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Monthly Notices of the Royal Astronomical Society
, Volume 476 (2) – May 1, 2018

8 pages

/lp/ou_press/formation-and-evolution-of-substructures-in-tidal-tails-spherical-dark-gj2qvEF571

- Publisher
- The Royal Astronomical Society
- Copyright
- © 2017 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society
- ISSN
- 0035-8711
- eISSN
- 1365-2966
- D.O.I.
- 10.1093/mnras/stx2900
- Publisher site
- See Article on Publisher Site

Abstract Recently a theory about the formation of overdensities of stars along tidal tails of globular clusters has been presented. This theory predicts the position and the time of the formation of such overdensities and was successfully tested with N-body simulations of globular clusters in a point-mass galactic potential. In this work, we present a comparison between this theory and our simulations using a dwarf galaxy orbiting two differently shaped dark matter haloes to study the effects of a cored and a cuspy halo on the formation and the evolution of tidal tails. We find no difference using a cuspy or a cored halo, however, we find an intriguing asymmetry between the leading arm and the trailing arm of the tidal tails. The trailing arm grows faster than the leading arm. This asymmetry is seen in the distance to the first overdensity and its size as well. We establish a relation between the distance to the first overdensity and the size of this overdensity. methods: numerical, galaxies: dwarf, galaxies: haloes, galaxies: kinematics and dynamics – galaxies: structure 1 INTRODUCTION An image of the globular cluster Palomar 5 (Pal 5) and its tidal tails has been published by Odenkirchen et al. (2001). In this image, the tidal tails of Pal 5 along with some overdensities within them are clearly visible. Besides Pal 5, there are also other satellites of the Milky Way (MW) which have elongated structures, for example the Sagittarius (Sgr) dwarf spheroidal galaxy (dSph), whose tidal tails were described by Majewski et al. (2001), the Virgo stellar stream (Vivas et al. 2001), the new Aquarius stream (Williams et al. 2011), and all the streams listed in the Field of Streams (Belokurov et al. 2006). Tidal tails could be useful to constrain the properties of the MW like the mass and the shape of the dark matter (DM) halo. Koposov, Rix & Hogg (2010), using a 6D space map of the GD-1 stream in Andromeda, found that an eccentric orbit in a flattened isothermal potential (qΦ = 0.87$$^{+0.07}_{-0.04}$$) fits the parameters of distance, line-of-sight velocity, and proper motion of the stars in the stream. Fellhauer et al. (2006) demonstrated that if the tails of the Sagittarius dSph are wrapping around twice the MW, causing the visible bifurcation (Belokurov et al. 2006), then the MW DM halo has to be close to spherical. To recover the Galactic potential using tidal tails, the first step is to determine the orbit of the satellite by looking at the elongation of its tails. The next step is to perform simulations using different Galactic potentials and assuming properties of the satellite like the initial mass, the eccentricity of the orbit, or the distance to its apo-centre or peri-centre. Finally, one has to compare the final properties of the stars in the debris like the velocity dispersion, line-of-sight velocity, and proper motion with data from observations. The bias from this method comes from the fact that one has to assume properties of the satellite and/or the MW. However, the study of the formation of tidal tails could give us some clues about the initial properties of the disrupting satellites, thus improving the methods to constrain Galactic properties. Following this line, Küpper, Macleod & Heggie (2008) published a theory that explains how tidal tails are formed. In their work, they propose that the stars that escape from a globular cluster move in epicycles inside a galactic potential. They found good agreement between their theory and N-body simulations of globular clusters under the influence of a point-mass galaxy (Küpper, Macleod & Heggie 2008; Küpper et al. 2010; Küpper, Lane & Heggie 2012). In their theory, Küpper et al. (2008) make the approach that all the particles escaping from the satellite have small velocities with respect to the satellite and they escape always through the Lagrangian points L1 and L2, and therefore two symmetrical tails are formed, a leading arm where particles orbit ahead of the satellite and a trailing arm where particles orbit behind the satellite. With these assumptions they derive equations that predict the position and the time of the formation of overdensities along both tails assuming that if D is the distance to the first overdensity in the leading arm, then −D is the distance to the first overdensity in the trailing arm where the reference frame is in the centre of the satellite. Our goal is to check the theory of Küpper et al. (2008) comparing their predictions with simulations performed by superbox (a collisionless particle mesh code; see Fellhauer et al. 2000) using a dwarf satellite and modelling the MW DM halo either with a logarithmic halo (cored profile) or with a Navarro–Frenk–White (NFW) halo (cuspy profile; Navarro, Frenk & White 1996). The paper is organized as follows. In Section 2, we describe the properties of the satellites used in our simulations, the DM potential, and the orbits of the satellites. In Section 3, we describe the analysis of the data and how we estimate our errors. In Section 4, we present our results. In Section 5, we discuss the results and present our conclusions. 