Problem about binary star system

[yoyoz41]

Problem about "binary star system"

Homework Statement

About half of the visible "stars" are actually binary star systems, two stars that orbit each other with no other objects nearby. Consider the motion of the center of mass of a binary star system. For a particular binary star system, telescopic observations repeated over many years show that one of the stars (whose unknown mass we'll call M1) has a circular orbit with radius R1 = 4 1011 m, while the other star (whose unknown mass we'll call M2) has a circular orbit of radius R2 = 11 1011 m about the same point.

This double star system is observed to complete one revolution in 41 years. What are the masses of the two stars? (For comparison, the distance from Sun to Earth is about 1.5 1011 m, and the mass of the Sun is about 2 1030 kg.) This method is often used to determine the masses of stars. The mass of a star largely determines many of the other properties of a star, which is why astrophysicists need a method for measuring the mass.

M1 = ?
M2 = ?

Homework Equations

Momentum principle
p(final)= p(initial) + F*t

Energu principle
E(final)= E(initial)+W+Q

The Attempt at a Solution

I have no clue how to solve this problem

 

[Shooting Star]

The centre of mass of the system lies on a line joining the stars. The stars are taken to be point masses, moving in circular orbits about the CM of the system. Obviously, the time period T of the orbits about the CM is the same. The distance of star m1 is r1 from the CM etc.

Equate the centripetal force acting on star m1 to the gravitational force on star m1 due to star m2:

Gm1m2/(r1+r2)^2 = m1*ω *(r1^2) and then put ω = 2π/T. You get m2 in terms of the things given.

 

[yoyoz41]

so m2 = (2π/T) *(r1^2)*(r1+r2)^2 / G , and I transfer 40 years into second

but it's not the right answer, can u show me where I did wrong?

 

[Shooting Star]

That's a typo on my part. It should be m1*(ω^2) *r1. Sorry for that, but you should derive the formula yourself, not copy me blindly. (The centripetal force is mv^2/r and v=rω.)

Now write the correct expression before plugging in the values and you'll get the correct answer.

 

[yoyoz41]

haha thanks a lot
now I know what to do

 

[jchojnac]

I have m1=5e11, m2=8e11, and t=47years. I plugged that into the formula to find m2:
m2 = ((2n/T)^2 * r1 * (r1+r2)^2) / G
m2 = 2.3067e28

...and I got the answer wrong. Am I doing it wrong or making calculation errors?

 

[fball558]

i see many people are having trouble on the binary star system question haha. ME TOO i posted a thread did not know this was here. i had a question for you about the
ω = 2π/T part. what is the n and the T?
is T just the time it takes to revolve 1 time? would this be in seconds or keep it in years?

 

[jchojnac]

i converted the 47 years to seconds, and used 1 for n, but I got it wrong so I'm not sure.

 

[fball558]

the n i think is really a (pi)
then T is Period
but i still get it wrong as well
who knows?

 

[jchojnac]

n = pi and t is the years converted to seconds

 

[jchojnac]

how would you compute m1 after getting m2?

 

[fball558]

i thought that is what i did.
what is your equation your using for M2?
i have M2 = (2n/T)*(R1)*(R1*R2)^2 / G
is this right? do you divide G only by (R1*R2)^2
i though it was by the whole thing.

 

[jchojnac]

never mind, that was a stupid question

 

[jchojnac]

((2n/T)^2*r1*(r1+r2)^2)/G

You had it right except you forgot to square the (2n/t)

 

[fball558]

ok now how do you get M1?
i used
Gm1m2/(r1+r2)^2 = m1*ω *(r1^2) plugged M2 value in and solved.
still says wrong. is one of my powers off or something?

 

[jchojnac]

to calculate m1 it's the same as m2 except you use the r2 value instead of r1 in the equation.

 

[fball558]

wow i feel dumb lol
thanks to all that helped!