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Orbital Mechanics Today, astronomers still use the elliptical orbits described by Kepler, and predicted by Newtonian physics, to describe the motions of the planets, comets, asteroids, and nearly every other object in the solar system.

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Orbital Elements

A Keplerian orbit can be defined by six mathematical quantities, called the orbital elements. The first two elements define the shape and size of the orbit; the remaining elements define the orbit’s orientation in space, and the object’s position in its orbit. The eccentricity (left) defines the shape of the orbit; the other elements (right) define its size and orientation.
  • The Eccentricity (e) defines the shape of the orbit. It is the ratio of the distance between the orbit’s foci to the length of its major axis, and it defines the shape of the orbit. For elliptical orbits, the eccentricity is less than 1; for a circular orbit, it is zero; for parabolic orbits, the eccentricity is exactly 1; and for hyperbolic orbits, the eccentricity is greater than 1.
  • The semimajor axis (a) is the size of the orbit. The semimajor axis is defined as half the length of the long axis of the orbit. The periapsis distance (q) is sometimes used instead of the semimajor axis. The periapsis is the point where the object is closest to the body it orbits.
  • The inclination (i) is the angle between the object’s orbital plane and the ecliptic plane.
  • The longitude of ascending node (Ω) is the angle in the Ecliptic plane from the vernal equinox to the ascending node. The ascending node is where the object’s orbit crosses the ecliptic plane from south to north.
  • The argument of periapsis (ω) is the angle in the orbital plane from the ascending node to the periapsis. The longitude of periapsis (π) is sometimes used instead of the argument of periapsis. If so, you can find the argument of periapsis by subtracting the longitude of the ascending node (Ω): in other words, ω = π - Ω.
  • The mean anomaly (M) describes the object’s position in its orbit at a particular point in time. When the object is at periapse, the mean anomaly is zero. Sometimes a different quantity, the mean longitude, is used instead of the mean anomaly. If so, you can find the mean anomaly by subtracting the longitude of perihelion (π). In other words, L = M + π = M + ω + Ω.
The orbital elements of the planets, comets, asteroids, and even other objects such as artificial earth satellites are usually expressed in this format. Using those elements, the object’s position can be computed using Kepler and Newton’s laws for years in advance to a very high degree of accuracy. Perturbations and Osculating Elements If an object and the object it orbits (called its primary) were the only two objects in the universe, then their trajectory would be a perfect Keplerian orbit such as described above. Hence, this kind of orbit is sometimes called a “two-body” orbit. But in reality, there are far more than two objects in the universe. Because of the gravitational perturbations of the other planets, the orbit of any object in the solar system is always changing. For earth-orbiting satellites, other forces - such as atmospheric drag - also apply, so a single set of orbital elements degrades much faster (weeks instead of years). A single set of orbital elements will gradually become less accurate over time. An object’s orbital elements at a particular moment in time (or epoch) are referred to as its osculating elements. At that one moment in time, the path predicted by those elements will exactly match, or “kiss”, the object’s true path through space (the word “osculating” derives from the Latin verb for “to kiss”.) Using only a single set of osculating orbital elements, the error in the predictedposition of as object (in this case the asteroid Ceres) grows over time. In fact, modern science can only predict the positions of the planets to “trustworthy” accuracy within about 10,000 years of the present. The fundamental problem is that the motions of the planets are chaotic - that is, essentially unpredictable - in the long term. We do not know the current positions, masses, and velocities of all planets and moons in the solar system to calculate their positions to arcsecond precision more than a few thousand years from the present. For instance, the positions of Saturn’s moons is affected by the gravity of Saturn’s rings, which in turn depends on the mass of Saturn’s rings - and that is an all-but-unknown quantity. While we do know that 500,000 years ago, the Earth was in an orbit roughly the same size and shape as it is today, we simply can’t predict which side of the Sun it was on.