Unperturbed planetary orbit

Consider a planet (say) of mass $m$ and relative position vector ${\bf r}$ that is orbiting around the Sun, whose mass is $M$. The planet's unperturbed equation of motion is written (see Section 4.16)

$$\ddot{{\bf r}} + \mu\,\frac{\bf r}{r^{\,3}} = {\bf0},$$ (I.1)

where $\mu= G\,(M+m)$. As described in Chapter 4, the solution to this equation is a Keplerian ellipse.

Let $x$, $y$, $z$ be a Cartesian coordinate system in a reference frame whose origin corresponds to the location of the Sun, and which is such that the planet's unperturbed orbit lies in the plane $z=0$, with the angular momentum vector pointing in the positive $z$-direction, and the perihelion situated on the positive $x$-axis. Let $r$, $\theta$, $z$ be a cylindrical coordinate system in the same reference frame.

We know from the analysis of Chapter 4 that

$${\bf r} = r\,{\bf e}_r,$$ (I.2)

$$\dot{\bf r} = \skew{5}\dot{r}\,{\bf e}_r + r\,\skew{5}\dot\theta\,{\bf e}_\theta,$$ (I.3)

where ${\bf e}_r=(\cos\theta,\,\sin\theta,\,0)$ and ${\bf e}_\theta = (-\sin\theta,\,\cos\theta,\,0)$. Moreover, the planet's mean orbital angular velocity is

$$n= \left(\frac{\mu}{a^{\,3}}\right)^{1/2};$$ (I.4)

its orbital energy per unit mass is

$${\cal E} = -\frac{\mu}{2\,a};$$ (I.5)

its orbital angular momentum per unit mass is

$${\bf h} = {\bf r}\times \dot{\bf r} = h\,{\bf e}_z,$$ (I.6)

where

$$h \equiv r^{\,2}\,\skew{5}\dot{\theta} = (1-e^{\,2})^{1/2}\,(\mu\,a)^{1/2};$$ (I.7)

and its eccentricity vector is

$${\bf e} \equiv \frac{\dot{\bf r}\times {\bf h}}{\mu} - \frac{{\bf r}}{r} = e\,{\bf e}_x.$$ (I.8)

Here, $a$ and $e$ are the planet's orbital major radius and eccentricity, respectively. Note that, for the unperturbed orbit, the quantities $a$, $e$, $n$, ${\cal E}$, ${\bf h}$, and ${\bf e}$ are all constant in time. We also have

$$r = \frac{a\,(1-e^{\,2})}{1+e\,\cos\theta}=a\,(1-e\,\cos E),$$ (I.9)

$$\skew{5}\dot{r} = \frac{h\,e\,\sin\theta}{a\,(1-e^{\,2})}=\frac{h\,e}{a\,(1-e^{\,2})^{1/2}}\,\frac{\sin E}{1-e\,\cos E},$$ (I.10)

$$\cos E = \frac{\cos\theta+e}{1+e\,\cos\theta},$$ (I.11)

$$\sin E = \frac{(1-e^{\,2})^{1/2}\,\sin\theta}{1+e\,\cos\theta},$$ (I.12)

$$E - e\,\sin E = {\cal M},$$ (I.13)

where $\theta$, $E$, and ${\cal M}$ are the planet's true anomaly, eccentric anomaly, and mean anomaly, respectively. (See Chapter 4.)