# Photometry: Determining a binary star system.

## Homework Statement

By performing CCD photometry on a pair of nearby stars A and B we obtain their relative magnitudes in the V filter and their colors:

Star A: mV = 8.70 , (B − V )= 1.30
Star B: mV = 11.90 , (B − V )= 1.81

Star A is known to be a of a main sequence K0V type, while no other information on Star B is available.

Argue if this consists of a visual binary system or if B is a background star.

## The Attempt at a Solution

$$m_{\lambda}(N stars) = -2.5log(\sum10^{.0.4m_{\lambda}_i})$$
Not sure what to do. Using the expression above, I found the apparent magnitude for the system as a whole and it came out brighter than A (as expected), but by very little: 8.645. What do I need to look for to determine if its a binary or not?

## Answers and Replies

cepheid
Staff Emeritus
Science Advisor
Gold Member
It seems to me that, based on the colour of star B, you can figure out it's spectral type. At the very least, you know that it is redder than K0. Now, if it's a binary, then you also know that both stars are at the same distance from Earth. Therefore the difference in their apparent magnitudes is the same as their difference in absolute msgnitudes. So, under this assumption, you know how much dimmer (intrinsically) star B is than star A. Ask yourself whether it makes sense that a star on the main sequence that is 0.5 mag redder than a K0 star would also be that much dimmer. If it does make sense, then this can be a binary. If it doesn't, then the only explanation must be that star B is in fact much farther away.

It seems to me that, based on the colour of star B, you can figure out it's spectral type. At the very least, you know that it is redder than K0. Now, if it's a binary, then you also know that both stars are at the same distance from Earth. Therefore the difference in their apparent magnitudes is the same as their difference in absolute msgnitudes. So, under this assumption, you know how much dimmer (intrinsically) star B is than star A. Ask yourself whether it makes sense that a star on the main sequence that is 0.5 mag redder than a K0 star would also be that much dimmer. If it does make sense, then this can be a binary. If it doesn't, then the only explanation must be that star B is in fact much farther away.

Thanks for the quick reply. How do I know that star B is redder than K0? Sorry if this sounds elementary, I'm just getting acquainted with calculating color indices, magnitudes etc.

I'm attaching the table provided for problem-solving, it claims different values for a K0V star(I think everything is shifted by one unit), and according to the table, star B is a G0V type (comparing (B-V) indices), which would imply it is brighter than K0V on the absolute scale. (am I doing this right?)

But on the apparent mag. scale star A is brighter, hence it is much closer than star B, am I on the right track? How far apart (or different in apparent magnitude) do they need to be in order to discard the possibility of it being a binary system?

#### Attachments

• table.pdf
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cepheid
Staff Emeritus
Science Advisor
Gold Member
Thanks for the quick reply. How do I know that star B is redder than K0? Sorry if this sounds elementary, I'm just getting acquainted with calculating color indices, magnitudes etc.

I know star B is redder because its B-V colour index is larger. Here is how to interpret the colour index:

B is the star's apparent magnitude in the B (blue) photometric band, which, if I recall correctly, uses filters centred on ~400 nm wavelength.

V is the star's apparent magnitude in the V (visual) photometric band, which, if I recall correctly, uses filters centred on ~550 nm wavelength.

If B-V is postiive, it means that B > V. Recall, that larger apparent magnitude = dimmer. Therefore, having B-V > 0 means that the star's observed B-band brightness is less than the its V-band brightness. The larger the colour index value is, the less emission is being received in the B-band relative to the V-band, and the (hence we infer) the redder the emission spectrum of the object must be. Since star B's colour index of 1.81 is larger than star A's colour index of 1.30, we conclude that star B has a redder spectrum (more emission at longer wavelengths, less at shorter wavelengths).

I'm attaching the table provided for problem-solving, it claims different values for a K0V star(I think everything is shifted by one unit), and according to the table, star B is a G0V type (comparing (B-V) indices), which would imply it is brighter than K0V on the absolute scale. (am I doing this right?)

No, I don't think you're doing this right. Even if everything were shifted by 1 unit, I have no idea how you'd get a spectral type G0V for star B, that just doesn't make any sense whatsoever. In any case, I don't think that everything is shifted by 1 unit. If it were, then the value of K0's B-V index in the table would be 0.30. It is not. The reason why the observed K0 colour index stated in the problem is greater than the theoretical one in the table is probably because of interstellar reddening. However, if both stars are in a binary, then they're at the same distance and hence you can assume they're both reddened by the same amount. So, although both of their colour indices will have changed, the difference between their colour indices will be the same as it was without the reddening. From what I just said in the previous sentence, since star B's colour index is about 0.5 mag larger than a K0 star, the table would seem to indicate that it is of type K7.

So, under the assumptions we've made, star B must be of type K7. Now, compare the absolute magnitudes of K0 and K7 main sequence stars. According to the table, a K7 is only 2.2 mag dimmer than a K0. Yet, the observations show that star B is a whole 3.2 mag dimmer than star A. It doesn't fit where it should on the main sequence for a K7 star. The only thing we can conclude is that our initial assumption about the stars being at the same distance must have been wrong.