Analytical estimates for heliocentric escape of satellite ejecta

TL;DR

The paper presents a general analytic framework to determine when satellite ejecta can escape the planet–satellite system and enter heliocentric space, using a patched-conics approach complemented by the circular restricted three-body problem (CRTBP). The key result is that escape thresholds depend on two dimensionless parameters: the satellite–planet mass ratio $\mu$ and the ratio of the satellite’s orbital speed to its surface escape speed $v_{orb,S}/v_{esc,S}$, with a clear expression for the required launch speed in the patched-conics picture and a necessary (but less restrictive) CRTBP bound via the Jacobi integral at $L_2$. The Earth–Moon system sits near a critical regime, enabling a narrow range of near-threshold ejecta to reach heliocentric orbits, while most other Solar System satellites require launch speeds well above $v_{esc,S}$; Moon migration does not qualitatively alter this prospect. The framework also informs expectations for binary asteroids and supports the plausibility of lunar material contributing to near-Earth objects and interplanetary space through long-term dynamical evolution. Overall, the work provides compact, physically grounded criteria to assess ejecta fates across planet–satellite systems and highlights the Earth–Moon system as a uniquely favorable case for generating heliocentric ejecta.

Abstract

We present a general analytic framework to assess whether impact ejecta launched from the surface of a satellite can escape the gravitational influence of the planet--satellite system and enter heliocentric orbit. Using a patched-conic approach and defining the transition to planetocentric space via the Hill sphere or sphere of influence, we derive thresholds for escape in terms of the satellite-to-planet mass ratio and the ratio of the satellite's orbital speed to its escape speed. We identify three dynamical regimes for ejecta based on residual speed and launch direction. We complement this analysis with the circular restricted three-body problem (CR3BP), deriving a necessary escape condition from the Jacobi integral at $\mathrm{L_{2}}$ and showing that it is consistent with the patched-conic thresholds. Applying our model to the Earth--Moon system reveals that all three outcomes--bound, conditional, and unbound--are accessible within a narrow range of launch speeds. This behavior is not found in other planetary satellite systems, but may occur in some binary asteroids. The framework also shows that the Moon's tidal migration has not altered its propensity to produce escaping ejecta, reinforcing the plausibility of a lunar origin for some near-Earth asteroids.

Figures (3)

• Figure 1: The threshold launch velocities required for escape to heliocentric orbit. Curves in different colors represent different cases of the satellite-to-planet mass ratio, $\mu$ (purple, yellow and green for $\mu_\mathrm{crit}=0.0147$, $10^{-3}$, and $10^{-5}$, respectively). The dashed curves represent the minimum launch speed to escape with a vertical launch from the leading side, and the solid curves represent the minimum launch speed to escape with a vertical launch from the trailing side. Launch velocities above the solid curves always escape, whereas launch velocities below the dashed curves do not; launch velocities in-between the solid and dashed curves may escape to heliocentric orbit depending upon launch location and direction of launch. The vertical lines indicate parameter values of several satellite--planet pairs.
• Figure 2: Ratio of residual speed at the satellite's Hill sphere and the satellite orbital speed, versus the satellite-to-planet mass ratio $\mu$. The curve shows the bounding case of particles launched with speed equal to the satellite's escape velocity. Shaded regions represent the regions of different outcomes for ejecta launched above escape velocity. The stippled region is the range of $\mu>0.2$ where the patched-conics approximation becomes less effective.
• Figure 3: Forbidden regions ( shaded) within the Earth--Moon system for $C_\mathrm{J} = C_{\mathrm{L_{2}}}$ ( left) and the location of the Lagrange equilibrium points (denoted by five-pointed stars). Decreasing the Jacobi integral below $C_{\mathrm{L_2}}$ ( right) opens the corridor or bottleneck around $\mathrm{L_2}$ enabling escape from the Earth--Moon system.