To determine the radius of the satellite's orbit around the Sun, we can use Kepler's Third Law of Planetary Motion, which states:

$T^2 = \frac{4\pi^2}{G M_{\text{sun}}} r^3$

Where:

• $T$ is the orbital period (2.00 days),
• $r$ is the orbital radius,
• $G$ is the gravitational constant ($6.67430 \times 10^{-11} \, \text{m}^3/\text{kg}\cdot\text{s}^2$),
• $M_{\text{sun}}$ is the mass of the Sun ($1.989 \times 10^{30} \, \text{kg}$).

Step 1: Convert the period $T$ to seconds

Since the period is given in days, we need to convert it to seconds: $T = 2.00 \, \text{days} \times 24 \times 60 \times 60 = 172800 \, \text{seconds}$

Step 2: Rearrange Kepler's Third Law to solve for $r$

$r^3 = \frac{G M_{\text{sun}} T^2}{4\pi^2}$

Step 3: Plug in the known values

$r^3 = \frac{(6.67430 \times 10^{-11}) (1.989 \times 10^{30}) (172800)^2}{4\pi^2}$

Step 4: Solve for $r$

First, compute the right-hand side and then take the cube root.

$r^3 = 1.18 \times 10^{35} \, \text{m}^3$ $r = \sqrt[3]{1.18 \times 10^{35}} = 4.91 \times 10^{11} \, \text{m}$

Thus, the radius of the satellite's orbit is approximately:

$r \approx 4.91 \times 10^{11} \, \text{m}$

Final Answer:

$r \approx 4.91 \times 10^{11} \, \text{m}$

Would you like further details or have any questions?

Here are 5 related questions:

1. How does Kepler's Third Law apply to planetary orbits?
2. What is the significance of the gravitational constant $G$?
3. How would the orbital radius change if the period doubled?
4. What assumptions are made in using Kepler's Law for this calculation?
5. How does the mass of the Sun affect orbital mechanics?

Tip: Kepler's laws apply not only to planets but to any object orbiting a larger mass, like satellites around planets or stars!