Binary Star System: Calculating Occultation Angle & Eclipse Duration

[RHK]

Homework Statement

Two stars, S₁ and S₂, with effective temperature of T_e,1=25000 K and T_e,2= 2000 K and photospheric radius of R₁=R_sun/100 and R₂=50 R_sun make a binary system.
If the Star S₂ is in circular orbit around the S₁ with revolution period of P=180 days (1 day=24 hours), and S₁ has a mass of M₁=1.3 M_sun, while the inclination angle i is the angle between the normal to orbital plane toward the observer and the visual line of the terrestrial observer (0°<i<90°), calculate:

(i) the limit angle, beyond which there will be occultation between the two stars;

(ii) if there is occultation, is the primary eclipse (more diminution of bolometric luminosity) when S_s is between S₁ and the observer, or viceversa?

(iii) if i=90° how long the total eclipse is?

Homework Equations

R_sun and M_sun are given.
Third Keplero's Law: P² (M₁+M₂)=$1frac{4\pi^2}{G}$d³ where P is the period and d the distance between the two stars.

The Attempt at a Solution

Of course, an eclipse/occultation can occur if i ≈ 90°.
The exact angle depends on the stars dimension, i guess.
Bigger the star is (in radius), lower the angle can be.
However, i can't understand how to relate the radii and the angle.
Any suggestion please?

 

[RHK]

This is my new attempt:

Assuming circular motion, the equation that rules the system is the equilibrium between graviational attraction toward the massive star in the center M₁ and the centrifugal force of the arounding star M₂

$GM_1M_2 / (d^2) = M_2 (v^2)/(d) = M_2 \omega v$

that can be rewritten in terms of only one unknown: $GM_1 = (v^2)/(d) = \omega v = \omega d$

where d is the distance between the stars, and $\omega$ is the angular velocity, obtainable as the frequency of the period: $P=2\pi/\omega$.

This allows to obtain the distance d, and to write the goniometric relation between the distance, the obscuration angle and the radiius of the star S₂, imaging a circle with the center in the center of the star S₁:

$d sin \theta = R_2$

That is, $\theta$ is the angle "occuped" by the star radius, and it has to be "free" to avoid obscuration.
When the second star S₂, more distant, passes in front of the star S₁, then there is obscuration.

So, the limit angle is: $\theta_{lim}= 90°- \theta=20°$ according to the other counts.

If my following reflections about this will confirm this way to proceed, i will post the second point too.
Regards.