Astrophysics - Apparent Magnitude of stars in a close binary system

[goldilocks]

Homework Statement

2 solar type stars are in a close binary system. Each indivual star has an apparent brightness of m = 10 mag.

Determine the apparent magnitudes of the combined system of the 2 stars assuming they cannot be resolved as individual objects. Determine the minimum brightness for this eclipsing binary when 100%, 75%, 50% and 25% of the stellar survace are covered during the eclipse.

Homework Equations

m₁ - m₂ = -2.5 log(f₁/f₂)

The Attempt at a Solution

m1 - m2 = -2.5 log (f1/f2)
10-10 = -2.5log(f1/f2)
f1/f2 =1
f1=f2

m1+2 - m2 = -2.5 log (f1+f2 \ f2) f1 = f2
m1+2 - m2 = -2.5 log (2f1 \ f1)
m1+2 - m2 = -2.5 log 2
m1+2 = -2.5log2 + 5 mag
m1+2=4.247425011
m1+2 = 4.25 mag

Then I'm not sure how to continue from here - any help would be most appreciated!

 

[mgb_phys]

One magnitude represents a change in brightness of a factor of around 2.5
so if you double the brightness (ie two stars) the apparent mag of the combination is only going to change by less than 1mag

 

[thomate1]

As an astrophysicist, I would first like to clarify that apparent magnitude is a measure of how bright a star appears to an observer on Earth, and is affected by the distance of the star from Earth. In a close binary system, the apparent magnitude of the combined system will depend on the relative brightness of the individual stars and their orbital positions.

Using the equation provided, we can determine the apparent magnitude of the combined system for different levels of eclipse. For a 100% eclipse, when both stars are completely covered, the combined brightness will be equal to the brightness of the fainter star. This means that the apparent magnitude of the combined system will be equal to the apparent magnitude of the fainter star, which is 10 mag in this case.

For a 75% eclipse, when 75% of the stellar surface is covered, the combined brightness will be equal to 75% of the brightness of the fainter star. This means that the apparent magnitude of the combined system will be equal to the apparent magnitude of the fainter star plus 2.5 log 0.75, which is approximately 9.72 mag.

Similarly, for a 50% eclipse, the apparent magnitude of the combined system will be equal to the apparent magnitude of the fainter star plus 2.5 log 0.5, which is approximately 9.25 mag.

And for a 25% eclipse, the apparent magnitude of the combined system will be equal to the apparent magnitude of the fainter star plus 2.5 log 0.25, which is approximately 8.72 mag.

It is important to note that these calculations assume that the two stars have the same intrinsic brightness, which is not always the case. If the two stars have different intrinsic brightness, this will have to be taken into account when calculating the apparent magnitude of the combined system.

In conclusion, the apparent magnitude of a close binary system can vary depending on the orbital positions of the stars and the level of eclipse. By using the provided equation and taking into account the relative brightness of the individual stars, we can determine the apparent magnitude of the combined system for different levels of eclipse.