2 SET-UP To integrate the particles forward in time, we use superbox (Fellhauer et al. 2000), a collisionless particle mesh code that uses three levels of grids, with the two high-resolution grids staying focused on the simulated and moving object. This makes the code faster by improving resolution only at the places of interest. To model the MW DM halo, we use analytic potentials. For the cored case, we use a logarithmic potential given by \begin{eqnarray} \Phi & = & \frac{1}{2} v_{\rm c}^{2} \ln (r^2 + d^2), \end{eqnarray} (1)where r is the distance to the centre of the galaxy, vc = 220 km s−1 and d = 12 kpc. Alternatively, to mimic a cusped potential, we use an NFW potential: \begin{eqnarray} \Phi & = & 4 \pi G \rho _{0} \frac{r_{\rm s}^3}{r} \ln \left( \frac{r_{\rm s}}{r_{\rm s} + r} \right), \end{eqnarray} (2)with \begin{eqnarray} \rho _{0} = \frac{M_{200}}{4 \pi G r_{\rm s}^{3} (\ln (1+c) - c/(1+c))}, \end{eqnarray} (3)using M200 = 1.8 × 1012 M⊙, r200 = 200 kpc, and rs = 32 kpc. We have chosen these values to produce similar rotation curves, which exhibit vrot, max values as expected for a galaxy like the MW (even though discs and bulges are omitted). The rotation curves for the two profiles can be seen in Fig. 1. The inclusion of non-spherical components and their influence on the tidal tails will be part of a future study. Figure 1. View largeDownload slide Rotation curves of the two potentials used to model the DM halo of the MW in our simulations. The red dashed line is the rotation curve of the logarithmic profile and the solid blue line is the rotation curve of the NFW profile. Note that both curves are very similar in the distance range we use to perform our simulations, in order to better compare the results. Figure 1. View largeDownload slide Rotation curves of the two potentials used to model the DM halo of the MW in our simulations. The red dashed line is the rotation curve of the logarithmic profile and the solid blue line is the rotation curve of the NFW profile. Note that both curves are very similar in the distance range we use to perform our simulations, in order to better compare the results. We investigate circular and elliptical orbits and the influence of satellite's mass in circular orbits, and therefore we use four different Plummer spheres (Plummer 1911): \begin{eqnarray} \rho (r) & = & \frac{3{M}_{\rm pl}}{4\pi R_{\rm pl}^3} \left( 1 + \frac{r^2}{R^2_{\rm pl}}\right)^{-({5}/{2})} \end{eqnarray} (4)with Mpl being the total mass and Rpl being the scale length of the Plummer profile, to model the satellites whose parameters are given in Table 1. The relation between the values of Mpl and Rpl is chosen to have approximately the same filling factor inside the tidal radius at a certain radius for the different satellites. The Plummer spheres are cut-off at 5Rpl. Table 1. Description of the four different satellites used in the simulations. Column 1 is the number to identify each satellite, columns 2 and 3 are the number of particles and the mass of the Plummer sphere and columns 4 and 5 are the scale radius of the Plummer profile and the half-mass radius of the satellite. N Mpl Rpl rh (M⊙) (kpc) (kpc) 1 106 0.5 × 108 0.2 0.34 2 106 108 0.35 0.595 3 106 2 × 108 0.45 0.765 4 106 4 × 108 0.6 1.02 N Mpl Rpl rh (M⊙) (kpc) (kpc) 1 106 0.5 × 108 0.2 0.34 2 106 108 0.35 0.595 3 106 2 × 108 0.45 0.765 4 106 4 × 108 0.6 1.02 View Large Now one can argue that, if we are using satellite galaxies, we should model their DM halo as well. The result would be that we would be discussing invisible DM tails. Instead, we are using a one-component model in which mass follows light. This could be identified by a satellite galaxy that has lost its outer DM halo already or a tidal dwarf galaxy that was born without DM in the first place. We place satellite 2 (as standard model) at six different distances from the centre of the Galaxy according to Table 2 and give them velocities to produce circular orbits and let them evolve for 5 Gyr in both DM haloes. Table 2. Velocity of satellite 2 at five different distances from the galactic centre to get circular orbits in the logarithmic potential (second column) and in the NFW potential (third column). R vlog vNFW (kpc) (km s−1) (km s−1) 15 171.79 174.36 20 188.65 187.75 25 198.34 196.90 30 204.27 203.34 35 208.11 207.93 50 218.43 214.66 R vlog vNFW (kpc) (km s−1) (km s−1) 15 171.79 174.36 20 188.65 187.75 25 198.34 196.90 30 204.27 203.34 35 208.11 207.93 50 218.43 214.66 View Large Then, we investigate the influence of eccentricity, so we keep a fixed apo-galacticon of 80 kpc and change the velocity of the satellite 2 according to Table 3 to produce elliptical orbits with eccentricities between 0.1 and 0.8. Table 3. Eccentricity of the orbit, velocity of the satellite at apo-galacticon (80 kpc) and peri-galactic distance for the eccentric orbits used in this study. e vapo rperi (km s−1) (kpc) 0.1 195.64 65.45 0.2 174.21 53.33 0.3 153.07 43.08 0.4 132.05 34.29 0.5 110.96 26.67 0.8 45.21 8.89 e vapo rperi (km s−1) (kpc) 0.1 195.64 65.45 0.2 174.21 53.33 0.3 153.07 43.08 0.4 132.05 34.29 0.5 110.96 26.67 0.8 45.21 8.89 View Large Finally, we investigate the influence of satellite's mass using the logarithmic halo only and satellites with a mass of 0.5 × 108, 1.0 × 108, 2.0 × 108, and 4.0 × 108 M⊙ orbiting at 25 kpc for 5 Gyr. 3 ANALYSIS To analyse the tidal tails, we use a code developed by Véjar (2013). This code divides both tails into bins of equal size and counts the number of stars in each bin. The size of each bin is given by the maximum length of tails that we want to analyse (L) and the number of bins we want to use (Nbin), and therefore each bin will have a size equal to Lbin = L/Nbin. To count the number of stars in each bin, the code finds the centre of density of the satellite at each time-step from an output file of superbox containing this information. Then, the code calculates the angular position of satellite's centre of density and the angular position of each star with respect to the x-axis, where the reference frame is centred on the centre of the Galactic potential and the initial position of the satellite is on the x-axis. Then, an array containing