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Château Rouge: a "Little Africa" in Paris?

The users and usages of a migrant commercial centrality

Marie Chabrol

The Château Rouge area of Paris is regarded as an "African neighbourhood". But what is the reality behind this image? Marie Chabrol shows that it has less to do with residents' backgrounds than with the commercial uses of the area, which put this district at the heart of large-scale mobilities.

Paris, like any major metropolis, has different neighbourhoods whose images are built and rebuilt around business activities linked to immigration. The Château Rouge area -situated in the shadow of Sacré-Coeur at the foot of the eastern slopes of Montmartre, in the 18 th arrondissement (city district) 1 of Paris -is often viewed as an "African neighbourhood". Although there are many foreign citizens from Sub-Saharan Africa and North Africa among its inhabitants (as in other areas of northern and eastern Paris 2 ), their presence alone does not explain the extraordinary numbers of Africans that frequent this neighbourhood. Rather, it is the number of specialist shops that creates an "African centrality" at the metropolitan level: the specificity of the products sold polarises the consumption practices of people of African or Caribbean origin, the vast majority of whom do not live here but give the area its image. Who are these "Africans of Château Rouge"? And how does this "migrant" or "minority" 3 commercial centrality, so typical of major world cities, function?

A detailed study combining observations and a questionnaire-based survey4 shows the diversity of users and practices in Château Rouge and reveals urban dynamics that are not immediately obvious. The extraordinary density of business activities in the neighbourhood masks large-scale daily mobility flows that connect it to other residential and commercial spaces, and which extend beyond the metropolitan area. Accordingly, a study of these mobilities is necessary in order to understand how neighbourhoods that are all too often apprehended only at local level actually function -especially when other phenomena, such as residential gentrification, come into play and create tensions with commercial activities. In all, 34.6% of the population of the Goutte d'Or neighbourhood are non-French, compared with 19.7% for the 18 th arrondissement as a whole and 13.3% for the city of Paris. For purposes of comparison, 25.3% of the population of Belleville (a neighbourhood with a similar population profile in the east of Paris) are non-French (Insee 2006).

3 "Migrant centrality" is a fairly neutral term, used in France in the 1980s and 1990s with regard to areas such as La Goutte d'Or, Belleville or the 13 th arrondissement of Paris [START_REF] Toubon | Centralité immigrée, le quartier de la Goutte d'Or[END_REF][START_REF] Guillon | Le Triangle de Choisy. Un quartier chinois à Paris[END_REF]. "Minority centrality", an expression used by the anthropologist Anne Raulin, has a broader meaning, as it applies to all minority groups (ethnic or sexual) compared with a given majority. In this context, "ethnic minority" refers as much to specific geographical origins (North African, Sub-Saharan, Caribbean, etc.) as to specific religious practices (e.g. halal or kosher meat) [START_REF] Raulin | Espaces marchands et concentrations urbaines minoritaires. La petite Asie de Paris[END_REF]).

An African commercial centrality in Paris

Since the 1990s, Château Rouge has been known for the specificity of the products that are sold there by traders from France, North Africa, China, Sub-Saharan Africa and the Indian subcontinent. The retail offer present (over a hundred shops) is both abundant and diverse, with products ranging from fresh and processed foods to textiles, clothing, cosmetics and cultural products (such as CDs, DVDs and religious items). The area is also home to numerous bars and restaurants, craftspeople (tailors, for example) and various service activities (telephony, freight agencies, hairdressers, etc.). The many posters plastered on walls and street furniture, advertising concerts, dances ("Senegalese night", "Congolese night") or evangelical preaching in the suburbs (very rarely in Paris and never in Château Rouge itself) testify to the centrality of the neighbourhood in terms of social networks. 5 New shops are regularly opened by entrepreneurs who seek to set up their business in this particular locality and niche, despite the policy of diversification of business activities that has been implemented by the authorities for many years now: the city council has actively sought to reintroduce local, neighbourhood stores in order to make the area a true "living space" for its residents and not just a place frequented by users from outside the district. 6 Nonetheless, Château Rouge retains its image as an "African" district, despite a growing disconnect between rapid gentrification of housing stock on the one hand and the continuity of these business activities, whose polarising effect extends well beyond the local neighbourhood, on the other.

Most users do not live in the neighbourhood

Of the respondents who participated in the questionnaire-based survey we conducted, 67% said they did not live in the neighbourhood. These non-residents share certain characteristics: over 70% of them were born outside mainland France, of which half in Sub-Saharan Africa; 7 men outnumber women; the most populous age group is 30-50 years; and a high proportion of respondents are in employment (70%). Finally, most of them (68%) use public transport to get to Château Rouge. Another interesting finding is that 87% of them have never lived in the neighbourhood.

A study of respondents' residential locations reveals five distinct geographical areas that illustrate the influence of the centrality of Château Rouge at different scales: first, other parts of the 18 th arrondissement (21%); second, other arrondissements of Paris (18%); third, the rest of the Îlede-France (Greater Paris) region, excluding the city of Paris (47%); fourth, other regions of France (11%); and, lastly, outside France (3%). The most populous group is the rest of the Île-de-France region (excluding the city of Paris). The presence of numerous bus and metro lines and the area's proximity to the Gare du Nord and the Gare de l'Est railway stations play a key role in these mobilities, as many users reside in inner-ring suburbs. However, the proportion of respondents that live in outer-ring suburbs or from outside the Paris region is not insignificant and adds a specific dimension to this centrality. Depending on one's starting point -the neighbouring Barbès district, a village in the outer eastern suburbs or a département in western France -a journey to Château Rouge does not have the same significance, bearing in mind the distances and costs involved. This objective guides different forms of public action that aim to regulate and diversify the retail offer of "single activity" areas in Paris. In Château Rouge, such retail-related action is one of the three key aspects of the urban renewal project under way in the neighbourhood since 2002 (the other two aspects being housing and public spaces).

7

In addition to French overseas territories and a number of Latin American, Asian and European countries, this included 24 African nations: Algeria, Angola, Benin, Cameroon, Cape Verde, Central African Republic, Chad, Comoros, Congo, Côte d'Ivoire, Democratic Republic of Congo, Egypt, Gambia, Guinea, Guinea-Bissau, Kenya, Madagascar, Mali, Mauritania, Morocco, Nigeria, Senegal, Togo, Tunisia.

