H W Wiesemeyer

C Thum

A Baudry

F Herpin

Precessing planetary magnetospheres in SiO stars ? First detection of quasi-periodic polarization fluctuations in R Leo and V Cam

Keywords: masers -polarization -stars: AGB and post-AGB, atmospheres, magnetic field, planetary systems -techniques: polarimetric H. W. Wiesemeyer et al.: Precessing planetary magnetospheres in SiO stars ?

Context. The origin of magnetism around asymptotic giant branch (AGB) stars is still uncertain. These stars may drive an important dynamo, but if the magnetic energy dissipates all in X-rays, the observed X-ray luminosities are too low for a strong, dynamically important global field. Other explanations for the circular polarization of SiO masers in the AGB atmospheres may thus be needed. Aims. The interaction of the AGB wind with previously ejected matter and with planets is expected to bear complex magnetohydrodynamic phenomena on a short time scale, such that strong magnetic fields can be maintained locally. Here we provide observational evidence for the corresponding magnetic fluctuations. Methods. We use the circular polarization of the v = 1, J = 2 -1 SiO masers as tracer for magnetic activity. A correlation polarimeter allows us to simultaneously record all Stokes parameters. An SiO maser survey of 80 AGB stars was performed, out of which eight sources with the strongest circular polarization were selected for further monitoring. Results. In two AGB stars, V Cam and R Leo, we find evidence of pseudo-periodic fluctuations of the fractional circular polarization on a timescale of a few hours, from which we infer magnetic fluctuations of ∼ 1 G. The phenomenon is rare and, if detected in an SiO star, restricted to a narrow range of velocities. It seems to be associated with planetary wake flows suggested by VLBI maps. Conclusions. While scenarios involving magnetic activity in the extended stellar atmosphere have problems to explain all observed features, precessing Jovian magnetospheres predict all of them without difficulty. For the case of R Leo, we constrain the orbit of the planet (estimated period 5.2 years), derive a stellar mass estimate of 0.7 M ⊙ from it, and discuss the impact of planetary magnetism on the survival of planets. Smooth velocity variations of the fluctuating circular polarization feature are predicted as the planet moves along its orbit.

Introduction

Towards the end of their lifetime, low-to intermediate mass stars undergo a phase in which they burn Helium in a shell on top of a Carbon-Oxygen core, and Hydrogen in another shell above the Helium shell. During this phase, when the stars appear in the Hertzsprung-Russell diagram in the upper asymptotic giant branch (AGB), they lose, due to pulsations, at least half of their mass, which forms a circumstellar envelope (see Herwig, 2005, for a review). The inner part of this envelope, also called extended atmosphere, is expected to bear complex magneto-hydrodynamic phenomena, due to the interaction of the wind with the previously ejected matter and with planets (Struck et al., 2004, 2002, Struck-Marcell, 1988). As in the solar system, where space weather changes on timescales of hours (e.g. Prangé et al., 2004), fluctuations of the magnetic field about a mean value can be expected due to this interaction, but the observational evidence is still lacking. Here we will show that for a narrow range of velocities the circular polarization of SiO masers, generally accepted as a tracer of the magnetic field in Send offprint requests to: H. Wiesemeyer ⋆ On leave to Instituto de Radioastronomía Milimétrica, Granada, Spain.

the extended atmosphere of AGB stars (Nedoluha & Watson, 1994, Elitzur, 1996), varies in two AGB stars with a period of a few hours. Previous multi-epoch observations (Diamond et al., 2003, Pardo et al., 2004, Kang et al., 2006) of SiO masers were not polarimetric, while the sampling of the polarization variability study of Glenn et al. (2003) was not dense enough to detect such intraday magnetic fluctuations.

Our knowledge about magnetism within the extended atmosphere of AGB stars relies on circular polarization measurements of SiO masers at 1.5 to 7 AU distance from a star with a typical radius of 0.7 to 1 AU. Maser theory implies molecular excitation in dense pockets of gas and amplification of light in narrow tubes. Assuming that the circular polarization is caused by the Zeeman effect in the non-paramagnetic SiO molecules in the tubes, the reported fractional circular polarizations of up to 9 % for the v = 1, J = 1 -0 transition (v and J are the vibrational and rotational quantum numbers, respectively) yield line-of-sight averaged magnetic flux densities of up to 100 G (Barvainis et al., 1987). More recent observations (Herpin et al., 2006) of the v = 1, J = 2 -1 masers (excited in gas layers very close to those where v = 1, J = 1 -0 is excited) also show the presence of circular and linear polarizations in many stars. These observations result in lower magnetic flux density estimates in the range 1 to 15 G, again assuming the circular polarization is due to the Zeeman effect. In the outer envelope, H 2 O and OH masers reside at ∼ 100 to 400 AU, respectively ∼ 1000 to ∼ 10000 AU, from the star. The Zeeman effect of H 2 O masers has been used (Vlemmings et al., 2002) to estimate, by extrapolation assuming a solar-type field topology, a magnetic flux density at the stellar surface of 100 G. In summary, it is now observationally evident that AGB stars maintain a magnetic flux throughout their envelope. Blackman et al. (2001) showed that AGB stars can generate a magnetic field via dynamo action at the interface between the rapidly rotating core and the extended convection zone. The importance of the magnetic field for global envelope dynamics is still a matter of debate, though (e.g. Soker, 2006).

Once the SiO maser spots are formed (often distributed along arcs around the star, hereafter referred to as maser shell, see e.g. Cotton et al., 2008), they are subject to magnetospheric events which are known to be rapidly variable if we refer to the solar system. Here we provide first evidence for such fluctuations in the atmospheres of two AGB stars by frequent polarization sampling. The polarization monitoring by Glenn et al. (2003) was only sensitive to slow variations and therefore to a longterm readjustment of the magnetic field (on a timescale of several months).

Observations and data analysis

Instrumentation and observations

The bulk of our observations was done in May 2006 with the IRAM 30m telescope at Pico Veleta, which is equipped with dual-polarization receivers. To confirm the rapid polarization variability discussed here, Mira-type star R Leo was observed during additional 4 hours on 2008 August 01. The receiver pair operating at 3mm wavelength was tuned to the v = 1, J = 2 -1 transition of SiO at 86.243 GHz and made coherent by using the same local oscillator reference. The signals were analyzed with XPOL (Thum et al., 2008), a correlation spectrometer enhanced with cross-correlation products for the signals from the orthogonally polarized receivers, at 39.0625 kHz or 0.136 km s -1 channel spacing within a 16.2 MHz bandwidth. Receiver temperatures varied between 36 K and 64 K, and system temperatures between 70 K and 235 K, resulting in an antenna temperature (T * A ) noise of σ rms = 36 -100 mK across the spectral baseline. The conversion factor from T * a to the flux density scale is 6 Jy/K. The phase difference between the receivers was measured with a polarizing grid mounted in front of the cold load (at about 70 -80 K effective temperature) of the calibration unit. Whenever the temperature scale was calibrated (every 10 to 20 minutes), the signal from the polarizer, with well known properties, was also observed and compared to the unpolarized calibration load at ambient temperature. The phase correction was then applied in order to attribute the measured real and imaginary part of the cross correlation product to the Stokes U parameter (in the reference frame of the telescope's Nasmyth focus) and to the Stokes V parameter (positive for a right hand circularly polarized signal, according to IAU convention). Small residual phase errors result in a leakage of the Stokes U signal into Stokes V, a subtle effect only worth worrying about when Stokes U is strong (which is the case here). The scaled (∼ 3 %) copy of the Stokes U signal thus contaminating the measured circular polarization can be removed, since it is subject to the parallactic rotation of the polarization vector, which easily shows in the data. The remaining Stokes V signal is intrinsic, the contri-bution of the telescope and the receiver cabin optics being negligible (-0.03 ± 0.12 % on the optical axis). 77 AGB stars were observed, out of which 62 are Mira stars, 10 are semiregular variable stars, and 2 red supergiants. Eight objects with the strongest circular polarization were retained for dense monitoring (twice per hour). They are all Mira stars, in agreement with the finding of Herpin et al. (2006) that this source class tends to have the strongest linear and circular polarizations, and is therefore well suited for our aims.