this information for each particle is sorted by its angular position. The code puts the centre of the first bin in the centre of density of the satellite and calculates the distance between the particles and the centre of the bin using the law of cosines. If the distance is smaller than half the size of the bin, then the particle is counted in that bin. To find a suitable place where to put the centre of the next bin the code calculates the average positions $$\hat{x}$$, $$\hat{y}$$, and $$\hat{z}$$ of the particles outside the bin. Then, the distance $$\hat{r}$$ between the centre of the previous bin and the average position of the particles outside the bin is calculated using the Pythagoras Theorem. If $$\hat{r} / L_{\rm bin} = 1,$$ then we have a place to set the centre of the next bin. This procedure is done until the code reaches the maximum length L defined by the user as we can see in Fig. 2. Figure 2. View largeDownload slide Path followed by the code (green line) to analyse tidal tails produced in a satellite orbiting at 15 kpc from the galactic centre. Figure 2. View largeDownload slide Path followed by the code (green line) to analyse tidal tails produced in a satellite orbiting at 15 kpc from the galactic centre. Once we have the data of the number of particles in each bin along both tails, we find that some bins have a number of particles larger than the average, these are the bins where the overdensities are located. However, identifying overdensities only by the number of particles is not very accurate, so we use another criterion. To detect overdensities properly, we pick pieces of 10 kpc of tails and calculate the average number of particles in each bin inside those 10 kpc, then we detect an overdensity by selecting the bins where the number of particles is larger than the average. By doing this, we have the information of the distance to each bin which carries an overdensity of stars. Usually there are therefore several bins in an overdensity to measure the distance to an overdensity we take the distance to the bin which is in the centre of the overdensity. We also define the size of an overdensity as the distance to the last bin in the overdensity minus the distance to the first bin in the overdensity. As the measurements of the distance and the size depend on the size of the bins, used to analyse the tails, we measure these quantities using bins of 0.4, 0.5, and 0.6 kpc and take the mean value in each case, then we can also calculate the error. The results of our simulations show that by choosing smaller bin-sizes, the results become unreliable because of low number statistics and larger bin-sizes will smear out the features. After counting stars in each bin at each time-step, the code presents the information in a colour density plot, in the x-axis is the time in intervals of the time-step chosen for the output of the simulation data (10 Myr in our case), in the y-axis it shows the longitudinal separation: at the centre (0 kpc) is the satellite, at positive distances is the leading arm, and negative distances represent the trailing arm. The colour scheme represents the logarithmic number densities of the tails. We can see two different evolutionary stages of satellite 2 orbiting at 25 kpc from the Galactic Centre in the cored halo in Fig. 3. After using the analysing code described in Section 3, we obtain the colour plot shown in the bottom panel. We clearly see how the tails grow with time. Furthermore, we see as yellow horizontal lines the overdensities that develop with time and that stay at constant distances to the satellite with time (on circular orbits). Figure 3. View largeDownload slide Lower panel: colour density plot of the number of stars along the tails for a dSph orbiting a logarithmic DM halo for 5 Gyr on a circular orbit in intervals of 100 Myr. The colour bar shows the logarithm of the particles per bin. Particles in the leading arm are assigned positive distances, whereas particles in trailing arm have negative distances. In the top left panel, we have the satellite and its tidal tails at 1.2 Gyr and in the right-hand panel, we have the same satellite at 2.2 Gyr. Figure 3. View largeDownload slide Lower panel: colour density plot of the number of stars along the tails for a dSph orbiting a logarithmic DM halo for 5 Gyr on a circular orbit in intervals of 100 Myr. The colour bar shows the logarithm of the particles per bin. Particles in the leading arm are assigned positive distances, whereas particles in trailing arm have negative distances. In the top left panel, we have the satellite and its tidal tails at 1.2 Gyr and in the right-hand panel, we have the same satellite at 2.2 Gyr. 4 RESULTS Once we have the data of the position and time of formation of overdensities, we can compare our results with the expected values from the theory. According to Küpper et al. (2008), the equations of motion for a particle that escapes from the tidal radius of a satellite with small velocity with respect to the satellite are given by \begin{eqnarray} x &=& \frac{4 \Omega ^2}{\kappa ^2}x_{\rm L} + \left( 1 - \frac{4 \Omega ^2}{\kappa ^2} \right) x_{\rm L} \cos \kappa t, \end{eqnarray} (5) \begin{eqnarray} y &=&- \frac{2 \Omega }{\kappa } \left( 1 - \frac{4 \Omega ^2}{\kappa ^2} \right) x_{\rm L}(\sin \kappa t - \kappa t), \end{eqnarray} (6) \begin{eqnarray} z &=& 0. \end{eqnarray} (7)With the reference frame in the centre of the satellite, the x-axis points towards the Galactic anticentre and the y-axis points in the direction of the motion of the satellite. κ is the epicyclic frequency, Ω is the angular velocity of the satellite, and xL is the tidal radius of the satellite which we calculate according to equation (8), r is the radius of the orbit, M is the mass of the satellite, and M(r) is the mass of the halo enclosed inside a radius r. κ is calculated according to equation (9): \begin{eqnarray} x_{\rm L}&=& r \left( \frac{M}{3 M(r)}\right)^{1/3}, \end{eqnarray} (8) \begin{eqnarray} \kappa ^2 & = & \frac{\mathrm{\partial} ^2 \Phi }{\mathrm{\partial} r^2} + 3\Omega ^2. \end{eqnarray} (9)If we assume a point-mass potential for the Galaxy, then κ = Ω. These equations are based on the standard potential and epicyclic theories (see e.g. Binney & Tremaine 2008). Escaped particles start to de-accelerate at the turnaround points of their epicyclic orbits, i.e. when they perform a complete oscillation of 2π at t = tcp = 2π/κ therefore if many particles escape from the satellite and decelerate at the same position, we will see an overdensity of stars at a distance y(tc) = ycp = 12πxL from the centre of the satellite. The predicted values of tc,p and yc,p from the point-mass approximation as well as if we use the real potential from our simulations tc and yc are presented in Tables 4 and 5 along with the values obtained from our simulations (ts and ys). Table 4. Predicted values of distance and time of formation of the first overdensity for leading arm according to the theory of Küpper et al. (2008) for a point-mass galactic potential(tc,p and yc,p) (columns 2 and 3), assuming a Logarithmic MW potential (tc and yc) (columns 4 and 5), the time of formation and position of the first overdensity from our logarithmic halo simulations ts and ys (columns 6 and 7). D tc,p yc,p tc yc ts ys (kpc) (Myr) (kpc) (Myr) (kpc) (Myr) (kpc) 15 536.798 38.961 322 3.416 477 ± 24 5.863 ± 0.285 20 651.774 44.342 410 5.403 580 ± 10 8.620 ± 0.430 25 774.927 49.765 503 7.369 647 ± 38 11.301 ± 0.133 30 902.917 55.104 599 9.224 760 ± 26 13.700 ± 0.254 35 1033.950 60.314 695 10.949 883 ± 18 15.839 ± 0.299 50 1436.906 75.111 989 15.460 1243 ± 27 20.610 ± 0.359 D tc,p yc,p tc yc ts ys (kpc) (Myr) (kpc) (Myr) (kpc) (Myr) (kpc) 15 536.798 38.961 322 3.416 477 ± 24 5.863 ± 0.285 20 651.774 44.342 410 5.403 580 ± 10 8.620 ± 0.430 25 774.927 49.765 503 7.369 647 ± 38 11.301 ± 0.133 30 902.917 55.104 599 9.224 760 ± 26 13.700 ± 0.254 35 1033.950 60.314 695 10.949 883 ± 18 15.839 ± 0.299 50 1436.906 75.111 989 15.460 1243 ± 27 20.610 ± 0.359 View Large Table 5. Predicted values of distance and time of formation of the first overdensity for leading arm according to the theory of Küpper et al. (2008) for a point-mass galactic potential(tc,p and yc,p) (columns 2 and 3), assuming an NFW MW potential (tc and yc) (columns 4 and 5), the time of formation and position of the first overdensity from our NFW simulations ts and ys (columns 6 and 7). D tc,p yc,p tc yc ts ys (kpc) (Myr) (kpc) (Myr) (kpc) (Myr) (kpc) 15 528.896 38.577 331 4.517 450 ± 10 7.196 ± 0.323 20 654.910 44.484 417 5.872 543 ± 15 9.974 ± 0.110 25 780.564 50.006 505 7.293 623 ± 38 11.484 ± 0.084 30 907.017 55.271 596 8.772 763 ± 18 13.524 ± 0.093 35 1034.834 60.349 688 10.302 900 ± 15 15.619 ± 0.297 50 1428.756 74.827 981 15.144 1277 ± 38 20.270 ± 0.407 D tc,p yc,p tc yc ts ys (kpc) (Myr) (kpc) (Myr) (kpc) (Myr) (kpc) 15 528.896 38.577 331 4.517 450 ± 10 7.196 ± 0.323 20 654.910 44.484 417 5.872 543 ± 15 9.974 ± 0.110 25 780.564 50.006 505 7.293 623 ± 38 11.484 ± 0.084 30 907.017 55.271 596 8.772 763 ± 18 13.524 ± 0.093 35 1034.834 60.349 688 10.302 900 ± 15 15.619 ± 0.297 50 1428.756 74.827 981 15.144 1277 ± 38 20.270 ± 0.407 View Large If we do not assume a point-mass Galaxy, then we need to solve equation (6) for t = tc = 2π/κ. Therefore, the position of the first overdensity along the y-axis and the time at which it is formed are given by \begin{eqnarray} y_{\rm c} &=& \frac{4 \pi \Omega }{\kappa } \left( 1 - \frac{4 \Omega ^2}{\kappa ^2} \right) x_{\rm L}, \end{eqnarray} (10) \begin{eqnarray} t_{\rm c} & = & \frac{2 \pi }{\kappa }. \end{eqnarray} (11) We are analysing the simulations using satellite 2 on circular orbits at different distances from the galaxy in both analytic potentials with respect to the location and formation time of the first overdensity and compare our results with the prediction formulas derived above. The values are displayed in Tables 4 and 5. Using the point-mass approximation, the time of formation of the first overdensity is overpredicted by a factor of ≈1.2 ± 0.1. If we use the equations for the real potential, the situation reverses and we underpredict the time of formation by a factor of ≈0.7 ± 0.1. This is true irrespective of the potential (logarithmic or NFW) we are using. A similar picture presents the predictions for the location (i.e. distance) of the first overdensity. Again the point-mass approximation overpredicts the location, now by a factor ranging between 6.6 and 3.6 with a decline roughly ∝1/r. The full equations again deliver an underprediction of the simulation values ranging from 0.58 to 0.75. Again, we see a 1/r dependence of the results, i.e. the further away we are from the centre of the galaxy the closer simulation values and their respective predictions get. We conclude that the point-mass approximation is not suitable to predict simulation results using more realistic Galactic potentials. That the predictions for the formation time, using the real potential, are somewhat smaller than the measured values can be explained that the theory gives the time when the first stars reach the point of the overdensity and we need some time more to establish a detectable signal (i.e. overdensity). This was already pointed out in Küpper et al. (2008). For the mismatch of the position, one could imagine that it is caused by the more extended object we are using in this study and that the Lagrangian points of escape are no longer almost coinciding with the distance of the orbit, i.e. that Δr = r ± xL is not negligible any longer. But this would mean that we see overpredictions and underpredictions depending if we analyse the leading arm or the trailing arm of our object. This is not the case in our simulations. We only detect a smaller