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Next: Orbital elements Up: Keplerian orbits Previous: Transfer orbits


Elliptic orbits

Let us determine the radial and angular coordinates, $ r$ and $ \theta $ , respectively, of a planet in an elliptical orbit about the Sun as a function of time. Suppose that the planet passes through its perihelion point, $ r=r_p$ and $ \theta=0$ , at $ t=\tau$ . The constant $ \tau$ is termed the time of perihelion passage. It follows from the previous analysis that

$\displaystyle r = \frac{r_p\,(1+e)}{1+e\,\cos\theta},$ (4.51)

and

$\displaystyle {\cal E} = \frac{\skew{3}\dot{r}^{\,2}}{2} + \frac{h^{\,2}}{2\,r^{\,2}} - \frac{G\,M}{r},$ (4.52)

where $ e$ , $ h = \sqrt{G\,M\,r_p\,(1+e)}$ , and $ {\cal E} = G\,M\,(e-1)/(2\,r_p)$ are the orbital eccentricity, angular momentum per unit mass, and energy per unit mass, respectively. The preceding equation can be rearranged to give

$\displaystyle \skew{3}\dot{r}^{\,2} = (e-1)\,\frac{G\,M}{r_p} - (e+1)\,\frac{r_p\,G\,M}{r^{\,2}} + \frac{2\,G\,M}{r}.$ (4.53)

Taking the square root, and integrating, we obtain

$\displaystyle \int_{r_p}^r\frac{r\,dr}{[2\,r + (e-1)\,r^{\,2}/r_p - (e+1)\,r_p]^{1/2}} = \sqrt{G\,M}\,\,(t-\tau).$ (4.54)

Consider an elliptical orbit characterized by $ 0<e < 1$ . Let us write

$\displaystyle r = \frac{r_p}{1-e}\,(1-e\,\cos E),$ (4.55)

where $ E$ is termed the eccentric anomaly. In fact, $ E$ is an angle that varies between $ -\pi$ and $ +\pi$ . Moreover, the perihelion point corresponds to $ E=0$ , and the aphelion point to $ E=\pi$ . Now,

$\displaystyle dr = \frac{r_p}{1-e}\,e\,\sin E\,dE,$ (4.56)

whereas

$\displaystyle 2\,r + (e-1)\,\frac{r^{\,2}}{r_p}- (e+1)\,r_p = \frac{r_p}{1-e}\,e^{\,2}\,(1-\cos^2 E)= \frac{r_p}{1-e}\,e^{\,2}\,\sin^2E.$ (4.57)

Thus, Equation (4.54) reduces to

$\displaystyle \int_0^E (1-e\,\cos E)\,dE = \left(\frac{G\,M}{a^{\,3}}\right)^{1/2} (t-\tau),$ (4.58)

where $ a = r_p/(1-e)$ . This equation can immediately be integrated to give

$\displaystyle E - e\,\sin E = {\cal M}.$ (4.59)

Here,

$\displaystyle {\cal M} = n\,(t-\tau)$ (4.60)

is termed the mean anomaly, $ n=2\pi/T$ is the mean orbital angular velocity, and $ T= 2\pi\,(a^{\,3}/GM)^{1/2}$ the orbital period. The mean anomaly is an angle that increases uniformly in time at the rate of $ 2\pi$ radians every orbital period. Moreover, the perihelion point corresponds to $ {\cal M}=0$ , and the aphelion point to $ {\cal M} = \pi$ . Incidentally, the angle $ \theta $ , which determines the true angular location of the planet relative to its perihelion point, is called the true anomaly. Equation (4.59), which is known as Kepler's equation, is a transcendental equation that does not possess a convenient analytic solution. Fortunately, it is fairly straightforward to solve numerically. For instance, when we use an iterative approach, if $ E_n$ is the $ n$ th guess then

$\displaystyle E_{n+1} = {\cal M}+ e\,\sin E_n;$ (4.61)

this iteration scheme converges very rapidly when $ 0\leq e\ll 1$ (as is the case for planetary orbits).