Food retail at the heart of a diverse range of usages

Although going to bars and restaurants, meeting friends or simply enjoying being in an African atmosphere contribute significantly the attractiveness of the area, it is above all the various food stores that explain the area's popularity. Despite this, 80% of respondents reported that they do a large proportion of their food shopping elsewhere (mainly in hypermarkets near their place of residence or on local markets) and come to Château Rouge to buy certain specific products only. Patronage of a commercial centrality such as Château Rouge is therefore not exclusive, but instead functions in connection with other retail spaces.

Food stores in Château Rouge: an attractiveness factor at different levels © Marie [START_REF] Chabrol | De nouvelles formes de gentrification ? Dynamiques résidentielles et commerciales à Château Rouge[END_REF] In order of importance, the arguments given to justify food purchases in Château Rouge are the specificity of the products, the low prices, the variety of products on offer, the quality and freshness of the products, and finally geographical proximity. The specificity of products was the key reason cited above all by respondents from the outer suburbs and from outside the Paris region, in towns and villages where this kind of retail offer simply does not exist, as well as by respondents living abroad, who are often present in Paris for some other reason (e.g. family, holidays, work). These three groups of users, who travel the greatest distances, are also those whose visits are, unsurprisingly, the least frequent. They typically make the trip to Château Rouge a few times a month, once a month or once every two to three months in order to do a bulk shop of African or tropical products. The populations concerned are almost exclusively of Sub-Saharan or Caribbean origin; they come to buy products for a month's worth of meals or, alternatively, to prepare for a party or a special meal. Most of these people are in employment, with a significant proportion of executive and mid-level occupations.

Their profile contrasts with that of "local" users, that is to say users who live in the 18 th arrondissement, elsewhere in Paris or in the inner suburbs. These users tend to come to the area every week, or even every day, to shop for smaller quantities of meat, fruit or vegetables. The reasons for their visits to Château Rouge are proximity and force of habit, the low prices and, lastly, the specificity of the products. Parisian respondents also come to meet friends, while low prices are the main reason cited by respondents from the suburbs. Of these very regular users, many are not in employment (retired, unemployed or housewives) and those who are in active employment tend to be manual or office workers.

One final user profile, independent of distance, also emerges: men aged 30 to 50, from Côte d'Ivoire, Congo and the Democratic Republic of Congo, who come not so much for the shops as to meet friends and enjoy the bars, restaurants and general atmosphere of the area. They belong to various socio-professional categories and come from each of the five geographical areas defined above.

A centrality at metropolitan level

The practices of these different user types have a major influence on the rhythm of daily life in Château Rouge. While on weekday mornings the area is relatively quiet and frequented by consumers wishing to shop with maximum efficiency, it is much more lively in the afternoons and even more so at weekends. On Saturday afternoon, crowds invade the busiest shopping streets; consumers, accompanied by children, friends and relatives, take their time in strolling along the streets, shopping and stopping to eat. The area is also a meeting place, and the practices that are deployed here resemble those that can be observed in other commercial centralities. These consumption practices play a role in maintaining the image of an "African neighbourhood"; however, beyond this clear-cut image and these practices of a shared space, there is little in common between, say, an embassy employee from Paris buying ingredients for a special dinner, a Haitian family from the outer suburbs doing a month's worth of shopping, or a family of Congolese origin that lives and works in a village in the Loire Valley, but which comes to Château Rouge just a few weekends a year to meet up with compatriots.

Furthermore, Château Rouge works in conjunction with other retail areas: not just small businesses in residential areas, but also other working-class or migrant commercial centralities. For example, Château d'Eau, a little over a mile to the south in the 10 th arrondissement of Paris, is another area of African centrality specialised in hairdressing activities, and so complements rather than competes with Château Rouge, with which it is well connected by public transport. Moreover, some retailers choose to invest in both areas. Even closer still, within very easy walking distance, the shops of Barbès (specialised in telephony, clothing, jewellery and accessories) are often points of call before or after food shopping in Château Rouge. At the metropolitan level, these retail dynamics reinforce one other, mainly because they are part of systems of places that are shared to a greater or lesser extent by a large number of users. Observing Château Rouge in terms of retail practices and identifying the influence of its centrality and its complementarity with other areas also makes it easier to appreciate of one of the current issues at play in this neighbourhood: the specialisation of its businesses and the considerable customer flows that they generate come into conflict with a trend of rapid residential gentrification, partly encouraged by the city council, whereby households of French or European origin, generally relatively well off, have moved into the area, leading to a demand for a little less animation and a few more European-style local shops. Understanding and tackling this issue, common to other areas with migrant centralities, means taking account of the different scales of practices present, and not just focusing on the same old local-level problems expressed in terms of social mix or public order.

1

  The 18 th arrondissement, in the north of Paris, covers (from west to east) the areas of Montmartre (including the northern part of Pigalle), Clignancourt, La Goutte d'Or (including Château Rouge and Barbès) and La Chapelle.

  2

5

  See sociologist Baptiste Coulmont's blog for his work on the collection and analysis of these posters and flyers (in French): http://coulmont.com/blog/2008/09/07/eglises-africaines.

  6

  Château d'Eau in the 10 th arrondissement of Paris: a commecial centrality specialised in hairdressing and cosmetics for black skin © Marie Chabrol, 2011.

As part of PhD fieldwork, repeated observations were made between 2006 and 2011, complemented in May 2009 by a questionnaire-based study among a sample of 600 individuals interviewed in the street (between 10 a.m. and 6 p.m. on a Thursday and a Saturday) with the assistance of students from the Université Paris-1 Panthéon-Sorbonne. The results of this study were published in[START_REF] Chabrol | De nouvelles formes de gentrification ? Dynamiques résidentielles et commerciales à Château Rouge[END_REF]. This study was inspired by exploratory research into non-residential spatial practices (Lévy and Dureau 2005).


