Eugenio Hamm

Jean-Christophe Géminard
email: christophe.geminard@ens-lyon.fr

The weight of a falling chain, revisited

A vertically hanging chain is released from rest and falls due to gravity on a scale pan. We discuss the various experimental and theoretical aspects of this classic problem. Careful timeresolved force measurements allow us to determine the differences between the idealized and the its implementation in the laboratory problem. We observe that, in spite of the upward force exerted by the pan on the chain, the free end at the top falls faster than a freely falling body. Because a real chain exhibits a finite minimum radius of curvature, the contact at the bottom results in a tensional force which pulls the falling part downward.

I. INTRODUCTION

Hold one end of a chain above a scale so that the other end lightly touches the pan. Now release it from rest [see Fig. 1(a)]. Theory predicts that during the fall, the scale will display an increasing weight equal to three times the weight of the deposited part of the chain and then a sudden drop to the chain weight once the chain lays completely motionless on the pan. 1 This simple experiment, which can be performed with basic laboratory equipment, has a few subtleties that make it worthwhile to do a thorough theoretical and experimental analysis.

This system belongs to the class of variable-mass problems usually considered in undergraduate physics textbooks. Among the various possible experimental configurations we cite a few that have been considered recent. For example, consider a chain attached by its two ends at the same height and separated horizontally by a certain distance [see Fig. 1(b)]. As one of the ends is released the acceleration of the chain tip is greater than g, the acceleration due to gravity. [2][3][4] Another configuration is a chain sliding off a table by a vertically hanging end [see Fig. 1(c)]. In this case the acceleration is only a fraction of g. [START_REF] Wong | The falling chain of Hopkins, Tait, Steele and Cayley[END_REF] Other configurations are discussed in Ref. 1.

In this paper we consider experimentally and theoretically the problem of a vertically falling chain that impinges on the pan of a scale. To our knowledge, the only experimental realization of this system has been reported by van den Berg. [START_REF] Van Den | Force exerted by a falling chain[END_REF] A theoretical description of the idealized problem, extended to the more general case of an additional constant vertical force acting on the upper end, was performed by Šíma and J. Podolský, [START_REF] Šíma | Buquoy's problem[END_REF] which also cites unavailable Czech and Russian bibliography, especially the role of Graf Georg von Buquoy, who was "the first who investigated mechanical systems with a varying mass." There is a mention in the solutions manual by Thornton and Marion 8 of the possibility of the free upper end falling faster than a freely falling mass, which is a puzzling effect. The authors report a personal communication from M. G. Calkin who ". . . has found that experimentally the time of fall for this problem is consistently less . . . by about 1.5%" and ". . . also finds evidence that suggests the free fall treatment is more valid if the table is energy absorbing."

We shall see that this effect is striking.

II. THEORETICAL BACKGROUND

The dynamics of the falling chain is usually treated as follows. Consider an inextensible and perfectly flexible chain of length L with a homogeneous linear mass density µ. At t = 0 the lower end of the chain just touches the scale pan while the upper end is released from rest [see Fig. 1(a)].

For t > 0 the free end of the chain has fallen a distance x = x(t) so that a part of the weight w(t) = µx g ( g is the acceleration due to gravity) lies at rest on the pan. At the same time, the falling part of the chain (length L -x) moves downward with velocity v (the chain is inextensible). The total force, F , exerted by the scale on the chain at time t opposes the weight w of the already deposited part of the chain and exerts the force f necessary to stop the chain links that collide with the pan. The latter contribution can be estimated by considering that during the time interval ∆t, a mass µv∆t of the chain (v = | v|) with velocity v at the beginning of the interval collides with the pan and comes to rest (assuming that the collision is perfectly inelastic). The change in the linear vertical momentum is thus

∆ p = -µv∆t v. From Newton's second law, f = d p/dt = -(µv) v.
The reading of the scale corresponds to the norm of F :

W ≡ | F | = µgx + µv 2 .
(1)

This result holds if µ is constant and the links that collide with the pan stop.

It is usually further assumed that the chain experiences free fall. In this case, x = gt 2 /2, v = ẋ = gt, and we obtain [xx usually we write y for the vertical displacement xx]

W = 3 2 µg 2 t 2 = 3w(t). (2)
During the fall, the apparent weight of the chain, W , increases quadratically in time and equals, at each instant, three times the weight w of the part of the chain already deposited on the scale pan. The duration of the free-fall over the length L equals T = 2L/g. At t = T , the last link touches the pan and a maximum value of 3Mg is attained (M denotes the mass of the entire chain). At this time the apparent weight W instantaneously drops to Mg, the weight of the chain which sits at rest on the scale pan.

In this treatment the chain is assumed to be homogeneous (constant mass density µ along its length) and has no limiting radius of curvature (it can be bent infinitely). In a real experiment these two assumptions are not fully satisfied. In particular, we shall see that the assumption of free-fall does not hold when the minimum radius of curvature is finite.

III. SETUP AND PROTOCOL

We shall report measurements obtained with two chains, namely a "ball chain" and a "loop chain" (see Fig. 2). The ball chain consists of stainless-steel identical links (length The experimental setup (see Fig. 3) consists of two parts. The part at the top holds the chain previous to its release and the part at the bottom measures the force exerted during its fall.

The system at the top [see Fig. 3(a)] makes it possible to simultaneously release the chain and a steel ball, so we can compare the motion of the chain with free-fall, and accurately determine the beginning of its motion. It is an adaptation of the system described in Ref. 4.

The upper end of the chain is initially attached with a nylon cord (fishing line, diameter 10 -4 m) to a steel ball such that the joining cord is held by a nickel wire (diameter 10 -4 m).

Two horizontal parallel rods are used to separate the falling paths of the two bodies. The whole system is adjusted so that initially the chain and the ball are at the same height. To initiate the fall, we quickly cut the nylon wire by injecting a large electric current (about 1 A) from a power supply in the metallic wire, during a few tenths of second. The mount holding the nickel wire is a slightly-flexible cantilever spring, and the release of the bodies results in a sudden upward acceleration of its free end. Measuring this acceleration by means of an accelerometer (Dytran, 3035BG) makes it possible to accurately determine the origin of time, t = 0.

The vertical component of the force applied by the falling chain (apparent weight W ) is measured by a force-sensor (Load cell, Testwell, KD40S, maximum force 10 N) which holds a vessel made from a horizontal disk (Dural, diameter 6 cm) surrounded by high walls which prevent the chain or the steel ball from escaping the system [see Fig. 3(b)]. As described in the following, the vessel can be filled with different materials to change the experimental conditions at the bottom. The material of the walls (paper or plastic) can also be changed.

The signals from the accelerometer and the force sensor are monitored using an acquisition board (National Instruments, PCI 6251). Given that the typical length of the chains used in the experiments is approximately 2 m, the duration of the free fall is consistently about 0.6 s (with g = 9.81 m/s 2 ). We acquired the signals at 10 kHz, which is sufficient to reveal the details of the force dynamics while avoiding the need to manipulate large data sets.

It is important to discuss the typical force-signal obtained for a ball chain falling in an empty vessel (see Fig. 4). Instead of the continuous quadratic increase in the apparent weight W with time, we observe large fluctuations whose amplitude increases in time until the end of the fall. The large peaks that characterize the force signal are a natural consequence of the discrete nature of the chain. The links hit the bottom of the vessel almost independently of each other and, as their velocity increases in time due to the acceleration of gravity, the amplitude and the frequency of the peaks increase as well. We could avoid the peaks by using a sensor with a long characteristic time. However, the frequency of the collisions ranges from about 30 to 1300 Hz during the fall. The choice of a characteristic time larger than 1/30 ≃ 30 ms would hide some relevant details of the fall. Thus, we chose a sensor with a resonant frequency of nearly 500 Hz, which makes it possible to obtain a temporal resolution of about 2 ms. The "hidden" ideal force can then be revealed using a numerical filtering procedure (see Fig. The signal from the force sensor obtained after addition of 1.5 g of cotton shows a monotonic increase in the force during the fall and a final drop to the weight of the entire chain when its motion ceases in the vessel (seeFig. 5). The low-frequency oscillations (at ∼ 15 Hz) at the tail of the signal are produced by restoring forces in the cotton bed. Our results show that the cotton layer constitutes a very efficient method of damping the force signal, and we therefore used it in our measurements. One advantage is that we are able to identify short-time features of the force signal, in particular, the time at which the last link of the chain hits the pan (maximum of the force), which otherwise would be smoothed out by a numerical filtering procedure.

