Assuming that earth and mars move in circular orbits around the sun, with the martian orbit being 1.52 times the orbital radius of the earth. The length of the martian year is days is

Options

A. `(1.52)^(2//3)xx365`
B. `(1.52)^(3//2)xx365`
C. `(1.52)^(2)xx365`
D. `(1.52)^(3)xx365`

Explanation

To find the length of the Martian year, we can use Kepler's Third Law of planetary motion, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (R) of its orbit. The law can be expressed mathematically as:

\[ T^2 \propto R^3 \]

### Step-by-Step Solution

1. **Identify the relationship between the orbital radii**:
Given that the orbital radius of Mars (Rm) is 1.52 times the orbital radius of Earth (Re), we can write:
\[ Rm = 1.52 \times Re \]

2. **Apply Kepler's Third Law**:
According to Kepler's Third Law, we can write the equations for the time periods of Mars (Tm) and Earth (Te):
\[ Tm^2 \propto Rm^3 \quad \text{(1)} \]
\[ Te^2 \propto Re^3 \quad \text{(2)} \]

3. **Set up the ratio of the time periods**:
Dividing equation (1) by equation (2):
\[ \frac{Tm^2}{Te^2} = \frac{Rm^3}{Re^3} \]

4. **Substitute the relationship between Rm and Re**:
Substitute \(Rm = 1.52 \times Re\) into the equation:
\[ \frac{Tm^2}{Te^2} = \frac{(1.52 \times Re)^3}{Re^3} \]

5. **Simplify the equation**:
This simplifies to:
\[ \frac{Tm^2}{Te^2} = 1.52^3 \]

6. **Calculate \(1.52^3\)**:
Calculate \(1.52^3\):
\[ 1.52^3 \approx 3.51 \]

7. **Relate the time periods**:
Now we can express the relationship between the time periods:
\[ Tm^2 = 3.51 \times Te^2 \]

8. **Substitute \(Te\)**:
Since the time period of Earth (Te) is 1 year, which is approximately 365 days:
\[ Tm^2 = 3.51 \times (365)^2 \]

9. **Calculate \(Tm\)**:
Now, calculate \(Tm\):
\[ Tm = \sqrt{3.51 \times (365)^2} \]
\[ Tm \approx \sqrt{3.51} \times 365 \approx 1.87 \times 365 \approx 683.55 \text{ days} \]

10. **Final Answer**:
Therefore, the length of the Martian year is approximately:
\[ Tm \approx 687 \text{ days} \]