A Plume

Numerical method overview

Simulation of the microwave propagation through the plume of a Hall thruster integrated on small spacecraft N. de Mejanes, 1 R. Pascaud, 1 V. Mazières, 1 A. Rossi, 2 V. Laquerbe, 2 L. Garrigues, [START_REF] Buchen | Small satellite market observations[END_REF] and O. Pascal 3 1) Université de Toulouse, ISAE-SUPAERO, 31055 Toulouse, France 2) Centre National d'Etudes Spatiales CNES, 31400 Toulouse, France 3) LAPLACE, Université de Toulouse, CNRS, 31062 Toulouse Cedex 9, France (*Electronic mail: romain.pascaud@isae-supaero.fr) (Dated: 1 July 2022) More and more CubeSats are being launched. On these small platforms, subsystems such as propulsion and communication ones have to coexist. This paper focuses on the electromagnetic interaction between these two critical subsystems. Hence, a numerical multi-physics method is proposed in order to quantify the perturbation caused by an electric thruster's plume on the antenna of a CubeSat type spacecraft. A plume simulation model has been coupled with an electromagnetic simulation software. As an example, the farfield radiation patterns and radioelectric characteristics of a 436 MHz dipole are presented when located near a Hall thruster's plume on a 6U type platform. Changes in radiation patterns are observed in the presence of the plume for the dipole antenna. This versatile method makes it possible to represent the microwave propagation through a plume for various antennas, thruster's plumes, or relative positions.

I. INTRODUCTION

Conquest of space was for a long time perceived as being only accessible to a few states, public institutions or major research centers. However, for more than twenty years, we have witnessed an evolution in the space sector which has brought us today to what is often called the "New Space" [START_REF] Sweeting | Modern small satellites-changing the economics of space[END_REF] . Not a renewal, the "New Space" aims to make space accessible to new private actors, paving the way for the development of a multitude of commercial applications [START_REF] Villela | Towards the thousandth cubesat: A statistical overview[END_REF][START_REF] Buchen | Small satellite market observations[END_REF] . To achieve this, this industry relies on the time-to-market and costs reduction that has been made possible through the emergence of a new standard, the CubeSat.

Developed in 1999 at California Polytechnic University and Stanford University, the CubeSat is a small platform originally designed for educational purposes [START_REF] Puig-Suari | Development of the standard cubesat deployer and a cubesat class picosatellite[END_REF] . The development of these projects had to meet a double constraint: be able to be carried out quickly in order to follow the university calendar with a limited budget. Miniaturization and simplicity thus prevailed. Nowadays CubeSat has become a category of small satellites made up of one or more standard cubes of 10 cm edge length, called unit "U". Today many CubeSats of different sizes (1U, 3U, 6U etc.) are launched each year 5 , not only by universities and research centers but also by companies that saw this low cost standard as a promising solution to adress new markets [START_REF] Denis | From new space to big space: How commercial space dream is becoming a reality[END_REF][START_REF] Heidt | Cubesat: A new generation of picosatellite for education and industry lowcost space experimentation[END_REF] .

Of course, there is no question of competition with conventional large satellites. The latter make it possible to provide information with very high precision thanks to large equipment. However, single CubeSats or as a constellation for applications such as earth observation [START_REF] Selva | A survey and assessment of the capabilities of cubesats for earth observation[END_REF] , disaster prevention [START_REF] Santilli | Cubesat constellations for disaster management in remote areas[END_REF] , Internet of Things (IoT) connectivity [START_REF] Akyildiz | The internet of space things/cubesats[END_REF] , or technology demonstrator [START_REF] Viquerat | Functional and qualification testing of the inflatesail technology demonstrator[END_REF] represent a cheaper solution than conventional satellites.

However, having combined all the subsystems into a small and compact platform creates new constraints [START_REF] Poghosyan | Cubesat evolution: Analyzing cubesat capabilities for conducting science missions[END_REF] , especially for the communication and propulsion subsystems. First, all of the satellite's subsystems have to deal with lower on-board power. Indeed, the volume being constrained, the solar pan-els and the power unit can only deliver a power of around a hundred watts [START_REF] Arnold | Energy budgeting for cubesats with an integrated FPGA[END_REF] unlike large satellites which could deliver kilowatts.

For the antennas, the low available power reduces the radiated power. In addition, the antenna system is constrained by the size of the satellite [START_REF] Gao | Antennas for modern small satellites[END_REF][START_REF] Chahat | Advanced cubesat antennas for deep space and earth science missions: A review[END_REF] . Indeed, for a miniature satellite, the available surface of the structure is considerably reduced, so using planar antennas will be at the expense of payload or solar panels. Of course, deployable antenna solutions exist [START_REF] Costantine | Cubesat deployable antenna using bistable composite tape-springs[END_REF][START_REF] Fragnier | Compact antennas for nano-and micro-satellites: development and future antenna needs at CNES[END_REF] but are sometimes constrained by a mechanical deployment system which can be difficult to set up, especially in a low cost production context. Regarding the propulsion system, the electric solution saves space and weight to be carried on board compared to chemical propulsion [START_REF] Levchenko | Space micropropulsion systems for cubesats and small satellites: From proximate targets to furthermost frontiers[END_REF] . There is a plethora of different electric propulsion system technologies [START_REF] Tummala | An overview of cube-satellite propulsion technologies and trends[END_REF][START_REF] Wright | Electric micropropulsion systems[END_REF][START_REF] Krejci | Space propulsion technology for small spacecraft[END_REF][START_REF] Sheehan | New low-power plasma thruster for nanosatellites[END_REF] : arcjet, ion or Hall thruster to name but a few. All these solutions are based on the same principle: electrical power is applied to a gas, a solid, or a liquid to create and accelerate a plasma in order to provide thrust. Although electric propulsion has an excellent specific impulse compared to chemical propulsion, electric propulsion also has its drawbacks [START_REF] Lev | The technological and commercial expansion of electric propulsion[END_REF][START_REF] Holste | Ion thrusters for electric propulsion: Scientific issues developing a niche technology into a game changer[END_REF] . This type of propulsion is based on its ability to convert electrical energy into kinetic energy; as a result, its performance depends on the power available on board. Also we can add that miniature propulsion technologies developed so far have reduced efficiency and lifetime due to the less favorable volume-to-surface ratio [START_REF] Levchenko | Space micropropulsion systems for cubesats and small satellites: From proximate targets to furthermost frontiers[END_REF] .

