Raverdy Bruno
email: bruno.raverdy@cea.fr

Simulation of two-phase air / water flows in bubbly flow in horizontal pipe

Keywords: two-phase flow, horizontal flow patterns, NEPTUNE_CFD, METERO-H

This study presents the two-phase CFD simulation of horizontal pipe flow experiments carried out on METERO-H facility in the frame of a Loss-Of-Coolant Accident (LOCA). The computational domain is based on conformal hexahedral mesh. A mesh sensitivity study was performed with a comparison with experimental data. A sensitivity study on number of field used in the modelling was performed. The three fields approach permits to well reproduce the experiments at high void fraction while the two fields approach underestimate the void fraction within the upper part of pipe. Results are in good agreement with experimental data.

Introduction

In Pressurized Water Reactors (PWRs), Loss of Coolant Accidents (LOCA) initiated by a breach in the primary circuit wall are the most studied accident situations for nuclear safety. The breach causes a leak of primary coolant and depressurization of the primary circuit with the steam produced by boiling, and thus the establishment of two-phase flow in horizontal pipe which is characterized by different regimes such as a Bubbly flow or stratified flow. This transition fundamentally changes the interactions between phases with notably the coefficient of interfacial friction, which decreases by several orders of magnitude, as well as the mass of water lost. The work presented here focuses on the simulation of two-phase flows of METERO-H experiment [START_REF] Bottin | Stratification d'un écoulement diphasique horizontal : modélisation et validation expérimentale[END_REF] under development in a horizontal pipe and more particularly the beginning of stratification with the buoyant bubbly flow and its transition to the intermittent flow which is the intermediate regime between bubbly flow and stratified bubbly flow.

Experimental Facility

The METERO-H experiment is dedicated to understand the responsible mechanisms of the transition between bubbly and stratified two-phase flows in a horizontal pipe. The METERO-H experiment allows measuring various twophase air/water flow quantities, such as void fraction, interfacial area, bubble sizes, velocity of the gas phase, liquid height, turbulence level, mean and fluctuating velocity of the liquid phase in order to analyse two-phase flows in details. Each simulation is characterised by different couple of superficial velocities of water (JL) and of air (JG) at the pipe inlet. These different inlet conditions are summarised in Table 1. These conditions correspond to different flow regimes experimentally observed at 40D from the inlet which are representative of buoyant bubbly flow, buoyant bubbly/ stratified bubbly transition flow and stratified bubbly flow. In the following, the computed void fraction profiles are compared to experimental void fraction profiles measured at 40 D.

Numerical description

NEPTUNE_CFD is a generalised multiphase code with an Eulerian approach. The numerical algorithm is based on a non-linear fractional step method called alpha-pressureenergy cycle. The two-field modelling consists of a continuous liquid water field carrying a dispersed air field. The three-field modelling [START_REF] Mimouni | CFD calculations of flow pattern maps and LES of multiphase flows[END_REF]) uses an additional field which is a continuous air field that exchanges mass with the dispersed air field [START_REF] Mimouni | Multifield Approach and Interface Locating Method for Two-Phase Flows in Nuclear Power Plant[END_REF]. Indeed, if the volume fraction of scattered air exceeds αcrit = 0.3, then mass from the dispersed air field is transferred to the continuous air field. The bubble size variation can be taking into account with the interfacial area modelling with the interfacial area transport equation given by [START_REF] Ruyer | A bubble size distribution model for the simulation of bubbly flows[END_REF]. Three mesh refinements are used which are Square 1 (3.10 5 cells), Square 2 (10 6 cells) and Square 3 (2.10 6 cells).

Results and Discussion

The two-field and three-field approaches were evaluated with the three grid levels. Figures 1 and2 show the comparison of the air volume fraction profiles at 40 diameter obtained with the 3 meshes and for the two approaches. In bubbly/stratified transition flow, the 3-field approach overestimates the void fraction with mesh level 2 while for the other mesh levels, the results are similar to the 2-field approach, which predicts a rather satisfactory void fraction. For stratified flow, the 3-field approach calculates a void fraction of the same order of magnitude as the experimental void fraction, whereas the 2-field approach underestimates the void fraction in the upper part of the pipe. For both approaches, there is no significant effect of the mesh on the results. For bubbly flow, the 2-field approach correctly predicts the void fraction distribution with the right order of magnitude in the pipe section. On the other hand, the 3-field approach overestimates the void fraction for all meshes. In cases of bubbly/stratified transition flow and bubbly flow, which are characterised by a moderate void fraction (< 0.3), the 2-field approach gives results that are in good agreement with experience. The 2-field approach does not allow to correctly calculating the void fraction in a stratified flow with a high void ratio.

Conclusion

The 2-field and 3-field approaches have been evaluated on bubbly flow, transition bubbly/stratified flow and stratified flow with a mesh sensitivity study. Concerning the bubbly configuration, the 2-field approach gives rather satisfactory results with all meshes. The 3-field approach gives results that are also satisfactory but still with a slight overestimation of the air volume fraction in the upper part of the test section. In case of stratified configuration, the 3-field are in better agreement with experimental data than the 2-field which strongly overestimates the air volume fraction in the upper part. This work is still in progress, a detailed analysis of the experimental data will be performed in order to determine which physical phenomena are taking place. The objective is to try to determine if fragmentation and coalescence phenomena occur and with what intensity respectively.

Figure 1 :

 1 Figure 1: Air volume fraction for bubbly / stratified transition flow at 40 D: 2 fields (top), 3 fields (bottom).

Figure 2 :

 2 Figure 2: Air volume fraction for bubbly / stratified transition flow at 40 D: 2 fields (top), 3 fields (bottom).

Table 1 :

 1 Test conditions selected.

	Flow regime	QL	JL	QG	JG
		(m 3 /h)	(m/s)	(m 3 /h)	(m/s)
	bubbly/stratified	125	4.42	30	0.068
	Stratified	125	4.42	60	0.136
	Bubbly	150	5.31	60	0.136

Acknowledgments

This work has been performed with the financial support of the French Alternative Energies and Atomic Energy Commission (CEA). The author would like to thank all the NEPTUNE_CFD team for providing technical support.


Herry Tanguy
email: tanguy.herry@cea.fr

Raverdy Bruno
email: bruno.raverdy@cea.fr

Mimouni Stéphane
email: stephane.mimouni@edf.fr

Evaluation of condensation models on TOPFLOW experiment

Keywords: Condensation, CFD, Steam-Water, Monodispersed approach

Some condensation models were evaluated in this work using TOPFLOW experiment, in which saturated steam bubbles are injected in sub-cooled water inside a vertical pipe. At different heights, the void fraction, velocity and bubble size distribution were measured as radial profiles. A monodispersed approach was used for CFD simulations with Neptune_CFD code. The results obtained with this code slightly overpredicts the void fraction at the first 1.5 meters of the pipe and underestimates it above 1.5m. The void fraction depends of heat and mass transfers between phases, which is linked to the interfacial area and so, to the fragmentation and coalescence phenomena. As a consequence, an analysis of the experimental results will be done to investigate the role of each phenomena. Then, the most appropriate condensation model will be used.

Introduction

The condensation play a significant role in nuclear field in case of sub-cooled boiling or bubbles entrainment caused by emergency core cooling (ECC) injection. For this reason, some models have been developed in order to predict as well as possible the condensation rate. The most frequently used is the Ranz-Marshall model [START_REF] Ranz | Evaporation from drops Part II[END_REF] and some others are presented in (Kim and Park 2011). The main advantage of working on this TOPFLOW experiment is that the walls are adiabatic so there is no boiling, which means that all heat and mass change are generated by the condensation phenomena.

Experimental Facility

The experiment is described in detail in [START_REF] Lucas | Experimental database on steam water flow with phase transfer in a vertical pipe[END_REF]. It consists of an injection of steam bubble in subcooled water at high pressure (65bar) inside a large vertical pipe with an inner dimater of 195.3mm and a length of about 8m. There are tests with injection orifices of 1mm and 4mm, which allows variation of initial bubble size distribution. The test case selected in the current work is the test 096 of K16 test series. The detail of this test case is provided in table 1. J l and J v are respectively the liquid and vapor superficial velocity, ∆T in the inlet subcooling and D orif ice the orifice diameter.

