For a satellite to maintain a stable circular orbit, the gravitational pull of the planet must provide exactly enough centripetal force to keep the satellite moving in its curved path.

The Physics of Circular Orbits

When an object orbits a planet, it is essentially in a state of constant free-fall. However, because it has enough horizontal (tangential) velocity, it constantly “misses” the ground, following the curvature of the planet.

Orbital Velocity ($v_t$)

By setting the Force of Gravity equal to the Centripetal Force, we can derive the speed required to stay in a circular orbit:

\[\frac{G M m}{r^2} = \frac{m v^2}{r}\]

Solving for $v$ gives the Orbital Velocity formula:

\[v = \sqrt{\frac{GM}{r}}\]

$M$: Mass of the central body (e.g., Earth).

$r$: Distance from the center of the planet to the satellite.

Note: The mass of the satellite ($m$) cancels out—it doesn’t matter how heavy the satellite is!

Orbital Period ($T$)

The period is the time it takes to complete one full revolution ($2\pi r$). Using the relationship $v = \frac{2\pi r}{T}$, we can derive the period:

\[T = 2\pi \sqrt{\frac{r^3}{GM}}\]

This confirms Kepler’s Third Law: the square of the period ($T^2$) is proportional to the cube of the radius ($r^3$).

Interactive: Orbital Mechanics Vocabulary

Match the variables and terms to their roles in defining satellite motion.

Match the terms and variables for orbital speed and period to their correct descriptions.

Interactive: Orbital Velocity Visualizer

Use the slider to change the orbital radius. Observe how the velocity vector (green arrow) shortens as you move further away, and the time to complete one orbit increases.

120 km (scaled)
Velocity: --
Period: --

Key Insights

Inverse Relationship: As the radius ($r$) increases, the required orbital velocity ($v$) decreases. This is why outer planets move much slower than inner planets.

Mass Independence: Notice that the mass of the satellite does not appear in the velocity formula. A grain of sand and a space station orbit at the same speed if they are at the same altitude.

💡 Quick Concept Check:

If a satellite moves to a higher orbit (increasing r), what happens to its orbital speed and its period?

Click to Reveal Answer
Its orbital speed decreases (since v is inversely proportional to the square root of r) and its period increases (since it has a longer path to travel and is moving slower).
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