I was attempting to do some calculations of apparent magnitude to help solidify my understanding of the topic, but have been running into some confusion.
According to Wikipedia, the apparent magnitude can be given as:
$m_x = -2.5\log_{10}(F_x/F^0_x)$
where $F_x$ is the observed flux and $F^0_x$ is a reference flux (in other words, this equation provides the difference of apparent magnitude between two observed values). Also, this is assuming that the same wavelength band is used in both flux measurements.
Flux, in turn, can be calculated as:
$F = \frac{L}{A}$
where $L$ is the star's luminosity and $A$ is the flux density. Since stars act as point sources, this can be simplified to:
$F = \frac{L}{4\pi r^2}$
where $r$ is the distance to the star.
Since, historically, Vega has been used as the reference zero-point (having an apparent magnitude around 0.03), I tried doing a simple calculation to find out the apparent magnitude of Fomalhaut using the values for luminosity and distance given in Wikipedia for both of them.
First, the flux of Vega:
$F_{Vega} = \frac{37\,L_\odot}{4\pi (25.3\,ly)^2}$
$F_{Vega} = 4.5999\times 10^{-3}\,L_\odot/ly^2$
Next, the flux of Fomalhaut:
$F_{Fomalhaut} = \frac{17.66\,L_\odot}{4\pi (25\,ly)^2}$
$F_{Fomalhaut} = 2.2485\times 10^{-3}\,L_\odot/ly^2$
Now, to calculate the apparent magnitude:
$m_{Fomalhaut} = -2.5\log_{10}(\frac{2.2485\times 10^{-3}\,L_\odot/ly^2}{4.5999\times 10^{-3}\,L_\odot/ly^2})$
$m_{Fomalhaut} = 0.7777$
Huh?? Fomalhaut's apparent magnitude is supposed to be 1.16. Even correcting for Vega's offset of 0.03, we still come up with 0.8077. Why are the calculations failing? I don't think I've made a mistake in the mathematics. Am I using the wrong values?
Best Answer
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