In the previous section we have shown that the perturbation schemes being used all appear to reduce the attractor to a matter of placing particles on infalling orbits. One of the dangers of placing a large population of particles on highly radial orbits is that the system becomes susceptible to the radial orbits instability (ROI) which can produce a set of very well constrained behaviours. In fact, these behaviours are so regular and predictable that much work has already been done to determine whether ROI is the explanation for the universality of the NFW profile in dark matter halos (Macmillan & Henriksen, 2006; Bellovary et al., 2008; Lapi & Cavaliere, 2011). As part of that endeavour, several universal profiles and convergent behaviours have been identified.
The following behaviours are not all completely exclusive to ROI but they are all symp- tomatic of highly radial systems and the kinds of radial infall that our models undergo. However, ROI is still the primary mechanism which relates these effects and as such we shall concentrate on their connection to it.
Before continuing it is important to note that the mechanism we cause is not identical to ROI. These models start off stable and have a wide variety of anisotropies and only become radial as a result of this forced collapse. In this sense, we ignore the initial suscep- tibility of the system to ROI and would thus expect the effect to emerge more gradually as the system is perturbed. We suggest that this is why our systems do not display the high degree of triaxiality that is normally associated with ROI-unstable systems (Antonov, 1973; Barnes et al., 2009). As we showed in§5.3 our models generally maintain equal axis ratios apart from the systems that are more dispersed. Interestingly the more dispersed system also undergo a more significant amount of collapse so it is also possible that the development of triaxiality inthesesystems is not caused by noise, as previously suggested, but ROI.
5.6.1 Universal anisotropy profiles
In many simulations of ROI unstable systems it was observed that the anisotropy profiles of the final systems would tend to cluster around a common profile. This profile was more complex than a pure power-law (Bellovary et al., 2008; Lapi & Cavaliere, 2011) and displayed an isotropic nucleus that rises smoothly to radial anisotropy in the outer
Figure 5.17:Evolution of the anisotropy profile in the Newtonian isotropic (black), radial (red), tangential (green) and extremely radial (orange) models as well as the MONDian isotropic (purple) and radial (blue) models from their initial states (dashed/top) to their final states (solid/bottom).
edges in a similar manner to the attractor. There is some tolerance within this, however, as Bellovary et al. (2008) found that the initial anisotropy profile did leave a lingering impression on the final profile.
We also find good convergence among the Newtonian simulations although we note that the MOND simulations stand out somewhat in Fig. 5.17. The very radial initial
conditions also stand apart from the others although this is still consistent with ROI.
5.6.2 Convergence ofρ/σ3
One important result, although not one exclusive to ROI, was that systems formed by mergers (such as DM halos) display convergence in a quantity that proxies for the phase- space density (Taylor & Navarro, 2001; Dehnen & McLaughlin, 2005). This quantity is
ρ/σ3 and it is found that it quite precisely follows a power-law:
ρ σ3
r
∝r−1.9 (5.12)
As noted in those papers, the peculiar thing about this convergence is that neither ρ
norσ3r are themselves individually convergent to a power-law. However, the significance of this quantity does suggest that there may be some connection between it and ourγ+κ
axis. In Fig. 5.18 we investigate whether our simulations develop the same power-law during convergence.
Fig. 5.18 shows us firstly that all the models start off with very similar phase-space density profiles with the exception of the extremely radial case. However, allthe New- tonian models evolve towards r−1.9 as per Eq. 5.12 and end up with profiles that are even closer together than when they began. Standing out from this close convergence is, once again, the extremely radial case which ends up with ar−1.9 power-law but a signifi- cantly offset from the other system. Also interesting is that this high degree of clustering is largely absent from the MONDian profiles, which retain good agreement with each other but barely evolve from their initial conditions.
This shows that our perturbation, developed from HJS’s attempt to simplify the mod- elling of halo mergers, does seem to display some characteristics of a system that has un- dergone repeated mergers. Accordingly, we may use this power-law as a possible avenue of investigation into our attractor formula, particularly in light of the notable connections to the next result.
Figure 5.18:Evolution of phase-space density in the Newtonian isotropic (black), radial (red), tangential (green) and extremely radial (orange) models as well as the MONDian isotropic (purple) and radial (blue) models from their initial states (dashed/top) to their final states (solid/bottom).
5.6.3 β - γ relation
Seeing that we now have two convergent properties both of which involve velocity disper- sions and density, it is not surprising that there are relations directly linkingβ andγ that also provide convergent results. Many studies have found evidence for a relationship be- tween the two quantities (Huss et al., 1999; Barnes et al., 2005; Hansen & Moore, 2006; Hansen et al., 2006; Macmillan & Henriksen, 2006; Bellovary et al., 2008; Lapi & Cava- liere, 2011) and in particular Hansen & Moore (2006) went as far as positing a specific relationship of β(r) = −0.15 + 0.2γ. Although this result is not explicitly connected to ROI, instead being found in every relaxed structure tested, it still applies and several of those papers have used it for such.
This might sound rather familiar when put in the context of the work from chapter 2 on the GDSAI. The difference here is that these results discuss a much more specific relationship between these quantities than just the general inequality provided by the GDSAI. Whether or not the attractor and the GDSAI have any relation to each other is not known and will hopefully be the subject of future investigation.