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

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<b>Lecture 18: Orbits III: Total Orbital Energy and Perturbed Orbits</b><br>
Thu 26 Mar 2026 15:00
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== Eccentricity ==

We can classify orbits by their eccentricity <nowiki>$\epsilon$</nowiki>:

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<nowiki>$\epsilon = 0$</nowiki>: Circular orbit.

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<nowiki>$0 < \epsilon < 1$</nowiki>: Elliptical orbit.

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<nowiki>$\epsilon = 1$</nowiki>: Parabolic orbit.

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<nowiki>$\epsilon > 1$</nowiki>: Hyperbolic orbit.

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

== Total Energy in an Elliptical Orbit w/ <nowiki>$1/r$</nowiki> Central Potential ==

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

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

We have total energy: <div class="pn-import-equation"><nowiki>$$E_\text{tot} = \frac{1}{2}m \dot{r}^2 + \frac{l^2}{2mr^2} - \frac{k}{r} = \text{const.}$$</nowiki></div>

Noting that <nowiki>$l$</nowiki> is a constant, and <nowiki>$k = GMm$</nowiki>. As the apogee and perigee are turning points, <nowiki>$\dot{r} = 0$</nowiki> here. Hence: <div class="pn-import-equation"><nowiki>$$E_\text{tot} = \frac{l^2}{mr_1^2} - \frac{k}{r_1} = \frac{l^2}{mr_2^2} - \frac{k}{r_2} \tag{1}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\frac{l^2}{2m} \left(\frac{1}{r_1^2} - \frac{1}{r_2^2}\right) = k\left( \frac{1}{r_1} - \frac{1}{r_2}\right)$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$\frac{l^2}{2m} \left(\frac{r_2^2 - r_1^2}{r_1^2 r_2^2}\right) = k \frac{r_2 - r_1}{r_1 r_2}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$l^2 = 2mk \frac{r_2 - r_1}{r_1 r_1} \frac{r_1^2 r_2^2}{r_2^2 - r_1^2}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$l^2 = \frac{2mk (r_2 - r_1) r_1 r_2}{(r_2 + r_1) (r_2 - r_1)}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$l^2 = \frac{2mk r_1 r_2}{r_1 + r_2} \tag{2}$$</nowiki></div>

Eqn (2) into (1): <div class="pn-import-equation"><nowiki>$$E_\text{tot} = \frac{2mk r_1 r_2}{r_1 + r_2} \frac{1}{2mr_1^2} - \frac{k}{r_1} = \text{ \dots for the RHS, but not required.}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$E_\text{tot} = \frac{r_2 K}{(r_1 + r_2) r_1} - \frac{k}{r_1}$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$= \frac{k}{r_1} \left(\frac{r_2}{r_1 + r_2} - 1\right)$$</nowiki></div> <div class="pn-import-equation"><nowiki>$$= - \frac{k}{r_1 + r_2}$$</nowiki></div>

This finally gives us: <div class="pn-import-equation"><nowiki>$$E_\text{tot} = \frac{-k}{2a}$$</nowiki></div> Where <nowiki>$a$</nowiki> is the length of the semimajor axis, and <nowiki>$2a$</nowiki> 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 <nowiki>$\Delta v$</nowiki> 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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