Effective Potential Method for Solar System Orbits

[orange]

Hey everyone!

I have an exam question, but I don't know how to approach it. The question is,


"Discuss orbits of bodies in the Solar System using the effective potential method."

I thought about every planet having a certain amount of kinetic and potential energy, showing how they balance out as a planet orbits the Sun. During lectures we havn't mentioned the effective potential, so hopefully someone here will enlighten me. I really hope so, and it would make me very grateful.

Orange

 

[Andrew Mason]

Hey everyone!

I have an exam question, but I don't know how to approach it. The question is,


"Discuss orbits of bodies in the Solar System using the effective potential method."

I thought about every planet having a certain amount of kinetic and potential energy, showing how they balance out as a planet orbits the Sun. During lectures we havn't mentioned the effective potential, so hopefully someone here

The energy of an orbiting body is:

$$E = KE + PE = \frac{1}{2}mv^2 + V(r) = constant$$

If you break the velocity into a radial and tangential component and use polar coordinates:

$$KE = \frac{1}{2}m(\frac{dr}{dt})^2 + \frac{1}{2}m(rd\theta/dt)^2 = \frac{1}{2}m(\frac{dr}{dt})^2 + \frac{1}{2}mr^2\omega^2$$

Substituting angular momentum $L = mr^2\omega$:

$$KE = \frac{1}{2}m(\frac{dr}{dt})^2 + \frac{L^2}{2mr^2}$$

Since the force is central (no torque) L is constant. If you let:

$$V_{eff}(r) = V(r) + \frac{L^2}{2mr^2}$$ you can write the energy equation as:

$$E = \frac{1}{2}m(\frac{dr}{dt})^2+ V_{eff}(r) = constant$$

Then you can think of the variable energy in terms of the rate of change of the radius.

Consider an oribit in which the V_eff = constant; where V_eff has a minimum and maximum; where V_eff has a minimum but no maximum.

AM

 

[orange]

Thanks a lot! And I mean a lot! :-)