Binary Star System: Calculating Occultation Angle & Eclipse Duration

In summary, the conversation discusses a binary star system with two stars, S1 and S2, with different effective temperatures and photospheric radii. It also mentions the revolution period, mass, and inclination angle of the stars, and asks for calculations related to eclipse and occultation. The equations used include the Third Kepler's Law and the equilibrium between gravitational attraction and centrifugal force. From these calculations, it is determined that the limit angle for occultation is 20° and that the primary eclipse occurs when S2 is between S1 and the observer. If the inclination angle is 90°, the total eclipse lasts for the entire revolution period.
  • #1
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Homework Statement



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?

Homework 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.

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?
 
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  • #2
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

[itex]GM_1M_2 / (d^2) = M_2 (v^2)/(d) = M_2 \omega v[/itex]

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

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

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:

[itex]d sin \theta = R_2 [/itex]

That is, [itex]\theta [/itex] 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: [itex]\theta_{lim}= 90°- \theta=20°[/itex] 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.
 
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FAQ: Binary Star System: Calculating Occultation Angle & Eclipse Duration

1. What is a binary star system?

A binary star system is a system of two stars that are gravitationally bound and orbit around a common center of mass. These stars can be of similar or different sizes and can have various orbital patterns.

2. How is the occultation angle calculated in a binary star system?

The occultation angle is the angle between the line of sight from the observer to the primary star and the line connecting the primary and secondary stars. It can be calculated using trigonometry and the orbital parameters of the system, such as the separation between the two stars and their orbital period.

3. What is an eclipse duration in a binary star system?

An eclipse duration is the length of time that one star in a binary system is blocked from view by the other star. This can occur during an occultation or when one star passes in front of the other, causing a total or partial eclipse.

4. How does the distance between the two stars affect the occultation angle and eclipse duration?

The distance between the two stars directly affects the occultation angle and eclipse duration. A smaller distance will result in a larger occultation angle and shorter eclipse duration, while a larger distance will result in a smaller occultation angle and longer eclipse duration.

5. Can we observe binary star systems from Earth?

Yes, binary star systems can be observed from Earth with the help of telescopes and other astronomical instruments. Many binary star systems have been discovered and studied, providing valuable insights into stellar evolution and the dynamics of multi-star systems.

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