8. Integrated magnitudes

Todo

Not finished.

To obtain the integrated magnitude for the cluster region cleaned from field stars contamination, the following equations are used.

First, assume:

(8.1)\[I_{cl+fl} = I_{cl} + I_{fl}\]

then:

\[V^*_{cl} - V^*_{cl+fl} = -2.5 \log(I_{cl}/I_{cl+fl})\]

using Eq. (8.1) we have:

(8.2)\[V^*_{cl} - V^*_{cl+fl} = -2.5 \log(1 - I_{fl}/I_{cl+fl})\]

and given:

(8.3)\[V^*_{fl} - V^*_{cl+fl} = -2.5 \log(I_{fl}/I_{cl+fl}) \Rightarrow \frac{I_{fl}}{I_{cl+fl}} = 10^{(V^*_{fl} - V^*_{cl+fl})/-2.5}\]

we can combine now Eqs. (8.2) and (8.3) to obtain:

\[V^*_{cl} = -2.5 \log(1 - 10^{(V^*_{fl} - V^*_{cl+fl})/-2.5}) + V^*_{cl+fl}\]