Another constraint linked to the reduction of the available volume within a CubeSat is the electromagnetic coupling due to the proximity of the various subsystems of the platform. Indeed, on large platforms of several meters, it was possible to physically separate the subsystems in order to avoid electromagnetic compatibility problems.

The problem of electromagnetic coupling on a reduced platform has already been studied in the context of the CubeStar satellite [START_REF] Tresvig | Investigating the coupling effects between cubestar's communication antenna and deployable payload sensors[END_REF] where deployable probes influenced the radiation patterns of the antenna system. Depending on the relative position of these two elements, the deployable probes could act as a passive reflector causing distortion of the antenna gain. To this day, one point has still not been addressed: the electromagnetic coupling that may exist between the plasma thruster plume and the communication system integrated on a small platform. The microwave communication system relies on antennas that can operate between one hundred megahertz and several tens of gigahertz to ensure positioning, telemetry or telecontrol. When the thruster in ON, a plasma called "plume" is expanded from the thruster's exit. The plume is an electromagnetic complex medium that may modify the closed environment of the antennas and disrupts their operation. First, the plume is a complex media where physical processes can lead to electromagnetic radiation in the bandwidth of the receivers [START_REF] Beiting | Time-domain characteristics of 0.5 -8 GHz pulsed emission from Hall thrusters[END_REF] . From the point of view of the receiving antenna, theses radiations behave as a noise source. Secondly, the plume can modify the microwave propagation channel [START_REF] Bandinelli | EM modelling of RF propagation through plasma plumes[END_REF] . Indeed, its permittivity is not that of vacuum, so the environment in the vicinity of the antenna can be modified, resulting in a change in its gain pattern and thus its received signal power. This second particular point, illustrated in Figure 1, is the subject of this work.

The electromagnetic compatibility between communication system and plume's thruster has already been studied but for large satellites. Sarri et al. measured the propagation perturbation of an electromagnetic wave through a plasma plume [START_REF] Sarri | A customized measurement system for the assessment of electromagnetic interference caused by plasma plumes[END_REF] . It has been concluded that the localized plasma flow modifies the electromagnetic wave in amplitude and phase in the 3-7 GHz band due to the permittivity gradient and the plasma conductivity. Bandinelli et al. measured at higher frequencies the electromagnetic interference caused by the plasma plume of an ion thruster firing through the beam of a reflector antenna in the 12 -18 GHz band (Ku) [START_REF] Bandinelli | EM modelling of RF propagation through plasma plumes[END_REF] . No change of the signal by the plume was observed in this frequency band. Zhang et al. measured the influence of the ion thruster plume by comparing the ON and OFF |S 21 | parameter amplitude in the 1 -5 GHz range [START_REF] Zhang | A test method for electromagnetic interference caused by plasma plumes[END_REF] . In this frequency band, an amplitude attenuation was observed. It was suggested by the authors that "the satellite is designed to avoid the communication electromagnetic wave propagating cross the plasma plumes". For CubeSats this is no longer possible as illustrated in Figure 2, where we note the much smaller distance between the thrusters and the antennas.

Electromagnetic compatibility issues thus become of major interest for small spacecraft. One of the difficulties encountered when investigating these effects comes from the multi-scale and multi-physics analysis required. In this work, a numerical multi-physics method is presented to investigate the propagation of a microwave signal through the plume of an electric thruster integrated on a small spacecraft. Another encountered difficulty comes from the plethora of electric thruster that can be employed for small satellites, each one possesses its own physics and thus its specific plasma plume. We here focus on the plasma plume of a Hall thruster, which is one of the primary candidates for small satellite applications [START_REF] Levchenko | Space micropropulsion systems for cubesats and small satellites: From proximate targets to furthermost frontiers[END_REF] , but the versatile method developped in this work can easily be applied to other kind of thrusters. This paper is organized as follows. In the second Section we present an overview of the multi-physics simulation method. In Sections III and IV, the plasma discretization method is detailed and applied for a test case, a UHF-band dipole antenna mounted on a 6U CubeSat (10 × 20 × 30 cm platform). In Section V radio electric and farfield radiation characteristics are presented for this test case and for other thruster's operating conditions (i.e., thruster's position and power).

II. NUMERICAL METHOD

The proposed method follows the steps shown in Figure 3. The input data required to treat this problem is the electron density distribution in the plume. The thruster's plume is then represented by its local permittivity and conductivity given by the Drude model [START_REF] Lieberman | Principles of plasma discharges and materials processing[END_REF] which is a function of the electron density. The numerical analysis of the integrated antennas with and without the plume is finally performed with an electromagnetic solver. In the following we detail each of these steps.

As mentioned earlier, there are many different technologies of electric thrusters [START_REF] Vavilov | Review of electric thrusters with low consumption power for corrective propulsion system of small space vehicles[END_REF] . Each of them creates and accelerates a plasma to provide thrust. However, depending on the engine, the plasma is not exactly the same in terms of composition, density, temperature, and shape [START_REF] Blanchet | 2D mapping of vacuum arc thruster plume plasma parameters[END_REF][START_REF] Kronhaus | Investigation of physical processes in CAMILA Hall thruster using electrical probes[END_REF] . Furthermore, some of these thrusters are operated in a pulsed regime, leading to a pulsed behavior of the plasma plume, making the investigation of its influence on the radio communication complex. Even for the thrusters operated in a stationary regime, they have been shown to exhibit instabilities and intrinsic plasma oscillations, leading to a temporal modulation of the plasma density, such as for example the instabilities of Hall thruster [START_REF] Mazouffre | Electric propulsion for satellites and spacecraft: established technologies and novel approaches[END_REF] , the electron instabilities in Magnetoplasmadynamic thruster [START_REF] Kirdyashev | Spectra of electron oscillations in magnetoplasmadynamic thruster[END_REF] and the "onset phenomena" [START_REF] Paganucci | MHD instabilities in magneto-plasma-dynamic thrusters[END_REF] , or the poorly understood instabilities in microwave discharge ion thrusters [START_REF] Nakagawa | Performance characterization of a miniature microwave discharge ion thruster operated with water[END_REF] . In order to investigate the effect of the plasma plume on the communication link in simulation, a spatio-temporal description of the plasma plume would be required. However, obtaining such desciption, either experimentally or numerically, is complicated. Hence, as a first approximation, in this paper the plasma plume is assumed to be stationary. This will give us information about the deterioration of the communication link due to a stationary plasma plume, but it is important to emphasize that the temporal modulations of the plasma density that are present in electric thrusters may deteriorate even more the communication link [START_REF] Yao | Experimental study on correlation between amplitude and phase of electromagnetic wave affected by time-varying plasma by amplitude-modulated radio frequency plasma generator[END_REF] .

