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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......