Francesco Di Gregorio

Abdoulaye Gamatié

Gilles Sassatelli

Arnaud Castelltort

Michel Robert

Exploration of Energy-Proportional Distributed Systems

Keywords: Distributed Systems, Smart Grid, Cloud Computing, Multicore, Energy Management

With the advent of Internet of Things (IoT) and the increase of the Internet Services, more and more datacenters are needed to manage the Internet traffic. Energy consumption related to these infrastructures have already reached more than 2% of the global energy consumption and it is estimated to increase in the next years. Trends such as a billion more people coming online in developing countries, IoT, driverless cars, robots, and artificial intelligence (AI) are major drivers of unrelenting growth. Our work is intended to explore new distributed architectures to deploy datacenters in order to reduce their energy consumption and reach the energy proportionality. An effort in the use of Renewable Energy Sources (RES) and Energy Storage Systems (ESS) is done in order to approach the zero-carbon emission objective. In this sense, microgrid, nanogrid and smart grid structures are analyzed in terms of stability, architecture, control, communication and energy management.

Introduction

With the advent of Internet, people and electronic systems have the possibility to exchange information by using the network. Chat, video chat, social networks, online gaming, video conference, streaming video, Cloud computing are some of the services Internet can offer today. Moreover, Internet of Things (IoT) is another application which is taking place nowadays and permits objects to be connected to Internet in order to enhance them with smart features. All of these services generate a huge traffic of data through the network.

Trends on datacenters' power consumption. Datacenters play a key role in managing and processing these data. Indeed, companies like Facebook, Google, Amazon, already have many of these high computing and storing centers to manage users requests all over the globe. In this way, energy consumption of datacenters have already reached more than 2% of the world's electricity [START_REF] Adams | Power consumption in data centers is a global problem[END_REF]. The Information and Communication Technology (ICT) sector predicts to use 20% of all the world's electricity by 2025 and emit up to 5.5% of all carbon emissions. That number could surge as high as 14% by 2040 as a torrent of internet-connected devices, high-resolution photos, streamed videos, emails, surveillance camera footage, cell phones, and smart TV require more compute power, greater amounts of network bandwidth, and ever-higher amounts of storage capacity. Moreover, low-efficiency of power supply and the need of cooling systems makes these centers very inefficient. It is estimated that datacenters waste more than 50% of the their total energy in power supply losses and cooling systems. Energy proportionality is reached when all the energy losses are eliminated. In this way, the overall power consumption of the system become proportional to its performance.

Motivations of the current study. Our work aims to explore new architectures allowing to increase the overall efficiency of this systems trying to reach the energy proportionality. In this sense, distributed architectures are good candidates because they reduce or eliminate the necessity of cooling systems. Moreover, distributed architectures offer other advantages like modularity, scalability, reliability etc. This permits to manage the increase of the demand, by simply adding new modules to the system. Faults would be managed by simply replacing the damaged module without compromising the Quality of Service (QoS) of the whole system which will be guaranteed by all the other modules.

On the other hand, distributed architectures find a good matching with renewable energies such as solar, wind, etc., because the distributed nature of the latter. Then, integration of Renewable Energy Sources (RES) with ICT will be considered by using microgrids and smart grids technology. In fact, by exploiting RES, the zero-carbon emission objective could be reached. Energy management policies and online optimization algorithm are also explored to increase the efficiency of the overall system. Integration of System of Chip (SoC) in these systems is foreseen to add these smart features directly on chip.

Distributed Systems

A distributed system is a set of independent components that appears to its users as a single coherent system or as a single system. Two examples of distributed systems are presented in this section: cloud computing infrastructure in data-domain and smart grid infrastructure in energy-domain.

Cloud computing infrastructure. A Cloud computing infrastructure consists of a set of interconnected computers, storage systems etc. which appear to the end-user like a single machine. This allows the end-user to take advantage of a powerful hardware infrastructure without buying it. In fact, these services can be rent on-demand. Of course, several software abstraction layers are needed which virtualizes resources and logically presents them to users through application program interfaces and APIenabled command-line or graphical interfaces.