IV. RESULTS AND DISCUSSION

We see in Fig. 5 that the measured apparent weight does not follow the theoretical curve and, in particular, the maximum force is much larger, about 5.5 times the weight of the chain. We might claim that the difference originates from the condition at the bottom but, even more striking than the value of the maximum force, we notice in Figs. 4 and5 that the force drops to its rest value earlier than predicted for free fall. Whatever the condition at the bottom, the free end of the chain falls faster than expected. The difference is significant, about 38 ms, corresponding to 12% of the total length of the chain.

To check that the ball chain does fall faster than a simple body in free fall, we photographed the simultaneous falls of the chain and of the steel ball. In this experiment the surface at the bottom is hard and flat (wood board covered by an acrylic layer) so that the dissipation is much less than for the vessel padded with cotton wool. The sequence in Fig. 6 is conclusive: the last link is clearly ahead of the ball during the entire fall, which is possible only if, in addition to its own weight, a tensional force Γ pulls the falling part of the chain downward. The origin of this force is surprising because the force exerted on the chain due to the pan is oriented upward.

It is interesting to consider several mechanisms that could alter the dynamics of the chain and explain the deviation from free fall.

Air friction. The falling chain is subjected to friction with the surrounding air. This friction is the same along the chain, and thus does not generate any tension, but rather reduces homogeneously the chain velocity. This drag, which increases the time of fall, is not responsible for the experimental observations. Initial tension of the chain. Previous to its release, the chain, which is hanging from its upper end, is subjected to a tension Γ as a consequence of its own weight. Mechanical equilibrium imposes Γ(s) = µg(L+s), where s denotes the position in a frame of coordinates attached to the chain, oriented upward; the origin is attached to the uppermost link. When the chain is released, a longitudinal wave propagates from top to bottom, accelerating the links downward. From the value of the elastic constant associated with the stretching, k = (5.5 ± 0.4) × 10 3 N (defined by Γ = k∆L/L, where ∆L is the elongation and L the initial length, and Γ is obtained by measuring the elongation due to known masses), we estimate that the wave propagates at the velocity c l = k/µ ≈ 600 m/s, assuming the chain to be continuous. Thus, the wave front reaches the lowermost link 3 ms after the release of the uppermost one. This relaxation leads to a contraction of about MgL/(2k) ≈ 50 µm of the chain and to an acceleration of the center of mass toward the bottom. It is interesting to consider the associated decrease in the fall time, which can be estimated by an energy argument. The total elastic energy initially stored in the chain is U = 1 6 (Mg) 2 (L/k). If all the elastic energy was transferred to kinetic energy of the whole chain, the latter would gain an initial velocity v 0 = gL/( √ 3c l ) ≈ 18 × 10 -3 m/s, derived from the condition U = 1 2 Mv 2 0 , which leads to a decrease in the fall time, v 0 /g, of about 2 ms, much smaller than the experimental value of 38 ms (see Sec. III). Therefore the initial tension of the chain does not explain the observed behavior, even for our unrealistic hypothesis that all the elastic energy goes into kinetic energy. In a process more consistent with momentum conservation the chain would rapidly dissipate the elastic energy because of the friction associated with the relative motion of the links. We can thus assume that the chain as a whole starts falling from rest at t = 0, in spite of its initial tension.

Finite bending limit of the chain. We have to abandon the infinite flexibility approximation for a real chain, which can fold only over a finite distance with a geometry that depends on its construction. Specifically, there exists a minimum accessible radius of curvature, equal to a few times the link length. We shall see that this fact is responsible for the deviation of the experiments from the simplistic theory assuming free-fall.

V. THEORETICAL ANALYSIS

What is the origin of the downward pulling force responsible for a net acceleration larger than g, explaining the earlier arrival of the free end? A similar phenomenon is well known in the context of another variable-mass problem, namely the partial fall of a chain that is held by its two ends at the same height and then released at one of its ends. 2,3 The free end is observed to fall faster than a simple body in free-fall. This behavior is easy to understand if we consider that the energy is conserved during the fall, 10 so that while more and more links come to rest, the potential energy is converted into the kinetic energy of the falling part, which contains fewer and and links. The continuous decrease in the mass of the moving part results in a continuous increase in the acceleration. The energy is likely not to be conserved, due to internal dissipation associated with the relative motion of the links. Morin 1 draws a distinction between a dissipative (loosely spaced links joined by a massless string) and a non-dissipative (closely spaced links) chain, and discusses the influence of the folding on the dynamics for each case.

In our experimental configuration the energy is not conserved because the chain remains at rest after the fall. Therefore we will discuss a model based on Newton's laws. We will see that the geometry of the folding at the bottom is crucial and that, to account for the dynamics of the chain, we must consider not only the momentum but also the angular momentum [xx you used kinetic moment, but that term is never used in any classical mechanics book I know of. I assume you mean angular momentum xx] of the system. We will not solve a set of coupled equations modeling the dynamics of each link. Such models exist, 3,9 but they would obscure the simple physics that is sufficient to understand the phenomenon. We thus consider a minimal model containing the basic elements that describe our observations. The main difference between the ideal case and the experiment is that the chain cannot be bent infinitely and, thus, exhibits a minimum accessible radius of curvature R. The ideal chain would fall vertically toward a single point O. Instead the real chain is straight and falls vertically in a large part of its length, but bends and curls close to the bottom in a region of typical size ∼ R. Based on the experimental observations (see Fig. 6), we approximate the shape of the falling chain according to the illustration in Fig. 7. We discard the details of the heap formation and the exact shape of the curled part, and consider that the chain hits the bottom at the point P which is a finite distance D (of the order of R) from O. The dynamical part f of the total force exerted by the bottom on the chain is applied at P. We denote by y, the vertical coordinate (oriented upward) of the upper free end. [xx here you use y, not x xx] We assume that y ≫ R; that is, the length of the falling part is much larger than the typical size of the curl. In this limit, we can neglect the momentum of the curl with respect to that of the vertical part and write the equation governing the change of momentum as

d dt µy ẏ = -µyg + f, (3)
where µy corresponds to the falling mass and ẏ to its vertical velocity.

In the same approximation, disregarding the rotation of the point P around O in Fig. 7, we can estimate the horizontal component of the angular momentum, L O , of the chain about O to which only the curl contributes, because the vertical part points toward O and the remaining part of the chain is at rest. If we assume that in the curl, the links move with a velocity of the order of ẏ at a typical distance R from O, we obtain L O ∼ µR 2 ẏ (where the sign is associated with a contact point on the right-hand side as sketched in Fig. 7).

The change in the angular momentum is due to two contributions. There is a change in the overall velocity of the chain ẏ, which leads to a term µR 2 ÿ. Also the links in the curl come to rest. In the curl the angular momentum goes from σ O to 0 in the typical time τ ≃ -R/ ẏ ( ẏ < 0). This contribution thus is estimated to be of the order of -L O /τ ≃ µR ẏ2 . According to Newton's laws, we obtain

dL O dt = Df ≃ µ(R 2 ÿ + R ẏ2 ). (4)
We combine Eqs. ( 3) and ( 4) and assume that R ≪ y [which leads to neglecting the term µR 2 ÿ in Eq. ( 4)] and obtain the differential equation for y(t):

ÿ = -g + (γ -1) ẏ2 y , (5)
where the geometrical factor γ accounts for the shape of the curl. Equation (4), with the same approximation, gives the dynamical force

f = γµ ẏ2 . (6)
Accordingly, the apparent weight measured at the bottom is

W = µ(L -y)g + γµ ẏ2 . (7)
If we write x = L -y, we recover Eq. (1) for γ = 1, the case of an ideal free-falling chain. For γ = 1, Eq. ( 5) must be solved numerically for the initial conditions y(0) = L and ẏ(0) = 0. It is useful to define the dimensionless quantities τ ≡ t/T , Y ≡ y/L, and Ω ≡ W/(Mg), where T ≡ 2L/g is the time of free-fall from the height L. Equation ( 5)

becomes Ÿ = -2 + (γ -1) Ẏ 2 Y , (8)
and must be solved with the initial conditions Y (0) = 1 and Ẏ (0) = 0. The apparent weight is given by

Φ = 1 -Y + γ 2 Ẏ 2 . (9)
The numerical solution of Eq. ( 8) best interpolates the measured values of the free-end position for γ = 0.83 (see Fig. 8). The associated duration of the fall is τ ≃ 0.94, which corresponds to the free end reaching the bottom 36 ms earlier than expected for free fall.