deviation for the leading arm than for the trailing arm but both directed in the same direction with respect to overpredictions and underpredictions. We suspect that the zero velocity assumption (i.e. stars leave the satellite with small velocities only) is no longer valid for the extended objects we use. The discrepancy between the point-mass approximation and the real simulation values was not visible in the results of Küpper et al. (2008), as they were actually using a point-mass potential to mimic the Galaxy together with the point-mass approximation to compare their results. In that sense, we agree with Küpper et al. (2008), just one has to use the prediction equations matching the used potential. After comparing the theory with our simulations, we analyse the rate of growth of the tails arising from the satellites. As the growth of the tails is linear with the time for satellites on circular orbits, we simply take two values of the length of the tails together with two values of the time and calculate the slope. We use the length at 3000 Myr and the length at 0 Myr, which is zero to calculate the growth rate. For satellites orbiting on eccentric orbits, the growth of the tails is not linear with time, tails are stretched and compressed as the satellite passes through peri- and apo-galacticon, respectively. To measure the rate of growth, we have taken the values for the length of the tails between the time after two consecutive peri-centre passages measured half-way to the next apo-galacticon. We find that the trailing arm always grows faster than the leading arm irrespective of circular or elliptical orbits (see Fig. 4). For circular orbits, the rates of growth are independent of the radius of the orbit and ∼0.02 kpc Myr−1 for leading arms and ∼0.025 kpc Myr−1 for trailing arms as shown in the top panel of Fig. 4. One can also see that these results are independent of the form of the potential, i.e. using a logarithmic or an NFW halo. Figure 4. View largeDownload slide Rates of growth of the leading and trailing arms depending on the radius of circular orbits (top), the mass of the satellite (middle), and the eccentricity of the orbit (bottom). We plot data from simulations using a logarithmic halo as squares and an NFW profile as triangles. Red filled symbols are data from the leading arm and blue open symbols are data from the trailing arm. Red and blue lines are the best linear fits for the leading and trailing arms, respectively. As we see no difference between the logarithmic and NFW halo results we fit both results simultaneously. The exception is the middle panel as for varying satellite mass we only used the logarithmic potential. Figure 4. View largeDownload slide Rates of growth of the leading and trailing arms depending on the radius of circular orbits (top), the mass of the satellite (middle), and the eccentricity of the orbit (bottom). We plot data from simulations using a logarithmic halo as squares and an NFW profile as triangles. Red filled symbols are data from the leading arm and blue open symbols are data from the trailing arm. Red and blue lines are the best linear fits for the leading and trailing arms, respectively. As we see no difference between the logarithmic and NFW halo results we fit both results simultaneously. The exception is the middle panel as for varying satellite mass we only used the logarithmic potential. This result shows that for the growth of the tidal tails, the underlying potential is of no importance. The velocity of the tail growth is given by the parameters of the satellite only. The fact that the trailing arm is growing slightly faster can be understood by the fact that stars are leaving with similar velocities through the Lagrangian points. For an extended object as used in our simulations, those stars have slightly different angular velocities compared to that of a circular orbit at this distance. These differences could be different in magnitude for the leading and trailing arms leading to this asymmetry of tail growth. In the middle panel of Fig. 4, we see clearly that the growth depends on the mass of the satellite in a linear fashion. The rate of growth for the leading arm varies from 0.0172 ± 0.0004 kpc Myr−1 for a satellite of 0.5 × 108 M⊙ up to 0.0329 ± 0.0009 kpc Myr−1 for a satellite of 4.0 × 108 M⊙. The best linear fits are given by equation (12) for the leading arm and equation (13) for the trailing arm: \begin{eqnarray} \dot{L_{\rm L}} &=& 0.0043 \pm 0.0007 \times M + 0.017 \pm 0.002, \end{eqnarray} (12) \begin{eqnarray} \dot{L_{\rm T}} & = & 0.0051 \pm 0.0010 \times M + 0.021 \pm 0.002. \end{eqnarray} (13)For these simulations, we have only used the logarithmic potential as Galactic potential. Here, we clearly see the dependence on the internal satellite parameters. Stars leaving the satellite should have escape velocity or slightly higher. We conclude that the assumption that stars leave the satellite with zero or negligible relative velocity may be valid for small star clusters but not for very massive and extended objects. In this respect, our study shows a disagreement with the assessment of Küpper et al. (2008). Finally, we find that the rate of growth of the tails measured between the peri-galacticon and the next apo-galacticon for satellites orbiting in eccentric orbits seems to depend linearly on the eccentricity of the orbit and varies from 0.0241 ± 0.0022 kpc Myr−1 for the leading and 0.0320 ± 0.0013 for trailing arm for an eccentricity of 0.1, up to 0.1157 ± 0.0019 and 0.1474 ± 0.0063 kpc Myr−1 for the leading and trailing arms, respectively, for an eccentricity of 0.8. The best linear fits are shown in equation (14) for the rate of growth of the leading arm and in equation (15) for the rate of growth of the trailing arm. \begin{eqnarray} \dot{L_{\rm L}}&=& 0.128 \pm 0.005 \times e + 0.010 \pm 0.002, \end{eqnarray} (14) \begin{eqnarray} \dot{L_{\rm