A1 - the magnitude system

stars can be classified using two properties:
- brightness
- colour
the apparent magnitude:

m = - 2.5 \log_{10} (F) + c

where, $F$ is the flux, and $c$ is a constant (not the speed of light)

flux is energy per unit time per unit time:

F = \frac{L}{4 π d^{2}}

where, $L$ is the luminosity, and $d$ is the distance

the negative sign is historical, so lower magnitude means greater apparent magnitude
the scale is set such that the star Vega has $m = 0$
since the scale is on $\log_{10}$ with a factor of 2.5, if flux changes by a factor of hundred, the apparent magnitude changes by a factor of 5
colour is the difference of magnitudes, or a ratio of fluxes

m_{1} - m_{2} = - 2.5 \log_{10} \frac{F_{1}}{F_{2}}

absolute magnitude is the magnitude as viewed at $10 pc$

\begin{matrix} M = - 2.5 \log_{10} F^{'} + c \\ F^{'} = \frac{L}{4 π d^{' 2}}, d^{'} = 10 pc \\ M = - 2.5 \log_{10} (\frac{L}{4 π d^{' 2}}) \\ M = - 2.5 \log_{10} F - 2.5 \log_{10} {(\frac{d}{d^{'}})}^{2} \\ M = m - 5 \log_{10} (\frac{d}{d^{'}}) \end{matrix}

distance modulus:

D M = m - M = 5 \log_{10} (\frac{d}{10 pc})

this links flux to luminosity, and hence able to derive distance