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Sample Problem 1

Prove that with a small change in the distance to the luminous object  (\Delta r/r<<1) its stellar magnitude changes on

 \Delta m = 2.17 \, \frac{\Delta d}{d} 

 Sol^n Method \: 1 :

From Taylor Series,
 ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - continue 
 \approx x \, [ \because x \ll 1 ]

So let ,  m_1 = m + \Delta m ;\: d_1 = d + \Delta d

 m_1 - m = 5\,  log \frac{d_1}{d}
  m + \Delta m - m = 5 \, log \frac {d + \Delta d}{d}
 \Delta m = 5\,  log (1+ \frac {\Delta d}{d} )

Now let,  \frac {\Delta d}{d} = x

 \Rightarrow \Delta m = 5 \, log (1+ x)
 \Rightarrow \Delta m = 5\,  log_e \times ln(1+x)
 \Rightarrow \Delta m = 5\,  log_e \times x
 \Rightarrow \Delta m = 5\,  \times 0.434 \times x

\centering \boxed { \Rightarrow \Delta m = 2.17\, \frac{\Delta d}{d}  }

(Proved)

Solution Method 2 coming soon......