T}} & = & 0.152 \pm 0.013 \times e + 0.012 \pm 0.006. \end{eqnarray} (15)In our simulations this is explained easily. The higher the eccentricity the deeper into the Galactic potential the satellite is orbiting. Stars get lost by tidal shocks at peri-galacticon passages. The closer to the Galactic centre the satellite is orbiting the smaller the instantaneous tidal radius will be and the more stars can get stripped, which have higher relative velocities with respect to the satellite. Again here, we see a clear deviation from the zero velocity assumption. In Fig. 5, we show the distance to the first overdensity in the leading arm and in the trailing arm. We find that the distance to the first overdensity in the leading and trailing arms depends on the radius of the orbit and the mass of the satellite as expected from the theory of Küpper et al. (2008) but does not depend on the potential used to model the MW. This can be seen in the top panel as the symbols for the logarithmic halo and the NFW profile basically overlap. Figure 5. View largeDownload slide Top panel: distance to the first overdensity in the leading and trailing arms as function of the radius of the orbit. Bottom panel: distance to the first overdensity as function of the satellite mass. We plot data from simulations using a logarithmic halo as squares and an NFW profile as triangles. Red filled symbols are data from the leading arm and blue open symbols are data from the trailing arm. Red and blue lines are the best linear fits for the leading and trailing arms, respectively. As we see no difference between the logarithmic and NFW halo results, we fit both results simultaneously. Again in the lower panel we only show simulations using the logarithmic halo for the varying satellite mass. Figure 5. View largeDownload slide Top panel: distance to the first overdensity in the leading and trailing arms as function of the radius of the orbit. Bottom panel: distance to the first overdensity as function of the satellite mass. We plot data from simulations using a logarithmic halo as squares and an NFW profile as triangles. Red filled symbols are data from the leading arm and blue open symbols are data from the trailing arm. Red and blue lines are the best linear fits for the leading and trailing arms, respectively. As we see no difference between the logarithmic and NFW halo results, we fit both results simultaneously. Again in the lower panel we only show simulations using the logarithmic halo for the varying satellite mass. We also find that in the leading arm the first overdensity is always at a smaller distance from the satellite than in the trailing arm. After the formation of the overdensities, the distance to them remains constant (see e.g. in Fig. 3). The difference in the distance between the two tails is a puzzling detail of our simulations. It could be explained by the fact that we have a much more extended object than used in the previous study. Again, the difference in distance between the two Lagrangian points should be taken into account. But, then we would expect that this difference is more pronounced the closer we are to the centre of the Galaxy. That we see the opposite trend in our results is therefore counter-intuitive and needs further investigation. Nevertheless, it is a fact that should be taken into account when analysing observational results. The fact that tidal tails of one and the same object have not the same length and that the overdensities appear at different locations is not necessarily a sign for an interaction with another object. We have studied the distance to the first overdensity in the leading and trailing arms. In reality, we have measured the distance to the centre of the overdensity, because the overdensities have a size. We calculate the mean value of all bins that present the overdensity. As the size of these overdensities may help us to distinguish further between Galactic and/or satellite properties, we now take a closer look at the sizes. We find that after the formation of the overdensities, they have a nearly constant size; however, the size is decreasing after ∼3600 Myr as we can see in Fig. 6. Figure 6. View largeDownload slide Size of the first overdensity as function of time for a satellite orbiting at 15 kpc (solid red line), 20 kpc (blue dotted line), 25 kpc (green short dashed line), 30 kpc (cyan long dashed line) and 35 kpc (magenta dot dashed line). We can note that after ∼3600 Myr the size of the overdensity is decreasing down to about 2 kpc. Figure 6. View largeDownload slide Size of the first overdensity as function of time for a satellite orbiting at 15 kpc (solid red line), 20 kpc (blue dotted line), 25 kpc (green short dashed line), 30 kpc (cyan long dashed line) and 35 kpc (magenta dot dashed line). We can note that after ∼3600 Myr the size of the overdensity is decreasing down to about 2 kpc. To study how the size depends on the properties of the satellite, we take an average of the size of the overdensities between 1 and 3 Gyr, i.e. a time-span, starting after the formation of the overdensities, where the size of them does not vary significantly. The results are shown in Fig. 7. Figure 7. View largeDownload slide Size of the first overdensity as function of the mass of the satellite (top panel). In the lower panel, we plot the size of the first overdensity as function of the radius of the orbit. We plot data from simulations using a logarithmic halo as squares and an NFW profile as triangles. Red filled symbols are data from the leading arm and blue open symbols are data from the trailing arm. Red and blue lines are the best linear fits for the leading and trailing arms, respectively. As we see no difference between the logarithmic and NFW halo results, we fit both results simultaneously. Again in the top panel we only show simulations using the logarithmic halo for the varying