# Astrophysics - Apparent Magnitude of stars in a close binary system

• goldilocks
In summary, the problem involves determining the apparent magnitudes of a binary system of two solar type stars. Using the equation m1 - m2 = -2.5 log(f1/f2), it was found that the combined apparent magnitude would be 4.25 mag. Further calculations would need to be done to determine the minimum brightness during an eclipse when different percentages of the stellar surfaces are covered.
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!

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

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.

## 1. What is the apparent magnitude of a star in a close binary system?

The apparent magnitude of a star in a close binary system refers to the brightness of the star as seen from Earth. This can vary depending on the distance between the two stars in the binary system, as well as the individual luminosity of each star.

## 2. How does the apparent magnitude of a star in a close binary system affect its perceived brightness?

The apparent magnitude of a star in a close binary system directly affects its perceived brightness. The higher the apparent magnitude, the dimmer the star will appear to us on Earth. A lower apparent magnitude indicates a brighter star.

## 3. What is the difference between apparent magnitude and absolute magnitude in a binary star system?

Apparent magnitude refers to the brightness of a star as seen from Earth, while absolute magnitude is a measure of the actual brightness of a star, regardless of its distance from Earth. In a binary star system, the apparent magnitude may change due to the changing distance between the two stars, while the absolute magnitude remains constant for each individual star.

## 4. How is the apparent magnitude of a star in a close binary system measured?

The apparent magnitude of a star in a close binary system is measured using a scale where the lower the number, the brighter the star appears. This scale is based on the ancient Greek astronomer Hipparchus' system, where he designated the brightest stars as magnitude 1 and the dimmest as magnitude 6. Modern astronomers use a more precise system with negative numbers for exceptionally bright objects.

## 5. Can the apparent magnitude of a star in a close binary system change over time?

Yes, the apparent magnitude of a star in a close binary system can change over time. This is because the distance between the two stars in the binary system can vary, causing the perceived brightness of the stars to change. Additionally, if one of the stars in the binary system undergoes a change in luminosity, it can also affect the apparent magnitude of the system.

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