# Apparent magnitude of two stars

• sanitykey
In summary, the apparent magnitude of the brighter star is 15.44 and the fainter star has an absolute magnitude of 4.50 and is 2.18Kpc away.f

## Homework Statement

Two stars have apparent magnitudes of V = 5.1 and V = 4.6 but are too close together to be resolved with the naked eye and appear to be a single object. What is its apparent magnitude?

## Homework Equations

I don't know what the relevant equations are if there are any.

## The Attempt at a Solution

Apparent magnitude = the sum of the two magnitudes = 9.7 but i know that's probably wrong, any help would be appreciated thanks!

Sorry if this is in the wrong place by the way.

Magnitudes don't add. Luminosities add. What's the relation between magnitude and luminosity?

m = -2.5*log(L/4*pi*d^2) + c

So L = 10^((m-c)/-2.5) * (4*pi*d^2)

So errm i think i was given the c for the visual in another question i think it's -21.58...

So L1+L2 = 10^((5.1+21.58)/-2.5)*(4*pi*d^2)+10^((4.6+21.58)/-2.5)*(4*pi*d^2) = 2.13*10^-11*(4*pi*d^2) + 3.37*10^-11*(4*pi*d^2)

= 5.50*10^-11*(4*pi*d^2)

m = -2.5*log(L/4*pi*d^2)+ c
m = -2.5*log(5.50*10^-11*(4*pi*d^2)/4*pi*d^2) -21.58
m = -2.5*log(5.50*10^-11) -21.58
m = 4.07

Is that right? I just thought woohoo i have a number which is different from the others but i don't know if it's right?

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It's right. You've got a lot of extra factors you don't really need in this particular problem, but I wouldn't worry about it.

Thanks for the help :)

I have two questions that are very very similar to the one above but I'm pretty sure I have no value for the constant.

Question 1, parts a & b

A binary star has a total apparent magnitude of 15.00. One component star is twice as bright as the other.

(a) Show that the apparent magnitude of the brighter star is 15.44. [2]

This one I can do, but don't understand why it works.

Ie - m + 2.5log(F) = mbright --> F= 1.5
15 + 2.5log(1.5) = 15.44

So why does that work?

(b) The fainter star has an absoloute magnetude of 4.50. How far away is the binary system in Kpc?

Then this next question I'm sure I could do if it wasn't for the confusion with part a but I have a mental block because of that. I know the distance equation D = 10 ^ (m-M)/5 x 10, but I need the apparent magnitude of the fainter star to work that out and I can't do it. I don't think you need to find the total apparent magnitude to find the answer but I could be wrong.

EDIT - Going on a complete guess is the apparent mag of the faint star 16.19 (using 15 + 2.5log(3) ) giving a distance of 2.18Kpc? Even if this is right, it's revision for an exam so I'd like to know how to do it rather than fluking my way there.

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You are getting the right answers but you don't know why? For problems like this you can simplify the luminosity magnitude relation to just M=(-2.5)*log(L). The real formula has an additive constant and the L is divided by some stuff, but that all cancels out. You have a bright star of luminosity 2L and a dimmer one at L. So the total magnitude of the system is 15=(-2.5)log(3L). The magnitude of the brighter is M2=(-2.5)log(2L). Take the difference of the two. M2-15=(-2.5)*(log(2)-log(3))=0.44 (use stuff like log(3L)=log(3)+log(L)). Similarly the magnitude of the dimmer is M1=(-2.5)log(L). Take the difference again, M1-15=(-2.5)*(-log(3))=1.19.

Thanks, that was really helpful. Knowing my luck it probably won't come up on the exam now I understand it :D.