω c (49) with the resonant velocity v z0 . (Upper panel) : The parameters are those of Fig. 2. (Lower panel) : The parameters are those of Fig. 6.

  This work was granted access to the HPC resources of IDRIS under the allocation 2017-A0010510106 made by GENCI. This work has been done within the LABEX Plas@par project, and received financial state aid managed by the Agence Nationale de la Recherche, as part of the programme "Investissements d'avenir" under the reference ANR-11-IDEX-0004-02. C.K. acknowledges the "Programme National Soleil Terre" (PNST) and the Centre National d'Etudes Spatiales (CNES, France). The work was carried out with the financial support of the Russian Foundation for Basic Research, project No. 16-52-16010-NTSNIL a.