P [bar] J l [m/s] J v [m/s] ∆T in [K] D orif ice [mm] 65 1.017 0.0898 5.0 1.0 Table 1: Description of the selected test case

Model setup

Neptune_CFD is a 3D multifield CFD code using Eulerian approach [START_REF] Guelfi | NEPTUNE: A New Software Platform for Advanced Nuclear Thermal Hydraulics[END_REF]). The solver is based on a pressure correction method simulating multi-component multiphase flows by solving the three balance equations for each phase as follows. In those equations, the overline represents the statistical average, and a weighted average is introduced:

ξ k = ρ k ξ k ρ k (1)
The equations are given in their primary form with k = v, l denoting vapor and liquid phases:

∂ ∂t (α k ρ k ) + ∇(α k ρ k .v k ) = Γ k (2) ∂ ∂t (α k ρ k v k )+∇(α k ρ k v k .v k ) = ∇.(α k τ m k + τ T k ) -α k ∇P + α k ρ k g + Γ k v k I + I ′ k (3) ∂ ∂t (α k ρ k H k ) + ∇.(α k ρ k H k v k ) = -∇.(α k Q k ) + α k ∂P ∂t + Π k + q wk (4)
Γ k represents the volumetric mass transfer rate across the interface. The interphase mass transfer due to condensation will intervene in this term as:

Γ l = ql h f g = h l,k (T sat -T l )A i h f g (5)
h f g is the latent heat, A i the interfacial area density. By assuming a spherical morphology of interfacial structure, which is a common assumption for small bubbles, the interfacial area density is given by:

A i = 6α g d b (6)
α is the void fraction and d b the bubble diameter. The heat transfer coefficient h l is provided by the Ranz-Marshall correlation:

h l = λ l d b 2 + 0.6Re 0.5 b P r 0.3 l (7)
Then, a monodispersed approach is used with the interfacial area transport equation given by [START_REF] Ruyer | A bubble size distribution model for the numerical simulation of bubbly flows[END_REF]).

Computational grid and boundary conditions

Since the geometry and the flow field inside the pipe is axially symmetrical, the simulations can be done in quasi 2D, for a slice of 4°as it is shown in Fig. 1. The inlet conditions of volume fraction, gas velocity and bubble size distribution as well as liquid temperature and mass flow rate were defined according to experimental data. The outlet pressure is also defined according to experimental data. The wall has no-slip conditions for liquid and is assumed adiabatic.

Results and Discussion

Overall, using Ranz-Marshall correlation, Neptune_CFD code predicted well the condensation rate in the test case treated as shown in Fig. 2. However, in first quarter of the tube, where condensation plays a dominant role (from 6% of void fraction to 1%), the code slightly underpredicts the condensation and therefore, overpredicts the void fraction. This is in accordance with the results obtained with a polydispersed approach and Ranz-Marshall correlation on different test cases for this experiment [START_REF] Liao | A baseline closure concept for simulating bubbly flow with phase change: A mechanistic model for interphase heat transfer coefficient[END_REF]. Then, the code overestimates the condensation from 1.5 m, since all the vapor is condensed at 3.5m, contrary to the experiment.

Conclusion and future work

The Ranz-Marshall correlation has been evaluated in this work on the TOPFLOW experiment where saturated steam bubbles were injected in subcooled water. For the test case K16-096, the correlation seems to be adequate and gives rather satisfactory results with a monodispersed approach. This work is still in progress, this correlation as well as other correlations presented in (Kim and Park 2011) will be used for the other test cases of this experiment and will be compared. In addition of the axial evolution of cross-section averaged volume fraction, some local parameters will be compared between the code and the experiment. Indeed, radial profiles of void fraction, velocity and bubble size will be compared at each measurement point.

Figure 1 :

 1 Figure 1: Calculation domain

Figure 2 :

 2 Figure 2: Evolution of cross-section averaged volume fraction in the axial direction between experimental measurements and simulation results

th International Conference on Multiphase Flow, ICMF 2023, Kobe, Japan, April 2-7, 2023

Acknowledgments

The authors would like to thank all the HZDR team who provided the data of this experiment.


Francois Hammer

Yanbin Yang

Jianling Wang

Frederic Arenou

Mathieu Puech

Hector Flores

Carine Babusiaux

Typeset using L A T E X twocolumn style in AASTeX62

Keywords: Galaxy, general -galaxies, dwarf -(cosmology, ) dark matter

This paper presents an alternative scenario to explain the observed properties of the Milky Way dwarf Spheroidals (MW dSphs). We show that instead of resulting from large amounts of dark matter (DM), the large velocity dispersions observed along their lines of sight (σ los ) can be entirely accounted for by dynamical heating of DM-free systems resulting from MW tidal shocks. Such a regime is expected if the progenitors of the MW dwarfs are infalling gas-dominated galaxies. In this case, gas lost through ram-pressure leads to a strong decrease of self-gravity, a phase during which stars can radially expand, while leaving a gas-free dSph in which tidal shocks can easily develop.

The DM content of dSphs is widely derived from the measurement of the dSphs self-gravity acceleration projected along the line of sight. We show that the latter strongly anti-correlates with the dSph distance from the MW, and that it is matched in amplitude by the acceleration caused by MW tidal shocks on DM-free dSphs. If correct, this implies that the MW dSphs would have negligible DM content, putting in question, e.g., their use as targets for DM direct searches, or our understanding of the Local Group mass assembly history. Most of the progenitors of the MW dSphs are likely extremely tiny dIrrs, and deeper observations and more accurate modeling are necessary to infer their properties as well as to derive star formation histories of the faintest dSphs.

INTRODUCTION

The MW halo within ∼ 300 kpc is populated by the two Magellanic Clouds and by ten to a million times fewer massive dwarf-spheroidal galaxies (dSphs). Several tens of MW dSphs (McConnachie 2012; Muñoz et al. 2018;Fritz et al. 2018) have now been discovered and their number may continue to increase. Furthermore, observations of dSphs revealed 1D, line-of-sight (los) velocity dispersions (σ los ) that are considerably larger than expectations from their stellar mass, which has led to assume the presence of an additional, dominant component of dark matter (DM). The total mass (M J ), including DM, was estimated by assuming that in each dSph a single, dispersion-supported stellar population is in dynamical equilibrium in the underlying Corresponding author: Francois Hammer francois.hammer@obspm.fr gravitational potential (Walker et al. 2009;Wolf et al. 2010). Further assuming spherical symmetry and neglecting the impact of the MW gravitation, the dynamical mass was deduced from dSph star kinematics using the Jeans equation (Walker et al. 2009;Wolf et al. 2010).

However, Hammer et al. (2018) have shown that dSph Jeans masses (M J ) and their ratio to luminosity can be derived from the half-light radius (r half ) and the gravitational acceleration exerted by the MW, which appears to be at odds with the assumption that dSph kinematics are not affected by the MW. Furthermore, it was found that MW tidal shocks provide enough kinetic energy to dSph stars to predict the observed σ los with a good accuracy. This is supported by the fact that MW tidal shocks are exerted along the radial direction, which coincides with the line of sight along which σ los are measured. This casts doubts about the validity of the dynamical mass estimate using the Jeans equation, hence on the estimates of the DM amount in dSphs.

Why tidal shocks appear to be the dominant effect of the MW gravitational tides remains to be clarified. In the meantime, Muñoz et al. (2018) have considerably improved the photometric measurements of most dSphs, improving the accuracy on r half by one order of magnitude.

Here, we demonstrate from data analysis that, instead of DM, the MW gravitation through tidal shocks can fully account for the dSph kinematics. In Sect. 2 we show that, assuming self-gravity and using the Jeans equation, one can only predict the projected mass density along the line of sight. We then find that this quantity is highly (anti)correlated 1 with the MW distance compared to other DM-related quantities (e.g., mass or 3D mass density). In Sect. 3 we demonstrate that this is fully explained by the effect of MW tidal shocks on DMfree dSphs. We also characterize the time-scales that warrant the impulse approximation regime necessary for tidal shocks to be efficient. Sect. 3 discusses an astronomical scenario in which gas removal by ram-pressure of gas-rich dSph progenitors ensures the predominance of tidal shocks in MW dSphs, then providing dSph properties very similar to that observed. The sample of 21 dSphs with kinematic data and after removing three outliers has been defined in the Appendix A in which scaling relations are analyzed, and which includes two Tables showing the data and the calculated quantities, respectively. In the following we assume M stellar /L V ratio of 2.5 for dSphs without young or intermediate age stars, and 1.5 for Carina, Fornax, and Leo I (see, e.g., Lelli et al. (2017) and references therein).