This result compares to the experimental value of 38 ms given in Sec. IV.

The apparent weight predicted by Eq. ( 9) for γ = 0.83 is compared with the experimental measurements in Fig. 9. For the same value of γ we can interpolate the kinematic curve (see Fig. 8) obtained for the fall onto a hard surface and the dynamic curve (Fig. 9) obtained for the fall on a cotton wool layer. Thus, γ is independent of the dissipation at the bottom.

Note that the simple model fails to account for the finite value of the force because the approximation R ≪ y is no longer satisfied when the free end reaches the bottom.

Let us determine the tension that acts, according to our model, on the falling part of the chain and which is responsible for the larger than g acceleration of the free end. For a length element ds of the chain and the tension Γ, we can write the change in the momentum as

µdsÿ = -(µds)g + ∂Γ ∂s dl. (10)
Thus, at a given time t, the tension in the chain Γ(s) = µ(ÿ + g)s, where the condition Γ(0) = 0 at the free end is already taken into account. From Eq. ( 5), we can determine the tension pulling the chain toward the bottom at s = -y:

Γ(-y) = µ(1 -γ) ẏ2 (11)
The chain cannot transmit a compressive stress, which leads to the condition that Γ ≥ 0 and, thus, that γ ≤ 1. Note that, in the limit γ = 1, we recover the ideal case in which no tension exists. Also whatever the value of γ, the total force applied to the curl equals (Γ(-y)+f ) = µ ẏ2 . Thus, in all cases the total force applied to the curl is correctly predicted by considering the change in the linear momentum of the links per unit time in that region

[xx what region? xx], as described in Sec. II. During a time ∆t a lengthẏ ∆t changes from the vertical velocity ẏ to rest, so that the total vertical applied force must be

µ(-ẏ ∆t) (0 -ẏ) ∆t = µ ẏ2 . ( 12
)
The difference between the measured apparent weight and µ ẏ2 is thus due to the internal tension of the chain at the bottom.

VI. DISCUSSION AND CONCLUSION

We have demonstrated that the existence of a minimum radius of curvature, which imposes a curl at the bottom, is responsible for the appearance of an additional tension in a falling chain, which pulls the falling part toward the bottom. However, the model discussed in Sec. V leaves a few open questions.

In accord with our observations, we assumed that the chain falls vertically, except in a small region of size R at the bottom. Nonetheless, the deformation of the chain at the bottom excites transverse waves that can eventually climb along the chain. No such waves were observed. To understand this behavior, we consider the velocity, c t = Γ/µ, of the transverse waves associated with the tension Γ. At the bottom, where Γ (and thus c t ) is maximum, we have, from Eq. ( 11), c t = √ 1 -γ| ẏ| < | ẏ|. The velocity c t , even at the bottom, is always smaller than the fall velocity so that the perturbations at the bottom cannot propagate upward, which is the reason why the chain remains straight and vertical and does not move laterally.

As mentioned, the energy is not conserved. The total energy of the chain is E = µy ẏ2 /2+ µy 2 g/2. If we differentiate E with respect to time and use Eq. ( 5), we one obtain the rate of energy dissipation:

dE dt = γ - 1 2 µ ẏ3 . (13)
As a consequence, because ẏ < 0, we must have γ > 1/2. We know that γ ≤ 1 from the condition that the chain cannot transmit a compressive stress. Thus, γ must be in the range [1/2, 1] and therefore the fall time ranges from the time of free fall T for γ = 1 to 0.847 T for γ = 1/2, in the limit where energy is conserved. The experimental value γ ≃ 0.83 obtained for the ball chain is in [1/2, 1].

The dimensionless parameter γ in our model t accounts for the geometry of the curl at the contact with the bottom. It might be tempting to interpret γ as characterizing the dissipation at the bottom. However, the values of γ measured for a hard or a soft surface are the same, in contrast with Ref. 8, but in agreement with our theoretical interpretation.

If γ characterizes the geometry of the curl, it must depend on the structure of the chain.

For instance, the ball chain exhibits a minimum radius of curvature of about 2 links. In contrast, the classical loop chain used in previous experiments [START_REF] Van Den | Force exerted by a falling chain[END_REF] does not exhibit a welldefined minimum radius of curvature and can apparently be bent without any resistance. In a similar experiment with the loop chain we obtain γ ≃ 0.95, again in the interval [1/2, 1],

and we find that the free end reaches the bottom 8 ms earlier than the body in free fall. For the loop chain, the force is applied at a point almost aligned with the vertical falling part (P is close to O in Fig. 7), and the effect is weaker than for the ball chain.

Our model does not give an estimate the maximum apparent weight. W is predicted to diverge at the end of the fall, when y vanishes, which is not physically acceptable. The discrepancy between the model and the experiment is not surprising in this limit, because the assumption R ≪ y does not hold at the end of the fall. The analysis developed in Sec. V can be refined to include this effect, but we would not learn much more. Our analysis correctly describes the temporal behavior of the apparent weight during most of the fall and explains why the free end reaches the bottom earlier than a mass in free fall. The line joining the open circles corresponds to the integration of Eq. ( 8) with γ = 0.83.

ℓ

  = (4.46 ± 0.01) 10 -3 m) [xx better to use ℓ since l is difficult to distinguish from 1 xx] that are each made from one rod and a hollow sphere (diameter φ = (3.26 ± 0.01) × 10 -3 m) attached to each other. The total length of the chain is L = 1.898 ± 0.001 m, the number of links is N = 413, and the total mass M = (29.6 ± 0.1) × 10 -3 kg. The mass of one link is thus m = M/N ≈ 7 × 10 -5 kg. The loop chain is made of equal concatenated [xx what do you mean by concatenated ? xx] links (length ℓ = (10.22 ± 0.01) × 10 -3 m, width (5.50 ± 0.01) × 10 -3 m, thickness (1.15 ± 0.01) × 10 -3 m). Its total length, number of links, and mass are respectively L = 1.890 ± 0.001 m, N = 235, and M = (41.8 ± 0.1) × 10 -3 kg.

  4, inset) that might wipe out relevant features of the signal. [xx I'm confused. if the force can be revealed, what is being wiped out? xx] Instead of using a numerical filter, we look for a cleaner force signal by changing the experimental conditions at the bottom. To efficiently reduce the amplitude of the oscillations, we filled the vessel successively with various viscous and non-newtonian fluids, a granular material, diluted modeling clay, and cotton wool. The force always exhibited violent oscillations except when a thick layer of cotton wool (typically a few centimeters) was placed at the bottom of the vessel [see Fig. 3(b)].

FIG. 1 .FIG. 3 .FIG. 4 . 6 FIG. 5 .FIG. 7 .

 134657 FIG. 1. Falling chain geometries. (a)The chain is falling vertically on a scale pan. During the fall, the apparent weight is predicted to be three times the deposited weight. (b) The chain is initially attached by its two ends at the same height. As one of the ends is released the acceleration of the chain tip is greater than g, the acceleration due to gravity. (c) The chain slides off a table by a vertically hanging end. In this case the acceleration is predicted to be g/2.

FIG. 8 .

 8 FIG. 8. The vertical position y [xx note change xx] versus the time τ . In the experiment the ball chain (open circles) and the steel ball (full circles) fall simultaneously (data obtained from Fig. 6).

FIG. 9 .

 9 FIG.9. Experimental force (black points) compared to the prediction of the model (continuous line) with γ = 0.83. Also shown is the ideal force for a chain with the same length as the one used in the experiment (dashed line).

ACKNOWLEDGMENTS

The authors would like to thank R. Freund for introducing them to the problem, and H. Gayvallet and B. Castaing for fruitful discussions. E. H. thanks the ENS de Lyon for having invited him as a professor in July 2009, during which this work was done.


