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A binary star is observed to consist of a blue star B (Peak wavelength 800 nm) and a red star R (Peak wavelength 400 nm) orbiting each other. As observed from the earth, B and R appear bright. Assuming that the stars radiate as perfect black bodies, it follows that the ratio of volumes VB/VR of the two stars is_________.
Correct answer is '64'. Can you explain this answer?
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A binary star is observed to consist of a blue star B (Peak wavelength...
For blackbody, According to wien’s displacement law

and energy per unit area per unit time is E = 

So, that the volume 
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A binary star is observed to consist of a blue star B (Peak wavelength...
Explanation:

Step 1: Use Wien's displacement law to determine the temperature of each star.

- Wien's displacement law states that the peak wavelength of a black body radiation curve is inversely proportional to its temperature.
- λmax = b/T, where λmax is the peak wavelength, b is Wien's constant (2.898 x 10^-3 m K), and T is the temperature in Kelvin.
- For star B, λmax = 800 nm = 8 x 10^-7 m. Therefore, T_B = b/λmax = 2.898 x 10^-3 m K / 8 x 10^-7 m = 3622 K.
- For star R, λmax = 400 nm = 4 x 10^-7 m. Therefore, T_R = b/λmax = 2.898 x 10^-3 m K / 4 x 10^-7 m = 7256 K.

Step 2: Use the Stefan-Boltzmann law to determine the ratio of the surface areas of the two stars.

- The Stefan-Boltzmann law states that the total energy radiated per unit time per unit surface area by a black body is proportional to the fourth power of its temperature.
- E = σT^4, where E is the energy radiated per unit time per unit surface area, σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2 K^4), and T is the temperature in Kelvin.
- For star B, E_B = σT_B^4 = 5.67 x 10^-8 W/m^2 K^4 x (3622 K)^4 = 2.73 x 10^7 W/m^2.
- For star R, E_R = σT_R^4 = 5.67 x 10^-8 W/m^2 K^4 x (7256 K)^4 = 1.97 x 10^9 W/m^2.
- The ratio of the surface areas is therefore A_B/A_R = E_B/E_R = (2.73 x 10^7 W/m^2)/(1.97 x 10^9 W/m^2) = 0.0139.

Step 3: Use the ratio of the surface areas to determine the ratio of the volumes.

- The ratio of the surface areas is proportional to the square of the ratio of the radii, which is proportional to the cube of the ratio of the volumes.
- Therefore, (A_B/A_R)^(1/2) = (V_B/V_R)^(1/3).
- Substituting the value of A_B/A_R, we get (0.0139)^(1/2) = (V_B/V_R)^(1/3).
- Solving for V_B/V_R, we get V_B/V_R = (0.0139)^(3/2) = 0.0003.
- Finally, V_B/V_R = (r_B/r_R)^3, where r_B and r_R are the radii of stars B and R, respectively.
- Therefore, (r_B/r_R)^3 = 0.0003, or r_B/r_R = 0.064.
-
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A binary star is observed to consist of a blue star B (Peak wavelength 800 nm) and a red star R (Peak wavelength 400 nm) orbiting each other. As observed from the earth, B and R appear bright. Assuming that the stars radiate as perfect black bodies, it follows that the ratio of volumes VB/VR of the two stars is_________.Correct answer is '64'. Can you explain this answer?
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