R P Muñoz
email: rmunoz@astro.puc.cl

V Motta

T Verdugo

F Garrido

M Limousin

N Padilla

G Foëx

R Cabanac

R Gavazzi

L F Barrientos

J Richard

Dynamical analysis of strong-lensing galaxy groups at intermediate redshift

Keywords: Galaxies, groups, general -Galaxies, kinematics and dynamics -Galaxies, distances and redshifts

We present VLT spectroscopic observations of 7 discovered galaxy groups between 0.3 < z < 0.7. The groups were selected from the Strong Lensing Legacy Survey (SL2S), a survey that consists in a systematic search for strong lensing systems in the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS). We give details about the target selection, spectroscopic observations and data reduction for the first release of confirmed SL2S groups. The dynamical analysis of the systems reveals that they are gravitationally bound structures, with at least 4 confirmed members and velocity dispersions between 300 and 800 km s -1 . Their virial masses are between 10 13 and 10 14 M , and so can be classified as groups or low mass clusters. Most of the systems are isolated groups, except two of them that show evidence of an ongoing merger of two sub-structures. We find a good agreement between the velocity dispersions estimated from the analysis of the kinematics of group galaxies and the weak lensing measurements, and conclude that the dynamics of baryonic matter is a good tracer of the total mass content in galaxy groups.

Introduction

Galaxy groups are the most common structures in the Universe, containing at least 50% of all galaxies at the present day (Eke et al. 2004), and cover the intermediate mass range between large elliptical galaxies and galaxy clusters. A wide array of methods have been used to identify groups at intermediate and high-z: Percolation algorithms based on optical photometric and spectroscopic data (Marinoni et al. 2002;Eke et al. 2004;Adami et al. 2005;Yang et al. 2007;Zapata et al. 2009), X-ray emission from hot intragroup gas (Brough et al. 2006;Finoguenov et al. 2009), bright arcs due to strong lensing (Cabanac et al. 2007;Limousin et al. 2009;More et al. 2012).

The first large sample of groups detected in redshift space was presented by Geller & Huchra (1983), who found 176

Based on observations collected at the European Southern Observatory, Paranal, Chile (Program P80.A-0610B); based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council (NRC) of Canada, the Institut National des Sciences de l'Universe of the Centre National de la Recherche Scientifique (CNRS) of France, and the University of Hawaii. This work is based in part on data products produced at TERAPIX and the Canadian Astronomy Data Centre as part of the Canada-France-Hawaii Telescope Legacy Survey, a collaborative project of NRC and CNRS. groups up to z = 0.03 by using the Center for Astrophysics (CfA) galaxy redshift survey. Nowadays, with the advent of large spectroscopic surveys such as the Two Degree Field Galaxy Redshift Survey (2dFGRS; Colless et al. 2001), the Sloan Digital Sky Survey (SDSS; York et al. 2000), and the Deep Extragalactic Probe 2 Redshift Survey (DEEP2; [START_REF] Davis | Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference[END_REF], well over 5,000 groups have been identified up to z ∼ 1. Eke et al. (2004) identified about 7, 000 groups and clusters in the 2dFGRS with at least 4 members, and found that they cover a wide range in mass between 10 12 and 10 15 M . At low redshift, Berlind et al. (2006) used a friends-of-friends (FOF) algorithm to identify groups in the SDSS Data Release 2 (DR2; Abazajian et al. 2004) and found about 8, 100 groups between 0.01 < z < 0.10. At higher redshift, Gerke et al. (2005) found about 900 groups with 2 or more members between 0.7 < z < 1.4 using the DEEP2 survey.

Recently, Knobel et al. (2012) presented a sample of about 1, 500 galaxy groups between redshifts 0.1 and 1.0 that were identified in the zCOSMOS-bright survey (Lilly et al. 2007). They detected a clear evidence for the growth of cosmic structure over the last seven billion years because the fraction of galaxies that are found in groups (in volume-limited samples) decreases significantly to higher redshifts.

The analysis of the galaxy content in groups and clusters is essential to understand the effects of the local environment on galaxy formation and evolution processes. For instance, galaxy collisions are expected to be most effective in less mas-sive groups, where the system velocity dispersions are comparable to the internal velocities of galaxies, leading to strong galaxy-galaxy interactions and therefore enhancing star formation (Mihos & Hernquist 1996). Furthermore, the feedback from supernovae and supermassive black holes on the hot intragroup gas is expected to be relevant in suppressing the onset of catastrophic cooling of the hot gas (eg. Churazov et al. 2001), since the energy input associated with these sources is comparable to the binding energies of these systems (McCarthy et al. 2010).

The evolution of the stellar and gas content of galaxies strongly depends on the properties of their host galaxy cluster (Oemler 1974). For instance, Hansen et al. (2009) studied a large sample of groups and clusters in the Sloan Digital Sky Survey (SDSS; York et al. 2000), and found that the fraction of redsequence galaxies increases with cluster mass and it decreases with cluster-centric distance (see also Padilla et al. (2010)).

Ir order to obtain reliable conclusions about the relative importance of different physical processes in driving galaxy evolution in groups, it is necessary to build composite samples of galaxy groups and then use the virial mass as the mass normalization. Biviano et al. (2006) studied the accuracy of the virial mass estimate from numerical simulations using both the darkmatter particles and simulated galaxies in 67 synthetic clusters. To analyze how the observational strategy and sample sizes affect the cluster mass estimates, they used these synthetic clusters to select a sample of galaxies and estimate the cluster dynamical mass by using two different estimators.. They found that the total mass of clusters can be estimated with an accuracy of 10 to 15 percent when using 400 cluster members, and that these figures become twice as large when the available number of members is 20. In this paper, we follow Biviano et al. (2006) to analyze our lens galaxy groups.

In this paper we introduce the first spectroscopically confirmed groups of the SL2S survey that were observed at the Very Large Telescope (VLT). The SL2S survey is a small and well defined sample of groups of galaxies selected by their strong lensing features (Cabanac et al. 2007;More et al. 2012). The description of the spectroscopic observations, their reduction and calibrations are presented in Section 2. In Section 3 we present the membership determination and velocity dispersion estimation for the confirmed groups. The numerical simulations used to study the accuracy of the velocity dispersion estimations are presented in Section 4. The discussion of the main results is presented in Section 5, and the conclusions are summarized in Section 6.

We assume a ΛCDM cosmology with Ω M = 0.25, Ω Λ = 0.75, and a Hubble constant of H 0 = 73 km s -1 Mpc -1 .

Observations and Data Reduction

The groups studied in this work were selected from the Strong Lensing Legacy Survey (SL2S; Cabanac et al. 2007), a large systematic search for strong-lensing systems in the Canada-France-Hawaii Telescope Legacy Survey (CFHTLS) 1 . The detection and classification of group candidates is explained in Cabanac et al. (2007), but it basically consisted in running the ARCFINDER algorithm by Alard (2006) on the stacked CFHTLS images and then doing a visual inspection to reject spurious candidates.

Recently, More et al. (2012) has published a catalog of 127 strong-lensing systems detected in the SL2S survey with photometric redshifts between 0.2 and 1.2. They found a system-1 http://www.cfht.hawaii.edu/Science/CFHTLS atic alignment of the giant arcs with the major axis of the baryonic component of the putative lens, and more important, they were able to probe the average density profiles of groups using the image separation distribution. Several SL2S systems presented in Cabanac et al. (2007) and More et al. (2012) have been followed-up with optical observations at the Hubble Space Telescope (HST), near-infrared observations at the CFHT and optical spectroscopy at the ESO Very Large Telescope (VLT).

