Question on binary stars & binary stars

[mikky_t]

Hi.
What a great forum, so many interesting things

I am facing a problem with the working out for my last question in my practice paper, for my mid-year examinations.

i don't know how to write the notation here, so i have just attatched a screenshot, i hope u don't mind.

i just have to derive it. The hint says, use Kepler's third Law and that it is a description of the orbits of binary stars around their common centre of mass.

Thank You.

 

[mikky_t]

Wooooops!

Sorry about the name of the thread, I don't know where the extra, "& binary stars" came from.



Also, where is the picture? I uploaded it.

I just read Tom's post at the start so i will write up what I have. Its all on a piece of a paper:

mA = mass of star A
mB = mass of star B

Keplers third law,
r^3/T^2 = GM/4(pi^2)

Would it be sufficient to say that since in a binary system, they would robit according to Keplers third Law,

mA + mB = r^3/T^2...(1)

The text states formula (1) only, without any explanations.

Even from here if I try to prove that,

mA + mB = 4(pi^2)(r^3)/G(T^2)

It doesn't work out.

As you can probably see, i am a bit confused. Sorry about the messy-ness.

 

[M98Ranger]

Hi there,

Glad to hear that you find this forum interesting! Binary stars are definitely a fascinating topic to learn about.

In regards to your question, it looks like you are being asked to derive Kepler's third law for binary stars. Kepler's third law states that the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. In the case of binary stars, this law can be applied to the orbits of the two stars around their common center of mass.

To derive this law, you can start by writing out the equation for Kepler's third law: T^2 = k * a^3, where T is the orbital period, a is the semi-major axis, and k is a constant. In the case of binary stars, we can rewrite this equation as follows: (T1 + T2)^2 = k * (a1 + a2)^3, where T1 and T2 are the orbital periods of the two stars and a1 and a2 are their respective semi-major axes.

Next, we can use the fact that the total mass of the system, M, is equal to the sum of the masses of the two stars, m1 and m2. We can also use the definition of the center of mass, which states that the distance from the center of mass to each star is equal to the product of its mass and its distance from the center of mass. This can be written as: m1 * a1 = m2 * a2 = M * R, where R is the distance between the two stars.

Now, we can substitute this relationship into our equation and simplify to get: (T1 + T2)^2 = k * M^3 * R^3. Since M^3 * R^3 is a constant, we can rewrite this as: (T1 + T2)^2 = k' * M^3, where k' is a new constant. Finally, taking the square root of both sides, we get: T1 + T2 = k'' * M^(3/2), where k'' is another constant. This is the derived form of Kepler's third law for binary stars!

I hope this helps and good luck on your mid-year examinations! Remember to always check your units and use the correct notation when writing out equations. Let me know if you have any further questions or if you need clarification on any of the steps