Calculations of apparent magnitude

Thanks for asking this question. It is something we all assume to be obviously trivial and often skip. Your question made me think and I wasn't sure whether the values for luminosities listed in Wikipedia were in the optical range, or the bolometric luminosity i.e. the luminosity over all wavelengths.

A little bit of googling led me to this page, where this question seems to have been discussed well and also resolved.

Updated link

I think the real problem will be in the errors in measurements. Recall that the error in a function of $n$ variables, $f(x_1, x_2, ..., x_n)$ with associated errors for each variable $\sigma_1$, $\sigma_2$, ... , $\sigma_n$ is given by $$\sigma_{f}=\sqrt{\sum_i^n \left(\sigma_i\frac{df(x_1, x_2, ..., x_n)}{dx_i}\right)^2}$$

In your case, the function is $F(L, r)=\frac{L}{4\pi r^2}$, so the error propagation formula is

$$\sigma_{F}=\sqrt{\left(\frac{\sigma_{L}}{4 \pi r^2}\right)^2 + \left(\frac{-3\sigma_{r}}{4 \pi r^3}\right)^2}$$

For Vega, $\sigma_{L}=3L_{\odot}$, $\sigma_r=0.1LY$. That gives $$\sigma_{F_{Vega}}=3.77\times10^{-4}$$

For Fomalhaut, it was a bit trickier to track down since wikipedia doesn't give the error in luminosity, but in the article it's given: $\sigma_{L}=0.82L_{\odot}$, $\sigma_r=0.1LY$. That gives $$\sigma_{F_{Fomalhaut}}=1.05\times10^{-4}$$

Using the standard error propagation formula again, the error associated with the apparent magnitude is

$$\sigma_{m}=\sqrt{\left(\frac{-2.5\sigma_{F_{Fomalhaut}}}{F_{Fomalhaut}}\right)^2 + \left(\frac{2.5\sigma_{F_{Vega}}}{F_{Vega}}\right)^2}$$

If you plug in all the values used above, I get $$\sigma_{m}=0.237$$ which means that your estimate is off by only $1.5\sigma$. That's pretty good, considering that these luminosities are probably bolometric rather than the visual band alone, yet your apparent magnitude is only in the V-band which makes it only a fraction of all the light from the star.