Chapter MPS, Problem 8.6MPS

To determine

To Find: The difference in brightness of cresent moon and the Sirius star.

Answer to Problem 8.6MPS

  $594-247.5{\log }_{10}\frac{L}{L_o}$

Explanation of Solution

Given:

  $\begin{array}{l}m=\text{Brightness or apparent magnitude of a star or planet}\\ L=\text{Amount of light coming to the Earth from star or planet}\\ L_o=\text{Amount of light from sixth magnitude star}\text{.}\\ \text{The cresent moon is 100 times bright than Sirius}\end{array}$

Formula Used:

  $m=6-2.5{\log }_{10}\frac{L}{L_o}$

Calculation:

  $\begin{array}{l}m=6-2.5{\log }_{10}\frac{L}{L_o}\\ Let,\\ \text{Brightness of moon be }{m}_1\\ \text{Brightness of crescent be }{m}_2\\ \Rightarrow m_1=100m_2\\ \Rightarrow m_2=6-2.5{\log }_{10}\frac{L}{L_o}\\ \Rightarrow m_1=100\left(6-2.5{\log }_{10}\frac{L}{L_o}\right)\\ \Rightarrow m_1=600-250{\log }_{10}\frac{L}{L_o}\\ \Rightarrow m_1-m_2=\left(600-250{\log }_{10}\frac{L}{L_o}\right)-\left(6-2.5{\log }_{10}\frac{L}{L_o}\right)\\ \Rightarrow m_1-m_2=600-250{\log }_{10}\frac{L}{L_o}-6+2.5{\log }_{10}\frac{L}{L_o}\\ \Rightarrow m_1-m_2=594-247.5{\log }_{10}\frac{L}{L_o}\end{array}$

Conclusion:

Thus the difference in magnitude of cresent moon and the Sirius is $594-247.5{\log }_{10}\frac{L}{L_o}$ .