# Relationship between absolute magnitude of a star and its luminosity?

• Why does this relationship involves the absolute magnitude of the sun and its luminosity?
• How to derive an expression relating the absolute magnitude of a star to its luminosity?

So according to the magnitude equation, $$m - M = 2.5\log\left(\frac{d^2}{d_0^2}\right)$$ $$\Rightarrow M = m - 2.5\log\left(\frac{d^2}{d_0^2}\right)\ \ (eq1)$$

and Luminosity is $$L = 4\pi(d^2) \times f$$
$$\Rightarrow d^2 = \frac{L}{4\pi \times f}\ \ (eq2)$$

Plugging eq2 to eq1 would have seem reasonable, but how is absolute magnitude of the sun and its luminosity used?

• It's only used because we compare other star's luminosity to the Sun. In theory, you could measure luminosity in watts (en.wikipedia.org/wiki/Luminosity tells us the Sun's total brightness is 3.846×10^26 watts), but it's the same reason we use light years and astronomical units: to avoid numbers from getting too large. – barrycarter Feb 12 '16 at 18:27
• This site has TeX enabled. You can format your equations using TeX – James K Feb 13 '16 at 21:37

Absolute bolometric magnitude is an analogous magnitude system considering all the energy emitted by the star. The formula relating absolute bolometric magnitude with luminosity is as follows: $$L_\text{star} = L_0 10^{-0.4 M_\text{Bol}}$$
where $$L_\text{star}$$ is the star's luminosity, $$M_{\text{Bol}}$$ is the bolometric magnitude of the star, and $$L_{0}$$ is the zero-point luminosity (the luminosity of a star with $$M_{\text{Bol}} = 0$$) arbitrarily defined as $$L_{0}=3.0128×10^{28}$$ watts by the IAU. We can approximately convert between the two magnitudes via a bolometric correction: $$M_{\text{bol}}\approx M_V + BC$$ The bolometric correction term $$BC$$ is empirically determined for the spectral class and evolutionary stage (although it can be modeled; see Torres (2010)). A table of values is given on the Wikipedia page.
The idea is that stars of the same spectral class and evolutionary stage have a similar distribution of wavelengths, so the discrepancy between all emitted wavelengths ($$M_{\text{bol}}$$) and visible wavelengths ($$M_V$$) should be similar.
Thus, the conversion between luminosity and absolute visual magnitude is approximately $$L_\text{star} \approx L_0 10^{-0.4 (M_V + BC)}$$ Of course, this is only an approximation, but it can give us a pretty decent idea of what to expect. Trying it on 9 Pegasi, I predicted it would be around $$1000 L_\odot$$, which is still in the ballpark of the actual value $$1950 L_\odot$$. Mind you, the approximation would be better suited for less extreme stars than 9 Pegasi.