Kepler's three laws are empirical, meaning that he determined all of them from an examination of the patterns in experimental data. Kepler was a contemporary of Galileo, both of whose work predated that of Newton. Thus, for example, Kepler didn't have at his disposal Newton's three laws, the law of gravitation, and the relationship for the acceleration of an object in circular motion. These laws and relationships can be used to theoretically derive Keplers empirical laws but, Kepler was unaware of that.

Figure 1	Figure 2
Figure 3	Figure 4
Figure 5	Figure 6

Kepler's 1st Law

Ellipses as mathetical objects

Before getting into the orbits of planets, we provide some information about the properties of ellipses as mathematical entities. Refer to Figure 1. We define these terms:

• C is the center of the ellipse and bisects the major axis, AE, and the minor axis, BD.

• F₁ and F₂ are the focal points of the ellipse and are equidistant from point C. The symbol c represents the distance from a focal point to the center of the ellipse.

• AC is termed the semi-major axis and is represented with the symbol, a.

• BC is termed the semi-minor axis and is represented with the symbol, b.

• The eccentricity of an ellipse is given by ε = c/a. The eccentricity is the property of an ellipse that describes how nearly circular it is. Note that if c = 0, the two focal points converge to the center, and the ellipse becomes a circle. Thus, ε = 0 is the eccentricity of a circle. The other limiting value is obtained for c = a, in which case ε = 1 and the figure is no longer an ellipse but rather a parabola.

• The area of an ellipse is equal to πab.

Operational definition of an ellipse: See Figure 2. For any point P on an ellipse with focal points, F₁ and F₂, the sum, F₁P + F₂P, is a constant. Thus, you can draw an ellipse by tacking down the ends of a string to the focal points, stretching the string in the middle with a pencil, and tracing the ellipse with the tip of the pencil while keeping the string taut. This method is shown in Figure 12-7 of the text.

Algebraic definition of an ellipse: See Figure 3. The ellipse is situated in an x-y plane with the center of the ellipse at the origin and the major axis coincident with the x-axis. For any point (x,y) on the ellipse, this algebraic relationship holds between the coordinates and the values of the semi-major and semi-minor axes: x²/a² + y²/b² = 1.

For the special case of a circle, a = b = r and x²+ y² = r², the equation of a circle of radius, r. This is shown in Figure 4. Note that x²+ y² = r² is simply an expression of the Pythagorean theorem here.

Physical application: All of the above has to do with the mathematical properties of ellipses. We make application to the physical world in Kepler's 1st Law, which states: The orbits of the planets are ellipses with the Sun at one focus. We now know that this law applies to other planetary systems, but Kepler only had data for the planets of our solar system that were visible to the naked eye.

Refer to to Figure 5 for a diagram of a planet orbiting the Sun in our solar system. Note that the eccentricity of the ellipse is exaggerated as most of the planets in our solar system have more nearly circular orbits. Note also that the Sun is not truly fixed but has a small orbit of its own. In fact, the Sun and the planet both orbit the center of mass of the system. Due to the Sun's much larger mass, the CM is actually inside the Sun, and the Sun executes a very small orbit. The situation is even more complicated than this, as all of the planets have an influence on the Sun's motion and vice versa. For the sake of simplicity, we will take the Sun to be fixed at a focal point in what follows.

Referring again to Figure 5, the acceleration of the planet is due to gravitational force only, and this is always directed toward the Sun. The velocity of the planet is always tangent to the path; hence, the velocity is in general not perpendicular to the acceleration as would be the case for circular motion. Morevoer, the magnitudes of the velocity and of the acceleration are not constant. The acceleration changes magnitude, because the distance between the planet and the Sun changes. This must result in a changing velocity as well.

We define two special points of the orbit. The perihelion is the point of closest approach of the planet to the Sun, and the aphelion is the greatest distance between the planet in the Sun. These are the endpoints of the major axis. The velocities at these points are labeled v_p and v_a. There's an inverse relationship between the perihelion and aphelion distances and the corresponding velocities. We'll take this up in the discussion of Kepler's 2nd Law.

