Section 7–1 Planetary motions

(Law of gravitation / Nicolaus Copernicus / Tycho Brahe)

It may seem strange that the lecture on Newton’s law of universal gravitation is delivered before Newton’s three laws of motion. However, this chapter on the law of gravitation is further improved in the first Feynman’s Messenger lecture delivered at Cornell University. In this section, the three interesting points discussed are the law of gravitation as well as the contributions of Nicolaus Copernicus and Tycho Brahe.

1. Law of gravitation:

“…every object in the universe attracts every other object with a force which for any two bodies is proportional to the mass of each and varies inversely as the square of the distance between them (Feynman et al., 1963, section 7.1 Planetary motions).”

Feynman states the law of gravitation as every object in the universe attracts every other object with a force which for any two bodies is proportional to the mass of each and varies inversely as the square of the distance between them. Mathematically, the law of gravitation can be expressed by the equation, F = Gmm′/r². Importantly, Newton (1687) writes that the force “will be reciprocally proportional to the square of the distance of the centers (p. 158).” Thus, some physicists prefer the term distance to be more precisely stated as the “distance between centers of mass of the two objects.” Alternatively, Newton’s law of universal gravitation can be stated as “[e]very particle attracts any other particle with a gravitational force of magnitude F = GMm/r² (Halliday, 2005, p. 331).” Essentially, this law is applicable to particles.

Historically, Ismaël Bullialdus (1605-1694) published a book titled Astronomia Philolaica in 1645 and suggested an inverse square force law before Hooke and Newton. Bullialdus argued that Kepler’s planetary force if it existed, would diminish according to the inverse square of the distance just like Kepler’s law of light propagation. That is, Bullialdus did not agree with the existence of this force because of his metaphysical tenet that a moving object must contain in itself a principle of its motion (Jammer 1999). Importantly, Newton (1687) gave a mathematical justification for applying the inverse square law to large spherical objects as if they are particles.

2. Nicolaus Copernicus:

“…The story begins with the ancients observing the motions of planets among the stars, and finally deducing that they went around the sun, a fact that was rediscovered later by Copernicus (Feynman et al., 1963, section 7.1 Planetary motions).”

Feynman explains that the ancients observing the planetary motions among the stars, and finally deducing that they orbited around the sun. In fact, Nicolaus Copernicus was aware that Aristarchus of Samos (c. 310 – c. 230 BC) had already proposed the heliocentric theory (Sun-centered universe) earlier. Although Copernicus rediscovered this theory later, there were still debates as to whether the planets really orbited around the sun in the fifteenth century. More important, the “exact” planetary motions around the sun were deduced by having more astronomical observations. They are elaborated in the next section as Kepler’s three laws of planetary motions.

Copernicus (1473 – 1543) proposed a heliocentric theory that positioned the Sun at the center of the universe in which the Earth and the other planets rotating around it in circular paths at uniform speeds. In essence, Copernican theory is different from Ptolemy’s geocentric theory (Earth-centered universe) in placing the Earth at the center of the universe. A weakness of geocentric theory is the need of using epicycles (or cycles within cycles) to explain the “retrograde motions” of planets. By using astronomical instruments such as a triquetrum, Copernicus achieved more accurate observations and concluded there are “epicycles on epicycles” in the geocentric theory. However, the heliocentric theory is not quite correct because the Sun is also not stationary.

3. Tycho Brahe:

“Tycho Brahe had an idea … that these debates about the nature of the motions of the planets would best be resolved if the actual positions of the planets in the sky were measured sufficiently accurately (Feynman et al., 1963, section 7.1 Planetary motions).”

Feynman elaborates that Tycho Brahe (1546 – 1601) had an idea that was different from anything proposed by the ancient philosophers. Brahe’s main idea was that the debates about the nature of the orbital motions of the planets would be resolved if the actual positions of the planets in the sky were measured more accurately. In other words, Brahe proposed an empirical viewpoint in which the planetary motions should be determined by what we measure instead of what we think. This was a revolutionary idea because ancient philosophers prefer using deep philosophical arguments instead of performing careful experiments.

Tycho Brahe built large astronomical instruments, such as a triangular sextant and a revolving steel quadrant, that help to achieve accurate planetary observations. Tycho’s naked eye measurements of planetary parallax were accurate to the arcminute, or about 1/30 width of the full moon. However, he disagreed with Copernicus and proposed a “geo-heliocentric” model (or Tychonic system) in which the Sun orbiting the Earth, while the other planets orbiting the Sun. Curiously, Tycho (as an empiricist) maintained his belief of perfectly circular orbits for all celestial bodies despite his advocate of accurate astronomical observations.

Although Tycho opined that the planetary motion should be based on accurate astronomical observations, he believed that the Earth was just too heavy to be continuously in motion. In Tycho’s words, “This innovation expertly and completely circumvents all that is superfluous or discordant in the system of Ptolemy. On no point does it offend the principle of mathematics. Yet it ascribes to the Earth, that hulking, lazy body, unfit for motion, a motion as quick as that of the aethereal torches, and a triple motion at that (Gingerich, 1993, p. 181).”

Questions for discussion:

1. How would you state Newton’s law of universal gravitation?

2. How is the Copernican model different from Ptolemy’s geocentric model?

3. Was Tycho’s belief of circular orbits based on accurate astronomical observations?

The moral of the lesson: every object attracts every other object with a gravitational force that is proportional to the mass of each and varies inversely as the square of the distance between centers of mass of the two objects.

References:

1. 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.

2. Gingerich, O. (1993). The eye of heaven: Ptolemy, Copernicus, Kepler. New York: American Institute of Physics.

3. Halliday, D., Resnick, R., & Walker, J. (2005). Fundamentals of Physics (7th ed.). New York: Wiley.

4. Jammer, M. (1999). Concepts of Force. New York: Dover Publications.

5. Newton, I. (1687/1995). The Principia. Translated by Andrew Motte. New York: Prometheus.