Computer simulation of planetary orbits

Hahn calculus book coverThis simulation ran on Flash Player which is no longer available as of January 1, 2021.

There are several other similar simulations available:

 Information on the Original Simulation

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

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

This simulation relied on a modern version of the mathematics of Kepler as developed and illustrated in Section 10.4 of the text.

Here is a section of code that shows some of the math that was used in the original simulation.  

The sizes of the orbits and the orbital speeds are all to scale, but the sizes of the Sun and planets are not. As we know, Jupiter and Saturn dwarf all the other planets and the Sun is huge compared to everything else. To provide a sense of scale for the outer planets, the orbiting Earth appears in the center of the display.

The simulations correctly capture the fact that all the planets orbit the Sun in the same direction. But they model all the orbits on the orbital plane of the Earth and ignore the fact that the planes of the orbits differ from the plane of the Earth from about 1 to 3.5 degrees (except for Mercury’s orbit which deviates by 7 degrees). These differences are much greater for some comets and asteroids (some orbital planes are essentially perpendicular to the plane of the Earth).

Information for some of the more important comets and asteroids follows next. As before, a, epsilon, and T represent the semimajor axis in AUs, the astronomical eccentricity, and the period in Earth years, respectively. Kepler's third law tells us that a3/T2 is the same constant for any body in orbit around our Sun. Since a = 1 and T = 1 for the Earth, it follows that T = a3/2 for all planets, comets and asteroids. In some cases, the gravitational perturbations caused by Saturn and Jupiter result in irregularities, so that only averages or ranges for the orbital parameters can be given. The parameter i represents the angle between the plane of the orbit of the comet or asteroid and the plane of the Earth's orbit. As an illustration, consider Halley’s comet. The angle between the plane of Halley's orbit and that of the Earth is 17.8o with Halley moving in the direction opposite to that of the Earth. Since 180 - 17.8 = 162.2, this information is captured by i = 162.2o. In particular, i = 180o would mean that the orbital plane is the same as that of the Earth, but that the motion around the Sun is in the opposite direction.

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.

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. The fact that this proved to be correct was a glorious triumph for Newtonian mechanics. Today's information is more precise. Every orbit is slightly different and there is a variation of 17.62< a < 18.26. This corresponds to 74 < T < 78. The average values are a = 17.94 AUs and T = 76 years. The astronomical eccentricity is epsilon = 0.97. As already noted, the angle of Halley's orbital plane is i = 162.2o.  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 of its parameters are a = 2.22, epsilon = 0.85, T = 3.31, and i = 12o. 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 either in the range 3.5 < a < 4.5 AU (just inside Jupiter’s orbit) or in the range 6.0 < a < 8.0 AU (just outside Jupiter’s orbit). The eccentricity was in the range 0.05 < epsilon < 0.3 and the angle of inclination was in the range 0o < i < 6o. 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 = 187.48, epsilon = 0.99, T = 2567, and i = 89.4o. The most recent closest approach to Earth was on March 22, 1997. It gained extraordinary notoriety at that time in connection with the mass suicide of the 39 members of a religious sect.

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 = 25.74, epsilon = 0.99, T = 130.6, and i = 124.9o. This comet strains the simulation. Why do you think this is?

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.

Ceres: On January 1, 1801, the Italian astronomer Giuseppe Piazzi (1746-1826) pointed his telescope into the night in Palermo, Sicily, and noticed a very faint moving object. At that time, the Sun, the planets (with the exception of Neptune and Pluto), and some comets were the only known travelers in our solar system, so this was something new. Over a period of 41 days Piazzi observed the object move through an arc of 3 degrees across the sky. (This corresponds to about six Moon diameters.) But then it disappeared in the Sun’s glare. Was it possible to predict where it would reappear from the meager information that existed? To the 24 year old Carl Friedrich Gauss this question presented an exciting challenge, and he focused his extraordinary mathematical powers on it. Assuming that Piazzi’s object circumnavigated the Sun on an elliptical course, and using only three observed positions, Gauss estimated its orbit. The observations resumed, and on December 7, 1801, Piazzi’s object was relocated only a short distance away from where Gauss had predicted. This computational triumph brought Gauss immediate recognition as Europe's top mathematician and he became an instant celebrity. For more of the details, see

D. Teets and K. Whitehead, The discovery of Ceres: How Gauss became famous.
Mathematics Magazine, 1999, 72 (April), 83,
This first asteroid to be observed now bears the name Ceres. Its average orbital parameters are a = 2.77, epsilon = 0.08, T = 4.60, and i = 10.6o.

Eros: A small asteroid in orbit near the Earth. Studied by the space probe NEAR. ). Its average parameters are a = 1.46, epsilon = 0.22, T = 1.76, and i = 10.8o.

Icarus: This asteroid has a highly eccentric orbit and 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 = 1.08, epsilon = 0.83, T= 1.12, and i = 22.9o.

Chiron: This asteroid/comet’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 = 13.63, epsilon = 0.38, T = 50.7, and i = 6.9o.

Encke's Comet and the asteroids Ceres, Eros, and Icarus can be modeled using either the Inner Planets display or the Outer Planets display. The other comets and Chiron are best modeled with the Outer Planets display. Hale-Bopp will quickly go "off stage" in either display.

 

For additional information, see Orbital Mechanics and other resources at the National Aeronautics and Space Administration (NASA) and the Jet Propulsion Laboratory (JPL).


This simulation was created at the University of Notre Dame. Kevin Barry produced the original in 1997 and Michael Federico (ND '04) made the current update in 2003.