A satellite orbits the sun with a period of 2.00 days. What is the radius of its orbit? Express your answer with the appropriate units.

To solve for the radius of the satellite's orbit around the Sun, we can use **Kepler's Third Law of Planetary Motion**, which relates the orbital period of a satellite to the radius of its orbit around the Sun. Kepler's Third Law is given by:

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

Where:
- \( T \) is the orbital period of the satellite (in seconds),
- \( G \) is the gravitational constant (\(6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2}\)),
- \( M_{\odot} \) is the mass of the Sun (\(1.989 \times 10^{30} \, \text{kg}\)),
- \( r \) is the radius of the orbit (which we want to find).

We can rearrange the equation to solve for \( r \):

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

### Step 1: Convert the orbital period \( T \) to seconds.
Given that the period \( T \) is 2.00 days, we need to convert this to seconds:

\[
T = 2.00 \, \text{days} \times 24 \, \text{hours/day} \times 3600 \, \text{seconds/hour}
\]
\[
T = 2.00 \times 86400 \, \text{s} = 172800 \, \text{s}
\]

### Step 2: Substitute known values into the equation.
Now, substitute \( T = 172800 \, \text{s} \), \( G = 6.674 \times 10^{-11} \, \text{m}^3\text{kg}^{-1}\text{s}^{-2} \), and \( M_{\odot} = 1.989 \times 10^{30} \, \text{kg} \) into the rearranged Kepler's Third Law:

\[
r^3 = \frac{(6.674 \times 10^{-11})(1.989 \times 10^{30})(172800)^2}{4\pi^2}
\]

### Step 3: Calculate the value of \( r \).
Let's compute this value.The radius of the satellite's orbit around the Sun is approximately \(4.65 \times 10^9\) meters, or 4.65 million kilometers.

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. How would the radius change if the satellite's period was 5 days instead of 2 days?
2. How is Kepler's Third Law applied to objects orbiting planets like Earth?
3. How does the mass of the central object (the Sun in this case) affect the orbit of a satellite?
4. What assumptions are made in Kepler’s Laws about the shapes of orbits?
5. How does the gravitational force between two objects depend on their distance?

**Tip:** The orbital period and the radius of an orbit are strongly related. Increasing the orbital period significantly increases the orbital radius due to the cubic relationship in Kepler's Third Law.

Learn how to calculate the radius of a satellite's orbit with an orbital period of 2.00 days. This physics problem uses Kepler's Third Law and gravitational constants to determine the orbital radius, providing a detailed step-by-step solution. Suitable for advanced high school and college-level students.

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