## TyPes Of Orbits

In the absence of planetary perturbations and nongravitational forces, a comet will orbit the Sun on a trajectory that is a conic section with the Sun at one focus. The total energy E of the comet, which is a constant of motion, will determine whether the orbit is an ellipse, a parabola, or a hyperbola. The total energy E is the sum of the kinetic energy of the comet and of its gravitational potential energy in the gravitational field of the Sun. Per unit mass, it is given by E = %v2 - GMr-1, where v is the comet's velocity and r its distance to the Sun, with M denoting the mass of the Sun and G the gravitational constant. If E is negative, the comet is bound to the Sun and moves in an ellipse. If E is positive, the comet is unbound and moves in a hyperbola. If E = 0, the comet is unbound and moves in a parabola.

In polar coordinates written in the plane of the orbit, the general equation for a conic section is r = q(1+e)/(1 +e cos 0), where r is the distance from the comet to the Sun, q the perihelion distance, e the eccentricity of the orbit, and 0 an angle measured from perihelion. When 0 < e < 1, E < 0 and the orbit is an ellipse (the case e = 0 is a circle, which constitutes a particular ellipse); when e= 1, E = 0 and the orbit is a parabola; and when e > 1, E > 0 and the orbit is a hyperbola.

In space, a comet's orbit is completely specified by six quantities called its orbital elements. Among these are three angles that define the spatial orientation of the orbit: i, the inclination of the orbital plane to the plane of the ecliptic; Q, the longitude of the ascending node measured eastward from the vernal equinox; and w, the angular distance of perihelion from the ascending node (also called the argument of perihelion). The three most frequently used orbital elements within the plane of the orbit are q, the perihelion distance in astronomical units; e, the eccentricity; and T, the epoch of perihelion passage.

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