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`

Step-by-step 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 = (Mv^2)/(r)`

The gravitational force acting on these stars is mathematically represented as

`F_g = (GM^2 )/(d^2)`

So
`F_c = F_g`

=>
`(mv^2)/(r) = (Gm_1 * m_2 )/(d^2)`

=>
`(v^2)/(r) = (GM)/((2r)^2)`

=>
`(v^2)/(r) = (GM)/(4r^2)`

=>
`M = (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= (2.592*10^(11))/(2 \pi )`

=>
`r= 4.124 *10^(10) \ m`

So

`M = (v^2*4r)/(G)`

=>
`M = ((240000)^2*4*(4.124*10^(10)))/(6.67*10^(-11))`

=>
`M =1.43 *10^(32) \ kg`