# What is the relationship between the absolute magnitude and power output of a star?

I can't seem to find a straight answer online.

Is the absolute magnitude of a star equal to / linearly proportional to / or unrelated to the power output of the star.

When I say absolute magnitude I mean the one found in the equation m - M = 5log(d/10). When I say power output I mean the value calculated from Stefan's law. (I don't mean to sound patronising with these clarification I'm worried that I might be using the wrong terms or an overly-vague terms because I have very little knowledge of astrophysics)

It's a little tricky. You're thinking of absolute visual magnitude; how bright an object would appear to our eyes if the object were 10 parsecs away from us. However, you must remember that stars emit a wide spectrum of light — much of it isn't visible to the naked eye.

The luminosity is dependent on the total energy emitted by the star per unit time — across all wavelengths. So you're not going to get an exact answer, although there is an approximation we can do.

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 $$M_{\text{Bol}} = 0$$).

The zero-point luminosity is $$3.0128×10^{28}$$ watts. According to the IAU, this value was set so that the Sun's luminosity ($$3.828 × 10^{26}$$ watts) corresponds closely to absolute bolometric magnitude of 4.74, which is an arbitrary value that's commonly used in modern literature.

The conversion between absolute bolometric magnitude and absolute visual magnitude is made via a bolometric correction: $$M_V=M_{\text{bol}} - BC$$ The bolometric correction term $$BC$$ is an empirical estimate that depends on the spectral type and evolutionary stage of the star. A table is given on the Wikipedia page. Thus: $$L_\text{star} \approx L_0 10^{-0.4 (M_V + BC)}$$

Additionally, by using the formulae here, it’s easy to show that the conversion between apparent magnitude and flux is quite similar: $$F_\text{star} \approx F_0 10^{-0.4 (m + BC)}$$ where I've denoted $$F_0=\frac{L_0}{4\pi (10 \mathrm{pc})^2}\approx 2.5180 \times 10^{-8} \mathrm{\frac{W}{m^2}}$$ as the flux of a zero-point luminosity star at 10 parsecs.