Question

op

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 (see figure below). 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. More Information solar masses

Answer 1

The idea is to use two concepts:

circular motion:     F = m v² /r

and gravitational force:    F = G mm / d²    

Notice that in this case,   d =2r     so we can write

      G m m / (2r)² = m v² / r

        G m /4r = v²         

but what is r?      well, you aregiven the speed and time time to orbit once, so...

   time = 12.5 days = 1080000 seconds

   speed =   240000 m/s

distance for one orbit is    2pi r      which isalso    speed * time    so

     2pi r = 240000 *1080000

       r = 4.127 x10¹⁰ meters

And now...

   m = 4 r v² / G  = 4 * 4.127 x 10¹⁰ * 240000² / 6.67 x10^-11 =

        =   1.425x 10³²   kg

Our sun has mass    1.99 x10³⁰ kg so this star is

    142.5x 10³⁰ / 1.99 x10³⁰ =    71.64 timesmore massive than our sun

Answer 2

Orbital period = 12.5*24*60*60 s = 1080000 s

Speed = 240 km/s

So, R = V*t / 2*pi = 41252961249.42 m

So, M*v^2/R = GM^2/(2R)^2

So, M = 4(v^2)R/G = 1.425*10^32 kg

Answer 3

G*M^2/(2R)^2=Mv^2/R

==> M=4v^2*R/G

and R=v*t/2*pi