Theoretical Constraints

The properties of dark halos in cosmological models have been studied extensively, enabling in principle a test of consistency with our tidal tail arguments. The main theoretical uncertainty is calculating how disks settle within these halos - hydrodynamic simulations are still plagued with difficulties due to incomplete treatments of star formation and feedback, while semi-analytic methods are not well-tested. The claim from DMH that the dark:luminous mass ratio of galaxies must be less than 10 was based on a restricted set of halo models; it is useful to expand and refine the arguments using the simulation survey done here in order to set better constraints.

Most recently, Navarro, Frenk, & White (1997) have analyzed halo formation in a variety of cosmological models, showing how halo properties depend on the cosmological parameters (e.g., $\Omega, \Lambda, n$). Their main result is that the properties of dark halos can be quantified by a single parameter, the halo concentration, c=r₂₀₀/r_s, which is the ratio of the ``virial radius'', r₂₀₀, to the scale-radius of the fitted NFW profiles. In general, the concentration various with mass scale in different cosmological models with the mean value depending on the cosmological parameters and choice of normalization of the power spectrum. In principle, the measurement of the concentration then can constrain a cosmological model as exemplified by Navarro's (1998) attempts to estimate c from rotation curves.

The structure and kinematics of tidal tails are similarly sensitive to halo concentration. We can calculate the formal concentration for our models by finding the effective r₂₀₀ assuming the galaxies are scaled to galactic dimensions. The concentration parameter is found by solving:

$$0.047 \left( \frac{v_s}{H_o r_s} \right)^2 = \frac{c^3}{\log(1+c) - c/1+c}$$ (8)

The formal values of c based on a galaxy with Milky Way dimensions (length units of 4 kpc, velocity units of 220 km/s) and H_o=50 km/s/Mpc are given in Table 2 for comparison. (When assuming H_o = 75 km/s/Mpc all concentrations are systematically lowered).

As a first comparison, we look at halos from standard CDM cosmology, which are predicted to have c=10-20. Neglecting dissipative effects on the halo, this range of concentrations is represented in Figure 5 by the band of models with r_s=4.8 and models with r_s=6.9 and v_s > 0.8. Our original $\Omega = 1$ CDM-like mass models C and D lie near the bottom of the diagram where tails are hard to make, consistent with the original conclusions in DMH. We see, however, that some models with halos having CDM concentrations yet which are disk-dominated in their inner regions (Region II models) are more forgiving. The reason for this is simply that the circular velocities of the disk and halo are mismatched, so that there is a decline in the rotation curve at the disk-halo transition. To manufacture long tidal tails, in essence we must include a massive disk component to create a falling rotation curve within a CDM halo. These disk-dominated models are consistent with ``maximum disk'' models or even Bottema (1993, 1997) disk models, in which approximately 60% of the total rotation curve velocity is attributed to the disk component. However, such models are marginally inconsistent with the satellite data of Zaritsky et al. (1994) for Milky Way-type spirals and completely incompatible with recent rotation curve decompositions of LSB disk galaxies (e.g., de Blok & McGaugh 1996; McGaugh & de Blok 1998), which show flat rotation curves with little disk contribution.

In contrast to the predictions for standard CDM halos, the dark halos in low-density CDM models as well as flat CDM+$\Lambda$ models are predicted to have somewhat smaller concentrations, $c\sim 7$, when normalized to COBE (Navarro 1998). Models with low concentration halos are found at the right edge of Figure 5. The only way such galaxies can produce long tidal tails is if the rotation curve falls at the disk-halo transition, such as the models in the upper right of Figure 5. Rotation curves which are solely supported by low concentration halos have potential wells too steep to eject long tidal tails. Following the trends, we expect only disk-dominated models with smaller concentrations to make good tails and perhaps this kind of model is the most consistent theoretical choice. These arguments suggest that the galaxies which gave rise to the Toomre Sequence were predominantly luminous HSB galaxies with modestly declining rotation curves due to the disk-halo transition.

