Physics › 🌌 Astrophysics & Space

Orbital Motion & Kepler’s Laws

Every orbit is just falling forever — moving sideways fast enough to keep missing the ground.

High schoolUni Year 1
Orbital Motion & Kepler’s Laws — illustration
Illustrative hero image.
💡
The big idea: Gravity is the only force at work in every orbit: it isn't holding planets up, it's making them fall continuously while their sideways motion keeps carrying them past the point they'd otherwise hit. Kepler's three laws describe the shape, pace, and scale of that endless fall with remarkable precision, and Newton's law of gravitation explains why they work.
🎯 By the end, you'll be able to
  • explain why an orbit is really a continuous state of free-fall rather than something 'floating' above gravity
  • state Kepler's three laws and describe what each one tells you about a planet's or satellite's path
  • calculate the orbital speed and period of a circular orbit from its radius and the central body's mass
  • use Kepler's third law to relate a planet's or satellite's orbital period to its distance from the body it orbits
📎 You should already know
  • Newton's Laws of Motion
  • Circular Motion & Centripetal Force
  • Basic algebra (square roots, exponents, and rearranging equations)

Falling Forever

Picture Isaac Newton's thought experiment: fire a cannonball horizontally from a very tall mountain. Fire it gently, and gravity curves its path down to the ground nearby. Fire it harder, and it lands farther away. Fire it fast enough, and something wonderful happens — the ground curves away from the cannonball exactly as fast as the ball falls toward it. It never lands. That is an orbit.

The International Space Station, the Moon, and every planet around the Sun are all doing the same thing: falling continuously, but moving sideways fast enough that they keep missing whatever they're falling toward.

🔑 The Core Idea

An orbit isn't something held up against gravity — it's a state of constant free-fall. Gravity supplies exactly the sideways-pulling (centripetal) force needed to bend a straight-line path into a closed loop. No orbit needs an engine to keep going; an engine is only needed to get into that state, or to change it.

\[ F = \dfrac{GMm}{r^2} \]
Newton's law of universal gravitation: every mass M pulls every mass m toward it with a force that fades with the square of the distance r between them.
\[ v = \sqrt{\dfrac{GM}{r}} \]
Setting gravity equal to the centripetal force needed for a circle gives the speed a body needs to hold a stable circular orbit at radius r around mass M.
\[ T^2 = \dfrac{4\pi^2}{GM}\,a^3 \]
Kepler's third law, derived from Newton's gravitation: the orbital period squared scales with the cube of the orbit's semi-major axis a — bigger orbits take disproportionately longer.
🎮 Interactive: Build Your Own Orbit LIVE
Drag to set a starting speed and watch gravity bend the path — too slow and it spirals in, too fast and it escapes, and somewhere in between you get a stable ellipse.

Why Ellipses, Not Circles

A perfect circular orbit is possible, but it's a special case — one exact speed at one exact radius. Give a body slightly more or less than that speed and gravity still holds onto it, but now the path stretches into an ellipse: the body swings in closer at one point (perihelion) and farther out at another (aphelion), trading speed for distance the whole way around.

This is Kepler's first law: every orbit is an ellipse with the central body sitting at one focus, not the center. Kepler's second law describes the trade-off precisely — a line from the Sun to the planet sweeps out equal areas in equal times, so the planet moves fastest when closest (perihelion) and slowest when farthest (aphelion). Nothing pushes it faster; it's the same conservation of angular momentum that speeds up a spinning skater when she pulls her arms in.

📝 Worked example: A geostationary satellite must have an orbital period equal to Earth's rotation, 24 hours (86,400 s), so it appears to hover over the same spot. Using \(GM_{Earth} \approx 3.986\times10^{14}\ \text{m}^3/\text{s}^2\), find the radius of its orbit and its altitude above Earth's surface (Earth's radius \(\approx 6{,}371\) km).
  1. Start from Kepler's third law solved for r: \(r^3 = \dfrac{GM\,T^2}{4\pi^2}\).
  2. Plug in \(T = 86{,}400\ \text{s}\), so \(T^2 \approx 7.465\times10^{9}\ \text{s}^2\).
  3. \(r^3 = \dfrac{(3.986\times10^{14})(7.465\times10^{9})}{4\pi^2} \approx 7.54\times10^{22}\ \text{m}^3\).
  4. Taking the cube root: \(r \approx 4.22\times10^{7}\ \text{m} = 42{,}200\ \text{km}\).
  5. Altitude above the surface: \(42{,}200 - 6{,}371 \approx 35{,}800\ \text{km}\).
✓ About 42,200 km from Earth's center — roughly 35,800 km above the surface — matching the real geostationary belt (about 35,800 km) where communication satellites park.
📝 Worked example: For anything orbiting the Sun, Kepler's third law simplifies beautifully if you measure time in years and distance in astronomical units (AU, Earth's orbital radius): \(T^2 = a^3\). An object orbits the Sun at an average distance of 4 AU. What is its orbital period?
  1. Apply the simplified law: \(T^2 = a^3 = 4^3 = 64\).
  2. Solve for T: \(T = \sqrt{64} = 8\).
✓ 8 years — four times farther out than Earth, but the period is eight times longer, not four, because period scales with distance to the 3/2 power.
⚠️ Common Trap

It's tempting to think a satellite "escapes gravity" once it's in orbit — it hasn't. Gravity at the International Space Station's altitude is only about 10% weaker than at Earth's surface; astronauts float because they and their station are falling together, not because gravity switched off. The other common slip: assuming faster orbits are farther out. It's the opposite — the closer an orbit, the faster the object must move to keep from falling in, which is why the Moon creeps along on a month-long orbit while the ISS circles Earth in about 90 minutes.

Check your understanding

1. According to Kepler's first law, where does the Sun sit relative to a planet's elliptical orbit?
Kepler's first law states orbits are ellipses with the central body (like the Sun) at one focus, not the geometric center — that's why perihelion and aphelion distances differ.
2. A planet orbits its star at 9 AU, using the Sun-centered simplification \(T^2 = a^3\) (T in years, a in AU). What is its orbital period?
\(T^2 = 9^3 = 729\), so \(T = \sqrt{729} = 27\) years.
3. Where does a planet move fastest along its elliptical orbit?
Kepler's second law (equal areas in equal times) means a planet must move fastest when closest to the Sun (perihelion) to sweep out the same area it sweeps more slowly when farther away.
4. If a satellite's circular orbital radius is doubled, what happens to its orbital speed \(v = \sqrt{GM/r}\)?
Since \(v \propto 1/\sqrt{r}\), doubling r multiplies v by \(1/\sqrt{2} \approx 0.71\) — the satellite moves slower, not faster, at a higher altitude.
✅ Key takeaways
  • Orbits are continuous free-fall: gravity supplies the exact sideways pull needed to curve a path into a closed loop.
  • Kepler's three laws describe the shape (ellipse, one focus), the pacing (equal areas, fastest at perihelion), and the scaling (T² ∝ a³) of every orbit.
  • Orbital speed \(v=\sqrt{GM/r}\) and period both depend only on the central mass and orbital radius, not on the orbiting object's own mass.
  • Closer orbits are faster, not slower — the Moon takes weeks to circle Earth while the ISS takes about 90 minutes.