2. On the stability of galaxies in MOND
2.1. The Toomre parameter and dwarf galaxies
It is widelly accepted that the Toomre parameter Q in galactic discs must lie in the range in order to be dynamically cool enough to develop spiral structure but not too cold to be violently unstable to radial instabilities (e.g. Toomre 1981). This constraint has been used, for example, to put limits on the value of the of stellar discs (e.g. Athanassoula et al. 1987; Quillen & Sarajedini 1998; Fuchs et al. 1998). We can ask what are the implications of this constraint in MOND dynamics.
Milgrom (1989) deduced the Toomre condition for local stability of gaseous discs under modified dynamics, , where is related to that of Newtonian dynamics, , by
where is the gas surface density, is the one-dimensional velocity dispersion of the gas, the epicyclic frequency and is the logarithmic derivative of µ just above the disc, where µ is the MOND function that connects the acceleration in modified dynamics, , with the Newtonian one, :
Here cm s-2 is a constant of the theory. The interpolating function must satisfy that for and for .
In the case where the density of the disc and the velocity dispersion are fixed, then disc galaxies with the smallest values of will be the most unstable, i.e. dwarf galaxies. In fact, we may express the critical surface density as
Just only for illustrative purposes we will assume here that the stability criterion in MOND is obtained from the Newtonian criterion under the substitution . Using the same value for than Zasov & Bizyaev (1996), the critical surface density expressed in is
where is the rotation curve in km/s, R is the galactocentric radius in kpc, , and in the deep MOND limit. To derive Eq. (4) it has been assumed that (Milgrom 1988) and that . It is apparent from Eq. (4) that, in order to maintain, at least, a similar degree of stability for small galaxies with low circular velocity, the surface density of gas should be smaller for them. However, the observed behaviour is the opposite. Small disc galaxies are usually gas-rich galaxies and they do not present any clear spiral pattern or evidence for the existence of density waves.
The stability of the small galaxies IC 2574 and NGC 1560 is studied in detail in the next sections. We show that the values of the Toomre parameter under MOND are very low for these galaxies, even though a very favourable value for stability, km/s, is assumed. These galaxies, however, should not be considered as exceptional cases. There are some other galaxies which may present an extraordinary level of axisymmetric instability under modified dynamics, such as NGC 1560, NGC 3109, NGC 55, DDO 9, F561-1 and F565V2. Published HI observations for these galaxies can be found in Broeils (1992), Jobin & Carignan (1990), Puche et al. (1991), Swaters (1997) and de Blok et al. (1996) for the last two, respectively.
Due to the explicit linear dependence of on the velocity dispersion of the gas, , it is worthwhile reviewing the observational and expected trends of for different galaxies. This discussion is given in a separate subsection.
2.2. On the HI velocity dispersion in dwarf galaxies
There are many theoretical and observational support to believe that the HI interstellar medium is turbulent even in the outer part of the discs (e.g. Scalo 1987; Sellwood & Balbus 1999). In that case, the assumption used in Eq. (1) that the interstellar medium is homogeneous and uniform breaks down and, consequently, the scale-dependences of the density and velocity dispersion of the gas should be taken into account. However, since the typical scalelength of the instability is greater than the size of the largest eddies, , ( is expected to be of the order of the semi-thickness of the disc, ), we may extend the validity of Eq. (1) to turbulent discs.
More dramatic is the dependence of on the velocity dispersion; it is clear that stability is recovered by increasing . From an observational point of view, spiral galaxies in different environments show a remarkably uniform HI velocity dispersion of km/s in the outer disc, and a few km/s higher in the bright optical disc (Sellwood & Balbus 1999, and references therein). Different processes to supply the requisite energy to maintain turbulent motions have been proposed, such as supernova heating, differential rotation, gravitational instabilities or MHD driven turbulence.
One could argue that since small galaxies in modified dynamics are very responsive to self-gravity perturbations, these gravitational instabilities are stirring the gas layer to maintain the level of turbulence of the gas for which the condition is fulfilled. It is easy to see from simple scaling arguments that this is not the expected situation for low-mass galaxies.
Let with km/s fixed. For some galaxies, it happens that between two radius and (for IC 2574 see next section). These galaxies require a higher than to ensure stability. Suppose, in addition, that these galaxies are self-gravitating discs of almost pure gas, as it is deduced from their mass decompositions in Sanders (1996). If the input energy into turbulence, , is driven by gravitational instabilities, the corresponding typical time scale is expected to scale essentially as the dynamical one, i.e. the reciprocal of the angular velocity, , so that . Furthermore, the dissipation rate of turbulent energy per surface area is , with the scale height of the disc. Equating the input and dissipation energy rates
where we have used for a stable self-gravitating and isothermal disc. Thus the expected in terms of is given by
This simple argument shows that the expected value for in turbulent discs with low should be even smaller that the standard value , in contradiction with the requirement . In other words, the gravitational instability promotes the formation of large gas concentrations instead of being a source of turbulence.
In the unlikely case that the constant internal HI line-width represents the thermal temperature the above arguments are no longer valid. We therefore prefer to present a study of a gas-rich galaxy with measured HI velocity dispersions, IC 2574, to avoid further speculations.
© European Southern Observatory (ESO) 1999
Online publication: April 12, 1999