LIMITATIONS ON THE USE OF THE JEANS

EQUATION TO ESTIMATE DM Wolf et al. (2010) derived the dSph total mass within r 1/2 =4/3 × r half (r 1/2 and r half are the 3D and 2D halflight radii, respectively, see also Walker et al. 2009). Assuming self-equilibrium and using the Jeans equation:

M J /M = 930 × (r half /[1pc]) × (σ los /[1km × s -1 ]) 2 (1)
The above mass estimate (see, e.g., Walker et al. 2009;Wolf et al. 2010) further assumes that the DM is spherically distributed, although attempts have been made to minimize the impact of the unknown value of the velocity dispersion anisotropy (Wolf et al. 2010). Since the 1 Along the manuscript we have used a Spearman's rank correlation ρ that does not assume any shape for the relationship between variables; the significance of ρ has been tested using t= ρ (n -2)/(1 -ρ2 ), which is distributed approximately as Student's t distribution with n -2 degrees of freedom under the null hypothesis.

Figure 1. Left: logarithmic anticorrelation between the gravitational acceleration in kms -2 that would be due to the DM according to Eq. 3 and the distance to the MW center in kpc. The best fit for the 21 dSphs is represented by the solid line, while the dashed line represents the best fit after excluding Leo I and Leo II (magenta dots), since they do not obey to the impulse approximation. Right: same after replacing the MW distance by √ 2 × αMW × gMW, which is the expected acceleration due to the MW in the frame of the impulse approximation (see Eq. 5). The MW mass profile is coming from (Sofue 2015, top) and from (Bovy 2015, bottom), respectively. determination of DM is based on a los velocity dispersion (e.g., σ los ), only the mass density projected along the line of sight can be determined without any assumption on anisotropy 2 . Under the spherical symmetry assumption, it is also a surface mass density (Σ J ) proportional to M J × r -2 half . The los projected mass density is proportional to the gravitational acceleration caused by the total mass M J , whose projection along the line of sight is assumed to induce the observed los velocity dispersion σ los . One may try to isolate the effect of the DM in Eq. 1 by removing the small and known contribution of the stellar mass. Within r= r 1/2 , the Jeans mass is the sum of the DM and stellar masses, and the DM surface-mass density is then:

Σ DM = M DM × r -2 half = (M J -M stellar /2) × r -2 half , (2)
In the following, we estimate 3 the 1D los velocity dispersion (σ stars ) associated with the sole stellar mass by inverting Eq. 1 and replacing M J (r 1/2 ) by M stellar /2. By multiplying Eq. 2 by the gravitational constant G it leads to the acceleration caused by the DM (a DM = GM DM × r -2 half ) for which only its projection along the line of sight is robustly known, which is:

a DM = GM DM × r -2 half = (σ 2 los -σ 2 stars ) × r -1 half . (3)
Left panel of Figure 1 shows that the acceleration due to the DM is strongly anticorrelated (ρ = 0.76 and t = 5.1, see Appendix C) with the distance to the center of the MW (D MW ). Could this be due to a combination of correlations, e.g., between surface mass density (Σ DM ) and radius, and between radius and MW distance? Appendix A.2 indicates that σ los correlates well (ρ=0.80 for the 21 dSph sample) with r half , following: log(σ los ) = 0.37 × log(r half ) + 0.028 (4)

Combining Eq. 4 with Eq. 1, it results that M J scales as r 1.75 half . If spherical symmetry is assumed, Σ DM = M J ×r -2 half should depend weakly on radius, i.e., ∝ r -0.25 half . Such a behavior is verified by the data (logarithmic slope=-0.33 with ρ= 0.49). The radius is weakly correlated with the distance (see Appendix A.3, logarithmic slope=0.87 with ρ= 0.45). Combining a weak anticorrelation with a weak correlation can neither provide the strong anticorrelation nor the slope (-0.62) of the relation shown in the left panel of Figure 1.

Here we have discovered that the most accurately determined DM quantity, Σ DM (or a DM ), is unexpectedly anticorrelated to the MW distance. This indicates that MW dSphs cannot be further considered as isolated, which questions the validity of the DM mass estimates provided by self equilibrium deduced from the Jeans equation.

The left side of Figure 2 is made up of three panels showing the slope, correlation strength, and Student t parameter of a generalized correlation between (σ 2 los -σ 2 stars ) × r power half and D MW . By replacing r -1 half by r power half in Eq. 3, it identifies the DM mass and its 1D to 3D-mass densities (see indications in the bottom panel), after neglecting the effect of MW tides. As for Eq. 1, the above equations assume spherical symmetry for the dSphs and DM-related quantities are calculated at r= 3 When calculating the 1D velocity dispersion of a Plummer body made of stars only, we cannot assume a constant sigma profile as done in Eq. 1; Eq. 31 of Evans & An (2005) Wolf et al. (2010). It leads to a value of σ starsonly = 0.9127 × σstars.

yields 4.80/G × σ 2 starsonly × r half , instead of 4/G × σ 2 stars × r half in Eq. 2 of
r 1/2 = 4/3 × r half .
The right side of Eq. 3 illustrates dSph observed quantities (σ los , r half , M stellar ) without any dependence on modeling or on the nature of dSphs, and Figure 2 is only based on them. Concerning the DM content of dSphs, Figure 2 is illustrative because:

• Data show significant anticorrelations for power ≤ -1, with a prominent peak at r power= -1 (Σ DM ), i.e., for the quantity robustly estimated from observations; it suggests that the anticorrelation between the DM 3D-mass density, ν DM , and the MW distance is only due to that between Σ DM with D MW ;

• Data confirm the absence of correlation between σ los and distance (r power = 0), as well as between the assumed total DM mass (r power = 1) derived from Eq. 1 and the MW distance.

This may also explain why the "DM in isolated dSph" hypothesis received so much credit, given that its associated mass appears to not be correlated with the host galaxy properties. Reproducing Figures 1 and 2 is likely a challenge for the standard scenario of DM-dominated MW dSphs. For example, why by multiplying Σ DM (or a DM ) by r half , which is weakly correlated with the MW distance, the strong anticorrelation found in Fig. 1 disappears (at r power= 0 in Fig. 2)?

In fact an anticorrelation between DM 3D-mass density (ν DM ) and distance to the MW has been found by Diemand et al. (2008) and by Moliné et al. (2017), in the frame of a tidal stripping of pure DM subhalos, which would be denser when lying near the MW center. Could such an anticorrelation be at the root of the relation shown in Figure 1? Top left panel of Figure 2 reveals that ν DM shows a logarithmic slope of -2.9, implying that at 30 kpc from the MW, sub-haloes would be 340 times denser than at 250 kpc. However, stellar densities stay unchanged within the same MW distance range, and it would require a considerably more efficient stripping of DM4 than that of stars while they lie within a similar volume in the dSph (see details in Appendix B).

WHY IS THE ACCELERATION CORRELATING WITH THE MW DISTANCE?

Indeed the anticorrelation between (σ 2 los -σ 2 stars )×r -1 half and D MW hints to Eqs. ( 2) and (B16) of Hammer

s 2 r s 2 s 2 /r s 2 /r 2 M DM M DM /r S DM n DM -1 -1 1 1
Correlation of s 2 x r power with D MW s 2 x r power with 2 1/2 a MW g MW (see Eq. 5)

Figure 2. Summary of logarithmic correlations (or absence of correlations) between dynamical quantities and DMW (left) and √ 2 × αMW × gMW (right), respectively. From top to bottom, the figure shows the slope, correlation strength, and Student t parameters when Eq. 3 is replaced by (σ 2 los -σ 2 stars ) × r power half (dubbed σ 2 r power ). Left: It shows how the correlation changes with the power index on r half . Observed physical quantities are indicated in the middle panel (right side) of Eq. 3. Calculations of DM-related quantities require the assumption of isotropy and absence of significant effects from MW tides (see the text), and they are indicated in the bottom panel. They include the mass and its 1D, 2D, and 3D density, MDM, MDM/r, ΣDM and νDM respectively. Correlation strength and t peak at power = -1, which represents the DM los acceleration (see Eq. 3). Black and green curves are based on the dSph galaxy sample of 21 dwarfs defined in the Appendix A, while the blue curve is for the same sample but Leo I and Leo II. The green curve illustrates a case for which the M stellar /LV ratios are multiplied by 2. Right: same but for the correlation with √ 2 × αMW × gMW, for which we use the MW model from Bovy (2015). Using the MW model from Sofue (2015) provides very similar graphs.

et al. (2018) established in the frame of the impulse and distant-tide approximations, and assuming DM-free dSphs. They show that MW tidal shocks bring sufficient kinetic energy to generate an additional term to the los velocity dispersion, which is σ MW = σ 2 los -σ 2 stars . Note that both accelerations, which are due to MW tidal shocks on DM-free dSphs and that are assumed to be due to DM in the frame of self-gravity, are indeed projected along the MW radial direction that coincides with the line of sight. Eq. B16 of Hammer et al. (2018) can be written as:

σ 2 MW = σ 2 los -σ 2 stars = √ 2 α MW g MW r half ( 5
)
where g MW = GM MW /D 2 MW is the gravitational acceleration of the MW and

α MW = 1 -∂log(M MW )/∂log(D MW ) (6)
characterizes the shape of the MW mass profile, taking values close to 1 in the Galactic outskirts (∂M MW /∂D MW ∼ 0) and close to 0 when the slope of the MW mass profile is steep.

