# Mass Transfer in a Binary Star System

• Kelli Van Brunt
In summary, the homework statement states that a binary star system consists of two stars that are orbiting around a common center of mass. The stars rotate with an angular velocity w and this causes the orbital period and the distance between the stars to change slowly over time. Over a time duration of dt, a mass transfer between the two stars results in a change in mass of dM1 in star M1.
Kelli Van Brunt
Homework Statement
A binary star system consists of M1 and M2 separated by a distance D. M1 and M2 are revolving with an angular velocity w in circular orbits about their common center of mass. Mass is continuously being transferred from one star to the other. This transfer of mass causes their orbital period and their separation to change slowly with time. Assume the stars are like point particles and that the effects of the rotation about their own axes are negligible. In a time duration dt, a mass transfer between the two stars results in a change of mass dM1 in star M1. Find the quantity dw in terms of w, M1, M2, and dM1.
Relevant Equations
Total angular momentum of system: L = (wM1M2D^2) / (M1+M2)
Relation between angular velocity and distance between stars: w^2 = G(M1+M2) / D^3
Homework Statement: A binary star system consists of M1 and M2 separated by a distance D. M1 and M2 are revolving with an angular velocity w in circular orbits about their common center of mass. Mass is continuously being transferred from one star to the other. This transfer of mass causes their orbital period and their separation to change slowly with time. Assume the stars are like point particles and that the effects of the rotation about their own axes are negligible. In a time duration dt, a mass transfer between the two stars results in a change of mass dM1 in star M1. Find the quantity dw in terms of w, M1, M2, and dM1.
Homework Equations: Total angular momentum of system: L = (wM1M2D^2) / (M1+M2)
Relation between angular velocity and distance between stars: w^2 = G(M1+M2) / D^3

My strategy for solving this problem was to consider that angular momentum remains constant and thus to take ##dL/dt = 0##. Plugging in the expression for L, I obtained $$\frac{d}{dt} \frac {wM_1M_2D^2} {M_1+M_2} = 0$$
I plugged in the given expression for D in terms of w, M1, and M2 and used the quotient rule to take the derivative of this expression with respect to time, assuming that ##dM_2/dt = -dM_1/dt##. I got $$0 = \frac {\frac {dM_1}{dt} (M_2-M_1)(M_2+M_1)w - \frac{1}{3} (M_1+M_2) \frac {dw}{dt}}{(w(M_1+M_2))^{4/3}}$$
Thus, solving for dw, I obtained for my final answer $$dw = 3w(M_2-M_1) dM_1$$
However, this answer is obviously wrong - the units don't match up, and the answer given in the text is ##dw = 3w(M_2-M_1) dM_1 / M_1M_2##. The authors of the text provide a step-by-step solution which I understand and I can provide to anyone who wishes to see it, but it uses a completely different method which does not involve taking the derivative at all, and I am not sure why my method did not work. Was it a simple math error, a fault in my initial assumptions, etc? Any help would be much appreciated! (It is also my first time using LaTeX or posting on this site, so if I have done anything wrong please notify me.)

Delta2
Something went wrong with the derivative, it has mismatching units already.

Instead of assuming the angular momentum stays the same it might be easier to assume the square of the angular momentum stays the same. That is equivalent, of course, but then you get ##\omega^2## which you can replace using the relation between ##\omega## and D, that way you avoid fractional exponents.

jim mcnamara
mfb said:
Something went wrong with the derivative, it has mismatching units already.

Instead of assuming the angular momentum stays the same it might be easier to assume the square of the angular momentum stays the same. That is equivalent, of course, but then you get ##\omega^2## which you can replace using the relation between ##\omega## and D, that way you avoid fractional exponents.

Thank you so much! I cubed it instead so that I could replace D with the expression for D in terms of w and it worked out. I'll keep this in mind in the future!

mfb

## 1. What is mass transfer in a binary star system?

Mass transfer in a binary star system refers to the exchange of mass between two stars that are orbiting each other. This can occur through various mechanisms such as stellar winds, accretion, or gravitational interactions.

## 2. Why does mass transfer occur in a binary star system?

Mass transfer occurs in a binary star system due to the gravitational pull between the two stars. As they orbit each other, one star may pull material from the other, resulting in the transfer of mass.

## 3. How does mass transfer affect the evolution of binary star systems?

Mass transfer can significantly impact the evolution of binary star systems. It can alter the masses and compositions of the stars, leading to changes in their evolution and eventual fate. It can also influence the orbital dynamics and potentially cause the stars to merge.

## 4. What are some consequences of mass transfer in a binary star system?

One consequence of mass transfer is the formation of accretion disks around the stars, which can give rise to phenomena such as X-ray emissions and novae. It can also lead to the formation of exotic objects like black holes and neutron stars.

## 5. Can mass transfer in a binary star system be observed?

Yes, mass transfer in binary star systems can be observed through various methods such as spectroscopy, photometry, and interferometry. These techniques allow scientists to study the properties of the stars, their orbital dynamics, and the effects of mass transfer on their evolution.

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