Answer to Question #131134 in Mechanics | Relativity for Khethiwe

Question

Sirius is a binary star system consisting of two stars (one bright and the other very small and dim). The mass of the bright star is 2.1 Mo and the mass of the small dim star is 1.05 Mo. The semi-major axis ‘a’ for Sirius system is about 20 AU. Calculate the period of such system as the stars rotate about their center of mass. (Note : Mo is the Solar Mass and 1 AU is the Earth-Sun distance)

Answer 1

Explanations & Calculations

• According to the Kepler's first law the periodic time (T) of a circular motion around another planet in an elliptical path of semi major axis (a) is given by,

$\qquad\qquad \begin{aligned} \small T^2 &= \small \frac{4\pi^2}{GM}a^3 \end{aligned}$

• We need to find the corresponding values for a & M to calculate the periodic time of each& assume circular paths instead of elliptical paths.
• For these kind of systems a is defined as the semi major axis of the system if one of the two planets is made virtually stationary ; that is the distance between the two stars then.
• There are several equations related to binary system of planetary motion.
• The M for this system is total mass = 3.15M_o

• And the period of the system is given by

$\qquad\qquad \begin{aligned} \small T^2_{system} &= \small \frac{4\pi^2}{G(m_1+m_2)}a^3\\ &= \small \frac{4\pi^2}{G(3.15M_o)}(20A_u)^3\\ &= \small (2539.683)\frac{4\pi^2}{GM_0}A_u^3\\ \small T&= \small \sqrt{(2539.683)\frac{4\pi^2}{GM_0}A_u^3} \end{aligned}$

• If $\sqrt{\frac{4\pi^2}{GM_0}A_u^3}$ is considered in years then the period of the Sirius system can be calculated to be 50.395 years.