Year 12 / Gravity and motion
Gravity supplies the turn.
For a circular satellite orbit, equate GMm/r² to mv²/r. Satellite mass cancels: = GM/r. Combining this with v = 2πr/T gives /r³ = 4π²/(GM).
Kepler’s laws describe elliptical planetary orbits with the central body at one focus, equal areas swept in equal times, and the period–semimajor-axis relationship for objects around the same dominant central mass. The simple circular formula uses the orbital radius.
Orbit size and period
Explore the relationship
Circular-orbit scaling T² ∝ r³/M. Satellite mass cancels.
Period / reference period
Tap or focus a plotted point to read its values.
View plotted data as a table
| Series | Orbital radius / reference radius | Period / reference period |
|---|---|---|
| Period / reference period | 0.5 | 0.353553 |
| Period / reference period | 0.558333 | 0.417196 |
| Period / reference period | 0.616667 | 0.484257 |
| Period / reference period | 0.675 | 0.554569 |
| Period / reference period | 0.733333 | 0.627989 |
| Period / reference period | 0.791667 | 0.704391 |
| Period / reference period | 0.85 | 0.783661 |
| Period / reference period | 0.908333 | 0.865701 |
| Period / reference period | 0.966667 | 0.950419 |
| Period / reference period | 1.025 | 1.03773 |
| Period / reference period | 1.08333 | 1.12757 |
| Period / reference period | 1.14167 | 1.21986 |
| Period / reference period | 1.2 | 1.31453 |
| Period / reference period | 1.25833 | 1.41154 |
| Period / reference period | 1.31667 | 1.51082 |
| Period / reference period | 1.375 | 1.61233 |
| Period / reference period | 1.43333 | 1.71601 |
| Period / reference period | 1.49167 | 1.82183 |
| Period / reference period | 1.55 | 1.92973 |
| Period / reference period | 1.60833 | 2.03969 |
| Period / reference period | 1.66667 | 2.15166 |
| Period / reference period | 1.725 | 2.2656 |
| Period / reference period | 1.78333 | 2.38149 |
| Period / reference period | 1.84167 | 2.49929 |
| Period / reference period | 1.9 | 2.61897 |
| Period / reference period | 1.95833 | 2.7405 |
| Period / reference period | 2.01667 | 2.86386 |
| Period / reference period | 2.075 | 2.98901 |
| Period / reference period | 2.13333 | 3.11593 |
| Period / reference period | 2.19167 | 3.2446 |
| Period / reference period | 2.25 | 3.375 |
| Period / reference period | 2.30833 | 3.5071 |
| Period / reference period | 2.36667 | 3.64087 |
| Period / reference period | 2.425 | 3.77631 |
| Period / reference period | 2.48333 | 3.91338 |
| Period / reference period | 2.54167 | 4.05208 |
| Period / reference period | 2.6 | 4.19237 |
| Period / reference period | 2.65833 | 4.33425 |
| Period / reference period | 2.71667 | 4.4777 |
| Period / reference period | 2.775 | 4.62269 |
| Period / reference period | 2.83333 | 4.76921 |
| Period / reference period | 2.89167 | 4.91725 |
| Period / reference period | 2.95 | 5.06679 |
| Period / reference period | 3.00833 | 5.21782 |
| Period / reference period | 3.06667 | 5.37032 |
| Period / reference period | 3.125 | 5.52427 |
| Period / reference period | 3.18333 | 5.67967 |
| Period / reference period | 3.24167 | 5.8365 |
| Period / reference period | 3.3 | 5.99475 |
| Period / reference period | 3.35833 | 6.1544 |
| Period / reference period | 3.41667 | 6.31544 |
| Period / reference period | 3.475 | 6.47787 |
| Period / reference period | 3.53333 | 6.64166 |
| Period / reference period | 3.59167 | 6.80682 |
| Period / reference period | 3.65 | 6.97332 |
| Period / reference period | 3.70833 | 7.14115 |
| Period / reference period | 3.76667 | 7.31031 |
| Period / reference period | 3.825 | 7.48078 |
| Period / reference period | 3.88333 | 7.65256 |
| Period / reference period | 3.94167 | 7.82564 |
| Period / reference period | 4 | 8 |
Explore: Increase orbital radius. Track the period and compare with r to the power 3/2.
A second circular satellite orbits at four times the radius of the first.
- T₂/T₁ = (r₂/r₁)^().
- Its period is eight times as long.
Assumed knowledge
Centripetal acceleration and square roots.
Learning checkpoints & source
YOUR LEARNING CHECKPOINT- Connect gravity with circular orbital motion.
- Apply Kepler’s laws and period–radius scaling.
READY TO TRY IT?