Mathematics of Two Attracting Particles

General Setup

Note: Bold characters in equations denote vectors.

Central force law

We then use Newton's mass times acceleration=force law for the total momentum:

where x_i are the positions of the centers of the bodies.

R is the center of mass vector

Conservation of momentum:

Center of mass speed:

since the second time derivative of R is zero.

Center of mass position vector:

where t=time.

Particle relative displacement r:

Newton's ma=F law for 1-2 displacement

Reduced mass:

Mass sum:

Kepler Orbit Parameters: http://en.wikipedia.org/wiki/Kepler_orbit

where G is the gravitational force constant

For the non-vector r acceleration, we must take centripetal acceleration into account:

(1)

For gravitational forces:

Let:

where G is the gravitational constant

For charged particle forces.

where Z₁ and Z₂ are the charge numbers of the particles, e is the electronic charge, and e₀ is the permittivity of vacuum. The following Kepler orbit treatment will be valid only for particles of opposite charge.

The specific angular momentum, H, is a signed constant pointing along the z direction:

where r₀ and v₀ are the starting values for r and v and

The speed transverse to r is:

Let s=1/r:

Then from equation 1 we can write the following differential equation for s:

The solution for s is:

where q₀ is the usually chosen to be 0 at the major axis.

If we let q-q₀=0, then we can compute e in terms of the initial value, r₀, of r:

If H is large enough, |e| can be larger than 1 and that will cause a hyperbolic orbit.

Speed parallel to r:

Define a speed associated with the constants of motion:

To choose reasonable initial values for r₀ and v₀, see the Appendix.

We next obtain the initial angle using the above parameters:

where arctan2(y,x) computes the value of q₀ on the entire unit circle.

Orbit eccentricity e from above parameters:

Note that e can be negative as used in this document.

Coordinate system unit vectors, x₁,y₁, of the major and minor axis of the ellipse:

where the unit r and q vectors are defined in terms of the initial r as follows:

r(q) can now be expressed in terms of q₀ and qdot as:

where the rate of angle change:

We want to express the new vector r in terms of the x₁ and y₁ coordinate system so that at q=q₀ the values r and r₀ are the same. Obviously we can write the following vector expression for r any time t:

We only have to use the expressions for unit vectors x₁,y₁ to express r in terms of the original coordinate system.

The inverse relations for x₁ and x₂ are computed below:

Repeating the expressions for R and r:

Figure 1: Diagram of the elliptical orbit of the separation vector r showing the tilted ellipse angle q_r0 as well as the angle of r in the elliptical coordinate system . The diagrams of the individual particles will be different.

The value of q_r0:

Appendix: Finding the maximum relative velocity for an elliptical orbit of finite size:

Repeating the equation for the second time derivative of r:

(A1)

Rewrite A1 using the chain rule:

(A2)

Integrating both sides with respect to r we have:

(A3)

If we set r₂ equal to infinity and relative velocity at r₂ equal to zero then we can solve for the maximum allowable speed at r₁:

(A4)

The speed in this equation is often called the "escape velocity" but is more accurately referred to as the escape speed since its value doesn't depend on the direction of .

Equation A4 is useful conceptually but, for the animation purposes, we can't allow r₂ to be infinity. Therefore using equation A3 but this time setting r₂=r_max and = we have a more useful expression for r_max :

A reasonable choice for might be