Smart grid infrastructure. In the same way, a smart grid infrastructure consists of a set of interconnected energy sources, loads and Energy Storage Systems (ESS) which appear like a single, large energy bank. An energy source can connect to this energy bank to inject energy. Instead, a load can connect to this energy bank to take energy. Of course, this kind of network needs different control software layers in order to guarantee stability of the power network and QoS. In fact, stability, efficiency and maximum resource utilization are some of the challenges these systems are facing up. Distributed control algorithms, energy sharing policies, online optimization algorithms are some of the directions researchers are taking to solve these problems. All of these challenges have the common goal of creating an energy bank that can be used for the interconnection of an ever increasing set of heterogeneous sources and loads.

Towards an Energy-Proportional Distributed Computing

A novel approach is investigated in our work, which integrates the cloud computing technology with the smart grid technology. In this approach, a smart grid is used to feed a cloud computing infrastructure. More specifically, each server of the Cloud computing infrastructure can be fed on-demand by the smart grid as illustrated in Figure 1. This enables the integration of zero-carbon emission RES with conventional Fossile Energy Sources (FES). Moreover, intrinsic advantages of distributed systems make this architecture more reliable, scalable and efficient. The smart grid is composed by a grid-connected DC MicroGrid (DCMG) whose intelligence is given by the integration of SoCs.

DC MicroGrid. A DCMG is an electrical system that can efficiently distribute, consume and potentially create and store, Direct Current (DC) electricity to feed a wide variety of electrical devices. Several DCMG architectures have been proposed in the last decades [START_REF] Kumar | A review on overall control of dc microgrids[END_REF]. These can be classified into six main categories: singlebus DCMG, Multi-Bus DCMG, Multi-Terminal DCMG, Ring-Bus DCMG, Ladder DCMG, Zonal DCMG. The choice of the DCMG architecture influences many factor such as the cost of the project, robustness, resiliency, controllability and hierarchy of the system, reliability, availability, resource utilization, and flexibility.

Communication and control are also important in order to have an efficient operation and guarantee for stability. Several control strategies can be applied to DCMG such as voltage and current control, load sharing, power flow control, economic dispatching, minimization of transmission losses etc. In order to obtain these objectives, centralized, decentralized and distributed control methods are investigated. Moreover, energy management policies and online optimization techniques are explored in order to increase the overall efficiency of the system and guarantee the QoS. Machine Learning-based techniques can be one of the possible directions.

Conclusions

This work proposes an Energy-proportional Distributed Computing infrastructure to solve the problem of energy waste in datacenters. This infrastructure merges two recent technologies: Cloud Computing and Smart Grid. DCMG architectures are explored and integrated with SoC to add smart features. Distributed Control algorithms and Distributed communication techniques are investigated to better control this system in data-domain and energy-domain. Integration of RES is also considered to reach the zero-carbon emission objective. Finally, energy management policies and online optimization algorithms will be explored in order to increase the energy efficiency of the proposed system.

Figure 1 .

 1 Figure 1. Proposed approach.


First evidence of quasi-periodic magnetic intraday activity from SiO emission in

 [START_REF] Habing | Circumstellar envelopes and Asymptotic Giant Branch stars[END_REF]. 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 expulsed matter and, possibly, with Jovian or terrestial planets [START_REF] Struck | Models of Planets and Brown Dwarfs in Mira Winds[END_REF][START_REF] Struck-Marcell | SiO masers in late-type giant stars -Jovian planets engulfed[END_REF] . As in the solar system, where space weather changes on timescales of hours [START_REF] Prangé | An interplanetary shock traced by planetary auroral storms from the Sun to Saturn[END_REF] , fluctuations of the magnetic field about a mean value can be expected, but the observational evidence is still lacking.

