Energy Diagrams and Orbits

Let's write the total energy of a particle moving in a gravitational field in a clever way:

$$E_{\rm tot} = \frac{1}{2}mv^2 - \frac{GMm}{r}$$ (20)

$$= \frac{1}{2}mv_r^2 + \frac{1}{2}mv_\perp^2 - \frac{GMm}{r}$$ (21)

$$= \frac{1}{2}mv_r^2 + \frac{1}{2mr^2}(m v_\perp r)^2 - \frac{GMm}{r}$$ (22)

$$= \frac{1}{2}mv_r^2 + \frac{L^2}{2mr^2} - \frac{GMm}{r}$$ (23)

$$= \frac{1}{2}mv_r^2 + U_{\rm eff}(r)$$ (24)

Where

$$U_{\rm eff}(r) = \frac{L^2}{2mr^2} - \frac{GMm}{r}$$ (25)

is the (radial) potential energy plus the transverse kinetic energy (related to the constant angular momentum of the particle). If we plot the effective potential (and its pieces) we get a one-dimensional radial energy plot.

By drawing a constant total energy on this plot, the difference between $E_{\rm tot}$ and $U_{\rm eff}(r)$ is the radial kinetic energy, which must be positive. We can determine lots of interesting things from this diagram.

In this figure, we show orbits with a given angular momentum $\vec{\bf L} \ne 0$ and four generic total energies $E_{\rm tot}$. These orbits have the following characteristics and names:

1. $E_{\rm tot} > 0$. This is a hyperbolic orbit.
2. $E_{\rm tot} = 0$. This is a parabolic orbit. This orbit defines escape velocity as we shall see later.
3. $E_{\rm tot} < 0$. This is generally an elliptical orbit (consistent with Kepler's First Law).
4. $E_{\rm tot} = U_{\rm eff,min}$. This is a circular orbit. This is a special case of an elliptical orbit, but deserves special mention.

Note well that all of the orbits are conic sections. This interesting geometric connection between $1/r^2$ forces and conic orbits was a tremendous motivation for important mathematical work two or three hundred years ago.