# Binary star system

1. Oct 8, 2011

### RHK

1. The problem statement, all variables and given/known data

Two stars, S1 and S2, with effective temperature of Te,1=25000 K and Te,2= 2000 K and photospheric radius of R1=Rsun/100 and R2=50 Rsun 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 Msun, 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 Ss is between S1 and the observer, or viceversa?

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

2. Relevant equations

Rsun and Msun 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.

3. 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.

2. Oct 10, 2011

### 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.

Last edited: Oct 10, 2011