## Orbital Elements

We have derived a set of integrals convenient for studying the dynamics of orbital motion. We now turn to another collection of constants more appropriate for describing the geometry of the orbit. The following six quantities are called the orbital elements (Fig. 6.5):

- semimajor axis a,

- eccentricity e,

gives the eccentricity), and the perihelion time t. (c) The third method best describes the geometry of the orbit. The constants are the longitude of the ascending node Q, the argument of perihelion w, the inclination i, the semimajor axis a, the eccentricity e and the time of perihelion t

Fig. 6.5a-c. Six integration constants are needed to describe a planet's orbit. These constants can be chosen in various ways. (a) If the orbit is to be computed numerically, the simplest choice is to use the initial values of the radius and velocity vectors. (b) Another possibility is to use the angular momentum k, the direction of the perihelion e (the length of which

Fig. 6.5a-c. Six integration constants are needed to describe a planet's orbit. These constants can be chosen in various ways. (a) If the orbit is to be computed numerically, the simplest choice is to use the initial values of the radius and velocity vectors. (b) Another possibility is to use the angular momentum k, the direction of the perihelion e (the length of which gives the eccentricity), and the perihelion time t. (c) The third method best describes the geometry of the orbit. The constants are the longitude of the ascending node Q, the argument of perihelion w, the inclination i, the semimajor axis a, the eccentricity e and the time of perihelion t

- longitude of the ascending node Q,

- argument of the perihelion u>,

- time of the perihelion t .

The eccentricity is obtained readily as the length of the vector e. From the equation of the orbit (6.14), we see that the parameter (or semilatus rectum) of the orbit is p = k2/p. But the parameter of a conic section is always a 11 — e21, which gives the semimajor axis, if e and k are known:

By applying (6.13), we get an important relation between the size of the orbit and the energy integral h:

-p/2h , if the orbit is an ellipse , p/2h , if the orbit is a hyperbola.

For a bound system (elliptical orbit), the total energy and the energy integral are negative. For a hyperbolic orbit h is positive; the kinetic energy is so high that the particle can escape the system (or more correctly, recede without any limit). The parabola, with h = 0, is a limiting case between elliptical and hyperbolic orbits. In reality parabolic orbits do not exist, since hardly any object can have an energy integral exactly zero.

However, if the eccentricity is very close to one (as with many comets), the orbit is usually considered parabolic to simplify calculations.

The orientation of the orbit is determined by the directions of the two vectors k (perpendicular to the orbital plane) and e (pointing towards the perihelion). The three angles i, Q and w contain the same information.

The inclination i gives the obliquity of the orbital plane relative to some fixed reference plane. For bodies in the solar system, the reference plane is usually the ecliptic. For objects moving in the usual fashion, i. e. counterclockwise, the inclination is in the interval [0°, 90°]; for retrograde orbits (clockwise motion), the inclination is in the range (90°, 180°]. For example, the inclination of Halley's comet is 162°, which means that the motion is retrograde and the angle between its orbital plane and the ecliptic is 180° - 162° = 18°.

The longitude of the ascending node, Q, indicates where the object crosses the ecliptic from south to north. It is measured counterclockwise from the vernal equinox. The orbital elements i and Q together determine the orientation of the orbital plane, and they correspond to the direction of k, i. e. the ratios of its components.

The argument of the perihelion w gives the direction of the perihelion, measured from the ascending node

in the direction of motion. The same information is contained in the direction of e. Very often another angle, the longitude of the perihelion m (pronounced as pi), is used instead of a>. It is defined as m = ¡¡2 + œ .

which gives directly the mean anomaly M (which will be defined in Sect. 6.7).

where er is a unit vector parallel with r (Fig. 6.6). If the planet moves with angular velocity f, the direction of this unit vector also changes at the same rate:

This is a rather peculiar angle, as it is measured partly along the ecliptic, partly along the orbital plane. However, it is often more practical than the argument of perihelion, since it is well defined even when the inclination is close to zero in which case the direction of the ascending node becomes indeterminate.

We have assumed up to this point that each planet forms a separate two-body system with the Sun. In reality planets interfere with each other by disturbing each other's orbits. Still their motions do not deviate very far from the shape of conic sections, and we can use orbital elements to describe the orbits. But the elements are no longer constant; they vary slowly with time. Moreover, their geometric interpretation is no longer quite as obvious as before. Such elements are osculating elements that would describe the orbit if all perturbations were to suddenly disappear. They can be used to find the positions and velocities of the planets exactly as if the elements were constants. The only difference is that we have to use different elements for each moment of time.

Table C.12 (at the end of the book) gives the mean orbital elements for the nine planets for the epoch J2000.0 as well as their first time derivatives. In addition to these secular variations the orbital elements suffer from periodic disturbations, which are not included in the table. Thus only approximate positions can be calculated with these elements. Instead of the time of perihelion the table gives the mean longitude where ef is a unit vector perpendicular to er. The velocity of the planet is found by taking the time derivative of (6.19):

The angular momentum k can now be evaluated using (6.19) and (6.21):

where ez is a unit vector perpendicular to the orbital plane. The magnitude of k is k = r2 f .

The surface velocity of a planet means the area swept by the radius vector per unit of time. This is obviously the time derivative of some area, so let us call it A. In terms of the distance r and true anomaly f, the surface velocity is

r x \ a/\/ \ | |

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