Question about calculating the ratio of volumes of two components of an eclipsing binary star

I'm trying to solve this task:

At the maximum, the eclipsing binary star has a brightness of 6^m, and at a minimum of 8^m. Considering the eclipse to be central and the companion to be dark, find the ratio of the volumes of the components of this pair.

This probably has something to do with this formula but I'm not sure:

E = L/S = L/4πr2

Edit: I've understood that I can find the radius ratio of the star and it's component and then find the volume ratio. I've started calculating but got weird result. I have a square root with a negative value inside.

Could you please help me find mistake? Thanks for any help.

• If the eclipsing star is dark, then the ratio of brightness is going to be proportional to the ratio of visible emitting areas (ignoring limb darkening). Aug 5 at 6:50

1 Answer

You are using the formula in an incorrect way. The meaning of the formula is the following: the flux we receive on Earth is $$F = \frac{L}{4\pi d^2}$$ where $$L$$ is the total luminosity of the star and $$d$$ is the distance from the star to Earth, not the radius of the star.

You may see that this formula is not particularly useful to solve your problem, since it doesn't contain the radius of the star.

You might instead consider that the total luminosity is proportional to the visible surface of the star, that appears to us as a circle not as a sphere. Therefore the visible surface will be $$\pi R^2$$, not $$4\pi R^2$$.

The ratio of the visible surfaces is equal to the ratio of luminosities, that you can then convert in magnitudes.