Data analysis

The temperature and phase calibration was done with the MIRA raw data reduction software 1 . The subsequent processing of the Stokes spectra, described in the following, was done with the CLASS and GREG software 1 . Due to the irregular sampling, a determination of the spectral power density (SPD) of the time series using Fourier transform techniques would yield poor results, and preference was given to the SPD estimate provided by the Lomb technique (Lomb, 1976, see also [START_REF] Press | Numerical Recipes in C. The Art of Scienctific Computing[END_REF]. This method only uses the measured data without any prior interpolation to a regular sampling function, and does a least-square fitting to the harmonic contents of the time series. A detection of quasi-periodic fluctuations of the fractional circular polarization (hereafter p C ) needs to pass three critical tests. First, we do not expect a significant fluctuation in the Stokes U residuals left after the subtraction of the stationary (within the timescales considered here) linear polarization feature. Second, the oscillation of p C was tested against the null hypothesis that the data are not periodic, but random noise with a Gaussian distribution; this test yields a false-alarm probability. Third, because a restricted number of random samples cannot be strictly Gaussian, Monte-Carlo simulations, with random samples contemporaneous with the observed features, were used to confirm the significance of the false-alarm probabilities.

Results

Only two stars, the Mira variables V Cam and R Leo (Fig. 1) passed the three tests and show clear evidence for a quasiperiodic modulation of their fractional circular polarization p C (Fig. 2), of peak-to-peak amplitude 2.8 % and 3.7 % for V Cam and R Leo, respectively (Tab. 1). The observations were made ∼ 40 days after the optical maximum of V Cam (pulsation period 522 d) and ∼ 30 days after that of R Leo (period 310 d), in an expansion phase of the SiO maser shell. Within a ∼ 0.4 km s -1 narrow range of radial velocities centred at 7.5 km s -1 and 4.4 km s -1 , we detect periods of 5. h 4 ± 0. h 1 and 6. h 3 ± 0. h 3, for V Cam and R Leo, respectively (Tab. 1). The phenomenon is both rare and localized in velocity space (and thus in circumstellar space, due to the velocity structure). The likelihood of misinterpretation of a time series of Gaussian noise is ≤ 4 % and ≤ 16 % for V Cam and R Leo, respectively (Fig. 3, left panels). We note that the Lomb periodograms of the Stokes U residuals do not show any significant periodic signal (Fig. 3, right panels).

The Monte Carlo tests confirm the significance of the results. For 10 5 runs, only 0.28 % of the Lomb periodograms mimic a periodic signal with ≤ 4 % false-alarm probability (V Cam), respectively 2.9 % a periodic signal with ≤ 16 % false-alarm probability (R Leo). We therefore conclude that it is unlikely (for Table 1. Observational Results

V Cam

R Leo Phase of optical lightcurve (1) 0.08 0.10 Oscillation period of circular polarization (2) (5.4 ± 0.1) h (6.3 ± 0.3) h False-alarm probability (3) 4 % 16 % Radial velocity of maser spots (4) 7.5 km s -1 4.4 km s -1 Mean circular polarization -0.3 % +2.0 % Mean magnetic flux density in maser (5) ∼ 100 mG ∼ 1 G Peak-to-peak variation of fractional polarization 2.8 % = 5.6 σ rms 3.7 % = 7.4 σ rms Fluctuation of magnetic field (5) ∼ 1 G ∼ 1 G

(1) With respect to the maximum of the optical lightcurve, (2) for V Cam the second harmonic of the modelled rotation period, (3) with respect to the null hypothesis (data are Gaussian noise), (4) with respect to the local standard of rest, (5) order-of-magnitude estimate based on the Zeeman hypothesis, and for a linewidth of 1 km s -1 . This corresponds to a maser at 3 r J and a planetary magnetic dipole field eight times stronger than Jupiter's (Wiesemeyer, 2008). V Cam extremely unlikely) that random noise creates the observed fluctuations. If they were produced by an oscillation of the telescope tracking, moving the polarized sidelobes across the line-of-sight towards the source, all velocity channels would show fluctuations with the same period, which clearly is not observed and can therefore be safely excluded.

Variations of the circular polarization of 4 % peak-to-peak amplitude were found again in August 2008 (Fig. 4), but now within two and wider velocity intervals, from v lsr = -3.6 to -2.7 km s -1 and from -0.5 to 0.7 km s -1 . A meaningful Lomb analysis as in the case of the 2006 data was not possible for these data, because the observing interval was too short to measure a full fluctuation period. The data (Fig. 4 left and middle) suggest a period of about 2 hours with a guessed uncertainty of about 0.5 hours. It is interesting to note that in channels of relatively strong Stokes I emission, no significant polarization fluctuation was found (Fig. 4, right), while now also the linear polarization seems to fluctuate at about v lsr = 0.6 km s -1 , where Stokes I is strong. Again, this rules out artifacts as the reason for the fluc-tuations. If they were of instrumental origin, the whole spectral band would be affected.

Discussion

The fractional circular polarization is a more reliable magnetic tracer than linear polarization, especially for spatially unresolved observations: our telescope beam averages the polarization signal from all maser spots at the same radial velocity, hence the orientation of the linear polarization is either parallel or perpendicular to the maser shell (e.g. Diamond & Kemball, 2003, Desmurs et al., 2000). This tends to decrease the beam-averaged linear polarization, except if it is dominated by few, very strong maser spots close together (which may explain the linear polarization fluctuations showing up in the August 2008 observations of R Leo, Fig. 4, middle). Furthermore, linear polarization can be entirely produced without magnetic fields by anisotropic pumping (Western & Watson, 1983, Asensio Ramos et al., 2005), though polarization angle swings by 90 • in the afore mentioned resolved observations are explained by changes of the magnetic Fig. 2. Time series of polarization measurements. From top to bottom: Stokes I in Jansky, the fractional polarizations p L (linear) and p C (circular), and the polarization angle (i.e. the position angle of linear polarization, in degree E from N), for the SiO maser spots from V Cam at v lsr = 7.5 km s -1 (left) and R Leo at v lsr = 4.4 km s -1 (right). The continuous heavy lines in the plots for p C show model results (Wiesemeyer, 2008) for a saturated maser in the equatorial plane, at 3 r J from the planet, with a magnetic dipole field of eight times Jupiter's and misaligned with the rotation axis (in the sky plane) by 10 • , and rotation periods of 10. h 8 (V Cam, left) and 6. h 3 (R Leo, right). field direction with respect to the propagation direction of radiation in the maser (Goldreich et al., 1973). By contrast, circular polarization is enhanced in magnetized maser spots, either directly via the Zeeman splitting (e.g. Watson & Wyld, 2001), or indirectly via birefringent conversion of linear into circular polarization due to changes of the magnetic field topology along the maser slab (Wiebe & Watson, 1998, for unsaturated masers), or due to maser saturation, rotating the quantization axis from the direction of the magnetic field to that of the radiation propagation in the maser (Nedoluha & Watson, 1994). In the latter case, pump conditions would be required not only to vary in time but also quasi-periodically, an event which we cannot exclude but seems unlikely on short time scales. In turn, this would produce a varying Stokes I flux density, which is not observed. We therefore suggest that the intraday fluctuations of p C are of magnetic origin. They are quasi-periodical, not stochastic, and rare and only at one velocity, which means that an ordered magnetic field structure with smooth gradients in time and space is locally maintained within the dense AGB winds of R Leo and V Cam.

A case for precessing Jovian magnetic fields ?

The magnetopheres of Jovian planets have already been proposed to explain several features of SiO maser polarization (Struck et al., 2002(Struck et al., , 2004)), thus providing a local explanation for the strong magnetism in the atmosphere of AGB stars. If the magnetic fields in which SiO masers are embedded, of typically 10 G, were global, one would assume, at the stellar surface, a magnetic field of about 100 to 1000 G, depending on assumptions on the gradient of the magnetic flux density. Such a field could be generated by a dynamo between the rotating core and the stellar convection zone (Blackman et al., 2001). However, we now have to explain (see item 3 below) why Mira stars were not detected (Haisch et al., 1991) by the Rosat all-sky soft X-ray survey, which is sensitivity-limited to X-ray luminosities above typically 2.4 × 10 29 erg s -1 . As a matter of fact, Kastner & Soker (2004a) find that the X-ray luminosity of the single Mira stars TX Cam and T Cas is below the threshold expected for a dynamically important magnetic field. Their conclusion relies on the assumption that the X-ray luminosity equals the kinetic luminosity of the stellar wind powered by magnetic fields. Since there may be ways to dissipate magnetic energy other than by X-ray emission from accelerated particles, this is still an open issue.

We now summarize the arguments supporting the plausible planetary origin of our detections.