In this work, we present medium-resolution spectroscopy of 8 SL2S systems observed at the ESO VLT telescope. They were selected for showing extended arcs with Einstein radius (R E ) lower than 8 , and having photometric redshifts (z phot ) between 0.3 and 0.7. The selection criteria is based on the predicted angular separations from N-body numerical simulations of dark matter haloes by Oguri (2006), where they obtained that strong-lensing arcs with 3 < R E < 8 are likely generated by galaxy-group scale dark matter haloes.

Several of the systems presented in this work have weak lensing mass estimates from Limousin et al. (2009). They measured the weak lensing signal for 13 SL2S systems between 0.3 < z phot < 0.8, and were able to estimate weak lensing masses for 6 of them. Furthermore, the gravitational potential of the system SL2S02140-0535 presented in this work was studied in detail by Verdugo et al. (see 2011) by combining strong-lensing, weak-lensing and dynamic measurements.

Imaging

The groups have u, g, r, i and z-band photometry as part of the CFHTLS survey, a major photometric survey in five bands that covers a total area of 159 deg2 (T0006 release of the CFHTLS Deep and Wide surveys; see more details in Goranova et al. 2009 2 ). The CFHTLS survey observations were obtained at the 3.6m CFHT telescope with the MEGACAM camera, a wide field imager that consists of 36 2048x4612 pixels CCDs of pixel size 0.186 .

The images and photometric catalogs used in this work are based on the T0005 release of the CFHTLS survey (November, 2008), and were built at the TERAPIX data processing center at the Institut d'Astrophysique de Paris (IAP; see more details in Mellier et al. 2008 3 ). The 50% completeness magnitude of point-like sources in these catalogs are u = 25.34, g = 25.47, r = 24.82, i = 24.48 and z = 23.60.

Spectroscopy

We obtained medium resolution spectra of group galaxies with the Focal Reducer and low dispersion Spectrograph 2 (FORS2; Appenzeller et al. 1998) at the VLT telescope. The FORS2/VLT observations were carried out during the ESO observing programme P80.A-0610B (P.I. Motta) and consisted of multi-object spectroscopy (MOS) of 8 SL2S systems. We used a medium resolution grism (GRIS 600RI, 0.83 Å/pix) since we wanted to measure the internal velocity dispersion of the brightest group galaxies, and adopted a 2 × 2 binning in order to improve the signal-to-noise ratio (SNR) of the spectra.

Depending on the number of group member candidates, we used one or two FORS2 masks to do MOS of each group. One FORS2 mask allowed us to take spectra of ∼ 40 targets simultaneously within a field of view of 4.25 × 4.25 . The criteria used to select the galaxies that entered in the MOS masks was based on the magnitudes and colors of galaxies. We defined as candidates those galaxies with magnitudes i < 22.0 and colors within (gi) lens -0.15 < gi < (gi) lens + 0.15, where (gi) lens is the color of the brightest lens galaxy within the R E . As the masks could not be filled only with group candidates, we randomly selected galaxies within the field of view with i < 20.0 and no color restrictions.

For all the masks we obtained two exposures of 1400 s each, except for the SL2SJ08591-0345 mask where we used only one exposure because of time constraints. We found that two exposures were enough to remove most of the cosmic rays from the 2D spectra, although a couple of them were not removed and had to be manually masked in the 1D spectra. The number of masks and total exposure time for each group is given in Table 1.

The MOS masks were reduced using the standard ESO data reduction procedures4 and the Optimal Spectrum Extraction Package (OSEP) for IDL5 . The basic data reduction steps consisted on bias subtraction, flat-fielding and wavelength calibration, which were done using the ESO Recipe Execution Tool (EsoRex; http://www.eso.org/sci/software/cpl/ esorex.html) and the Common Pipeline Library (CPL; http: //www.eso.org/sci/software/cpl). The advanced steps consisted on the removal of cosmic rays, the background subtraction from the 2D spectra, the 1D spectra extraction and the average of multiple spectra for each source, and they were done using the OSEP IDL procedures inspired in the optimal extraction algorithm by Horne (1986).

Analysis & Results

Redshift measurements

The spectroscopic redshifts were determined using the Radial Velocity SAO package (RVSAO; Kurtz & Mink 1998) within the IRAF software6 . We first identified several emission and absorption lines by doing visual inspection of the galaxy spectra, and then we determined the redshifts by cross-correlating a spectrum against template spectra of known velocities.

The galaxy spectra cover the wavelength range 5200 Å < λ < 8400 Å, and the SNR per resolution element varies from ∼ 5 to ∼ 30. For most of the galaxy spectra we were able to identify the Ca II K+H, G-band and Mg I absorption lines, and for few of them we also identified the O II, Hβ, O III and Hα emission lines. The errors in the redshift measurements is affected by the instrumental resolution and the RVSAO template fit, and it is δz = 0.001.

The redshifts were classified in three types: secure, questionable and unknown. Secure redshifts correspond to spectra having at least three identified lines, between absorption and emission lines; questionable redshifts to spectra having only one or two identified lines; and unknown redshifts to spectra having no identified lines. The success ratio of secure redshift determination is between 50% and 70%, and correspond to groups SL2SJ02140-0535 and SL2SJ08544-0121, respectively. We found that this ratio strongly depends on the total exposure time and magnitude of the targets.

For all the SL2S group candidates we were able to measure the redshift of the brightest galaxy within the Einstein radius, hereafter called main lens galaxy. The spectra of the main lens galaxy of SL2S groups are shown in the top panel of figures in Appendix A, and the main absorption and emission lines have been identified.

The galaxy redshift distributions in the direction of the SL2S group candidates are shown in Figure 1. For five of the eight group candidates, we detected a strong peak in the redshift distribution around the redshift of the main lens galaxy. The red dashed lines correspond to the redshift of the group center of mass (see Section 3.2).

Group membership and velocity dispersions

We adopted the formalism by Wilman et al. (2005) to determine the group membership of the SL2S systems. For the systems that showed a single peak in the redshift distribution around the redshift of the main lens galaxy, we identified the group members as follows: the group was initially assumed to be located at the redshift of the main lens galaxy, z lens , with an initial observed-frame velocity dispersion of σ(v) obs = 500(1 + z lens ) km s -1 . Then, we computed the maximum redshift shell, δz max , and the maximum spatial distance, δθ max , as following,

δz max = 2σ(v) obs c , (1)
δθ max = 206, 265 c δz max b(1 + z lens )H(z)D θ (z) , ( 2
)
where c is the speed of light, H(z) is the Hubble constant at z, D θ (z) is the angular diameter distance at z, and b is the axis ratio of the cylindrical linking volume. In N-body numerical simulations of dark matter halos, the cylindrical linking volume is a cylinder oriented along the line of sight with a radius equal to the projected linking length. We adopted a value of b=3.5 in this work.

The initial guess for the velocity dispersion was based on the estimated velocity dispersion of the strong-lening galaxy group B2108+213 measured by McKean et al. (2010). They obtained a mean value of 555 km s -1 for the galaxy group using three different linking velocity kernels. The factor (1 + z lens ) is used to account for the cosmlogical expansion of the Universe.

Upon identifying potential members as those galaxies located inside the maximum redshift shell and the maximum projected distance, we computed the observed velocity dispersion of the group σ(v) obs . For those groups with more than 10 members we used the biweight estimator (Beers et al. 1990) in order to compute the velocity dispersion, and for those with less than 10 members we used the gapper algorithm (Beers et al. 1990). The new computed value of σ(v) obs was then used to compute new values of δz max and δθ max . Finally, we defined as confirmed group members those galaxies located within these limits.