In this work, we have chosen to study the plume of a miniature Hall thruster. Nevertheless, it should be noted that the following work can be carried out for any type of thruster as long as its that the electron density distribution within the plume is known.

In this work, the JET2D simulation code developed by Garrigues et al. [START_REF] Garrigues | Modeling of the plasma jet of a stationary plasma thruster[END_REF] was used to obtain the electron density distribution within the plume. JET2D is a software based on a plume of a Hall type thruster with xenon gas. It is a hybrid model where ions are processed in a particular way by the Montecarlo particle in cell model and the electrons in a fluid way by the local electron density linked to the local potential by Boltzman's law. The electric field is calculated at each time step by assuming that the electrons respond immediately to electric potential variation. The potential is then deduced from Boltzman's law considering a quasineutral plasma where the electron density is assumed to be equal to the ionic density. JET2D performs several diagnostics on the plume such as the spatial distribution of electron density or the angular distribution of the ion current density.

The simulation conditions are function of a certain number of parameters concerning the choice of the dimensions of the simulation domain, the mesh, and the initial distributions of un-ionized neutrals and ions. JET2D was originally developed and experimentally validated for standard Hall type SPT100 thrusters used on satellites larger than those of our interest. But thanks to the adjustable input parameters like gas flow rate, specific impulse, discharge current, ion current to total current ratio, and ejection angle we can approach the conditions of a representative plume of a miniature Hall thruster.

Although there are measurements in the literature on plumes from engines such as the SPT100, very little data has been published on plumes from small thrusters. However, Hallouin and Mazouffre have published angular distribution measurements of ion current density for a 100 W class Hall thruster which is suitable for propulsion for miniaturized satellites [START_REF] Hallouin | Far-field plume characterization of a 100-W class Hall thruster[END_REF] . Hence, the input parameters of JET2D, as the angular distribution, have been adapted so that the simulated current density is the closest to the current density of Ref. [START_REF] Hallouin | Far-field plume characterization of a 100-W class Hall thruster[END_REF] . Specific impulse I sp , gas flow rate ṁXe and discharge current I d have been fixed as in Ref. [START_REF] Hallouin | Far-field plume characterization of a 100-W class Hall thruster[END_REF] , namely I sp = 1740 s, ṁXe = 0.50 mg/s, and I d = 0.44 A. Figure 4 shows the comparison between the current density simulated by JET2D and measured in Ref. [START_REF] Hallouin | Far-field plume characterization of a 100-W class Hall thruster[END_REF] .

A reasonable agreement between the two curves can be observed, in particular for the intermediate angles although the current density simulated by JET2D is underestimated. On the wings (i.e., for |θ | > 50 • ), the simulated ion current is overestimated compared to the measurement. However, for these angles the electron density is very low and the permittivity remains close to that of vacuum in the microwave range. It thus does not have an impact on the electromagnetic model. Likewise for low angles (i.e., θ closed to 0 • ), the simulated current density is overestimated which is potentially more problematic because it is a zone of high density and therefore strongly contributes to the permittivity of the plasma. Although the agreement is not perfect over the whole angular distribution, JET2D simulation remains a fairly good estimate. Figure 5 shows the distribution in a half plane of the electron density obtained from JET2D for the parameters described previsously. The origin x = 0 cm and y = 0 cm represents the center of the half Hall thruster. The electron density is maximum at the thruster exit and decreases along the x and y direc- tions. On the simulation domain, the electron density varies between n e = 6.86 × 10 10 cm -3 close to the thruster exit and n e = 5.30 × 10 5 cm -3 far from it. The electron density gradient is strong especially in the y direction, indeed for y = 0 cm and y = 20 cm in the plane of the thruster exit, the electron density is respectively equal to n e = 6.86 × 10 10 cm -3 and n e = 2.6 × 10 7 cm -3 , i.e., a decrease higher than three orders of magnitude.

B. Drude model

From the microwave point of view, the Hall thruster plume simulated with JET2D is represented as an isotropic dispersive medium with a complex equivalent local relative permittivity given by the Drude model [START_REF] Lieberman | Principles of plasma discharges and materials processing[END_REF] :

ε p (⃗ r) = 1 - ω 2 p,e (⃗ r) ω(ω -jν e-n )
with ω the angular frequency of the wave that interacts with the plasma, ν e-n the electron-neutral collision frequency, and ω p,e (⃗ r) = n e (⃗ r)e 2 m e ε 0 the angular plasma frequency where n e (⃗ r) is the local electron density, e the elementary charge, m e the electron mass, and ε 0 the vacuum permittivity. Only the response of the electrons to the microwave electromagnetic field is taken into account since in the high frequency range, ions are too heavy to respond to the instantaneous electric field. Indeed, for a frequency of 436 MHz, and an ion density equal to n i = 6.86 × 10 10 cm -3 , that is to say the highest density within the plume, the ratio ω p,i ω , where ω p,i is the ion angular frequency, is equal to 0.01.