1. The rareness of the phenomenon can be explained by the number of Jovian planets around AGB stars, and by the requirement that the gain region of the maser needs to cross the magnetosphere where it is dominated by the dipole component. At present we cannot distinguish which explanation prevails -the possible rareness of Jovian planets in AGB atmospheres due to infall of planets into the star (Villaver & Livio, 2007, Nordhaus & Blackman, 2006), or the un- . The horizontal dashed lines indicate the spectral power density at which a periodic signal can be mimicked by Gaussian noise with a probability given by the labels ("false alarm probability"). For a given source, the plot scale for the Lomb periodograms of p C and the Stokes U residuals is the same, to make the comparison of the significance of the peaks easier.

favourable location of the SiO maser with respect to the precessing magnetosphere. 2. Both in V Cam and R Leo, the signature is statistically significant only at one velocity. We interprete this velocity as the current line-of-sight velocity of a planet on its Keplerian orbit. 3. Dedicated models (Wiesemeyer, 2008) of saturated SiO maser emission in a precessing magnetic dipole field show that a dipole moment eight times stronger than that of Jupiter's magnetic field is sufficient to provide the ∼ 1 G magnetic flux density at 3 r J distance, required to explain the peak-to-peak variation of p C (Tab. 1). This reconciles maser polarization and the lack of an X-ray luminosity expected for a stellar corona. Saturated maser action is not required here, if the weak circular polarization of unsaturated masers is enhanced by the combined effect of anisotropic pumping and magnetorotation (Wiesemeyer, 2008).

4. The fluctuations are quasi-periodic, not stochastic. The fluctuation periods derived from the Lomb periodograms of the p C fluctuations in R Leo and V Cam can be modelled as the first and second harmonic, respectively, of the planetary rotation period which is 6. h 3 and 10. h 8, respectively (cf. Jupiter's rotation period of 9. h 9, and Saturn's of 10. h 5). A misalignment of the dipole axis with the planetary rotation axis (10 • in the models, 9. • 6 for Jupiter) naturally explains both the amplitude and the period of the fluctuation, due to the ordered structure of the magnetic field. This implies that the magnetosphere is protected by a Chapman-Ferraro type magnetopause, where ram pressure equals magnetic pressure, at about 10 r J (Jupiter radii) substellar distance (according to Struck-Marcell, 1988). To date it is still uncertain why our R Leo data from 2008 yield a different period. This may be partly due to the incompleteness of the data, partly due to the proper motion of the maser spots, which may cross 10 r J in 10 hours and thus modify the pattern expected for a station- ) of SiO maser spots (with ∼ 0.1 AU diameter) across a quasi-stationary magnetic perturbation, this would result in periods which are two orders of magnitude longer than the observed ones. In turn, this means that the magnetic field diffuses through the gas of the maser slab, i.e. the ambipolar diffusion speed exceeds the maser velocity. A Jovian magnetic field both provides the required strength and curvature radius of the field lines (Struck et al., 2002), unlike a magnetic field of stellar origin. 6. Spatially resolved observations (Cotton et al., 2008) of the v = 1, J = 1 -0 SiO masers in R Leo show, at ∼ 4 -7.9 km s -1 radial velocity (with the largest redshifted velocity closest to the star), a remarkable elongated feature suggestive of a planetary wake flow as modelled by Struck et al. (2004). The corresponding feature in the v = 2, J = 1 -0 transition is even double-lobed. Similar features were found by Cotton et al. (2004) and Soria-Ruiz et al. ( 2007) at various circumstellar positions and different epochs.

Kinematical and astrometric evidence for a planet orbiting around R Leo

We deliberately consider here the presence of elongated features (see item 6 above) as the possible signature of a planetary wake although explanations involving the theory of maser formation cannot be ruled out. In the frame of this working hypothesis we now will use the available kinematic and astrometric information to illustrate the potential interest of combining SiO polarization monitoring observations with contemporaneous VLBI imaging of SiO masers. The VLBA observations of (2008). Remarkably, the v = 1, J = 1 -0 map of Soria-Ruiz et al. shows, at about v lsr = -7 km s -1 (with respect to the local standard of rest, hereafter LSR), a similar elongated maser feature pointing radially away from the star towards the South-East, and another one towards North-East, with velocities peaking at v lsr = 2.5 km s -1 and -2 km s -1 . This suggestion is confirmed by inspection of the VLBA map taken in August 2001 by Cotton et al. (2004), again showing a quasi-radial alignment of maser spots, now towards the north. The repeated occurence of such features, systematically rotated counter-clockwise from North, strengthens the suggestion of a wake flow associated with the same planet. However, the combined use of v = 1, J = 2 -1 polarization monitoring with the v = 1, J = 1 -0 VLBI astrometry raises new questions. Soria-Ruiz et al. (2007) show that in R Leo the former maser spots do not coincide with those corresponding to the latter transition. The characteristic ring radii of the respective maser shells are 29.2 mas and 33.8 mas, for the J = 1 -0 and J = 2 -1 transitions, whereas collisional and radiative pumping models (see Soria-Ruiz et al. for further references) suggest that for our purposes the maser spots of these transitions should be sufficiently close to each other. With this inconsistency in mind a lower limit to the astrometric error in the comparison of our polarization data with the VLBA positions can therefore be estimated from the difference between the ring radii, 4.6 mas or 0.5 AU (for 113.5 pc distance, Fedele et al., 2007, further references therein). The actual astrometric error may be higher, because the absolute astrometry of these VLBI data from different epochs is far less well constrained than the relative astrometry for a single observing date. Despite this limitation we use here the available combined kinematics and astrometric information to constrain the planetary orbit. The velocity in the wake flows is usually decelerated, the maximal blue-respectively redhifted velocity should therefore be closest to the planet. A sine fit (Fig. 5, top) yields a period of 5.2 years and a stellar velocity of v lsr = 1.7 km s -1 , but more observations are needed in order to assess the significance of the fit. Our 2008 data are best described by the velocity of the more blueshifted polarization fluctuation. The fitted stellar velocity is between the value derived by Bujarrabal et al. (1989) from thermal molecular lines in the circumstellar envelope (-0.5 km s -1 ) and the catalogue value of 13.4 km s -1 (GCRV, Wilson, 1953). Using the method discribed in Appendix A, this velocity fit allows us to determine the radius of the planetary orbit to 23.8 mas which corresponds to 2.7 AU, at a distance of 113.5 pc. The inclination of the orbital plane with respect to the plane of the sky is i = 34 • . The result of the least-square fit to the available astrometric information is shown in Fig. 5 As for the August 2008 data in R Leo which show polarization fluctuations at two different velocities, contemporaneous VLBI observations would have been required to perhaps further suggest that the second velocity interval could be due to a wake flow and therefore to another planet. However, it seems too speculative at this stage to suggest that each elongated VLBI maser feature could be associated with a planet while jet-like features could as well be interpreted as ejections of matter (see e.g. Cotton et al., 2008). Our fit leads to a stellar mass estimate of 0.7 M ⊙ , below, but not unreasonably far from the value inferred from measurements of stellar parameters by infrared Kband interferometry (Fedele et al., 2005), namely 1.0 -1.2 M ⊙ , for an equivalent non-pulsating star of the same luminosity. The discrepancy can either be due to observational error or model uncertainties. As a matter of fact, the R Leo maps of Cotton et al. (2008) mentioned above show several elongated maser features, and it is unlikely that each is associated with a planet (Cotton et al. rather suggest jet-like ejections of matter, see section 4.3.6 below). The conclusion that the polarization fluctuations observed at two velocities in R Leo in 2008 are due to the presence of two planets would therefore be speculative, even if contemporaneous VLBI data were at hand.

Are alternative scenarios viable ?

Boldly extrapolating the rich magnetic phenomena in the solar system to late-type stellar systems, alternative scenarios have to be seriously examined. One might suggest magnetic activity due to stellar companions, coronal loops, magnetic clouds or Parker instabilities in shocks forming in response to stellar pulsations.

Binarity

Here the mass transfer from the AGB star to a close companion generates flares and related magnetic fluctuations, which are not necessarily located in the atmosphere of the former (e.g. the Mira AB system, Kastner & Soker, 2004b). However, neither V Cam nor R Leo are known for having stellar companions, otherwise their X-ray luminosities would be higher. As a matter of fact, Tatebe et al. (2006, IR interferometry at 11.5 µm) do not see any obvious asymmetry in the envelope of R Leo that could hint at a companion (see also Gatewood, 1992). Cotton et al., 2004). Other references: Soria-Ruiz et al., 2007(2002 Dec. 07), Cotton et al., 2008(2004Sep. 05), this work (2006May 22 and 2008 Aug. 01). Bottom: available astrometric information and least-square fit of the derived planetary orbit (Appendix A), for d = 113.5 pc stellar distance. VLBI positions are indicated by black dots. The planetary orbit is indicated by the black line, crosses mark the fitted position at the VLBI epochs. Empty circles give the positions of the planet at the epochs of our polarization measurements. The linear polarizations measured by us and their position angles are shown as bars (the bar in the top left corner indicates a fractional linear polarization of 30 %). The grey-shaded disk indicates the measured λ 2.2 µm uniform-disk diameter at stellar phase 0.08 (in April 2001, Fedele et al., 2005).