satellite mass. Figure 7. View largeDownload slide Size of the first overdensity as function of the mass of the satellite (top panel). In the lower panel, we plot the size of the first overdensity as function of the radius of the orbit. We plot data from simulations using a logarithmic halo as squares and an NFW profile as triangles. Red filled symbols are data from the leading arm and blue open symbols are data from the trailing arm. Red and blue lines are the best linear fits for the leading and trailing arms, respectively. As we see no difference between the logarithmic and NFW halo results, we fit both results simultaneously. Again in the top panel we only show simulations using the logarithmic halo for the varying satellite mass. We find that the size of the first overdensity in the leading arm depends only weakly on the radius of the orbit and does not depend on the potential used to model the MW. The size varies from 1.33 ± 0.16 kpc for a satellite orbiting at 15 kpc up to 3.55 ± 0.05 kpc for a satellite orbiting at 50 kpc. In the trailing arm, we see a clear dependence of the size of the first overdensity with the radius of the orbit. The size of the first overdensity increases with increasing radius of the orbit from 1.35 ± 0.16 kpc for a circular orbit at 15 kpc up to 6.47 ± 0.30 kpc for a circular orbit at 50 kpc. The size of the first overdensity depends on the mass of the satellite. In the trailing arm, the size of the first overdensity increases from 2.49 ± 0.20 kpc for a satellite with a mass of 0.5 × 108 M⊙ up to 7.62 ± 0.22 kpc for a satellite with a mass of 4.0 × 108 M⊙. In the leading arm, the size of the first overdensity depends only weakly on the mass of the satellite ranging from 1.27 ± 0.04 kpc for a satellite with a mass of 0.5 × 108 M⊙ up to 2.71 ± 0.02 kpc for a satellite with a mass of 4.0 × 108 M⊙. Apparently, the size of the first overdensity in leading arms depends linearly with the mass of the satellite; however, for trailing arms it could be a power law. We show in Fig. 7 the best linear fits to the data of the size of the first overdensity in the leading arm and in the trailing arm depending on the radius of the orbit in kpc (equations 16 and 17) and the mass of the satellite in M⊙ (equations 18 and 19): \begin{eqnarray} S_{\rm L}&=& 0.058 \pm 0.005 \times r + 0.4 \pm 0.1, \end{eqnarray} (16) \begin{eqnarray} S_{\rm T} & = & 0.15 \pm 0.01 \times r - 1.1 \pm 0.2. \end{eqnarray} (17) \begin{eqnarray} S_{\rm L} &=& 0.37 \pm 0.08 \times M + 1.3 \pm 0.2, \end{eqnarray} (18) \begin{eqnarray} S_{\rm T} & = & 1.5 \pm 0.2 \times M + 1.7 \pm 0.5. \end{eqnarray} (19) Again, we see a clear asymmetry of the two tails. The size of the first overdensity in the trailing arm grows about three times faster with mass and distance than the one of the leading arm. This cannot be explained by the use of an extended object alone. Even though we do not have a valid explanation for this finding, it is a fact that should be taken into account when observing real objects. Finally, we combine the data from the distance to the first overdensity in the leading and trailing arms with the size of the first overdensity for simulations of satellites orbiting a logarithmic halo in circular orbits. The data are shown in Fig. 8. The reason for this exercise is that in this way we might be able to distinguish between internal and external effects, i.e. mass of the satellite and the potential strength. Unluckily, we do not see such a trend in Fig. 8 as all data points follow the same relation, no matter the mass of the satellite. Figure 8. View largeDownload slide Size of the first overdensity as function of the distance to the first overdensity for trailing arms (blue symbols) and leading arms (red symbols) using simulations of a satellite orbiting a logarithmic halo on circular orbits and the best linear fits in red for the leading arm and blue for the trailing arm. Figure 8. View largeDownload slide Size of the first overdensity as function of the distance to the first overdensity for trailing arms (blue symbols) and leading arms (red symbols) using simulations of a satellite orbiting a logarithmic halo on circular orbits and the best linear fits in red for the leading arm and blue for the trailing arm. Nevertheless, we can establish a relation between the distance to the first overdensity and the size of the first overdensity for the leading and trailing arms separately, which could be observationally investigated. A simple linear fit to the data from our simulations gives the following relations: \begin{eqnarray} S_{\rm L} &=& 0.15 \pm 0.01 \times D + 0.2 \pm 0.2, \end{eqnarray} (20) \begin{eqnarray} S_{\rm T} & = & 0.31 \pm 0.04 \times D - 1.7 \pm 0.6. \end{eqnarray} (21)Here, SL is the size of the first overdensity in the leading arm and ST is the size of the first overdensity in the trailing arm and D is the distance (in kpc) to the overdensity from the centre of the satellite. 5 DISCUSSION AND CONCLUSIONS From simulations of dwarf galaxies orbiting a spherical logarithmic and an NFW DM halo (instead of a point-mass potential), we find that the theory of Küpper et al. (2008), giving theoretical estimates for the time of formation and the distance to the first overdensities in tidal tails, is overpredicting the results we are obtaining with our simulations. In the case of the location of the first overdensity, this can be up to factor of 6. We have shown that this is not due to the fact of the different-sized objects the two studies use but that Küpper et al. (2008) were right with their estimates as they also used a point-mass potential in their simulations. The solution is that one cannot use the simple equations from their study but has to go back and solve the original epicyclic equations for the potential one actually uses. We still see some discrepancies but those could be explained by the real set-up in