Can Eq. 5 be at the root of Figure 1? To verify this we have to compare (σ 2 los -σ 2 stars ) × r -1 half with √ 2×α MW ×g MW . The later term requires the knowledge of the MW potential and we have considered two models reproducing the MW kinematics (Bovy 2015) and extended rotation curve (Sofue 2015) from observations of distant massive stars and globular clusters. Right panels of Figure 1 show that the correlation between the two quantities is as strong as the one shown in the left panel. Moreover, for most dSphs, within their error bars, the los gravitational acceleration attributed to the DM self-gravity equals that caused by MW tidal shocks on DM-free dSphs. Why would the acceleration caused by the DM be precisely what it is expected from MW tidal shocks on DM-free dSphs? In other words, why does the latter predict that the DM mass-to-light ratio of Segue 1 is several thousand, while that of Fornax is only 10?

Figure 3 shows the comparison between predictions from Eq. 5 to the observed los velocity dispersions (σ los ) in dSphs. In the left panel, MW model from Sofue 2014) by adopting a softening radius of 0.5 pc, i.e., warranting that almost no stellar particles are affected by non-Newtonian motions (see Figure 4). Right: same after optimizing the MW mass profile.

(2015) is found to predict with good precision the observations, with slope = 0.9 and a correlation coefficient ρ = 0.78. Similar values (slope = 1.17 and ρ = 0.89) are found when using the MW model from Bovy (2015). We notice that the two models are based on the Navarro et al. (1997) density profile that does not converge at large distances. It indeed excludes values of α MW larger than 0.7, which allows significant room to use more appropriate MW profiles and optimize them. Although it is beyond the scope of this paper to derive the MW total mass and its profile, the right panel of Figure 3 illustrates the result of a very first attempt, which leads to a total MW mass and a mass profile in very good agreement with mass and extended rotation curve profiles from Bhattacharjee et al. (2014) and Huang et al. (2016).

The right panel of Figure 2 illustrates the correlation between (σ 2 los -σ 2 stars ) × r power half and √ 2 α MW g MW , the former term involving only observed properties, the later term involving only the MW gravitation and mass density profile. It is essentially similar to the left panel, although the slope of the correlation is exactly 1 for power = -1, which indicates that Eq. 5 fully accounts for energy exchanges between the MW and the dSphs along the line of sight, and that within an acceleration range of almost 2 decades.

Eq. 5 applies to DM-devoid dSphs orbiting into the MW halo, and it is based on the fact that the sizes of the dSphs are much smaller than their distances to the MW (the distant-tide approximation) allowing one to account for smooth variations of the MW poten-tial across dSphs (see Eq. 8.34 in [START_REF] Binney | Galactic Dynamics: Second Edition[END_REF]. Furthermore, Eq. 5 requires that dSph stars undergo a fluctuation of the MW gravitational potential over time scales that are smaller than their presentday crossing-time scales (t cross = r half /σ los ). It has been demonstrated by Aguilar & White (1985) that such an impulse approximation holds even for slow encounters (or those with rather low-eccentricities), i.e., if the encounter is as long as, or a few times longer, than t cross (see also [START_REF] Binney | Galactic Dynamics[END_REF], 2008).

This situation might seem similar to that of globular clusters (GCs) progressively destroyed by tidal shocks during their passages near the MW bulge (Aguilar 1993;Gnedin et al. 1999;Gnedin & Ostriker 1999). However, dSphs have central densities 10,000-100,000 times smaller than GCs, and their interaction with the MW potential is much more destructive during a single dynamical time t dyn = r half /σ starsonly (see footnote 3), which ranges from 5 × 10 7 to 9 × 10 8 yr (see the second table of Appendix A). Gnedin & Ostriker (1999) showed that it takes more than 3 (resp. 10-15) dynamical times for a system to virialize (resp. relax) after a perturbation. In the course of their long orbits5 within the halo, dSphs are affected by changes of the energy brought by the MW tides. The encountering time (t enc ) can be evaluated by assuming that these changes have to be larger than the kinetic energy associated with the self-gravity of the DM-free dSphs (1/2 × σ 2 starsonly ). Differentiating Eq. 5 yields:

∆(σ 2 MW )/∆(D MW ) = -2 √ 2α MW ×GM MW /D 3 MW r half , (7)
in which we have neglected the (slow) variation of α MW with D MW . Then:

∆(σ 2 MW ) = -2 × σ 2 MW × ∆(D MW )/D MW , (8)
hence a distance change of ∆(D MW ) induces a change of ∆(σ 2 MW ) equal to twice the kinetic energy due to DMfree dSph self-gravity (1/2σ 2 starsonly ), if:

∆(D MW ) = σ 2 starsonly /σ 2 MW × D MW /2 (9)
Given the small dSph stellar masses, the ratio σ 2 starsonly /σ 2 MW is usually so small (see Appendix A) that changing D MW by only a few percent warrants a kinetic energy change larger than 1/2 × σ 2 starsonly . In the impulse approximation, the perturbation causes an instantaneous change in the velocity of each star that depends only on its position (Aguilar & White 1985). It holds when t cross is larger than the encountering time t enc =∆(D MW )/V GSR , and eventually leads to Eq. 5. Another way to establish the validity of Eq. 5 is to consider the stars that are in resonance with the MW gravitational perturbation: following Weinberg (1994) and Gnedin et al. (1999) the fraction of resonant stars follows (t enc /t cross ) -1 , which is 100% for most dSphs. Values of t enc /t cross are below 1 for most dSphs, but larger than 10 for Leo I and Leo II (see Appendix A). Intermediate values for Carina and Fornax are likely yielding results consistent with the impulse approximation even if the conditions are not strictly satisfied (Aguilar & White 1985;[START_REF] Binney | Galactic Dynamics[END_REF]. Moreover, stars in these dSphs would not have time to come back to equilibrium (virialization), since the ratio between t enc and the virializing time (3 ×t dyn ) is always below or very close to 1, except for Leo I and Leo II. It is noteworthy to see that Leo I and Leo II are also outliers in the relations shown in Figure 1 (see magenta dots).

TIDAL SHOCKS MAY EXPLAIN THE MW DSPH KINEMATICS

It is widely accepted that dSph progenitors have probably lost their gas through ram-pressure effects caused by the diffuse MW halo hot gas (Mayer et al. 2001;Grcevich & Putman 2009). This is supported by a strong dichotomy in dwarf gas content, those beyond 300 kpc being gas-rich, the others being gas-free (Grcevich & Putman 2009). The old stellar ages and low metallicities of most of the dSphs have been often considered as evidence for an ancient infall of primordial dwarf galaxies. However, there is mounting evidence for a more recent infall of most dSphs. 74% of them (Fritz et al. 2018) have orbital motions along a gigantic plane perpendicular to the MW disk suggesting a common infall (Pawlowski et al. 2014) with the Magellanic Clouds (MCs), which also belong to this gigantic plane and are at their first passage (Kallivayalil et al. 2013). The MCs show large orbital eccentricities, a property shared by most dSphs as inferred from Gaia measurements (Fritz et al. 2018), with half of them having apocenters in excess of 300 kpc. This is especially true for MW mass6 and profile consistent with most MW structural parameters (Bovy 2015). In the following we show how the above astronomical context leading to Eq. 5 can fully account for the large measured values of dSph velocity dispersions.