Figure 4.5: Eccentric anomaly.
\begin{figure}
\epsfysize =3.25in
\centerline{\epsffile{Chapter03/fig3.05.eps}}
\end{figure}

Equations (4.51) and (4.55) can be combined to give

$\displaystyle r\,\cos\theta = a\,(\cos E-e).$ (4.62)

This expression allows us to give a simple geometric interpretation of the eccentric anomaly, $ E$ . Consider Figure 4.5. Let $ PGA$ represent the elliptical orbit of a planet, $ G$ , about the Sun, $ S$ . Let $ ACP$ be the major axis of the orbit, where $ P$ is the perihelion point, $ A$ the aphelion point, and $ C$ the geometric center. It follows that $ CA=CP=a$ and $ CS=e\,a$ (see Section A.9), where $ a$ is the orbital major radius and $ e$ the eccentricity. Moreover, the distance $ SG$ and the angle $ GSQ$ correspond to the radial distance, $ r$ , and the true anomaly, $ \theta $ , respectively. Let $ PRA$ be a circle of radius $ a$ centered on $ C$ . It follows that $ AP$ is a diameter of this circle. Let $ RGQ$ be a line, perpendicular to $ AP$ , that passes through $ G$ and joins the circle to the diameter. It follows that $ CR=a$ . Let us denote the angle $ RCS$ as $ E$ . Simple trigonometry reveals that $ SQ=r\,\cos\theta$ and $ CQ=a\,\cos E$ . But, $ CQ=CS+SQ$ , or $ a\,\cos E = e\,a+r\,\cos\theta$ , which can be rearranged to give $ r\,\cos\theta = a\,(\cos E-e)$ , which is identical to Equation (4.62). We, thus, conclude that the eccentric anomaly, $ E$ , can be identified with the angle $ RCS$ in Figure 4.5.

Equations (4.51) and (4.55) can be combined to give

$\displaystyle \cos\theta = \frac{\cos E - e}{1-e\,\cos E}.$ (4.63)

Thus,

$\displaystyle 1+\cos\theta = 2\,\cos^2(\theta/2) = \frac{2\,(1-e)\,\cos^2( E/2)}{1-e\,\cos E},$ (4.64)

and

$\displaystyle 1-\cos\theta = 2\,\sin^2(\theta/2) = \frac{2\,(1+e)\,\sin^2 (E/2)}{1-e\,\cos E}.$ (4.65)

The previous two equations imply that

$\displaystyle \tan (\theta/2) = \left(\frac{1+e}{1-e}\right)^{1/2}\tan (E/2).$ (4.66)

The eccentric anomaly, $ E$ , and the true anomaly, $ \theta $ , always lie in the same quadrant (i.e., if $ 0\leq E\leq \pi/2$ then $ 0\leq\theta\leq\pi/2$ , etc.) We conclude that in the case of a planet in an elliptical orbit around the Sun the radial distance, $ r$ , and the true anomaly, $ \theta $ , are specified as functions of time via the solution of the following set of equations:

    $\displaystyle {\cal M}$ $\displaystyle = n\,(t-\tau),$ (4.67)
    $\displaystyle E - e\,\sin E$ $\displaystyle = {\cal M},$ (4.68)
    $\displaystyle r$ $\displaystyle =a\,(1-e\,\cos E),$   $\displaystyle ,$ (4.69)
and   $\displaystyle \tan(\theta/2)$ $\displaystyle =\left(\frac{1+e}{1-e}\right)^{1/2} \tan (E/2).$ (4.70)

Here, $ n=2\pi/T$ , $ T=2\pi\,(a^{\,3}/G\,M)^{1/2}$ , and $ a = r_p/(1-e)$ . Incidentally, it is clear that if $ t\rightarrow t+T$ then $ {\cal M}\rightarrow {\cal M}+2\pi$ , $ E\rightarrow E + 2\pi$ , and $ \theta\rightarrow
\theta+2\pi$ . In other words, the motion is periodic with period $ T$ .


next up previous
Next: Orbital elements Up: Keplerian orbits Previous: Transfer orbits
Richard Fitzpatrick 2016-03-31