Doubt in Gravitation: Understanding the Orbital Motion of Two Stars

[Vibhor]

Homework Statement

Suppose two stars are orbiting each other in circular orbits with angular speed $\omega$ .M1 is at distance r1 from CM wheras M2 is at distance r2 such that r1+r2=d where d is the distance between them . Now i have a little doubt whether the stars are orbiting around their common CM or they are orbiting each other . If we consider them orbiting CM then for M1 $\frac{GM_1M_2}{d^2}=M_1\omega^2 r_1$ .But it is wrong to write $\frac{GM_1M_2}{d^2}=M_1\omega^2 d$ .Could someone help me understand what is wrong with the latter expression ? Why can't we write centripetal acceleration to be $M_1\omega^2 d$.Please pardon me for missing something obvious . Many thanks !

Homework Equations

The Attempt at a Solution

 

[ehild]

Orbiting around the common CM means the same as orbiting each other.
The force of gravity is inversely proportional to the square of the distance between the stars. There is nothing at the CM to attract any of them.
But you can write the centripetal acceleration of the stars as M1ω²r₁ and M2ω²r₂. Both are equal to GM1M2/d². Also, r1+r2=d, and r1=dM2/(M1+M2) and r2=dM1/(M1+M2).

ehild

 

[Vibhor]

I agree with what you have said . But i still don't understand why is it correct to have r1 in the expression for centripetal acceleration and not d . Sorry if I am sounding dumb .

 

[rude man]

I agree with what you have said . But i still don't understand why is it correct to have r1 in the expression for centripetal acceleration and not d . Sorry if I am sounding dumb .

Because rotation of each star is about the CM, thus r1 and r2 for the radii of rotation, not d. Just imagine one star at a time rotating about the CM.

 

[Vibhor]

Because rotation of each star is about the CM, thus r1 and r2 for the radii of rotation, not d. Just imagine one star at a time rotating about the CM.

I understand how M1 is orbiting CM. But M1 does have an angular velocity about M2 which means M1 is rotating about M2. This in turn means that the expression for centripetal acceleration should have 'd'. I still can't convinve myself what is wrong in this.

 

[rude man]

M1 is not rotating circularly about M2. M1 is rotating circularly about the CM. If you fix the position of M2, then the trajectory of M1 is not a circle. Only a circle has constant centripetal force.

 

[Vibhor]

Ok . I have realized the flaw in my reasoning.Sorry for putting up a real bad question . Another thing i would like to know is what is the trajectory of M2 as seen from M1 ? How would M2 move as seen from the reference frame of M1? Thanks !

 

[dean barry]

Two body data

This two body data sheet attachment might come in handy.
Dean

 

[ehild]

Both stars move along the same circle with the same angular velocity. In a co-rotating frame of reference, they are in rest, both of them. So M1 sees M2 in rest, with respect to itself - the distance does not change. But M2 seems to move along a circle of radius d with respect to the far-away stars.
(If M1 rotates also around its axis, the situation is different. Think of the Earth and Sun. You see the Sun rise and set, and going along a circle on the sky - why? )

ehild

 

[rude man]

Ok . I have realized the flaw in my reasoning.Sorry for putting up a real bad question . Another thing i would like to know is what is the trajectory of M2 as seen from M1 ? How would M2 move as seen from the reference frame of M1? Thanks !

Put M2 at the center of a polar coordinate system and the orbit of M1 would still be a circle.

The general solution can include circle, ellipse (e ≠ 0), parabola or hyperbola, depending on the kinetic energy of the system. In your case though it's a circle. The equation in this coordinate system is

r= k²/K = your "d"
k = r² dθ/dt
K = G M2

This is not trivial math!