Solution to Binney and Merrifield Problem 2.1

Surface brightness is defined as the radiative flux per unit solid angle of the image. If we are given a surface brightness $I_{1} = 19 \, \text{mag}\,\mathrm{arcsec}^{-2}$ at a given point in a galaxy, we know that point emits as much as light as a star of magnitude 19. We know that apparent magnitudes are related by,

\begin{equation} m - m_{\mathrm{ref}} = -2.5\log_{10}\left(\frac{F}{F_{\mathrm{ref}}}\right) \end{equation}

and we wish to determine the combined surface brightness of $I_1$ which corresponds to the surface brightness from the point in M31 and from $I_2$ from the center of a globular cluster. Let's choose Vega as our reference.

\begin{equation} m_{\mathrm{M31}}-m_{\mathrm{Vega}} = -2.5\log_{10}\left(\frac{F_{\mathrm{M31}}}{F_{\mathrm{Vega}}}\right) \end{equation}

Since Vega is defined to have a magnitude of 0 in all bands,

\begin{equation} m_{\mathrm{M31}}= -2.5\log_{10}\left(\frac{F_{\mathrm{M31}}}{F_{\mathrm{Vega}}}\right) \end{equation}

We can do the same for the globular cluster:

\begin{equation} m_{\mathrm{GC}}= -2.5\log_{10}\left(\frac{F_{\mathrm{GC}}}{F_{\mathrm{Vega}}}\right) \end{equation}

We cannot add magnitudes to get a total magnitude but we can add fluxes to get a total flux. If we solve for the flux of both M31 and the globular cluster andd add them together, we get:

\begin{equation} F_{\mathrm{tot}} = F_{\mathrm{M31}} + F_{\mathrm{GC}} = F_{\mathrm{Vega}}(10^{-\frac{m_{\mathrm{M31}}}{2.5}} + 10^{-\frac{m_{\mathrm{GC}}}{2.5}}) \end{equation}

Now we apply (1) with our $F_{\mathrm{tot}}$ and $F_{\mathrm{Vega}}$.

\begin{equation} m_{\mathrm{tot}} - m_{\mathrm{Vega}} = -2.5\log_{10}\left(\frac{F_{\mathrm{Vega}}(10^{-\frac{m_{\mathrm{M31}}}{2.5}} + 10^{-\frac{m_{\mathrm{GC}}}{2.5}})}{F_{\mathrm{Vega}}}\right) \end{equation}

Now, because $m_{\mathrm{Vega}} = 0$ and the $F_{\mathrm{Vega}}$ will cancel from the numerator and denominator, we can substitute $m_{\mathrm{M31}} = 19$ and $m_{\mathrm{GC}} = 20$ and get:

\begin{equation} \boxed{m_{\mathrm{tot}} = 18.6\, \text{mag}\,\mathrm{arcsec}^{-2}} \end{equation}

This makes sense, as the combined light should be brighter than either of the two light sources individually.