We finish this section with a note about the data that Kepler used in determining his first law. At the time of his work, people believed that the orbits of the planets were circles; thus, it took courage on Kepler's part to posit that the orbits were elllipses instead. He had confidence in his results, because the data he used was obtained with the best instruments of the day in the observatory of Tycho Brahe. His sighting instruments weren't telescopic, but they had precision sighting mechanisms that allowed Kepler to say that, within the uncertainty of the measurements, the data supported the orbits being ellipses rather than circles. Fortuitiously, the orbit of Mars is the most eccentric of any of the planets for which Kepler had good data.

Kepler, and any astronomer for that matter, had to deal with another complication in interpreting data. All of this data was, of course, taken from the point of view of the Earth, which was also in revolution around the Sun. From this point of view, the orbits of the planets actually appear to have loops against the background of fixed stars, doubling back on themselves at some point before returning to their original direction of motion. This motion is termed retrograde and is illustrated in this animation. We mention this only to make clear that the task of determining the shapes of orbits from data taken on the Earth was a difficult task for Kepler. He spent many years in determining his three laws.

Kepler's 2nd Law

Kepler's 1st Law deals only with the shape of the planetary orbits. His second law deals with the dynamics. Here's what Kepler's 2nd Law says: A line from the Sun to the planet sweeps out equal areas in equal times.

See Figure 6 for an illustration. The filled-in yellow regions represent areas swept out by the position vector from the Sun to the planet in equal time intervals, Δt. The area swept out near the aphelion is (r_av_aΔt)/2 while that swept out near the perihelion is (r_pv_pΔt)/2. These expressions come from treating the yellow regions as being approximately triangular and calculating the areas of the triangles. The altitude of each triangle is r, and the base is (vΔt), the distance traveled by the planet in time Δt. The error in treating what is actually a sector of an ellipse as a triangle decreases as the time interval is made shorter.

Applying Kepler's 2nd law, we can say that (r_av_aΔt)/2 = (r_pv_pΔt)/2. This simplifies to r_av_a = r_pv_p, since the time intervals are equal. This can be rearranged as v_a/v_p= r_p/r_a. Note that this is the same statement that we made previously that the aphelion and perihelion distances and velocities have an inverse relationship to each other.

We demonstrated the use of Kepler's 2nd Law for the special case of the extremes of the planet's motion. However, we can generalize to any point on the orbit. Consider the sector swept out by position vector, r, in Figure 6. If arc length Δs swept out by r is small enough, we can say that the velocity v is approximately constant over this arc length. In that case, Δs = vΔt. Treating the sector as a triangle, its area is ΔA = rΔs/2 = rvΔt/2. Taking the Δt to the other side of the equation, we have: ΔA/Δt = rv/2. The left-hand side of the equation is the rate at which the area is swept out by the position vector. Thus, by Kepler's 2nd Law, we see that the product rv must be a constant. This is, in fact, what Kepler discovered for the orbit of Mars and what led him to his second law.

The relationship ΔA/Δt = rv/2 becomes exact in the limit as Δt approaches 0. Let's look at the result using physics from the last chapter. Taking the planet as a particle, its angular momentum about the Sun is L_p = mrv, where m represents the mass of the planet. Since m is a constant and rv is constant by Kepler's 2nd Law, we see that the second law is simply an application of the law of conservation of angular momentum to the system. Note that we've taken the system to be the planet only, assuming the Sun to be fixed. For a complete treatment, we would consider the motion of the Sun as well in its small orbit around the CM of the system. For work in this class, we will take the Sun to be fixed as mentioned before.

Kepler's 3rd Law

Keper's 3rd Law states that the squares of the periods of the planets in their revolution about the Sun are directly proportional to the cubes of their average distances to the Sun. For an elliptical orbit, the average distance is the simple average of the aphelion and perihelion distances. Kepler didn't have the theoretical tools that we now have in order to show that the constant of proportionality is 4π²/(GM_s), where M_s is the mass of the Sun.

More generally, we can say the 3rd law applies to other planetary systems orbiting a central star and to moons orbiting planets. In that case, we replace M_s with M to represent the mass of the central object. For the moons of Jupiter, for example, M would be the mass of Jupiter.

The textbook gives a thorough discussion of the 3rd law, so we won't go into greater detail here.

Generalization

The previous discussions have used a planet-Sun system as an example. Kepler's Laws are more general than that, though. They apply to any orbital system governed by a central force law. The latter is one for which the force always points to an orbital center.