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