Planetary Orbits

Figure: Keplerian orbital elements for the elliptical orbit of the point $\mathbf{r_m}$ around the focus $F$ with true anomaly $\nu$. Parameters of the ellipse are the axes $a$ and $b$, the focal distance $c = ae$, the semi-latus rectum $p$ and the point of periapsis $P_a$ at distance $r_a$ from $F$. Also shown are the concentric circles for the eccentric motion of the point $\mathbf{r_q'}$ with eccentric anomaly $E$, and mean motion of the point $\mathbf{r_q}$ with mean anomaly $M$ (dashed).

As pointed out in the introduction transformations based on classical Keplerian elements can only achieve a limited precision. But for many applications it is useful to have approximate positions available. For this reason we describe in the following the calculation of position and velocity of objects in Keplerian orbits. There are many textbooks on this subject - we recommend Murray and Dermott (2000) but e.g. Bate et al. (1971),Danby (1988) or Heafner (1999) are also very useful. There are also some good web sites devoted to the subject¹⁹.

The gravitational motion of two bodies of mass $M$ and $m$ and position vectors $\mathbf{r_M}$ and $\mathbf{r_m}$ can be described in terms of the three invariants: gravitational parameter $\mu=\gamma_0(M+m)$, specific mechanical energy $\mathcal{E} = \frac{v^2}{2} - \frac{\mu}{r}$, and specific angular momentum $h = \vert\mathbf{r} \times \mathbf{v}\vert$, where $\mathbf{r} = \mathbf{r_M}-\mathbf{r_m}$, $r = \vert\mathbf{r}\vert$ and $\mathbf{v} = \mathbf{\dot{r}}$. $\gamma_0$ is the constant of gravitation whose IAU1976 value is determined by [A. K6]:

$$\gamma_0 = k^2 \mbox{ with } k=0.01720209895$$ (24)

when masses are given in solar masses, distances in AU [1 AU = 149 597 870 km], and times in days.

The elements of the conical orbit (shown in Fig.2) are then given as semi-major axis $a = -\mu/\mathcal{E}$ and semi-minor axis $b = h/\sqrt{-\mathcal{E}}$, or alternatively as semi-latus rectum $p = b^2/a = h^2/\mu = a(1-e^2)$ and eccentricity $e = \sqrt{1 - b^2/a^2} = \sqrt{1 + \mathcal{E} h^2/\mu^2}$. Let the origin be at the focus $\mathbf{r_M}$, the vector $\mathbf{r}$ then describes the motion of the body $\mathbf{r_m}$. The true anomaly $\nu$ is the angle between $\mathbf{r}$ and the direction to the closest point of the orbit (periapsis) and can be determined from

$$r = \frac{p}{1 + e \cos\nu}.$$ (25)

If there are two focal points (ellipse, hyperbola) their distance is given by $c = e\cdot a$, the distances of the periapsis and apoapsis are given by $r_p = a(1-e)$ and $r_a = a(1+e)$. An elliptical orbit has the period $P = 2\pi a\sqrt{a/\mu}$.

Mean elements of a body in an elliptical orbit ($e<1$) are defined by the motion of a point $\mathbf{r_q}$ on a concentric circle with constant angular velocity $n = \sqrt{\mu/a^3}$ and radius $\sqrt{ab}$, such that the orbital period $P = 2\pi a\sqrt{a/\mu}$ is the same for $\mathbf{r_q}$ and $\mathbf{r_m}$. The mean anomaly $M = \sqrt{\mu/a^3}(t-T)$ is defined as the angle between periapsis and $\mathbf{r_q}$. Unfortunately there is no simple relation between $M$ and the true anomaly $\nu$. To construct a relation one introduces another auxiliary concentric circle with radius $a$ and defines $\mathbf{r_q'}$ as the point on that circle which has the same perifocal x-coordinate as $\mathbf{r_m}$. The eccentric anomaly $E$ is the angular distance between $\mathbf{r_q'}$ and the periapsis measured from the centre and is related to the mean and true anomalies by the set of equations:

$$M = E - e\sin E \hspace{1cm} \mbox{\it (Kepler equation)}$$ (26)

$$\cos\nu = \frac{e-\cos E}{e\cos E -1} \hspace{0.5cm} \cos E =... ...cos\nu}=\frac{r}{p}(e+\cos\nu) \hspace{0.5cm} r = a(1-e\cos E)$$ (27)

Thus, if the orbital position is given as an expansion in $t_0$ of the mean longitude $\lambda = \Omega + \omega +M$, the true longitude $\lambda_0 = \Omega + \omega +\nu$ can be found by an integration of the transcendental Kepler equation. In most cases a Newton-Raphson integration converges quickly (see Danby (1988) or Herrick (1971) for methods). For hyperbolic orbits ($e>1$) one can as well define a mean anomaly $M_h = \sqrt{\mu/\vert a\vert^3}(t-T)$ but this quantity has no direct angular interpretation. The hyperbolic eccentric anomaly $E_h$ is related to $M_h$ and the true anomaly $\nu$ by

$$M_h = e\sinh E_h - E_h \hspace{0.5cm} \cos\nu = \frac{e-\cos... ...frac{e+\cos\nu}{1+e\cos\nu} \hspace{0.5cm} r = a(1-e\cosh E_h)$$ (28)

Figure: Orientation of a Keplerian orbit of the point $\mathbf{r_m}$ around the focus $O$ with respect to the ecliptic plane. Symbols are given for the equinox $\vernal$, the ascending node $\ascnode$ and its longitude $\Omega$, the periapsis $\mathbf{r_p}$ and its argument $\omega$, the inclination $i$, and the true anomaly $\nu$. The perifocal system is denoted by (X',Y',Z').

The orientation of an orbit with respect to a reference plane (e.g. ecliptic) with origin at the orbital focus is defined by the inclination $i$ of the orbital plane, the longitude of the ascending node $\Omega$, and the argument of periapsis $\omega$ which is the angle between ascending node and periapsis $\mathbf{r_p}$ (see Fig.3). The position of the body on the orbit can then be defined by its time of periapsis passage $T$, its true anomaly $\nu_0$ at epoch $t_0$, or its true longitude $\lambda_0 = \Omega + \omega +\nu_0$ at epoch $t_0$. The perifocal coordinate system has its X-axis from the focus to the periapsis, and its Z-axis right-handed perpendicular to the orbital plane in the sense of orbital motion. In this system the position and velocity vector are given by

$$\mathbf{r} = r (\cos\nu,\sin\nu,0) \hspace{1cm} \mathbf{v} = \sqrt{\mu/p}(-\sin\nu,e+\cos\nu,0).$$ (29)

These might directly be expressed by the eccentric anomaly $E$:

$$\mathbf{r} = a (\cos E -e,\sqrt{1-e^2}\sin E,0) \hspace{1cm}... ...thbf{v} = \frac{\sqrt{\mu/a}}{r}(-\sin E, \sqrt{1-e^2}\cos E,0)$$ (30)

In the hyperbolic case replace $\cos$ by $\cosh$ and $\sin$ by $\sinh$. The transformation from the reference system to the perifocal system is given by the Eulerian rotation $E(\Omega,i,\omega)$ as defined in the Appendix. The ecliptic position of a planet is then given by $\mathbf{r_e}=E(\Omega,i,\omega)\mathbf{r}$.