Here we show that for a narrow range of velocities the circular polarization of SiO masers, generally accepted as a tracer of the magnetic field in the extended atmosphere of AGB stars [START_REF] Nedoluha | Intensity-dependent circular polarization and circumstellar magnetic fields from the observation of SiO masers[END_REF][START_REF] Elitzur | Polarization of Astronomical Maser Radiation. III. Arbitrary Zeeman Splitting and Anisotropic Pumping[END_REF] , varies in two AGB stars with a period of a few hours. All our knowledge about the magnetism within the extended atmosphere of AGB stars relies on circular polarization measurements of SiO masers at 1.5 to 7 AU distance from the star, which typically has a radius of 0.7 to 0.8 AU. Assuming that the circular polarization is caused by the Zeeman effect in the non-paramagnetic SiO molecule, fractional circular polarizations of up to 9 % have been reported [START_REF] Barvainis | Evidence for strong magnetic fields in the inner envelopes of late-type stars[END_REF] for the v=1, J=1-0 transition (v and J are the vibrational and rotational quantum numbers, respectively), yielding magnetic flux densities of up to 100 G. These observations have been recently confirmed [START_REF] Herpin | Full polarization study of SiO masers at 86 GHz[END_REF] for the spatially coincident v=1, J=2-1 masers, though resulting in lower magnetic flux densities (up to 20 G), depending on theories of maser polarization [START_REF] Deguchi | Circular polarization of astrophysical masers due to overlap of Zeeman components[END_REF][START_REF] Elitzur | Astrophysical Spectropolarimetry[END_REF] . In the outer envelope, H 2 O and OH masers reside at ~ 100 to 400 AU, respectively ~1000 to ~10.000 AU, from the star. The Zeeman effect of H 2 O masers has been used [START_REF] Vlemmings | Circular polarization of water masers in the circumstellar envelopes of late type stars[END_REF] 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 can maintain a magnetic flux throughout their envelope which later may shape the gas expelled by the planetary nebulae (though its dynamical importance is still a matter of debate). Indeed, AGB stars are expected to drive a powerful dynamo at the coreenvelope interface [START_REF] Blackman | Dynamos in asymptotic-giant-branch stars as the origin of magnetic fields shaping planetary nebulae[END_REF] . As for the nature of the magnetized environment in which SiO masers reside, several scenarios exist, and there are two crucial tests for them: they need to explain the observed lifetime [START_REF] Glenn | A Polarization Survey of SiO Maser Variability in Evolved Stars[END_REF] of linearly polarized SiO maser features (though the linear polarization may be of non-magnetic origin), but they also predict observable magnetic fluctuations on top of the stellar magnetosphere.

