Velocities of Free Floaters in a Sea of Stars

TL;DR

This work analyzes how gravitational scatterings with background stars modify the velocities of free-floating planets and interstellar objects. Using Chandrasekhar diffusion in an infinite Maxwellian sea, it derives a nontrivial equilibrium velocity $v_{\mathrm{eq}} \approx \sqrt{2}\,\sigma\sqrt{\ln(m/m_p)}$ for $m_p\ll m$ and shows the approach to this state is extremely slow, with significant mass-dependent behavior. Early, mass-independent acceleration can still boost slow floaters by several $\sigma$ within a few relaxation times, and the velocity distribution evolves away from Maxwellian, even when birth distributions are Maxwellian. The results imply that in the Galactic disk, the kinematics of low-mass free floaters may retain imprints of their parent stars and ejection histories, though in dense clusters scatterings can dominate the evolution; the analysis relies on an idealized infinite-sea model and highlights the long timescales required to reach true equipartition for $m_p\ll m$.

Abstract

We investigate the velocity evolution of free-floating planets and interstellar objects (``free floaters'') through gravitational scatterings by field stars (with the stellar mass $m$ much larger than the mass of the floater, $m_p$). We show that the equilibrium velocity -- where dynamical friction balances stochastic acceleration -- is given by $σ\sqrt{2\ln(m/m_p)}$ (where $σ$ is the velocity disperson of the field stars), diverging from the standard energy equipartition scaling. While the timescale to reach this equilibrium is prohibitively long, we find that slow floaters ($v \lesssim σ$) undergo mass-independent acceleration, doubling their velocities within a few relaxation times. Consequently, free floaters initially following the Maxwellian distribution of their parent stars develop distinctly non-Maxwellian velocity distributions on a relaxation timescale. Since the relaxation time of the Galactic disk is longer than the age, our results suggest that the kinematics of low-mass free floaters in the disk may preserve signatures of their parent stars and ejection history.

Figures (3)

• Figure 1: The time evolution of the velocity (in units of $\sqrt{2}\sigma$) of a free floater due to gravitational scatterings by background stars. The time is in units of the relaxation time (see Eq. \ref{['eq:t_r']}). The different colored curves correspond to different initial velocities $(\tilde{v}_0 = 0, 0.5, 1,2)$. The solid lines are for $m_p/m = 10^{-4}$, and the dashed lines for $m_p/m = 10^{-8}$. Note that the dashed lines overlap the solid lines before they diverge at $\tilde{t} \sim 10^4$.
• Figure 2: The 1D velocity distribution of $F(\tilde{v},\tilde{t})$ at different times assuming that at $\tilde{t} = 0$ an initial population is born with $f_0 \propto \exp(-\tilde{v}_0^2)$.
• Figure 3: The velocity distribution of the "observed" free floaters $\bar{F}(\tilde{v},\tilde{T})$ at several different time ${\tilde{T}}$ (as labeled), assuming that the floaters were produced during $\tt\in (0,{\tilde{T}})$ at a constant rate with the same initial distribution at birth. The velocity is in units of $\sqrt{2}\sigma$, and time in units of $t_r$ (see Eq. \ref{['eq:t_r']}).