Planetary Motion

Kepler's greatest contribution to astronomy is summarized in the three laws of planetary motion that bear his name. In this section we will explore each of these laws.

The First Law - Planets Move in Elliptical Orbits

Figure 4.14 provides a movie clip of the motion of the inner or Terrestrial planets. All three of Kepler's Laws are illustrated here. You can easily see that Mercury's orbit is a bit "off". Not quite centred and not quite round. Mercury has the most elliptical of any of the 8 planets.
	Figure 4.14 Motion of the Terrestrial planets
Kepler's first law is the statement that planetary orbits are elliptical. With the advantage of a "God's-eye-view" this may seem like a rather obvious statement. However, Kepler had no such privileged view and it took he mathematical genius and the detailed observations of Brahe to unlock this secret. The applet shown in figure 4.15 will be used to help investigate elliptical orbits as well as the other laws.

Facts about Ellipses

An easy way to generate an ellipse is shown in the video clip in Figure 4.15. The string moves around two fixed points inside the ellipse. These are the focal points (or focii) of the ellipse. This is the shape that Kepler needed in order to explain planetary motion. In Kepler's first law the sun occupies one of the focal points. There is nothing at the other focal point. Eccentricity is the measure of how much the shape of an orbit departs from a circle. To get some feeling for this number, use the applet in Figure 4.16 and generate ellipses with the eccentricities given in Table 4.2

Figure 4.15 Clip showing how to create an ellipse

Eccentricity is given by the formula:

where r_a is the aphelion distance and r_p is the perihelion distance.

	Object	Orbital Eccentricity
	Mercury	0.206
	Venus	0.007
	Earth	0.017
	Mars	0.093
	Jupiter	0.048
	Saturn	0.056
	Halley's Comet	0.967
	Table 4.2 Orbital eccentricities for selected solar system objects
	Orbital eccentricity is denoted by the small letter "e" and for closed orbits is a number somewhere between 0 and 1. An eccentricity greater than 1 means that the orbit is not a closed curve but is either a hyperbola or a parabola. Another important feature on an ellipse is the semi-major axis, denoted "a". This is one-half the length of the long axis of the ellipse.
Figure 4.16 Applet showing how eccentricity affects shape of a planetary orbit

Example 4.3 What is the "radius" of an orbit for which e = 0 and semi-major axis a = 1.0 AU. How would your answer change if e = 0.65 (or any other non-zero number)?

Solution If e = 0 then the orbit is circular. In that case the radius is the same as the semi-major axis. When e is not zero then the term "radius" becomes ambiguous because it now represents a continuously changing value. In general an ellipse does not have a unique "radius".

In example 4.3 several new ideas are introduced. Rather than speak about the radius of an elliptical orbit, it is more common, when orbitting the Sun, to identify: aphelion: farthest point from theSun perihelion: nearest point to the Sun average radius = length of semi-major axis or average of aphelion and perihelion.
	Figure 4.17 Apogee and perihelion for an orbit e = 0.65

Example 4.4 Halley's comet has a closest approach to the Sun (perihelion) of about 0.6 AU. Use the applet in 4.16 to estimate the farthest Halley's comet travels from the sun.

Solution: You can scale the values for perihelion and aphelion that are given on the applet in figure 4.16. The applet indicates that if the semi-major axis is 1 AU then with e = 0.967 the perihelion is 0.03 AU. The scale factor is found by a simple calculation: 0.6 AU = (scale) X 0.03 or the scale = 0.6/0.03 = 20. So, the aphelion of the orbit should be the aphelion distance times this scale factor. Using the applet you find aphelion distance = 20 X 1.97 AU = 39.4 AU. If you would like a bit more precise a calculation follow the "take a closer look" link given above.

The Second Law - "Equal Areas in Equal Times"

If you inspect the video clip in Figure 4.15 you will note that, especially with Mercury, the speed of the planets is not constant. Mercury speeds up close to the sun and slows down when farther from the sun. Kepler's Second Law of planetary motion provides a way to quantify this behaviour. To better visualize the second law use the applet shown below in figure 4.18. Be sure to select the "Kepler's 2nd Law" tab. The law is called the "Equal Areas" law because it states that if you draw a line from the sun to the planet and then let the line "sweep out" an area in a given period of time, that area will be the same now matter where you create it along the orbit. If you are close to the sun then the planet must move a greater distance along the orbit to still sweep out the same area. But - since it had to complete this in the same time period it must move faster to accomplish this. Figure 4.19 illustrates this for a highly elliptical orbit.

Figure 4.18 Kepler's Laws applet (applet produced by and used with permission Nebraska Astronomy Applet Project)

Kepler's 2nd Law solved a vexing puzzle in astronomy. The Ptolemaic theory had a very difficult time explaining the non-constant motion of the planets and could only do this (approximately) by introducing a complicated set of circles moving within circles.

Figure 4.19 Equal areas swept out in equal time intervals.

The elliptical nature of orbits, and as we will see in the next section the varying force of gravitational attraction, makes the speeding up and slowing down of planets a very natural effect.

Example 4.5 Why is the analema shown in Chapter 3.1, figure 3.5 "asymmetric"? In particular - why is it fatter in the winter months than in the summer months?

Solution: Recall that one of the contributing factors to the analema is the Earth's motion around the sun. In the winter months we are closer to the Sun by a few percent. According to Kepler's 2nd Law, the Earth must be moving faster at this time and this accounts for the asymmetry in the shape of the analema.

The Third Law - "Period Squared Equals Radius Cubed"

If you return to the video clip in Figure 4.14 it is easy to see that the planets closer to the Sun travel faster than ones farther from the sun. Since their orbits are also smaller the amount of time required to make one orbit around the Sun is correspondingly smaller. The time required to make one complete orbit as called the Period of the orbit. Kepler's 3rd Law is a precise mathematical statement that relates the average orbital radius to the period. If you consistently measure the period in years and the average orbital radius in AU (astronomical units) then Kepler's 3rd Law can be expressed in a very compact mathematical formula:

Example 4.6 Use Kepler's 3rd Law to predict the period of a comet with an average orbital radius of 10 AU and an eccentricity of e = 0.98.

Solution: Using Kepler's 3rd Law write:

The comet has a period of 31.6 years. (Note that the period does not depend on the eccentricity - only on the average radius).

Figure 4.20 Kepler's 3rd Law graphing calculator

Figure 4.20 provides a Kepler's 3rd Law graphing calculator that you can use to simplify calculations.

Example 4.7 A small comet is observed to return every 3.6 years. What is the average radius of the comet's orbit?

Solution: Use Kepler's 3rd Law to wrtite:

You can confirm this with the Kepler's 3rd Law applet. The comet has an average orbital radius of 2.3 AU

To understand Kepler's Three Laws of Planetary Motion

Chp 3.1