\begin{table}%t10
\caption{\label{tab:second_order_airmass_corrections}Synthetic second order airmass corrections.}
\small%\centerline
 {
\begin{tabular}{cccccc}
\hline
\hline
Band  & Color  & $k'$    & $\langle {\rm color} \rangle$    & $\langle {\rm color} \rangle$   & Bias$^{a}$\\
      & index  & (synth) & (Landolt) & (tertiaries) &      \\
\hline 
$g_M$ &  $g-r$      & $-0.014$  &   $+0.735$  &  $+0.950$    &  ${<}0.001$    \\
$r_M$ &  $r-i$      & $-0.005$  &   $+0.408$  &  $+0.618$    &  ${<}0.001$    \\
$i_M$ &  $r-i$      & ${<}0.001$  &   $+0.408$  &  $+0.618$    &  ${<}0.001$    \\
$z_M$ &  $i-z$      & ${<}0.001$  &   $+0.192$  &  $+0.272$    &  ${<}0.001$    \\
\hline
\end{tabular}}
\medskip
$^{a}$ Effects on the tertiary magnitudes of neglecting the second order correction. The resulting bias is smaller than 0.001~mag in all bands.
\end{table}