Online material

Appendix A: Determination of spatial velocities

We update the transformation matrix T defined in Johnson & Soderblom (1987) (following Green (1985)) to equinox J2000: The matrix B of Johnson & Soderblom (1987) remains defined as: where are the equatorial coordinates of the generic dwarf galaxy at the present time t = t₀. In our case, we get for the Carina dwarf galaxy. Now we have to consider a reflection of the velocity axis that we want pointing away from the Galactic centre and obtain the correction for the rotation of the galaxy, as well as the motion of the sun relative to the local standard of rest: where is the velocity of the dwarf galaxy at the instant t = t₀, v_r is the radial velocity, (μ_α,μ_δ) the proper motions in arcsec   yr^-1, d the distances that have to be assumed in pc to apply the conversion values k = 4.74; v _{⊙ ,   LSR} = {10.0,5.2,7.2}   km   s^-1 as in Binney & Merrifield (1998) and V_c = 220   km   s^-1 as adopted by the IAU (1986). For the distances, we also adopt a different reference system from Johnson & Soderblom (1987) in order to obtain a right-handed reference system. This is achieved by imposing a double reflection for the positive X-axis originating in the Galactic centre and the positive Y-axis pointing along decreasing Galactic longitude: where x = (x,y,z) is the generic position of the dwarf galaxy, x_⊙ = {x_⊙,y_⊙,z_⊙} ≅ {8.5,0.0,0.0} is the Sun’s position assumed for simplicity to lie in the plane of the MW and are the Galactic coordinates that can be consistently derived for a cross check from

The selection of an aligned reference system between the configuration space and the velocity space will permit us to treat, in a simpler form, the velocity vector as a derivative of the position vector for times different from the present. Similarly, we can derive the errors as where p is the parallax and B² the matrix with elements ∀i,j. With these equations and the proper motion and errors in the text, we can derive our estimate value for the orbital energy of the dwarf galaxy we are analysing.

Appendix B: Barycentre determination

Once we try to move from the crude point mass determination to the full N-body description and include many different astrophysical aspects, we encounter the necessity of analysing the system properties of the many N-body system realizations by detecting the position of the centre of mass of the system. The determination of the not inertial centre of mass of a galaxy moving within an external force field can be performed in different ways but it is clear that we must evaluate it automatically to speed up analysis of the large number of simulations we performed for finding the best match with the observational constraints. In this Appendix, we present an original approach based on the downhill simplex method (Nelder & Mead 1965), which is a direct search method that works moderately well in low-dimensional stochastic problems. Our task is to apply the method to find the barycentre of the nucleus of our dwarf galaxy, limiting ourselves to the knowledge of the position of the galaxy at a given snapshot for every particle, ∀i, and at every moment, ∀t, during the evolution of our dwarf. We refer the reader to books such as Numerical Recipes by Press et al. (1986) which uses the method in the section dedicated to the minimization already back in 1986. Here we limit ourselves to showing a further possible application where this method is suitable. In the practical implementation, we refer to the matrix formalism developed in the original work of Nelder and Mead. The simplex, a convex hull of a tetrahedron for our 3D space surrounding our MW galaxy, represented by a time-dependent matrix, whose columns are the vertex where . For any simplex α ∈ R³ we can define the matrix, , as the 3 × 3 matrix whose columns represent the edges of : and ê = (1, 1, 1)^T. In this way we can, as a first step, check the degenerate character of the simplex by ensuring that the 3D volume of , . As a consequence, in a Euclidean geometry the reflection, expansion, inside/outside-contraction and shrinkage computed by the algorithm will always produce a non-degenerate tetrahedron. We then define the diameter of the simplex as ∅(α) = maxi ≠ j ∥ x_i − x_j ∥ ∀t where ∥.∥ is the standard norm. Finally we define a suitable function for finding the best point representing a barycentre of the dwarf galaxy. If we call the distance of the ith-star from the guess value of the barycentre x_b at the instant under consideration, then we need to maximize the function (B.1)where r _∗  is a characteristic radius of the system we want to analyse. Our experience shows that it does not have to be physically related to the system, but a suitable choice of a few kpc can definitely improve the convergence velocity and the stability of the barycentre value if related to the convergence criteria. If f is a bounded function then for every non-degenerate case it can be proved for the downhill simplex algorithm (we indicate with k the iteration of the algorithm) that

• the sequence always converges;

• at every non-shrinking iterations k, for , with strict inequality for at least one variable of I;

• if there is only a finite number of shrink iterations, then

  • each sequence converges as k → ∞ for ;

  • if we call then and ;

  • .

• if there is only a finite number of non-shrink iterations, then all simplex vertexes converge to a single point.

All this ensures that the down hill simplex method applied to the function (B.1) will always converge if the number of shrink iteration is small (which is in our experience). We adopted a variable diameter as indicated by the subset of the stars for which we computed the barycentre, , e.g. with a physical radius converging to zero as the number of successive times that the down hill simplex method is applied, j, increases the assigned . In practice, we of course chose r _∗  to be slightly higher than the softening length of the gravitational potential r _∗  > ε_star.

Appendix C: Encounter with the Magellanic Clouds

In this Appendix we discuss the probability for a fly by interaction with the Magellanic Clouds to investigate the possibility suggested by Muñoz et al. (2006). We relegate these arguments to this Appendix because they simply concern probability considerations that are derived from dynamics. The suggestion presented in the paper of Muñoz et al. (2006) comes from the analysis of only 15 stars in the local universe, too small to justify a further set of full simulations. In particular, we are interested in the recent determination of surprisingly large proper motion for the Large Magellanic Cloud (LMC) and the Small Magellanic

Cloud (SMC) (Kallivayalil et al. 2006b,a), which caused interest due to the possible implications for their evolutionary history (e.g., Kallivayalil et al. 2009; Piatek et al. 2008; Besla et al. 2007; Olsen & Massey 2007). As explained in Kallivayalil et al. (2009), the most relevant problem faced by the integration of the orbits of the Magellanic Clouds with an expected apocentre so far from the MW barycentre, is the unknown mass distribution of the MW for distances larger than 100–200 kpc (e.g., Besla et al. 2007). With our MW galactic potential model presented in Appendix B, we cannot extend the integration of the LMC orbit much further as required to overlap the entire time with the orbit integration spanned by the minimization of the action for Carina. Thus we limit ourselves to the last 3 Gyr of evolution in look-back time because in the potential of MW, the LMC after 3 Gyr of look-back time integration will already be more distant than 200 kpc from the MW barycentre, thereby falsifying any phase-space derivation. Thus we accept the conclusion already presented in Besla et al. (2007) in favour of a single/first pericentre passage for the MC not more than a few hundred million years ago, and we proceed by minimizing the distance function d_Car−MC:R⁷ → R ⁺ in the 7-dimensional space of the initial values for v_0,   Car ∈  [v_0,   Car ± δv_0,   Car] , v_0,   MC ∈  [v_0,   MC ± δv_0,   MC]  computed as in Appendix A and t ∈]−3,0] Gyr. We integrated the equation of motion here for the Magellanic Clouds, taking into account an extra term due to the dynamical friction as ascribed in Eq. (8) caused by the dependence of the force on the square of the mass of the Magellanic Clouds (e.g. m_LMC ≅ 2. × 10⁹   M_⊙) that makes the dynamical friction much more relevant than for Carina. The result is that the closest distance approach permitted between the two galaxies in the range of the possible phase-space observational errors is more than 50 kpc, completely ruling out any possible interaction within the currently suggested phase-space error range deduced from the observations. The same consideration holds for any interaction with the SMC.

© ESO, 2010