Eq. 5 has been established under the impulse approximation regime (tidal shocks), which is robust for all dSphs but Leo I and II. The factor √ 2 in Eq. 5 results from the assumed Plummer model for the stellar density in dSphs, assuming further a spherical symmetry and using a radius that is the geometric mean between the major and minor half-light radii. Adopting different models does not affect Eq. 5 by more than a few percent (Hammer et al. 2018). However, Eq. 5 does not account for tidal stripping. The global instantaneous energy change ∆E caused by the MW tides on an individual star with velocity v is:

∆E = v • ∆v + 1/2(∆v) 2 (10)
where ∆v is the change in velocity (see Eq. 7-41 of [START_REF] Binney | Galactic Dynamics[END_REF]. If dSphs are spherically symmetric, the first term (called "tidal stripping" or "diffusion term," see Binney & Tremaine 2008, P. 663) vanishes when averaged over all stars letting the second term (called "tidal shocking" or "heating," here approximated to 1/2(∆σ) 2 ) lead to Eq. 5 (Hammer et al. 2018). The first comprehensive simulations of DM-devoid dwarfs falling into the gravitational potential of the MW has been done by Piatek & Pryor (1995), Kroupa et al. (1997), and Klessen & Kroupa (1998), all assuming gas-free progenitors. During their approach, progenitors are initially at large distances from the MW, which results in weak gravitational field and | ∆v | | v |, leading to a dominant first term in Eq. 10 as described by the diffusive regime (see Binney & Tremaine 2008, P. 663). This led Piatek & Pryor (1995) to model dSphs that experience large star losses by tidal stripping. However, these modeled dSphs were too elongated along the line of sight and had kinematics that were not consistent with observations (see also Klessen et al. 2003;Read et al. 2006) Figure 4 illustrates the overall process from the gas removal by ram-pressure to the expansion of the stars, and later on, how tidal shocks exerted by the MW dynamically heat the stars along the MW direction, which coincides with the line of sight. Figure 3 (see also Fig. 5 of Hammer et al. 2018) shows how hydrodynamical simulations from Yang et al. (2014) based on gas removal by ram-pressure and then tides from MW can reproduce expectations from Eq. 5. These can reproduce the morphologies (see Figs. 9 and 10 of Hammer et al. 2018), the stellar density (see Appendix A.2), and the flat velocity dispersion profiles along the dSph radii (see Figs. 5 and 6 of Yang et al. 2014, see also calculations in Hammer et al. 2018). The prominence of tidal shocks in the case of gas-rich progenitors has been shown by Kazantzidis et al. (2017), who also found that most of the very elongated shapes or bar-like features obtained in gas-free simulations ( Lokas et al. 2012) then vanish.

Properties of 'tidally-shocked,' DM-devoid dSphs are likely undistinguishable from those observed. Tidal features are then not expected to be observed in DM-devoid dSph at recent infall, because MW tidal shocking dominates tidal stripping. This is consistent with the two following observations: (1) there are no obvious tidal features in the relatively bright, so-called classical dSphs except perhaps for Carina (Battaglia et al. 2012), which is likely near its pericenter (Fritz et al. 2018); and (2) wide-field observations of several dSphs have revealed stars beyond their DM tidal radius (Carina: McMonigal et al. 2014;Draco: Kleyna et al. 2001;Ségall et al. 2007, Fornax: Battaglia et al. 2006). Conversely to Aguilar & White (1985), Gnedin et al. (1999), and Gnedin & Ostriker (1999), we have not examined the theoretical evolution with time of the DM-devoid dSphs, although our modeling indicates also mass losses, in particular, for the smallest ones. Tidal shocking could lead to disrupt these objects, and large parts of their stellar content can be expelled toward their surroundings. Hercules is likely in such a stage (Roderick et al. 2015;Garling et al. 2018) and [START_REF] Küpper | [END_REF] recently modeled it as an exploding stellar system after two passages, concluding to an absence of a DM dominant component. This supports our conclusions, since Hercules lies on the relations shown in Figures 1 to 3.

DISCUSSION: THE DM-FREE MW DSPHS IN THE NEAR-FIELD COSMOLOGY CONTEXT

We propose a reasonable scenario that successfully explains the MW dSph properties and that excludes DM as being their main mass component. However, before accepting such a major change of paradigm in near field cosmology, one has to verify whether (1) most MW dSphs can be at their first passage, (2) what are their progenitors, and (3) how this could be consistent with their star formation histories.

A first passage for most dSphs?

A very massive MW (e.g., ∼ 2×10 12 M ) would lead to more circular orbits for the dSphs after considering their Gaia DR2 proper motions. For example, Fritz et al. (2018) found that one-third of them have orbits with eccentricities larger than 0.66 for their heavy MW mass model (1.6×10 12 M ), a fraction that increases to twothirds for their light mass model (0.8×10 12 M , see also Bovy 2015). Hence for a less massive MW, typically smaller than, e.g., 1.3×10 12 M , most dSphs including the numerous ultra-faint dwarfs, would have large orbital eccentricities suggesting a first infall. This is consistent with the fact that most of them lie in a vast polar structure (Pawlowski et al. 2014) that also includes the Magellanic Clouds, which are at their first passage.

Gaia DR2 has also considerably improved our knowledge of the MW mass distribution up to 20-50 kpc, by establishing a more accurate rotation curve (Eilers et al. 2019;Mróz et al. 2019), better constraints on the Globular Cluster motions (Eadie & Jurić 2019) and on the estimates of the escape velocity (Deason et al. 2019, see also Grand et al. 2019). Even if they are still using extrapolations of the mass profile to the outer halo, these studies provide MW masses ranging from 0.7 to 1.3×10 12 M , i.e., allowing a scenario of first passage for most dSphs.

What are the progenitors of DM-free dSphs?

Due to their proximity, MW dSphs are by far the tiniest galaxies for which detailed properties can be derived. Their progenitors are likely gas-rich galaxies, for which Lelli et al. (2016a) provided the most insightful report of their properties, including their rotation curve, their gas content, and the 3.6µm photometry for deriving their stellar masses. However, among the sample of 21 dSphs studied here, only Fornax possesses a sufficiently large mass (3×10 7 M ), to be compared to, e.g., galaxies belonging to the lowest stellar mass range of the Lelli et al. (2016a) study, all of which are dIrrs.

Here we consider the nine dIrrs for which the distance is accurately determined (from the tip of the Red Gi- 3. It has been improved by adopting a gravitational force softening of 0.5 pc instead of 50 pc for stellar particles (17 M /particle, 500,000 particles) in dwarf galaxies. Given the stellar density in dSphs, there should be no or very few stellar particles that can be affected by the undesirable effects due to the softening. These simulations are based on the hydrodynamical / N-body code Gadget2 (Springel 2005), and use simulated DM-free galaxies with initial mass and gas fraction (within a 3 kpc projected radius) of 1.35×10 8 M and 71%, respectively. The simulation in the video begins at TPERI = -1.635 Gyr and ends at 0.839 Gyr. The video duration is 71 s.

ant Branch) and with L 3.6µm smaller than 10 8 L , corresponding to a stellar mass of 5×10 7 M according to Lelli et al. (2016a). Among them, two (UGC 07577 and CamB) show no evidence for an excess of mass from their rotation curves, while for four (WLM, D564-8, ESO0444-G084, and NGC 6789) DM seems mandatory to explain their rotational velocity within R 2.2 (= to 2.2 times the disk scalelength). For the remaining three galaxies (UGC 0443, NGC 3741, and DDO154) the necessity for DM within the R 2.2 radius is rather marginal, especially when accounting for systematic errors due to the assumed inclination. We verify that the above results are unchanged by increasing the L 3.6µm limit to 2×10 8 L .

It seems quite puzzling at a glance that only DM-free dIrrs are reaching the MW halo, while others would have avoided such an infall. However:

• A shallow-distributed DM could be preferentially stripped lowering its fraction in the transformed galaxy (Kazantzidis et al. 2011;Lokas 2016), including perhaps in their central region if DM particles have radially biased orbits compared to that of stars.

• Alternatively, the MW dSph progenitors could have come within a group (D'Onghia & Lake 2008) at first infall, for which, e.g., the brightest galaxies could be DM-free, which cannot be excluded from the small statistics discussed above. Progenitors might also be gas-rich, tidal dwarf galaxies that are devoid of DM as suggested by Metz & Kroupa (2007). For example, tidal dwarf galaxies have been used with success to reproduce the Magellanic Stream, the Leading Arm, and the disks of satellites surrounding both MW and M31 (Fouquet et al. 2012;Hammer et al. 2013Hammer et al. , 2015;;Yang et al. 2014;Wang et al. 2019).