S Peirani

On the coldness of the local Hubble flow: the role of baryons

Keywords: Galaxies, Local Group -Galaxies, haloes -Dark matter -Methods, N-body simulations

Our aim is to investigate whether the presence of baryons can have any significant influence on the properties of the local Hubble flow which has proved to be "cold". We use two cosmological zoom simulations in the standard ΛCDM cosmology with the same set of initial conditions to study the formation of a local group-like system within a sphere of ∼ 7 h -1 Mpc. The first one is a pure dark matter simulation (run DM ) while a complete treatment of the physics of baryons is introduced in the second one (run baryons ). A simple algorithm based on particles identity allows us to match haloes from the two runs. We found that galaxies identified in run baryons and their corresponding dark matter haloes in run DM have very similar spatial distributions and dynamical properties on large scales. Then, when analyzing the velocity field and the deviation from a pure Hubble flow in both simulations, namely when computing the dispersion of peculiar velocities of galaxies σ * (R) and those of their corresponding dark matter haloes σ DM (R) in run DM , we found no particular differences for distances R =1 to 8 Mpc from the local group mass center. This suggests that the presence of baryons have no noticeable impact on the global dynamical properties of the local Hubble flow within such distances. Then, the results indicate that the "true" σ * (R) values can be estimated from the pure dark matter simulation with a mean error of 3 km/s when dark matter haloes are selected with maximum circular velocities of V c ≥30 km/s, corresponding to a population of dark matter haloes in run baryons that host galaxies. By investigating the properties of the Hubble flow at distances R ∼0.7 to 3 Mpc, we also found that the estimation of the total mass enclosed at the radius of the zero-velocity surface R 0 , using the spherical infall model adapted to ΛCDM, can be underestimated by at least 50%.

INTRODUCTION

The study of dynamical and photometric properties of galaxies in the local universe represents an ideal framework to confront predictions of the standard Λ cold dark matter (ΛCDM) model with observations. In particular, an interesting feature of the dynamical properties of the local universe is that the Hubble flow is rather "cold", e.g. the dispersion in the peculiar velocities within the Hubble flow is quite small, namely ≤ 100 km/s (Sandage & Tammann 1975;Giraud 1986;Schlegel et al. 1994; ⋆ E-mail: peirani@iap.fr Ekholm et al. 2001, Karachentsev et al. 2003;Macciò, Governato & Horellou 2005;Tikhonov & Klypin 2009).

The presence of the dark energy was proposed as a possible explanation for the smoothness of the local Hubble flow, first argue by Baryshev et al. (2001), Chernin et al. (2001Chernin et al. ( , 2004Chernin et al. ( , 2007) ) and Teerikorpi et al. (2005) and supported by Macciò, Governato & Horellou (2005) using a set of N-body simulations. However, Hoffman et al. (2007) compared results from their own simulations of CDM and ΛCDM cosmologies with identical parameters, apart from the presence or not of the cosmological constant term. They claimed that no significant differences were noticed in the velocity flow around c 0000 RAS galaxies having properties similar to those observed in the neighborhood of the Milky Way. A similar conclusion was obtained more recently by Martinez-Vaquero et al. (2009) using a range of N-body simulations in different cold dark matter scenarios. They conclude that the main dynamical parameter that can affect the coldness of the flow is the relative isolation of the Local Group. Another approach was proposed by Peirani & de Freitas Pacheco (2006, 2008) who derived a velocity-distance relation by modifying the Lemaître-Tolman model (TL) with the inclusion of a cosmological constant term. They found that this new relation, which describes the behavior of the Hubble flow near the central dominant objects (≤2-3 Mpc), in addition to that derived from the "canonical" model (Ωm = 1), provide equally acceptable fits to the existing available data. As a consequence, no robust conclusion about the effects of the cosmological constant on the dynamics of groups could be established. Moreover, Axenides & Perivolaropoulos (2002) studied the dark energy effects in the growth of matter fluctuations in a flat universe. They concluded that the dark energy can indeed cool the local Hubble flow but the parameters required for the predicted velocity dispersion to match the observed values are out by observations that constrain either the present dark energy density or the equation of state parameter w(=Px/εx).

Other cosmological models were proposed to study the local Hubble flow. Dark matter simulations by Governato et al. (1997) for cosmological models with Ωm = 1 or Ωm = 0.3 are, according to these authors, unable to produce systems embedded in regions of "cold" flows, i.e., with 1-D dispersion velocities of approximately 40-50 km/s. Recently, Tikhonov et al. (2009aTikhonov et al. ( , 2009b) have compared the observed spectrum of minivoids in the local volume with the spectrum of minivoids determined from the simulations of CDM or WDM models. They found that model predictions and observations match very well provided that galaxies can only be hosted by dark matter haloes with circular velocities greater than 20 km/s (for WDM) and 35 km/s (for ΛCDM). They have also derived rms deviations from the Hubble flow which seem to be consistent with observational values.

All those past numerical works have used collisionless N-body simulations to investigate the puzzle of coldness of the local Hubble flow and thus, the velocity dispersions have been derived by considering the dynamical properties of dark matter haloes supposed to host galaxies. However, it is nonetheless not obvious that positions and velocities of galaxies and their host dark matter haloes are identical. A first element of answer was given by Weinberg et al. (2008) who have studied the subhalo and galaxy populations in a galaxy group simulated either by a collisionless cosmological simulation or a hydrodynamics simulation with the same initial conditions. They found that positions and masses of large subhaloes are very similar in both runs while they can be different for low mass subhaloes whose orbits can be more easily modified by the host halo potential. Past works also suggest the existence of bias between the star and dark matter components (see for instance, [START_REF] Carlberg | L29 c[END_REF][START_REF] Zhao | [END_REF]Sousbie et al. 2008). In the present work, we study pos-sible effects of the presence of baryons in the dynamical properties of the local Hubble flow by using high resolution simulations that include most of the relevant physical processes that lead to the formation of galaxies inside dark matter haloes. This paper is organized as follows: in Sect. 2, we summarize the numerical methodology; in Sect. 3 we present results on the dispersions of peculiar velocities around the mean Hubble local flow derived from our numerical models and, finally, in Sect. 4, our main conclusions are given.

NUMERICAL MODELING

Initial conditions and simulation methodology

The cosmological simulations analysed in this work use the technique of "zoom" [START_REF] Tormen | [END_REF]. We summarize the main steps here. First, we have performed a cosmological simulation for a ΛCDM universe using WMAP5 parameters (Komatsu et al. 2009), namely ΩM = 0.274, ΩΛ = 0.726, ΩB = 0.0456, H0 = 70.5 km/s/Mpc, n = 0.96 and σ8 = 0.812. The simulation was performed in a periodic box of side 100 h -1 Mpc with 512 3 dark matter particles (e.g. with mass resolution of ∼ 5.6 × 10 8 h -1 M⊙). The simulation started at z ∼ 40 and ended at the present time z = 0 where we have selected one local group (LG) type halo (see section 2.2). In a second step, we have resimulated this LG candidate by using an equivalent resolution of 2048 3 particles within a sphere of 7 h -1 Mpc. The particle mass resolution is progressively degraded around this region of interest to reach the lowest resolution of 256 3 effective particles. Initial conditions have been generated from the MPgrafic code (Prunet et al. 2008), a parallel (MPI) version of Grafic (Bertschinger 2001).

In order to study effects of baryons, we have run 2 simulations using the same set of initial conditions: in the first one, only the dark matter component is considered (referred to as runDM ) whereas baryons are included in the second one (referred to as run baryons ) with a full treatment of the physics of baryons (see below). In these cases, the highest mass resolution are mDM ≈ 8.8×10 6 h -1 M⊙ (runDM ) and mDM ≈ 7.4×10 6 h -1 M⊙, mgas = mstars ≈ 1.5 × 10 6 h -1 M⊙ (run baryons ). The Plummer-equivalent force softening adopted for the high mass resolution particles were 1 and 0.5 h -1 kpc for dark matter and gas particles respectively and were kept constant in comoving units. An intermediate pure dark matter zoom simulation with a lower resolution (e.g. 1024 3 DM effective particles in the central part) has been realized in order to check the convergence of the dynamical and physical properties of the cental main haloes (masses, relative distance of haloes, etc...).

The simulations were performed using GADGET2 (Springel 2005). As far as the run baryons is concerned, we have added prescriptions for cooling, star formation, feedback from Type Ia and II supernovae (SN), UV background and metal enrichment. Most of this implementation have been tested (effects of mass resolution on the star formation rate for instance) and used in previous works (see, for instance, Peirani et al. 2009a,b). For the sake of clarity, we summarize the main points in the following.