The case of the 2003 soft X-ray outburst in the Mira AB system is, interestingly, not related to the binarity of the system. The spatially resolved Chandra observations (Karovska et al., 2005) clearly show that it occured within the envelope of Mira A, and it would be worthwhile to estimate whether it was triggered by burst-like accretion onto a Jovian planet, as suggested by Struck et al. (2004). As a matter of fact, the position of the X-ray flare may coincide with a linear SiO feature seen by Cotton et al., (2004).

Coronal loops

Coronal loops are generated by the differential rotation in the sub-photosphere, together with the dynamo (Blackman et al., 2005) at the interface with the core. If the convection zone extends to at least 80 % of the stellar radius (Holzwarth & Schüssler, 2001), magnetic flux loops are trapped there, because magnetic stresses at the tips of the loops act against convective buoyancy. This also explains the lack of a coronal X-ray luminosity, and why the magnetic fluctuations are rare. One may suggest coronal loops to occasionally extend to the site of the SiO masers. If there were coronal flux loops, the timescale for magnetic fluctuations would be too long, though. This can be demonstrated in the following way: Assuming (by analogy with the sun) that the size of a loop corresponds to that of a super-granulation cell, this yields 0.2 AU, for 1 AU stellar radius and 400 convection cells in the stellar sub-photosphere [START_REF] Schwarzschild | [END_REF], in contrast to 2 × 10 6 for the sun. Interferometric K-band size measurements (Fedele et al., 2005) yield a Rosseland radius of 1.6 AU at phase 0.08, close to the brightness maximum in April 2001 (cf. our measurements at phase 0.08 after the brightness maximum in 2006). Our orbital radius estimate of 2.7 AU thus infers that the maser is 1.1 AU above the stellar photosphere. Assuming a photospheric rotation speed of ∼ 1 km s -1 (a higher rotation speed is only possible for AGB stars with at least a brown dwarf companion, Soker 2006), a corotating flux loop would need ∼ 100 days to cross a maser spot of 0.1 AU diameter. This is two orders of magnitude above our fluctuation periods, which makes it highly unlikely that coronal loops can generate the observed hour-scale magnetic fluctuations. The scenario involving trapped magnetic flux loops also demonstrates that from the weak X-ray luminosities one cannot infer the absence of an efficient stellar dynamo -the magnetically accelerated particles simply remain within the convective envelope and do therefore not show in X-ray observations.

Magnetic clouds

As coronal loops, magnetic clouds originate from the magnetic activity on the surface of the stellar photosphere. If coronal loops are absent, due to a lack of the latter, this also holds for magnetic clouds of the type encountered in the solar wind (e.g. Bothmer & Schwenn, 1998). Should magnetic clouds traverse the extended stellar atmosphere, we can provide a rough estimate of the time scale associated with the corresponding magnetic fluctuations. The size of magnetic clouds in the solar system is comparable to that of SiO masers (0.1 AU), and their magnetic field vector also rotates on a timescale of ∼ 10 hours. However, solar system magnetic clouds expand with the speed (∼ 400 km s -1 ) of the magnetically driven solar wind. Since the latter is an order of magnitude above that of an AGB wind and SiO maser proper motions, the timescale does not fit -the resulting polarization would vary too slowly (especially if the magnetic cloud is decelerated by the dense ambient gas).

Parker instabilities

In order to produce polarized maser features of the observed lifetime (months to years, Glenn et al., 2003), the latter needs to be shorter than the ambipolar diffusion timescale, i.e. the magnetic flux has to be coupled to the masing matter. The fragments forming as a consequence of Parker instabilities (Hartquist & Dyson, 1997) fulfil this requirement under the typical conditions in the extended atmosphere of AGB stars. Since Parker instabilities arise from a local indentation orthogonal to the field lines, some short-term variations of the magnetic field component along the line of sight may be expected if the ambipolar diffusion timescale of these fluctuations exceeds that of the maser lifetime and proper motion. The resulting variability of p C should globally affect all maser spots of an SiO star if they commonly arise from fragments formed by this mechanism, in contradiction with our observations. Furthermore, the mass of fragments forming in response to Parker instabilities scales with magnetic pressure, and the strongest fluctuations would be expected from the strongest masers, which has not been observed (cf. Figs. 2 and4).

Alfv én waves

Our discussion would not be complete without considering Alfvén waves (Alfvén, 1942, see e.g. [START_REF] Sturrock | Plasma Physics, An Introduction to the Theory of Astrophysical[END_REF] as a possible origin of rapid magnetic fluctuations. However, our knowledge of hydromagnetic wave propagation in the dense AGB atmospheres is poor. The relevant time scale here would be

t A = D/v A ( 1
)
where D is the diameter of a maser slab, and v A is the Alfvén speed for the propagation of a hydromagnetic wave, given by (cgs units)

v A = B 4πρ (2)
where in the weak-coupling regime ρ is the ion mass density, and in the strong coupling regime (i.e. when collisions between ions and neutrals are important) the total mass density of neutral and ionized particles (Zweibel, 1989). Which regime holds in the atmosphere of AGB stars is uncertain. If the ambipolar diffusion speed is close to the velocity of the AGB wind, as estimated by Soker (2006) and Hartquist & Dyson (1997), strong coupling holds, and we estimate the Alfvén speed to

v A = 10 G 4π 2.10 -14 g cm -3
∼ 200 kms -1 .

(3) for a 10 G magnetic field and a hydrogen number density of 10 10 cm -3 . Thus, the timescale for an Alfvén wave crossing a maser slab of 0.1 AU diameter is about 20 hours, not unreasonably far from the timescales given in Table 1. However, we would like to stress that the parameters used for the estimate of both the ambipolar diffusion and Alfvén speeds are much uncertain.

A precessing jet ?

The role of a planetary or brown-dwarf companion in the shaping of circumstellar envelopes has been modelled by Nordhaus & Blackman (2006). Since in their scenarios the companion is already engulfed in the convective envelope of the star, it cannot be detected anymore by polarimetry. The deposit of the planetary angular momentum leads to a spin-up of the star and by consequence to the ejection of matter by equatorial mass loss or, if assisted by the stellar magnetic field, to a bipolar jet. Is it conceivable that we observed rather the polarization signature due to the precession of such a jet ? If the precession is caused by the misalignment between the jet axis and the stellar rotation axis, we can estimate the timescale of the phenomenon as in the case of coronal loops (section 4.3.2.) except that we have to account for the spin-up of the star. Nordhaus & Blackman showed that an infalling brown dwarf of 0.05 M ⊙ may increase the stellar rotation rate by an order of magnitude. The fluctuation of the circular maser polarization due to a crossing jet would therefore occur on a timescale of at least 10 days if the cross-sectional area of the jet perpendicular to its axis is smaller than the size of the maser spot, or longer otherwise. This is still one to two orders of magnitude above the fluctuations observed by us. Furthermore, to leave a signature only in a few SiO masers the opening angle of the jet cannot be too large.

Conclusions

In summary, in our opinion none of the non-planetary alternatives satisfactorily explains the present observations of R Leo and V Cam, while a single scenario, namely precessing Joviantype magnetospheres, are consistent with all of them. Since the discovery of a planet around a star in the post-red giant phase (Silvotti et al., 2007), we know that planets can survive in such an environment. Whether a given planet survives the AGB phase of its parent star depends on several factors (Villaver & Livio, 2007). Evolution of the orbit is the main one -depending on its distance from the star, the planet is either dragged inwards by tidal friction and evaporated, or repelled due to stellar mass loss, while for Jupiter-sized planets the gain of mass due to Roche flow from the star or accretion from the stellar wind barely affect their orbits. If the suggested planet around R Leo, one of the best studied Mira stars, of solar mass and at 113.5 pc distance (Fedele et al., 2005), does not spiral in and evaporate during the remainder of the upper AGB phase, it will survive the subsequent planetary nebula stage against evaporation (Villaver & Livio, 2007) only if it has at least two Jupiter masses and orbits at least at ∼ 3 AU. We note that the fluctuating SiO maser feature in R Leo has an orbital radius of 2.7 AU, close to the critical distance. We have demonstrated that the orbital elements of its host planet can be determined by measuring the velocity of the fluctuation and the position of the corresponding maser spot. The orbital period of the planet is 5.2 years, which provides an estimate of the stellar mass of 0.7 M ⊙ . This agrees with mass estimates found in the literature, of about 1 M ⊙ for R Leo. On the long term, our method has the potential to reveal, together with an independent estimate of the stellar mass, sub-Keplerian orbital motion and thus migration of the planet through the atmosphere towards the star. This requires correspondingly accurate observations. The situation may be complicated by the presence of several planets, whose orbits could be constrained by our method provided that the polarization fluctuations are not affected by a spectral blend of independent features at the same radial velocity and that the VLBI observations are free from a spatial blend of independent features along the same line of sight. The estimated mass loss of R Leo, ∼ 10 -7 M ⊙ /y, is too weak to measure orbit evolution within a reasonable amount of time (∼ 3 µas in 1000 years).