It should be noted that, although the Drude model allows the study of plasma/microwave interactions [START_REF] Bandinelli | EM modelling of RF propagation through plasma plumes[END_REF][START_REF] Laroussi | Numerical calculation of the reflection, absorption, and transmission of microwaves by a nonuniform plasma slab[END_REF][START_REF] Gürel | Interaction of electromagnetic wave and plasma slab with partially linear and sinusoidal electron density profile[END_REF][START_REF] Zhu | Influence of time-varying plasma sheath on the lock condition of phase-locked loop for tt&c carrier tracking system[END_REF][START_REF] Zhao | Effects of reentry plasma sheath on GPS patch antenna polarization property[END_REF][START_REF] Bai | Effects of reentry plasma sheath on the polarization properties of obliquely incident em waves[END_REF][START_REF] Song | Intra-pulse modulation of the linear frequency modulated pulse caused by time-varying plasma[END_REF][START_REF] Yang | Propagation of phase modulation signals in time-varying plasma[END_REF] , it does not allow the non-local effects introduced by the electron density gradient to be accounted for. Thus, the Drude model only describes collisional energy transfers between the wave and the plasma without dealing with the resonance absorption that may occur in a weakly collisional inhomogeneous plasma around the critical plasma frequency. [START_REF] Niknam | Microwave ponderomotive action on the inhomogeneous collisionless and collisional plasmas[END_REF][START_REF] Kamal-Al-Hassan | Dynamic of ion density perturbations observed in a microwave-plasma interaction[END_REF][START_REF] Lee | Resonance absorption in a microwave plasma interaction[END_REF] A more accurate approach have to be developed to take these effects into account. Nevertheless, the Drude model remains a fairly good first approximation.

Concerning the Drude relative permittivity, its real part represents the dielectric constant and the imaginary part energy absorption via the electron-neutral collision frequency. For a plume, the collision frequency which is proportional to the neutral density, is very low compared to the frequency of the microwave signal since the neutral pressure is very low in space vacuum condition. For a neutral density on the order of 10 12 cm -3 and an electron temperature of 2 eV and, the electron-neutral collision frequency is around 3 × 10 5 s -1 . So the ratio between the collision frequency and the frequency of a microwave signal is very low and the plasma is therefore weakly collisional. The imaginary part is then negligible compared to the real part of the permittivity.

Figure 6 shows the distribution of the real part of the relative permittivity for several satellite applications frequencies namely 436 MHz, 1.2 GHz, and 2.15 GHz, for the electron density showed in Figure 5.

We can see a zone where the real part of the permittivity is negative (hatched area). In this area, electromagnetic wave becomes evanescent and therefore does not propagate. However, this area is quite small, being less than 20 cm long and less than 10 cm wide. We also notice a gradient in the dielectric constant and so in the index of refraction all along the plume.

C. Electromagnetic (EM) solver

Finally, in order to study the propagation of the microwave signal emitted or received by onboard antennas through the plume of a thruster integrated on a small satellite, we use a commercial electromagnetic numerical code that solves Maxwell's equations. More specifically, we use Ansys HFSS that is based on the finite element method (FEM) to solve It is well known that the dielectric permittivity of a plume is continuous and inhomogeneous, but it must be discretized in the solver to be simulated. Although the plume is already sampled in the JET2D model, it is necessary to choose a discretization scheme adapted to our electromagnetic interaction problem. Indeed, the discretization needed to represent the plume physics is not the same to represent the plume/wave interaction. In general, in the broader context of plasma wave interactions, what is done is that the inhomogeneous density profile is approximated in several homogeneous layers. There are different strategies to this aim. Firstly, one can make spatially distributed iso-layers, so that the distance between two consecutive layers is constant. For instance, Laroussi and Roth chose this strategy due to experimental considerations since twelve Langmuir probe points defined twelve plasma layers [START_REF] Laroussi | Numerical calculation of the reflection, absorption, and transmission of microwaves by a nonuniform plasma slab[END_REF] . Gürel and Öncü used a numerical criterion, as many iso-distributed layers were made until numerical convergence was achieved [START_REF] Gürel | Interaction of electromagnetic wave and plasma slab with partially linear and sinusoidal electron density profile[END_REF] . Zhu et al. made layers in such a way that the thickness of a layer is much smaller than the wavelength of the passing signal [START_REF] Zhu | Influence of time-varying plasma sheath on the lock condition of phase-locked loop for tt&c carrier tracking system[END_REF] . Another method of constructing plasma layers is based on a gradient criterion [START_REF] Zhao | Effects of reentry plasma sheath on GPS patch antenna polarization property[END_REF] . Here the density vari- It should be noted that in all these works no results of convergence have been given and no discussion on how to treat the boundary between what is considered to be plasma and vacuum is done.

In this work, particular attention is given to the plasma discretization, either on the method of discretization or on the choice of the plasma/vacuum boundary. This is presented in details in the following section for a test case, namely a UHF dipole mounted on a 6U CubeSat.

III. PLASMA DISCRETIZATION

In this section we present the plasma discretization and apply it to a test case as shown in Figure 7. The test case consists of a 436 MHz UHF half-wave dipole mounted on a 6U CubeSat (20 × 10 × 30 cm 3 platform) which is considered as a Perfect Electric Conductor (PEC). The thruster, and thus the plume, is located on the bottom side of the platform. Here, we have chosen to represent the CubeSat as a simplified structure in order to analyse the only interaction of the plume and the antenna and not the influence of the different elements integrated in the CubeSat such as solar panels. Note that Ansys HFSS allows us to model more complex structures if needed.

In contrast to previous discretization strategies based on the construction of layers, it was chosen to represent the plume by an assembly of toroids. Indeed, the electron density distribution obtained from JET2D consists of a 2D grid where in each grid cell the local electron density is given. Each grid cell in 2D is equivalent to a 3D toroidal object using axial symmetry. In each of these objects the local relative permittivity is fixed using the electron density of the cell in question and the Drude model. Thus, the spatial resolution of the plume simu-FIG. 8. Random Acces Memory (RAM) used and length of the simulated plume as a function of the limit permittivity ε max with two illustrations of the discretized plume for ε max = 0.3 and for ε max = 0.95. lation tool sets the spatial resolution of the discretized plume in the electromagnetic solver. Pratically, the axial and radial resolution ∆x, ∆y chosen in JET2D is a function of the wavelength λ of the signal such that ∆x, ∆y ≪ λ and with the aim of limiting the simulation time. In this test case where the frequency is 436 MHz, ∆x and ∆y are both equal to 1 cm so the ratio ∆x, y/λ = 0.0145. Note that a convergence analysis, not shown in this paper, have been performed to validate that such a discretization leads to a converged result for the electron density in JET2D.

