If the earth be one-half of its present distance from the sun, how many days will be in one year ?

Options

A. 183
B. 730
C. 129
D. 365

Answer

To solve the problem of how many days will be in one year if the Earth is at one-half of its present distance from the Sun, we can use Kepler's Third Law of Planetary Motion. Here's a step-by-step solution:

### Step 1: Understand the Current Situation
The current distance of the Earth from the Sun is denoted as \( R \). The current time period (one year) is \( T_1 = 365 \) days.

### Step 2: Determine the New Distance
If the Earth is at one-half of its present distance from the Sun, we need to clarify what "one-half" means. It means that the new distance \( R_2 \) will be:
\[ R_2 = \frac{1}{2} R \]

### Step 3: Apply Kepler's Third Law
According to Kepler's Third Law, the square of the time period \( T \) of a planet is directly proportional to the cube of the semi-major axis of its orbit (the distance from the Sun). This can be expressed as:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]
Where:
- \( T_1 \) is the current time period (365 days)
- \( T_2 \) is the new time period we want to find
- \( R_1 \) is the current distance (R)
- \( R_2 \) is the new distance (\( \frac{1}{2} R \))

### Step 4: Substitute the Values
Substituting the known values into the equation:
\[ \frac{(365)^2}{T_2^2} = \frac{R^3}{\left(\frac{1}{2} R\right)^3} \]

### Step 5: Simplify the Right Side
Calculating the right side:
\[ \left(\frac{1}{2} R\right)^3 = \frac{1}{8} R^3 \]
Thus, the equation becomes:
\[ \frac{(365)^2}{T_2^2} = \frac{R^3}{\frac{1}{8} R^3} = 8 \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ T_2^2 = \frac{(365)^2}{8} \]

### Step 7: Calculate \( T_2 \)
Taking the square root of both sides:
\[ T_2 = \frac{365}{\sqrt{8}} = \frac{365}{2\sqrt{2}} \approx \frac{365}{2 \times 1.414} \approx \frac{365}{2.828} \approx 129.9 \text{ days} \]

### Final Answer
Thus, if the Earth were at one-half of its present distance from the Sun, the number of days in one year would be approximately:
\[ T_2 \approx 129.9 \text{ days} \]