How to Calculate the Ratio of Power Output Between Two Stars Using Stefan-Boltzmann Law
How can we calculate the ratio of power output between star A and star B?
Given that star A has 2.5 times the temperature and 0.36 times the diameter of star B, calculate the ratio PA/PB with a margin of error of +/- 2.5%.
Calculation of Power Output Ratio Between Star A and Star B
To calculate the ratio of the power output of star A (PA) to star B (PB), we can use the Stefan-Boltzmann law, which relates the power emitted by a star to its temperature and surface area. The law states that the power output (P) of a star is proportional to its surface area (A) and the fourth power of its temperature (T).
The formula for calculating the power output ratio between two stars is:
PA / PB = (A_A / A_B) * (TA / TB)^4
Using the given data that star A has 2.5 times the temperature and 0.36 times the diameter of star B, we can calculate the ratio PA/PB as follows:
Since star A has 2.5 times the temperature and 0.36 times the diameter of star B:
A_A / A_B = (0.36)^2 = 0.1296
Substituting the values, we get:
PA / PB = 0.1296 * (2.5 / 1)^4 = 0.1296 * 39.0625 ≈ 5.06
Therefore, the ratio of the power output of star A to star B is approximately 5.06.
With a margin of error of +/- 2.5%, the range within which the ratio falls is:
Lower Bound = 5.06 - (2.5% * 5.06) ≈ 4.9335
Upper Bound = 5.06 + (2.5% * 5.06) ≈ 5.1865
Therefore, the ratio PA/PB falls within the range of approximately 4.9335 to 5.1865, with a margin of error of +/- 2.5%.