Section 7–2 Kepler's laws

(Kepler’s First law / Kepler’s Second law / Kepler’s Third law)

In this section, the three physical laws discussed are Kepler’s First law, Kepler’s Second law, Kepler’s Third law.

1. Kepler’s First Law (Law of orbits, 1609):

“Each planet moves around the sun in an ellipse, with the sun at one focus (Feynman et al., 1963, section 7.2 Kepler’s laws).”

Kepler found that each planet is orbiting around the Sun in an elliptical path with the sun at a focus of the ellipse. Feynman explains that an ellipse is not just an oval, but it is a curve that can be obtained by using a string and pencil method in which the sum of whose distances from two fixed points is a constant. In other words, Feynman defines an ellipse as a collection of points in a curve in which the sum of whose distances from the two foci is constant. In a “lost lecture” of Feynman on planetary motions, he provides another definition of ellipse: “if you wish, these two points are called the foci, and the focus means that light emitted from F will bounce to F’ from any point on the ellipse (Goodstein & Goodstein, 1996, p.150).”

In general, an ellipse can be mathematically represented by a polar coordinate equation: r(θ) = r₀/(1 + e cos θ) where r is the distance from a planet to the Sun, r₀ is the semi-latus rectum of the ellipse, e is the eccentricity of the ellipse, and θ is the angle with respect to the planet’s original position. (The term semi-latus rectum is a compound of the Latin: semi means half, latus means side, and rectum means straight.) Strictly speaking, the path of a planet is approximately elliptical because there are gravitational influences from other planets and celestial bodies such as comets and moons. Importantly, the first law provides a revolutionary idea based on astronomical observations that planetary orbits are elliptical instead of circular.

2. Kepler’s Second Law (Law of equal areas, 1609):

“The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time (Feynman et al., 1963, section 7.2 Kepler’s laws).”

Kepler’s second law refers to his observations that the Sun-planet line (an imaginary line connecting the Sun and a planet) is sweeping area at a constant rate; it can be mathematically represented as dA/dt = constant. This is based on Tycho’s data on the motion of Mars in which it moves faster when it is nearer to the Sun and slower when it is farther away from the Sun. This law provides another revolutionary idea: planets move around the sun at varying speeds. In essence, the orbital speed of a planet is related to how the Sun-planet line “sweeps out” equal areas in equal times.

Kepler initially ponders on whether a decrease in velocity of a planet is a cause of an increase in distance from the Sun, or vice versa. In his book titled The Cosmographic Mystery, Kepler (1596) conceived of a planet as being moved by a “motive soul.” Based on the astronomical data inherited, Kepler proposes that the planets are moving in their orbits due to a central force from the Sun. In addition, Kepler guesses that the central force is magnetic in nature because it was known that the Earth is a magnetic body. Note that the central force from the Sun does not provide a torque, and thus, the angular momentums of the planets remain constant.

Note: In Kepler’s laws, the Sun is assumed to be stationary. To be more accurate, the Sun and planets rotate about the center of mass of the solar system.

3. Kepler’s Third law (Law of periods, 1619):

“The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits (Feynman et al., 1963, section 7.2 Kepler’s laws).”

Feynman mentions that Kepler’s third law was discovered much later and this law is of a different category from the other two laws, because it does not focus on a single planet, but relates one planet to another. In essence, this law states that the square of the orbital period of a planet is proportional to the cube of the orbital size. Mathematically, it can be represented as T² µ d³ in which T is the time interval it takes a planet to move one round about its orbit and d is the major axis that can be expressed as the length of the greatest diameter of the elliptical orbit. Simply put, if the planets move in circles, as they nearly do, the time required to go around the circle is proportional to the 3/2 power of the orbital diameter (or radius).

Some physicists prefer to state the Kepler’s third law as a relationship of the semi-major axis of the orbit to its sidereal period. Interestingly, Kepler (1619) writes that “it is absolutely certain and exact that the proportion between the periodic times of any two planets is precisely the sesquialterate proportion of their mean distances, that is, of the actual spheres (p. 411).” Thus, one may write “mean distances” to the Sun instead of “semi-major axis.” More important, the orbital period of a planet is also dependent on its semi-minor axis. Furthermore, physicists should realize that this third law is based on Kepler’s calculation (or manipulation) of Tycho’s data (Donahue, 1988). However, Tycho’s data are not as superb as some scholars claimed because they were observed by naked eyes without the use of a telescope.

Questions for discussion:

1. What is a good definition of an ellipse for Kepler’s first law?

2. Could you explain how the durations of summer and winter are slightly different by using Kepler’s second law?

3. Should Kepler’s third law be stated in terms of mean distances or semi-major axis?

The moral of the lesson: the motions of planets are in elliptical orbits and do not have constant speeds; their orbital periods are dependent on the size of orbits.

References:

1. Donahue, W. H. (1988). Kepler’s fabricated figures: Covering up the mess in the New Astronomy. Journal for the History of Astronomy, 19(4), 217-237.

2. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

3. Goodstein, D. L., & Goodstein, J. R. (1996). Feynman’s Lost Lecture: the motion of planets around the sun. New York: W. W. Norton & Company.

4. Kepler, J. (1596). Mysterium cosmographicum (The Secret of the Universe). Tübingen: Georgius Gruppenbachius.

5. Kepler, J. (1619). Harmonice Mundi (The Harmony of the World). translated by E. Aiton, A. Duncan & J. Field. Philadelphia: American Philosophical Society.