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:

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

then:

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

using Eq. (1) we have:

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

and given:

(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. (2) and (3) to obtain:

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