Two planets `A` and `B` are revolving in circular orbits around a fixed sun. Time period of the planet `B` is `2sqrt(2)` times that of planet `A`. Find the ratio of solar constant for planet `A` to that of `B`. Neglect gravitational force between the planets.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the relationship between time period and radius
According to Kepler's third law of planetary motion, the square of the time period \( T \) of a planet is directly proportional to the cube of the radius \( r \) of its orbit around the sun. This can be expressed as:
\[ T^2 \propto r^3 \]
From this, we can derive the relationship:
\[ r \propto T^{2/3} \]

### Step 2: Relate the time periods of planets A and B
Let the time period of planet A be \( T_A \) and that of planet B be \( T_B \). According to the problem, we have:
\[ T_B = 2\sqrt{2} \cdot T_A \]

### Step 3: Find the radius of planet B in terms of radius of planet A
Using the relationship derived from Kepler's law, we can express the radius of planet B in terms of the radius of planet A:
\[ r_B \propto T_B^{2/3} = (2\sqrt{2} \cdot T_A)^{2/3} \]
Calculating this gives:
\[ r_B \propto (2\sqrt{2})^{2/3} \cdot T_A^{2/3} \]
Calculating \( (2\sqrt{2})^{2/3} \):
\[ (2\sqrt{2})^{2/3} = (2^{1} \cdot 2^{1/2})^{2/3} = 2^{(1 + 1/2) \cdot (2/3)} = 2^{(3/2) \cdot (2/3)} = 2^{1} = 2 \]
Thus, we have:
\[ r_B = 2 \cdot r_A \]

### Step 4: Find the ratio of the solar constants
The solar constant (intensity of sunlight) at a distance \( r \) from the sun is inversely proportional to the square of the distance from the sun:
\[ I \propto \frac{1}{r^2} \]
Thus, the ratio of the solar constants \( I_A \) and \( I_B \) for planets A and B can be expressed as:
\[ \frac{I_A}{I_B} = \frac{r_B^2}{r_A^2} \]
Substituting \( r_B = 2r_A \):
\[ \frac{I_A}{I_B} = \frac{(2r_A)^2}{r_A^2} = \frac{4r_A^2}{r_A^2} = 4 \]

### Final Answer
The ratio of the solar constant for planet A to that of planet B is:
\[ \frac{I_A}{I_B} = 4 \]