If the distance between the earth and the sun were half its present value, the number of days in a year would have been

Explanation

To solve the problem of how many days there would be in a year if the distance between the Earth and the Sun were half its present value, we can use Kepler's Third Law of Planetary Motion. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand Kepler's Third Law
Kepler's Third Law 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. Mathematically, this is expressed as:
\[ T^2 \propto R^3 \]
This means:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]

### Step 2: Define the Variables
Let:
- \( T_1 \) = current orbital period of Earth = 365 days
- \( R_1 \) = current distance between Earth and Sun = \( d \)
- \( T_2 \) = new orbital period we want to find
- \( R_2 \) = new distance between Earth and Sun = \( \frac{d}{2} \)

### Step 3: Apply Kepler's Law
Using the relationship from Kepler's Third Law:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]
Substituting the values:
\[ \frac{365^2}{T_2^2} = \frac{d^3}{\left(\frac{d}{2}\right)^3} \]

### Step 4: Simplify the Right Side
Calculating the right side:
\[ \frac{d^3}{\left(\frac{d}{2}\right)^3} = \frac{d^3}{\frac{d^3}{8}} = 8 \]
Thus, we have:
\[ \frac{365^2}{T_2^2} = 8 \]

### Step 5: Solve for \( T_2^2 \)
Rearranging gives:
\[ T_2^2 = \frac{365^2}{8} \]

### Step 6: Calculate \( T_2 \)
Taking the square root:
\[ T_2 = \sqrt{\frac{365^2}{8}} = \frac{365}{\sqrt{8}} = \frac{365}{2\sqrt{2}} = \frac{365}{2 \times 1.414} \approx \frac{365}{2.828} \approx 129 \]

### Conclusion
Thus, if the distance between the Earth and the Sun were half its present value, the number of days in a year would be approximately **129 days**.

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