Centre of mass of binary system calculation

[R_moor]

Homework Statement

Given a graph (see below) containing the velocities of two stars with respect to the sun, I am asked to calculate the velocity of the centre of mass of a binary system. I am not given the mass of either star, nor the shape of the orbit nor the velocity of the centre of mass.

Homework Equations

I know that the velocity of the centre of mass is given by :
V_cm = 1/M * [ m_1V_1 + m_2V_2] with M being the total mass of the system (m_1 + m_2) .

The Attempt at a Solution

So far I don't know how to start the problem, without more information, what I thought was that since the orbits are out of phase ( one orbits looks like a cosine function and the other like a sine function), one possible answer might be that the velocity of the centre of mass is just the average velocity at any given instant but I am honestly so lost I can't even write anything that make sense.
I would appreciate any help getting started.

 

[Orodruin]

I am not given the mass of either star, nor the shape of the orbit nor the velocity of the centre of mass.

You do not need this information. A hint is that you should be able to read this information off directly from the graph.

the average velocity at any given instant

This clearly cannot be correct as the average velocity of the objects change with time and the centre of mass velocity should not.

 

[R_moor]

Okay, thank you for your response I think it helped me to get started.

So I looked at the graphs and wrote an equation for each of the velocities :
1) V_1 = 150cos(t) +50
2) V_2 = -50cos(t) +50

(From here on I used: V_cm = 1/M * [ m_1V_1 + m_2V_2])

After this I evaluated both V's at t = 0 and t = π, so that I could express M only in terms of either m1 or m2. I choose m2 and found that M = 4m2.

Then I just plugged it in at t = 0 so that:
V_cm = 1/(4m_2) * (200) m_2
Canceling it out m_2 I found that V_cm = 50 km/s

Seems to me that this would only be right if the speed of the center of mass doesn't change (as you said), would you care to explain why this is true ?

Thank you !

 

[Alloymouse]

Since both stars are orbiting each other (about the CM), it will appear as if the CM is static w.r.t. both stars (think about how planets in the solar system are perceived by us to be orbiting about a (relatively static) sun.

Since the average velocity of both stars is 50km/s, that means that the binary system is moving away from you at 50km/s

 

[R_moor]

Since both stars are orbiting each other (about the CM), it will appear as if the CM is static w.r.t. both stars (think about how planets in the solar system are perceived by us to be orbiting about a (relatively static) sun.

Since the average velocity of both stars is 50km/s, that means that the binary system is moving away from you at 50km/s

I see thank you. Do you know how could I prove this mathematically ?

 

[Alloymouse]

I see thank you. Do you know how could I prove this mathematically ?

Well unfortunately I'm not sure what you're asking to prove in this question; I see that you have proven that Vcm is indeed 50km/s.

If you're asking about proving that CM is static, I'd like you to consider the conservation of energy. They cannot be in uniform circular orbits if distance from each star to the CM continuously varies :)

By "uniform" I mean constant tangential velocity - the sinosoidal equations you worked out has proven that mathematically too!

 

[conscience]

it will appear as if the CM is static w.r.t. both stars

No .

In the reference frame of a star , CM is rotating around that star .

 

[Alloymouse]

Sorry, intended to say that there is no horizontal movement observed by the observer on Earth if we see that the stars have horizontal velocity

 

[R_moor]

Thanks Everyone !