<p>Without information about stellar radii, I think it's reasonable to assume
<span class="math-container">$R_A \approx R_B$</span>.
Then your equation becomes</p>

<p><span class="math-container">$$ m_p - m_s = -2.5 \log \frac{F_A}{F_B} $$</span></p>

<p>and you can compute <span class="math-container">$k$</span> and the non-eclipsed total magnitude.</p>

CC BY-SA 4.0
2020-03-25T05:21:15.637Z
Thanks, that seems reasonable! To be clear, there's no way to exactly solve the problem without the information about stellar radii?
Alistair
CC BY-SA 4.0
2020-03-25T05:29:43.313Z
@Alistair Right. Don't put more thought into these questions than the contest author did.
Mike G
CC BY-SA 4.0
2020-03-25T05:10:02.157Z
2020-03-25T05:10:02.157Z
Mike G
<p>This is a question from the 2016 USAAAO national exam, for which no solutions are given:</p>

<blockquote>
  <p>An eclipsing binary system has a magnitude <span class="math-container">$m_p$</span> = 14.2 during the
  primary transit and <span class="math-container">$m_s$</span> = 13.7 during the secondary transit. Find
  the normal (non-eclipsed) apparent magnitude of the system.</p>
</blockquote>

<p>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.</p>

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

<p>The denominator in the argument of the logarithm, <span class="math-container">$F_A(1-(R_B/R_A)^2)+F_B$</span>, 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 <span class="math-container">$R_B &lt;&lt; R_A$</span>). </p>

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

CC BY-SA 4.0
2020-03-25T07:29:12.710Z
@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
<binary-star><mathematics><apparent-magnitude><eclipsing-star-systems>
Total apparent magnitude of eclipsing binary system
Alistair
# Total apparent magnitude of eclipsing binary system

(Asked by: username_0 on 2020-03-25)

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!

**Tags:** binary-star, mathematics, apparent-magnitude, eclipsing-star-systems
> **Comments:**
> **username_0**: @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.  

---

## 1 answers

### Answer by username_1 on 2020-03-25. Score: 6 | Accepted answer

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.

> **Comments:**
> **username_0**: Thanks, that seems reasonable! To be clear, there's no way to exactly solve the problem without the information about stellar radii?  
> **username_1**: @Alistair Right. Don't put more thought into these questions than the contest author did.  

---

