Question: Instruction: Set up 4–5 different orbits by changing the initial distance ($r$) or preset planet configurations. Record the orbital period ($T$) and average orbital radius ($r$), then calculate $r^3$, $T^2$, and the ratio $\frac{T^2}{r^3}$.
1. **Problem Statement:** We are given data for orbits with semi-major axis ($a$) in astronomical units (AU) and orbital period ($T$) in years. We need to calculate $a^3$, $T^2$, and the constant ratio $k = \frac{T^2}{a^3}$ for each orbit.
2. **Formula Used:**
- Cube of radius: $$a^3 = a \times a \times a$$
- Square of period: $$T^2 = T \times T$$
- Constant ratio: $$k = \frac{T^2}{a^3}$$
3. **Given Data for Orbit 1:**
- Semi major axis: $a = 2.25$ AU
- Period: $T = 3.42$ years
4. **Calculate $a^3$ for Orbit 1:**
$$a^3 = 2.25 \times 2.25 \times 2.25 = 11.390625$$
5. **Calculate $T^2$ for Orbit 1:**
$$T^2 = 3.42 \times 3.42 = 11.6964$$
6. **Calculate constant ratio $k$ for Orbit 1:**
$$k = \frac{11.6964}{11.390625} \approx 1.027$$
7. **Summary for Orbit 1:**
| Orbit | $a$ (AU) | $a^3$ | $T$ (years) | $T^2$ | $k = \frac{T^2}{a^3}$ |
|-------|----------|-------|-------------|-------|-----------------------|
| 1 | 2.25 | 11.3906 | 3.42 | 11.6964 | 1.027 |
8. **Instructions for Orbits 2 to 5:**
- Repeat steps 3 to 6 for each orbit by changing $a$ and $T$ values.
- Calculate $a^3$, $T^2$, and $k$ similarly.
This process helps verify Kepler's third law where $k$ should be approximately constant for all orbits.