Trajectory of a Test Particle in a Semidetached Binary

[M87]

Hello,

I am currently trying to figure out how I can use Microsoft Excel to calculate the ballistic trajectory of a test particle released from the inner Lagrange point in a semidetached binary. To keep things simple, I am using a two-dimensional coordinate system, and I am assuming that the initial conditions of the test particle have very little impact on the trajectory.

So far, I have the conditions of the semidetached binary determined: in the xy plane I have the accreting star located at the origin, and I have the center of mass of the binary and Lagrange Point L1 correctly positioned on the x axis. I have also graphed the radius of the accreting star. All other parameters are also available.

I understand the forces at work, although I am not too familiar with how the directions should be specified for the Coriolis and centrifugal forces on the test particle. I am familiar with the Roche potential of the system, yet I do not know how to apply it to this particular case in determining trajectories. I have also read several publications, including one which has several graphs (the ones on the left side) of what I would like to achieve with this spreadsheet: http://www.astro.psu.edu/~mrichards/research/tomog.gif" The problem is that none of them have gone into detail on how such graphs were derived. So this is as far as I have gotten with my spreadsheet. I would be thankful of someone could lead me in the right direction in calculating these trajectories.

 

[M87]

All right. I apologize for being vague in my first post. I have done more reading, and I am now familiar with the required equations, but I am still a little confused on how I should approach the situation.

In the restricted problem of three bodies the potential is given by:

$$\Psi=\frac{1-\mu}{r_1}+\frac{\mu}{r_2}-\mu x+\frac{1}{2}(x^2+y^2)$$

(Kruszewski 1966). Here, μ is the mass of the secondary star divided by the total mass of the binary system. x and y are the coordinates of the test particle, while r1 and r2 are the distances of the primary and secondary star from the origin, respectively.

The equations of motion are given by:

$$\ddot{x}=(2A+1)x+2\dot{y}$$

$$\ddot{y}=-(A-1)y-2\dot{x}$$

$$A=\frac{\mu}{|X_{L1}-1+\mu|^3}+\frac{1-\mu}{|X_{L1}+\mu|^3}$$

(Lubow and Shu 1975). Here, x and y refer to the coordinate system with the Lagrange point L1 centered at the origin. So the positive x-axis points to the accreting star (the mass-accepting star). XL1 is the distance from L1 to the center of mass of the binary system.

Finally, the constant of motion for the restricted three-body problem is given by the Jacobi constant:

$$C=\frac{1-\mu}{r_1}+\frac{\mu}{r_2}-\mu x+\frac{1}{2}\mu^2$$
$$-\frac{1}{2}\left [\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right]+\frac{1}{2}(x^2+y^2)$$

What I wanted to know is how do I use these equations to solve for the trajectory of a test particle originating at L1 as it "free falls" onto the accreting star? The publications where I have obtained these equations mention that the Runge-Kutta method of 7th order was used to calculate the trajectories of the test particle. I have heard of this method before, but I am unfamiliar with how and where I should start. Any help is appreciated.