Hill sphere

Remarks

Origin of the formula: in the circular restricted three-body problem with mass parameter $\mu = m / (M + m)$, the Lagrange points $L_1$ and $L_2$ lie at distance $a \left( (\mu / 3)^{1/3} + O(\mu^{2/3}) \right)$ from the small body, as $\mu \to 0$. Some authors write $m / (3 (M + m))$ instead of $m / (3M)$, which agrees to first order when $m \ll M$. For an elliptic orbit with semi-major axis $a$ and eccentricity $e$, the radius at pericentre is usually taken as $a (1 - e) \sqrt[3]{m / (3M)}$.

Example: for the Earth around the Sun, $m / M \approx 3.0 \times 10^{-6}$ and $a \approx 1.496 \times 10^8$ km. Then $\sqrt[3]{m / (3M)} \approx \sqrt[3]{1.0 \times 10^{-6}} = 0.01$, and $r_H \approx 1.5 \times 10^6$ km. The Moon, at about $3.84 \times 10^5$ km, lies well inside.

Non-example: for two stars of equal mass the condition $m \ll M$ fails, and the formula does not give a meaningful region of dominance.

The Hill sphere is only an approximation. Numerical studies show that orbits of satellites are typically stable only well inside it, out to a fraction of $r_H$.