Star A has 4 times the surface temperature and 0.7 times the radius of Star B. What is the ratio of the luminosity of Star A to the luminosity of Star B? (Indicate your answer to an accuracy of 0.1).
To find the ratio of the luminosity of Star A to the luminosity of Star B, we need to use the Stefan–Boltzmann law, which states that the luminosity of a star is proportional to the fourth power of its temperature and the square of its radius.
Let's assign some variables:
- Let T_A be the surface temperature of Star A.
- Let T_B be the surface temperature of Star B.
- Let R_A be the radius of Star A.
- Let R_B be the radius of Star B.
- Let L_A be the luminosity of Star A.
- Let L_B be the luminosity of Star B.
According to the given information:
- T_A = 4 * T_B (Star A has 4 times the surface temperature of Star B)
- R_A = 0.7 * R_B (Star A has 0.7 times the radius of Star B)
Using the Stefan–Boltzmann law, we have the following relationships:
L_A ∝ T_A^4 * R_A^2
L_B ∝ T_B^4 * R_B^2
Now we can form the ratio of L_A to L_B:
L_A / L_B = (T_A^4 * R_A^2) / (T_B^4 * R_B^2)
Substituting the given values:
L_A / L_B = (4 * T_B)^4 * (0.7 * R_B)^2 / (T_B^4 * R_B^2)
= 256 * T_B^4 * 0.49 * R_B^2 / (T_B^4 * R_B^2)
= 256 * 0.49
= 125.44
Therefore, the ratio of the luminosity of Star A to the luminosity of Star B is approximately 125.4 (rounded to one decimal place).