Modeling a binary star system

In summary: This concludes the proof that the center of mass is at (mr,mr').In summary, the program tasked with simulating a binary star system with two unequal masses is having difficulty understanding the logic of the problem and how to proceed.
  • #1
besjbo
15
0
I am tasked with modeling a binary star system using VPython. The language itself is relatively irrelevant, as I can deal with the syntax. My problem is with the logic of it and how I should go about structuring the necessary calculations.

Homework Statement



A binary star system consists of two unequal mass stars that orbit about their center of mass. One star has a mass of 2e30 kg and the other has mass of 1e30 kg. Take the orbital period of the system to be 15 days. Your task is to simulate the binary-star system numerically, assuming that the stars are moving in the x-y plane.

The mass of one star does not dominate the total mass of the system, so you cannot make the approximation that the more massive object does not accelerate due to the force placed on it by the less massive one.

Homework Equations



Calculate initial velocities for the two stars assuming zero total momentum for the two-body system and a period of 15 days. Evolve the system for 45 days. Use two spheres of different sizes to represent the two stars, and track their orbits.

Make a graph of the x-component of its momentum vs. time. Is either of these graphs constant? Now add the x-components of the momentum of the two particles together and graph the time behavior of the x-component of the total momentum of the binary system. Is this constant in time? Why or why not? Compute the magnitude of the total system momentum and plot it versus time. How close is it to remaining constant? Should it change with time?

The Attempt at a Solution



Unfortunately, I'm not quite sure how to approach the problem beyond the elementary basics. I am aware that there is a gravitational force between the two stars that must be continuously calculated. However, I suspect center of mass of the two-star system must also be considered, but I don't quite know how, in terms of calculating position.

With any guidance, I'll try my best to proceed and continue as far as I can, but I currently feel stuck at a fairly early stage of programming this simulation. Any assistance will be much-appreciated.
 
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  • #2
Since there are only two bodies, an analytical solution is possible. You can just get the positions and momenta as functions of time. But initial conditions are a problem: you aren't given any. So you are probably supposed to assume circular orbits, in which case, given the period and the masses, you can establish the orbit and start it anywhere.
 
  • #3
Does center of mass play any role? How would I go about calculating the initial velocities?
 
  • #4
Assuming the orbits are circular, the center of mass is the center of the orbits.

If the motion is not circular, then it is elliptic, and the the center of mass is at a mutual focus of both orbits.

With the condition that the total momentum is zero, what is the motion of the center of mass?
 
  • #5
I would think the COM is not moving at all.
 
  • #6
Correct - but only if the total momentum is zero.
 
  • #7
So how would I find the initial velocities of the two stars?
 
  • #8
First you need to make sure you understand the geometry of the problem. If the smaller star (weight = m) is at displacement r (this is a vector!) from the C. M., what can be said about the displacement of the other star from the C.M.?
 
  • #9
voko said:
First you need to make sure you understand the geometry of the problem. If the smaller star (weight = m) is at displacement r (this is a vector!) from the C. M., what can be said about the displacement of the other star from the C.M.?

Intuitively, I would say the star that is twice as massive would be closer to the CM, but I'm not quite sure if its displacement would be 1/2*r in magnitude.
 
  • #10
What is the definition of the center of mass?
 
  • #11
voko said:
What is the definition of the center of mass?

It is basically a point where all of the mass of a system could be considered to be located.

Mathematically, the x-coordinate, for example, of a two-body system (containing mass M and mass m) would be (Mx1 + mx2)/(M+m), where x1 and x2 are the x-coordinates of of M and m, respectively.

I'm sure you asked this question to get me to realize something, probably in regards to the displacement, but I may have missed the point nevertheless.
 
  • #12
So, in the vector form, taking r' and R' to be the masses' displacements from some fixed point, the displacement of the C.M. is c = (mr' + MR')/(m + M).

Now let r = r' - c, and R = R' - c, in other words, measure the masses' displacement from the C.M. Then, mr + MR = mr' - mc + MR' + Mc = mr' + MR' - (m + M)c = 0. Thus

R = -(m/M)r.

r and R are vectors, so R is always opposite of r w.r.t. the C. M., and its magnitude is m/M.

What does that mean w.r.t. the radii of the orbits then?
 
  • #13
besjbo said:
It is basically a point where all of the mass of a system could be considered to be located.

Mathematically, the x-coordinate, for example, of a two-body system (containing mass M and mass m) would be (Mx1 + mx2)/(M+m), where x1 and x2 are the x-coordinates of of M and m, respectively.

I'm sure you asked this question to get me to realize something, probably in regards to the displacement, but I may have missed the point nevertheless.
You want to keep the CM stationary. The coordinates of the CM are as you stated. The same equation holds for the velocities: vx(CM)=(vx1m1+vx2m2)/(m1+m2) and vy(CM)=(vy1m1+vy2m2)/(m1+m2)


Vx(CM)=0, Vy(CM)=0. So you need to give the initial velocities accordingly.


ehild
 

1. What is a binary star system?

A binary star system is a system that consists of two stars orbiting around their common center of mass. These stars are gravitationally bound to each other and can either be in a close orbit (where they are very close together) or a wide orbit (where they are farther apart).

2. How do you model a binary star system?

To model a binary star system, you need to use the laws of gravity and motion to calculate the orbits of the two stars around each other. This involves taking into account the masses of the stars, their distance from each other, and their velocities.

3. What tools are used for modeling a binary star system?

There are several tools that can be used for modeling a binary star system, including computer simulations, mathematical equations, and observations from telescopes. Each tool has its own advantages and limitations, and scientists often use a combination of these tools to get a more accurate model.

4. What are the benefits of modeling a binary star system?

Modeling a binary star system allows scientists to understand the complex dynamics and interactions between two stars. It can also provide insights into the formation and evolution of binary star systems, as well as the effects of one star on the other.

5. What are some real-life applications of modeling a binary star system?

Modeling a binary star system has many practical applications, such as predicting the behavior of eclipsing binary stars, which are used as standard candles for measuring cosmic distances. It can also help in studying the effects of binary stars on their surrounding environments, such as the formation of planetary systems.

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