San José State University
& Tornado Alley
The Classical Analysis of a More|
General Case of the Two Body Problem
Consider two particles, one of mass m and the other of a mass M. Let the distance of the first particle from the center of mass of the system be denoted by r and that of the second particle by R. Let s designate the separation distance of the two particles. The center of the mass of the system is such that
This means that
The expression mM/(m+M) is equal to the reduced mass μ of the system. The reduced mass is defined by
Let the force between the particles be given as
where Q is the product of the force constant and the charges of the two particles.
The angular momentum of the system is:
The expression (L/μ) plays a crucial role in the further analysis and will be denoted as λ. Thus ω=λ/s².
The radial kinetic energy E of the system is
and therefore the above expression for radial kinetic energy may rewritten as
Now consider the tangential kinetic energy ½m(rω)² + ½M(Rω)². This can be rewritten as
Note that since s²ω=λ, sω is equal to λ/s. Thus the tangential kinetic energy can be expressed as
Thus the total energy E can be expressed as
The above equation can be solved for (ds/dt)²; i.e.,
This latter formula will be used later.
Now let u=1/s so
The dynamic equations for the two particles are
Since mr=MR=μs the above two equations are the same when expressed in terms of s; i.e.,
When ω is replaced with λ/s² and the result divided through by μ the above equation becomes
The ratio Q/μ occurs often enough to justify it having its own symbol, say q.
By defining w equal to (u−q/λ²) the above equation reduces to
which has the solution
This latter solution can be put into the form
where ε turns out to be the eccentricity of the elliptical orbit;.
By a suitable choice of the polar coordinate system; θ0 can be made equal to zero. Thus the solution for the system is
Note for later use that
The next step is to derive the velocities of each particle. vm and vM. where
From the solution for s
Since (1+ε·cos(θ))²s² is equal to (λ²/q)².
The tangential velocity is rω, but rω is equal to (M/(m+M))sω and sω is equal to λ/s. Thus
Combining the two terms for v² gives
The probability density per unit path length for both particles are proportional to
This gives a probability density function with one maximum and one minimum. The constants of proportionality disappear in the process of normalization. The probability density has one point at which it is a maximum and another point at which it is a minimum. Here is an example for the case of eccentricity equal to 0.25. The horizontal axis is in terms of θ measured in radians.
(To be continued.)
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