The above hypotheses might appear too radical and one may need to investigate further the progenitor problem. For example, the identification of gas-rich progenitors of the MW dSphs is possibly far more complex than that provided by the above short analysis. In an infalling DM-free dIrr scenario, the fraction of stars that have escaped the initial galaxy may be significantly high for some orbits, possibly reaching values up to 90%, and also depending on a detailed description of the gas removal by ram-pressure exerted by the MW halo gas. Under these conditions even Sculptor (M stellar = 5×10 6 M ) could have a progenitor within the lowest mass range of Lelli et al. (2016a) galaxies. More problematic is the fact that it becomes unclear to which radius DM has to be investigated in the progenitors, perhaps in a region more central than R 2.2 .

Another limitation is due to the fact that stellar masses of field dIrrs have been determined through stellar population synthesis models, i.e., in a very different way than for star-resolved dSphs. Detailed studies of the star formation and chemical enrichment histories lead to M stellar /L V values in excess of 2.1 for Fornax, and up to 4.3 for Sculptor (de Boer et al. 2012a,b). Quoting [START_REF] Kroupa | Galactic Structure and Stellar Populations[END_REF], "the chemical evolution modeling of the Fornax dwarf-spheroidal satellite galaxy demonstrates that this system must have produced stars up to at most about 25 M in agreement with the prediction of the integrated galactic initial mass function (IGIMF) theory given the low star formation rate ∼ 3×10 -3 M yr -1 deduced for this system when it was forming stars in the past." Since de Boer et al. ( 2012a,b) assumed a Kroupa (2001) IMF that is extending to large star masses (120 M ), the stellar mass of Fornax is perhaps underestimated. This can also apply to the Fornax progenitor as well as to most dIrrs, i.e., their IMF could be rather bottom-heavy according to the IGIMF theory [START_REF] Kroupa | Galactic Structure and Stellar Populations[END_REF], which could lead to large values of M stellar /L 3.6µm . For the later, larger values than 0.5 are not excluded by the baryonic Tully Fisher relation according to Lelli et al. (2016b), and, for example, a value of 1 would be consistent with an absence of DM in the R 2.2 radius of UGC 0443, NGC 3741, and DDO154.

Even if one may succeed in identifying the progenitor of Fornax (and perhaps of Sculptor), it leaves unknown the properties of the progenitors of the 19 remaining dSphs studied in this paper. There have been pioneering studies of very tiny dIrrs including the SHIELD (McNichols et al. 2016), the LITTLE THINGS (Oh et al. 2015), and the VLA-ANGST (Ott et al. 2012) surveys, reaching galaxies with stellar masses from 3×10 4 to 5×10 6 M , i.e., below the smallest dIrr mass values of Lelli et al. (2016a). Lelli et al. (2016a) have excluded the LITTLE THINGS dIrrs because of irregularities in their rotation curves possibly due to noncircular events. Some attempts (see Oh et al. 2015) have been done to estimate their rotation curves, using the titled ring method that assumes an infinitesimally thin HI layer (see [START_REF] Bosma | Outskirts of Galaxies[END_REF]. However, such low-mass objects have almost a spheroidal geometry (Sánchez-Janssen et al. 2010) and a velocity amplitude similar to that of their dispersion, the latter being mostly associated to star formation and turbulence (Stilp et al. 2013). This likely hampers a robust determination of their rotation curve and hence of their DM content. Stilp (2013) concluded that "rotation curves for the smallest dwarfs in our (VLA-ANGST) sample may be difficult to obtain due to their complex velocity fields" (see also Ball et al. 2018 for Coma P, Oh et al. 2015for DDO 210, McNichols et al. 2016for AGC 748778, and Bernstein-Cooper et al. 2014 for LeoP). This does not mean that there is no DM in the tiniest dIrrs, though it points out the need for a significantly more robust (3D) modeling of individual objects to disentangle effects due to star formation from those due to gravitation. 5.3. Are DM-free MW dSphs consistent with the star formation history?

The above scenario might also appear at odds with the star formation histories of dSphs, which are often dominated by a single, very old population (Weisz et al. 2014). However, star formation histories could be complex and difficult to interpret, especially if unknown amounts of initial gas have been removed by ram-pressure effects leading to a subsequent decrease of self-gravity and then to an expansion of stars within a spherical symmetry. For example, during the last few hundreds of megayears, the star formation history of Fornax (de Boer et al. 2013) has been consistent with a recent gas removal by ram-pressure and then with the tidal shock scenario. While this also applies to Carina, Leo I, and perhaps to Leo II, the cases of Sculptor, Sextans, UMi and Draco (Weisz et al. 2014) are perhaps more problematic. For example, why did the star formation in the Sculptor progenitor stop about 5 Gyr ago (de Boer et al. 2012a)? Only a full hydrodynamical simulation with a well determined orbital history for Sculptor could help us to verify this potential inconsistency. Recently, FIRE2 simulations (Garrison-Kimmel et al. 2019) have shown that gas-rich dwarfs with stellar mass ranging from 10 6 to 10 7 M have similar star formation histories to Sculptor. Beside this, most of dSphs are much fainter than classical dSphs and they show large orbital eccentricities that may be consistent with a first infall (Simon 2018), which might be difficult to reconcile with a single old stellar population (Weisz et al. 2014). Perhaps their progenitors were so tiny that these systems were unable to form stars for very long periods. Alternatively, one needs to verify whether or not the faintest dSphs may also contain an intermediate age population, and this is likely out of our reach with the present data because of the small number of giant stars in these very scarcely populated galaxies (V. Hill 2019, private communication).

CONCLUSION

The DM content of dSphs is calculated through the DM projected density along the line of sight. Here we demonstrate that the recently improved data from Muñoz et al. (2018) and Fritz et al. (2018) reveal that the gravitational acceleration attributed to the DM selfgravity is strongly anticorrelated to the MW distance. Assuming a mass modeling of the MW consistent with its kinematics and rotation curve, we find that the los gravitational acceleration attributed to the DM selfgravity is precisely equal to that caused by the MW tidal shocks on DM-free dSphs.

This leads us to conclude that all the observed properties of the dSphs can be explained by the action of the MW only, without having to include any DM in dSphs. In DM-dominated models of dSphs the DM is somewhat shielding the action of the MW, and the DM total mass appears not to be correlated to the MW distance or its gravitational action. The last point is in contradiction with the observed anticorrelation (ρ = 0.76 for 21 dSphs, 0.83 without Leo I and Leo II) between the MW distance and the los projected acceleration, which is the only quantity related to observations. The probability that this is just a coincidence is only 3×10 -4 , which can be conservatively considered as the chance that DM impacts the kinematics of dSphs. This result questions the search of DM in these evanescent stellar systems, as well as our understanding of dwarfs in the Local Group and their role in cosmology.

To further investigate about the origin of DM-free MW dSphs would require (1) the detection and the analysis of their progenitors that could be extremely low-mass (M stellar from 10 4 to 10 6 M ) gas-rich galaxies in the field, (2) a significant progress in estimating their stellar ages through the turn-over of their main-sequence stars, and (3) an accurate determination of dSph orbits and of the MW mass distribution to perform detailed and case-by-case hydrodynamical simulations.

We are very grateful to Hadi Rahmani for a careful reading of this paper. We warmly thank Pavel Kroupa for his very useful comments on an early version of the manuscript and Paolo Salucci for his suggestions. We are grateful to the referee for the very insightful comments that have considerably improved the paper's content. This work was granted access to the HPC resources of TGCC/CINES/IDRIS made by GENCI, 2018) provide one of the most complete and up-to-date list of 39 dSphs with spectroscopic data on individual stars that can be used to estimate their line-of-sight velocity dispersions (σ los ). The list also includes objects whose nature is still under debate (e.g., Crater I). Our goal is to test the DM estimate in dSphs lying in the virial halo of the MW, i.e., within 300 kpc, leading us to exclude Phoenix (which is indeed not a dSph) and Eridanus II. Objects with only upper limits on σ los must also to be rejected either because their nature as a dSph or a star cluster is under debate (Crater I) or because they cannot lead to DM mass estimates (Segue II, Hydra II, Triangulum II, Tucana III, and Grus I). Velocity dispersion estimates are likely contaminated by the presence of binary stars, which particularly affects galaxies with only a few stars spectroscopically measured (see, e.g., Spencer et al. (2017)). In order to keep the number of dSphs sufficiently high we chose to keep targets having eight or more stars to measure σ los , i.e., excluding Bootes II, Carina III, Horologium, and Pisces II. Finally, since we aim at estimating the influence of the MW on the DM measurements, we chose to further exclude Carina II, Reticulum II, and Hydrus, which lie in between the MW and the LMC.

