Laws of Orbital Motion
Over the course of the 17th century, the German astronomer Johannes Kepler realized that the planets do not describe circular orbits around the earth. He developed what came to be known as Kepler's Laws of Planetary Motion. Stated simply:
- A planet orbits the sun, and its path is an ellipse with the sun at one focus of the ellipse.
- The line segment joining the sun and any planet sweeps out equal areas in equal intervals of time as the planet moves in its orbit.
- The square of the period of a planet varies directly as the cube of its mean distance from the sun—or to be more specific, the semi-major axis.
Sir Isaac Newton would later formulate the physical basis for all of Kepler's laws. This was his Law of Universal Gravitation, which states that:
|“||Every particle in the universe attracts every other particle with a force inversely proportional to the square of the distance between them and directly proportional to the product of their masses.||”|
Or in algebra:
where G is a constant, M and m are the masses of the larger and smaller objects respectively, and R is the distance between their centers.
Generalized concepts of orbits
Kepler developed his laws in the special context of closed orbits, or orbits that begin and end at the same point. But not all orbits are closed. Technically, the path of any object whose motion is subject almost entirely to gravity is an orbit. This would include even the path of an artillery shell. In fact, an artillery shell—or to be more specific, a "Newtonian cannonball" fired from a cannon mounted high above the earth's atmosphere—is a common metaphor for teaching the orbit concept to science students.
An orbit is a conic section, which is the section that a plane makes with a right circular conical surface. A closed orbit is generally an ellipse—or, if the foci of the ellipse coincide, a circle. An open orbit is either a parabola or a hyperbola. Even in the case of an open orbit, Kepler's first and second laws still apply: the more-massive object, or primary, is at a focus (actually, the focus of a parabola), and the line segment joining the primary to the orbiting object sweeps equal areas in equal times. (The third law has no meaning in the case of an open orbit, because the "period," as such, is infinite.)
An object in a parabolic orbit moves at escape speed as it makes its closest approach to the primary. If the object is in fact moving faster than this critical speed at the moment of its closest approach, then its orbit will be hyperbolic.
No description of orbital elements is complete without defining some other terms used to define them.
- An orbit system is any combination of a massive body and all other bodies or objects in orbit around it. The central massive body is called the primary.
- A direction is a ray having the geometric center of the orbit as origin and passing through an arbitrary point in the orbit.
- A reference direction is an arbitrary direction chosen to be roughly the same for all bodies or objects in a given orbit system. For the solar system, the reference direction is the vernal point, or the direction of Earth at the vernal equinox. For a system of a planet and its moons, the reference direction is the ray parallel to the reference direction of the ecliptic. For an extrasolar planet, the reference direction is "galactic north."
- A reference plane is an arbitrary plane in the space local to the orbit system. For planetary or dwarf planetary orbits, the usual reference plane is the ecliptic (the orbit of Earth around the Sun). For lunar orbits, the reference plane is the one containing the equator of the primary.
- A node is a point at which the orbit passes through a reference plane. The ascending node is the point at which the orbiting object moves from "south" to "north" with respect to the reference plane. The descending node is the point at which the object moves from "north" to "south."
- The semi-major axis (symbol: a), which is the line segment that joins the geometric center of the orbit to either of the two points of the ellipse that lie along the major axis (the line that passes through the foci). This is the measure of the size of an orbit.
- The orbital eccentricity (symbol: e), a special dimensionless number that defines the deviation of the orbit from a perfect circle. For a circle, e = 0. For an ellipse, e is the quotient of the distance from the geometric center to either focus, divided by the semi-major axis. Hence 0 < e < 1. Open orbits begin with e = 1 (parabola) and continue with e > 1 (hyperbola). Eccentricity determines the shape of an orbit.
- The orbital period (symbol: T), the time the orbiting body takes to revolve around the primary. By Kepler's Third Law as modified by Newton, this varies inversely as the square root of the primary's mass. Thus the measurement of the semi-major axis and period of any orbit can accurately predict the mass of its primary. (In fact, this is how astronomers calculated the mass of the dwarf planet Eris and determined that Eris was significantly heavier than Pluto.)
- The inclination (symbol: i) of the orbit is the angle that its plane makes with a reference plane.
- The argument of periapsis (symbol: ω) is the angle that the periapsis makes with the direction (see above) of the ascending node. See apsis for the definitions of periapsis and apoapsis.
- The longitude of ascending node (symbol: Ω) is the arc, in the direction of the orbit, from the reference direction to the direction of the ascending node.
- Orbit by Wikipedia
- Walker, John. "Orbits in Strongly Curved Spacetime." Fourmilab Switzerland, December 25, 1998. Accessed January 23, 2008.
- "Newtonian Gravitation and the Laws of Kepler." Astronomy 161, University of Tennessee, Knoxville, TN. Accessed January 23, 2008.
- "Entry for 'Orbit'." The Columbia Encyclopedia, 6th ed., 2001-2007. Accessed January 23, 2008.
- "Solar System Data." <http://www.nineplanets.org/> Accessed January 23, 2008.
- Nave, C. R. "Kepler's Laws." HyperPhysics, Department of Physics and Astronomy, Georgia State University. Accessed January 23, 2008.
- Williams, James G., PhD. "Orbit (astronomy and physics)." Microsoft Encarta Online Encyclopedia, 2007. Accessed January 23, 2008.