The evolution of the thermodynamic properties of the gas depends on a range of physical processes. First we use a standard UV-background prescription described in Haardt & Madau (1996). The cooling and star formation (SF) recipes follow the prescriptions of Thomas & Couchman (1992) and Katz, Weinberg & Hernquist (1996), respectively. Gas particles with T > 10 4 K cool at constant density with a given metallicity for the duration of a timestep. Gas particles with T < 2 × 10 4 K, number density n > 0.1 cm -3 , overdensity ∆ρgas > 100, and ∇.υ < 0 form stars according to the standard SFR prescription: dρ * /dt = c * ρgas/t dyn , where ρ * refers to the stellar density, t dyn is the dynamical timescale of the gas, and c * is the SF efficiency. Instead of creating new (lighter) star particles, we implement the SF prescription in a probabilistic fashion. Assuming a constant, dynamical time scale across the timestep, the probability p that a gas particle forms stars in a time ∆t is p = 1 -exp(-c * ∆t/t dyn ). For each eligible gas particle, we select a random number (r) between 0 and 1 and convert it into a star if r < p.

Next, star particles are treated as evolving stellar populations that inject energy into the inter-stellar medium, through SN explosions, as a function of time. For this, we consider stellar lifetimes in the mass ranges of 0.8 M⊙ < m < 8.0 M⊙ and 8.0 M⊙ < m < 80.0 M⊙ for Type Ia and Type II progenitors, respectively. Using a Salpeter initial mass function for Type II SN, we find that the rate of energy injection is:

HSN II = 2.5×10 -18 m * M⊙ ESN 1300 τ (Myr) -3 0.24 erg.s -1 , (1)
where ESN = 10 51 erg, m * is the mass of the stellar population, and 3.53 < τ < 29 Myr. For Type Ia SN, the heating is delayed, since they appear t0 = 0.8 -1.0 Gyr after the onset of star formation. Following de Freitas Pacheco (1998), the probability of one event on a timescale τ after the onset of star formation is given by:

HSN Ia = 4.8 × 10 -20 m * M⊙ ESN t0 τ 3/2 erg.s -1 . (2)
Equations ( 1) and ( 2) are used to compute the energy released by SN derived from a star particle i (Ei). A fraction γ of this energy is deposited in the j th neighbour gas particle by applying a radial kick to its velocity with a magnitude ∆vj = (2wj γEi/mj), where wj is the weighting based on the smoothing kernel and mj is the mass of gas particle j. We note that all gas neighbours are located in a sphere of radius RSN , centered on the SN progenitor, to avoid spurious injection of energy outside the SN's region of influence. In the following, we use the following standard values: γ = 0.1, RSN = 0.4 kpc, and c * = 0.02.

Supernovae do not only inject energy to the ISM but also diffuse heavy metals leading to a progressive enrichment of their nearby environment. To estimate the stellar metallicity, we assume that the first generation of stars will have "zero-metallicity" and the subsequent generations will be formed in a medium enriched in metals by the previous generations. To estimate the metal abundance, we have considered two elements: Mg (magnesium), representing the "α-abundance" and Fe (iron), representing all heavy elements of the "iron-peak". The former is produced essentially by the type II events whereas the later is produced mainly by type Ia supernovae.

The amounts of Fe and Mg produced by type II supernovae originating from each stellar cluster are taken from Umeda et al. (2002):

MF e,M g = mF e,M g SN II m * λSN II M⊙, (3)
where m * is the mass of star particle in the simulation. All Fe (or Mg) is produced after only 29 Myr. Using a Salpeter IMF, the fraction by mass of these events is λSN II = 5.72 × 10 -3 M -1 ⊙ and the average Fe and Mg masses produced by type II supernovae are < mF e >SN II = 0.068 M⊙ and < mMg >SN II = 0.12 M⊙. In the case of type Ia supernovae, the amount of Fe and Mg ejected after a given age of a given generation is

MF e,M g = mF e,M g SN Ia m * λSN Ia 1- t0 τ 1/2
M⊙, (4) where λSN Ia = 7.2 × 10 -4 M -1 ⊙ , < mF e >SN Ia = 0.85 M⊙ and < mMg >SN Ia = 0.026 M⊙ (from model W7 by Nomoto et al. 1997).

The mass of Fe and Mg injected in the medium follows the supernovae rate estimated previously, being redistributed according to the smoothing kernel among the gas particles in the sphere of radius RSN . Then, at each timestep, one can estimate the [Fe/H], [Mg/H] and [Mg/Fe] abundances of each gas particle as well as a global metallicity Z by using the formula logZ = [F e/H] -1.70 + log(0.362 + 0.638fα) derived in Salaris, Chieffi & Straniero (1993), where fα = 10 [α/F e] is the enhancement factor of α-elements with respect to iron. In our model, the [Mg/Fe] abundance is used as a proxy of [α/Fe]. The inclusion of metal enrichment processes in the present study allows us to improve the treatment of the gas cooling using metal-dependent cooling functions adapted from Surtherland & Dopita (1993). Note that a similar methodology was adopted in previous works (see for instance, Kobayashi 2004;Tornatore et al. 2004;Scannapieco et al. 2005).

Properties of simulated Local Group type objects

In this study, we focus on pairs of galaxies with physical characteristics similar to the Milky Way-Andromeda pair (MW-M31). Each LG group candidate must satisfy the following criteria:

(a) The group contains two main haloes of comparable mass (∼ 0.7-1.2×10 12 h -1 M⊙), are separated from a distance 0.8 ≤ D ≤ 1.0 Mpc and their relative radial velocity is negative.

(b) There is no object with mass greater than 5 × 10 11 h -1 M⊙ and 10 13 h -1 M⊙ within the distance 3 and 8 Mpc from the center of mass of the main pair of haloes respectively.

(c) There is only one Virgo-like halo (M ≥ 10 14 h -1 M⊙) at the distance of 8 to 13 Mpc.

Such selection criteria are quite similar to those used in previous works (Macciò, Governato & Horellou 2005;Hoffman et al. 2008;Tikhonov & Klypin 2009;Martinez-Vaquero et al. 2007, 2009) and then make the comparison of the results easier. Several candidates of pairs of dark matter haloes that satisfy criterion (a) can be easily found but almost all of them are ruled out by criteria (b) and (c). In the end, only one candidate that satisfy all of the above criteria has been found. This latter system has been resimulated with and without baryons and using higher resolutions. The simulations were performed on local parallel supercomputers at IAP (France) and the CPU times for runDM and run baryons were ∼ 2 days and ∼ 2.5 months respectively using 32 processors. Table 1 summarizes the main properties of the central objects in both runs and the number of particles considered for each component. In the following, we will mostly characterize dark matter haloes by their maximum circular velocity which can defined by Vmax = max( Gm(≤ r)/r ), where m(≤ r) is the mass enclosed at radius r. Vmax can be estimated without any accurate estimate of the physical boundary of the objects which can be proved to be difficult especially for subhaloes.

Fig. 1 shows the projected density of dark matter from runDM and run baryons as well as the projected density of stars within a sphere of physical radius 4 Mpc. The distributions of dark matter look very similar between the two simulations. However, the dark matter haloes seem to be more concentrated when baryons are included in the simulation. This can be explained by the fact that dissipation of the gas (from radiative cooling processes) and subsequent star formation lead to a steeper DM density profile due to adiabatic contraction. This process can also be observed in dark matter substructures which become less sensitive to tidal torques and will survive a longer time in their host haloes, as has been extensively discussed in the past literature (Blumenthal et al. 1986;Gnedin et al. 2004;Duffy et al. 2010 and references therein).

Mock catalogs of dark matter haloes and galaxies

One of the main purposes of this work is to study how the inclusion of baryons may affect the dynamical properties of the simulated local universe. Then, we have used a simple algorithm which allows a direct comparison between the two simulations. First, we have prepared mock catalogs of dark matter haloes and galaxies at z = 0 for each simulation. To do this we have used the public code AMIGA † (Knollmann & Knebe 2009) which is an efficient halo and substructure finder by identifying local density maxima in an adaptively smoothed density field. The use of this code is also attractive since subhaloes are separated form their parent haloes and each (sub)halo center is estimated accurately by an iterative algorithm, which is crucial in this study. For each object, we define the virial radius RV (and thus the virial mass MV ) as the radius where the enclosed mean density MV /(4πR 3 V /3) is ∆c times the critical density, ρc(z) = 3H(z) 2 /8πG, where H(z) = H0 Ωm(1 + z) 3 + ΩΛ. ∆c can be computed from the spherical top-hat collapse model, and in the case of a flat cosmology with Ωm + ΩΛ = 1, one can use the fitting formula from Bryan & Norman (1998) which depends on both cosmology and redshift:

∆c = 18π 2 + 82x -39x 2 ,
(5)

x = Ωm(z) -1, ( 6)

Ωm(z) = Ωm(1 + z) 3 H0 H(z) . ( 7
)
In the cosmology used for the present study, ∆c = 97.6 at z = 0. Then, only bound structures with at least 100 dark matter particles (e.g. a mass higher than 8.8 × 10 8 h -1 M⊙ in runDM ) were retained in the different catalogs. In the following, SPH haloes refers to objects identified in run baryons which consists of a single dark matter halo containing a fraction of gas and (or not) a galaxy, namely a bound structure with at least 80 star particles (corresponding to a mass of 1.2 × 10 8 h -1 M⊙).