Previous multi-epoch observations

Here we provide observational evidence for short-term fluctuations of the magnetic field in the extended atmosphere of two AGB stars by simultaneous monitoring of all Stokes parameters of the v=1, J=2-1 maser transition of SiO with IRAM's 30m telescope. Theory of the SiO phenomenon invokes molecular excitation in dense pockets of gas and amplification of radiation in narrow tubes. Once the maser spots are formed (often distributed along arcs around the star, hereafter referred as the maser shell), they are subject to magnetospheric events known to be rapidly variable in the solar system, hence dense time sampling is important. 80 AGB stars were observed with a digital correlation polarimeter, out of which 8 objects with the strongest circular polarization were retained for a densely sampled monitoring (twice per hour). Only two AGB stars, the Mira variables V Cam and R Leo show clear evidence (Fig. 1 ) for a quasi-periodic modulation of their fractional circular polarization p C , at a narrow range of radial velocities of ~ 0.4 km/s centerd at 7.5 km/s and 4.4 km/s. The period is 5.4 h and 6.3 h, for V Cam and R Leo, respectively (Table 1). We note that the phenomenon is both rare and localized in velocity space (and thus presumably in real space, due to the velocity structure). The likelihood of misinterpretation of a time series of Gaussian noise is < 4 %, respectively < 16 % for the two sources (Fig. 3a,c). The observations were made in May 2006, ~40 days after the optical maximum of V Cam (pulsation period 522 d) and ~30 days after that of R Leo (pulsation period 310 d), at a phase when the SiO maser shell was expected to be in expansion [START_REF] Diamond | A Movie of a Star: Multiepoch Very Long Baseline Array Imaging of the SiO Masers toward the Mira Variable TX Cam[END_REF] . The peak-to-peak variation of p C implies a magnetic field of ~1 G (ref. 7, 12), if we rely on the Zeeman hypothesis to explain the origin of the Stokes V signal. It is not surprising that only the circular polarization exhibits this clear quasi-periodic fluctuation. If one considers that our telescope beam averages the polarization signal from all maser spots in the circumstellar shell, the different polarization angles -either parallel or perpendicular to the maser shell [START_REF] Desmurs | VLBA observations of SiO masers: arguments in favor of radiative pumping mechanisms[END_REF] -tends to strongly decrease the observed average linear polarization. Furthermore, linear polarization can be entirely produced without magnetic fields by anisotropic pumping [START_REF] Western | Linear polarization of astronomical masers by anisotropic pumping and its enhancement due to geometry[END_REF] (though polarization angle swings [START_REF] Diamond | A Movie of a Star: Multiepoch Very Long Baseline Array Imaging of the SiO Masers toward the Mira Variable TX Cam[END_REF][START_REF] Desmurs | VLBA observations of SiO masers: arguments in favor of radiative pumping mechanisms[END_REF] by 90° are explained by changes of the magnetic field direction with respect to the propagation direction of radiation in the maser [START_REF] Goldreich | Astrophysical Masers. II. Polarization Properties[END_REF] or the Hanle effect [START_REF] Asensio Ramos | Dichroic masers due to radiation anisotropy and the influence of the Hanle effect on the circumstellar SiO polarization[END_REF] ). By contrast, circular polarization is naturally enhanced in magnetized maser spots, either directly via the Zeeman splitting [START_REF] Deguchi | Circular polarization of astrophysical masers due to overlap of Zeeman components[END_REF][START_REF] Elitzur | Astrophysical Spectropolarimetry[END_REF] , or indirectly via birefringent conversion of linear into circular polarization due to changes of the magnetic field topology along the maser slab (ref. 22 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 [START_REF] Nedoluha | Intensity-dependent circular polarization and circumstellar magnetic fields from the observation of SiO masers[END_REF] . In the latter case, pump conditions varying with time are to be generated, 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 (Fig. 1 ). We therefore suggest that the intraday fluctuations of p c are of magnetic origin.

Monitoring of the Stokes I emission of SiO masers has reported variations on time

