Calculation of the orbital period for a binary star system

[simphys]

Homework Statement

15. A binary star system consists of two stars of masses m1 and m2 orbiting
about each other. Suppose that the orbits of the stars are circles of radii r1 and r2
centered on the center of mass (Figure 9.42). What is the period of the orbital motion?

Relevant Equations

gravitational attraction

Hello guys,

Would it be possible to get some help on how to approach this problem? I don't really understand it. do I need to look at the orbital motion of the center of mass here or? If so how should I start?

Thanks in advance.

 

[PeroK]

Homework Statement: 15. A binary star system consists of two stars of masses m1 and m2 orbiting
about each other. Suppose that the orbits of the stars are circles of radii r1 and r2
centered on the center of mass (Figure 9.42). What is the period of the orbital motion?
Relevant Equations: gravitational attraction

Hello guys,

Would it be possible to get some help on how to approach this problem? I don't really understand it. do I need to look at the orbital motion of the center of mass here or? If so how should I start?

Thanks in advance.

Could you do this question if $m_1$ were much larger than $m_2$? E.G. if $m_2$ was a planet.

 

[haruspex]

Start the usual way, with a diagram. Assign variable names as needed. Think about forces and write some equations.

 

[simphys]

Could you do this question if $m_1$ were much larger than $m_2$? E.G. if $m_2$ was a planet.

well no that is what I'm puzzled about. Analyzing the orbit around a planet I do understand, but looking at two planets orbiting about its center of mass.. don't know. I know that two planets will both orbit around the center of mass and not f.e. one planet around the other but that's about it.

 

[simphys]

Start the usual way, with a diagram. Assign variable names as needed. Think about forces and write some equations.

I tried relating vcm to v1 and v2 + using the newton's 2nd law then putting it together in $T = 2pi*r_(cm)/v_(cm)$ but I don't see how that is going to help unfortunately

 

[haruspex]

I tried relating vcm to v1 and v2 + using the newton's 2nd law then putting it together in $T = 2pi*r_(cm)/v_(cm)$ but I don't see how that is going to help unfortunately

What about forces? How far apart are the stars?

 

[simphys]

What about forces? How far apart are the stars?

let me show you what I have done here:

for m1:
newton's law:
$Gm_1*m_2/(r_1+r_2)^2 = m*v_1^2/r1$ --> $v_1 = square root of m_2*G*r_1/(r_1 + r_2)^2$

for m2:
netwon's law:
$Gm_1*m_2/(r_1+r_2)^2 = m*v_2^2/r2$ --> $v_2 = square root of m_2*G*r_2/(r_1+r_2)^2$

then $vcm = (m1v1 + m2v2) / m(tot)$ and $rcm = (m1r1 + m2r2) / m(tot)$
and then $T = 2*\pi*rcm / vcm$
but I don't see how this will give me the necessary answer.

 

[haruspex]

let me show you what I have done here:

for m1:
newton's law:
$Gm_1*m_2/(r_1+r_2)^2 = m*v_1^2/r1$ --> $v_1 = square root of m_2*G*r_1/(r_1 + r_2)^2$

for m2:
netwon's law:
$Gm_1*m_2/(r_1+r_2)^2 = m*v_2^2/r2$ --> $v_2 = square root of m_2*G*r_2/(r_1+r_2)^2$

then $vcm = (m1v1 + m2v2) / m(tot)$ and $rcm = (m1r1 + m2r2) / m(tot)$
and then $T = 2*\pi*rcm / vcm$
but I don't see how this will give me the necessary answer.

You are calculating velocities as scalars, but to add them they need to be vectors.
You can get there that way, but you might find it easier to think about angular velocities.

 

[simphys]

You are calculating velocities as scalars, but to add them they need to be vectors.
You can get there that way, but you might find it easier to think about angular velocities.

do you think that using vectors will fix the problem?
And what do you mean by using angular velocities? like v = omega * r or?

 

[haruspex]

do you think that using vectors will fix the problem?

You don’t have to use vectors, but you must add the velocities in a way that takes their directions into account. What can you immediately say about the directions of the two velocities?

And what do you mean by using angular velocities? like v = omega * r or?

Yes. What can you immediately say about the two angular velocities?

 

[kuruman]

Another way of looking at it is this. The two stars rotate about the CM with period $T.$ Say $r_1<r_2$. Imagine point P on orbit 1 that is always diametrically opposed to star 1. It too rotates with period $T$ about the CM but is also always on $r_2$, the radius joining the CM with $m_2.$ Does this help?

 

[simphys]

You don’t have to use vectors, but you must add the velocities in a way that takes their directions into account. What can you immediately say about the directions of the two velocities?

Yes. What can you immediately say about the two angular velocities?

apologies, went on vacation. well.. that is the only thing that I don't understand. How do we determine that the angular velocities of the two starts are the same??

 

[PeroK]

apologies, went on vacation. well.. that is the only thing that I don't understand. How do we determine that the angular velocities of the two starts are the same??

Think about where the two stars must be relative to the centre of mass. Draw a diagram if you need to.

 

[kuruman]

Here is a diagram. What do you see? Refer to post #11.

 

[haruspex]

How do we determine that the angular velocities of the two starts are the same?

Adopt a polar coordinate system with the common mass centre at the origin. If the stars are at $(r_1,\theta_1)$ and $(r_2,\theta_2)$, what is the relationship between the two angles?

 

[simphys]

Adopt a polar coordinate system with the common mass centre at the origin. If the stars are at $(r_1,\theta_1)$ and $(r_2,\theta_2)$, what is the relationship between the two angles?

thanks guys I understand it now I think. $\Delta \theta_1 = \Delta \theta_2$ which basically makes the angular velocity equal.

 

[haruspex]

thanks guys I understand it now I think. $\Delta \theta_1 = \Delta \theta_2$ which basically makes the angular velocity equal.

Right.

 

[PeroK]

thanks guys I understand it now I think. $\Delta \theta_1 = \Delta \theta_2$ which basically makes the angular velocity equal.

If you drew a diagram, you would see that the centre of mass lies on a straight line between the stars, which are, therefore, always directly opposite each other relative to the centre of mass. This implies that they orbit with the same angular velocity about the centre of mass.

 

[kuruman]

If you drew a diagram, you would see that the centre of mass lies on a straight line between the stars, which are, therefore, always directly opposite each other relative to the centre of mass. This implies that they orbit with the same angular velocity about the centre of mass.

Ahem $\dots$ post#14?

 

[PeroK]

Ahem $\dots$ post#14?

If you drew a diagram (or looked at the one Kuruman drew for you!), you would see that the centre of mass lies on a straight line between the stars, which are, therefore, always directly opposite each other relative to the centre of mass. This implies that they orbit with the same angular velocity about the centre of mass.