It results that the sample of dSphs inhabiting the virial halo of the MW and having robust kinematics data for evaluating the DM estimates includes 24 galaxies, enabling us to study the correlation between their fundamental parameters, i.e., their luminosities, radii, velocities,and distances.

A.2. Correlations between L V , r half , and σ los Table 1 lists the properties of the sample of 24 dSphs having robust kinematics data. Half-light radii (r half ) and V luminosities (L V ) are taken from Muñoz et al. (2018), assuming Plummer density profiles. Following McConnachie (2012) we calculate the geometric mean half-light radius between the major and minor axes to account for the presence of highly elliptical systems. Left panel of Figure 5 shows the very strong correlation for the 24 dSphs but Crater II (ρ= 0.87, t = 8.1) between L V and r half , leading to log(L V ) = 2.93 log(r half ) -1.48. Such a correlation can also be Fornax Sag CraterII BootesI Hercules Figure 6. Logarithmic correlation between half-light radius (r half ) and line-of-sight velocity dispersion (σ los ). Solid lines indicates the least-square fit, and the dashed-dotted line indicates the fit after combining the relations between (LV , r half ) and (LV , σ los ). The difference between the slopes calculated in both ways is taken as the uncertainty.

seen in Fig. 6 of McConnachie (2012). This means that the visible luminosity scales as r 3 half . Surface-brightness limit may affect the scaling. However, only the discovery of many Crater II type galaxies could change it significantly.

Assuming a Plummer profile, dSphs have an average central density of ν 0 = 0.0079 L pc -3 , a value about 20 times smaller that of the stellar density in the solar neighborhood. While the scatter between dSph values is quite large (the minimal value is ν 0 = 0.00025 for Aquarius, a value ∼ 350 times smaller than the largest one, for Leo I), the range of L V values is huge (a factor of 21,000). Also, note that Crater II is definitively an outlier with an ultra-low density, 33,000 lower than the average of other dSphs.

Figure 5 (right) shows that L V scales with σ 8.88 los (ρ = 0.68, t = 4.2), i.e., with more than twice the power value expected from the baryon Tully Fisher relationship. This discrepancy has already been noted by McGaugh & Wolf (2010). Combined together, the two relations displayed in Figure 5 lead to the relationship between r half , and σ los shown in Figure 6 (see the dashed-small dashed line). The latter has also been found by Walker et al. (2009, see their Fig. 6), though the improved data measurements from Muñoz et al. (2018), as well as our choice of a geometrical mean for r half (see, e.g., the Hercules location that is no more discrepant), have considerably reduced the scatter and increased the significance (ρ= 0.76, t = 5.4 for 23 galaxies). We also notice that the slope of the relation (0.37±0.04, see Eq. 4) is about twice that from Walker et al. (2009).

A.3. Correlations with D MW

Figure 7 shows the correlations between L V , r half , and σ los with the MW distance. After discarding the outlier Sagittarius, there is a weak correlation between the V luminosity and the distance (ρ = 0.56, t = 3.1), which can be explained by the fact that only the brightest dSphs have been fully detected in the MW halo. The strength of the L V -r half correlation is sufficient to explain the very weak correlation between r half and distance (ρ = 0.42, t = 2.2). Note that σ los does not correlate at all with distance, as it is also shown in Figure 2.

It results that both Sagittarius and Crater II are outliers in the relations between the MW dSph fundamental properties. This is not surprising since the former was never considered for estimating the DM content as it is evidently dominated by MW tidal forces, which is further evidenced by the associated gigantic stream (Ibata et al. 2003;Majewski et al. 2003). Crater II was discovered recently (see Caldwell et al. (2017)) and was immediately considered as an outlier, due to its extremely small stellar density. From examination of Figures 5 to 7 we find that while being less compelling, Bootes I is also an outlier when it is compared to the properties of the 21 remaining dSphs. Indeed Bootes I has two two distinct components with very discrepant velocity dispersions, and Simon (2019) advised against using these data to derive Bootes I dynamical properties. The 21 remaining dSphs will be considered as the sample of ordinary dSph galaxies, and all Figures in the main manuscript are based on this sample. Table 2 gives the quantities calculated in the paper assuming the MW model of Sofue (2015).

Sag Sag Sag

B. CHALLENGES TO THE SCENARIO OF DM-DOMINATED MW DSPHS

DSphs made of stars only are consistent with the fact that L V (∼ M stellar ) scales with r 3 half (Figure 5). It is unexpected that tidal shocks affect the slope of this relation, because the energy provided by the MW tides (1/2∆σ 2 M W ) is approximately balanced by losses due to (too energetic) stars leaving the system (Binney & Tremaine 1987, see pages 435-436). In other words the reaction of the system to the MW tidal shocks would be to shrink and to loose stars, i.e., to decrease both M stellar (or L V ) and r half . Strong deviations from the baryon Tully Fisher (see Figure 5) have led McGaugh & Wolf (2010) to suggest a role for tidal effects. Combining L V ∝ σ 8.88 los with the weaker correlation between L V and D 5 MW , is consistent with a weak, or the absence of correlation of, between σ los and D MW . This suggests that DM-free dSphs are consistent with all data and relationships presented in this paper.

Can these relationships be reproduced by models of DM-dominated MW dSphs? Pure DM subhalos in the MW potential (Diemand et al. 2008;Moliné et al. 2017) could yield an anticorrelation, such as that shown in the left panel of Figure 1. This is because tidal stripping dominates in the diffusive regime (see Binney & Tremaine 2008, P. 663) and leads to the evaporation of particles having the lowest binding energy, which results in higher concentrations and densities (Aguilar 1993). This particularly affects early infall, and dSphs that inhabit the inner part of the halo.

Figure 8 shows that the DM density of dSphs is indeed anticorrelated with the MW distance, however, with a far much steeper slope (54 times higher) than that derived from Fig. 4 of Diemand et al. (2008). In these simulations, dSphs are infalling into the MW halo at different epochs. If tidal stripping caused the correlation shown in the top panel of Figure 8, then stars present at early epochs should be as affected by tides as DM, except if DM particles have much more radial motions than stars. According to Navarro et al. (2010) and Taylor (2011), DM particle motions may not be too anisotropic in the central regions of a DM halo. The bottom panel of Figure 8 shows that on average the stellar density does not change with MW distance, which makes unlikely an explanation based on DM tidal stripping, or, alternatively, that additional investigations are necessary, e.g, by considering the effect of baryons to the Diemand et al. (2008) prediction. It remains unclear how simulations (presence of gas in progenitors and ram pressure stripping, stellar feedback and winds, radiative cooling, cosmic UV background affecting early star formation, see, e.g., Kazantzidis et al. 2017) of DMdominated dSphs could reproduce the above relations, including Figures 1 and2 and the scaling relations presented in the Appendices A.2 and A.3. In spite of our efforts to investigate many simulations that have been performed to reproduce the Local Group content, we find none of them able to reach the properties of the lower half of dSph (Diemand et al. 2008) as shown by the blue line. Bottom: star density (νstars) calculated at r= r 1/2 assuming a Plummer profile for dSphs (Muñoz et al. 2018). It shows no correlation with DMW, with ρ= 0.15 and t= 0.66, the dotted line indicating the average value.

mass range, impacting their ability to reproduce the relationships discovered in this paper. The main limitation is the resolution, e.g., in one of the highest resolution APOSTLE simulations (Fattahi et al. 2018), the softening length is 134 pc leading to a spatial resolution of ∼ 400 pc, i.e., insufficient for most dSphs (see Table 1). However, this limitation should soon be overcome, e.g., by extreme resolution simulations (Wheeler et al. 2018). that might be used in the future to test our relations between host and dSph galaxies. For the moment, even Wheeler et al. (2018, see their Fig. 7) have not fully reproduced the relation shown in the left panel of Figure 5.