contrast to an idealized, approximated theoretical framework. In that respect, we verify the results of Küpper et al. (2008). In our study, we have shown that the growth rate of tidal tails (on circular orbits) does not depend on the potential strength of the Galactic potential as the tails grow with the same rate, independent of the potential used (logarithmic halo or NFW halo) and the distance to the centre of the Galaxy. The tails grow faster if the satellite is more massive. We conclude that this is because the simple zero velocity approximation for the epicyclic equations is in fact not completely valid. In reality in larger, more massive objects, stars escape with higher relative velocities. This can be seen as well, when we investigate the growth rate as a function of the eccentricity of the orbit. The further in an object orbits, the stronger is the decrease of the tidal radius at peri-centre. Therefore, more stars are able to leave the satellite and escape into the tails. Naturally, these stars have higher relative velocities, and therefore we grow tidal tails faster when we orbit with higher eccentricity. Even though we could calculate the rates of growth for both tails, using eccentric orbits, we do not have sufficient data of the overdensities, because the code of Véjar (2013) does not consider the possibility of having more than two tidal tails, a feature that occurs naturally if new stars are lost at peri-galacticon. The new and old tails will align with each other when the satellite is approaching apo-galacticon. We find a relation between the distance to the first overdensity in the leading and trailing arms and the radius of the orbit and/or the mass of the satellite. As the rate of growth does not depend on the radius of the orbit, we conclude that the relation between the distance of the first overdensity and the radius of the orbit must be due to the change in the tidal radius alone. This is verified when only changing the mass of the satellite at the same distance to the galaxy. A change in the epicyclic frequency κ and the angular velocity Ω in affecting the results cannot be excluded but are of secondary order. A very particular result of our study is the visible asymmetry between the leading and the trailing arm, with the trailing arm growing faster and being larger than the leading arm. The most obvious explanation would be the difference in distance between the two Lagrangian points in large objects as used in our study. This is of course an effect that plays a role but it cannot explain why this trend is more pronounced at larger radii from the galaxy, where this difference should become more insignificant. Furthermore, we find a relation between the size of the first overdensity and the radius of the orbit or the mass of the satellite. This relation is very clear for trailing arms but not as clear for leading arms, where the size of the first overdensity is very close to 2 kpc in all simulations. A possible explanation for this behaviour is that particles not always escape with the same velocity from the satellite but around a central value vesc and with a small velocity dispersion σ. According to the theory of Küpper et al. (2008), this spread in the escape velocity will lead to a spread in the value of yc at which the particles turn around. Therefore, we will have an extended overdensity with a size S. We expect that satellites with higher masses produce larger σ values and therefore larger overdensities. Why this is so strongly visible in trailing arms only is a study on its own and should be dealt with in a future investigation. The dependence with the radius is similar. The sizes of overdensities in the trailing arms grow about three times faster than the respective sizes in the leading arm. Again this obvious asymmetry is not easy to explain and deserves further studies. Finally, we combine the results of the distance and the size of the overdensities and find a single linear relation between the distance to the first overdensity and its size for the leading and trailing arms separately but independent of the mass of the satellite, the radius of the orbit and its eccentricity. This relation could be investigated with detailed observations of tidal tails. Acknowledgements MF acknowledges financial support of Fondecyt grant No. 1130521, Conicyt PII20150171, and BASAL PFB-06/2007. BR acknowledges funding through Fondecyt grant No. 1161247. BR thanks A. Alarcon Jara and D.R. Matus Carrillo for their help with the code superbox and useful discussions during the realization of this work. REFERENCES Belokurov V. et al. , 2006, ApJ , 642, L137 CrossRef Search ADS Binney J., Tremaine S., 2008, Galactic Dynamics , 2nd edn. Princeton Univ. Press, Princeton, NJ Fellhauer M., Kroupa P., Baumgardt H., Bien R., Boily C. M., Spurzem R., Wassmer N., 2000, New Astron. , 5, 305 CrossRef Search ADS Fellhauer M. et al. , 2006, ApJ , 651, 167 CrossRef Search ADS Koposov S., Rix H.-W., Hogg W., 2010, ApJ , 712, 260 CrossRef Search ADS Küpper A., Macleod A., Heggie D., 2008, MNRAS , 387, 1248 CrossRef Search ADS Küpper A., Kroupa P., Baumgardt H., Heggie D., 2010, MNRAS , 401, 105 CrossRef Search ADS Küpper A., Lane R., Heggie D., 2012, MNRAS , 420, 2700 CrossRef Search ADS Majewski S., Skrutskie M. F., Weinberg M., Ostheimer J., 2001, ApJ , 548, L165 CrossRef Search ADS Navarro F., Frenk S., White D. M., 1996, ApJ , 462, 563 CrossRef Search ADS Odenkirchen M. et al. , 2001, ApJ , 548, L165 CrossRef Search ADS Plummer H. C., 1911, MNRAS , 71, 460 CrossRef Search ADS Véjar R., 2013, Thesis(Titulo) , Universidad de Concepción Vivas K. et al. , 2001, ApJ , 554, L33 CrossRef Search ADS Williams M. E. 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Monthly Notices of the Royal Astronomical Society – Oxford University Press

**Published: ** May 1, 2018

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