The situation may change if the opposite effect due to tidal fric-tion, neglected in this estimate, is by several orders of magnitude more important than that of stellar mass loss. Without further observational evidence at hand, one can only speculate on whether the suggested Jovian magnetic fields are decisive for evaporation or survival of the planets. They suppress turbulent viscosity and therefore the tidal friction induced by the latter (Zahn, 1977), and also lead to spin angular momentum loss due to magnetic braking. This contrasts with scenarios where the coupling between the stellar magnetic field and the planetary one lead to an increased drag area and therefore infall of the planet as compared to the case of an unmagnetized planet. The finding that our observations most likely imply a planetary magnetopause may turn out to be important here and motivate studies of planetary evolution in late-type stars that include magnetic star-planet interaction (omitted by Villaver & Livio, 2007, but included by Nordhaus & Blackman, 2006 who predict stellar outflows). The case for exoplanetary magnetism proposed here can only be strengthened by a combination of extensive simultaneous polarization monitoring and repeated spatially resolved observations. The experimentum crucis for an unambiguous evidence of planets will be the detection of orbital motion in the proper motion of maser spots showing the magnetic fluctuations, by contemporaneous direct imaging of the suggested wake flows at the same velocity as the fluctuations. Such a double evidence would rule out ambiguities: there may be planets without wake flows, if the wind density is not sufficient to produce observable features, and there are radial alignments of maser spots not related to wake flows. The coincidence of a persisting pseudoperiodic magnetic fluctuation with a radial alignment of maser spots strongly suggests that these phenomena are associated and naturally explained by a precessing Jovian magnetosphere and a planetary wake flow, respectively.

Appendix A: Determination of orbital parameters

Here we summarize the determination of orbital parameters, which follows the classical method of analysing spatially and spectrally resolved binary systems. Without further observational evidence at hand, we only treat the case of a circular orbit here. The coordinate systems applied and the orbital parameters are defined in Fig. A.1. The transformation of the radius vector (pointing from the star to the planet) between the orbital plane which defines the system ξηζ and the xyz coordinate system (where the x axis is the line of sight to the observer, and where the yz plane is the sky plane) is given by

- → r =         X Y Z         = r.         + cos Θ cos Φ cos (Ω∆t) -sin Φ sin (Ω∆t) -cos Θ sin Φ cos (Ω∆t) -cos Φ sin (Ω∆t) -sin Θ cos (Ω∆t)         (A.1)
where r is the distance between star and planet, Θ is the angle between the orbital angular momentum vector and the direction pointing from the star northwards (taken positive if the positive ζ axis is tilted in direction of the positive ξ axis), and Ω = 2π/T is the angular velocity of the planet's orbit (with period T ). The angles Φ, Θ and Ω∆t are the three Euler angles of the transformation between the ξηζ and xyz coordinate systems. They are related to the inclination i of the orbital angular momentum vector with respect to the line-of-sight to the observer by cos i = sin Θ cos Φ (A.2) ∆t = tt 0 is the time lapse between the epoch of the observation and the reference time t 0 when the radius vector is along the e. the drawing plane, is the plane of the sky), the ξηζ coordinate system by the dashed arrows (the orbit is in the ξη plane). The η axis is the node line about which the orbital plane is inclined by angle Θ and lies in the xy plane. Φ is the longitude of the ascending node. The star, the planet and its wake flow are schematically indicated, as well as the position of the ascending and of the descending node.

vector pointing from the star to the ascending node (the node line defines the axis about which the orbital plane is inclined with respect to the line of sight). The xyz coordinate system is stationary with respect to the local standard of rest (LSR). Applying the same coordinate transformation to the velocity vector of the planet -→ v P yields

- → v P = - → v * + v K        
cos Θ cos Φ sin (Ω∆t) -sin Φ cos (Ω∆t) -cos Θ sin Φ sin (Ω∆t) -cos Φ cos (Ω∆t) + sin Θ sin (Ω∆t)

        (A.3)
where -→ v * is the proper motion vector of the star with respect to the LSR (in astronomical velocity definition, i.e. negative velocity corresponds to motion towards the observer). Here v K is the Keplerian speed of the planet on the assumed circular orbit, i.e. where M * is the stellar mass. For the precision required here, it is sufficient to assume that the center of gravity of the star-planet system is undistinguishably close to the center of the star. The analysis of the observations is straightforward now: We need to identify a predominantly radial alignment of SiO maser spots at the velocity where the fluctuations of circular polarization are observed, suggesting that they are located in the wake flow of a planet whose magnetic dipole axis is misaligned with its rotation axis. The velocity at which the fluctuations occur is interpreted as the line-of-sight velocity of the planet in the LSR, ( -→ v P ) x . A sine wave fit to the measurements of -→ (v P ) x = ( -→ v * ) x + v K .(cos Θ cos Φ sin (Ω∆t) -sin Φ cos Ω∆t = ( -→ v * ) x + V. sin (Ω∆t + ψ) (A.5) at several epochs yields the stellar LSR velocity ( -→ v * ) x and the angular frequency Ω, as well as the parameters V = v K 1 -cos 2 Φ sin 2 i and tan ψ = tan Φ sec i .

(A.6)

These parameters are used together with the positions of the wake flows in the VLBI maps to constrain the radius of the orbit, r, and therefore the Keplerian velocity v K = Ωr, as well as the longitude of the ascending node, Φ, and the inclination of the orbit with respect to the sky plane, from Eq. A.1. The stellar mass is then determined from A.4. As positions, we use the head (i.e. the substellar tip) of the wake flow which we identify with the planet's position. The corresponding velocity is the maximum speed detected at the head of the wake flow, since the AGB wind is decelerated downstream the wake flow.

The main uncertainty of this combined kinematical and astrometrical method is the association between a magnetic fluctuation and a precessing planetary magnetosphere, and between the planet's position and the head of the wake flow. It is therefore advisable to over-determine the velocity and position fit, in order to minimize the observational uncertainties inherent to the method.

Fig. 1 .

 1 Fig. 1. Spectra of total power (Stokes I, i.e. polarized plus unpolarized flux density) and of fractional circular polarization. Greyshaded histogram: Stokes I flux density in Jansky vs. v lsr (the radial velocity with respect to the local standard of rest). Dots with errorbars: fractional circular polarization p C , in % with respect to Stokes I. Left: Mira-type star V Cam. Right: Mira-type star R Leo.

Fig. 3 .

 3 Fig.3. Lomb periodograms of the polarization time series (estimates of the spectral power density by least-square fits of their harmonic contents,Lomb, 1976), normalized by the variance of the samples. Top left: Lomb periodogram of p C from V Cam at v lsr = 7.5 km s -1 . Top right: same for the residuals of Stokes U (normalized by Stokes I, after subtraction of the stationary linear polarization). Bottom left and right: as top panels, but for R Leo at v lsr = 4.4 km s -1 . The horizontal dashed lines indicate the spectral power density at which a periodic signal can be mimicked by Gaussian noise with a probability given by the labels ("false alarm probability"). For a given source, the plot scale for the Lomb periodograms of p C and the Stokes U residuals is the same, to make the comparison of the significance of the peaks easier.

Fig. 4 .

 4 Fig. 4. Same as Fig 2, but for R Leo at three selected velocities (observations from 2008 August 01). Left: v lsr = -2.8 km s -1 . Middle : v lsr =0.6 km s -1 . Right: v lsr = -5.3 km s -1

Fig. 5 .

