We estimate that for typical parameters of elliptical galaxies residing in large clusters, a centroid shift of a few hundreds of parsecs is expected in MOND, as compared to the centroid of the outer isodensity contours of the galaxy. While tides can also cause some lopsidedness in Newtonian and MONDian gravity, there is a major distinction with the pure MOND effect under scrutiny here. The tidal effects are small inside the tidal radius, and are traditionally neglected when dealing with dense elliptical centers. The fact that these effects can be neglected in our analysis is confirmed by the fact that the average density of our model galaxy inside the saddle point where the external field starts to dominate is larger than the cluster density. The saddle point is estimated to be 15 kpc from the Plummer sphere (total mass 1011M⊙) center (Fig. 4.10 of Knebe et al. 2009). Here the
4.2. Cluster galaxies are non-axisymmetric in MOND
Figure 4.12: Color maps of xz-projected density on the scale of 10 kpc show the twist of isophotes between and inner and outer density distribution. The arrows indicate the direction of the external field.
Figure 4.13: The offset of nuclei in dwarf elliptical galaxies in Virgo cluster. The coloured empty circles are the 78 dwarf ellipticals (Binggeli et al. 2000) on the sky, and the red cross in the center is the position of Virgo cluster center. The different colours on the left panel show the centroids’ offsetsδrin unit of arcsecond (1”∼100 pc), and on the right panel the different colours show the
relative offsets δr
Chapter 4. Stability and Evolution of Galaxies in External Fields
average density enclosed is∼4×10−3M⊙pc−3, typically more than one order of magnitude larger than the typical cluster density at that same radius (1000×ρcrit ∼10−4M⊙pc−3). Note that Zhao (1995) argues that such tidal criteria holds in MOND as well in Newton. Thus, we conclude that if real elliptical galaxies in rich clusters show perfectly symmet- ric light with nosignificant offsets between the centroids of the inner and outer contours, the classical version of MOND is likely excluded. This predicted lopsidedness of galax- ies inside distant rich clusters should be falsifiable with photometry only, using, e.g. the VLTI. To resolve a centroid offset of 200pc in an elliptical galaxy in a rich cluster of typ- ical internal gravity ∼ 10−8cms−2 at a distance of 160-210 Mpc (e.g., Abell 1983, Abell 2717, MKW9 Pointecouteau et al. 2005, Sanders 2003) would require a minimum angular resolution of 0.5 arcsec. The offsets at the centroids of galaxies are between 10 pc and 500 pc: for an elliptical in Coma (100 Mpc far away) this corresponds to 0.02 to 1 arcsec, while for an elliptical in Virgo (20 Mpc far away) it corresponds to 0.1 to 5 arcsec. In Binggeli et al. (2000) they observed such centroid offsets in a sample of dwarf ellipticals in Virgo Cluster, of which 20% are significantly lopsided, and the offsets are typically 1”. On the other hand, observing such a lopsidedness would not consitute a direct falsifi- cation of the CDM paradigm. Indeed, CDM simulations predict that isolated CDM halos are typically lopsided due to a lack of relaxation, so this should be the case inside clusters too. Macci`o et al. (2007) studied the shapes of a large sample of ten thousand pure CDM halos, and found that any offset between the bary-center (CoM) and the density-center (potential minimum) is correlated with a massive satellite inside an unrelaxed halo, i.e., the offset is a measure of unrelaxedness in the CDM context. This is characterized by an offest of order of 1 or 2 percent between the central dark matter density maximum and the center of mass at the virial radius. Let us note that if this can be translated into a predicted lopsidedness for the light distribution of isolated ellipticals in CDM, this would be a clean test since MOND does not predict such an offset (except at extremely large radii) for isolated galaxies. For galaxies inside clusters, these CDM offsets would however not be expected to be aligned with the direction of the local external field pointing to the cluster center. If a statistically relevant preferred direction is found for galaxies residing in clusters, it would thus be in support of MOND, although it would not constitute a direct falsification of CDM.
4.2. Cluster galaxies are non-axisymmetric in MOND A caveat is that our simulations were based on a constant external field of∼10−8cm s−2. However, in reality, while galaxies are orbiting inside clusters, the external field acting on them varies with time, both in direction and amplitude. So the next step will be to make the external field vary as a function of time in our simulations, to check how this affects the predicted lopsidedness.
Finally, we also note that the flattened and lopsided potential created by the external field would also be valid in disk galaxies, although most of them are in the field or at the outskirts of galaxy clusters. Still, the flattened MOND potential would create a differential force with a component normal to the disk, hence a specific torque. This could cause differential precession of the disk angular momentum vector, and could lead to the formation of warps, even in a very low external field for isolated galaxies (e.g., Combes & Tiret 2009).
5
Conclusion
5.1
Summary
In this thesis, we investigated the dynamics of MONDian galaxies:
• We studied on the external field effect and the escape speeds for spiral galaxies in Chapter 2.1 including the Milky Way Galaxy. The external field makes the potential finite at infinity, enabling stars to escape. While MOND fits well with the rotation curve of the Milky Way, it also fits well with the escape speed (RAVE observations)at the solar neighbourhood. The rotation curves of Milky Way-like spiral galaxies re predicted to have a truncation if those galaxies are in the central regions of clusters. The High Surface Brightness (HSBs) in a strong external field will be distorted, and the Low Surface Brightness (LSBs) cannot have normal velocity dispersion like other field LSBs.
We further studied on the potentials of the Milky Way in four CDM models and compared with MOND Besan¸con model in§2.2, with a weak external field. We com-
Chapter 5. Conclusion
pared the escape velocities as a function of position for those models. We showed that MOND predicts deeper potential than CDM, while both gravities fit well with observations and circular velocity for all radii and escape speed at the solar neigh- bourhood. MOND predicts a “phantom” dark matter distribution that can have negative density in places where the external field is comparable to internal fields. • We constructed models with the Hernquist density profile for the field (§3) and
clustered (§4) ellipticals. We showed that all of our models are stable during the N- body simulations. We further studied the asymmetric profiles of ellipticals embedded in an external field§2.2, there is an offset of centroids defined by the inner and outer contours.