What is the apparent magnitude of the binary system?

[PaulWright]

Homework Statement

Two stars are in a circular visual binary system. The orbital
period of the binary is 30 years. The distance to the binary is 20
parsecs. The angular radius of the orbit of each star is 1". What
are the masses of the two stars?

each star works out to be 4.4 solar mass

2. From your result above, what should be the apparent magnitude of
the binary as a whole? You may assume that both stars are on the main
sequence, and that the standard mass-luminosity relationship applies
here. You may further assume that the absolute magnitude of the Sun
is 5.7. (Note: full credit for this question will be given for
answers that follow the right prodcedure, but start with an incorrect
answer to question 1).

Homework Equations

m=M+5(log(d)-1)

The Attempt at a Solution

mass of the whole binary system is 8.8M(sun)

I have no idea what the mass luminosity relationship is.

I just need some help getting started.

Cheers,
Paul

 

[cepheid]

You'll learn about it if you take a course in stellar astrophysics. For now, this article has everything you need to know:

http://en.wikipedia.org/wiki/Mass-luminosity_relation

This relation will tell you the luminosity of each star in units of solar luminosities. Then, since you have the ratio of the combined luminosity of both stars to the luminosity of the sun, you can figure out the difference in absolute magnitude between two of these stars and the sun. Since you are given the abs. magnitude of the sun, this in turn tells you the absolute magnitude of the two binaries. You can then use that to determine the apparent magnitude of the system using the formula you posted.

 

[PaulWright]

You'll learn about it if you take a course in stellar astrophysics. For now, this article has everything you need to know:

http://en.wikipedia.org/wiki/Mass-luminosity_relation

This relation will tell you the luminosity of each star in units of solar luminosities. Then, since you have the ratio of the combined luminosity of both stars to the luminosity of the sun, you can figure out the difference in absolute magnitude between two of these stars and the sun. Since you are given the abs. magnitude of the sun, this in turn tells you the absolute magnitude of the two binaries. You can then use that to determine the apparent magnitude of the system using the formula you posted.

L/L_sun = 2022
-25.9=-2.5log\frac{F_{sun}}{F_0}\\
m=-2.5log\frac{F}{F_0}\\
equating
\frac{10^{10.36}=F_{sun}}{10^{\frac{m}{-2.5}}=F}\\
\frac{10^{10.36}}{10^{\frac{m}{-2.5}}} = 2022
therefore
m=-17.64

Can you please advise me on what I have done wrong here?
(I was having problems inserting the latex in the tex tags, sorry)
Cheers.

 

[cepheid]

1. Your luminosity is wrong. You should compute the luminosity of each star separately using the M-L relation and THEN add those L's together. This is not the same as adding up the masses and then applying the M-L relation (which is wrong). A single 8.8 solar mass star is significantly more luminous than two 4.4 solar mass stars. This should be obvious, because the M-L relation is NON-linear (just look at it).

I'll get back to you on some of the later steps in a bit.

 

[PaulWright]

ah that does make a lot more sense. So the ratio of the L/Lsun = 357.4

 

[cepheid]

2. Okay, for the later steps of the problem, I think that what you're doing wrong is that you first have to compute the absolute magnitude of the binary system, and then use the distance modulus formula to convert that into an apparent magnitude.

To compute the absolute magnitude, it suffices to use the luminosities. You shouldn't have to deal with anything related to fluxes. After all, absolute magnitude IS luminosity in the sense that it is just another way of representing the same quantity -- namely the intrinsic energy output rate of a body (in the form of light). So let's say you have computed the luminosity (L) of a single component of the binary in units of solar luminosities. In other words, you have computed:

$$\frac{L}{L_\odot}$$​

You can get the difference in the absolute magnitudes of the binary system and the sun as follows:

$$M_{\textrm{bin}}-M_{\textrm{sun}}= -2.5\log\left(\frac{L_{\textrm{bin}}}{L_\odot}}\right) = -2.5\log\left(\frac{2L}{L_\odot}\right)$$​

Here, $M_{\textrm{sun}}$ is the absolute magnitude of the sun (as opposed to $M_\odot$, which is the mass of the sun). So, differences in absolute magnitude measure ratios of luminosities. In general for two objects:

$$M_1-M_2= -2.5\log\left( \frac{L_1}{L_2}\right)$$​

If you don't believe me, you should work out this result yourself. Hint: start with the definition of absolute magnitude as "the apparent magnitude that an object would have if it were at a distance of 10 pc."

 

[PaulWright]

2. Okay, for the later steps of the problem, I think that what you're doing wrong is that you first have to compute the absolute magnitude of the binary system, and then use the distance modulus formula to convert that into an apparent magnitude.

To compute the absolute magnitude, it suffices to use the luminosities. You shouldn't have to deal with anything related to fluxes. After all, absolute magnitude IS luminosity in the sense that it is just another way of representing the same quantity -- namely the intrinsic energy output rate of a body (in the form of light). So let's say you have computed the luminosity (L) of a single component of the binary in units of solar luminosities. In other words, you have computed:

$$\frac{L}{L_\odot}$$​

You can get the difference in the absolute magnitudes of the binary system and the sun as follows:

$$M_{\textrm{bin}}-M_{\textrm{sun}}= -2.5\log\left(\frac{L_{\textrm{bin}}}{L_\odot}}\right) = -2.5\log\left(\frac{2L}{L_\odot}\right)$$​

Here, $M_{\textrm{sun}}$ is the absolute magnitude of the sun (as opposed to $M_\odot$, which is the mass of the sun). So, differences in absolute magnitude measure ratios of luminosities. In general for two objects:

$$M_1-M_2= -2.5\log\left( \frac{L_1}{L_2}\right)$$​

If you don't believe me, you should work out this result yourself. Hint: start with the definition of absolute magnitude as "the apparent magnitude that an object would have if it were at a distance of 10 pc."

Cheers.
Can I confirm that the absolute magnitude is 0.6830?

 

[PaulWright]

I mean -0.6830

 

[cepheid]

I mean -0.6830

That's also what I get, assuming the 4.4 solar masses is right.

 

[PaulWright]

That's also what I get, assuming the 4.4 solar masses is right.

Cheers.