For the groups that showed a bimodal redshift distribution, we identified the group members of each component as follows: the first component was initially assumed to be located at the higher redshift peak, z peak,high , with an initial observed-frame velocity dispersion of σ(v) obs = 250(1+z peak,high ) km s -1 . Then, we applied the same procedure we detailed before for groups with a single component, and finally determined the group members of the higher redshift component.

To determine the membership of the lower redshift component, first we excluded the galaxies linked to the first component. Then, we repeated the same procedure used for the first component, but this time using the lower redshift peak as a first guess.

We classified as groups those structures having at least 4 confirmed members in order to reduce the contamination by spurious structures in the catalogs. The final list of SL2S groups is shown in Table 2, and consists of five structures with a single component in redshift space, and two structures with a double component. The system SL2SJ02132-0743 was excluded from the group list since only two galaxies have a redshift consistent with the redshift of the main lens galaxy and also no overdensity in redshift space was found.

The group redshifts, z spec , and the number of spectroscopically confirmed members, N spec , are shown in columns 4 and 6 of Table 2. The group redshift was computed by taking the mean of the redshift of the spectroscopically confirmed members (presented in Table B.1 of Appendix B), and then removing the peculiar motion of the Sun with respect to the CMB. In order to study the accuracy of the velocity dispersions and mass estimates presented in this work (see Section 3.3), we also computed the number of confirmed members with colors consistent with the observed E/S0 ridgeline of galaxy groups and clusters [START_REF] Dressler | [END_REF] and located inside a group-centric distance of 1 h -1 M pc, and denoted it by N spec,RS in Table 2.

The group velocity dispersions were estimated using the recessional velocity of their members. The measured redshift of a galaxy member, z, can be related to its peculiar velocity with respect to the group center of mass by the following,

1 + z = (1 + z O )(1 + z R )(1 + z G ) ;
where z O is the local observer O comoving with the expanding Universe, z R is the cosmological redshift of the structure as measured by O relative to a comoving observer R in the vicinity of the structure, and z G is the peculiar velocity of the galaxy with respect to the center of mass of the structure. The z O contribution is negligible for groups at z > 0.02.

Substituting z R by the spectroscopic redshift of the group, we obtain the following equation for the line-of-sight velocity of a galaxy with respect to the group center of mass,

v los = c(z -zspec ) 1 + zspec ( 3
)
where zspec is the group redshift as shown in Table 2.

The group velocity dispersion is related to the sum in quadrature of the v los of all group members, and this value is affected by the recessional velocity errors. We computed the rest-frame velocity dispersion of each group by applying the biweight estimator of scale (Beers et al. 1990) to the v los of its members. In order to remove the recessional velocity errors from the velocity dispersion measurements, we followed the prescription by Danese et al. (1980), i.e., we subtracted in quadrature the mean v los errors from the velocity dispersion. The corrected line-ofsight velocity dispersions, σ(v) los , of the SL2S groups are shown in Table 2. The upper and lower errors in σ(v) los were estimated by using a bootstrap technique of 10, 000 repetitions (see Beers et al. (1990) for details on the methodology).

Mass Estimates

The total mass of the SL2S groups was estimated by using the virial theorem. We assumed that the groups are in hydrostatic equilibrium, have spherical symmetry and have isotropic velocity distributions. It is important to note that the first assumption could be wrong for the youngest and less massive groups, at which the internal velocity dispersions of the galaxies are comparable to that of the group and, therefore, galaxy mergers are favored (Hickson 1997).

For distant galaxy clusters and groups, it is only possible to measure their projected velocity dispersions and galaxy separations. We computed the projected virial radius for each group using the sky angular distances between all its members, following the formalism by Girardi et al. (1998). The projected virial radius and mass were computed as follows,

R PV = D θ (z)N(N -1) 1 N i=1 N j=i+1 1 θ i j , (4)
M V = 3π 2 σ(v) 2 los R PV G , (5)
where R PV is the projected virial radius, M V is the virial mass, D θ (z) is the angular diameter distance at redshift z, N is the number of confirmed members, θ i j is the sky angular distance between galaxies i and j, and G is the gravitational constant.

The projected virial radii and virial masses of SL2S groups are shown in Table 3, in units of Mpc and 10 14 M , respectively.

Accuracy of velocity dispersion and mass estimates

The dynamical masses estimated for the lensed clusters can be subject to statistical and systematic biases. In this section we use GALFORM semi-analytic galaxies from the Bower et al. (2006) version of the model, which populate the Millennium simulation (Springel et al. 2005).

--

Notes. The columns show the name of the identified SL2S groups; J2000.0 coordinates of the brightest group member with spectroscopic redshift; spectroscopic redshift of the group, zspec ; line-of-sight velocity dispersion of the group in units of km s -1 ; number of spectroscopically confirmed members, N spec ; number of confirmed members within the E/S0 ridgeline (RS) and inside a group-centric distance of 1 h -1 M pc, N spec,RS ; estimated total number of group members within the E/S0 ridgeline and inside 1 h -1 M pc, N gal (see Section 5).

Table 3. Virial radii and masses of SL2S groups. Notes. The columns show the name of the SL2S group; spectroscopic redshift of the group, zspec ; projected virial radius of the group, R PV , in units of Mpc; virial mass of the group in units of 10 14 M ; projected mass derived from weak lensing as computed within a circular aperture of radius 2 Mpc, in units of 10 14 M . (a) M WL inside a projected radius of 2 Mpc taken from Limousin et al. ( 2009) (b) R E is below 3 arcsec and is within galaxy lensing regime. (c) The weak lensing signal is low, and only an upper limit could be established. (d) The system is located at the edge of the field of view.

This simulation adopts a flat ΛCDM cosmology with z = 0 dark-matter and baryon density parameters Ω dm = 0.205, Ω b = 0.045, a dimensionless Hubble constant of h = 0.73, rms linear mass fluctuations in spheres of 8 h -1 M pc of σ 8 = 0.9, and a n = 1 slope for the primordial power spectrum. The simulation followed 2160 3 particles from z = 127 to z = 0 in a comoving periodic volume of 500 h -1 M pc a side. The resulting galaxy population after applying the Bower et al. (2006) model can be considered complete down to an absolute magnitude in the rband of M r = -15.

In order to check the presence of biases in the method that was used to compute the SL2S group masses (Section 3.3), it is necessary to first determine the observational selection and completeness effects. The dominant selection effect is the fraction of group members that were observed and classified as secure members of each SL2S group. Coupon et al. (2009) computed the photometric redshifts for galaxies in the CFHTLS survey, and obtained a mean photometric redshift error of σ z/(1+z) ∼ 0.038 and an outlier rate of η ∼ 3% using a sample of 1,532 galaxies (from W1 field) with secure spectroscopic redshifts. We estimated the total number of red-sequence galaxies for each group, N gal , using a method similar to the one used by Koester et al. (2007) for building the MaxBCG cluster catalog, but we added photometric redshift information to reduce the contamination by background and foreground galaxies. We estimated N gal for each group by counting the number of galaxies within a radius of 1 Mpc, having magnitudes brighter than R = 22.5 and colors within |g -R| < 0.24 (equivalent to 2σ δ(g-R) ) with respect to the E/S0 ridgeline, and having |z photzspec | ≤ 0.038 * (1+z spec ), i.e. 1σ z . We found that the number of red-sequence galaxies in the SL2S groups goes between 2 and 64 galaxies (see Table 2). The fraction of group members with measured recessional velocities was estimated as the ratio between N spec,RS and N gal , and it ranges from 0.25 to 0.85 (not considering the bimodal groups).

