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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!

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    $\begingroup$ We've had several questions about solving USAAAO problems recently, and I'm not getting a positive impression of the quality of the USAAAO. $\endgroup$ – antispinwards Mar 25 at 7:17
  • $\begingroup$ @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. $\endgroup$ – Alistair Mar 25 at 7:29
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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.

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  • $\begingroup$ Thanks, that seems reasonable! To be clear, there's no way to exactly solve the problem without the information about stellar radii? $\endgroup$ – Alistair Mar 25 at 5:21
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    $\begingroup$ @Alistair Right. Don't put more thought into these questions than the contest author did. $\endgroup$ – Mike G Mar 25 at 5:29

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