Why Is 2r Used in Calculating the Masses in Plaskett's Binary System?

[unknown_2]

hey
i've been having trouble understanding a part in this question. i'll just map out the basics to the question: 2 stars with a velocity of 220km/s are orbiting at equal distance. the masses of the stars are equal. period = 14.4 days it asks to find the mass of each star.

i understand that u can use kepler's law to get the answer.

GMm/r² = mv²/r

-which is what i used to come up with my answer seeing that i found the mass relative to a center mass. but according to the solution it's suppose to be:

GMm/(2r)² = mv²/r

my question is, why is r denoted 2r in order to arrive at a correct answer?

 

[Redbelly98]

On the left hand side, you use the distance between the 2 stars.

On the right hand side, you use the distance from a star to the center of the circular orbit.

 

[thomate1]

I understand your confusion and I am happy to provide some clarification on the Plaskett's binary system. First, let's discuss the basics of this system. Plaskett's binary system is a binary star system, meaning it consists of two stars orbiting around a common center of mass. In this case, both stars have equal mass and are orbiting at the same distance. The period of their orbit is 14.4 days, meaning it takes 14.4 days for the stars to complete one full orbit.

Now, to answer your question about the use of "2r" in the equation. This is because in Kepler's third law, the distance between the two bodies (r) is measured from the center of one body to the center of the other. However, in Plaskett's binary system, the distance between the two stars is measured from the center of one star to the center of the other, which is twice the distance between the two bodies. Therefore, to accurately calculate the mass of each star, we need to use "2r" in the equation instead of just "r". This ensures that we are taking into account the distance between the two stars and not just the distance between their centers.

I hope this explanation helps you better understand the solution and the use of "2r" in the equation. If you have any further questions, please don't hesitate to ask. Keep up the good work in your studies!