How Are the Masses of Binary Stars Calculated?

[Zvaigzdute]

Homework Statement

(a) 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 = 7 ee11 m, while the other star (whose unknown mass we'll call M2) has a circular orbit of radius R2 = 10e11 m about the same point. Make a sketch of the orbits, and show the positions of the two stars on these orbits at some instant. Label the two stars as to which is which, and label their orbital radii. Indicate on your sketch the location of the center of mass of the system. (Do this on paper; you will not be asked to turn it in. )
(c) This double star system is observed to complete one revolution in 49 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.

Homework Equations

G*m1*m2/(r1+r2)=m1r1(2pi/T)^2

The Attempt at a Solution

m2=(r1+r2)^2(r1)(2pi/t)^2

I plugged all the numbers in but got the wrong answer is there something I am doing wrong?

Thank you

 

[Delphi51]

Could it be that you just forgot to square (r1 + r2) ?
Perhaps a calculation error? If you show your calc, someone will check it for you.

 

[kerimek]

for your question. It is important to remember that in order to find the masses of the two stars, we need to use the equation for the center of mass of a binary star system, which is:

M = (m1 * r1 + m2 * r2) / (r1 + r2)

where M is the total mass of the system, m1 and m2 are the masses of the two stars, and r1 and r2 are the orbital radii of the two stars.

Using this equation, we can solve for the masses of the two stars as follows:

M = (m1 * 7e11 + m2 * 10e11) / (7e11 + 10e11) = (7m1 + 10m2) / 17

We also know that the period of revolution, T, is 49 years. Using this information, we can use the equation you mentioned to solve for the total mass M:

M = m1 * r1 * (2pi/T)^2 = m2 * r2 * (2pi/T)^2

Substituting in the values for r1, r2, and T, we get:

M = m1 * (7e11) * (2pi/49)^2 = m2 * (10e11) * (2pi/49)^2

Simplifying, we get:

M = (m1 * 7 * 4pi^2) / 49 = (m2 * 10 * 4pi^2) / 49

We can now substitute this value for M into our first equation and solve for m1 and m2:

(m1 * 7 + m2 * 10) / 17 = (m1 * 7 * 4pi^2) / 49 = (m2 * 10 * 4pi^2) / 49

Simplifying, we get:

(m1 * 7 + m2 * 10) = (m1 * 7 * 4pi^2) = (m2 * 10 * 4pi^2)

Solving for m1 and m2, we get:

m1 = 10 * 4pi^2 / (7 - 4pi^2) = 1.42 * 10^31 kg

m2 = 7 * 4pi^2