Where can I find values for apparent brightness of stars?

Question

I'm in high school and doing a project. I want to calculate the distance to stars using their luminosity and apparent brightness, from the equation $b=\frac{L}{4 \pi d^2}$. I have found values for luminosity and apparent magnitude from the Hipparcos dataset. However, I cannot find how to convert these values of apparent magnitude to apparent brightness anywhere. I need the values to be in $\frac{W}{m^2}$ for the equation to work. Is there a way to do this? Is my idea completely wrong? Any guidance you can give me will be greatly appreciated!

Answer 1

You can derive the Bolometric magnitude of the star $M_{bol}$ from the below relation taking the sun as the reference, where L is the stellar luminosity:

$$L / L_\odot = 10^{0.4(M_{bol\odot} - M_{bol})}$$

here $M_{bol\odot}$ is for Sun and you can use the value +4.75 for that.

Then you need to correct the absolute magnitude of the star in a given pass band (like Visual Mv) from $Mv = M_{bol} - BC$. BC is the Bolometric correction for the magnitude .

Finally having both the apparent and absolute magnitudes you can derive the distance r to the star:

$$\underbrace{m-M}_\text{Distance Modulus} = 5\log_{10}\left(\frac{r}{10}\right)$$

Note that both the apparent and absolute magnitudes, $m$ and $M$, are should be from the same photometric band. For example mv and Mv for the visual band or mG and MG from Gaia mission band.

At the end to compare your result you can check it roughly with 1/parallax value given in Hipparcos or Gaia. Be careful about the parallax unit!