If the radius of a star is `R` and it acts as a black body, what would b the temperature of the star, in which the rate of energy production is `Q`?

Options

A. `((Q)/(4piR^2sigma))^(1//4)`
B. `(Q)/(4piR^2sigma)`
C. `((Q)/(4piR^2sigma))`
D. `(4piR^(2)Q//sigma)^(1//4)`

Explanation

To find the temperature of a star acting as a black body with a given rate of energy production \( Q \) and radius \( R \), we can use Stefan-Boltzmann's law. Here’s the step-by-step solution:

### Step 1: Understand the Stefan-Boltzmann Law
The Stefan-Boltzmann law states that the power radiated by a black body is proportional to the fourth power of its absolute temperature \( T \). The formula is given by:

\[ Q = \sigma A T^4 \]

where:
- \( Q \) is the rate of energy production (power),
- \( \sigma \) is the Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \, \text{W/m}^2 \text{K}^4 \)),
- \( A \) is the surface area of the black body,
- \( T \) is the absolute temperature.

### Step 2: Calculate the Surface Area of the Star
The surface area \( A \) of a sphere (which we assume the star is) is given by:

\[ A = 4 \pi R^2 \]

where \( R \) is the radius of the star.

### Step 3: Substitute the Surface Area into the Stefan-Boltzmann Law
Substituting the expression for surface area into the Stefan-Boltzmann equation, we get:

\[ Q = \sigma (4 \pi R^2) T^4 \]

### Step 4: Rearrange the Equation to Solve for Temperature \( T \)
To find the temperature \( T \), we rearrange the equation:

\[ T^4 = \frac{Q}{\sigma (4 \pi R^2)} \]

### Step 5: Take the Fourth Root to Find \( T \)
Now, take the fourth root of both sides to solve for \( T \):

\[ T = \left( \frac{Q}{4 \pi R^2 \sigma} \right)^{1/4} \]

### Final Result
Thus, the temperature of the star is given by:

\[ T = \left( \frac{Q}{4 \pi R^2 \sigma} \right)^{1/4} \]