We select haloes from the z = 0.509 simulation output and repeat as closely as possible the observational procedure to measure cluster dynamical masses. The simulation cube consists of (X,Y,Z) spatial coordinates and (v X ,v Y ,v Z ) velocities . A first step consists of choosing the Z coordinate axis in the simulation cube as the line of sight, and defining the recessional velocity by Z × 100 h km s -1 Mpc -1 + v Z . Since all the groups studied in this work are bona fide gravitational lensing systems, we assume the sample to be free from spurious groups and clusters, and therefore use the full sample of haloes to do these tests.

For each individual dark-matter halo we select galaxies in the red sequence (defined using empirical color cuts) in a cylinder with depth ∆v = 500 km s -1 and width ∆θ = 220 , transformed into comoving coordinates at the redshift corresponding Fig. 2. Comparison between estimated gapper mass and underlying halo mass in the simulation. This shows the results when using a 30% of the group members brighter than M r = -18, randomly selected. Dots represent individual measurements; the solid lines correspond to the median in bins of halo mass, and the errorbars enclose 68 percent of the individual measurements.

to the selected output (z = 0.509). These values of ∆v and ∆θ are iteratively corrected once the velocity dispersion and harmonic projected radius of the halo are obtained from the possible members of the halo. Their final values are used to calculate the gapper mass of the halo. Figure 2 shows a comparison between the recovered and simulated masses of group-size dark matter haloes (the latter being simply the number of dark-matter particles per halo multiplied by the particle mass), where it can be seen that when the member galaxies are those brighter than M r = -18 (which corresponds to the observed i = 22 mag for a group at z ∼ 0.4), the gapper method introduces important uncertainties in the recovered masses of about of 20% and 50%, for masses of ∼ 10 14 h -1 M and ∼ 10 15 h -1 M respectively; notice the significantly larger values compared to the results by Biviano et al. (2006), mainly due to the low number of members available in our observational samples.

A more detailed interpretation of the simulation tests is shown in Figure 3. The top left panel shows the ratio between the gapper and simulated mass of the dark matter haloes, as function of the number of group members that were selected to estimate their respective group virial masses. We use the entire sample of galaxy groups from the simulation, and on average the number of members is 60. As can be seen, the statistical errors shown by the error bars (enclosing 68 percent of the individual results) is significantly larger than the systematic bias. In the case of the larger sample of galaxies (selected with M r < -18), the bias is almost null, with a slight tendency of recovering a lower value for the estimated mass as the fraction increases. The same conclusion applies for the analysis using the brighter sample of members, with the difference that regardless the fraction of members used in the analysis, the estimated group masses are always lower than their simulated counterparts. It is important to note that biases in the mass estimate are lower than 20 percent for all the cases. Also, we note that there is an important drop in the error bars as soon as the fraction of group members used in the calculation of the mass increases above 20%. It is important to notice that the average number of true members is constant across the xaxis, i.e. we only change the number of spectroscopic members used to estimate the group mass. The top right panel shows the mass ratio as a function of the total number of members, which spans between 20 and 300 members on average. As discussed in Section 2, the mean fraction of group members with spectroscopic measurements for the SL2S groups is about 30%; as can be seen, using a larger fraction would not have resulted in much of an improvement (and would require significantly more telescope time).

In order to account for the sources of uncertainty in the estimation of the virial mass, in the bottom left panel of Figure 3 we show the ratio between the group velocity dispersions estimated by using the gapper method and the ones obtained from the full numerical simulation, and in the bottom right panel we show the ratio between the estimated harmonic radii, r gap , and the average three-dimensional radii of simulated dark matter haloes, r 3D . Our results suggest that the velocity dispersion is always underestimated, and that it gets closer to the underlying value as the fraction of observed group members increases (with little change above a fraction of 0.3). Furthermore, we found that the virial radius is also underestimated and it departs from the actual value as the fraction of members increases. We note that the ratio r gap /r 3D can not be compared with the ratios in columns 9 and 10 of Table 1 from Biviano et al. (2006), since we use a different method to estimate the virial radius.

The previous tests were applied to bound systems with masses larger than 10 14 M , but could also be applied to a subset of simulated groups of masses similar to any of the observed SL2S groups. We chose the group SL2SJ02140-0535 as a particular case for applying the test, since this group has the largest number of confirmed members and a virial mass properly covered by the numerical simulation (Verdugo et al. 2011). We selected haloes from the numerical simulation with velocity dispersions within a narrow range around the measured value for SL2SJ02140-0535 (264 < σ(v) los < 464 km s -1 ), and restricted to haloes having a total number of M r < -21 galaxies between 20 and 40, which brackets the estimated number of members for this group. The main aim of studying this restricted sample of groups was to obtain the systematic and statistical uncertainties in the estimated mass of SL2SJ02140-0535. The ratio between the estimated and actual masses for this subsample of haloes is shown in Figure 4, where as can be seen, when using galaxies brighter than M r = -18 the resulting biases are comparable to those found for the full sample of haloes in the numerical simulation.

The main conclusions are that the expected biases in the virial mass estimation are lower than the statistical uncertainties of group properties in a narrow range of mass, and that the virial mass is underestimated by about ∼ 15 percent when using 30% of the group members. According to Figure 4, the bias appears to worsen for higher fractions of members, but this effect is likely due to the small sample of haloes resulting from the cuts applied to mimic SL2SJ02140-0535.

Discussion

The formation and evolution history of galaxies has been shown to be strongly dependent on the properties of the environment they inhabit. The existence of this environmental dependance Fig. 3. Top left: Ratio between the estimated and actual masses of group-size dark matter haloes as a function of the fraction of group members with recessional velocity measurements, as obtained from the numerical simulations presented in this work. Top right: Ratio between the estimated and actual masses of group-size dark matter haloes as a function of the number of member galaxies. Bottom left: Ratio between the velocity dispersion as estimated with the gapper method and as obtained from the actual group members in the simulation. Bottom right: Ratio between the deprojected virial radius and the mean 3D radius of actual groups members in the simulation, as a function of the fraction of members. The solid red lines show the results when using only members with absolute magnitudes brighter than M r = -18, and the dotted blue lines for members brighter than M r = -19. Errorbars enclose 68% of the measurements in each bin of fraction.

has been confirmed both in the local (Goto et al. 2003;Helsdon & Ponman 2003;Bamford et al. 2009) and intermediate redshift Universe (Dressler et al. 1997;Treu et al. 2003;Pannella et al. 2009), and the observational results suggest that star formation and galaxy merging processes are accelerated in high density environments such as galaxy groups and clusters.

Therefore, galaxy groups represent natural laboratories to study the relative importance of the different astrophysical pro-cesses occurring in dense regions of the Universe. For instance, galaxy collisions are expected to be most effective in groups because the dynamical friction timescale is similar to the orbital timescale of galaxies within the group. Since galaxy groups contain a low number of members (less than hundred within 1 Mpc) and cover a wide range in mass, velocity dispersion and hot gas content, it is necessary to characterize their properties in detail in Fig. 4. Ratio between the estimated and actual masses of simulated groups with total number of members and velocity dispersions similar to the ones measured for SL2SJ02140-0535, i.e. number of members between 20 and 40, and velocity dispersion between 260 and 460 km s -1 . The number of groups in each bin of the plot remains constant. The x-axis corresponds to the fraction of members used for the calculation of the mass. order to build representative samples of groups and to properly study the relative importance of the different mechanisms.