Another parameter to set is the maximum limit of the relative permittivity, that is to say the limit between what we consider to be the plasma plume and vacuum. In other words, we need to specify a given threshold from which any element coming from JET2D will be simply represented by vacuum and not as plasma, in the electromagnetic solver. The objective is also to limit the computational time because the more elements to simulate, the longer it takes. But we still need to represent correctly the problem so that this choice does not influence the final results. Figure 8 shows the occupied Random Access Memory (RAM) by the EM solver and the length of the plume as a function of the maximum permittivity chosen for the test case shown in Figure 7. The size of the plume increases exponentially with the chosen limit. As a result the number of plume elements, the volume of the computational box for the EM simulation increase. Indeed, for a threshold of ε max = 0.4, the plume is made up of 95 toroids whereas for ε max = 0.97 the plume is made up of 1833 toroids.

In order to choose the best value for the threshold ε max , we performed a convergence study on the linear gain distribution of the test case, i.e., a dipole at 436 MHz. Figure 9 shows the normalized deviation ∆G norm of the gain G ε from a reference simulation G re f as a function of several values of maximum permittivity limit ε max , as described by:

∆G norm (%) = abs G ε -G re f G re f ,max × 100
The reference value G re f corresponds to a plume constructed considering ε max = 0.99. We observe a decrease in the maximum deviation as the permittivity get closer to the reference value. For ε max = 0.2, the maximum deviation ∆G max observed is 16.5% and for ε max = 0.95, ∆G max is 4.5%. These results show that the choice of the plasma vacuum limit is an important parameter to take into account.

Considering the computation time, the RAM used, and the convergence study, we defined ε max = 0.95 as permittivity limit value in the case of a 436 MHz dipole and for a Hall thruster under the operating conditions described above. Figure 10 shows the simulated and discretized plume when considering a limiting permittivity of 0.95 and a spatial resolution of 1 cm. The plume under these conditions is 71 cm long and 54 cm wide. The left side of the figure shows several permittivity contours. The permittivity ranges from -28 in the higher density area near the thruster exit plane to a maximum of 0.95. Beyond this, the medium is considered to be vacuum. In the area where the relative permittivity is negative, electromagnetic waves become evanescent and therefore do not propagate. The right-hand side of the figure shows the discretization scheme used to construct the torroids in the electromagnetic solver. The plume is the assembly of 1097 torroids of different relative permittivities.

IV. RESULTS AND DISCUSSION

In this section the simulated characteristics of the UHF halfwave dipole antenna described above are studied with the Hall thruster plume and the discretization conditions described before. First, we present the results obtained for the test case, then we apply the same methodology for other cases (i.e. several relative positions and thruster power) to highlight the versatility of the method.

A. Test case

In this subsection, the characteristics of the UHF half-wave dipole antenna are studied for the test case described before. The antenna is positioned at the center of the side face of a 6U CubeSat and the discretized plume is at the center of the bottom face as seen in Figure 11. Also the two main cut planes for the radiation patterns of this antenna namely φ = 0 • and 90 • are also shown in Figure 11.

Figure 12 shows the reflection coefficient |S 11 | in dB which is equal to the reflected power over the incident power. It also presents the radiation efficiency of the antenna, which is the ratio of the radiated power to the accepted power at the antenna terminals. These two quantities are compared on the 406 -466 MHz band when the thruster is OFF (simulation without the plume) and when the thruster is ON (simulation with the plume). We observe that the presence of the plume has little or no influence. We can add that for the OFF case, for an injected power of 1 W, the radiated power by the antenna is 1 W and for the ON case, it is 0.99 W. Indeed, in the Drude model, energy losses of the electromagnetic waves are only collisional. These losses are related to the imaginary part of the permittivity describing the plume which is extremely low due to the ratio between the collision frequency and the wave frequency ν (e-n) /ω.

Figure 13 shows the antenna gain radiation patterns for the OFF and ON conditions, for the φ = 0 • and φ = 90 • cutplanes as defined in Figure 11. For θ = 180 • (i.e. towards the plume), the gain is reduced by 1 dB. Since there is no sig- nificant losses in the plume, the observed gain reduction is associated with radiation pattern distortion. The plume behaves as a refracting medium that changes the antenna radiation patterns: decreasing radiation in some directions, increasing it in others.

In addition to causing changes in the gain pattern, the presence of the plume can lead to variations of the phase of the microwave signal. Figure 14 shows the phase delay difference ∆Φ of the electric field between the OFF and ON conditions as a function of θ , for the φ = 0 • and φ = 90 • cut-planes as defined in Figure 11. Only the phase difference in the main radiation directions are shown, i.e. in the angular ranges where the gain is greater than the maximum gain minus 10 dB. We observe a non-zero phase difference between the OFF and ON condition, with a maximum of -7 • for θ = 0 • . A change in the phase delay may affect the communication signals. Firstly, it may lead to wave depolarization between the transmitting and receiving antennas [START_REF] Bai | Effects of reentry plasma sheath on the polarization properties of obliquely incident em waves[END_REF] . Secondly, a modulation of the phase at the ignition of the thruster, or for non-stationary thrusters, may lead to error in the frequency measurement and so the rotation of complex modulation constellation that may influence the bit-error rate (BER) [START_REF] Song | Intra-pulse modulation of the linear frequency modulated pulse caused by time-varying plasma[END_REF][START_REF] Yang | Propagation of phase modulation signals in time-varying plasma[END_REF] . FIG. 14. Phase delay difference between the (ON) and (OFF) condition in the two cut planes φ = 0 • and φ = 90 • , the hatched area corresponds to the angular range where the gain is inferior to the maximum gain minus 10 dB.

B. Method versatility

As said before, this method can be used in a wide variety of cases, for example to study the effect of the position of the thruster on the farfield characteristics of an antenna. Figure 15 shows different cases of thruster position. The position of the thruster is where the discretized plume is located, namely in the middle of the three following faces: bottom, side, and top. The plume remains the same as the one described previously. The chosen antenna is still the same, i.e. a 436 MHz dipole, this time placed in the middle of the upper face of the 6U CubeSat. We have chosen to keep the antenna at a fixed position so that the influence of the platform (i.e. the 6U) on the radiation pattern of the antenna remains the same whatever the different cases.

