Year 1 Sem 2/LC Classical Mechanics and Relativity 2/Lecture 18

Lecture 18: Orbits III: Total Orbital Energy and Perturbed Orbits
Thu 26 Mar 2026 15:00

Eccentricity

We can classify orbits by their eccentricity $\epsilon$:

$\epsilon = 0$: Circular orbit.
$0 < \epsilon < 1$: Elliptical orbit.
$\epsilon = 1$: Parabolic orbit.
$\epsilon > 1$: Hyperbolic orbit.

Earth orbits very close to a circular orbit, with $\epsilon \approx 0.017$. Bound comets with very large orbits have $\epsilon$ close to 1. Unbound comets have $\epsilon > 1$.

Total Energy in an Elliptical Orbit w/ $1/r$ Central Potential

Consider a planet with mass $m$ orbiting a star with much larger mass $M$, in a $1/r$ central potential.

At perigee, the planet has velocity $v_1$ and away at apogee the planet has velocity $v_2$. The $M \gg m$ condition allows us to neglect motion of the star.

We have total energy:

$$E_\text{tot} = \frac{1}{2}m \dot{r}^2 + \frac{l^2}{2mr^2} - \frac{k}{r} = \text{const.}$$

Noting that $l$ is a constant, and $k = GMm$. As the apogee and perigee are turning points, $\dot{r} = 0$ here. Hence:

$$E_\text{tot} = \frac{l^2}{mr_1^2} - \frac{k}{r_1} = \frac{l^2}{mr_2^2} - \frac{k}{r_2} \tag{1}$$

$$\frac{l^2}{2m} \left(\frac{1}{r_1^2} - \frac{1}{r_2^2}\right) = k\left( \frac{1}{r_1} - \frac{1}{r_2}\right)$$

$$\frac{l^2}{2m} \left(\frac{r_2^2 - r_1^2}{r_1^2 r_2^2}\right) = k \frac{r_2 - r_1}{r_1 r_2}$$

$$l^2 = 2mk \frac{r_2 - r_1}{r_1 r_1} \frac{r_1^2 r_2^2}{r_2^2 - r_1^2}$$

$$l^2 = \frac{2mk (r_2 - r_1) r_1 r_2}{(r_2 + r_1) (r_2 - r_1)}$$

$$l^2 = \frac{2mk r_1 r_2}{r_1 + r_2} \tag{2}$$

Eqn (2) into (1):

$$E_\text{tot} = \frac{2mk r_1 r_2}{r_1 + r_2} \frac{1}{2mr_1^2} - \frac{k}{r_1} = \text{ \dots for the RHS, but not required.}$$

$$E_\text{tot} = \frac{r_2 K}{(r_1 + r_2) r_1} - \frac{k}{r_1}$$

$$= \frac{k}{r_1} \left(\frac{r_2}{r_1 + r_2} - 1\right)$$

$$= - \frac{k}{r_1 + r_2}$$

This finally gives us:

$$E_\text{tot} = \frac{-k}{2a}$$

Where $a$ is the length of the semimajor axis, and $2a$ is the whole length of the major axis.

Perturbed Orbits

An impulsive force is applied for a very short time to create an instantaneous change in velocity $\Delta v$ at some point in the orbit. This changes the total energy of the system, and hence the shape of the orbit.

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