Gravity and the Orbits of the Planets I

The Development of the Heliocentric Theory

• Introduction
• The History of Astronomy
• Practice with Concepts
• Discussion Questions
• Optional Website Reading

Introduction

Today, most astronomers would find Osiander's argument offensive. Scientists are supposed to seek the truth and give us an explanation of what is actually happening in the physical universe. But in 1543, at the time that Osiander adds this preface to Copernicus' great work (probably without Copernicus' knowledge), the debate was raging over whether astronomy described real physical objects and part of natural philosophy or whether it described hypothetical constructs, and was thus merely mathematical in nature. We are not sure why Osiander felt the Copernican thesis needed to be presented as a mathematical model, when some of the phrases Copernicus uses to describe his circles and epicycles suggest that he thought of them as physical objects, but within a century of its publication, several scientists and philosophers run into opposition to their insistance that the "Copernican model" accurately describes the physical movements of real planets.

It is worth keeping in mind that there was no direct observational evidence for either the rotation of the Earth on its axis, or its revolution around the sun, until the nineteenth century. The conclusions of nineteenth century geologists that the Earth rotated, based on the apparent polar flattening and equatorial bulge measured by maritime explorers, was not supported by direct observational evidence until Foucault set up his pendulum at the Pantheon in 1851, and showed that the Earth turned beneath the swinging ball. Likewise, there was no direct observational evidence for the Earth's revolution around the sun prior to Friedrich Bessel's measurement of the parallax of Cygni 61 in 1838. For nearly three hundred years, astronomers had to choose Ptolemy over Aristotle and Copernicus over Ptolemy based on something other than observational evidence. Many chose "Occam's Razor": that the simplest explanation, the one with the fewest assumptions, was the best, even if there was no way to show whether it was true in the sense of being an accurate representation of the physical universe. Others based their choice on whether a theory produced accurate predictions of planetary positions, since this ability had practical applications for calendar-making and astrological medicine, still prevalent into the eighteenth century.

History of Astronomy

The early astronomers in Babylon, Egypt, and Greece were unable to use accurate instruments. At best, they could predict the positions of the planets, sun, and moon to within 1/2 degree—about the diameter of the moon itself. Even so, they managed to put together a cosmogony, a model of planetary motion, which served them within that range of error.

The movie below shows predicted positions of Mars every 6 hours between April 1 and December 16, 2018 as seen from Earth against the background sky. The yellow line marks the plane of the ecliptic. What do you notice about the "path" of Mars in the sky?

Any theory about planetary motion has to account for the following phenomena:

• the planets, sun, moon, and stars all move from east to west daily
• all of these stay within a narrow range of declination, no more than 15 degrees to the north or south of the celestial equator
• Mercury is never more than 29 degrees from the sun
• Venus is never more than 47 degrees from the sun
• the other naked-eye planets (Mars, Jupiter, and Saturn) can be in the opposite part of the sky from the sun (overhead at midnight)
• most of the time, the planets move from west to east against the background sky (prograde motion)
• part of the time, all the planets move from east to west against the background sky (retrograde motion)
• the planets, sun, and moon speed up and slow down in their orbits; they do not travel with "uniform motion"
• all the planets grow brighter and dimmer over time

While it seems rather odd from our modern point of view, Ptolemy's epicycle and eccentric circle system actually can account for all of these phenomena with the epicycle/eccentric, which is why people used it for about 1200 years, and gave it up, reluctantly, only when they were convinced that the system replacing it would be better.

The plant Mars (red dot) travels once clockwise around its epicycle, while the epicycle travels once counterclockwise around the deferent circle centered on the Earth. To an Earth Observer, the planet appears to speed travel eastward against the background sky most of the time, speeding up and slowing down, and traveling with retrograde motion during a short period. It also moves closer and further from Earth, changing size and brightness.

Theories also had to fit the concepts of matter laid down by the Greek philosophers, in particular, Aristotle, who claimed that

• the four terrestrial elements, earth, fire, water, and air, moved with linear motion
• the celestial element or quintessence was not subject to generation or destruction
• celestial objects were perfect spheres
• celestial objects moved in perfect circles around a center that coincided with Earth

Obviously, some of these premises contradict assumptions that Ptolemy made. The conflicting theories provided for controversy and debate but continued to exist side by side, much like our modern theory of quantum mechanics and relativity, because each predicted or explained phenomena that the other theory could not.