C. ACCOUNTING FOR UNCERTAINTIES IN THE ANALYSIS OF THE ANTICORRELATION BETWEEN ACCELERATION AND MW DISTANCE

The validity of the correlation analysis done in Sect. 2 and shown in the left panel of Figure 1 may be questioned since both axes (here called x and y) suffer from observational errors, which are both heteroschedastic and sometimes large. For each dSphs i, we note x,i the uncertainties of the measurement on x i , y,i the uncertainties of the measurement on y i and R i = y i -(ax i + b) the residuals from the fitting process. However, the observed dispersion on y originates not only from the measurement errors, but also from some (unknown) intrinsic scatter σ int , which increases the observed dispersion, σ 2 y,i = σ 2 int + 2 y,i . To account for this, we checked the significance of our correlation analysis by two different methods.

First, we made a weighted regression (Ripley & Thompson 1987) accounting for errors in both axes, x,i and σ y,i . Initially starting with a null intrinsic scatter, we iterate twice the regression for the estimation of this intrinsic scatter: we compute the residuals, and we estimate the variance of the intrinsic scatter with the variance of the residuals minus the average variances of the measurement errors, σ 2 int ≈ R 2 i /(n -2) -2 y,i /n, as it is a consistent estimator of the intrinsic scatter when x,i can be neglected. We then found that the slope is significantly negative with a p-value = 3 ×10 -4 .

Second we used an MCMC based Bayesian hierarchical model to fit the data, which derives the posterior distribution function from the data, and considers the observation uncertainties and intrinsic scatter simultaneously 7 . The slope is -0.54 ± 0.15, i.e. with a significance similar to what has already been found.

These results show that the correlation found at Figure 1 does not occur by chance only.

(km s -1 ) (kpc) (M /L ) (km s -1 ) (deg) (1) (kpc)
(3)

(5) (1)

(7) (8) (6)
(2)

(3) (4)

(5) (6)

Figure 3 .

 3 Figure 3. Correlation between the observed σ los and that predicted from Eq. 5, i.e., σ pred = σ 2 MW + σ 2 stars . Left: It assumes the MW mass and its profile from Sofue (2015). The blue line indicates the track of a simulated dwarf falling into the MW halo, adapted from Yang et al. (2014) by adopting a softening radius of 0.5 pc, i.e., warranting that almost no stellar particles are affected by non-Newtonian motions (see Figure4). Right: same after optimizing the MW mass profile.

  T PERI = -0.28 Gyr T PERI = -0.69 Gyr T PERI = -1.07 Gyr

Figure 4 .

 4 Figure4. Transformation of a DM-devoid and gas-rich dwarf into a dSph after a first passage into the MW halo. An animation is available that shows a close-up view of the dwarf galaxy in the orbital plane that includes the MW; distance to the dwarf and direction (see orange arrows) are indicated in the top left of each panel. The initially gas-rich dwarf galaxy (H I gas in green, stars in black) infalls into the MW halo, where hot gas content exerts ram-pressure onto the HI gas content as seen in the top-left panel where the dwarf motion is almost along the direction of the MW (orange arrow). The top-right panel is a snapshot taken after the HI gas has been fully removed from the dwarf, and since the gas accounted for 71% of the total initial mass, it left the stars free to expand over all directions due to the decrease of self-gravity. The bottom left panel shows the dwarf closely approaching its pericenter (time to pericenter is indicated in the bottom left of each panel) and expanded stars are mostly affected by MW tidal shocks that let them elongate along the line-of-sight direction that matches the direction of the MW gravitational force. The bottom right panel shows the dwarf after pericenter, at which time stars are continuously elongated along the line of sight, providing the large observed values of σ los (see the value and red arrow). This simulation is adapted from one of the simulations published inYang et al. (2014), i.e., the model T DG3 -rp28 in their Table3. It has been improved by adopting a gravitational force softening of 0.5 pc instead of 50 pc for stellar particles (17 M /particle, 500,000 particles) in dwarf galaxies. Given the stellar density in dSphs, there should be no or very few stellar particles that can be affected by the undesirable effects due to the softening. These simulations are based on the hydrodynamical / N-body code Gadget2(Springel 2005), and use simulated DM-free galaxies with initial mass and gas fraction (within a 3 kpc projected radius) of 1.35×10 8 M and 71%, respectively. The simulation in the video begins at TPERI = -1.635 Gyr and ends at 0.839 Gyr. The video duration is 71 s.

Figure 5 .

 5 Figure 5. Logarithmic correlation between visible luminosity (LV ) and half-light radius (r half , left panel) and line-of-sight velocity dispersion (σ los , right panel). Solid lines indicate the least-square fit of each relation, after excluding Crater II. In the left panel, small dots indicate the location of globular clusters from Moreno et al. (2014), the dotted line shows a detection limit in surface brightness of 31 mag arcsec -2 , and the arrow indicates the track of the simulated dSph (same simulation than in Figures3 and 4). In the right panel, the dotted line shows the correlation for the 21 dSphs of the sample after removing the three outliers (red points).

Figure 7 .

 7 Figure 7. From left to right: distance to the MW center against visible luminosity (LV ), half-light radius (r half ), and line-of-sight velocity dispersion (σ los ).

Figure 8 .

 8 Figure 8. Variations of dSph densities taken at r= r 1/2 with the MW distance for the sample of 21 ordinary MW dSphs. Top: DM density (νDM) calculated according to Eqs. 1 and 2. It is logarithmically anticorrelated with DMW, with ρ= -0.60, t= -3.3, and a logarithmic slope of -2.86 (full line) as indicated in the left panel of Figure2, and which contrasts with the prediction by the Via Lactea II simulation(Diemand et al. 2008) as shown by the blue line. Bottom: star density (νstars) calculated at r= r 1/2 assuming a Plummer profile for dSphs(Muñoz et al. 2018). It shows no correlation with DMW, with ρ= 0.15 and t= 0.66, the dotted line indicating the average value.

  (2) and (3) V luminosity and its uncertainty; (4) and (5) Half-light radius and its uncertainty; (6), (7), and (8) los observed velocity dispersion and its uncertainty; (9) & (10) Distance to the Sun and its uncertainty; (11) and (12) Local Standard of Rest velocity and its uncertainty; (13) Distance to the MW center; (14) Stellar mass to V luminosity ratio: 2.5 for dSphs without young or intermediate age stars, and 1.5 for Carina, Fornax, and Leo I (see, e.g., Lelli et al. 2017 and references therein); (15) Galactic Standard of Rest velocity; (16) Angle between two lines passing by the Galactic center and the Sun and crossing at the dSph; (17) = 1-b/a, a and b being the major and minor axis of the dSph.

Table 2 .

 2 Quantities calculated in the paper assuming the MW model of Sofue (2015). Note. (1) dSph name; (2) Twice the kinetic energy associated to the velocity dispersion of a Plummer body made of stars only (see footnote 3); (3) Same for the excess of velocity dispersion due to the MW tidal shocks (see Eq. 5); (4) ∆DMW /DMW : relative variation of DMW that would provoke a change of ±1/2 σ 2 starsonly in the kinetic energy of a dSph dominated by MW tidal shocks; (5) αMW = 1 -(∂MMW/∂DMW)×(DMW/MMW); (6) Predicted velocity dispersion that is σ 2 stars + σ 2 M W , in which σMW is the dispersion caused by MW tidal shocks (see Eq. 5); (7) For comparison, the los observed velocity dispersion; (8) Crossing time approximated by r half /σ los ; (9) Encountering time approximated by ∆(DMW )/vGSR; (10) Dynamical time approximated by r half /σ starsonly (11) Orbital time approximated by DMW /(2 × vGSR); (12) & (13) ratios of the encountering time to the crossing time, and to the virialization time (3 ×t dyn ), respectively.

Table 1 .

 1 Observed Parameters and Their Error Bars. Note. (1) dSph name;

To illustrate this, let us consider a mass distribution very elongated along the line of sight: the induced σ los would indeed be much larger than if the (same) mass was spherically distributed.

If DM particles have strongly radially biased orbits (but see Taylor 2011) they will be more affected by tidal stripping than stars.

For example, with t orbit ∼ D MW /(2 × V GSR ), where V GSR is the galactocentric projection of the radial velocity. Replacing V GSR by V rad calculated byFritz et al. (2018) does not change the results.

Fritz et al. (2018) have also considered another mass model, which is like that ofBovy (2015) but for which they multiply the halo DM mass by a factor 2 (their M M W = 1.6 × 10 12 M model). It is however discrepant with the extended rotation curve of the MW.