Calculating magnitude

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Hey guys.

I'm struggling with a question I need practice with. Spent ages on this now and even asked for help from a friend with a first degree in maths and we're still not too sure. Hopefully someone might be able to help.. Here we go.

Two stars in a binary system. S1 and S2. Combined light totals abvout 5.3 apparent magnitude. S2, the smaller star has a magnitude of 8.01 and contributes 8% to the combined magnitude.

Using (m1 - m2) = -2.5 log (b1/b2), prove that the apparent magnitude of S2 is in fact 8.01....

Any ideas?

Thanks a bunch!

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OK, here's a go:

m1 - m2 = -2.5 log10 (b1/b2)

m1 = m2 - 2.5 log10 (b1/b2)

now write bc = b1 + b2 for the combined brightness

so m1 = m2 - 2.5 log10  [(bc - b2)/b2 ]

= m2 - 2.5 log10 (bc/b2 - 1)   (1)

so we just need the brightness ratio of the combined brightness to the mag of S2

bc/b2 = 2.5^(s2-sc) = 2.5^(8.01 - 5.3) = 11.9789

Plugging this into (1) we get

m1 = 8.01 - 2.5 log10 (10.9789) = 5.41

Now a case of checking if the brightness ratio is as you expected...

Martin

Edited by Martin Meredith
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My reply didn't seem to post.....

I'm very grateful for this, but I'm soooooo lost. I've actually done something completely different with help frm someone else.

Here's what I have....

Using Fig 1, the magnitude of S1 and S2 combined is approximately 5.3.

So the difference in magnitudes of S1 and S2 is:

(m1 – m2) = - 2.5 log (b1/b2)

So, 5.3 – m2 = -2.5 log (100/8)

= 2.74 magnitude

8.01 magnitude – 2.74 magnitude

= 5.27 magnitude

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If S2 contributes 8% to the total then S1 contributes the rest, so the ratio of b1/b2 is 92/8 rather than 100/8, I imagine. That way you end up with a magnitude for S1 that is smaller than the combined magnitude.

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The other rule you can use is the one to compute the combined magnitude of two stars, namely

mtotal = m2 - 2.5 log10 (10^(0.4(m2-m1)) + 1)

Doing so with m1 = 5.4 and m2 = 8.01 gives mtotal of just over 5.3

cheers

Martin

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Hey Martin, just wanted to say thanks for your help. I don't even think I know how to type that into the calculator to get mtotal. Guess I'm in a little over my head.

This is what I ended up with, but no idea if I'm right.

(m1 – m2) = - 2.5 log (b1/b2)

The ratio of b1 / b2 = 92/8 as we know S2 is only 8% of the combined magnitude = 11.5 So,

(m1 – m2) = -2.5 log (11.5)

= -2.65

We can then go further and calculate the apparent magnitude of S1 as we know its absolute magnitude to be 4.85 and its distance to be 12.82 pc.

m1 = M + 5 log (d/10pc) So,

m1 = 4.85 + 5 log (12.82/10pc)

= 5.39 magnitude

So we now know both apparent magnitudes and can prove 8.01 is in fact correct for S2 if it contributes only 8% to the combined magnitude of 5.25.

(m1 – m2) = -2.5 log (b1/b2) So,

5.39 – m2 = -2.65

In other words,

5.39 = -2.65 + m2

m2 = 5.39 + 2.65 = 8.04 magnitude which is close enough to our value of 8.01

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