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Answer :
3.33. Magnitude is a unit of measurement used in astronomy to describe the brightness of celestial objects such as stars. The lower the magnitude, the brighter the star. Star A has a magnitude of 5 while Star B has a magnitude of 10. Thus, we need to use the magnitude scale formula:
$$\Delta m = 2.5 \log_{10}\left(\frac{I_2}{I_1}\right)$$Where Δm is the difference in magnitude between two stars, I1 is the intensity of the fainter star, and I2 is the intensity of the brighter star.Using the formula, we have:$$\Delta m = 2.5 \log_{10}\left(\frac{I_{StarB}}{I_{StarA}}\right)$$Plugging in the values we have, we get:$$\Delta m = 2.5 \log_{10}\left(\frac{10}{1}\right)$$$$\Delta m = 2.5 \times 1$$$$\Delta m = 2.5$$So Star B is 2.5 magnitudes fainter than Star A. The difference in brightness between two stars with a magnitude difference of 2.5 is given
by:$$\text{Brightness ratio} = 2.5^{\frac{-\Delta m}{2.5}}$$Plugging in the values, we have:$$\text{Brightness ratio} = 2.5^{\frac{-2.5}{2.5}}$$$$\text{Brightness ratio} = 2.5^{-1}$$$$\text{Brightness ratio} = 0.4$$Thus, Star A is 0.4 times brighter than Star B. So, the ratio of brightness of Star A to Star B is:$$\frac{Brightness_{Star A}}{Brightness_{Star B}} = 1:0.4 = 2.5:1$$Hence, Star A is 3.33 times brighter than Star B .Star A is 3.33 times brighter than Star B.:Star A is 0.4 times brighter than Star B. So, the ratio of brightness of Star A to Star B is 2.5:1. Therefore, Star A is 3.33 times brighter than Star B.
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