How Do You Calculate the Center of Mass in a Three-Body Celestial System?

[Ertosthnes]

Homework Statement

A space system consists of two visible stars, one is a blue giant with a mass of 11M and the other is a red dwarf with a mass of 0.5M. The system also has a black hole with a mass of 2M but we don't know where it is located. The blue giant is 700 gigameters away from you along the x-axis and the red dwarf is 825 gigameters away from you 14 degrees below the x axis. The blue giant is moving in the +y direction and the dwarf moves 45 degrees clockwise of the +y direction.

We're looking for the system's center of mass, and the location of the black hole.

We also assume the following about the system:
1) Orbits are approximately circular about the system's center of mass
2) All lie in the same plane
3) All orbit in the same direction (e.g., clockwise or counterclockwise)

Homework Equations

The relevant equations are uses of algebra, trigonometry, and the center of mass equation, as far as I can tell.

The Attempt at a Solution

So far I've mapped out the locations of the two planets; the blue giant's coordinates are (700, 0) and the red dwarf's coordinates are (800,-200). I have no idea how to continue.

 

[D H]

Given that the stars have circular orbits about the system center of mass, what is the angle between the vector from the center of mass to a star and the star's velocity vector? What does this mean in terms of the inner product between vector from the center of mass to a star and star's velocity vector?

 

[m2003]

I would approach this problem by first using the given information to calculate the individual center of mass for the blue giant and the red dwarf. This can be done using the equation for center of mass, which takes into account the mass and distance of each object.

Once the individual center of masses are calculated, I would then use the fact that the system's center of mass must lie along the line connecting the two individual center of masses. Using trigonometry, I would then be able to determine the coordinates of the overall center of mass for the system.

To determine the location of the black hole, I would use the fact that all objects in the system orbit around the center of mass. This means that the black hole must also orbit around the center of mass, and its location can be determined by using the equation for circular motion and the known masses and distances of the other objects in the system.

It is also important to consider the direction of rotation of the objects in the system and how that may affect the location of the black hole. For example, if all objects are orbiting in a counterclockwise direction, the black hole must also be located in that same direction from the center of mass.

Overall, this problem requires a combination of algebra, trigonometry, and understanding of center of mass and circular motion. It is important to carefully analyze the given information and use appropriate equations and concepts to arrive at a solution.