Copernicus knew that the synodic period and the sidereal period of Mars and Venus were different from the rest of the planets. The synodic period is the period between phases of the planet as seen from earth, for example, western elongation to the subsequent western elongation). The sidereal period is the time required for the planet to make one complete orbit of the sun, regardless of the earth's position. The formulae are different for inferior and superior planets, because the motion of the earth around the sun changes our perception of the planetary motions.

Sidereal Period	Earth's Period	Synodic Period	Type of Orbit
1/P	= 1/E +	1/S	INFERIOR
1/P	= 1/E -	1/S	SUPERIOR

By placing the earth in orbit between Venus and Mars, Copernicus could account for this fundamental relationship.

Copernicus' heliocentric theory also had epicycles to handle the different rates of motion, but got rid of the equant and centered his eccentric circles on the sun. Conceptually, the overall heliocentric system was simpler, but was not able to make predictions that were any more accurate than Ptolemy's until improvements in instruments and measurements gave astronomers better information to base their calculations on.

Some of the improvements came before the invention of the telescope, with the instrument improvements made by Tycho Brahe. Brahe had a couple of things going for him: he was independently wealthy, the emperor commissioned his astronomical observations, he had craftsmen who could make his instruments for him, and he had the good fortune to make observations of a spectacular comet and a Milky Way supernova. To give you an idea of how rare supernova within the Milky Way are, the most recent one in 1987 (supernova 1987A in Dorado, an area in the Lesser Magellanic Cloud visible from the southern hemisphere) was the first such supernova since the two in 1577 (Tycho's) and 1604 (Kepler's), and the first to be observed with telescopes. So Tycho had two sets of evidence with which to attack Aristotle's premises:

• the comet demonstrably orbited the sun, and therefore violated the principle of circular motion around the earth
• the nova appeared suddenly, changed in brightness and eventually disappeared, contradicting the idea that celestial objects could not change.

Brahe's assistant took over the business when Brahe died. After some legal tussles with Brahe's heirs (one of the earliest battles for intellectual property), Johannes Kepler managed to get the data for the observations he had made in Brahe's observatory, and used them to show that Mars' orbit was an ellipse. He went on to establish three laws describing the orbits of all planets:

• planets move in elliptical orbits about the sun, with the sun at one of the ellipse's foci
• planets move with varying speed, such that a line from the sun to the planet sweeps out equal areas in equal times
• planetary sidereal periods P(the time to do one complete orbit) are directly proportional to their distance A from the sun, P² = kA³, and k can be determined based on the units used to measure P and A.

Galileo is one of the most controversial of all scientific figures. We won't go into his history of conflict with the Roman Catholic Church, which has been the subject of much recent debate with the release of new documents about his trial by the Vatican. For our purposes, it is enough to note that Galileo established the experimental method in his studies of falling bodies, investigated many areas of physics, and when he got his hands on an early telescope, used it with good result to show that

• The phases of Venus proved that Venus moved around the sun and not around the earth.
• The surface of the moon was not perfect, but covered with craters.
• The surface of the sun was not perfect, but covered with dark marks (sunspots).
• The moons of Jupiter orbited Jupiter and neither the earth nor the sun.

While these observations showed that the planets and their moons did not orbit the earth, they did not conclusively prove that the earth revolves about the sun. But combined with Kepler's earlier observations, Galileo's provided the final blows to the Aristotelian idea of the earth as the center of a cosmic dance in which all the participants moved in perfect and uniform circles.

Practice with the Concepts

Two planets have periods of 5 and 8 years, respecively. What is the ratio of their distances from the sun?

Discussion Questions

• What evidence led to the abandonment first of Aristotle's geocentric system, then of Copernicus' circular orbits?
• How did Tycho's improvement in instrument accuracy lead to new discoveries and theories?

Optional Readings

There is a nice summary page at NASA with the planetary orbit facts for when you don't have your book nearby. Follow the links for information about the planets themselves.

There is a nice biography of Kepler (and the rest of the astronomy gang) at MacTutor. Follow the biographies Index link to get to Galileo, Newton, and Brahe.

Check out this explanation of Kepler's Three Laws at The Physics Classroom.

---

© 2005 - 2024 This course is offered through Scholars Online, a non-profit organization supporting classical Christian education through online courses. Permission to copy course content (lessons and labs) for personal study is granted to students currently or formerly enrolled in the course through Scholars Online. Reproduction for any other purpose, without the express written consent of the author, is prohibited.