Orbits & Effective Potential: E vs. r Graph Analysis

[stunner5000pt]

attached is the effective potential plotted on E vs r graph. Discuss the nature of the orbits are the various values for r
also $$V_{eff} (r) = - \frac{\alpha}{r} - br^2 - \frac{L^2}{2mr^2}$$
where b << alpha
L is the angular momentum and is not zero

for R > R5 the orbit is a hyperbola because the value of r will go on increasing infinitely
for R2 <= R <= R3 the orbit is an ellipse because the orbit is bounded

for R = R4 or R = R5 then the orbit is a circle special case where there is no external torque

for R< R1 orbit is a parabola? (not sure)
im not usre how to interpret R = 0 and R1< R < R2

 

[Jhero]

The nature of the orbits in this system can be best understood by looking at the effective potential, which is a combination of the gravitational potential and the centrifugal potential. The effective potential is plotted on an E vs r graph, where E is the total energy of the system and r is the distance from the center of the gravitational field.

At large values of r (R > R5), the effective potential is dominated by the inverse square law term, -\frac{\alpha}{r}, where \alpha is a constant. This results in a hyperbolic shape for the effective potential, indicating that the orbit will be a hyperbola. This means that the particle will approach the center of the gravitational field but will never reach it, instead moving away to infinity.

In the range of R2 <= R <= R3, the effective potential is a combination of both the inverse square law term and the quadratic term, -br^2. This results in an elliptical shape for the effective potential, indicating that the orbit will be an ellipse. This means that the particle will have a bounded orbit, continuously moving between its closest and farthest distance from the center of the gravitational field.

At R = R4 or R = R5, the effective potential becomes a constant, indicating that there is no external torque acting on the particle. This results in a circular orbit, where the particle will continuously move at a constant distance from the center of the gravitational field.

For R < R1, the effective potential becomes a parabolic shape, indicating that the orbit will be a parabola. This means that the particle will approach the center of the gravitational field and then move away to infinity, but at a slower rate than in the hyperbolic case.

At R = 0, the effective potential becomes undefined as the term \frac{L^2}{2mr^2} approaches infinity. This means that the particle will have an unbounded orbit, moving away from the center of the gravitational field at an infinite rate.

For R1 < R < R2, the effective potential has a minimum point, indicating that the particle will have a bounded orbit with a closest and farthest distance from the center of the gravitational field. However, the shape of the orbit will depend on the specific values of R1 and R2 in relation to the other parameters in the equation. Further analysis would be needed to determine the exact nature of the orbit in this range.

In summary, the nature