 5 Fig. 5. Top: Keplerian velocity fit of maser features associated with the suggested planet (LSR velocities at the tip of the planetary wake flow, and velocities of polarization fluctuations). Time is in years since 2001 Aug. 04 (observations ofCotton et al., 2004). Other references:Soria-Ruiz et al., 2007 (2002 Dec. 07),Cotton et al., 2008(2004 Sep. 05), this work (2006 May 22 and 2008 Aug. 01). Bottom: available astrometric information and least-square fit of the derived planetary orbit (Appendix A), for d = 113.5 pc stellar distance. VLBI positions are indicated by black dots. The planetary orbit is indicated by the black line, crosses mark the fitted position at the VLBI epochs. Empty circles give the positions of the planet at the epochs of our polarization measurements. The linear polarizations measured by us and their position angles are shown as bars (the bar in the top left corner indicates a fractional linear polarization of 30 %). The grey-shaded disk indicates the measured λ 2.2 µm uniform-disk diameter at stellar phase 0.08 (in April 2001,Fedele et al., 2005).

Fig. A. 1 .

 1 Fig. A.1.Definition of the orbital parameters. The xyz coordinate system is shown by solid arrows (the yz plane, i.e. the drawing plane, is the plane of the sky), the ξηζ coordinate system by the dashed arrows (the orbit is in the ξη plane). The η axis is the node line about which the orbital plane is inclined by angle Θ and lies in the xy plane. Φ is the longitude of the ascending node. The star, the planet and its wake flow are schematically indicated, as well as the position of the ascending and of the descending node.

  (bottom). The linear polarization measured by us is predominantly parallel to the radius vector calculated from the model fit. Due to uncertainties in our method and in the astrometric positions of the VLBI spot maps (no absolute uncertainties are derived in the works ofCotton et al. and Soria-Ruiz et al.), it is not too surprising to observe a discrepancy between the observed and modelled positions, especially for the 2002 Dec 07 VLBA observation(Soria-Ruiz, 2007).

GILDAS collection, see http://www.iram.fr/IRAMFR/GILDAS

Acknowledgements. Based on observations with the IRAM 30m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). Gabriel Paubert, supported by the IRAM backend group, built VESPA which provides simultaneous measurements of all Stokes parameters. We used the valuable optical lightcurves of the AAVSO (http://www.aavso.org) and acknowledge support from the telescope staff. We thank the anonymous referee for his constructive and helpful comments. We dedicate this paper to the memory of Matthew Carter, who passed away in November 2008. Matt was a valued colleague and friend who made many important contributions to the success of the IRAM receivers.


Vincent Zoma

Wendpanga Jacques

Ismaël Tarama

Sébastien Kiema

Typology of farms and application of the assisted natural regeneration strategy in southwestern Burkina Faso

Keywords: assisted natural regeneration, typology of farms, Burkina Faso, West Africa

Faced with the multiple challenges of natural resource management, Assisted Natural Regeneration (ANR) is one of the strategies recommended especially in Africa for farmers. The present research therefore aims to study the characteristics of farms in relation to the strategy of application of Assisted Natural Regeneration (ANR) in agroecosystems in the southern Sudanian agro-climatic zone of Burkina Faso. The study took place on the outskirts of the Classified Forest and Partial Reserve of faun of Comoé Léraba located in the southwestern part of Burkina Faso. The methodological approach consisted of collecting socioeconomic data from 102 farms through a survey. The analysis of the results reveals that producers interested in ANR can be divided into three (03) distinct classes. It highlights the socio-economic differences as well as the divergences in the conduct of the work of the ANR.

Introduction

Natural resources constitute the main economic and food source for the populations of many African states with an agro-pastoral vocation. However, some of these countries are faced with a situation of degradation of their renewable natural resources, so that conventional production efforts very often have unsatisfactory results. Indeed, we are witnessing a saturation of agricultural space and an abandonment of the practice of fallowing in several localities in Africa and this would be partly linked to the phenomenon of climatic variability observed everywhere in recent decades. Improving the management of agricultural land is therefore an emergency for the protection of the environment and for an increase in production. Aware of this reality and anxious to reverse the trend, the actors involved in the management of natural resources are organizing themselves everywhere. The usual reaction of decision makers and funders is to plant trees. However, E. Botoni and C. Reij (2009) [START_REF] Botoni | The silent transformation of the environment and production systems in the Sahel: impact of public and private investments in the management of natural resources[END_REF] show that this reaction is not necessarily the best one and that other alternatives such as Assisted Natural Regeneration (ANR), water harvesting technics could be promoted and supported. To solve the deterioration of the environment while improving the living conditions of rural populations. Moreover, agricultural extension and development institutions have instead begun to work for the maintenance and regeneration of trees in the fields (J. Boffa, 2000) [START_REF] Boffa | Agroforestry Parks in Sub-Saharan Africa[END_REF] . According to A. Diouf (2002) [START_REF] Diouf | Typology of farms and study of the profitability of fodder crops in the production systems of the Groundnut Basin of Senegal[END_REF] , the introduction of a new agrarian technique in an environment is always an undertaking that requires the consideration of several factors. These are essentially socio-cultural, economic and ecological factors. Indeed, the introduction of the practice of ANR in a region consists in encouraging active participation of producers in order to protect and manage the regrowth of woody vegetation in their field, and to recreate a woody stratum. Previous studies have concluded that its practice has had numerous socio-economic, biophysical and climatic impacts in the Sahelian and northern Sudanian zones of Burkina Faso and elsewhere in West Africa. However, the introduction of Assisted Natural Regeneration raises questions about the real impacts of its application in the relatively more humid southern Sudanian agro-climatic zone of Burkina Faso. Thus, like the Sahel and the northern Sudanese zone, is the application of ANR of interest to all producers in the southern Sudanese zone of Burkina Faso? In other words, what profile of producers is interested in introducing and carrying out the practice of ANR in this area? The present research therefore aims to study the characteristics of farms in relation to the strategy of application of Assisted Natural Regeneration (ANR) in agro-ecosystems in the southern Sudanian agro-climatic zone of Burkina Faso.

Methodological approach of the research

The fieldwork for this research was conducted in the extreme south-west of Burkina Faso as shown in Map 1.

Map 1: Location of the study area

The study area shown on the map above is between latitude 9°35' and 10°10' North and longitude 4°15' and 5°00' West. It lies along the Ivorian border with Burkina Faso. It is located around the confluence of the Comoé and Léraba rivers. The villages concerned by this research are located between the departments of Mangodara and Niangoloko in the province of Comoé. The investigation area belongs to the southern Sudanian agro-ecological zone with an annual rainfall that varies between 1000 and 1200mm (S. Guinko, 1997) [START_REF] Guinko | Characterization of vegetation units and assessment of faunal diversity in the intervention area of the GEPRENAF project[END_REF] . Indeed, S. Guinko (1997) [START_REF] Guinko | Characterization of vegetation units and assessment of faunal diversity in the intervention area of the GEPRENAF project[END_REF] , Beckeropsis uniseta (Ness) K Schum. The population density is around 8.7 to 20.3 inhabitants per km². 90% of the population is engaged in agriculture (S. Guinko, 1997) [START_REF] Guinko | Characterization of vegetation units and assessment of faunal diversity in the intervention area of the GEPRENAF project[END_REF] . In addition to agriculture, animal husbandry is also practiced here. Hunting is one of the traditional practices deeply rooted in local cultures (D. Sirima, 2010) [START_REF] Sirima | Small sport hunting in the Classified Forest and Partial Wildlife Reserve of Comoé Léraba and in the communal conservation areas of the intervention zone of the Forest Resources Management Association of Comoé Léraba: status places and opportunities for increasing the income of grassroots communities[END_REF] . Concerning the collection of field data, in particular for the choice of the farms studied, the study consisted in carrying out a survey which focused on a group of 230 households having adopted Assisted Natural Regeneration since 2012 and listed in a database from the Comoé Léraba Natural Resources and Wildlife Management Association (AGEREF/CL). Consultation of the database made it possible to take stock of the number of producers concerned by the introduction of the technology in the area. It also made it possible to locate them and organize the implementation of data collection. This is why, concerned about spatial representativeness, a fixed number of 06 operators per village was retained. These 06 operators were chosen by their peers at the village level in view of their availability. Thus, for the conduct of the investigation, an exploration trip first made it possible to inform the leaders of the villages concerned. This outing made it possible to collect general information on the study area from the personnel of various technical services. All the villages bordering the Comoé-Léraba classified forest and partial wildlife reserve (17 in number) were concerned. The information was collected using semi-structured interviews. The observation unit of the survey was the agricultural holding of the agricultural household type. To do this, the questions were addressed to the heads of operations. Field observations supplemented the comments of the respondents. Indeed, based on observations and surveys, the analysis of farm diversity leads to the construction of typologies (MAEF, 2002). For the choice of segregation criteria and variables, the approach consisted first of all in identifying the basic variables for the constitution of the typology. A Pearson correlation analysis was performed to select inter-correlated variables. This analysis made it possible to retain a reduced number of variables (08 in total) for the classification analyses. A survey of the correlation coefficients then made it possible to verify the interest of carrying out a principal component analysis (PCA) of the data. The analysis of the correlations between the variables led to the conservation of 08 variables for the PCA (Table 1). Total cost of inputs used during the crop year of the survey period; 6. Income from the production of plots under assisted natural regeneration; 7. Costs saved thanks to the use of organic manure; 8. Quantity of organic manure used. Many correlation coefficients (r) are quite strong and are above a fixed threshold (r>0.4), which suggests that the analysis is relevant For the processing of the data collected, they were entered in Microsoft Excel and then processed using XLSTAT software version 7.5.2 and version 16.3.01 from Addinsoft and SPSS (Statistical Package for Social Science). After checking the correlation, a classification by principal component analysis (PCA) was carried out with the selected variables. The method of moving centers, also called dynamic clusters or K-means, was used to group together individuals who are close. This is the technic best suited to large tables of data. It has the advantage of being efficient and very fast. It is based on the principle of classification based on the criterion of the nearest neighbors. This principle means that each individual is assigned to a class if it is very close to its center of gravity. Moreover, its particularity is that the number of classes must be specified beforehand (M. Hadd 1999). The last step in processing the collected data consisted of validating the types formed by a discriminant analysis (AFD). In this analysis, the different classes (types) to which individuals belong are used as variables to be explained to characterize the types of farms established. The present methodological approach described in this part made it possible to obtain results which are presented and discussed in this work.

