Find the position of the center of mass of the binary star system

In summary, the conversation discusses the calculation of the position of the center of mass for a binary star system consisting of two stars, P and Q, with masses of 4.0 x 10^10 kg and 2.0 x 10^10 kg respectively, separated by a distance of 6.3 x 10^9 m. The conversation also mentions the use of gravitational field strength and moments in solving this problem, and clarifies that the position of the center of mass is independent of forces and depends only on the masses and their positions. The solution is derived using the equation for total momentum and the distance between the stars.
  • #1
csharsha
4
0

Homework Statement



A binary star system consists of a star P and a star Q, of mass 4.0 x 10^10 kg and 2.0 x 10^10 kg respectively, separated 6.3 x 10^9 m apart. Star P and Star Q move in circular orbits with their centers at the center of mass which remains at rest.

Find the position of the center of mass of the binary star system.


Homework Equations



g = GM/r^2 (I used this and equated the field strengths from both)

But I have some worked solution to this from my teacher, and he used moments to solve this so I am just adding that equation here as well.

Moments = F x Perpendicular distance (This is just what my teacher used, but in many similar questions I have done, I never used this moments for such a question)

Or is it because this is a binary system that moments have to be used?

And in space normally, with no binary system or anything, we must use the g-field?

Also, I don't use calculus or vectors, so please don't use that for this question.



The Attempt at a Solution



Let x be the distance of the Center of Mass from Star P

At that point,

Gravitational Field Strength of P = Gravitational Field Strength of Q

g (p) = g (q)

(G(4 x 10^10))/(x^2) = (G(2 x 10^10))/(6.3 x 10^9 - x)^2

2 = (x^2)/(6.3 x 10^9 - x)^2

2(6.3 x 10^9 - x)^2 = x^2

solving the quadratic equation, I get x = (3.7 x 10^9) m


I would appreciate if someone could help me with this. And also, if someone could tell me if this is the way I am supposed to work it out for binary star systems and normal masses in space at a distance, or does it differ for this type of binary star system questions?
 
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  • #2
The position of the center of mass is independent of all forces that act, it depends only on the masses and their positions. Your teacher used momentums to find a point with respect to the sum of the momentums is zero:

m1v1+m2v2=0.

Both stars orbit along concentric circles so as the straight line connecting them goes through the common center of the circles (the center of mass, CM) and their angular velocities are the same, ω. The total momentum of the binary star system is zero. See the figure: when one star moves up, the other down, so the direction of their momentums are opposite. If m1 is at distance r1 from the center and m2 is at distance r2, the velocities are

v1=m1r1ω and

v2=-m1r1ω .

The total momentum is

m1v1+m2v2=0, that is

m1r1ω -m1r1ω =0

m1r1=m2r2. *

You are given the distance between the stars,

d=r1+r2. **

Find r1 and r2
from equations * and **.


See also :

http://hyperphysics.phy-astr.gsu.edu/hbase/cm.html


ehild
 

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1. How do you calculate the position of the center of mass of a binary star system?

To calculate the position of the center of mass of a binary star system, you will need to know the masses and positions of both stars. The formula for calculating the center of mass is: CM = (m1 * x1 + m2 * x2) / (m1 + m2), where m1 and m2 are the masses of the stars and x1 and x2 are their respective positions. This will give you the position of the center of mass along the line connecting the two stars.

2. Why is it important to know the position of the center of mass in a binary star system?

The position of the center of mass is important because it gives us information about the overall motion of the system. It allows us to determine the orbital period and distance between the stars, as well as understand their gravitational interactions. It also helps us to study the evolution and dynamics of the system.

3. Can the position of the center of mass change over time in a binary star system?

Yes, the position of the center of mass can change over time in a binary star system. This is because the stars are constantly moving and their positions and masses can also change due to various factors such as mass transfer, tidal interactions, and gravitational influences from other celestial bodies.

4. How does the position of the center of mass affect the orbits of the stars in a binary star system?

The position of the center of mass affects the orbits of the stars in a binary star system by determining the orbital period and distance between the stars. If the center of mass is closer to one star, it will have a larger influence on the orbit of that star. This can also lead to orbital eccentricity and other complex dynamics within the system.

5. Can the position of the center of mass be observed or measured?

Yes, the position of the center of mass can be observed and measured using various techniques such as astrometry, radial velocity, and interferometry. These methods allow us to track the positions and motions of the stars and determine their center of mass. The accuracy of the measurement depends on the precision of the instruments and the observational data.

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