Interpretation of the result

Let us collect now what we derived.

$${\bf L}= (L_1, L_2, L_3) = (\mu e, 0, 0)$$

and

$$\dot{{\bf L}} = (\dot{L_1}, \dot{L_2}, \dot{L_3}) = (0, \frac{3\epsilon\mu A}{a^4} e, 0)$$

The derivative of ${\bf L}$ is in the ``$(x,y)$" plane, it is perpendicular to ${\bf L}$ and it is a constant. Therefore, ${\bf L}$ will traverse a circle of radius $\mu e$. The angular speed is the velocity in the ``$y$" direction divided by the radius of the circle. (If the speed is $a$ and the radius is $b$, then the time it takes to go around the circle is $2\pi b/a$ and the angular speed is $2\pi$ divided by how long it takes to traverse a full circle, i.e., $2\pi/(2\pi b/a) = a/b$.)

Hence, we have

$$\frac{\dot{L_2}}{L_1} = \frac{3\epsilon A}{a^4} = \frac{3\epsilon \sqrt{\mu a (1-e^2)}}{a^4}$$

Remember, this is an approximation, valid only for small eccentricities. In particular, the longitude of pericenter, $\varpi$, measured from the $x$ axis, has a secular perturbation

$$\varpi = \varpi_0 + \frac{3\epsilon \sqrt{\mu a (1-e^2)}}{a^4} t$$

We can rewrite this, with the mean motion $n=\sqrt{\mu a^{-3}}$, as

$$\varpi = \varpi_0 + \frac{3\epsilon n \sqrt{1-e^2}}{a^2} t$$

In textbooks you will find similar expressions. What we derived is the secular change for the longitude and not the argument of pericenter. When you compare our results to those in books, keep this in mind.

Geometrically speaking, we can think that the sattelite moves along an ellipse which rotates about its focus at an angular velocity of $a/b$. This is of course only true for small eccentricities.

Exercise: Compute the secular part of $\dot{\varpi}$ without expansions in eccentricity and compare the result to those in textbooks. You will need the computation in the next section in order to identify terms in the perturbations.