Once the two catalogs are built, we match the SPH haloes to their equivalent DM haloes of runDM by using the following algorithm. Since each particle in the simulation can be identified, it is possible to obtain the constitution of each SPH and DM halo at z = 0. If more than 50% of dark matter particles of a given SPH halo is found in a dark matter halo of runDM , we assume that both objects are the same. To illustrate this, Fig. 2 shows the distribution of all galaxies that have a corresponding dark matter halo in runDM within a distance of 4 Mpc from the LG mass center. However, it is worth mentioning that a small number of SPH haloes have no corresponding dark matter halo in runDM . These SPH haloes are mainly substructures belonging to the most massive objects such as the MW or M31 haloes. In runDM , they don't exist anymore, mainly because they have been completely disrupted by tidal torques. Such a statement has already been studied in great detail in previous works (see for instance, Nagai & Kravtsov 2005;Libeskind et al. 2010, Klimentowski et al. 2010).

It is now interesting to study the statistical differences between the radial distances (in the frame of the Local Group) and peculiar velocity of galaxies (rg, -→ Vg) identified from run baryons and those of their corresponding dark matter haloes in runDM (rDM , ---→ VDM ) at z = 0. The main results are shown in Fig. 3 where the probability distribution functions of the position ratio rDM /rg, † http://popia.ft.uam.es/AMIGA ---→ VDM ) with respect to positions rg. It appears clearly that most of the galaxies have very similar position and velocity properties to their corresponding dark matter haloes. However, in some cases, we notice important differences between the two runs. This happens mainly for substructures (∼ 5 × 10 9 M⊙) in more massive objects where their dynamics and orbits are more likely to be modified by the halo potential, as previously observed by Weinberg et al. (2008). Moreover, a detailed study by Libeskind et al. (2010) has also shown that subhaloes in hydrodynamic runs tend to be more radially concentrated than those in pure dark matter ones. Vg of galaxies identified in run baryons and those of their corresponding dark matter haloes in run DM (r DM ,V DM ) at z = 0: the first column shows the variations of the norm ratio r DM /rg (first line), the velocity norm ratio V DM /Vg (second line) and angle between velocity vectors (third line) as respect to rg while the probability distribution functions of r DM /rg (first line), V DM /Vg (second line) and the angle (third line) are plotted in the second column. The red crosses correspond to dark matter subhaloes with a mass lower that 5 × 10 9 M ⊙ .

Among the population of SPH haloes, there is a subpopulation of dark matter haloes that contain no stars and thus no galaxy. From a theoretical point of view, the existence of such dark halo populations can be explained by the fact that their star formation have been suppressed by feedback processes such as UVbackground (Thoul & Weinberg 1996;Bullock, Kravtsov & Weinberg 2000;Somerville 2002;Ricotti, Gnedin & Shull 2002;Benson et al. 2002;Read, Pontzen & Viel 2006;Hoeft et al. 2006;Okamoto, Gao & Theuns 2008). In particular, the presence of a photoionizing background suppresses the formation of galaxies with circular velocities Vcirc ≤ 30 km/s (Thoul & Weinberg 1996;Bullock, Kravtsov & Weinberg 2000) and a similar result has been derived by Tikhonov & Klypin (2009) in order to explain the observed size of local voids, using models with σ8 = 0.75. Moreover, Hoeft et al. (2006) have also studied the formation of dwarf galaxies in voids using high-resolution hydrodynamical simulations including UV-background. They derived a critical mass for haloes at z = 0, Mc = 6.5×10 9 h -1 M⊙, below which the cooling of gas is suppressed and similar results have been derived by Okamoto, Gao & Theuns 2008). Then, we have compare the properties of dark matter haloes of runDM that have either a corresponding galaxies or a dark halo in run baryons . In Fig. 4, the probability distribution function is represented for the maximum circular velocity Vc of DM haloes from runDM and the variations of Vc as respect to their total halo mass. The results suggest that there is a critical value (Vc ∼ 30 km/s) where the star formation is suppressed, in good agreement with past works. Moreover, dark matter haloes that have corresponding starless SPH haloes in run baryons are characterized by Vc ≤ 35 km/s and have a mass lower than ∼ 7 × 10 9 h -1 M⊙. Note that, the circular velocity of dark matter haloes of runDM can be well approximated by Vc = 39.08 km s -1 × (M/10 10 h -1 M⊙) 0.32 in good agreement with Hoeft et al. (2006) and Klypin, Trujillo-Gomez & Primack (2010).

THE LOCAL HUBBLE FLOW

The local Hubble flow has proved to be cold according to past observational analysis. As mentioned in the introduction, this puzzle has been studied in various numerical works but no one so far has investigated the impact of baryons, the main purpose of this work.

In the following, the peculiar velocity dispersion σH in the local volume is estimated by using the root mean square (RMS) of the residual from the global Hubble flow, defined by:

σ 2 H = 1 N N i=1 vi -vH (ri) 2 (8)
where vi and ri are radial velocities and radial distances of galaxies (or dark matter haloes) in the frame of the Local Group where the origin matches with the mass center of the MW-M31 pair, and vH is the global Hubble flow. In order to compare with recent works, and more specifically with the analysis of Tikhonov & Klypin (2009) and Martinez-Vaquero et al. (2009), we have estimated σH using different samples of dark matter haloes or galaxies with distances between 1 to D Mpc (where D=2, 3, 4, 5, 6, 7 and 8 Mpc) and considering vH (ri) = H0ri, with H0 = 70.5 km/s/Mpc. However, it is worth mentioning that for distances within ∼ 3 Mpc, due to the decelerating influence of the main central objects to nearby low mass objects, the Hubble flow is not well described by vH (ri) = H0ri (see Fig. 7) and one should use another relation such as the one derived in Peirani & de Freitas Pacheco (2008). Then, the study has been limited to a distance of 8 Mpc to avoid to be in the vicinity of the boundary of the highest resolution domain. First, sample A includes all galaxies identified in run baryons and represents therefore the fiducial sample. For this case only, σH has been estimated either by assuming a 10% error as the mean error of distance Table 2. The velocity dispersion around the Hubble flow v H (r) = H 0 r for different samples of selected galaxies (run baryons ) or dark matter haloes (run DM ) as a function of the distance from the mass center of the simulated MW-M31 pairs. Each number in parenthesis gives the number of objects that have been considered in the computation of σ H . g σ H computed from DM haloes identified in run DM with a maximum circular velocity Vc ≥20 km/s (Sample C 3 ).

R(Mpc

h rms = 1 7 σ 2 H (R) -σ 2 * (R)
measurements (referred to as σ * ,cor) or with no error (referred to as σ * ). Samples B1 and B2 regroup galaxies and their corresponding dark matter haloes in runDM respectively and have therefore the same number of objects which permits a direct comparison between the two runs. Finally, samples C1, C2 and C3 gather dark matter haloes extracted from runDM with circular velocities higher than 45, 30 and 20 km/s respectively. Fig. 5 shows the Hubble diagram for galaxies and dark matter haloes belonging to sample A, B1 and B2. As expected from results presented in the previous section, radial distances and velocities of galaxies and those of their corresponding dark matter haloes in runDM follow the same trend. Such a statement is also confirmed by Table 2 where all the σH values derived at different distances from the LG mass center and for different samples are shown. In particular, σH values derived from samples B1 and B2 are very close and suggests that the presence of baryons does not affect the velocities dispersion of satellites around the main pair at distances 1-8 Mpc. Moreover, results from samples C1, C2 and C3 show that estimations of σH from the pure dark matter simulation give similar values than those derived from the fiducial sample as it can been clearly seen in Fig. 6. For instance, when dark haloes are selected with Vc ≥ 30 km/s, the number of objects considered is quite close to the "true" number of galaxies and the mean error on σH is in order of ∼ 3 km/s. However, when haloes are selected with Vc ≥ 45 km/s and Vc ≥ 20 km/s, both cases lead to an estimation of σH with a mean error of ∼ 5 km/s as respect to σ * values although the number of objects taken into account for the σH computation are underestimated or overestimated respectively.

