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Let $f_*$ and $f_0$ be the observed flux of a star and a reference flux in a particular spectral band, and let $m_*$ and $m_0$ be their respective apparent magnitudes. Then the star's magnitude is given by $m_*-m_0=-2.5\log(f_*/f_0)$.

Encyclopædia Britannica maintains that

All magnitude systems must have a reference, or zero, point.

Why is this? It seems to me that one could just say $m_*=-2.5\log f_*$ and move on, no?

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    $\begingroup$ 1. It's $m_{\star} - m_{0} = -2.5 \log(f_{\star} / f_{0})$. 2. How can you convert a magnitude back into a flux if you don't have a zero point? 3. Just saying $m_{\star} = -2.5 \log f_{\star}$ is equivalent to having a zero point with $f_{0} = 1$ in whatever the flux units are. $\endgroup$ May 14 at 21:36
  • $\begingroup$ @PeterErwin Fixed, just a brain glitch. thanks $\endgroup$ May 14 at 21:37
  • $\begingroup$ 4. For historical reasons, magnitudes have usually defined so that the star Vega has an apparent magnitude of zero, or else gives values not too broadly different. This means you have to set $f_{0}$ in the appropriate units so that, e.g., an observation of Vega will indeed give you a magnitude close to 0. (This is generically useful because if I tell you an object has an apparently magnitude of 10, you'll have a generally correct idea of how bright it is without having to look up how the magnitude was generated.) $\endgroup$ May 14 at 21:45
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Because you can't take the logarithm of something with dimensions only the log of a number.

Your magnitude would depend on what you measured f in. In other words, by choosing a set of units, you choose a zero point! Whether that unit is Jy, W/m$^2$ or the flux of some other object.

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