Taylor 8.5-8.7 (today) and 8.7-8.8 (Monday).
The r equation can be rewritten in the form
where I've introduced the effective potential Ueff = U(r) + l²/2μr² in order to make this look like a one-dimensional problem of motion in r subject to this effective potential. If we multiply both sides of this expression by r[dot], then each side is a time derivative:
therefore
The quantity in parentheses is the energy in the CM system, and this says that the energy is constant.
Show Ueff versus r for several potentials, ½kr², -k/r
Now we will find the equation for the orbit. We continue to think in terms of a general central force, but the following technique is certainly motivated by the specific case of a 1/r² force. We begin with the equation of motion
and massage it into a more convenient form. Begin by making a change of variables to u=1/r (or r=1/u), and, because it is convenient to solve for the spatial shape of the orbit (r as a function of φ) rather than the coordinates as a function of time, transform the time derivative to a φ derivative using the chain rule
The time derivative of r becomes
and the second time derivative becomes
Substituting these changes into the equation of motion, it becomes
or
where the double prime is shorthand for the second derivative with respect to φ.
Find the orbit for a free particle (no force).
Letting F=0, the equation becomes
We've seen this equation before but in the form of a second time derivative rather than a φ derivative. The solution is the same, just use φ rather than t
Of course, we want to know r, not u, so change variables again
This odd looking function is nothing more than the equation for a straight line in polar coordinates. Sketch to demonstrate.
The interesting case is for an attractive 1/r² force, F = -k/r². For the gravitational force, k = Gm1m2, and the force is always attractive. For the coulomb force, k = k|q1q2| = |q1q2|/4φε0, and is attractive if the charges have opposite sign.
Changing to the variable u, the force is F(1/u) = -ku² and the equation of motion looks like
Notice that the term involving the force becomes constant. This is a special property of the 1/r² force and yields orbits that close on each rotation. To solve, change variables again to w(φ) = u(φ) - μk/l² so that w''(φ) = u''(φ) and the differential equation becomes w''(φ) = -w(φ). This is the same differential equation we saw above, and the solution is
The phase shift δ can be set to zero by choosing an appropriate axis from which to measure φ. With this, we can change variables back and find
where we've replaced the constant A by the constant ε. This change seems arbitrary at this point, but be patient. Finally, we get back to r
This is our solution for the orbit under a 1/r² force. We have an expression for r as a function of the angle φ and the constants ε and C. C depends on the angular momentum, reduced mass, and force constant. ε is related to the constant of integration, A, so depends on initial conditions. We'll explore the nature of these orbits.
Note that if ε<0, then we have the same result as for |ε| and φ going to φ+π. Therefore, we only need to consider the cases for &epsilon≥0. Also, if ε<1, then the denominator is never equal to zero. Physically this means that r never goes to infinity; therefore, solutions with &epsilon<1 correspond to bounded orbits. If ε≥1, then the denominator can become zero for certain values of φ and r goes to infinity; these solutions correspond to unbounded orbits. The case of ε=1 is the transition from bounded to unbounded orbits. We'll see that the bounded orbits have E<0, the unbounded orbits have E≥0, and E=0 is the transition energy. Let's begin by understanding the characteristics of the bounded orbits.
When ε<1, the orbits are ellipses, that is, the expression for the orbit can be cast in the form
This will be left as an exercise for you. There is a displacement, d, in the x direction because the ellipse is centered at (x,y) = (-d, 0) rather than at (0,0). But the origin is a focus of the ellipse.
The separation, r, oscillates in synch with cosφ, with a minimum value, rmin = c/(1 + ε) when φ = 0, and a maximum value rmax = c/(1 - ε) when φ = π. For orbits of planets, comets, or other bodies about the sun, the location of minimum separation is called the perihelion (peri-, Greek for around or near and -helion, Greek for sun) and the location of maximum separation is called the aphelion (ap- Greek for away). For orbits around the earth, these locations are called perigee and apigee (gee Greek for earth). If ε=0, then r = c is constant and the orbit is circular.