Star a has a radius of 200 000 km and a surface temperature of 6 000 k. star b has a radius of 400 000 km and a surface temperature of 3 000 k. the emissivity of both stars is the same. what is the ratio of the rate of energy radiated by star a to that of star b?

Explanation:

In order to solve this problem, the Stefan-Boltzmann law will be useful. This law establishes that a black body (an ideal body that absorbs or emits all the radiation that incides on it) "emits thermal radiation with a total hemispheric emissive power proportional to the fourth power of its temperature":  

(1)  

Where:  

is the energy radiated by a blackbody radiator per second, per unit area (in Watts).

is the Stefan-Boltzmann's constant.  

is the Surface of the body  

is the effective temperature of the body (its surface absolute temperature) in Kelvin.

However, there is no ideal black body (although the radiation of stars like our Sun is quite close).   Therefore, we will use the Stefan-Boltzmann law for real radiator bodies:  

(2)  

Where is the star's emissivity  

Knowing this, let's start with the

We have two stars where the emissivity  of both is the same:

Star A with a radius and a surface temperature  .

Star B with a radius and a surface temperature  .

And we are asked to find the ratio of the rate of energy radiated by both stars:

  (3)

Where  is the  rate of energy radiated by Star A and  is the  rate of energy radiated by Star B.

On the other hand, with the radius of each star we can calculate their surface area, using the formula for tha area of a sphere (assuming both stars have spherical shape):

  (4)

  (5)

Writting the Stefan-Boltzmann law for each star, taking into consideration their areas:

(6)  

(7)  

Substituting (6) and (7) in (3):

  (8)

  (9)

  (10)

Finally:

the correct answer is b