A satellite orbits the sun with a period of 2.00 days. What is the radius of its orbit?

To find the radius of the orbit of the satellite, we will use Kepler's third law of planetary motion. Kepler's third law states that the square of the orbital period \( T \) of a planet (or satellite) is proportional to the cube of the semi-major axis \( r \) of its orbit:

\[
T^2 = \frac{4\pi^2}{GM} r^3
\]

Where:
- \( T \) is the orbital period (in seconds),
- \( r \) is the radius of the orbit,
- \( G \) is the gravitational constant \( (6.67430 \times 10^{-11} \, \text{m}^3/\text{kg}\cdot\text{s}^2) \),
- \( M \) is the mass of the Sun \( (1.989 \times 10^{30} \, \text{kg}) \).

### Step 1: Convert the orbital period to seconds.
Given:
\[
T = 2.00 \text{ days}
\]
Convert days to seconds:
\[
T = 2.00 \times 24 \times 3600 = 172,800 \text{ seconds}
\]

### Step 2: Rearrange Kepler's third law to solve for \( r \).
Rearranging the formula:
\[
r^3 = \frac{GMT^2}{4\pi^2}
\]

### Step 3: Substitute the values.
\[
r^3 = \frac{(6.67430 \times 10^{-11})(1.989 \times 10^{30})(172,800)^2}{4\pi^2}
\]

Calculating this expression:

1. Calculate \( GM \):
   \[
   GM = (6.67430 \times 10^{-11})(1.989 \times 10^{30}) = 1.327 \times 10^{20} \text{m}^3/\text{s}^2
   \]

2. Calculate \( T^2 \):
   \[
   T^2 = (172,800)^2 = 2.985 \times 10^{10} \text{s}^2
   \]

3. Compute \( r^3 \):
   \[
   r^3 = \frac{(1.327 \times 10^{20})(2.985 \times 10^{10})}{4\pi^2} = 6.47 \times 10^{20} \text{m}^3
   \]

### Step 4: Take the cube root to find \( r \).
\[
r = \sqrt[3]{6.47 \times 10^{20}} \approx 8.67 \times 10^{6} \text{km}
\]

Thus, the radius of the orbit is approximately \( 8.67 \times 10^{6} \text{km} \).

Would you like further details or have questions about this calculation?

### Related Questions
1. How does Kepler's third law help in understanding satellite motion?
2. What factors can influence the period of a satellite's orbit around the Sun?
3. How do orbital radius and speed correlate for satellites orbiting the Sun?
4. What is the relationship between gravitational force and orbital motion?
5. How would the period change if the satellite were further from the Sun?

### Tip
Remember, Kepler's laws apply to any two-body system where one body is much more massive than the other, such as a planet and a satellite.

Find the radius of a satellite's orbit around the sun with a 2-day period using Kepler's Third Law of Planetary Motion. This solution involves orbital mechanics, gravitational force, and includes detailed steps for solving the equation. Suitable for advanced high school or undergraduate physics students.

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