If the distance of earth form the sun were half the present value, how many days will make one year?

Explanation

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

### Step 1: Understand Kepler's Third Law
Kepler's Third Law states that the square of the period of revolution (T) of a planet is directly proportional to the cube of the semi-major axis of its orbit (R). Mathematically, this can be 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:
- \( R_1 \) = current distance of Earth from the Sun (R)
- \( T_1 \) = current period of revolution of Earth (365 days)
- \( R_2 \) = new distance of Earth from the Sun (which is \( \frac{R}{2} \))
- \( T_2 \) = new period of revolution (which we need to find)

### Step 3: Set Up the Equation
Using Kepler's Third Law:
\[ \frac{T_1^2}{T_2^2} = \frac{R_1^3}{R_2^3} \]
Substituting \( R_1 \) and \( R_2 \):
\[ \frac{T_1^2}{T_2^2} = \frac{R^3}{\left(\frac{R}{2}\right)^3} \]

### Step 4: Simplify the Equation
Calculating \( R_2^3 \):
\[ \left(\frac{R}{2}\right)^3 = \frac{R^3}{8} \]
Thus, the equation becomes:
\[ \frac{T_1^2}{T_2^2} = \frac{R^3}{\frac{R^3}{8}} = 8 \]

### Step 5: Solve for \( T_2 \)
Taking the square root of both sides:
\[ \frac{T_1}{T_2} = \sqrt{8} \]
This implies:
\[ T_2 = \frac{T_1}{\sqrt{8}} \]
Substituting \( T_1 = 365 \) days:
\[ T_2 = \frac{365}{\sqrt{8}} \]

### Step 6: Calculate \( T_2 \)
Calculating \( \sqrt{8} \):
\[ \sqrt{8} = 2\sqrt{2} \approx 2.828 \]
Now, substituting this value:
\[ T_2 \approx \frac{365}{2.828} \approx 129 \text{ days} \]

### Final Answer
Thus, if the distance of the Earth from the Sun were half the present value, one year would be approximately **129 days**.

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