The virial mass estimation relies on the following assumptions: sphericity, kinematic isotropy and virialization. In our virial mass estimation we use the velocity dispersion, which we assume constant through the group, and the harmonic radius as an estimate of the virial radius (see Section 3.1). These assumptions depart slightly from observational results. For instance, i) groups and clusters usually show substructure (Riemer-Sørensen et al. 2009), ii) it is known that nearby clusters show a small gradient in the velocity dispersion (Kent & Gunn 1982), iii) some groups at high redshift show evidence of having merging events, and therefore are not in a virialized state (McKean et al. 2010). Furthermore, it is a well known fact that the harmonic radius depends strongly on the area covered by the spectroscopic observations and the number of confirmed members (Biviano et al. 2006;Girardi et al. 1998), biasing the measurement of the virial radius. From our simulations, it seems that the virial radius is underestimated when the number of members increases, contrary to the results obtained by Biviano et al. (2006). The difference between our results and those from Biviano et al. (2006) could be explained due to differences in the simulations, since the latter uses a N-body hydrodynamical simulation.

We ran numerical simulations to assess the bias introduced in the virial mass estimation (see Section 4). The results show that the method adopted in this work allows to recover the mass of the simulated groups within the error bars, showing no systematic deviations (see top panel of Figure 3). It is also important to note that: i) the mass estimation of a group with 40% of their members observed is as good as when using 90% of the members, ii) the mass estimation improves with the number of groups Fig. 5. Velocity dispersion of SL2S groups as estimated from galaxy dynamics (solid circles) and weak lensing measurements (open triangles). The red symbols show the groups with dynamical and weak lensing estimates, while blue symbols those groups with only dynamical estimates. Only one group at z > 0.6 has enough weak lensing signal to estimate an upper limit to its velocity dispersion. used, and justifies the stacking of groups of similar properties in order to obtain better estimations. However, for the particular case in which we applied a cut in velocity dispersion, the case of SL2SJ02140-0535, the simulation shows that the method underestimates the group mass by about 20% (see Figure 4). The analysis of simulated groups selected from the numerical simulations (see Section 4 and Figure 3) reveals that the virial mass of the SL2S groups is underestimated by 15%. Limousin et al. (2009) computed the weak lensing velocity dispersions and masses for several of the SL2S systems studied in this work. The mass estimate obtained from weak lensing measurements makes no assumption regarding the dynamic state of the systems, as opposed to the kinematical measurements presented in this work. Although weak lensing requires less assumptions, it has been shown that the estimated mass of galaxy clusters can be strongly affected by intervening large-scale structure along its line of sight (Hoekstra et al. 2011).

Figure 5 shows the line-of-sight velocity dispersions as estimated from dynamics and weak lensing measurements for the groups SL2SJ02140-0535, SL2SJ02215-0647, SL2SJ08544-0121 and SL2SJ09413-1100. We found that weak lensing estimates are systematically larger by 50% than dynamical estimates. This is in stark contrast with measurements from galaxy-galaxy lensing by SLACS (Treu et al. 2006a,b), where the ratio between the central stellar velocity dispersion and the velocity dispersion that best fit the lensing model is 1.01 ± 0.02. Although numerical simulations (see Section 4) suggest that the viral mass is underestimated by 20%, this is not enough to explain the inferred discrepancy. Thus, our results may indicate that the isothermal sphere model is not a good assumption for galaxy groups.

Conclusions

This paper presents the spectroscopic follow-up and dynamical analysis of 8 group candidates identified in the SL2S survey. Our analysis reveal that 7 of the systems correspond to gravi-R.P. Muñoz et al.: Dynamical analysis of SL2S galaxy groups tationally bound structures, where 5 have a single component in redshift-space and 2 have a double component (bimodal galaxy distribution). They span a wide range in redshift between 0.35 and 0.65, and a wide range in mass between 5 × 10 13 M and 1.5 × 10 14 M .

The main results of this paper are given as follows,

1. The success rate of the spectroscopic confirmation of groups identified in the SL2S survey is about 88%. It is similar to the success rate of clusters followed-up in the Red-Sequence Cluster Survey (RCS-1; Gladders & Yee 2005). 2. We found that weak lensing estimates of the group velocity dispersions are 50% larger than dynamical estimates. This discrepancy has been never reported before by other studies in groups and clusters, and is in stark contrast with measurements from galaxy-galaxy lensing. 3. From numerical simulations, we conclude that measuring redshifts for only 30% of the total galaxy population in groups is enough to recover the group velocity dispersion with less than 5% systematic error.

SL2SJ02141-0405 main lens galaxy

SL2SJ02215-0647 main lens galaxy

SL2SJ08544-0121 main lens galaxy

SL2SJ08591-0345 main lens galaxy

SL2SJ09413-1100 main lens galaxy

Fig. 1 .

 1 Fig. 1. Redshift distribution of galaxies in the group fields. The spectroscopic redshift of isolated groups is denoted by the quantity z, and for double components of bimodal groups, by z A and z B . The vertical dashed line corresponds to the redshift of the most massive component. The bin size of the histogram is δz = 0.006

Fig. A. 2 .

 2 Fig. A.2. Top panel: Optical spectra of the brightest confirmed member of SL2SJ02141-0405. The main absorption and emission lines used to determine the redshift of the main lens galaxy have been identified. Bottom panel: Spatial distribution of galaxies in the group field of SL2SJ02141-0405. Red rectangles show the position of the spectroscopically confirmed members. The dashed magenta circle shows a circular aperture of radius 1 M pc at z=0.61. The contours are the same as in Fig. A.1.

Fig. A. 4 .

 4 Fig. A.4. Top panel: Optical spectra of the brightest confirmed member of SL2SJ02215-0647. The main absorption and emission lines used to determine the redshift of the main lens galaxy have been identified. Bottom panel: Spatial distribution of galaxies in the group field of SL2SJ02215-0647. Red rectangles show the position of the spectroscopically confirmed members. The dashed magenta circle shows a circular aperture of radius 1 M pc at z=0.62. The contours are the same as in Fig. A.1.

Fig. A. 5 .

 5 Fig. A.5. Top panel: Optical spectra of the brightest confirmed member of SL2SJ08544-0121. The main absorption lines used to determine the redshift of the main lens galaxy have been identified. Bottom panel: Spatial distribution of galaxies in the group field of SL2SJ08544-0121. Red rectangles show the position of the spectroscopically confirmed members. The dashed magenta circle shows a circular aperture of radius 1 M pc at z=0.35. The contours are the same as in Fig. A.1.

Fig. A. 6 .

 6 Fig. A.6. Top panel: Optical spectra of the brightest confirmed member of SL2SJ08591-0345. The main absorption lines used to determine the redshift of the main lens galaxy have been identified. Bottom panel: Spatial distribution of galaxies in the group field of SL2SJ08591-0345. Red rectangles show the position of the spectroscopically confirmed members. The dashed magenta circle shows a circular aperture of radius 1 M pc at z=0.64. The contours are the same as in Fig. A.1.

Fig. A. 7 .

 7 Fig. A.7. Top panel: Optical spectra of the brightest confirmed member of SL2SJ09413-1100. The main absorption lines used to determine the redshift of the main lens galaxy have been identified. Bottom panel: Spatial distribution of galaxies in the group field of SL2SJ09413-1100. Red rectangles show the position of the spectroscopically confirmed members. The dashed magenta circle shows a circular aperture of radius 1 M pc at z=0.39. The contours are the same as in Fig. A.1.

