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Newtonian Gravity

The Keplerian Orbit Simulator

The basic behavior of the planets orbiting the Sun is displayed by the simulator below. This simulator is based on the Newtonian equations of motion under the assumption that the mass of a planet is negligible compared to the mass of the Sun. Under this assumption, the orbits are closed ellipses, with the square of the period of a planet proportional to the cube of the semimajor axis.

The simulator has three basic starting states that are selected with the radio buttons on the lower left corner of the simulator. The first state, which is the state when this page is loaded and is designated by the radio button marked Planets, simulates the orbits of Mercury, Venus, Earth, Mars, and Jupiter. The second starting state, designated by the radio button marked a Fixed, gives the planets the same semimajor axis but different eccentricities; this is the most interesting of the three basic states to watch. The third starting state, designated by the radio button marked e Fixed, gives the same eccentricity to all planets, but sets the semimajor axes so that the orbital periods fall in the ratios of 2:1, 3:2, 4:3, and 5:3.

The orbits are described by three variables: the semimajor axis a, the eccentricity e, and the perihelion angle . The perihelion angle is the angle between a planet's perihelion and the first Point of Aries. The semimajor axis is limited to values between 0.1AU and 6AU, the eccentricity is limited to values between 0 and 0.95, and the perihelion angle is limited to values between 0 and 360. You can try other values, but the simulator will complain and prevent the values from being entered.

Usage of the Simulator

The simulator is a Java applet, so it only runs in browsers that are Java enabled. General navigation and control of the applets on this web site are described on the Applet Control Guide page. The buttons can be controlled with either the mouse or the keyboard. The buttons labeled Start and Stop do what one expects. Only one of these two buttons is active, and the active button is the applet's default button: it can be selected with the Return or Enter key when the applet has the keyboard focus. With the mouse, click anywhere on the applet to give it the keyboard focus. When the simulator is stopped, the variables for the planets, which are listed in the table at the bottom of the simulator, can be modified by double-clicking on a table cell. The button marked Reset places the planets in their initial positions for the current values in the parameter table. To reset the parameter values to the model values, select the model radio button.

I would appreciate hearing from you if you encounter an error while running the simulator or if you have suggestions for improvement. Send your e-mail to the editor of the web site.

The Planets Simulation

The planet simulation shows the relationship between semimajor axis and period for the five innermost planets. For instance, Jupiter (planet 5) makes one orbit in the time Earth (planet 3) makes 11.89 orbits. Beyond this relationship, this simulation shows how slight are the eccentricities of the planets. Only Mercury has a pronounced eccentricity, although the eccentricity of Mars is noticeable.

The a Fixed Simulation

The a Fixed simulation shows that the eccentricity plays no role in the period. It also shows the consequence of the Keplerian law of equal areas being swept-out in equal times. This law means that the angular velocity of a planet relative to the Sun is inversely-proportional to the distance from the Sun. The high-eccentricity planets therefore travel with a high velocity at perihelion, the closest approach to the Sun, and they travel with a low velocity at aphelion, the farthest point from the Sun. These episodes of high and low velocity balance each other, so that the orbital period of a high-eccentricity planet is equal to the period of a low-eccentricity planet with the same semimajor axis.

The e Fixed Simulation

In the e Fixed simulation, the values of a are set to give resonant periods. The resonances are relative to planet 1. The orbit of planet 1 to planet 5 are in 2:1 resonance, so that planet 1 orbits the Sun twice for every orbit of planet 5. Planet 1 is in 3:2 resonance with planet 3, it is in 4:3 resonance with planet 2, and it is in 5:3 resonance with planet 4. Resonances are important because the gravitational attraction between the two planets at resonance can build over many orbits, so that they influence one another's orbits despite their small size compared to the Sun. These resonances appear in the orbits of the Kuiper objects, which are tiny planetoids outside of Neptune's orbit, many of which are in orbital resonance with Neptune, and in the gaps in the rings of Saturn, which fall at radii that are in resonance with one or another of Saturn's moons.

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