scales from days [START_REF] Pijpers | Short time scale monitoring of SiO sources[END_REF] to years [START_REF] Pardo | 28 SiO v=1 and v=2, J=1-0 maser variability in evolved stars. Eleven years of short spaced monitoring[END_REF][START_REF] Kang | Time Monitoring Observations of SiO J = 2-1 and J = 3-2 Maser Emission toward Late-Type Stars[END_REF] . To study magnetic fluctuations, the time variability of all Stokes parameters must be monitored. However, poor sampling can only be sensitive to a long-term readjustment of the magnetic field (ref. 17), while our dense sampling is sensitive to rapid magnetospheric events. As for the latter, scenarios such as (1) close binaries like the Mira AB system, (2) a global magnetic field, (3) flares and coronal loops, (4) magnetized clouds 24 , (5) Parker instabilities in shocks forming in response to stellar pulsations 25 , (6) Jovian magnetospheres, have been proposed, or could be proposed. In case (1), mass transfer from the AGB star to a companion may generate flares and related magnetic fluctuations, which are not necessarily located in the atmosphere of the former. V Cam and R Leo are not known for having lower mass companions, which rules out this case. As for case (2), the X-ray luminosities in two single AGB stars were found to be below the threshold expected for a dynamically important magnetic field [START_REF] Kastner | Constraining the X-ray luminosities of asymptotic giant branch stars: TX Camelopardis and T Cassiopeiae[END_REF] . This does not exclude episodic magnetic fluctuations, but in absence of a companion scenarios other than a global magnetic field need to be assumed if we want to explain the rareness of the observed fluctuations. In support to our case (3) we may suppose that the tips of flares and coronal loops directly reach the SiO maser zone. The differential rotation in the sub-photosphere of AGB stars, together with the dynamo beneath [START_REF] Blackman | Dynamos in asymptotic-giant-branch stars as the origin of magnetic fields shaping planetary nebulae[END_REF] , will lead to coronal loops (our case 3). For an upper limit, we take the rotation rate of the core from where the magnetic field emerges to be Ω c ~ 10 -5 s -1 (ref. 16). Assuming (by solar analogy) that the size of a coronal loop corresponds to that of a super-granulation cell, this yields 0.2 AU for 400 convection cells in the stellar subphotosphere [START_REF] Schwarzschild | On the scale of photospheric convection in red giants and supergiants[END_REF] (in contrast to 2x10 [START_REF] Nedoluha | Intensity-dependent circular polarization and circumstellar magnetic fields from the observation of SiO masers[END_REF] for the sun), taking the stellar radius to be 1 AU. Thus, at 5 AU distance from the stellar centre, a reversal of the magnetic field component due to the flux loop should cross the line of sight of an SiO maser in one to two hours. The speed of the maser spots away from the star does not contribute to this estimate -it is lower by at least two orders of magnitude [START_REF] Diamond | A Movie of a Star: Multiepoch Very Long Baseline Array Imaging of the SiO Masers toward the Mira Variable TX Cam[END_REF] . However, neither the size of the super-granules nor the stellar rotation profile are sufficiently well known to provide evidence for coronal loops crossing SiO masers, and the interaction with the pulsationdriven high-density wind certainly leads to an extremely complex physics. As in the solar system [START_REF] Burlaga | A magnetic cloud and a coronal mass ejection[END_REF] , coronal loops trigger mass ejections which subsequently form magnetic clouds (our case 4), which need about two days to cross a maser slab of 0.13 AU diameter (lower size limit from 86 GHz VLBI observations of Mira30), assuming a relative speed of 100 km/s (maser spots usually expand with a few 10 km/s, magnetic clouds in the solar system can reach a few 100 km/s). This is clearly too short to account for stable polarization features lasting for at least a month. It could, however, generate observable fluctuations with respect to the mean magnetic field.

The magnetic flux being coupled to the masing matter, the observed SiO maser lifetime must be shorter than the ambipolar diffusion timescale. The fragments forming as a consequence of Parker instabilities (our case 5, ref. 25) fulfil this requirement under the typical conditions in the extended atmosphere of AGB stars (T=1500 K, hydrogen density n H ~10 10 cm -3 , ionization fraction X i ~10 -6 , magnetic flux density B ~50 G).

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 are also expected, and should globally affect all maser spots if the latter commonly arise from fragments formed by this mechanism.