For each of these cases, the farfield gain pattern in the two cut planes of the antenna, φ = 0 • , 90 • , is shown in Figure 16. Regardless of the relative position, the gain is affected by the presence of the plume. How the gain is modified depends on the relative position between the plume and the antenna. When the thruster is located below the antenna, there is a decrease in gain for θ = 180 • . When the thruster is placed on top, and the plasma is above the dipole, the area where the gain decreases corresponds to the area where the plume is located, namely for angles close to θ = 0 • . Finally, when the thruster, and thus the plume is located on the side with respect to the antenna, there is a very small change in the gain pattern in the φ = 0 • plane, since this plane does not cross the plasma. On the other hand in the φ = 90 • plane, we observe a modification of the gain pattern, with in particular a decrease of the gain in the area where the plume is located namely around θ = 0 • . For each case, the area where the gain is decreased corresponds to the range of angles where the plume is located with respect to the antenna. Since there is no loss, an increase in gain in other directions is observed. The maximum gain decrease in the area where the plume is located is equal to 1 dB for each of these cases.

In addition to study the effect of propulsion and antenna positions, we can study the plume disturbance for different thruster operating conditions. Figure 17 shows the thruster mounted in the center of the CubeSat side face with a 436 MHz dipole positioned on the center of the satellite top face. Cases a, b, and c represent successively simulated plume for a Hall thruster operating power of 132 W, 159 W, and 201 W. Each of these plumes were simulated as described above, using the JET2D simulation software. For each plume the input parameters (i.e. total current, acceleration voltage, gas flow rate, ionic current to total current ratio) were chosen as given in the paper by Hallouin and Mazouffre [START_REF] Hallouin | Far-field plume characterization of a 100-W class Hall thruster[END_REF] . In Figure 17 we can see that depending on the power, the plume does not have the same characteristics as presented in Table I.

Table I shows the maximum electron density, volume, and number of toroids required to discretize the plume at a given thruster power. The higher the power of the engine, the more energy is available to ionize the gas and thus the higher the electron density. Thus the volume of the plume which corresponds here to the volume occupied such that the local relative permittivity remains below 0.95 at 436 MHz increases with the power. Consequently, the number of toroids needed to discretize the plume also increases.

Figure 18 shows the total gain radiation pattern in the φ = 90 • plane for the 436 MHz dipole presented earlier as a function of Hall thruster power. Regardless of power, the gain diagram is once again distorted by the presence of the plume. As shown previously when studying the position of the thruster, the gain is particularly disturbed in the angular range where the plume is located with respect to the antenna, i.e., around θ = 90 • . It can be noted that in this area, the distortion is all the more important as the power thuster is large. Indeed for a power P = 132 W, for θ = 90 • , the gain reduction is equal to 1 dB while for a power P = 201 W the gain reduction is more than 2 dB. As shown by Table 1, the higher the power thuster the larger the plume, the larger the refracting effect of the plasma plume on the electromagnetic propagation.

V. CONCLUSION

In this work, a numerical method has been presented to study the propagation of microwave signals emitted or received by antennas on a CubeSat and its electric thruster. This method requires the electron density distribution within the plume. In this work, a special effort was made to be represen- tative of a low-power Hall thuster plume. It was shown that the two key parameters of the simulation are the spatial resolution and the plasma vacuum interface. The method was applied to a 436 MHz dipole mounted on a 6U CubeSat. The antenna characteristics have been presented. No significant changes in the reflection coefficient nor the radiation efficiency were shown. However a distortion of the gain pattern was observed for all cases of study, especially for angular directions through the plume. Hence, the link budget is affected. Also in this work we have shown the versatility of the method. Indeed the influence of the plume on the dipole radiation pattern has been studied for several plume/antenna relative positions, and for several power thruster showing that this method can be used for a wide variety of cases. In a future work, this method will be applied for other antennas in other frequency band, for other geometries, for other thruster type and for other operating conditions. The ability offered by this method to investigate a large variety of configurations and thusters is of particular interest in the actual context of "New Space", where the electromagnetic compatibility between the thruster and the small spacecraft apparatus still remains an unresolved issue.

FIG. 1 .

 1 FIG. 1. Representation of the problem: plume representing a modification of the antenna environment.

FIG. 2 .

 2 FIG. 2. Representation of a large satellite (Eutelsat 172B from Airbus Defense and Space ©) next to a 6U CubeSat.

FIG. 4 .

 4 FIG. 4. Comparison between measured and simulated ion current density

FIG. 6 .

 6 FIG. 6. Half plane distribution of the relative permittivity for a frequency of a) 436 MHz, b) 1.2 GHz, and c) 2.15 GHz

FIG. 7 .

 7 FIG. 7. Test case with L the length of one pole of the dipole

FIG. 9 .

 9 FIG.9. Normalized simulated gain deviation from a reference value (ε max = 0.99) for three limit permittivity values.

FIG. 11 .

 11 FIG. 11. Ansys HFSS CAD of a UHF dipole (436 MHz) on a 6U CubeSat with the Hall thruster ON.

FIG. 12 .

 12 FIG. 12. Refection coefficient |S 11 | and radiation efficiency as a function of the frequency with (ON) and without (OFF) the plasma.

FIG. 16 .

 16 FIG.16. Antenna gain radiation pattern in the cut-planes of the dipole antenna for different thruster's positions.

FIG. 17 .

 17 FIG. 17. Ansys HFSS CAD of plumes as a function of the thruster power.

TABLE I .

 I FIG. 15. Ansys HFSS CAD of several plume/antenna relative positions. Characteristics of the plume as a function of the thruster power.

	Power	n e,max	Volume	Number of
	(W)	(cm -3 )	(m -3 )	toroids
	132	6.86 × 10 16	0.10	1097
	159	1.01 × 10 17	0.18	1534
	201	1.37 × 10 17	0.25	2000

ACKNOWLEDGMENTS

The authors would like to thank the Centre National d'Etudes Spatiales (CNES), and the Institut Supérieur de l'Aéronautique et de l'Espace (ISAE-SUPAERO) for their financial support. Also, the authors would like to thank reviewers for their valuable comments

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.