Inner Planets: Mercury, Venus, Earth, and Mars,

as well as the

Outer Planets: Jupiter, Saturn, Neptune, Uranus, and Pluto.

These simulations rely exclusively on the mathematics of Kepler as explained and illustrated in Sections 4.7 to 4.9 of Chapter 4.

In the display Orbital Motion The Inner Planets, the Sun (it is shown much larger than to scale) appears in the center. Simply click on one or more of the images of the four inner planets and the planet(s) will begin to orbit the Sun. The orbital parameters  (astronomical eccentricity), T (period of the orbit in years), and a (semimajor axis in AU), for the last planet clicked will appear in the displays. Click on The Inner Planets and all four inner planets will orbit. Mark the box Trails and the elliptical orbit(s) will be traced out. Click on clear and the planet(s) disappear. Click on The Outer Planets, and a similar display for Jupiter, Saturn, Neptune, Uranus, and Pluto will appear with the same features as the earlier one. The circle in the center of this display depicts the orbit of Mars.

One additional orbiting body, for example a comet or asteroid, can be launched by typing its  with 0 <  < 1 and a into the appropriate box. Click orbit and it will join the inner planets or outer planets already selected.

Information for some of the more important comets and asteroids follows below. Distances are given in astronomical units (AUs) and time in Earth years. As before, the parameters a, , and T represent respectively, the semimajor axis, the astronomical eccentricity, and the period of the orbit. The parameter i is the angle of the plane of the orbit with the plane of the Earth's orbit. Note that the orbital planes are often quite different from the Earth's orbital plane. Kepler's third law tells us that a³/T² is the same constant for any body in orbit around the Sun. Since a = 1 and T = 1 for the Earth, it follows that T = a^3/2 for all planets, comets and asteroids. (In the latter cases, the perturbations caused by Saturn and Jupiter can result in irregularities.)
 

Comets contain matter left over from the formation of the solar system. Consisting of ice, dust, and gases, they are commonly referred to as "dirty snowballs." They are studied by astronomers (among other reasons) for the information they reveal about the formation process of the solar system. According to an explanation developed by J.H. Oort in 1950, comets originate in a cloud of comets, dust, and gases that lies in the range of 40,000 AU to 50,000 AU from the Sun. Asteroids are minor planets. Most of them orbit the Sun in the region between the orbits of Mars (semimajor axis 1.52 AUs) and Jupiter (semimajor axis 5.20 AUs) in what is known as the asteroid belt. All are relatively small, Ceres with a diameter greater than 940 kilometers is the largest, and only one, Vesta, is visible to the naked eye. It is thought that before asteroids could form into full-fledged planets, their orbits were tilted and elongated (possibly due to the gravitational effects of Jupiter) and that this prevented them from growing into planets.

Halley's Comet:
The English astronomer Edmund Halley (1656-1742) published his work on comets in 1705 in which he claimed that the comet of 1682 had a period of about 76 years and that it would be seen again in 1758. He was correct and this comet was later named Halley in his honor. Today's information is more precise. Each orbit is slightly different with a variation of 17.626 < a < 18.256. This corresponds to 74 < T < 78. The average value is a = 17.94 and this corresponds to T = 76 years. The astronomical eccentricity is  = 0.97. The angle between the plane of Halley's orbit and that of the Earth is 17.8^o with Halley orbiting in the direction opposite to that of the Earth. Since 180 ^_ 17.8 = 162.2^o, the angle of inclination of the orbit is taken to be i = 162.2^o. The most recent closest approach occurred in 1986.

Encke's Comet:
This comet was discovered by the German astronomer Johann Frank Encke (1791-1865). After having fought in the Napoleonic wars, Encke became director of the Berlin Observatory in 1825. The comet is named in his honor because he established its periodicity. The average values are a = 2.22,  = 0.85, T = 3.31, and i = 12^o. So the plane of Encke's orbit is close to that of the Earth with the comet orbiting in the same direction. It has the shortest period of all known comets.

Shoemaker-Levy 9:
This comet was first detected on March 24, 1993 by Carolyn and Eugene Shoemaker and David Levy in orbit around Jupiter. The comet had broken up into at least 18 large fragments after a close approach to Jupiter on July 7, 1992. The fragments impacted Jupiter in a spectacular way at the next closest approach during the period July 16-22, 1994. The available data allowed a reconstruction of the essential history of the comet. Prior to its capture by Jupiter, the comet orbited the Sun with a semimajor axis either in the range 3.5 < a < 4.5 AU (just interior to Jupiter's orbit) or in the range 6.0 < a < 8.0 AU (just exterior to Jupiter). In either case, it had an eccentricity in the range 0.05 <  < 0.3 and an inclination in the range 0^o < i < 6^o. It is likely that the comet was captured by Jupiter during the years 1929 to 1939 and that it orbited unseen until its discovery.

Hale-Bopp:
This comet was discovered in July 1995 by Alan Hale and Thomas Bopp. The average values of the orbital parameters are a = 187.482,  = 0.995, T = 2567, and i = 89.4^o. The most recent closest approach to Earth was on March 22, 1997.

Hyakutake:
Discovered by Yuji Hyakutake in January 1996 with a pair of binoculars, this comet had its most recent closest approach to Earth on March 25, 1996. The average values of its orbital parameters are a = 25.741,  = 0.999, T = 130.6, and i = 124.9^o.

Ceres:
This largest of all asteroids was discovered in 1801 by the Italian astronomer Giuseppe Piazzi (1746-1826). Its average orbital parameters are a = 2.745,  = 0.079, T = 4.06, and i = 10.6^o.

Vesta:
Discovered in 1807 by the German astronomer Heinrich Olbers, it is the second largest of the asteroids. Its average orbital parameters are a = 2.36,  = 0.237, T = 3.63, and i = 7.1^o.

Hidalgo:
This asteroid was discovered in 1920 by the German astronomer Walter Baade (1893-1960). Hidalgo's average orbital parameters are a = 5.805,  = 0.657, T = 14.4, and i = 42.5^o.

Icarus:
This asteroid, discovered by Baade in 1949, travels as far as 1.97 AU from the Sun (beyond Mars) to as close as 0.19 AUs to the Sun (within the orbit of Mercury). Its average parameters are a = 1.8,  = 0.827, T = 1.12, and i = 22.9^o.

Chiron:
Discovered in 1977 by the American astronomer Charles Kowal from Mount Palomar, this asteroid's orbit lies well outside the asteroid belt. It spends most of its orbit between the orbits of Saturn and Uranus. Its average orbital parameters are a = 13.5,  = 0.378, T = 50.7, and i = 6.9^o.

Encke's Comet and the asteroids Ceres, Vesta and Icarus can be modeled using either the Inner Planets display or the Outer Planets display. (Some asteroids and comets will go "off stage" but will return.)  All other comets and asteroids are best modeled with the Outer Planets display.

For additional information, consult the web sites:

http://ssd.jpl.nasa.gov/sbdb.cgi?sstr=2P;orb=1;view=Far

You may also view the Lingo Scripting that generates these ellipses.

This site was produced by Kevin Barry, Office of Information Technologies, University of Notre Dame.