An Accurate Comprehensive Approach to Substructure: IV. Dynamical Friction

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.

💡 Research Summary

The paper extends the authors’ previous analytical framework for dark‑matter halo substructure (developed in Papers I–III) by incorporating dynamical friction (DF) in a fully analytic way. Earlier work combined the peak‑statistics of the CUSP formalism with a realistic model of tidal stripping and shock‑heating, but deliberately omitted DF, limiting the applicability to subhaloes less massive than ~10⁻⁴ M_h. Here, the authors derive simple yet accurate expressions for the subhalo mass function (MF) and radial distribution that explicitly include DF, without resorting to numerical integration of subhalo orbits.

The introduction reviews the importance of DF across many astrophysical systems (massive stars, spiral arms, globular clusters, black holes, etc.) and points out that current semi‑analytic models either ignore DF or must integrate orbits numerically, which defeats the purpose of having a transparent analytical description. The authors therefore set out to treat DF analytically.

Section 2 distinguishes two DF mechanisms: the classic Chandrasekhar “local wake” and a “global mode” arising from resonant interactions. The paper focuses on the local wake, which dominates the early orbital decay of massive subhaloes. Using Chandrasekhar’s formula, the DF coefficient A(v,r,M_s) is written as

A = 4π G² M_s ρ(r) f_dDM(r) ln Λ F(<v) / v³,

where ρ(r) and σ(r) are the host halo’s density and velocity‑dispersion profiles, f_dDM is the diffuse‑dark‑matter fraction, and ln Λ is taken as a constant 2.1 (a value calibrated against N‑body simulations of NFW haloes). The Coulomb logarithm is treated as a fudge factor, but the authors argue that a constant value suffices for the purposes of this work.

The core of the analytic treatment is to evaluate the energy loss ΔE and angular‑momentum loss ΔL over a single orbital period T without solving the full equation of motion with DF. By multiplying the equation of motion by the velocity and integrating, the authors obtain

ΔE = – M_s ∫₀ᵀ A v² dt,

ΔL = – L ∫₀ᵀ A dt,

where the integrals are evaluated over the “virtual” orbit that would exist in the absence of DF. Two approximations are presented: (i) a leading‑order estimate where the integrals are performed on the DF‑free orbit, and (ii) a fully analytic estimate that avoids any integration. Both are compared to exact numerical integrations of the full DF‑included orbit (Figures 1 and 2). The leading‑order approach reproduces the exact ΔE/E and ΔL/L to within a few percent for apocentric radii r ≳ 0.1 R_h, while the fully analytic version remains accurate enough for the purposes of the paper.

Using the first‑order changes in energy and angular momentum, the authors derive the final apocentric radius r_f and tangential velocity v_f after one orbit. Expanding the potential Φ(r) about the initial radius and solving the resulting cubic equation yields

Q_f ≡ r_f/r = 1 + k/(1–k)