3. Modeling the interaction
3.1. Order-of-magnitude estimates
We start by illustrating the effect of the passage of the dwarf galaxy through a thin gaseous disk using analytical order-of-magnitude arguments. Neglecting gas dynamics, we can write the momentum transferred to the volume element of the disk as
where and v are the mass and impact velocity of the dwarf galaxy and r is the distance of the disk element from the place of impact. After the collision, the gas element acquires velocity which should bring it to some elevation h above the disk. Assuming the model of the Galactic halo potential described below, the vertical gradient of the potential at the point of impact can be approximated well by the relation
where . The scale height of the perturbation caused by the dwarf will therefore be
where is the gravitational radius of Bondi-Hoyle accretion. In our scenario the orbital velocity of the dwarf is . With an extreme case of , this model gives a hole in radius, while material at the edge of this hole is displaced to . Since , any disk element with impact parameter, say, can be propelled up to above or beyond the disk. However, this order-of-magnitude estimate lacks essential physics, most importantly dissipation via shocks and subsequent radiative cooling.
3.2. Modeling the HI disk
A more careful treatment is required to predict quantitatively the effect of the collision on the gaseous disk. Walker et al. (1996) presented an N-body simulation to study the interaction of a disk-halo-satellite system in a merger which does not destroy the disk. After such an event, the disk appears to be of an earlier Hubble type than its progenitor, with a substantial amount of heating having been pumped into the disk. This energy will clearly alter the physical state of the gas in the disk resulting in more physical diffusion than in purely N-body simulations. To address this issue, we modeled the H Ias a continuous self-gravitating fluid, using a smooth particle hydrodynamics (SPH) tree-code kindly provided by M. Steinmetz (Steinmetz & Müller 1993).
For all of our models of the H Idisk, we assume an initially uniform temperature . We take an exponentially decreasing surface mass density beyond with scale length (our parameterization of the data of Burton & te Lintel Hekkert 1986), a Gaussian density distribution in the direction perpendicular to the Galactic plane with a scale height of (the measured value at - Burton (1992)), and a number density of H Iat of (so as to yield the surface mass density at measured by Burton & te Lintel Hekkert 1986). We chose to set the outer limit of the disk at . We also chose, for tractability, not to populate the H Idisk inside , to avoid having to simulate the complex interaction of the gaseous disk with the stellar disk. The total mass in H Iin the model is then .
Artificial viscosity in the simulation is corrected for shearing, as given by Benz (1990), which would otherwise result in a rapid transfer of mass from the outer to the inner regions of the disk.
One has to make sure that the surface number density of particles in the disk is high enough to resolve the impact area. Our criterion is to have at least 100 particles in the disk impact region corresponding to the size of the dwarf. We therefore chose to populate the annulus representing the H Igas with particles whose masses decrease exponentially from with a scale length of , giving a uniform surface number density.
3.3. Cooling of the atomic hydrogen
An appropriate thermal model for the atomic hydrogen is required, so as to mimic the existing temperature and density distributions of the gaseous disk as closely as possible. In nature, the temperature of the gas follows the equilibrium temperature set by the interplay of heating and cooling processes. However, this process is very complex: the cooling rate for a given temperature, density and chemical composition depends on the number density of free electrons, which in turn, depends on the flux of ionizing radiation (from cosmic rays, star formation, supernovae, and so on), which is poorly constrained in the outer regions of the Galaxy. The most problematic issue is that the ISM is likely to be optically thick to radiation in some important emission lines; developing a cooling rate algorithm with the necessary radiative transfer calculations is far beyond the scope of the present contribution.
To make the problem tractable, we instead assume an ISM equilibrium state given by Scheffler & Elsässer (1988), which is based on the local chemical composition and heating rate in the absence of hydrodynamical heating. The settling of the gas onto this equilibrium curve will depend on the amount of dissipational heating, which in our case, is given by the SPH algorithm. The cooling timescale is then just taken from the explicit cooling functions for atomic hydrogen of Scholz & Walters (1991), with the modification that the cooling time may not be longer than years. It was found that the results were not sensitive to the choice of this upper bound in the range , in the sense that the configuration of the disk at the end point of the simulations were qualitatively indistinguishable.
3.4. The Galactic potential
Ideally, one would prefer to model the Galaxy with a "live" halo, stellar disk, bulge and spheroid. However, the computational resources required for such a treatment are beyond our present capabilities. Instead, the SPH scheme was altered to include a fixed potential for the Milky Way and a moving potential for the dwarf galaxy. The potential of the Milky Way is derived from the mass model of Evans & Jijina (1994). In this model, the disk component, described by a double exponential disk, has radial scale length and a Solar neighborhood surface density of . We further assume that the vertical scale length of the disk is , and that the density falls to zero at . The potential corresponding to this density distribution is found by multipole expansion of the Poisson equation using an algorithm described in Englmaier (1997). The halo component is described by a `power-law' halo, so the potential has the following analytical expression:
where R and z are Galactocentric cylindrical coordinates, the core radius , , the exponent , and the oblateness parameter .
A shortcoming of this approach is that the fixed halo behaves differently to a "live" halo that is free to respond to the passage of the massive dwarf. A "live" halo would respond (in a way that is dependent on the density and kinematics of the halo, and the mass and velocity of the object) to the extra gravitational attraction, leaving a wake of halo material behind the object. This wake would also interact with the H Idisk. So, in reality, the interaction of the Sagittarius dwarf with the Galactic disk should be more violent than found here using a fixed halo model.
3.5. Modeling the dwarf galaxy
To obtain a good representation of the dwarf galaxy, one would ideally construct a set of spherical stellar models, evolve them in the Galactic potential for Gyr, and choose the model that best reproduces the observations. However, as yet no N-body model has been found that is able to survive Galactic tides and also give a reasonable approximation to the observed distance, radial velocity dispersion and radial velocity gradient. In the absence of such a self-consistent N-body model, we account for the Sagittarius dwarf by including a Plummer sphere potential , which progresses along a predetermined orbit. Here r is a radial distance from the guiding center of the orbit, and . Note that spherical models are not a good representation of the present shape of the Sagittarius dwarf, since it is significantly elongated (axis ratios 3:1:1; IWGIS), however the difference to the perturbation on the Galactic gas is likely to be small.
For a chosen model mass , we calculate the orbit in the above Galactic potential whose projected velocity best fits the kinematic data; dynamical friction is taken into account using the Chandrasekhar dynamical friction formula (see e.g. Binney & Tremaine 1987). So as to minimize the perturbations on the Galactic disk in the initial evolution of the simulation, at the beginning of the integration, the dwarf galaxy models are placed at the position where the center of mass of the model was years ago.
© European Southern Observatory (ESO) 1998
Online publication: July 7, 1998