(the astronomical eccentricity), and T (the period
of the orbit in Earth years) for the last planet clicked will be displayed at
the top (in red). Click Trails and the elliptical orbit(s)
will be traced out. This site provides computer simulations of the orbits of the
Inner Planets: Mercury, Venus, Earth, and Mars,
as well as the
Outer Planets: Jupiter, Saturn, Neptune, Uranus, and Pluto.
The simulations rely on the mathematics of Kepler as explained and illustrated in Sections 4.7 to 4.9 of Chapter 4. Only the data displayed in the Plot Times/Speeds box requires considerations that go beyond that (a velocity formula that is developed in the Additional Exercises Section for Chapter 8).
In the Inner Planets display below, the Sun is depicted in
the center. Simply click on one or more of the images of the four inner planets
and the planet(s) will appear in the system. Click on Start/Pause
and the planet(s) that you have selected will begin to orbit the Sun. The orbital
parameters a (the semimajor axis in AUs), (the astronomical eccentricity), and T (the period
of the orbit in Earth years) for the last planet clicked will be displayed at
the top (in red). Click Trails and the elliptical orbit(s)
will be traced out.
One additional orbiting body, for example a comet or asteroid, can be launched.
Enter its parameters a and with 0 < < 1 and a into the boxes in the upper right corner,
click the yellow comet/asteroid button and it will join the inner planets
already selected. Information about some of the more important comets and
asteroids, including their orbital parameters, is provided below.
To see how fast the planets (or the comet or asteroid) are moving, click the Plot Times/Speed button and the elapsed time in Earth days and all the speeds (in miles/second) at that instant will be displayed. Click on Clear Times/Speed and this information will disappear.
Click on Clear All and the planet(s) and additional orbiting body disappear.
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, the Sun is huge compared to everything else.
Click on Outer Planets in the display for the simulation of the orbits of Jupiter, Saturn, Neptune, Uranus, and Pluto. To provide a sense of scale, the orbiting Earth appears in the center of the display. To launch the planets and/or an additional orbiting body proceed as before. Again, the sizes of the orbits and the orbital speeds are to scale, but the sizes of the planets are not. We know that Jupiter and Saturn dwarf all the other planets.
The simulations of the inner and outer planets both 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, , 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 = 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, = 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 < < 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, = 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, = 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 (we saw him in action in Section 4.7) 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. See
D. Teets and K. Whitehead, The discovery of Ceres: How Gauss became famous. Mathematics Magazine, 1999, 72 (April), 83,for more of the details. This first asteroid to be observed now bears the name Ceres. Its average orbital parameters are a = 2.77,
= 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, = 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, = 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, = 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.
View the Actionscript that was used to generate the orbits.
For additional information, consult the web sites:
http://nssdc.gsfc.nasa.gov/planetary/factsheet/cometfact.html
http://nssdc.gsfc.nasa.gov/planetary/factsheet/asteroidfact.html
This site was produced by Kevin Barry (original
version) and Michael Federico
(updated version, 2003) of The Kaneb Center for Teaching and Learning at the
University of Notre Dame.