Study results

The results of this investigation focus on the one hand on the typologies and the main characteristics of the different exploitation groups and on the other hand on the strategies and the conduct of Assisted Natural Regeneration in the identified classes.

Typology and main characteristics of the different farm groups

After different analysis scenarios, 3 homogeneous groups of producers were selected. These groups present a certain proportionality between groups and in relation to all individuals. The factor components retained make it possible to explain 82.88% of the variations observed. Chart 1 shows the discriminated grouping of producers by axis.

Source : Field Works

Graph 1: Representation of producer classes in the factorial space Chart 1 shows the discriminated grouping of producers by axis. From this graph above, there three classes. The first relates to Small Family Farms (PEAF), the second is to Medium Family Farms (EAFM) and the third relates to Large Family Farms (GEAF). Regarding the class Class 1 (Small Family Farms (PEAF)), it forms 62.74% of the farms surveyed. Farms in this class are the most widespread and are poorly endowed with factors of production. Mechanization is relatively weak there. Half of the constituent producers of this group, i.e. 50% of the farms, have teams. Thus, this class alone concentrates the total number of manual operations. Farms in this class spend less than 2,500 FCFA to 1,600,500 FCFA for equipment and less than 2,500 FCFA to 193,000 FCFA for production inputs. They farm small areas for all of their annual production (on average around half a dozen hectares per farm). The minimum total area harvested in this class is 1.5 hectares and the maximum is 20 hectares. The areas under Assisted Natural Regeneration range from 0.25 ha to 2 ha, i.e. an average of 0.99 ha per farmer. The crops most exploited in these farms are peanuts, maize, cowpeas and sesame. As for class 2 (Medium Family Farms or EAFM), it represents 34.31% of the producers met during the survey. Farms in this class are moderately endowed with factors of production. The level of mechanization in this class is interesting. All farms (100%) are equipped with hitching. On average, they farm a total annual production area of 12.30 hectares per farm. The minimum total area is around 5 hectares and the maximum around 24 hectares. They dedicate between 1 to 5 ha of surface area for ANR (i.e. an average of 2.22 ha), and intensify the production of cotton and, to a lesser extent, that of cereals. The minimum cost mobilized for equipment in this class is 270,000 FCFA against a maximum of 3,162,500 FCFA for the same expenditure. With regard to expenses for inputs, the minimum remains identical to that of the class described above but the maximum invested for the same cause is 348,000 FCFA. Finally, class 3 (Large Family Farms or GEAF) represents only 2.94% of the sample surveyed. The producers have extensive experience in agricultural production. Better equipped with factors of production, they invest at least 2,409,000 FCFA and at most 10,010,000 FCFA to equip themselves. They also spend between 159,000 FCFA and 446,000 FCFA for the purchase of inputs. This explains the high level of mechanization in this class. Indeed, data analysis reveals that 70% of farmers in this group use at least 3 animal traction chains for their work in the field, and one farm is motorized. The highest average total agricultural area exploited (28.17 ha per farm) as well as the average area under ANR (4.08 ha/farm). The minimum total area under exploitation in this class is 9.50 hectares while the maximum is 40 hectares.

After having presented the results in relation to the typologies and the main characteristics of the different exploitation groups, we also present the strategies and the conduct of Assisted Natural Regeneration in the identified classes.

and conduct of Assisted Natural Regeneration (ANR) in identified classes

In terms of tree regeneration techniques, the survey essentially recorded two regeneration processes implemented in the localities surveyed. The first consists in sparing the natural rejections of foot. This technique is used by 100% of producers of all classes. It is essentially the only process used by all producers of Small Family Farms. In addition to natural shoots from tree trunks, producers of other classes sow or transplant wildlings of species that are not found on their plot in a timely manner. Thus, during the investigations, 41.66% of producers from Small Family Farms and 44.64% of producers from Medium Family Farms claimed to have already sown the species Tamarindus indica L. or Adansonia digitata L. in their field. This practice has the advantage of contributing to specific diversity in the fields. Those on the relatively more intensive Large Family Farms do not plant. The silvicultural assistance and care given to trees in the conduct of ANR in the fields varies from class to class. Producers of Small Family Farms are more involved in silvicultural operations to assist regeneration in the fields. In addition to the choice to spare tree bases, to practice thinning and pruning to allow them to grow better, they intervene in the protection against all forms of aggression (mainly men, animals and wind). For this, some producers (more than 3%) in this class said they had installed protection devices for spared feet in their plots. Producers of Medium Family Farms mainly practice thinning to select the plants to be saved and pruning to support their growth. No form of protection has been recorded in this class. Producers of Large Family Farms provide the trees with the care of thinning and pruning mentioned in the previous class, but they are less involved in maintenance operations than producers of other classes. All classes assist the regeneration of feet in the fields through the control of fires. Grazing control is carried out by all producers of Small Family Farms and Medium Family Farms. Producers of Large Family Farms allow grazing in their fields. This is probably linked to the fact that they themselves own cattle or because their fields are very large and therefore difficult to control. The productivity of the fields under RNA was measured through the calculation of yields per hectare for each speculation and by class. The results are shown in Table 2. 2 shows that yields per hectare are low and vary depending on the class. This is related to the depletion of land eluted for RNA. In addition, the performance of the farms was also analyzed in relation to the Non-Timber Forest Products (NTFP) harvested in the fields under AN. The distribution of average production at the level of each class is given in table 3. This table provides insight into the repair of average NWFP production by class. It presents the average production of Shea, Néré, Tamarind, Bombax and Detarim according to the three classes which are Small Family Farms (PEAF), Medium Family Farms (EAFM) and Large Family Farms (GEAF). After presenting the main results of our study, the last part is devoted to the discussion of these results in relation to especially the literature review concerning our research topic.

Discussion of study results

The discussion of the results addresses respectively the number of classes obtained, the difference in the socioeconomic characteristics of the producers and the low technicality of the producers.

