Doubt in Gravitation: Understanding the Orbital Motion of Two Stars

In summary: The trajectory of M2 is different depending on the frame of reference in which it is measured. In the frame of reference of M1, M2 is at rest. In a co-rotating frame of reference, M2 moves along a circle with radius d.
  • #1
Vibhor
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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

 
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  • #2
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
 
  • #3
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 .
 
  • #4
Vibhor said:
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.
 
  • #5
rude man said:
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.
 
  • #6
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.
 
  • #7
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 !
 
  • #8
Two body data

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

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  • #9
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
 
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  • #10
Vibhor said:
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!
 

Related to Doubt in Gravitation: Understanding the Orbital Motion of Two Stars

1. What is the concept of simple doubt in gravitation?

Simple doubt in gravitation refers to the questioning of the validity of the theory of gravity as proposed by Sir Isaac Newton. It suggests that there may be other factors or forces at play in the phenomenon of gravitation that have yet to be discovered or fully understood.

2. How is simple doubt in gravitation different from other theories of gravity?

Simple doubt in gravitation differs from other theories of gravity because it challenges the fundamental principles of Newton's theory, which has been widely accepted and used for centuries. It suggests that there may be limitations or inaccuracies in Newton's theory that need to be further explored.

3. What evidence exists to support the concept of simple doubt in gravitation?

One of the main pieces of evidence that supports the concept of simple doubt in gravitation is the observation of the anomalous precession of the orbit of Mercury. This phenomenon cannot be fully explained by Newton's theory and has led scientists to question its validity.

4. How does simple doubt in gravitation impact our understanding of the universe?

If simple doubt in gravitation is proven to be true, it could significantly impact our understanding of the universe and how it operates. It could lead to the development of new theories and explanations for gravity, which could in turn lead to a better understanding of other phenomena in the universe.

5. What research is currently being done to explore simple doubt in gravitation?

Scientists are currently conducting various experiments and studies to explore the concept of simple doubt in gravitation. These include observations of the orbits of other planets and objects in the universe, as well as experiments with gravitational waves and other potential factors that could affect the phenomenon of gravitation.

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