Combined magnitudes of two solar-type stars

[Taylor_1989]

I am having a issue with part a) of this question, I am not entirely sure if what I am saying is correct, could somone if possible check my logic

1. Homework Statement

Two solar type stars (all physical properties are the same as for the Sun) are in a close binary system. Each individual star has an apparent brightness of $m=10$mag.

1a) Determine the apparent magnitude of the combined system of two stars, assuming we cannot resolve them as individual objects.

1b) Assume the system is an eclipsing binary (i.e. from time to time one of the stars passes in front of the other and covers some of its light). Determine the minimum brightness for this eclipsing binary when $100\%, 75\%, 50\%$and $25\%$ of the stellar surface are covered during the eclipse.

2a) Two solar type stars $(S1$ and $S2)$ are in an apparent binary system (i.e. projected at the same position onto the sky but not physically connected). The first of the two stars $(S1)$ is at a distance of $10$pc and has an apparent magnitude of $m=4.83$ mag.

Homework Equations

[1] $m_2-m_1=2.5log(\frac{f_1}{f_2})$ apparent mag
[2] $m_{1+2}-m_1=2.5log(\frac{f_1}{f_1+f_2})$ combined apparent mag
[3] $\frac{f_1}{f_2}=\frac{L_1}{L_2}\left(\frac{d_2}{d_1}\right)^2$
[4] $10^{\frac{\left(m_2-m_1\right)}{2.5}}=\frac{f_1}{f_2}$

The Attempt at a Solution

[/B]
So I have included the first question to this as I have assumed what is said the q1 would be the same in q2 which is ''Two solar type stars (all physical properties are the same as for the Sun)". with this I thought that if the two stars are the same except that they are at different distances the using [3] I could say that it dose not matter what distance each star is, there luminosity would remain the same, which to me the units also indicated this as it is $js^{-1}$, with this in mind I then did the following:

$m_2-m_1=2.5log\left(\frac{f_1}{f_2}\right)$

$m_2-m_1=2.5log\left(\frac{d_2}{d_1}\right)^2$

$m_2-m_1=5log\left(\frac{d_2}{d_1}\right)$

$m_2=m_1+5log\left(\frac{d_2}{d_1}\right)$



using the given values $m_1=4.83$ and $d_2=5,50,500$pc

$m_2=3.32$ for $5$pc

$m_2=8.32$ for $50$pc

$m_2=13.32$ for $500$pc

these values seem reasonable as the further the star moves moves away the dimmer it will seem.

So using [4] to find the flux ratios, I get the following

$10^{\frac{\left(3.32-4.83\right)}{2.5}}=0.24$ for $5$pc

$10^{\frac{\left(8.32-4.83\right)}{2.5}}=24.8$ for $50$pc

$10^{\frac{\left(13.32-4.83\right)}{2.5}}=2488.9$ for $500$pc

so using these ratios I can use and rearrange [2] to find the combine apparent mag

$m_{1+2}=3.32+2.5log\left(\frac{1}{1.25}\right)=3.1$mag

$m_{1+2}=8.32+2.5log\left(\frac{1}{25.9}\right)=4.79$mag

$m_{1+2}=13.32+2.5log\left(\frac{1}{2489.9}\right)=4.83$mag

So here my thinking for why these number are what they are.

If I Imagen two stars at these give distances the at the 5pc I should see a brighter combined magnitude as the stars a relatively close together. But as the star moves further from the other then the combined magnitude should reduce to the magnitude of $S1$ due to $S2$ moving so far away it would not matter.

 

[phyzguy]

You're making part (a) way more complicated than it needs to be. You have two stars, each of apparent magnitude m=10. The two stars together have twice the flux of each star alone. So the apparent flux is doubled. So what happens to the apparent magnitude when you double the flux?

 

[Taylor_1989]

You're making part (a) way more complicated than it needs to be. You have two stars, each of apparent magnitude m=10. The two stars together have twice the flux of each star alone. So the apparent flux is doubled. So what happens to the apparent magnitude when you double the flux?

Sorry I am referring to 2a) the reason I have included question 1 is because for question 2 I do not see a away of calculating the value that have been asked for unless I assume that they are both sun like stars.

I have just see I said a) in my post which make it unclear, sorry I can't seem to edit this as too much time has passed.

 

[phyzguy]

Sorry I am referring to 2a) the reason I have included question 1 is because for question 2 I do not see a away of calculating the value that have been asked for unless I assume that they are both sun like stars.

OK, you are asking about question 2a). Your OP lists question 2a) as:

2a) Two solar type stars (S1 and S2) are in an apparent binary system (i.e. projected at the same position onto the sky but not physically connected). The first of the two stars (S1) is at a distance of 10pc and has an apparent magnitude of m=4.83.

I don't see a question here. What are you being asked to find?

 

[Taylor_1989]

Sorry for the confusion when I pasted in the question from the doc it seem to have not been pasted all in, here is the actual question below:

Question:
2a) Determine the combined apparent magnitudes of the system S1+S2 if the star S2 is at a distance of $5,50,500$pc

 

[phyzguy]

OK, now I see what you have done. It looks correct to me.

 

[Taylor_1989]

Ah okay thank you.