kepler's third law

Question

Two planets have orbital radii r₁ and r₂ = 4r₁. What is the ratio of their time periods T₁/T₂?

RankGuru · Accepted Answer

1. **Kepler's Rule**: The square of the orbital period is proportional to the cube of the semi-major axis.
   - $T^2 \propto r^3 \implies T \propto r^{3/2}$.
2. **Ratio Format**: $\frac{T_1}{T_2} = \left( \frac{r_1}{r_2} \right)^{3/2}$.
3. **Substitute $r_2 = 4r_1$**:
   - $\frac{T_1}{T_2} = \left( \frac{1}{4} \right)^{3/2}$.
4. **Simplify**:
   - $\left( \frac{1}{4} \right)^{1/2} = \frac{1}{2}$.
   - $(\frac{1}{2})^3 = \frac{1}{8}$.
5. **Result**: The second planet takes 8 times longer to complete an orbit.

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