Absolute magnitude of a star and its relation to luminosity
Apparent magnitude m does not show a star’s true brightness, since it depends on distance. Absolute magnitude M is the magnitude the star would have at a distance of 10 pc; it lets us compare stars at one distance. The relation is m – M = 5 lg d – 5 (d in parsecs); m – M is the distance modulus. So for d = 10 pc, m = M; for d = 100 pc, m – M = 5. The Sun’s absolute magnitude is M☉ ≈ +4.8, so from 10 pc the Sun would be a faint naked-eye star in a dark sky. For luminosity L: a magnitude difference of 5 means a factor of 100 in brightness, i.e. L / L☉ = 10^(0.4 · (M☉ – M)). For example a star with M = –0.2 is 5 magnitudes brighter than the Sun, i.e. ≈ 100 times more luminous. The most luminous stars have M ≈ –8 to –10 (≈ 10⁵–10⁶ L☉) and the faintest M ≈ +16 and fainter (≈ 10⁻⁵ L☉ and less).
In your notebook, take m and d for Sirius (m = –1.46, d = 2.64 pc), Vega and Polaris from a planetarium program and compute M; compare with the Sun’s M = 4.8.