Hill Sphere Gravitational Dominance Radius
Calculates the radius of the gravitational sphere of influence within which a planet or moon can permanently retain natural satellites despite the tidal pull of the parent star.
\[ r_H \approx a \cdot (1 - e) \cdot \sqrt[3]{\frac{m}{3 M}} \]
Plain Text Formula:
r_H = a * (1 - e) * ((m / (3 * M))^(1/3))
📐 Variable Definitions & Parameters
| Symbol | Variable Name | Standard Unit | Physical Definition |
|---|---|---|---|
| \[ r_H \] | Hill Sphere Radius | km or AU | Boundary of stable satellite orbits |
| \[ a \] | Semi-Major Axis | km or AU | Orbital distance from primary star |
| \[ e \] | Orbital Eccentricity | dimensionless | Eccentricity of planetary orbit |
| \[ m \] | Secondary Body Mass | kg | Mass of planet or moon |
| \[ M \] | Primary Body Mass | kg | Mass of central star or host planet |
💡 Worked Engineering Example
Problem Statement:
Estimate Earth's Hill Sphere radius orbiting the Sun (a = 1.496e8 km, m/M = 3.003e-6, e = 0.0167).
Step-by-Step Solution:
- Cube root term: (3.003e-6 / 3)^(1/3) = (1.001e-6)^(1/3) = 0.010003
- Multiply by a*(1-e): 1.496e8 * (1 - 0.0167) * 0.010003
- r_H ≈ 1,471,000 km = 0.0098 AU
🎯 Result: Earth's Hill sphere extends to approximately 1.5 million kilometers (well beyond the Moon's 384,400 km orbit).