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The N-body simulations that we will discuss were tailored specifically for the understanding of bar formation and evolution in a gas-less disk embedded in a spherical spheroid. That is, the initial conditions were built so as to exclude, in as much as possible, other instabilities, thus allowing us to focus on the bar. Such initial conditions are often called dynamical (because they allow us to concentrate on the dynamics), or simplified, controlled, or idealised (because they exclude other effects so as to focus best on the one under study). They allow us to make `sequences' of models, in which we vary only one parameter and keep all the others fixed. For example, it is thus possible to obtain a sequence of models with initially identical spheroids and identical disk density profiles, but different velocity dispersions in the disk.

The alternative to these simulations is cosmological simulations, and, more specifically, zoom re-imulations. In such re-simulations a specific halo (or galaxy), having the desired properties, is chosen from the final snapshot of a full cosmological simulation. The simulation is then rerun with a higher resolution for the parts which end up in the chosen galaxy or which come to a close interaction with it, and also after having replaced a fraction of the dark matter particles in those parts by gas particles.

Zoom simulations are more general than the dynamical ones because the former include all the effects that dynamical simulations have, deliberately, neglected. However, they do not allow us to build sequences of models and also have less resolution than the dynamical ones and necessitate much more computer time and memory. Furthermore, some care is necessary because cosmological simulations are known to have a few problems when compared with nearby galaxy observations, concerning, e.g., the number and distribution of satellites, the inner halo radial density profile, the formation of bulge-less galaxies, or the Tully-Fisher relation (see, e.g., Silk & Mamon 2012 for a review). Thus, the zoom re-simulations could implicitly contain some non-realistic properties, which are not in agreement with what is observed in nearby galaxies, and therefore reach flawed results. Moreover, since many effects take place simultaneously, it is often difficult to disentangle the contribution of each one separately, which very strongly hampers the understanding of a phenomenon. For example, it is impossible to fully understand the bar formation instability if the model galaxy in which it occurs is continuously interacting or merging with other galaxies. A more appropriate way would be to first understand the formation and evolution of bars in an isolated galaxy, and then understand the effect of the interactions and mergings as a function of the properties of the intruder(s).

Thus, zoom simulations should not yet be considered as a replacement of dynamical simulations, but rather as an alternative approach, allowing comparisons with dynamical simulations after the basic instabilities has been understood. A few studies using cosmological zoom simulations have been already made and have given interesting results on the formation and properties of bars (Romano-Díaz et al. 2008; Scannapieco & Athanassoula 2012; Kraljic et al. 2012).

A non-trivial issue about dynamical N-body simulations is the creation of the initial conditions. These assume that the spheroid and the disk are already in place and, most important, that they are in equilibrium. This is very important, since a system which is not in equilibrium will undergo violent relaxation and transients, which can have undesirable secondary effects, such as spurious heating of the disk or altering of its radial density profile. At least three different classes of methods to create initial conditions have been developed so far.

  1. A wide variety of methods are based on Jeans's theorem (e.g., Zang 1976; Athanassoula & Sellwood 1986; Kuijken & Dubinski 1995; Widrow & Dubinski 2005; McMillan & Dehnen 2007), or on Jeans's equations (e.g., Hernquist 1993). In the case of multi-component systems, e.g., galaxies with a disk, a bulge and a halo, the components are built separately and then either simply superposed (e.g., Hernquist 1993), or the potential of the one is adiabatically grown in the other before superposition (e.g., Barnes 1988; Shlosman & Noguchi 1993; Athanassoula 2003, 2007; McMillan & Dehnen 2007). The former can be dangerous, as the resulting model can be considerably off equilibrium. The latter is strongly preferred to it, but still has the disadvantage that the adiabatic growing of one component can alter the density profiles of the others, which is not desirable when one wishes to make sequences of models. It also is not trivial to device a method for assigning the velocities to the disk particles without relying on the epicyclic approximation (but see Dehnen 1999). Last but not least, this class of methods is not useful for complex systems such as triaxial bulges or haloes.
  2. The Schwarzschild method (Schwarzschild 1979) can also be used for making initial conditions, but has been hardly used for this, because the application is rather time consuming and not necessarily straightforward.
  3. A very promising method for constructing equilibrium phase models for stellar systems is the iterative method (Rodionov et al. 2009). It relies on constrained, or guided, evolution, so that the equilibrium solution has a number of desired parameters and/or constraints. It is very powerful, to a large extent due to its simplicity. It can be used for mass distributions with an arbitrary geometry and a large variety of kinematical constraints. It has no difficulty in creating triaxial spheroids, and the disks it creates do not follow the epicyclic approximation, unless this has been imposed by the user. It has lately been extended to include a gaseous component (Rodionov & Athanassoula 2011). Its only disadvantage is that it is computer intensive, so that in some cases the time necessary to make the initial conditions is a considerable fraction of the simulation time.

I would also like to stress here a terminology point which, although not limited to simulations, is closely related to them. In general the dynamics of haloes and bulges are very similar, with of course quantitative differences due to their respective extent, mass and velocity dispersion values. For this reason, I will use sometimes in these lecture notes the terms `halo' and `bulge' specifically, while in others I will use the word `spheroid' in a generic way, to designate the halo and/or the bulge component. The reasons for this are sometimes historic (i.e., how it was mentioned in the original paper), or quantitative (e.g., if the effect of the halo is quantitatively much stronger that that of the bulge), or just for simplicity. The reader can mentally interchange the terms as appropriate.

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