The rapid magnetic fluctuations reported here are first rare and second strongly localized in velocity space. These findings cast doubt on the scenarios (3) and ( 4) , coronal loops respectively magnetic clouds, expected to be ubiquitous in the stellar atmosphere. The latter finding, namely fluctuations restricted to a small velocity space, makes scenarios ( 2) and ( 5) unlikely because in such cases fluctuations of SiO features should be observed over a broad range of maser spot velocities. We therefore tend to favour scenario (6), addressing the possible presence of Jovian planets in the AGB atmosphere. Their magnetospheres and magnetotails were proposed [START_REF] Struck | Models of Planets and Brown Dwarfs in Mira Winds[END_REF] for being responsible for the Zeeman effect in SiO masers, and their evaporating Galilean moons may provide an interesting reservoir to fabricate the SiO molecules [START_REF] Struck-Marcell | SiO masers in late-type giant stars -Jovian planets engulfed[END_REF] . Jovian magnetospheres provide a natural explanation for periodic fluctuations of maser magnetization: the rotation period of e.g. Jupiter is 9.9 hours, and its magnetic axis is tilted by 11° with respect to its rotation axis. The magnetic field, if not frozen in the maser slab, can then be expected to have a precessing component, leading to observable fluctuations of the magnetic flux density. Thus, Jovian magnetospheres provide the most attractive explanation. Even if Jovian planets in the SiO maser zone would not be rare, we need SiO masers that cross their magnetospheres, explaining why the phenomenon is rare and both restricted in velocity space and requiring a relatively strong (~ 1 G) magnetic field. As a matter of fact, R Leo shows, at the same radial velocity as our p C fluctuation, a linear topology of spatially resolved SiO maser spots perpendicular to the archetypical ring topology of the maser shell (ref. 30), a feature lending support to our case (6). Boldly extrapolating the rich magnetic phenomena observed in the solar system to the AGB atmospheres (where the situation is far more complex due to the high-density wind) and based on the present results, we conclude that extended and densely sampled polarization monitoring of SiO masers should satisfactorily confirm which of the scenarios (1) to ( 6) is predominantly at work. The case of Jovian magnetospheres will require a spatially fully resolved polarization monitoring, a difficult undertaking motivated by the rapidly increasing number of Jovian-type planets detected around main-sequence stars.

Methods summary

The IRAM 30m telescope at Pico Veleta is equipped with dual-polarization receivers.

The pair operating at 3mm wavelength was tuned to the v=1, J=2-1 transition of SiO at 86.243 GHz, using the same local oscillator reference. The signals were analyzed with XPOL [START_REF] Thum | XPOL -the correlation polarimeter at the IRAM 30m telescope[END_REF] , a correlation spectrometer enhanced with cross-correlation products for the signals from the orthogonally polarized receivers, at 39.0625 kHz channel spacing. 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 (ref. 31). 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 the Stokes V parameter (positive for a RHC polarized signal, according to IAU convention). The polarized calibration load fills the receiver beam, whereas the SiO masers are unresolved. The cross-correlation products therefore remain sensitive to the different phase apodization across the receivers´ aperture when an unresolved source is observed. The corresponding phase error results 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 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 contribution of the telescope and the receiver cabin optics being negligible (-0.03 ± 0.12 % on the optical axis). Due to the irregular sampling, a determination of the spectral power density (SPD) of the time series using Fourier transform techniques yields poor results, and preference was given to the SPD estimate provided by the Lomb technique [START_REF] Press | Numerical Recipes in C[END_REF][START_REF] Lomb | Least-squares frequency analysis of unequally spaced data[END_REF] , which only uses the measured data without any prior interpolation to a regular sampling function. Our detection of quasiperiodic fluctuations of p C passed three critical tests. First, we note that the Lomb periodogram of Stokes U, after subtraction of the stationary linear polarization feature, does not show any periodic signal (Figs. 3b,d). 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 the false-alarm probability given in Table 1 and Fig. 3. Third, because a restricted number of random samples cannot be strictly Gaussian, a Monte-Carlo simulation, with random samples contemporaneous with the observed features, shows, for 100.000 runs, that 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 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. by Gaussian noise with a probability given by the labes ("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.

