Question

Plaskett's binary system consists of two stars that revolve In a circular orbit about a center of mass midway between them. This statement implies that the masses of the two stars are equal . Assume the orbital speed of each star is |v | = 240 km/s and the orbital period of each is 12.5 days. Find the mass M of each star. (For comparison, the mass of our Sun is 1.99 times 1030 kg Your answer cannot be understood or graded.

Answer 1

Complete Question

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Answer:

The mass is    $M =1.43 *10^{32} \ kg$

Explanation:

From the  question we are told that

       The mass of the stars are $m_1 = m_2 =M$

        The orbital speed of each star is  $v_s = 240 \ km/s =240000 \ m/s$

         The orbital period is $T = 12.5 \ days = 12.5 * 2 4 * 60 *60 = 1080000\ s$

The centripetal force acting on these stars is mathematically represented as

      $F_c = \frac{Mv^2}{r}$

The gravitational force acting on these stars is mathematically represented as

      $F_g = \frac{GM^2 }{d^2}$

So  $F_c = F_g$

=>        $\frac{mv^2}{r} = \frac{Gm_1 * m_2 }{d^2}$

=>      $\frac{v^2}{r} = \frac{GM}{(2r)^2}$

=>      $\frac{v^2}{r} = \frac{GM}{4r^2}$

=>    $M = \frac{v^2*4r}{G}$

The distance traveled by each sun in one cycle is mathematically represented as

     $D = v * T$

      $D = 240000 * 1080000$

      $D = 2.592*10^{11} \ m$

Now this can also be represented as

      $D = 2 \pi r$

Therefore

                  $2 \pi r= 2.592*10^{11} \ m$

=>   $r= \frac{2.592*10^{11}}{2 \pi }$

=>    $r= 4.124 *10^{10} \ m$

So  

       $M = \frac{v^2*4r}{G}$

=>    $M = \frac{(240000)^2*4*(4.124*10^{10})}{6.67*10^{-11}}$

=>    $M =1.43 *10^{32} \ kg$