Circular Orbit Velocity⚠ unverified

Understanding the Parameters

• Radius $r$ — measured from the centre, not the surface — a common trap. LEO at 400 km altitude has $r = 6371 + 400 = 6771\ \text{km}$ from Earth's centre. The default encodes exactly this.

• Higher orbits are slower: $v \propto 1/\sqrt{r}$ — counter to intuition, a bigger orbit means a lower speed. LEO ($\sim7.7\ \text{km/s}$) is fast; a geostationary satellite ($r \approx 42{,}164\ \text{km}$) circles at only $\sim3.07\ \text{km/s}$; the Moon ($r \approx 384{,}400\ \text{km}$) at just $\sim1.0\ \text{km/s}$. The weaker gravity at altitude needs less speed to balance.

• The gravitational parameter $\mu = GM$ — everything depends on the central body only; the satellite's own mass cancels (see the derivation). $\mu$ is measured directly and precisely from tracking, which is why astrodynamics uses it rather than $G$ and $M$ apart.

• Relation to the other key speeds — escape velocity is exactly $\sqrt2$ times circular ($\sqrt{2\mu/r}$, see Escape Velocity); the circular-orbit period follows from $T = 2\pi r/v$, recovering Kepler's third law (Orbital Period). Circular speed is the anchor the other orbital quantities are measured against.

• A circle is a knife-edge — a circular orbit requires the speed and a purely horizontal (tangential) velocity. Any speed error or radial component turns it into an ellipse with the same or a different energy; perfectly circular orbits are an idealisation approached, never exactly held.

Derivation (Approaching a Proof)

For a circular orbit the geometry is simple: the body moves at constant speed $v$ on a circle of radius $r$, so it needs a constant inward (centripetal) acceleration $v^2/r$ (Centripetal Force). The only force available is gravity, pulling inward with the inverse-square law. Newton's second law (Newton's Second Law) in the radial direction sets gravity equal to the mass times the centripetal acceleration:

$$\underbrace{\frac{\mu m}{r^2}}_{\text{gravity}} = \underbrace{\frac{m v^2}{r}}_{\text{centripetal}}.$$

The satellite's mass $m$ cancels from both sides — the reason all objects orbit at the same speed regardless of mass:

$$\frac{\mu}{r^2} = \frac{v^2}{r}.$$

Multiply through by $r$ and take the square root:

$$v^2 = \frac{\mu}{r} \quad\Longrightarrow\quad v = \sqrt{\frac{\mu}{r}}. \qquad\blacksquare$$

Consistency with vis-viva. A circular orbit has $r = a$ (the radius is the semi-major axis). Substituting into the vis-viva equation $v = \sqrt{\mu(2/r - 1/a)}$ gives $v = \sqrt{\mu(2/r - 1/r)} = \sqrt{\mu/r}$ — the same result, confirming the circular case is vis-viva's simplest instance.

Dimensional check. $$\sqrt{\frac{\mu}{r}} = \sqrt{\frac{\text{m}^3/\text{s}^2}{\text{m}}} = \sqrt{\frac{\text{m}^2}{\text{s}^2}} = \frac{\text{m}}{\text{s}}. \checkmark$$

History and Development

• Newton's cannonball (1687). In A Treatise of the System of the World, Isaac Newton imagined firing a cannonball horizontally ever faster from a mountaintop: too slow and it falls to Earth; at just the right speed it "falls" around the curve of the Earth forever — a circular orbit. This thought experiment, and the force-balance above, first showed orbits are just projectile motion that never hits the ground.

• Kepler's third law recovered. Circular speed plus $T = 2\pi r/v$ yields $T^2 = 4\pi^2 r^3/\mu$, exactly Kepler's $T^2 \propto a^3$ (Orbital Period) — Newton's gravitation reproducing Kepler's empirical harmony.

• The first satellites. Reaching $\sim7.9\ \text{km/s}$ (surface circular speed) was the barrier the early space programmes had to clear; Sputnik 1 (1957) achieved it, and every satellite since is a Newtonian cannonball fired fast enough to keep missing the Earth. Geostationary orbit — the radius where the circular period equals one sidereal day — was foreseen by Herman Potočnik (1928) and popularised by Arthur C. Clarke (1945), and is now the crowded home of communications satellites.

Related Concepts: Orbital Velocity, Escape Velocity, Orbital Period, Circular Orbital Velocity, Centripetal Force, Gravitational Force, Newton's Second Law, Vis Viva Energy

Notes: $v \propto 1/\sqrt{r}$ — higher orbits are slower (LEO 7.7, GEO 3.07, Moon 1.0 km/s). Radius is from the centre (LEO 400 km alt → $r=6771$ km). Satellite mass cancels. Escape is $\sqrt2\times$ this. Cross-category duplicate of the Physics Circular Orbital Velocity (which uses $G$,$M$ separately; $\mu=GM$). The $r=a$ special case of Orbital Velocity.