Dark Halo Orbits , Tidal Stripping , and the Spirographic Approximation

This paper examines the problem of a small dark matter halo orbiting in a larger dark matter halo with a Hernquist potential. Undamped orbits can be obtained from numerically integrating the equation for force in a Hernquist potential. They can also be modelled analytically using a spirographic approximation (Adams and Bloch 2005). Damped orbits can be obtained by adding a damping term to the force equation before integration. Finally, the small halo loses mass as it falls into the larger halo through tidal stripping. This paper examines the accuracy of the spirographic approximation for a few different orbits and explores the effect of different parameters on the orbits as well as different damping functions. Next, tidal stripping in these different scenarios is examined, and finally the asymptotic behaviour of orbits and asymptotic value of the tidal radius is explored. Background Dark matter halos consist of the dark matter portion of galaxies, which is believed to form a large percentage of the total galactic mass. They were originally hypothesized as a solution to the ”missing mass” problem in the galaxy. The halos are ellipsoidal in shape and since they dominate the mass of a galaxy, they play a significant role in the dynamics of the galaxy as a whole. Thus, understanding their properties and interactions is key to understanding the formation and behaviour of galaxies. Dark matter halos fall into the category of extended mass distributions, for which there are several different density profile models. The Hernquist potential (Hernquist 1990) used here is one such model, and describes the structure of dark matter halos as seen in numerical simulations fairly accurately. Using this potential and standard Newtonian kinematics, we can study the dynamics of objects orbiting inside of dark matter halos. This paper focuses on the mergers of small halos with larger ones. When a small galaxy merges with a larger one, its dark matter halo orbits through the larger galaxy. This orbit decays as a result of dynamical friction (Binney and Tremaine 2008) until the core of the small halo approaches the center of the large halo. As the small halo orbits through the larger halo, some of its mass is slowly stripped away by the larger halo and is absorbed into the large halo’s density profile. Thus, only a small core of the small halo remains when it reaches the center. An example of this type of scenario is the orbit of the Large Magellanic Cloud in the Milky Way. The LMC will eventually be absorbed by the Milky Way, and follows an orbit like the ones described in this paper (Adams and Bloch 2005). N-body simulations of galaxy formation indicate that all dark matter halos are largely formed by a series of many mergers of the type described above. Furthermore, all dark halos simulated converge to a very similar density profile (Dekel, Devor, and Hetztroni 2001). Numerical simulations done by Dekel et. al. have indicated that the density profile is due to the way mass is deposited in the large halo by different types of satellites; satellites with a lower density deposit mass further from the center and tend to flatten the large halo’s profile, but satellites with more density fall to the center and create the cusp that is characteristic of dark matter halos. In this paper, some supporting analytical results are derived. Introduction The process of halo mergers has been studied primarily through large scale numerical simulations. However, the mergers can also be studied using some analytical models. While analytical models cannot provide the level of detail that numerical simulations can achieve, it is possible to use them to obtain good approximations to some of the numerical results. As stated previously, the potentials of galactic dark matter halos are well modelled by a Hernqist potential (Hernquist 1990). The orbit of a smaller halo within this potential can