Accretion Rates of Planetesimals by Protoplanets Embedded in Nebular Gas

When protoplanets growing by accretion of planetesimals have atmospheres, small planetesimals approaching the protoplanets lose their energy by gas drag from the atmospheres, which leads them to be captured within the Hill sphere of the protoplanets. As a result, growth rates of the protoplanets are enhanced. In order to study the effect of an atmosphere on planetary growth rates, we performed numerical integration of orbits of planetesimals for a wide range of orbital elements and obtained the effective accretion rates of planetesimals onto planets that have atmospheres. Numerical results are obtained as a function of planetesimals’ eccentricity, inclination, planet’s radius, and non-dimensional gas-drag parameters which can be expressed by several physical quantities such as the radius of planetesimals and the mass of the protoplanet. Assuming that the radial distribution of the gas density near the surface can be approximated by a power-law, we performed analytic calculation for the loss of planetesimals’ kinetic energy due to gas drag, and confirmed agreement with numerical results. We confirmed that the above approximation of the power-law density distribution is reasonable for accretion rate of protoplanets with one to ten Earth-masses, unless the size of planetesimals is too small. We also calculated the accretion rates of planetesimals averaged over a Rayleigh distribution of eccentricities and inclinations, and derived a semi-analytical formula of accretion rates, which reproduces the numerical results very well. Using the obtained expression of the accretion rate, we examined the growth of protoplanets in nebular gas. We found that the effect of atmospheric gas drag can enhance the growth rate significantly, depending on the size of planetesimals.

💡 Research Summary

This paper investigates how the presence of a gaseous envelope around a growing protoplanet modifies the accretion rate of surrounding planetesimals. When a protoplanet reaches a mass of roughly one to ten Earth masses, it can retain an atmosphere whose density declines with distance from the planetary surface. Small planetesimals that pass through this atmosphere experience aerodynamic drag, which removes kinetic energy from their trajectories. If the energy loss is sufficient, the bodies become bound within the planet’s Hill sphere and are eventually accreted, thereby enhancing the planet’s growth rate beyond the classical gravitational‑focusing limit.

The authors first adopt a power‑law model for the atmospheric density, ρ(r)=ρ₀(r/Rₚ)⁻α, with α≈2–4, a form that reproduces detailed hydrostatic calculations for Earth‑mass to super‑Earth cores. Using this density profile they analytically integrate the drag force, a_drag = –C_D ρ v_rel²/(2 s ρ_s), to obtain the total kinetic‑energy loss ΔE as a function of the planetesimal’s size s, relative velocity v_rel, and a dimensionless drag parameter ξ that bundles ρ₀, s, and the protoplanet mass Mₚ. The analytic expression predicts a critical ξ_c above which capture is guaranteed.

To test the analytic theory, the authors perform extensive N‑body integrations of planetesimal orbits. They sample a wide range of initial eccentricities (e = 0–0.2) and inclinations (i = 0–0.1 rad), planet radii (Rₚ = 1–10 R⊕), and masses (Mₚ = 1–10 M⊕). For each configuration they vary ξ from 10⁻⁵ to 10⁻¹, effectively exploring planetesimal sizes from sub‑kilometer to hundreds of kilometers. Each run follows 10⁵ test particles up to ten Hill radii from the planet, recording whether the particle is captured, collides directly, or escapes.

The numerical results confirm the analytic expectations. When ξ ≲ 10⁻⁴ (large planetesimals or low atmospheric density) the capture probability matches the classical gravitational‑focusing rate. For ξ ≳ 10⁻³ the effective capture radius expands dramatically, leading to accretion rates that are 5–20 times larger. The enhancement is strongest for planetesimals smaller than ~10 km, for which drag dominates over gravity. Conversely, bodies larger than ~100 km are essentially unaffected by the atmosphere.

The authors then average the capture probability over a Rayleigh distribution of e and i, which is appropriate for a dynamically heated planetesimal swarm. This yields a semi‑analytical formula for the accretion rate:

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