Nowadays it is pretty common to find in the literature studies of the formation and evolution of galaxies in a cosmological context, meaning that initial conditions consist of a scale-free or nearly scale-free spectrum of Gaussian fluctuations as predicted by cosmic inflation and with cosmological parameters determined from observations of the cosmic microwave background radiation obtained by spacecrafts such as WMAP [275, 123]. However, the most detailed and sophisticated cosmological simulations to date, such as the Millennium-II simulation [30] and the Bolshoi simulation [119] have force resolutions of the order of 1 kpc. This is barely enough to resolve large galaxies, but it is clearly insufficient to solve in detail DGs, whose optical radii are some times smaller than that. A lot in resolution can be gained by zooming in and re-simulating small chunks of a large cosmological box [284, 61, 286]. This method is gaining pace and has been applied by various groups to DGs [169, 245, 213]. Still, at the present time the best way to accurately simulate a DG is by numerically studying it as a single isolated entity [180, 287, 233, 252, 256, 306].

Numerical studies of galaxies in isolation assume some initial configuration of gas density, temperature and stellar distribution. This initial configuration is an equilibrium status of the system. Starting from an equilibrium condition is clearly necessary in order to pin down the effect of perturbing phenomena (star formation, environmental effects, AGN feedback, and so on).

A common strategy is to consider a rotating, isothermal gas in
equilibrium with the potential generated by a fixed distribution of
stars and/or of dark matter
[315,
261,
159].
Rotating gas configurations are usually better described by means of a
cylindrical coordinate system (*R*, ϕ, *z*). Often, axial
symmetry is assumed. The relevant equation to solve in order to find the
density distribution of gas
(*R*,
*z*) is thus the steady-state (time
independent) Euler equation

(1) |

where *P* is the pressure, **v** is the bulk velocity of the
gas and is the total
gravitational potential. In this equation, only the component
*v*_{ϕ} of the velocity must be
considered because it gives centrifugal support against the gravity.
Eq. 1 in fact implies that the gravitational pull is
counter-balanced by the combined effect of pressure gradient and
centrifugal force.

Most of the authors assume
to be independent of
. This
means that the self-gravity of the gas is not considered. A typical
justification of this choice is "The omission of self-gravity is
reasonable, given that the baryonic-to-dark matter ratio of the
systems is ~ 0.1."
[79].
However, even if the total
mass of a DG is dominated by a dark matter halo, within the Holmberg
radius (the radius at which the surface brightness is 26.5 mag
arcsec^{-2}), most of the galaxy is made of baryons
[206,
297],
so the inclusion of gas self-gravity in the central part of
a DG appears to be important. I will come back to this point later in
this section. For the moment it is enough to take note of the fact
that the assumption that
is independent of
greatly
simplifies the calculation of the steady-state density configuration.
Furtermore, a barotropic equation of state *P* =
*P*()
and a dependence of the azimuthal velocity
*v*_{ϕ} with known quantities is commonly assumed.

A widely used strategy is to assume that
*v*_{ϕ} = *ev*_{circ}, where
*v*_{circ}
= √*R*
*d*
/ *dR* is the circular velocity and *e*
is the spin parameter that determines how much the galaxy is supported
against gravity by rotation and how much it is supported by the
pressure gradient. A typical value for *e* is 0.9, independent on
the height *z*
[314,
295].
[293]
assume that *e* = 0.9 in
the plane of the galaxy, but it drops exponentially with height in
order to have non-rotating gas halos. It is however important to
remark that, according to the Poincare'-Wavre theorem
[147,
302,
13],
the rotation velocity of any barotropic gas
configuration (including thus also isothermal configurations) in
rotating equilibrium must be independent of *z*. In other words, it
is possible to construct a centrifugal potential to add to
in
Eq. 1 only if the circular velocity is independent on *z*.

Other authors [58] solve instead the equilibrium equation in the plane:

(2) |

and assume the azimuthal velocity to be independent of *z*, in
compliance with the Poincare'-Wavre theorem. The density at any *z*
is then found integrating the *z*-component of the hydrostatic
equilibrium equation, for any *R*. Some authors then
[109,
67,
306]
set the gas in rotation around the *z*-axis, using the
average angular momentum profile computed from cosmological simulations
[37].

A different approach is followed by [7]. Initially, there is no balance between gravity and pressure and the gas collapses into the midplane. Supernovae (SNe) go off, principally along the disk and this drives the collapsed gas upwards again. Eventually, upward and downward flowing gas come into dynamical equilibrium. Some multi-phase simulations [94, 155] adopt a similar approach for the diffuse component, i.e. the distribution of diffuse gas starts far from equilibrium. Then, it relaxes on a few dynamical time scales to a quasi-equilibrium state, which represents the initial conditions for the simulation.

One should be aware of the limitations of an equilibrium model without
gas self-gravity. Most of the numerical simulations treat
self-consistently the process of star formation. Since star formation
occurs when the gas self-gravity prevails over pressure, neglecting
the gas self-gravity in the set up of the model is clearly
inconsistent. Moreover, without self-gravity, there is the risk of
building gas configurations which would have never been realized if
self-gravity were taken into account. In order to solve these
problems, Vorobyov et al.
[331]
explicitly took into account gas self-gravity to build initial
equilibrium configurations. The gravitational potential
is composed of two
parts, one is due to
a fixed component (dark matter and eventually also old stars), one
(_{g}) is due to
the gas self-gravity. The gas gravitational
potential _{g}
is obtained by means of the Poisson equation

(3) |

The gas density distribution is thus used to calculate the potential, but this potential is then included in the Euler equation to find the gas distribution. Clearly, an iterative procedure, analogous to the classical self-consistent field method [200], is necessary to converge to an equilibrium solution.

For a given mass *M*_{DM} of the dark matter halo, many
solutions are possible, according to the initial assumption about the
density distribution of the gas. However, the self-gravitating equilibrium
configurations always have a maximum allowed gas mass
*M*_{max},
unlike the case of non-self-gravitating equilibria which can realize
configurations with unphysically high gas masses. Moreover, only for
some of the solutions, star formation was found to be permissible by
Vorobyov et al. (two star formation criteria based on the surface gas
density and on the Toomre parameter were assumed). The minimum gas
mass *M*_{g}^{min} required to satisfy the star
formation criteria was found to be mainly dependent on the gas temperature
*T*_{g}, gas spin parameter *e* and degree of
non-thermal support. *M*_{g}^{min} was then
compared with *M*_{b}, the amount
of baryonic matter (for a given *M*_{DM}) predicted by the
CDM theory of
structure formation. Galaxies with *M*_{DM}
10^{9}
*M*_{} are
characterised by *M*_{g}^{min}
*M*_{b},
implying that star formation in such objects is surely possible as the
required gas mass is consistent with what is available according to the
CDM theory. On the
other hand, models with *M*_{DM}
10^{9}
*M*_{} are
often characterised by
*M*_{g}^{min} ≫ *M*_{b},
implying that they need much more
gas than available to achieve a state in which star formation is
allowed. In the framework of the
CDM theory, this
implies the existence of a critical dark matter halo mass below which the
likelihood of star formation drops significantly
([331],
see also
[347,
208,
263]).