PX158 - D2 - brightness and magnitudes

apparent magnitude (m)

circa $150 B C E$ , hipparchus of nicaea catalogued the night sky using a system in which the brightest stars is a magnitude $1$ , and the faintest stars is a magnitude $6$
in the $1850 s$ , norman robert pogson worked out that the eye is a logarithmic detector, and that $Δ m a g n i t u d e$ is $\times 100$ in flux
consider two stars, star 1 and star 2, with fluxes, $f_{1}$ and $f_{2}$ , and magnitudes, $m_{1}$ and $m_{2}$

\begin{aligned} \frac{f_{2}}{f_{1}} & = {(\frac{1}{100})}^{\frac{m_{2} - m_{1}}{5}} \\ \log_{10} (\frac{f_{2}}{f_{1}}) & = - \frac{2}{5} (m_{2} - m_{1}) \\ ∴ (m_{2} - m_{1}) & = - 2.5 \log_{10} (\frac{f_{2}}{f_{1}}) \end{aligned}

pogson selected vega as a reference star due to its brightness:

m_{v e g a} = 0

magnitudes of other stars can be determined by:

∴ m = - 2.5 \log_{10} (\frac{f}{f_{v e g a}})

these are apparent magnitudes - depends upon where the observer is

absolute magnitude (M)

the apparent magnitude if the object was located at a distance of $10 p c$

\begin{aligned} M - m & = - 2.5 \log_{10} (\frac{\frac{L}{4 π (10 p c)^{2}}}{\frac{L}{4 π d^{2}}}) \\ = - 2.5 \log_{10} {(\frac{d}{10 p c})}^{2} \\ ∴ m - M & = 5 \log_{10} (\frac{d}{10 p c}) \end{aligned}

eg: how far away could the sun be seen with a naked eye?
$m = 6$
$M_{⊙} = + 4.8$

∴ d \approx 1.4 p c

eg: what is the absolute magnitude of a $100 W$ light bulb and how far away can it be seen with a naked eye?
- $L_{⊙} = 3.82 \times 10^{26} W$ , $M_{⊙} = + 4.8$

\begin{aligned} M_{b} - M_{⊙} & = - 2.5 \log_{10} (\frac{f_{b}}{f_{⊙}}) \\ M_{b} & = + 4.8 - 2.5 \log_{10} (\frac{L_{b}}{L_{⊙}}) \\ = + 4.8 - 2.5 \log_{10} (\frac{100}{3.82 \times 10^{26}}) \\ ∴ M_{b} & \approx + 66 \end{aligned}

- for bulb to be seen by a naked eye in a dark sky, $m = 6$

\begin{aligned} m - M & = 5 \log_{10} (\frac{d}{10 p c}) \\ \frac{6 - 66}{5} & = \log_{10} (\frac{d}{10 p c}) \\ 10^{- 12} & = \frac{d}{10 p c} \\ ∴ d & = 10^{- 11} \times 3.09 \times 10^{16} m \\ \approx 300 k m \end{aligned}