Number of classes obtained

The analysis of the survey data led to the constitution of 3 main classes. The same number of classes is also that found by A. Diouf (2002) [START_REF] Diouf | Typology of farms and study of the profitability of fodder crops in the production systems of the Groundnut Basin of Senegal[END_REF] in the groundnut of Senegal when making the typology of farms that have adopted fodder technology in their production system. However, he then proceeds to the dissection of one of the large classes into 2 subclasses to better characterize it. It is also the same number of classes that was found in Burkina Faso by M. Ouedraogo (2008) [START_REF] Ouedraogo | Analysis of the economic performance of family-type and agribusiness-type farms: the case of western Burkina Faso. End-of-study dissertation in sociology and rural economy[END_REF] in the west and by O. Ouattara (2010) [START_REF] Ouattara | Analysis of the financial profitability of farms managed by agricultural entrepreneurs in the Loop of Mouhoun region (Burkina Faso)[END_REF] in the Loop of Mouhoun region. But, in a study of two villages in the cotton zone of Burkina, Mr. Sy (1992) obtained 5 classes of producers in one of the villages and 04 classes in the other village. The authors M. Sy (1992) and M. Ouedraogo (2008) [START_REF] Ouedraogo | Analysis of the economic performance of family-type and agribusiness-type farms: the case of western Burkina Faso. End-of-study dissertation in sociology and rural economy[END_REF] established their typology empirically while A. Diouf (2002) [START_REF] Diouf | Typology of farms and study of the profitability of fodder crops in the production systems of the Groundnut Basin of Senegal[END_REF] and O. Ouattara (2010) [START_REF] Ouattara | Analysis of the financial profitability of farms managed by agricultural entrepreneurs in the Loop of Mouhoun region (Burkina Faso)[END_REF] used factorial analyses. The present study is therefore from a methodological point of view close to the last two authors cited. In addition to the number of classes, it is important to analyze the socio-economic characteristics of the producers.

Different socio-economic characteristics of producers

The producers concerned by this study differ from one class to another by their own socio-economic and technical characteristics. These results are similar to those of O. Ouattara (2010) [START_REF] Ouattara | Analysis of the financial profitability of farms managed by agricultural entrepreneurs in the Loop of Mouhoun region (Burkina Faso)[END_REF] regarding the characteristics of Small Family Farms and Medium Family Farms. Indeed, Small Family Farms are similar to type 1 "small family farms" (PEAF), while type 2 called "traditional market family farms" (EFMT) has similarities with Medium Family Farms of this study. However, the fundamental distinction between the two studies has to do with the targetting of farms. O. Ouattara (2010) [START_REF] Ouattara | Analysis of the financial profitability of farms managed by agricultural entrepreneurs in the Loop of Mouhoun region (Burkina Faso)[END_REF] had targeted family farms while the present study concerned family farms that are not oriented towards entrepreneurship. The choice to adopt and lead the saving and the planting of trees in the fields is affected by the 08 main variables which contributed to constitute the classes. Since these variables are inter-correlated, they influence each other in a reciprocal manner. From the point of view of age, for example, the youngest and oldest producers are relatively poor in means of production (production area, labour, equipment, inputs, etc.). They are small family farmers. However, they are more respectful of the prescriptions of the technical supervisors and conduct the ANR better in their fields than the producers of the other classes. They are grouped together in Small Family Farms. Furthermore, the different analyzes show that the large family farmers (GEAF) of class 3 are well supplied with factors of production and obtain better yields than those of the other classes. However, these producers seem to neglect certain silvicultural operations in the conduct of ANR. The intensive nature of their production system means that they do not plant trees in their fields. The results obtained by F. Kagne (2012) [START_REF] Kagne | Socio-economic impact of assisted natural regeneration (ANR) in strategies for improving the livelihoods of agricultural producers: the case of Gnagna and Gourma[END_REF] in the eastern region of the Sudanian zone of Burkina Faso corroborate ours. This author had counted 06 variables that influence the choice of adoption of ANR. Finally, after having discussed the number of classes, the socio-economic characteristics of the producers, we approach the result relating to the low technicality of the producers.

Low technicality of producers

The study noted in some producers a weak application of agricultural technical prescriptions and in others the persistence of shifting cultivation practices on slash and burn. The same forms of technical deficiencies have already been reported by F. Kagne (2012) [START_REF] Kagne | Socio-economic impact of assisted natural regeneration (ANR) in strategies for improving the livelihoods of agricultural producers: the case of Gnagna and Gourma[END_REF] in the eastern region of Burkina Faso among producers practicing ANR. In general, the functioning and structuring observed in the farms studied is similar to the type of agriculture that J. F. Belieres and al. (2002) [START_REF] Belieres | iied[END_REF] qualify as family farming. These characteristics were also observed by R. Blein and al. (2008) [START_REF] Blein | The agricultural potential of Africa (ECOWAS)[END_REF] on 80 to 90% of farms in the entire West African region. According to these authors, these are types of exploitation whose areas vary between 3 to 5 ha. Our results indicate the commitment of a diversity of producers in the adoption of FMNR, but also in the conduct of this agroforestry technique. The commitment to the application of FMNR is therefore not linked to the uniformity of the socio-economic conditions of producers.

Conclusion

This research is a study on the characteristics of farms in relation to the strategy of application of Assisted Natural Regeneration (ANR) in agro-ecosystems in the southern Sudanian agro-climatic zone of Burkina Faso. The research shows that the actors involved in the application of this strategy in the peripheral zone of the Classified Forest and Partial Wildlife Reserve of Comoé-Léraba correspond to a wide variety of farmers. They are made up of small, medium and large family-type farmers characterized by significant differences. These differences mainly concern the available surface areas and the technical means with, in some cases, a high level of investment. These producers also diverge in terms of the technical conduct of ANR. The typology carried out revealed the existence of 03 distinct classes of producers. The practice of ANR therefore interests different categories of producers in the area studied.

  identified the well-represented species there as Daniellia oliveri (Rolfe) Hutch. & Dalz, Isoberlinia doka Craib & Stapf, Pterocarpus erinaceus Poir., Khaya seneglensis (Desr.) A. Juss., Detarium microcarpum Guill. & Perr., Burkea africana Hook., Vitellaria paradoxa Gaertn.f. In dense, tall gallery forests, species Berlinia grandiflora Hutch. & Dalz., Cola laurifolia Mast., Diallium guineense Willd., Pterocarpus santalinoides L'Her. Ex DC, Manilkara obovata (Sabine & G.Don) JHHemsl., Syzygium guineense (Willd.) DC… are dominant while Khaya senegalensis (Desr.) A. Juss., Cola gigantea A. Chev., Cola cordifolia (Cav.) RBR, Erythrophleum guineense G. Don, Anogneissus leiocarpus G. & Perr., Diospyros mespiliformis Hochst. ex A. Rich…are commonly observed. In developed wooded to tree savannas, on lowlands and on deep soils, the dominant species are Isoberlinia dalzielli Craib & Stapf., Isoberlinia doka Craib. And Stapf., Afzelia africana Sm., Pterocarpus erinaceus Poir., Erythrophylum guineense G. Don. The herbaceous cover appears discontinuous and is essentially composed of Anchomanes welwitschii Rentle, Sapium grahamii (Stapf) Prain, Cissus populnea Guill. & Perr., Andropogon tectorum Schumach. & Thonn.

Table 1 :

 1 Correlation matrix of the variables retained for the typology of producers

	Nbpers 1	1	0,382	0,431	0,539	0,341	0,476	0,280	0,443
	Suptot 2	0,382	1	0,528	0,395	0,492	0,433	0,305	0,538
	SupRNA 3	0,431	0,528	1	0,363	0,587	0,607	0,490	0,399
	Coutotéq 4	0,539	0,395	0,363	1	0,419	0,773	0,354	0,478
	Coutotin 5	0,341	0,492	0,587	0,419	1	0,709	0,218	0,465
	Revprod 6	0,476	0,433	0,607	0,773	0,709	1	0,428	0,544
	Fraiseco 7	0,280	0,305	0,490	0,354	0,218	0,428	1	0,226
	Qtfumorg 8	0,443	0,538	0,399	0,478	0,465	0,544	0,226	1
	Source: Fieldwork

Table 1

 1 above presents the correlation matrix of the variables retained for the typology of producers. In this table, the significant values (off the diagonal) at the alpha=0.050 threshold (two-tailed test) are in bold. The figures after allow an understanding of this table:1. Number of people living on the farm; 2. Total area exploited during the survey period; 3. Area under assisted natural regeneration; 4. Total cost of farm equipment; 5.

Table 2 :

 2 Breakdown of yields per hectare by speculation and by class (in kg)

	Class	Maize	Sorghum	Sesame	Yield (kg/ha) Cowpea	Cotton	peanut	Cassava
	PEAF	974	382	201	166	1100	546	0
	EAFM	1849	400	213	156	1000	2000	2250
	GEAF	2612	0	80	0	0	0	0
	Total	1692	387	199	160	1009	712	2250
	Source : Field Works
	In general, Table

Table 3 :

 3 Breakdown of average NWFP production by class (in kg)

	Classe	Shea (almond)	Néré (seed)	Average production (kg) Tamarind (fruit) Bombax (flower)	Detarium (fruit)
	PEAF	330	94	41	36	0
	EAFM	437	76	67	24	8
	GEAF	1253	16	70	5	0
	Total	394	85	53	32	1
	Source : Field Works