Accretion discs - Orbital angular momentum

[Sheepattack]

Homework Statement

There is a binary system accreting via Rcohe lobe overflow, where mass is transferred fromthe donor to the accretor and then lost in an isotropic wind.

State an expression for the orbital angular momentum of the accretor, J₁ and then use keplers third law to find an expression for the orbital angular speed

The Attempt at a Solution

am really stuck on this one. I know that the total angular mometum of the system can be written J = M1M2(Ga/M)^1/2 but i don't know how to split this into a term for the accretor. At the moment i have come up with J1 = M1a1^2w

keplers third law can be written a³w²=GM

any help or pointers will be greatly appreciated!

 

[mark726]

Hi there,

Firstly, let's define the variables in this system. M1 is the mass of the accretor, M2 is the mass of the donor, a1 is the distance from the accretor to the center of mass, and w is the angular speed of the system.

The total angular momentum of the system can be written as J = M1M2(Ga1/M)^(1/2). To find the expression for the orbital angular momentum of the accretor, we can use the fact that the mass of the accretor is much larger than the mass of the donor (M1 >> M2). Therefore, we can approximate the total angular momentum as J ≈ M1(Ga1/M)^(1/2). Now, to find the expression for the orbital angular momentum of the accretor, we can simply subtract the contribution of the donor's angular momentum, which is given by J2 = M2(Ga2/M)^(1/2) where a2 is the distance from the donor to the center of mass. So, the expression for the orbital angular momentum of the accretor, J1, is:

J1 = J - J2 ≈ M1(Ga1/M)^(1/2) - M2(Ga2/M)^(1/2)

Next, we can use Kepler's third law, which states that the square of the orbital period is proportional to the cube of the orbital distance, to find an expression for the orbital angular speed, w. Since we are dealing with a binary system, we need to use the reduced mass, μ = M1M2/(M1+M2), in Kepler's third law. So, we have:

a1^3w^2 = G(M1+M2) ⇒ w^2 = G(M1+M2)/a1^3

Substituting this expression into our equation for J1, we get:

J1 = M1(Ga1/M)^(1/2) - M2(Ga2/M)^(1/2) ≈ M1(Ga1/M)^(1/2) - M2(Ga2/M)^(1/2) = M1(Ga1/M)^(1/2) - M2(Ga2/M)^(1/2) = M1(Ga1/M)^(1/2) -