To finish, it is instructive to compare the velocity dispersion values derived from our numerical model to σH values obtained from the analysis by Tikhonov & Klypin (2009) who have considered the most complete observational data using an updated version of the Karachentsev et al. (2004) sample of galaxies. The number of galaxies used to compute σ * (first column of Table 2) is in quite good agreement with number of galaxies considered within 1-8 Mpc (see their Table . 3). However, we found that σH values derived in this work tend to be (∼ 10%) smaller within 1-8 Mpc. We have also compared in Fig. 7 our simulated data to updated data on distances and velocities of galaxies for distances 0.7 ≤ D ≤ 3 Mpc without an additional dipole component (Karachentsev et al. 2009). The nice agreement between simulated data and observations lead to similar values of the dispersion of peculiar velocities. Indeed, using in this case vH (r) = Vr(r) derived in Peirani & de Freitas Pacheco (2008), we found σH = 29.3 km/s from observational data whereas σ * = 33.0 km/s and σ DM,Vc≥30 = 29.9 km/s respectively for galaxies from run baryons and dark matter haloes from runDM with Vc ≥ 30 km/s. But an important feature of the Hubble flow at such distances is the decelerating influence of the central main objects on nearby galaxies which lead the radial velocities-distances regression to cross the line of zero velocity at the turnaround radius R0. In this regard, Lynden- Bell (1981) and Sandage (1986) proposed an alternative method to the virial relation in order to estimate the mass of the Local Group, which can be extended to other systems dominated either by one or a pair of galaxies. Based on the Lemaître-Tolman (LT) model (Lemaître 1933;[START_REF] Tolman | Proc. Nat. Acad. Sci[END_REF], they show that, if the velocity field close to the main central body (probed by satellites) allows the determination of R0, then the total central mass can be calculated straightforwardly from the relation

M = π 2 R 3 0 8GT 2 0 , (9)
where T0 is the age of the universe and G is the gravitational constant. In a flat cosmological model with a cosmological constant Λ, equation ( 9) needs to be modified as follows (Karachentsev et al. 2007):

V c [km/s] P(V c ) M/[h -1 M o ] V c [km/s]
M = π 2 R 3 0 H 2 0 8GT 2 0 f (ΩΛ) , (10)
where

f (ΩΛ) = 1 ΩΛ - 1 -ΩΛ 2Ω 3/2 Λ cosh -1 2 1 -ΩΛ -1 (11)
We have estimated R0 from our two mock catalogs of galaxies and haloes by fitting the Vr = Vr(r) formula derived in Peirani & de Freitas Pacheco (2008) to our simulated data. From the best fit solutions, we obtained R0 = 0.98 Mpc (galaxies sample) and R0 = 1.03 Mpc (for dark matter haloes sample with Vc ≥ 30 km/s).

Using ΩM = 0.274 and H0 = 70.5 km/s/Mpc, one can derive from Eq. ( 10) the total masses enclosed at radius R0, namely M = 2.0 × 10 12 M⊙ (galaxy sample) and M = 2.3 × 10 12 M⊙ (dark matter haloes sample) respectively. However, the total mass enclosed in a radius of 1 Mpc is ∼ 3.4 × 10 12 M⊙ from both simulations and suggest that masses derived from Eq. ( 10) are underestimated by a factor 1.5-1.7.

DISCUSSION AND CONCLUSIONS

In the present paper, we have used cosmological N-body simulations with and without a complete treatment of the physics of baryons to study the formation of a local group type halo. We have first identified a LG system in a simulation with a low mass particle resolution, using standard criteria. Then, the best candidate has been re-simulated with higher resolution using the technique of zoom. This approach differs from the technique of constrained simulations used for instance in the CLUES project [START_REF] Yepes | American Institute of Physics Conference Series[END_REF], which include the correct motion and position of objects in a large volume such as the Local Supercluster, the Virgo and the Coma cluster and the Great Attractor (see also Lavaux 2010). We focus on the dynamical properties of the local Hubble flow and in particular, the influence of baryons. From the two runs, we built mock catalogs of either galaxies or dark matter haloes and computed their velocities as respect to the mass center of the central main pair (Milky Way-M31). We found that the dispersions of peculiar velocities around the mean local Hubble flow of galaxies in run baryons and those of their corresponding dark matter haloes in runDM are very close for distances D = 1 to 8 Mpc. Moreover, similar σH values (with mean errors ≤ 5 km/s) are also obtained for samples of dark matter haloes extracted from runDM with constraint of their maximum circular velocity. These results suggest that the global dynamical properties of the Hubble flow is not affected by the presence of baryons. Such a result is not very surprising since galaxies are expected to form in the gravitational potential well of dark matter haloes. Therefore, their properties are expected to be closely related to those of their host haloes. This statement is confirmed on large scales as shown in Figs. 2 and3. However, some discrepancies between positions and dynamical properties of galaxies and those of their corresponding haloes may raised in the case of low mass objects in denser environment. For instance, the presence of baryons is supposed to favor the survival of substructures inside massive haloes against tidal striping (see Duffy et al. 2010 and references therein) and can affect their distribution (Weinberg et al. 2008;Libeskind et al. 2010, Knebe et al. 2010). But these effects seem to have no particular consequences in the estimation of the global σH values. It should also be pointed out that run baryons does not include any prescriptions of strong feedback such as galactic winds or Sedov shock waves that are expected to lead to a significant ejection of gas from the central region of forming galaxies. As a result, the fraction of baryons (in the form of stars and cold gas) in the central region of simulated massive galaxies such as the Milky Way or M31 is close to the cosmological baryonic fraction (see Table 1) whereas observations suggest lowest values (Fukugita, Hogan & Peebles 1998;Bell et al. 2003). However, no one has solved this missing baryon problem yet and recent studies even suggest that theoretical sources of strong winds such as AGN and supernovae are insufficient to explain the missing baryons in Milky Way type galaxies (see for instance Anderson & Bregman 2010;Silk & Nusser 2010). Although the present study mainly focus on lower mass galaxies (≤ 10 10 M⊙) in which supernovae feedback is supposed to have a significant impact in their formation, the other sources of strong feedback should be taken into account as well. Then, a natural extension of this work would be to add these processes while increasing the mass resolution in the simulation (see for instance [START_REF] Ceverino | [END_REF] to see whether the conclusions presented here (especially the distribution of galaxies as respect to their corresponding dark matter haloes) are significantly affected of not.

In good agreement with previous theoretical and numerical works, we found that the halo population in the pure dark matter run can be divided into two groups according to their maximum circular velocity Vc. There is indeed a critical value, Vc ∼ 30-35 km/s (corresponding to haloes with mass lower than ∼ 7 × 10 9 h -1 M⊙ in runDM ), below which the corresponding SPH haloes don't host any galaxy. Such a phenomenon may be explained by the fact that the cooling of gas is expected to be suppressed by feedback processes such as a UVphotoionisation in low mass haloes (see references in paragraph 2.3). As regards this point, we cannot exclude the fact that the numerical resolution used in this work may affect the results especially for low mass systems. However, haloes with mass of ∼ 7 × 10 9 h -1 M⊙ are composed by ∼ 800 particles and this seems reasonable for the estimation of the thermodynamic properties of the gas component. Then, table 2 indicates that the number of dark matter haloes in runDM selected with Vc ≥ 30 km/s is actually quite close to the real number of galaxies selected run baryons . Therefore, one can use this single criteria to select dark matter haloes in a collisionless simulation to have an good estimation of σH . The existence of a dark halo population in the Local Group can also have important consequences in understanding the discrepancies between the large number of subhaloes present in simulations but not observed (Kauffmann, White & Guiderdoni 1993;Moore et al. 1999;Klypin et al. 1999). This problem is beyond the scope of this paper and will be investigated in detail in a forthcoming paper.

