Orbital Velocity

For a satellite in a circular orbit of radius $r$ around mass $M$, gravity provides the centripetal force.

Concept

$\frac{GMm}{r^2} = \frac{mv_{\rm orb}^2}{r} \Rightarrow v_{\rm orb} = \sqrt{\frac{GM}{r}}$

For a low Earth orbit ($r \approx R$): $v_{\rm orb} \approx 7.9\,\mathrm{km/s}$.

Period $T = \frac{2\pi r}{v_{\rm orb}} = 2\pi\sqrt{r^3/GM}$ — Kepler's 3rd law.

Worked Example

ISS at 400 km altitude: $r=6771\,\mathrm{km}$, $v_{\rm orb}\approx 7.66\,\mathrm{km/s}$, $T\approx 92$ min.

Common Confusions

• Orbital velocity DECREASES with radius — high orbits are slower.
• $v_{\rm orb}$ is a SPEED tangent to orbit; direction matters for stability.

Key Takeaways

• $v_{\rm orb}=\sqrt{GM/r}$, $T \propto r^{3/2}$.
• $v_{\rm orb}=v_e/\sqrt 2$.
• Higher orbits ⇒ slower satellites.