Determining inclination angle of interacting binaries

Question

Consider an interacting binary system. If the plane of their orbits is inclined by some angle $\theta$ around its semi major axis $a$ and some angle $\phi$ around its semi minor axis, is there a situation that when observed from earth it is not possible to determine whether the orbital plane is inclined or not?

As far as I know inclination angle is determined by determining the displacement in the foci of the ellipse relative to the center of ellipse therefore I thought if $ae$ is not changed then it is not possible to determine inclination angle, using the following relation in ellipses:$$(ae)^2 = b^2 + a^2$$ $$a^2 + b^2 = a^2\cos^2(i) + b^2\cos^2(n)$$ $$\therefore \frac{a}{b} = \frac{\sin(n)}{\sin(i)}$$

I conclude if the above condition holds then it is not possible to determine whether the observed ellipse is inclined or not, I was wondering whether this conclusion is valid or not.

Also I wanted to know if Kepler's second and third law are affected by inclination angle and if they are can one determine the inclination angle using Kepler's laws?

Answer 1

I think you are referring to the orbits of visual binaries, since you are asking about parameters measured from an ellipse.

If the orbital plane were at right angles to the line of sight, then the focus of the ellipse defined by the motion of the secondary star would be coincident with the primary star. When you change the inclination, this is not the case, and the displacement of the measured focus from the position of the primary star will tell you what the inclination is to within an ambiguity of whether it is bigger or smaller than 90 degrees. This latter ambiguity is resolved from the direction of revolution of the secondary.