The velocity dispersion obtained from our fiducial model (e.g. from galaxies) is in quite good agreement with observational expectations. For instance, with added corrections for distance errors, we obtained σ * ,cor = 90.4 for D = 1 -8 Mpc, a value which tend to be even lower that the one derived from observational analysis, namely 99.3 km/s (Tikhonov & Klypin 2009) and with a reasonable number of galaxies considered. For distances from 0.7 to 3 Mpc, we also found a nice agreement between our simulated data and observational ones from Karachentsev et al. (2009). Then, these results suggest that there is no particular problem with the ΛCDM model in reproducing the velocity dispersion derived from observational analysis. A similar conclusion was obtained from Martinez-Vaquero et al. (2009) who found that a non-negligible fraction of LG like objects simulated form various cosmological models present a σH value close to (and even smaller) than the observed value. Moreover, similar results have been also obtained from the analysis of Tikhonov & Klypin (2009). Since the present study has been lim-ited to only one realization, due to the high computational cost when the physics of baryons is included in the simulations, general conclusions cannot be drawn. However, one of our main criteria to select LG candidates was their relative isolation (no massive galaxies within 3 Mpc). We stress that we found only one candidate (among 16 MW-M31 pair candidates) that was satisfying all the criteria. This raised the question whether the local group is located in a particular place of the universe or not.

To finish, we have also compared our simulated data to observational ones for R = 0.7 -3 Mpc. The nice agreement between them allows us to test models, in particular the estimation of the total mass enclosed at the zero-velocity surface radius R0 using the Lemaître-Tolman model adapted to ΛCDM. It appears that the estimation of the LG mass using this approach is underestimated by at least 50%. This is probably due to the fact that, on the one hand, the hypothesis of a spherical infall collapse is not valid anymore in this specific case. On the other hand, the mass enclosed within R0 is not constant in the time, as assumed by the TL model (e.g. no mass accretion), and this effect may have an important influence in the estimation of the final total mass.

Figure 1 .

 1 Figure 1. The projected density of dark matter from run DM (left panel) and run baryons (middle panel) within a sphere of radius 4 Mpc. The right panel represents the projected density of stars in the same volume.

Figure 2 .

 2 Figure 2. Distributions of galaxies (upper panel) and their corresponding dark matter haloes in run DM (lower panel) and using the same color code. For clarity reason we have limited to a distance of 4 Mpc from the LG mass center.

Figure 3 .

 3 Figure3. Comparison between the radial distances rg (in the frame of the Local group) and the peculiar velocities Vg of galaxies identified in run baryons and those of their corresponding dark matter haloes in run DM (r DM ,V DM ) at z = 0: the first column shows the variations of the norm ratio r DM /rg (first line), the velocity norm ratio V DM /Vg (second line) and angle between velocity vectors (third line) as respect to rg while the probability distribution functions of r DM /rg (first line), V DM /Vg (second line) and the angle (third line) are plotted in the second column. The red crosses correspond to dark matter subhaloes with a mass lower that 5 × 10 9 M ⊙ .

Figure 4 .

 4 Figure4. Dark matter haloes from run DM : the probability distribution functions of their maximum circular velocity Vc (upper panel) and the variations of their mass as respect to Vc (lower panel). Blue color refers to DM haloes that have corresponding starless DM haloes in run baryons whereas red color refers to DM haloes that match SPH haloes with a galaxy. In the lower panel, the solid line represents our best fit (see the text for more details) while the dashed lines show Vc = 35 km/s and M = 7 × 10 9 h -1 M ⊙ .

Figure 5 .

 5 Figure5. Simulated radial velocity-distance data for galaxies of sample B 1 (green dots) and dark matter haloes from sample B 2 (red dots). The blue dots correspond to velocitydistance data for galaxies that don't have a corresponding halo in run DM . The solid line represents the H 0 r Hubble flow and the dashed ones show H 0 r ± 98.9 km/s.

Figure 6 .

 6 Figure6. The velocity dispersion around the Hubble flow v H (r) = H 0 r as a function of the distance from the mass center of simulated MW-M31 pairs for galaxies identified in run baryons (σ * , black line), and for dark matter haloes from run DM with a maximum circular velocity defined by Vc ≥ 20 km/s (blue circles), Vc ≥ 30 km/s (green circles), Vc ≥ 45 km/s (red circles). The dashed lines show σ * ± 5 km/s.

Figure 7 .

 7 Figure7. Upper panel: radial velocities and distances for observed (green circles) and simulated (red circles) galaxies in the neighborhood of the LG with respect to its centroid. In the lower panel, the velocity and distance data derived from dark matter haloes in run DM with Vc ≥ 30 km/s is shown. In both panels, the inclined dashed line represents the linear Hubble relation with H 0 = 70.5 km/s while the solid line represents the best fit to the vr = vr(r) relation derived inPeirani & de Freitas Pacheco (2008).

Table 1 .

 1 Physical properties of the two central main objects in both runs. Numbers in parenthesis correspond to number of particles of each component. D M W -M 31 specifies the physical distance between the two central objects and the unit of mass is h -1 M ⊙ .

		run DM		run baryons
	Galaxy	Mtot	Mtot	M DM	Mstar	Mgas
	MW	8.85 × 10 11	8.96 × 10 11	7.58 × 10 11	1.08 × 10 11	3.02 × 10 10
		(100 097)	(196 947)	(102 846)	(73 600)	(20 501)
	M31	8.84 × 10 11	8.65 × 10 11	7.43 × 10 11	8.44 × 10 10	3.81 × 10 10
		(100 052)	(183 926)	(100 856)	(57 223)	(25 847)
	D M W -M 31	1.01 Mpc		1.03 Mpc

  σ H computed from galaxies identified in run baryons (Sample A). b σ H computed from galaxies identified in run baryons (Sample A) but corrected from distance errors. c σ H computed from galaxies identified in run baryons that have a corresponding DM halo in run DM (Sample B 1 ). d σ H computed from DM haloes identified in run DM that have a corresponding galaxy in run baryons (Sample B 2 ). e σ H computed from DM haloes identified in run DM with a maximum circular velocity Vc ≥45 km/s (Sample C 1 ). f σ H computed from DM haloes identified in run DM with a maximum circular velocity Vc ≥35 km/s (Sample C 2 ).

	)	σ * a	σ * ,cor b	σ * ↔DM	c	σ DM ↔ *	d	σ DM,Vc≥45	e	σ DM,Vc≥30	f	σ DM,Vc≥20	g
	1-2	51.4 (17)	50.1 (17)	51.4 (17)	52.8 (17)	48.5 (7)		52.0 (20)		59.1 (59)
	1-3	45.2 (37)	42.4 (37)	45.1 (36)	44.3 (36)	39.0 (14)		44.1 (41)		47.3 (126)
	1-4	58.5 (66)	54.8 (66)	57.5 (64)	56.2 (64)	48.2 (21)		53.9 (72)		57.0 (217)
	1-5	66.9 (125)	61.4 (125)	61.9 (114)	61.6 (114)	69.0 (48)		64.8 (134)		65.0 (372)
	1-6	75.1 (188)	68.2 (188)	72.3 (172)	71.1 (172)	74.3 (69)		72.8 (192)		70.1 (532)
	1-7	83.5 (267)	75.2 (267)	81.4 (246)	80.7 (246)	82.7 (92)		79.9 (269)		77.6 (724)
	1-8	98.9 (331)	90.4 (331)	98.5 (308)	97.9 (308)	101.8 (120)	94.7 (332)		91.5 (904)
	rms h	0.0		2.3		3.0		4.9		3.0		5.1

a

c 0000 RAS, MNRAS 000, 000-000

Acknowledgment I acknowledge support from the "Agence National de la recherche" ANR-08-BLAN-0222-02. It is a pleasure to thank the anonymous referee for his/her useful comments which have significantly improved this paper. I warmly thanks C. Alard, S. Colombi, J. Devriendt, R. Gavazzi, M. Hudson, Y. Kakazu, I. D. Karachentsev, A. Klypin, G. Mamon, R. Mohayaee, J. A. de Freitas Pacheco, C. Pichon, S. Prunet, J. Silk and T. Sousbie for interesting discussions. I also thanks D. Munro for freely distributing his Yorick programming language (available at http://yorick.sourceforge.net/) which was used during the course of this work. This work was carried within the framework of the Horizon project (http://www.projet-horizon.fr).