Question

Two binary stars, mass 10-kg and 2x10 kg respectively rotate about
their common centre of mass with an angular velocity. Assuming
that only force on a star is the mutual gravitational force between
them, calculate angular velocity of rotation if the distance between
the stars is 10° Km. ​

Answer 1

Answer:

Deriving Kepler's Formula for Binary Stars

Explanation:

Your astronomy book goes through a detailed derivation of the equation to find the mass of a star in a binary system. But first, it says, you need to derive Kepler's Third Law.

Consider two bodies in circular orbits about each other, with masses m1 and m2 and separated by a distance, a. The diagram below, shows the two bodies at their maximum separation. The distance between the center of mass and m1 is a1 and between the center of mass and m2 is a1.

Diagram showing two bodies in circular orbits about their center of mass.

Answer 2

Answer:

Solution,

mass of 1st star(m1)=〖10〗^20 kg

Distance between the centers(r)=〖10〗^6 m

G = 6.67 ×(〖10〗^(-11) Nm^2)/(kg^2 )

Let the centre of masses be at a distance r_1 and r_2 from 1st and 2nd star respectively. Then,

r_1+r_2=r…………..(i)

Again, centripetal force required for 1st star to move about the common centre of mass(F_1 )=F_1 ω^2 r_1

Centripetal force required for 2nd star to move about the common centre of mass(F_2 )=F_2 ω^2 r_2

Since F_1 and F_2 are equal .

m_1 ω^2 r_1=m^2 ω^2 r_2

or,m_1 r_1=m_2 r_2

or,〖10〗^20 r_1=2×〖10〗^20 r_2

〖or,       r〗_1=2r_2

or,〖 r〗_1-2r_2=0……………(ii)

From equation (i) and (ii)

r_1-2r_2=0

r_1+r_2=r

- -       -

¬-3r_2=-r

or,r_2=r/3

∴r_1=2r/3

Since F_1 and F_2 is provided by gravitational force of attraction between the two stars,

F_2=(Gm_1 m_2)/r^2  

or,m_2 ω^2 r_2=(Gm_1 m_2)/r^2  

or,ω^2×r/3=(Gm_1)/r^2  

or,ω^2=3G×m_1/r^3  

∴ω=1.41×〖10〗^(-4) rads^(-1)

Explanation: