tokens: half of the mass that has become unbound during the preceding orbit. The simulations used by Fujii et al. ( 2006 ) followed the orbital evolution of a self-consistent satellite galaxy (without dark matter). As they point out in their paper, since dark matter subhaloes are more extended than satellite galaxies, they typically experience more mass loss and thus should be susceptible to even stronger self-friction. In this paper we use idealized numerical experiments to study the nature of self-friction acting on dark matter substructure. Our main goal is to characterize the relative contribution of self-friction to the total drag force experienced by subhaloes as a function of orbital properties and the mass ratio between the sub- and host halo. The rest of the paper is organized as follows: in Section 2 , we discuss our numerical methods and simulations. Our results are presented in Section 3 and then discussed and summarized in Section 4 . Throughout, we adopt a Hubble parameter , which corresponds to a Hubble time of 13.97 Gyr for a flat universe with Ω Λ = 0.73. 2 METHODS To investigate the effects of self-friction, we perform numerical simulations of an N -body subhalo orbiting within an analytical host halo potential. We provide a brief description of the methods here and refer the reader to van den Bosch Ogiya ( 2018 , hereafter BO18 ) for full details on the numerical simulations and analysis methods. The host- and subhaloes are both assumed to initially be spherical NFW profiles (Navarro, Frenk White 1996 ) with concentrations c host and c sub , respectively. The host halo is represented by a static analytical potential. Throughout this paper, R and r are used to indicate radii with respect to the centre of the host halo and subhalo, respectively. The virial masses and radii of the two systems 2 are indicated by M host and R host , in the case of the host halo, and M sub and r sub in the case of the subhalo. Note that for the latter, these relate to the initial, unperturbed subhalo, prior to being introduced to the tidal field of the host halo. Positions and velocities of particles in the subhalo are initialized assuming an isotropic distribution function, f ( E ), using the method of Widrow ( 2000 ). In order to roughly account for the tidal mass loss that the subhalo would have experienced when approaching its initial position from an infinite distance, the initial distribution of particles is truncated at the instantaneous tidal radius of the subhalo (King 1962 ), given by Here, R is the (initial) distance from the centre of the subhalo to the centre of the host halo, m ( r tidal ) is the mass of the subhalo within r tidal , M ( R ) is the mass of the host halo within R , and Ω is the (initial) angular speed of the subhalo (Taylor Babul 2001 ; Zentner Bullock 2003 ; Peñarrubia Benson 2005 ; van den Bosch 2017 ). Note that we do not account for this truncation in the distribution function, which instead is computed under the
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