An Accurate Comprehensive Approach to Substructure: IV. Dynamical Friction

TL;DR

This paper addresses the analytic modeling gap posed by dynamical friction (DF) in CDM subhalo evolution. Building on the peak model and the CUSP formalism from earlier papers, it introduces a fully analytic treatment of DF via a local wake coefficient $A(v,r,M_{ m s})$ and simple orbit-averaged quantities, enabling explicit expressions for subhalo abundance and radial distribution across arbitrary halo mass, redshift, and formation history. The authors derive how DF shifts subhalo energies and angular momenta on individual orbits, then couple these with tidal stripping and shock-heating to obtain evolving final masses, radii, and concentrations, both for single orbits and concatenated orbits, and finally apply the results to the whole subhalo population. They show that DF makes the radial distributions steeper for massive subhaloes and modestly alters the subhalo mass function, with good agreement to simulations once resolution effects are included. The framework provides a powerful, parameter-light tool to predict substructure properties in diverse cosmologies, with potential applications to galaxy bias and broader self-gravitating systems.

Abstract

In three previous Papers we analysed the origin of the properties of halo substructure found in simulations. This was achieved by deriving them analytically in the peak model of structure formation, using the statistics of nested peaks (with no free parameter) plus a realistic model of subhalo stripping and shock-heating (with only one parameter). However, to simplify the treatment we neglected dynamical friction (DF). Here, we revisit that work by including it. That is done in a fully analytic manner, i.e. avoiding the integration of subhalo orbital motions. This leads to simple accurate expressions for the abundance and radial distribution of subhaloes of different masses, which disentangle the effects of DF from those of tidal stripping and shock-heating. This way we reproduce and explain the results of simulations and extend them to haloes of any mass, redshift and formation times in the desired cosmology.

Figures (7)

• Figure 1: Approximate relative orbital energy $E$ and angular momentum $L$ increments (positive and negative, respectively) of subhaloes of mass $M_{\rm s}=10^{-2}M_{\rm h}$ as a function of their initial apocentric radius $r$ (scaled to the virial radius $R_{\rm h}$ of the halo) and the parameter $k$ measuring the square of their initial tangential velocity scaled to the circular velocity (dashed lines) in current haloes with MW-mass $M_{\rm h}$, compared to the exact results obtained by integration over real orbital motions with DF (solid lines). The black dot-dashed line marks the zero baseline. In the bottom panel we plot the relative differences between the approximations (dotted lines for $\Delta E/E$ and dot-dashed lines for $\Delta L/L$) and the exact solutions (the zero baseline). Left panel: Approximate $\Delta E/E$ and $\Delta L/L$ values obtained by integration over virtual orbits with no DF. Right panel: Approximate purely analytic $\Delta E/E$ and $\Delta L/L$ values (i.e. obtained with no integration). (A colour version of this Figure is available in the online journal.)
• Figure 2: Ratios of final to initial radii (curves below unity) and tangential velocities (curves above unity) at apocentre in one orbit found by numerical integration of the orbital motion of subhaloes (solid lines) and obtained to first order in the relative energy and angular momentum increments, $\Delta E/E$$\Delta L/L$ (dashed lines), as a function of $r$ for several $k$ values and the same $M_{\rm s}$, $M_{\rm h}$ and $t_{\rm h}$ as in Figure \ref{['f0']}. Again, the black dot-dashed line marks the zero value. In the bottom panel we plot the relative differences between the approximations (dotted lines for apocentric radii and dot-dashed lines for tangential velocities) and the exact solutions (the zero baseline). Left panels: Results obtained using the leading-order-approximate $\Delta E/E$ and $\Delta L/L$ increments found by integration over virtual orbits with no DF. Right panels: Results obtained using the approximate purely analytic $\Delta E/E$ and $\Delta L/L$ values. (A colour version of this Figure is available in the online journal.)
• Figure 3: Radial abundance of stripped subhaloes of mass $M_{\rm s}^{\rm tr}$ scaled to their total abundance (the scaled quantity is independent of subhalo mass) in current purely accreting MW-mass haloes calculated using the $M$-$c$ relations derived by Sea23 using CUSP (solid red line) and found by Gea08 (dashed red line) in the Millennium simulation with a limited halo mass resolution. Both solutions are essentially proportional to $r^{1.3}\rho(r)$ (dashed black line) at radii larger than $r/R_{\rm h}= 0.08$ (vertical dotted line) as found by Hea16 in the MW-mass AqA1 halo in that simulation. (A colour version of this Figure is available in the online journal.)
• Figure 4: Differential subhalo MF with DF in current purely accreting MW-mass haloes in the WMAP7 cosmology derived using the CUSP (solid green line) and Gao et al.'s (dashed green line) $M$-$c$ relation, compared to the predictions without DF (dashed black line). (A colour version of this Figure is available in the online journal.)
• Figure 5: Scaled number density profiles of subhaloes of several masses in the same haloes as in Figure \ref{['f2']} (coloured lines) derived using the CUSP (solid lines) and Gao et al.'s (dashed lines) $M$-$c$ relations, compared to the profiles found without DF, which overlap in one single curve (black line) and with the profiles with DF of subhaloes of $M_{\rm s}^{\rm f}\la 10^{-4}M_{\rm h}$. (A colour version of this Figure is available in the online journal.)