  

Figure 7: Same as Figure 5 but with R_p=2.0 and taken to the stage when the galaxies have just merged, as for NGC 7252.

We note also that when the power spectrum is normalized instead to the number density of rich clusters in the local universe the concentration has virtually no dependence on $\Omega$ with a constant value of $c\approx 10$ in all cosmological models (Navarro 1998). If this is the correct way to normalize, then tidal tails (as well as rotation curve shapes) cannot be used to constrain $\Omega$. However, the constant value of $c\approx 10$ is at least marginally consistent with tidal tails with this normalization. Stronger constraints on cosmological models must await a more thorough understanding of the normalization of the cosmological power spectrum and how the formation of disk galaxies inside dark halos proceeds in different cosmological models.

While there is some room for consistency between our tidal tail models and the cosmological halo formation models of NFW, the problem of LSB galaxies still remains. Attempts to fit NFW models to LSB rotation curves have been made by McGaugh & de Blok (1998) and Navarro (1998); while these attempts largely failed, the ``best'' matches yielded concentration indices c<10, where our models show that tidal tail formation is completely inhibited in halo-dominated models. Because LSBs probably trace the dark matter component most faithfully, these results suggest that ``bare'' dark matter halos with low mass disks prohibit the formation of tidal tails; it is only the inclusion of a (relatively dense) baryonic component at the center to make a falling rotation curve necessary to produce observed tidal tails. Yet it is a very fine line we walk with these arguments; any significant gradient is inconsistent with many observed rotation curves and the satellite galaxy velocity dispersion measurements of Zaritsky et al. (1994).

Recently, Springel & White (1998; hereafter SW) have also examined the consistency of long tidal tails within standard CDM halos against observations, and it is useful to compare their results and interpretations to our study. SW examined disk+CDM halo models constructed using the prescription of Mo et al. (1998). According to this approach, halos are adjusted adiabatically in a self-consistent manner to the presence of an exponential disk whose length scale is determined by the initial angular momentum. SW examined a sequence of models with the same initial halo model ( $v_c \sim 250$ km/s) but different baryon fractions and dimensionless spin parameter, $\lambda$. Their survey differs from ours in that SW fixed the halo mass profile and varied the disk properties, while we have fixed the disk+bulge mass profile and varied the halo properties. It is encouraging that SW find the same range of behavior in tail making that we see in our parameter survey for models with similar mass profiles. Their models cover mass profiles in the bottom of our Region I and left-hand area of Region II. All of their models are halo-dominated with falling rotation curves beyond 5 scale lengths. Their results emphasize that there is not necessarily a one-to-one correspondence between the disk scale and the halo properties, making a range of behaviors during galaxy interaction possible (since varying $\lambda$ results in a range of R_d).

The models of Springel & White span a range in disk scale-lengths between 1.5-6.9 h^-1 kpc. Those models with the smallest scale lengths of 1.5-3.3 h^-1 kpc (their models A,B,D) have difficulty in making long tails because disk material is tightly bound in the inner portions of the galaxies where the model rotation curve is not yet falling. Conversely, their models with large disk scale-lengths (their models C,E,F) > 5 h^-1 kpc have no difficulty making tails - these galaxies are endowed with a significant fraction of their disk material at large radii, where the rotation curve is dropping and the potential well is shallow. Based on the success of the large scale-length models, Springel & White claim that there is no inconsistency between observed galaxies and those produced in the CDM model. While this is true in principle, it is important to note that their successful models are large scale-length, high angular momentum, halo-dominated systems - the perfect description of a low surface brightness disk galaxy. However, the rotation curves of these models are declining, whereas LSB rotation curves remain flat out to very large distances (r > 50 kpc; McGaugh & de Blok 1998). While these models do produce long tidal tails, it is unclear whether or not they have any real analogues in the observed galaxy population. In addition, if only large-scale length high angular momentum systems give rise to long tidal tails, this model would predict that the Toomre Sequence arose predominantly from LSB progenitors. Whether or not this is compatible with observations of the Toomre Sequence is questionable.