# Total apparent magnitude of eclipsing binary system

This is a question from the 2016 USAAAO national exam, for which no solutions are given:

An eclipsing binary system has a magnitude $$m_p$$ = 14.2 during the primary transit and $$m_s$$ = 13.7 during the secondary transit. Find the normal (non-eclipsed) apparent magnitude of the system.

I assumed that, as is usual for eclipsing binary systems, the higher magnitude/primary transit refers to the complete eclipse of a smaller, hotter star (say Star B), and the lower magnitude/secondary transit refers to Star B passing in front of a larger, cooler Star A.

To find the normal, non-eclipsed apparent magnitude of the system, usually we achieve a relation like $$F_A = kF_B$$, where k is a constant. Then, as we know the apparent magnitude corresponding to $$F_A$$ alone, we could easily calculate that for $$F_A + F_B$$. However, this is where I ran into trouble. I set up the following equation to attempt to find a relation between $$F_A$$ and $$F_B$$: $$m_p - m_s = -2.5log(\frac{F_A}{F_A(1-(R_B/R_A)^2)+F_B})$$

The denominator in the argument of the logarithm, $$F_A(1-(R_B/R_A)^2)+F_B$$, expresses the flux observed at the secondary transit. But I have no way that I can see to figure out the ratio between radii, and I can't think of any reasonable assumption that would simplify the equation further (such as $$R_B << R_A$$).

Should this problem be attacked from a different angle? Any help would be much appreciated!

• We've had several questions about solving USAAAO problems recently, and I'm not getting a positive impression of the quality of the USAAAO. – antispinwards Mar 25 at 7:17
• @antispinwards Pre-2018 USAAAO exams are often riddled with mistakes, but it's hard to find other sources to prepare, as even international astro olympiad exams from before 2016 can be quite unclear/poorly translated/etc. – Alistair Mar 25 at 7:29

Without information about stellar radii, I think it's reasonable to assume $$R_A \approx R_B$$. Then your equation becomes
$$m_p - m_s = -2.5 \log \frac{F_A}{F_B}$$
and you can compute $$k$$ and the non-eclipsed total magnitude.