Table 1 .

 1 VLT/FORS2 spectroscopic data

	Target		Date	Mask	Exposure
	SL2SJ02132-0743 2008-08-11	M013	2 × 1400
	SL2SJ02140-0535	    	2007-10-19 2008-02-01	M012 M010	2 × 1400 2 × 1400
	SL2SJ02141-0405 2007-10-19 M014 1 2 × 1400
	SL2SJ02180-0515	    	2008-02-07 M014 2 2 × 1400 2008-08-26 M014 3 2 × 1400
	SL2SJ02215-0647	    	2008-08-24 M016 1 2 × 1400 2008-09-10 M016 2 2 × 1400
	SL2SJ08544-0121	    	2007-12-12 2008-02-06	M005 M005	1 × 1200 2 × 1400
	SL2SJ08591-0345	    	2007-12-12 M002 1 1 × 1400 2007-12-18 M002 2 1 × 1400
	SL2SJ09413-1100 2007-12-18	M001	2 × 1400
	Notes. The columns show the name of the target; UT date of observa-
	tions; name of the multi-object spectroscopy mask; number of individ-
	ual exposures and their corresponding exposure time in seconds.

Table 2 .

 2 Summary of confirmed SL2S groups.

	Group		RA	DEC	zspec	σ(v) los	N spec N spec,RS	N gal
	SL2SJ02140-0535 02:14:08.03 -05:35:32.3 0.445 364 +60 -137	16	11	40
	SL2SJ02141-0405 02:14:11.21 -04:05:02.8 0.611 478 +48 -178	7	5	6
	SL2SJ02180-0515	    	02:18:10.09 -05:15:33.5 0.645 131 +29 -61 02:18:07.29 -05:15:36.2 0.649 148 +37 -56	6 5	8	8
	SL2SJ02215-0647 02:21:51.17 -06:47:33.7 0.618 234 +76 -40	4	2	2
	SL2SJ08544-0121	    	08:54:46.55 -01:21:36.9 0.351 185 +30 -62 08:54:47.10 -01:21:35.6 0.356 341 +43 -109	8 10	10	64
	SL2SJ08591-0345 08:59:14.46 -03:45:14.2 0.642 507 +107 -336	5	3	6
	SL2SJ09413-1100 09:41:34.99 -11:00:55.0 0.384 350 +57 -210

  Table B.1. continued.

	Group	Mask	Chip	Slit	RA	DEC	z spec
		M005	CHIP1	4 133.708832 -1.363143	0.353
		M005	CHIP1	7 133.699173 -1.361177	0.351
		M005	CHIP1	8 133.693802 -1.361807	0.352
	SL2SJ08544-0121A	M005	CHIP1	16 133.703308 -1.342510	0.353
		M005	CHIP1	21 133.694809 -1.337374	0.353
		M005	CHIP2	12 133.709595 -1.387739	0.352
		M005	CHIP2	17 133.700745 -1.380287	0.353
		LRIS	-	-133.693954 -1.360260	0.352
		M005	CHIP1	2 133.681519 -1.383398	0.357
		M005	CHIP1	6 133.702240 -1.361963	0.357
		M005	CHIP1	10 133.698608 -1.356661	0.359
		M005	CHIP1	11 133.687943 -1.361066	0.358
	SL2SJ08544-0121B	M005	CHIP1	25 133.680481 -1.334525	0.360
		M005	CHIP1	33 133.651199 -1.322610	0.356
		M005	CHIP2	9 133.726212 -1.384477	0.356
		M005	CHIP2	11 133.742172 -1.371531	0.355
		M005	CHIP2	13 133.728638 -1.373227	0.354
		M005	CHIP2	16 133.698975 -1.384341	0.358
		M002 1 CHIP1	5 134.812363 -3.751204	0.644
		M002 1 CHIP1	6 134.810410 -3.753887	0.645
	SL2SJ08591-0345	M002 1 CHIP1	9 134.805069 -3.747538	0.642
		M002 2 CHIP1	1 134.817093 -3.740146	0.644
		M002 2 CHIP1	19 134.792923 -3.766603	0.639
		M001	CHIP1	5 145.394897 -11.015464 0.386
		M001	CHIP1	16 145.393402 -10.982292 0.386
	SL2SJ09413-1100	M001	CHIP2	5 145.425903 -11.059779 0.383
		M001	CHIP2	8 145.401794 -11.050273 0.385
		M001	CHIP2	10 145.402603 -11.041045 0.386

http://terapix.iap.fr/cplt/T0006-doc.pdf

http://terapix.iap.fr/cplt/oldSite/Descart/ CFHTLS-T0005-Release.pdf

Very Large Telescope Paranal Science Operations FORS data reduction cookbook, v1

5 http://physics.ucf.edu/ ˜jh/ast/software/ optspecextr-0.3.1/doc

IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.

R.P. Muñoz et al.: Dynamical analysis of SL2S galaxy groups, Online Material p 8

Acknowledgements. We thank Ricardo Salinas and Anupreeta More for enlightening conversations. We thank the referee for their useful and constructive comments to improve this paper. The authors acknowledge support from Fondo ALMA-CONICYT N • 31090019, Comité Mixto ESO-Gobierno de Chile and FONDECYT through grant 1090637. R.P. Muñoz acknowledges support from CONICYT CATA-BASAL and FONDECYT through grant 3130750. V. Motta gratefully acknowledges support from FONDECYT through grant 1090673 and 1120741. T. Verdugo acknowledges support from FONDECYT through grant 3090025 and CONACYT through grant 165365. M. Limousin acknowledges the Centre National de la Recherche Scientifique for its support and the Dark Cosmology Centre is funded by the Danish National Research Foundation. N. Padilla acknowledges support from FONDAP "Center for Astrophysics", CONICYT CATA-BASAL and FONDECYT through grant 1110328. G. Foëx acknowledges support from FONDECYT through grant 3120160. R. Gavazzi acknowledges support from the Centre National des Estudes Spatiales.

This work made use of the Geryon cluster at the Centro de Astro-Ingeniería UC.

Appendix A: Presentation of each group SL2SJ02140-0535 main lens galaxy The main absorption lines used to determine the redshift of the main lens galaxy have been identified. Bottom panel: Spatial distribution of galaxies in the group field of SL2SJ02140-0535. Red rectangles show the position of the spectroscopically confirmed members. The dashed magenta circle shows a circular aperture of radius 1 M pc at z=0.44. The contours in green show the luminosity contours equal to 3 × 10 5 , 10 6 , 3 × 10 6 and 10 7 L kpc -2 from outermost to innermost, as computed by Limousin et al. (2009).


Romaric Guillerm
email: guillerm@laas.fr

Hamid Demmou
email: demmou@laas.fr

Nabil Sadou
email: nabil.sadou@supelec.fr

Illustration of the information model for complex system modeling: from requirement to V&V

Keywords: system engineering, safety, requirements, SysML, braking system

This paper presents an illustration of the utilization of an information model through a complex system. The information model is in support of a model driven methodology of complex system design. It allows addressing requirements definition and their traceability towards the solution and the Verification and Validation (V&V) elements. The work considers especially an important system propriety which is safety. SysML language is used to establish the information model thanks to the different available diagrams which make SysML as the language for systems engineering. The system considered is the braking system of an aircraft.

