If earth describes an orbit round the sun of double its present radius, what will be the year on earth ?

Answer

To solve the problem of determining the year on Earth if it were to describe an orbit around the Sun with double its present radius, we can utilize 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 (r) of its orbit. Mathematically, this can be expressed as:
\[ T^2 \propto r^3 \]
This can also be written as:
\[ \frac{T_1^2}{T_2^2} = \frac{r_1^3}{r_2^3} \]

### Step 2: Define the Variables
Let:
- \( T_1 \) = current time period of Earth's orbit (1 year)
- \( r_1 \) = current radius of Earth's orbit (1 astronomical unit, AU)
- \( T_2 \) = new time period of Earth's orbit
- \( r_2 \) = new radius of Earth's orbit (2 AU)

### Step 3: Set Up the Equation
Using the values defined:
- \( r_1 = r \)
- \( r_2 = 2r \)

Substituting these into Kepler's Third Law gives:
\[ \frac{T_1^2}{T_2^2} = \frac{r^3}{(2r)^3} \]

### Step 4: Simplify the Equation
This simplifies to:
\[ \frac{T_1^2}{T_2^2} = \frac{r^3}{8r^3} = \frac{1}{8} \]

### Step 5: Solve for \( T_2 \)
Taking the square root of both sides:
\[ \frac{T_1}{T_2} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} \]

Rearranging gives:
\[ T_2 = T_1 \cdot 2\sqrt{2} \]

### Step 6: Substitute the Value of \( T_1 \)
Since \( T_1 \) is 1 year:
\[ T_2 = 1 \cdot 2\sqrt{2} \]

### Step 7: Calculate \( T_2 \)
Calculating \( 2\sqrt{2} \):
\[ 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828 \text{ years} \]

### Final Answer
Thus, if the Earth were to describe an orbit around the Sun with double its present radius, the year on Earth would be approximately 2.828 years.

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