Table 1 Observational Results

V Cam R Leo

Phase of optical lightcurve [START_REF] Herwig | Evolution of Asymptotic Giant Branch Stars[END_REF] 0.08 0.10

Oscillation period of circular polarization

(5.4±0.1) h (6.3 ± 0.3) h

False-alarm probability [START_REF] Habing | Circumstellar envelopes and Asymptotic Giant Branch stars[END_REF] % 16 % radial velocity of maser spots [START_REF] Struck | Models of Planets and Brown Dwarfs in Mira Winds[END_REF] 7.5 km/s 4.4 km/s

Mean circular polarization -0.3 % +2.0 % Mean magnetic flux density in maser [START_REF] Struck-Marcell | SiO masers in late-type giant stars -Jovian planets engulfed[END_REF] ~ 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 [START_REF] Struck-Marcell | SiO masers in late-type giant stars -Jovian planets engulfed[END_REF] ~ 1 G ~ 1 G [START_REF] Herwig | Evolution of Asymptotic Giant Branch Stars[END_REF] With respect to the maximum of the optical lightcurve, [START_REF] Habing | Circumstellar envelopes and Asymptotic Giant Branch stars[END_REF] with respect to the null hypothesis (data are Gaussian noise) , [START_REF] Struck | Models of Planets and Brown Dwarfs in Mira Winds[END_REF] with respect to the local standard of rest , [START_REF] Struck-Marcell | SiO masers in late-type giant stars -Jovian planets engulfed[END_REF] order of magnitude estimate based on the Zeeman hypothesis, and for a linewidth of 1 km/s (ref. 7, 12).

1

  the atmosphere of two Mira stars H.W. Wiesemeyer 1* , C. Thum 1 , A. Baudry 2 , F. Herpin 2 Institut de Radioastronomie Millimétrique, 300 rue de la piscine, 38406 Saint Martin d'Hères, France 2 Université de Bordeaux, Laboratoire d'Astrophysique de Bordeaux, 33000 Bordeaux, France & CNRS/INSU, UMR 5804, BP 89, 33270 Floirac, France * Present address: IRAM Granada, Spain 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 1 . During this phase, where the stars appear in the Hertzsprung-Russell-Diagram in the upper asymptotic giant branch (AGB), they loose, due to a wind driven by pulsations, at least half of their mass, which forms a circumstellar envelope

8 , 9 ,

 89 10 of SiO masers were neither polarimetric nor critically sampled to detect such intraday magnetic fluctuations. Because statistically significant fluctuations are seemingly rare and localized in the extended atmosphere, they are expected to be due to a variety of phenomena. Coronal flux loops, magnetic clouds or Jovian magnetospheres 3 provide suitable explanations. Our study opens the way to future observations combining intensive full polarization monitoring of SiO masers, sampling at least once per hour, with high spatial resolution. This will ultimately allow us to distinguish between the proposed scenarios and to investigate the fate of inner planetary systems around solar-type stars entering their AGB phase.

Figure 1 |

 1 Figure 1 | Spectra of total power (Stokes I, i.e. polarized plus unpolarized) flux density and fractional circular polarization. Grey-shaded histogram: Stokes 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. Top (a): Mira star V Cam. Bottom (b): Mira star R Leo.

Figure 2 |

 2 Figure 2 | Time series of polarization measurements. From top to bottom: time series of 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 (left, a) and R Leo at v lsr = 4.4 km/s (right, b).

Figure 3 |

 3 Figure 3 | Lomb periodograms of the polarization time series. Estimate of the spectral power density of the polarization time series by least-square fits of its harmonic contents (Lomb periodogram 32,34 ), normalized by the variance of the samples. (a) Lomb periodogram of the fractional circular polarization p C of V Cam at v lsr = 7.5 km/s , (b) same for the residuals of Stokes U (normalized by Stokes I, after subtraction of the stationary linear polarization). (c) and (d): as (a) respectively (b) but for R Leo at v lsr =4.4 km/s. The horizontal dashed lines indicate the spectral powerdensity at which a periodic signal can be mimicked

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.

Author Information Correspondence and requests for materials should be addressed to H.W.

(wiesemeyer@iram.es).