Introduction

Modern systems are inherently complex [START_REF] Bar-Yam | About engineering complex systems: Multiscale analysis and evolutionary engineering, Engineering self-organizing systems: methodologies and applications[END_REF]. Complexity implies that different parts of the system are interdependent so that changes in one part may have effects on other parts of the system. In this case system engineering processes are critical and addressing safety proprieties (considered as emergent) needs a global approach. In [START_REF] Guillerm | Information model for model driven design of complex system based on system engineering approach[END_REF] we proposed an approach that integrates safety [START_REF] Avizienis | Basic Concepts and Taxonomy of Dependable and Secure Computing[END_REF] evaluation of complex systems in the system engineering process. This approach allows considering safety not in small but in large. The proposed approach is based on system engineering standard EIA-632.

The requirement engineering process is generally considered as the most critical process within the development of complex systems [START_REF] Juristo | Is the European Industry Moving Toward Solving Requirements Engineering Problems[END_REF]. In the support of the integration approach, we have defined an information model [START_REF] Guillerm | Information model for model driven design of complex system based on system engineering approach[END_REF] based on SysML language to address requirements definition and their traceability [START_REF] Sahraoui | Requirements Traceability Issues: Generic Model, Methodology and Formal Basis[END_REF] towards the solution elements and the V&V (Verification and Validation) elements. In this paper we illustrate the utility of the information model through a complex system which is the brake system of an aircraft.

Integration approach 2.1 System engineering approach

System engineering is an interdisciplinary approach which develops concepts that allow building new applications. It is a collaborative process for problem resolution, supporting knowledge, methods and techniques resulting from the sciences and experiment in order to define a system which satisfies identified needs and is acceptable for the environment, while seeking has to balance total economy of the solution, on all the aspects of the problem in all the phases of the development and the life of the system. In SE it is a good practice to describe processes adopting the EIA standard. There are thirteen processes covering the management issues, the supply/acquisition, design and requirement and verification validation processes [START_REF] Sahraoui | Issues in systems engineering research[END_REF].

Integration approach

Managing requirements, and specially safety requirements, at the early stages of system development becomes more and more important as system complexity is continuously growing. Safety of complex systems relies heavily on the emergent properties that result from the complex interdependencies that exist among the involved systems or sub-systems and their environments. System Engineering (SE) is the ideal framework for the design of complex system. The need for systems engineering arose with the increase in complexity of systems and projects. A system engineering approach to safety starts with the basic assumption that safety properties can only be treated adequately in their entirety when taking into account all the involved variables and the relations between the social and the technical aspects. This basis for system engineering has been stated as the principle that a system is more than the sum of its parts. The Safety management must follow all the steps of SE from the requirements definition to the verification and the validation of the system. The integration of safety must concern all system engineering processes. In our approach [START_REF] Guillerm | Information model for model driven design of complex system based on system engineering approach[END_REF] we have identified all sub-processes concerned by the safety evaluation and we have indicated how they must be considered. In other words, the sub-processes of EIA-632 standard [START_REF]Processes for engineering systems, Electronic Industries Alliance standard[END_REF] are translated or refined in terms of safety and included in system design process.

Information model

Requirements management

Requirements management is a crucial activity for the success of a project [START_REF] Juristo | Is the European Industry Moving Toward Solving Requirements Engineering Problems[END_REF]. It is necessary to ensure that each requirement is properly declined, allocated, monitored, satisfied, verifiable, verified and justified. Figure 1 presents an overview of the requirements management of the EIA-632 standard. The proposed information model is inspired from this pattern. We see that other stakeholder requirements, when added to the acquirer requirements, make up a set of stakeholder requirements that are transformed into system technical requirements.

The logical and physical solution representations are derived from technical requirements. Design solution and specified requirements are defined by completing the solution definition Process.

Fig.1.Requirements management of the EIA-632

An information model can be used to guide the design, to manage requirements changes, or to evaluate project progress. Indeed, modeling is important for the following reasons: it is a support for system analysis and design, it can be used for sharing knowledge and it is used to capitalize knowledge. Our information model is compatible with the requirements of the EIA-632 standard, while adding aspects of safety and risk management. We use SysML [START_REF] Friedenthal | A Practical Guide to SysML: The Systems Modeling Language, 576 The MK[END_REF] language to establish this information model thanks to the different available diagrams which make SysML as the language for systems engineering.

Requirements modeling and management for safety

In this part, we propose a system approach to improve requirements management for safe system design. This approach is based on a SysML information model, following the SE process of the EIA-632 standard. This information model is the "system" knowledge basis of the design project, allowing data sharing between all expertise trades (mechanical, hydraulic, thermal, electrical...). Therefore, the model is intended to model the "system" level, showing the interactions with the environment and the connections between the various subsystems.

The information model must be seen as a means of knowledge sharing, including the 3 components: requirements, design solution and V&V. It is considered as the interconnection level between all the different trades.

The requirements, the design solution and the V&V parts are developed independently. We distinguish clearly these different concepts.

SysML allows mixing all the concepts in a single diagram. We proposed an extension of SysML and information meta-model that allows structuring the elements of the design project with respect of the separation concepts. In other words, our approach allows a rigorous organization of the project design. Indeed, different diagrams manage different concepts.

The information model

The information model (Figure 2) that we propose is adapted to the EIA-632 standard and makes a clear distinction between different requirements classes (acquirer, other stakeholders, technical or specified).

To achieve this meta-model in SysML, we have extended the language. Firstly, we define new stereotypes to requirements, while adding new attributes to our requirements. Then we define a new link type (specify) linking the specified requirements to model elements.

Fig.2.Information model

In this model, we have simplified the number of requirements classes. Indeed, we consider that the "systemTechnicalReqt" represents the system technical requirements, but also the system technical requirements non-allocated to the logical solution and the derived technical requirements coming from the logical and the physical solutions.

The acquirer and other stakeholders' requirements are represented, knowing that the field 'requirement source' must be consistent with the stereotype and indicates better the concerned stakeholders in the case of "OtherStakeholderReqt".

Note:

We invite readers who are not necessarily friendly with all the above concepts (system technical requirements, logical solution, physical solution …) to refer themselves to the EIA-632 standard [START_REF]Processes for engineering systems, Electronic Industries Alliance standard[END_REF] or the overview article [START_REF] Martin | Overview of the EIA-632 Standard: Processes for Engineering a System[END_REF].

All traceability links requested by the EIA-632 are considered in this model, and the distinction between logical solution (functional part) and physical solution (component part) appears.

In this model, we highlight the definition of interfaces, which are components themselves and which links several components together. The concept of interface is essential for a proper system design. Indeed, it is one source of problems encountered during development.

The last important element that is included in this model, neither a requirement nor a design solution element, is the "TestCase". These elements of V&V are included in the model to be directly connected to the requirements they satisfy.

Concerning safety requirements and the consideration of safety in design, which can be derived from risk analysis, a block risk is defined and is linked to safety requirements. In fact, identification of risks is the starting point for many studies about security, but also reliability. Thus, defining a block "risk" in the information model and its link with the safety requirements, allows on one way to improve the system understanding and justifying the requirement, and on the other way to show that all the identified risks are taken into account.

Impact analyses also derive benefit from the presence of risks in the information model, because the risks, which could be challenged by model element (requirement, function, component…) changing, can be viewed directly.

Application: the "S18 aircraft" system

In this part we present a case study to illustrate the interest of the SysML information model on a complex example. It consists on an S18 aircraft that the specifications are provided by the standard ARP-4761 [START_REF]Guidelines and methods for conducting the safety assessment process on civil airborne systems and equipment[END_REF]. We will consider a hierarchical decomposition using the notion of the building block defined by the standard EIA-632 and on which is based our approach of safety integration in system engineering processes. However, due to limited space in the paper, we will present only one building block (one level).

Presentation

The S18 aircraft is a four engine airliner used to transport passengers. It can carry out 300 to 500 passengers on 5,000 nautical miles at Mach 0.86. The average flight time is 5 hours.

Functions

Various functions are performed by this system: "control thrust", "control flight path", "determine orientation", "determine position and heading", "control aircraft on the ground", "control cabin environment"… These functions can be decomposed into sub-functions, like "control aircraft on the ground" whose sub-functions are: "determine air/ground transition", "decelerate the aircraft on the ground", "control aircraft direction on the ground"…

In this paper, we focus on only one the sub-function "decelerate the aircraft on the ground".

Sub-systems

The aircraft system consists of several sub-systems, like: structure, thrust system, control surface system, spoiler system, thrust reverse system, wheel brake system… Thanks to the interaction between these sub-systems, the above listed functions can be performed.

Sub-systems involved in the "decelerate the aircraft on the ground" function are: the "thrust reversers", the "spoilers" and the "wheel brakes".

Thrust reversers are associated with reactors and can reverse the direction of thrust. They can be used only above a certain speed (otherwise hot gases are injected again in the reactors which can cause damage).

Spoilers are movable surfaces on the wings which are used to reduce lift and increase drag. Consequently, they slow down the speed of the aircraft, preventing to take-off again by transferring more weight on the wheels. They are efficient only above a certain speed.

Finally, the wheel brakes are used at all speeds and are located on the main landing gear (not on the front). They can be used asymmetrically, to counteract against a crosswind or to make tight turns.

SysML Modeling

The following section presents the SysML modeling of the building block corresponding to the level presented previously: the aircraft system.

This modeling will respect the information model defined in paragraph 3.3 and its goal is to share engineering data through a common basis. This information will be shared between the various participants in the design, particularly between system engineers and safety engineers. In fact, most of the modeling presented here focuses on the definition of requirements and traceability links.

A structure in three packages organizes the modeling following the separation of concepts (requirements, solution design and verification and validation). Thus, the three packages are: "Design Solution", "Requirements", and "V&V". The tool TauG2 of IBM is used to generate the different diagrams.

"Design solution" package

The first package presents the design solution. It contains two diagrams: 1. A use case diagram, showing the system functions, 2. A block diagram for the system structure.

System functions

Figure 3 shows the system functions through a use case diagram. Showing also the system's actors, the use cases represent the functions. A hierarchical structure between these functions is performed using "include" links. For example, the "decelerate aircraft on the ground" function is a sub-function of the "control aircraft on the ground" function, which is a sub-function of the "pilot the aircraft" function.

Fig.3. S18 aircraft -System functions

System structure

Parts of the "S18 aircraft" system structure are represented in a blocks diagram in Figure 4. It shows that the "S18 aircraft" is composed of a structure, a thrust system, a surface control system, a spoiler system, a thrust inverser system, and a wheel brake system.

"Requirements" Package

The second presented package is the "Requirements" package. There are four requirement diagrams for:

1. Top-level requirements, to present the requirements at the system level, "S18 Aircraft" for this first building block, and to link those requirements to the risks they deal with the "treat" link 2. Requirement declinations, to show the declination of the system requirements into subsystem requirements through the composition link, 3. Requirement satisfaction, to indicate the system elements (system or subsystems) that meet the requirements by the "satisfy" link, 4. Requirement verification, to denote elements of V&V which satisfy the requirements by the "verify" link.

"Top-level requirements" diagram

Figure 5 shows the top-level requirements of the S18 aircraft. Only two risks appear "high speed overrun" and "offside excursion from the runway". Their severities are between two values; this is due to a modeling simplification to not multiply the number of risks according to the landing, taxi, or aborted takeoff phases.

In this diagram, we find for example the system safety requirement "unannounciated loss of deceleration capacity must have a frequency <5.10-9/flt" identified during the FMEA of the S18 aircraft system [START_REF] Guillerm | Combining FMECA and Fault Trees for declining safety requirements of complex systems[END_REF]. This requirement, whose identifier is "R2", can treat the risk "Risk_1": "the aircraft do a high speed overrun and can hit obstacles" identified during the preliminary risk analysis.

Requirement declinations diagram

Figure 6 shows the declination of top-level requirements (S18 aircraft system) in sub-systems requirements. We find here two declinations concerning the system requirements "unannounciated loss of deceleration" and "inadvertent deceleration after V1", which are declined on the sub-systems: thrust inversers, wheel brakes and spoilers.

Requirement satisfaction diagram

Figure 7 shows the allocations of requirements to the elements of the design solution, through the "satisfy" link of SysML. These elements can be any elements other than the SysML requirements, but in our model they correspond to blocks (representing systems) or use cases (representing functions).

Regarding the two above requirement declinations, system requirements are related to system elements, which correspond to "S18 aircraft" system functions (modeled with use cases). For the subsystem requirements, they are associated to blocks representing sub-systems (wheel brakes, thrust inversers, spoilers).

Fig.5. S18 aircraft -"Top-level Requirements" diagram

On the bottom of the diagram (figure 7), a set of requirements was simply linked to the "S18 aircraft" block representing the system. For a complete modeling, these requirements should also be declined. But to limit the size of the example, we do not study in depth these requirements.

Requirement verification diagram

Figure 8 shows the verification links between operations of V&V (which are stereotyped by "TestCase") and system requirements. In this paper, only two basic system requirements, roots of the declinations, are considered in this diagram. Then, operations of V&V correspond to safety analysis. They are defined in the "V&V" package presented below. The last package presented is the "V&V" package, which contains the test cases. It contains the diagram of Figure 9, which defines the two operations of V&V presented in the previous paragraph. For these operations, we associate comments in the diagram to specify them:

 For the operation "loss of deceleration safety analysis": analyze with the fault tree of the root event "unannunciated loss of deceleration".  For the operation "inadvertent deceleration after V1 safety analysis": analyze with the fault tree of the root event "inadvertent deceleration after V1".

Fig.9. S18 aircraft -Test case diagram

In this paper, the two lower levels: the "wheel brake system" and the "BSCU calculator" are not presented. The same organization as the first level is considered.

Conclusion

Different concepts are handled in complex system design: requirements, design solution and V&V. An information model aims to help engineers in the expression of these concepts, and the creation of traceability links between them in order to facilitate the comprehension and/or the impacts analysis. It also helps to formalize the practice of systems engineering through the use of models. The objective is to improve quality/productivity and to reduce risk by introducing rigor, precision, and communications among system/project stakeholders and managing complexity.

The present paper show how the information model can be used through a complex system which consists on braking system of an aircraft. The objectives are to demonstrate the contribution and the added value of the information model and the scalability of the approach. Some additional work is needed for the validation of this information model and will be considered in future work. It is, for example, interesting to define automatism to ensure consistency with the meta-model.

Fig. 4 .

 4 Fig.4. S18 aircraft -System structure

Fig. 6 .

 6 Fig.6. S18 aircraft -Requirement declinations diagram

Fig. 7 .

 7 Fig.7. S18 aircraft -Requirement satisfaction diagram

Fig. 8 .

 8 Fig.8. S18 aircraft -Requirement verification diagram 4.1.2 V&V package: test cases